Theory of Set Differential Equations in Metric Spaces

Theory of Set Differential Equations

in Metric Spaces

V. LakshmikanthamT. Gnana BhaskarJ. Vasundhara Devi

Department of Mathematical SciencesFlorida Institute of Technology

Melbourne FL 32901USA

2

Contents

Preface 1

1 Preliminaries 51.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2 Compact Convex Subsets of Rn . . . . . . . . . . . . . . . . . . . 61.3 The Hausdorff Metric . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Support Functions . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5 Continuity and Measurability . . . . . . . . . . . . . . . . . . . . 141.6 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181.7 Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.8 Subsets of Banach Spaces . . . . . . . . . . . . . . . . . . . . . . 231.9 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Basic Theory 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Comparison Principles . . . . . . . . . . . . . . . . . . . . . . . . 282.3 Local Existence and Uniqueness . . . . . . . . . . . . . . . . . . . 322.4 Local Existence and Extremal Solutions . . . . . . . . . . . . . . 362.5 Monotone Iterative Technique . . . . . . . . . . . . . . . . . . . . 402.6 Global Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.7 Approximate Solutions . . . . . . . . . . . . . . . . . . . . . . . . 502.8 Existence of Euler Solutions . . . . . . . . . . . . . . . . . . . . . 512.9 Proximal Normal and Flow Invariance . . . . . . . . . . . . . . . 562.10 Existence, Upper Semicontinuous Case . . . . . . . . . . . . . . . 592.11 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . 63

3 Stability Theory 653.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.2 Lyapunov-like Functions . . . . . . . . . . . . . . . . . . . . . . . 663.3 Global Existence . . . . . . . . . . . . . . . . . . . . . . . . . . . 683.4 Stability Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.5 Nonuniform Stability Criteria . . . . . . . . . . . . . . . . . . . . 743.6 Criteria for Boundedness . . . . . . . . . . . . . . . . . . . . . . . 793.7 Set Differential Systems . . . . . . . . . . . . . . . . . . . . . . . 83

3

4 CONTENTS

3.8 The Method of Vector Lyapunov Functions . . . . . . . . . . . . 863.9 Nonsmooth Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 893.10 Lyapunov Stability Criteria . . . . . . . . . . . . . . . . . . . . . 933.11 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . 95

4 Connection to FDEs 974.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 974.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.3 Lyapunov-like functions . . . . . . . . . . . . . . . . . . . . . . . 1014.4 Connection with SDEs . . . . . . . . . . . . . . . . . . . . . . . . 1094.5 Upper Semicontinuous Case Continued . . . . . . . . . . . . . . . 1164.6 Impulsive FDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1204.7 Hybrid FDEs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1244.8 Another Formulation . . . . . . . . . . . . . . . . . . . . . . . . . 1294.9 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . 136

5 Miscellaneous Topics 1395.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1395.2 Impulsive Set Differential Equations (SDEs) . . . . . . . . . . . . 1405.3 Monotone Iterative Technique . . . . . . . . . . . . . . . . . . . . 1505.4 Set Differential Equations with Delay . . . . . . . . . . . . . . . . 1635.5 Impulsive Set Differential Equations with Delay . . . . . . . . . . 1735.6 Set Difference Equations . . . . . . . . . . . . . . . . . . . . . . . 1805.7 Set Differential Equations with Causal Operators . . . . . . . . . 1855.8 Lyapunov-like Functions in Kc(Rd

+) . . . . . . . . . . . . . . . . . 1955.9 Set Differential Equations in (Kc(E), D), . . . . . . . . . . . . . 1975.10 Notes and Comments . . . . . . . . . . . . . . . . . . . . . . . . . 199

References 201

Index 206

Preface

The study of analysis in metric spaces has gained importance in recent times.It is realized that many results of differential calculus and set valued analysis,including the inverse function theorem do not really rely upon the linear struc-ture and therefore can be adapted to the nonlinear case of metric spaces andexploited. Moreover, the concept of the differential equation governing evolutionin metric spaces has been suitably formulated.

Multivalued differential equations (now known as set differential equations(SDEs)) generated by multivalued differential inclusions have been introducedin a semi-linear metric space, consisting of all nonempty, compact, convex sub-sets of an initial finite or infinite dimensional space. The basic existence anduniqueness results of such SDEs have been investigated and their solutions havecompact, convex values. Also, these generated SDEs have been employed as atool to prove the existence of solutions, in a unified way, of multivalued differ-ential inclusions. The multifunctions involved in this set up are compact, butnot necessarily convex, subsets of the base space utilized.

Because of the fact that fuzzy set theory and its applications have been ex-tensively investigated, due to the increase of industrial interest in fuzzy control,the initiation of the theory of fuzzy differential equations (FDEs) in an appro-priate metric space has recently been accomplished. In view of the inherentdisadvantage resulting from the fuzzification of the derivative employed in theoriginal formulation of FDEs, an alternative formulation based upon a family ofmultivalued differential inclusions derived from the fuzzy maps involved in theFDEs, is recently suggested to reflect the rich behaviour of the correspondingordinary differential equation before fuzzification.

The investigation of the theory of SDEs as an independent discipline, hascertain advantages. For example, when the set is a single valued mapping, it isclear that the Hukuhara derivative and the integral utilized in formulating theSDEs reduce to the ordinary vector derivative and the integral, and thereforethe results obtained in the framework of SDEs become the corresponding resultsof ordinary differential systems if the base space is Rn. On the other hand, if thebase space is a Banach space, we get from the corresponding SDE’s the differ-ential equations in a Banach space. Moreover, one has only a semilinear metricspace to work with in the SDE set up, compared to the complete normed linearspace that one employs in the usual study of an ordinary differential system. Asindicated earlier,the SDEs that are generated by multivalued differential inclu-

1

2 PREFACE

sions when the needed convexity is missing, form a natural vehicle for provingthe existence results for multivalued differential inclusions. Also, one can utilizeSDEs profitably to investigate FDEs. Consequently, the study of the theory ofSDEs has recently been growing very rapidly and is still in the initial stages.Nonetheless, there exists sufficient literature to warrant assembling the existingfundamental results in a unified way to understand and appreciate the intrica-cies and advantages involved, so as to pave the way for further advancement ofthis important branch of differential equations as an independent subject area.It is with this spirit we see the importance of the present monograph. As aresult, we provide a systematic account of recent development, describe the cur-rent state of the useful theory, show the essential unity achieved and initiateseveral new extensions to other types of SDEs.

In Chapter 1, we assemble the preliminary material providing the necessarytools including the calculus for set valued maps relevant to the later develop-ment. Chapter 2 is devoted to the investigation of the fundamental theory ofSDEs such as various comparison principles, existence and uniqueness, contin-uous dependence, existence of extremal solutions suitably introducing a partialorder in the metric space, monotone iterative technique using lower and uppersolutions and global existence under the continuity assumption for SDEs. Wealso discuss, utilizing the method of nonsmooth analysis, existence and flowinvariance results without any continuity assumption, in terms of Euler solu-tions. Finally, we consider the case of upper semicontinuity in the frameworkof Caratheodory and prove an existence result in a general set up.

In Chapter 3, we extend Lyapunov stability theory to SDEs, employingLyapunov-like functions, proving first suitable comparison results in terms ofsuch functions. The stability and boundedness criteria are obtained by choosingappropriate initial values in terms of the Hukuhara difference to eliminate theundesirable part of the solutions of SDEs, so that the rich behaviour of the cor-responding ODEs, from which SDEs are generated, is preserved. The methodsof vector Lyapunov-like functions and the perturbing Lyapunov-like functionsare discussed in detail. Also, employing lower semicontinuous Lyapunov-likefunctions and utilizing nonsmooth analysis, stability results are described un-der weaker assumptions.

Chapter 4 deals with the interconnection between SDEs and fuzzy differentialequations(FDEs). For this purpose, necessary tools are provided for formulatingFDEs, and basic results are proved, including the stability theory of Lyapunov.Then the interconnection between FDEs and SDEs is explored via a sequenceof multivalued differential inclusions, suitably generating SDEs as describedearlier. The impulsive effects are then incorporated in FDEs and then it isshown how impulses can help to improve the qualitative behaviour of solutionsof FDEs. Hybrid fuzzy differential equations are introduced and their stabilityproperties are discussed. Another concept of differential equations in metricspaces is considered which can be applied to the study of FDEs.

Chapter 5 is devoted to initiate several topics in the setup of SDEs suchas impulsive SDEs, SDEs with time delay, set difference equations, and SDEsinvolving causal maps, which cover several types of SDEs including integro-

PREFACE 3

differential equations. Some important basic results are provided for each typeof SDEs. We then introduce Lyapunov-like functions whose values are in somemetric space, prove suitable comparison results and study stability theory inthis general set up. This study includes the methods of single, vector, matrixand cone-valued Lyapunov-like functions by an appropriate choice of the metricspace. Since the basic space utilized to define the metric space (Kc(Rn), D) isrestricted, for convenience of understanding, to Rn, we indicate how one canextend most of the results described when we choose a Banach space E insteadof Rn, so that we have the corresponding metric space (Kc(E), D) to work with.Finally, notes and comments are provided for each chapter.

Some of the important features of the monograph as follows:

1. It is the first book that attempts to describe the theory of set differentialequations as an independent discipline.;

2. It incorporates, the recent general theory of set differential equations,discusses the interconnections between set differential equations and fuzzydifferential equations and uses both smooth and nonsmooth analysis forinvestigation.

3. It exhibits several new areas of study by providing the initial apparatusfor further advancement.

4. It is a timely introduction to a subject that follows the present trend ofstudying analysis and differential equations in metric spaces.

This monograph will be very useful to those experts and their doctoral stu-dents who work in Nonlinear Analysis, in general. It will also be a good referencebook to Engineering and Computer Scientists since it also covers fuzzy dynamicsas a subset.

We are immensely thankful to Professors Alex Tolstanogov, Donta Satya-narayana and S. Leela for their valuable suggestions and help and to Mrs. DonnMiller-Kermani for the excellent CRC typing of the manuscript. We are pleased,indeed, to offer our heartfelt thanks to Ms. Janie Wardle for her interest andcooperation in our projects.

4 PREFACE

Chapter 1

Preliminaries

1.1 Introduction

Recently the study of set differential equations (SDEs) was initiated in a metricspace and some basic results of interest were obtained. The investigation ofset differential equations as an independent subject has some advantages. Forexample, when the set is a single valued mapping, it is easy to see that theHukuhara derivative, and the integral utilized in formulating the SDEs reduceto the ordinary vector derivative and the integral and therefore, the resultsobtained in this new framework become the corresponding results in ordinarydifferential systems. Also, we have only a semilinear complete metric space towork with in the present setup, compared to the normed linear space that oneemploys in the usual study of ordinary differential systems.

Furthermore, SDEs that are generated by multivalued differential inclusions,when the multivalued functions involved do not possess convex values, can beused as a tool for studying multivalued differential inclusions. Moreover, onecan utilize SDEs indirectly to investigate profitably fuzzy differential equations,since the original formulation of fuzzy differential equations suffers from gravedisadvantage and does not reflect the rich behavior of corresponding differentialequations without fuzziness. This is due to the fact that the diameter of anysolution of a fuzzy differential equation increases as time increases because ofthe necessity of the fuzzification of the derivative involved.

In order to formulate the set differential equations in a metric space, weneed some background material, since the metric space involved consists ofall nonempty compact, convex sets in finite or infinite dimensional space. InSection 1.2, we define the necessary ingredients of such sets restricting ourselvesto the Euclidean n-space Rn. Since the difference of any two sets in Kc(Rn)(set of all nonempty, compact, convex sets in Rn) is not defined in general,conditions for the existence of the difference are provided in this section. Section1.3 introduces the Hausdorff metric D[·, ·] for Kc(Rn) and lists its properties.Support functions are defined in Section 1.4, where they are utilized to create

5

6 CHAPTER 1. PRELIMINARIES

a mapping that makes it possible to embed the metric space (Kc(Rn), D) intoa complete cone in a specified Banach space.

In Section 1.5, the continuity and measurability properties of mappings intothe metric space are dealt with. Section 1.6 investigates the concept of differ-entiation of such mappings and its behavior. In Section 1.7, we consider thetheory of integration of these mappings and the needed properties. Section 1.8summarizes the corresponding situation when the elements of the metric spaceconsidered are from a Banach space. Notes and comments are listed in Section1.9.

1.2 Compact Convex Subsets of Rn

We shall consider the following three spaces of nonempty subsets of Rn, namely,

(i) Kc(Rn) consisting of all nonempty compact convex subsets of Rn;

(ii) K(Rn) consisting of all nonempty compact subsets of Rn;

(iii) C(Rn) consisting of all nonempty closed subsets of Rn.

Recall that a nonempty subset A of Rn is convex if for all a1, a2 ∈ A and allλ ∈ [0, 1], the point

a = λa1 + (1 − λ)a2 (1.2.1)

belongs to A. For any nonempty subset A of Rn, we denote by coA its convexhull, that is the totality of points a of the form (1.2.1) or, equivalently, thesmallest convex subset containing A. Clearly

A ⊆ co A = co(co A) (1.2.2)

with A = coA if A is convex. Moreover, coA is closed (compact) if A is closed(compact).

Let A and B be two nonempty subsets of Rn and let λ ∈ IR. We define thefollowing Minkowski addition and scalar multiplication by

A+ B = a+ b : a ∈ A, b ∈ B (1.2.3)

andλA = λa : a ∈ A. (1.2.4)

Then we have the following proposition.

Proposition 1.2.1 The spaces C(Rn), K(Rn) and Kc(Rn) are closed underthe operations of addition and scalar multiplication. In fact, the following prop-erties hold:

(i) A+θ = θ+A = A, θ ∈ Rn, is the zero element of Rn, treated as a singleton.

(ii) (A +B) + C = A+ (B +C)

1.2 COMPACT CONVEX SUBSETS OF RN 7

(iii) A+ B = B +A

(iv) A+ C = B + C implies A = B

(v) 1 ·A = A

(vi) λ(A+ B) = λA + λB

(vii) (λ + µ)A = λA + µA

where A,B,C ∈ Kc(Rn), λ, µ ∈ IR+.

Proof We only give the proof of (iv), the rest being simple to prove.Let A, B, C ∈ Kc(Rn). We show that A 6= B implies A + C 6= B + C.

Suppose, for example, that there exists a point a ∈ A which does not belongto B. Through a pass hyperplanes which are disjoint from B. Let one of thesehyperplanes be P. Let P ′ be the support hyperplane of C, which is parallel toP and such that, if we move P ′ parallel to itself onto P, C moves on a compactconvex set which is located on the same side of P as B. If c is a point of C ∩P ′,then a+ c /∈ B + C. Hence the proof.

In general, A+(−A) 6= θ. This fact is illustrated by the following example.

Example 1.2.1 Let A = [0, 1] so that (−1)A = [−1, 0], and therefore

A+ (−1)A = [0, 1] + [−1, 0] = [−1, 1].

Thus, adding (−1) times a set does not constitute a natural operation of sub-traction.

This leads us to the following definition.

Definition 1.2.1 For a fixed A and B in Kc(Rn) if there exists an elementC ∈ Kc(Rn) such that A = B +C then we say that the Hukuhara Difference ofA and B exists and is denoted by A− B.

When the Hukuhara difference exists it is unique. This follows from (iv) ofProposition 1.2.1.

The following example explains the above definition.

Example 1.2.2 From Example 1.2.1, we get

[−1, 1]− [−1, 0] = [0, 1] and [−1, 1]− [0, 1] = [−1, 0].

Note that the Hukuhara difference A− B is different from the set

A + (−B) = a+ (−b) : a ∈ A, b ∈ B.

The next proposition provides the necessary and sufficient condition for theexistence of the Hukuhara difference A −B.


Proposition 1.2.2 Let A,B ∈ Kc(Rn). For the difference A−B to exist, it isnecessary and sufficient to have the following condition. If a ∈ ∂A, there existsat least a point c such that

a ∈ B + c ⊂ A. (1.2.5)

Proof Necessity: Suppose the difference A − B exists. Let C = A− B. Then,A = B + C. If a ∈ ∂A, a ∈ B + C, that is, a = b + c where b ∈ B and c ∈ C.Also, if z ∈ B, then z + c ∈ A and therefore (1.2.5) is satisfied.

Sufficiency: Suppose (1.2.5) holds. Consider the set C = x : B + x ⊆ A.Clearly C is compact and we have B + C ⊆ A. Now, if d and d′ ∈ C, then wehave B + d ⊆ A and B + d′ ⊆ A, from which we obtain

(1 − λ)(B + d) + λ(B + d′) ⊂ A, for 0 ≤ λ ≤ 1. (1.2.6)

We can write the L.H.S of (1.2.6) as B+ z with z = (1−λ)d+λd′. Hence z ∈ Cand C is convex.

Let u ∈ A. A straight line through u meets ∂A at two points a and a′.By hypothesis there exist elements d and d′ in C such that a ∈ B + d, anda′ ∈ B + d′. We can write u = (1 − λ)a+ λa′ with 0 < λ < 1. Then u ∈ B + x,where x = (1 − λ)d + λd′ ∈ C. Hence A ⊆ B + C. Thus A = B + C and theproof is complete.

We note that a necessary condition for the Hukuhara difference A − B toexist is that some translate of B is a subset of A. However, in general, theHukuhara difference need not exist as is seen from the following example.

Example 1.2.3 0 − [0, 1] does not exist, since no translate of [0,1] can everbelong to the singleton set 0.

1.3 The Hausdorff Metric

Let x be a point in Rn and A a nonempty subset of Rn. The distance d(x,A)from x to A is defined by

d(x,A) = inf‖x− a‖ : a ∈ A. (1.3.1)

Thus d(x,A) = d(x, A) ≥ 0 and d(x,A) = 0 if and only if x ∈ A, the closure ofA ⊆ Rn.

We shall call the subset

Sε(A) = x ∈ Rn : d(x,A) < ε (1.3.2)

an ε−neighborhood of A. Its closure is the subset

Sε(A) = x ∈ Rn : d(x,A) ≤ ε. (1.3.3)

In particular, we shall denote by

Sn1 = S1(θ), (1.3.4)

1.3 THE HAUSDORFF METRIC 9

which is obviously a compact subset of Rn. Note also that

Sε(A) = A+ εSn1 , (1.3.5)

for any ε > 0 and any nonempty subset A of Rn. We shall for conveniencesometimes write S(A, ε) and Sε(A).

Now, let A and B be nonempty subsets of Rn. We define the Hausdorffseparation of B from A by

dH (B,A) = supd(b, A) : b ∈ B (1.3.6)

or, equivalentlydH(B,A) = infε > 0 : B ⊆ A + εSn

1 .

We have dH(B,A) ≥ 0 with dH(B,A) = 0 if and only if B ⊆ A. Also, thetriangle inequality

dH(B,A) ≤ dH (B,C) + dH(C,A),

holds for all nonempty subsets A, B and C of Rn. In general, however

dH(A,B) 6= dH(B,A).

We define the Hausdorff distance between nonempty subsets A and B of Rn by

D(A,B) = maxdH(A,B), dH(B,A), (1.3.7)

which is symmetric in A and B. Consequently,

(a) D(A,B) ≥ 0 with D(A,B) = 0 if and only if A = B;(b) D(A,B) = D(B,A);(c) D(A,B) ≤ D(A,C) +D(C,B), (1.3.8)

for any nonempty subsets A, B and C of Rn.If we restrict our attention to nonempty closed subsets of Rn, we find that

the Hausdorff distance (1.3.7) is a metric known as the Hausdorff metric. Thus(C(Rn), D) is a metric space.

In fact, we have

Proposition 1.3.1 (C(Rn), D) is a complete separable metric space in whichK(Rn) and Kc(Rn) are closed subsets. Hence, (K(Rn), D) and (Kc(Rn), D) arealso complete separable metric spaces.

The following properties of the Hausdorff metric will be useful later.We start by stating a proposition dealing with the invariance of the Hausdorff

metric.

Proposition 1.3.2 If A, B ∈ Kc(Rn) and C ∈ K(Rn) then,

D(A +C,B +C) = D(A,B). (1.3.9)


We need the following result which deals with the law of cancellation to proceedfurther.

Lemma 1.3.1 Let A, B ∈ Kc(Rn) and C ∈ K(Rn) and A+ C ⊆ B +C, thenA ⊆ B.

Proof Let a be any element of A. We need to show that a ∈ B. Given anyc1 ∈ C, we have a + c1 ∈ B + C, that is, there exist b1 ∈ B and c2 ∈ C witha+c1 = b1+c2. For the same reason, b2 ∈ B and c3 ∈ C with a+c2 = b2+c3 mustexist. Repeat the procedure indefinitely and sum the first n of the equationsobtained. We get

na +n∑

i=1

ci =n∑

i=1

bi +n+1∑

i=2

ci

which implies

na+ c1 =n∑

i=1

bi + cn+1.

Then,

a =1n

n∑

i=1

bi +cn+1

n− c1n.

Set xn = 1n

∑ni=1 bi. Thus

a = xn +cn+1

n− c1n.

We observe that xn ∈ B for all n, because B is convex and cn+1n

− c1n

→ 0 as Cis compact. Thus xn converges to a. But since B is compact, a ∈ B. Thus, ifA +C = B + C then A = B. This completes the proof of the lemma.Proof of Proposition 1.3.2. Let λ ≥ 0 and S denote the closed unit sphereof the space. Consider the following inequalities

(1) A+ λS ⊃ B,

(2) B + λS ⊃ A,

(3) A+ C + λS ⊃ B +C,

(4)B + C + λS ⊃ A +C.

Put d1 = D(A,B) and d2 = D(A + C,B + C). Then d1 is the infimum of thepositive numbers λ for which (1) and (2) hold. Similarly, d2 is the infimum ofthe positive numbers λ for which (3) and (4) hold. Since (3) and (4) follow from(1) and (2) respectively, by adding C, we have d1 ≥ d2. Conversely, since byLemma 1.3.1, canceling C is allowed in (3) and (4), we obtain d1 ≤ d2, whichproves the proposition.

1.3 THE HAUSDORFF METRIC 11

Proposition 1.3.3 If A,B ∈ K(Rn)

D(co A, co B) ≤ D(A,B). (1.3.10)

If A, A′, B, B′ ∈ Kc(Rn) then

D(tA, tB) = tD(A,B) for all t ≥ 0, (1.3.11)

D(A +A′, B + B′) ≤ D(A,B) +D(A′, B′), (1.3.12)

Further,D(A −A′, B − B′) ≤ D(A,B) +D(A′, B′), (1.3.13)

provided the differences A−A′ and B−B′ exist. Moreover with β = maxλ, µ,we have

D(λA, µB) ≤ βD(A,B) + |λ− µ|[D(A, θ) +D(B, θ)] (1.3.14)

andD(λA, λB) = λD(A −B, θ), if A −B exists. (1.3.15)

Proof Since (1.3.10) is obvious, we begin with the proof of (1.3.11). For alla ∈ A and u ∈ A′, compactness of B and B′ ensures that there exist b(a) ∈ Band v(u) ∈ B′ such that

infb∈B

|a− b| = |a− b(a)|; infv∈B′

|u− v| = |u− v(u)|. (1.3.16)

From the relation

|a+ u− b(a) − v(u)| ≤ |a− b(a)| + |u− v(u)|

and (1.3.16), it follows that

supa∈A, u∈A′

infb∈B, v∈B′

|a+ u− b− v| ≤ supa∈A

infb∈B

|a− b|+ supu∈A′

infv∈B′

|u− v|.

From the above and the analogous inequality obtained by interchanging theroles of A with B and A′ with B′, we obtain (1.3.11).

We now prove (1.3.13).Using Proposition 1.3.2, we find that

D(A − A′, B −B′) = D(A − A′ +A′ + B′, B − B′ +B′ +A′)= D(A + B′, B + A′)≤ D(A,B) +D(A′, B′),

which follows from (1.3.11).To prove (1.3.14), consider, for λ − µ ≥ 0,

D(λA, µB) ≤ µD(A,B) + (λ− µ)D(A, θ),


and if λ − µ ≤ 0, then

D(λA, µB) ≤ λD(A,B) + (µ− λ)D(B, θ).

The relations above put together prove (1.3.14).The proof of (1.3.15) follows from Proposition 1.3.2.Next, we define the magnitude of a nonempty subset of A of Rn by

‖A‖ = sup‖a‖ : a ∈ A, (1.3.17)

or equivalently,‖A‖ = D(θ, A). (1.3.18)

Here, ‖A‖ is finite, and the supremum in (1.3.17) is attained when A ∈ K(Rn).From (1.3.10) it obviously follows that

‖tA‖ = t‖A‖, for all t ≥ 0. (1.3.19)

Moreover, (1.3.8) and (1.3.18) yield

|‖A‖ − ‖B‖| ≤ D(A,B), (1.3.20)

for all A,B ∈ K(Rn).We say that a subset U ∈ K(Rn) or Kc(Rn) is uniformly bounded if there

exists a finite constant c(U) such that

‖A‖ ≤ c(U), for all A ∈ U. (1.3.21)

We then have the following simple characterization of compactness.

Proposition 1.3.4 A nonempty subset A of the metric space (K(Rn), D) or(Kc(Rn), D), is compact if and only if it is closed and uniformly bounded.

Set inclusion induces partial ordering on K(Rn). Write A ≤ B if and only ifA ⊆ B, where A,B ∈ K(Rn). Then

L(B) = A ∈ K(Rn) : B ≤ A, U(B) = A ∈ K(Rn) : A ≤ B, (1.3.22)

are closed subsets of K(Rn) for any B ∈ K(Rn). In fact, from Proposition 1.3.4,U(B) is compact subset of K(Rn).

Proposition 1.3.5 U(B) is a compact subset of K(Rn).

This assertion remains true with Kc(Rn) replacing K(Rn) everywhere.Sequences of nested subsets in (K(Rn), D) have the following useful inter-

section and convergence properties.

Proposition 1.3.6 Let Aj ⊂ K(Rn) satisfy

· · · ⊆ Aj ⊆ · · ·A2 ⊆ A1.

Then A = ∩∞j=1Aj ∈ K(Rn) and

D(An, A) → 0 as n→ ∞. (1.3.23)

On the other hand, if A1 ⊆ A2 ⊆ · · ·Aj ⊆ · · · and A = ∪∞j=1Aj ∈ K(Rn), then

(1.3.23) holds.

1.4 SUPPORT FUNCTIONS 13

1.4 Support Functions

Let A be a nonempty subset of Rn. The support function of A is defined for allp ∈ Rn by

s(p,A) = sup< p, a >: a ∈ A, (1.4.1)

which may take the value +∞ when A is unbounded. However, when A is acompact, convex subset of Rn the supremum is always attained and the supportfunction s(·, A) : Rn → R is well defined. Indeed,

|s(p,A)| ≤ ‖A‖‖p‖, (1.4.2)

for all p ∈ Rn, and|s(p,A) − s(q, A)| ≤ ‖A‖‖p− q‖, (1.4.3)

for all p, q ∈ Rn.Further, for all p ∈ Rn,

s(p,A) ≤ s(p,B), if A ⊆ B, (1.4.4)

ands(p, co(A ∪B)) ≤ maxs(p,A), s(p,B). (1.4.5)

The support function s(p,A) is uniquely paired to the subset A ∈ Kc(Rn) inthe sense that s(p,A) = s(p,B) for all p ∈ Rn if and only if A = B when Aand B are restricted to Kc(Rn). It also preserves set addition and nonnegativescalar multiplication. That is, for all p ∈ Rn,

s(p,A+ B) = s(p,A) + s(p,B), (1.4.6)

which, in particular reduces to

s(p,A + x) = s(p,A)+ < p, x >, (1.4.7)

for any x ∈ Rn, ands(p, tA) = ts(p,A), t ≥ 0. (1.4.8)

For a fixed A ∈ Kc(Rn), s(p,A) is positively homogeneous

s(tp, A) = ts(p,A), t ≥ 0, (1.4.9)

for all p ∈ Rn, and subadditive:

s(p1 + p2, A) ≤ s(p1, A) + s(p2, A), (1.4.10)

for all p1, p2 ∈ Rn. Moreover, combining (1.4.13) and (1.4.14) we see that s(·, A)is a convex function, that is, it satisfies

s(λp1 + (1 − λ)p2, A) ≤ λs(p1, A) + (1 − λ)s(p2, A), (1.4.11)

for all p1, p2 ∈ Rn and λ ∈ [0, 1].The nonempty compact convex subsets of Rn are uniquely characterized by

such functions.


Proposition 1.4.1 For every continuous, positively homogeneous and subad-ditive function s : Rn → R there exists a unique nonempty compact convexsubset

A = x ∈ Rn :<p, x> ≤ s(p) for all p ∈ Rn,

which has s as its support function.

The Hausdorff metric is related to the support function for A,B ∈ Kc(Rn),since we have

D(A,B) = sup|s(p,A) − s(p,B)| : p ∈ Sn−1, (1.4.12)

where Sn−1 = p ∈ Rn : ‖p‖ = 1 is the unit sphere in Rn.Let C(Sn−1) denote the Banach Space of continuous functions f : Sn−1 → R

with the supremum norm

‖f‖ = sup|f(p)| : p ∈ Sn−1.

One can use the support function to embed the metric space (Kc(Rn), D) iso-metrically as a positive cone in C(Sn−1).

For this, define j : Kc(Rn) → C(Sn−1) by j(A)(·) = s(·, A), for each A ∈Kc(Rn).

From the properties of the support function, j is a univalent mapping satis-fying

j(A +B) = j(A) + j(B), (1.4.13)

andj(tA) = tj(A), t ≥ 0, (1.4.14)

with‖j(A) − j(B)‖ = D(A,B), (1.4.15)

for all A,B ∈ Kc(Rn).The desired positive cone is the image j(Kc(Rn)) in C(Sn−1). Obviously j

is continuous, as is its inverse

j−1 : j(Kc(Rn)) → Kc(Rn).

1.5 Continuity and Measurability

We consider mappings F from a domain T in Rk into the metric space (Kc(Rn), D).Thus F : T → Kc(Rn) or equivalently,

F (t) ∈ Kc(Rn), for all t ∈ T. (1.5.1)

We shall call such a mapping F a (compact convex) set valued mapping from Tto Rn.

1.5 CONTINUITY AND MEASURABILITY 15

The usual definition of continuity of mappings between metric spaces applieshere. We shall say that a set valued mapping F satisfying (1.5.1) is continuousat t0 in T if for every ε > 0 there exists a δ = δ(ε, t0) > 0 such that

D[F (t), F (t0)] < ε, (1.5.2)

for all t ∈ T with ‖t− t0‖ < δ.Alternatively, we can write (1.5.2) in terms of the convergence of sequences,

that islim

tn→t0D[F (tn), F (t0)] = 0, (1.5.3)

for all sequences tn in T with tn → t0 ∈ T.

Using the Hausdorff separation dH and neighborhoods, we see that (1.5.2)is the combination of

dH(F (t), F (t0)) < ε, (1.5.4)

that isF (t) ⊂ Sε(F (t0)) ≡ F (t0) + εSn

1 , (1.5.5)

anddH(F (t0), F (t)) < ε, (1.5.6)

that isF (t0) ⊂ Sε(F (t)) ≡ F (t) + εSn

1 , (1.5.7)

for all t ∈ T with ‖t − t0‖ < δ. As before, Sn1 = x ∈ Rn : ‖x‖ < 1 is the

open unit ball in Rn. If the mapping F satisfies (1.5.4), (1.5.5) we say thatit is upper semicontinuous at t0, or that it is lower semicontinuous at t0, if itsatisfies (1.5.6), (1.5.7). Thus, F is continuous at t0 if and only if it is both lowersemicontinuous and upper semicontinuous at t0. A set valued mapping can belower semicontinuous without being upper semicontinuous, and vice versa.

Example 1.5.1 The set valued mapping F from R into R defined by

F (t) =0, for t = 0,[0, 1], for t ∈ IR \ 0,

is lower semicontinuous, but not upper semicontinuous, at t0 = 0. On the otherhand, F : R → R defined by

F (t) =

[0, 1], for t = 0,0, for t ∈ R \ 0,

is upper semicontinuous, but not lower semicontinuous, at t0 = 0.

Example 1.5.2 A single valued mapping f : R → R+ is upper (lower) semicon-tinuous if the set valued mapping F defined by F (t) = [0, f(t)] is upper (lower)semicontinuous.


If the mapping is continuous, upper semicontinuous or lower semicontinuousat every t0 ∈ T, we shall replace the qualifier ‘at t0’ with ‘on T ’, or omit italtogether.

We say that a set valued mapping F from T into Rn is Lipschitz continuouswith Lipschitz constant L if

D [F (t′), F (t)] ≤ L‖t′ − t‖, (1.5.8)

for all t′, t ∈ T. A Lipschitz continuous mapping is obviously continuous.The distance d(x, F (t)) of F (t) from a point x ∈ Rn satisfies

|d(x, F (t))− d(y, F (t′))| ≤ ‖x− y‖ +D [F (t), F (t′)] , (1.5.9)

for all x, y ∈ Rn and t′, t ∈ T. Thus d(·, F (·)) : Rn × T → R+ is continuouswhenever F is continuous, and Lipschitz continuous whenever F is Lipschitzcontinuous. Since the magnitude ‖F (t)‖ = D[F (t), θ], a similar assertion holdsfor ‖F (·)‖ : T → R+ with the same Lipschitz constant when F is Lipschitzcontinuous.

We saw in Section 1.4 that the support function s(·, A) of an element A ∈Kc(Rn) can be used to form an isometric embedding j : Kc(Rn) → C(Sn−1)with j(A)(·) = s(·, A).

Thus, if F : T → Kc(Rn), the support function s(·, F (t)), for each t ∈ Tdefines a mapping j(F (·)) : T → C(Sn−1). From (1.4.12) we have

sup|s(p, F (t′)) − s(p, F (t))| : p ∈ Sn−1 = D [F (t′), F (t)] . (1.5.10)

So j(F (·)) is continuous or Lipschitz continuous (with the same Lipschitz con-stant) whenever F is continuous or Lipschitz continuous.

Combining (1.4.3) and (1.5.10) we obtain

|s(x, F (t′)) − s(y, F (t))| ≤ ‖F (t)‖‖x− y‖ +D [F (t′), F (t)] , (1.5.11)

for all t′, t ∈ T and x, y ∈ Sn1 = x ∈ Rn : ‖x‖ ≤ 1.

Thus the support function s(x, F (t)), considered as a mapping s(·, F (·)) :Sn

1 × T → R is continuous or Lipschitz continuous whenever F is continuous orLipschitz continuous.

Let B(Rk) and B(Kc(Rn)) denote the σ− algebras of Borel subsets of Rk and(Kc(Rn), D) respectively. Adopting the usual definition of Borel measurabilityof a mapping between metric spaces, we shall say that a mapping F : T →Kc(Rn) is measurable if

t ∈ T : F (t) ⊂ B ∈ B(Rk) for all B ∈ B(Kc(Rn)). (1.5.12)

We shall write F−1(A) = t ∈ T : F (t) ∩A 6= ∅ for any subset A of Rn.Then we have

Proposition 1.5.1 The following assertions are equivalent:

(i) F : T → Kc(Rn) is measurable;

1.5 CONTINUITY AND MEASURABILITY 17

(ii) F−1(B) ∈ B(Rk) for all B ∈ B(Rk);

(iii) F−1(O) ∈ B(Rk) for all open subsets O of Rn;

(iv) F−1(C) ∈ B(Rk) for all closed subsets C of Rn;

(v) d(x, F (·)) : T → R is measurable for each x ∈ Rn;

(vi) ‖F (·)‖ : T → R is measurable;

(vii) s(x, F (·)) : T → R is measurable for each x ∈ Rn.

In (v)-(vii) we mean measurability of the single valued mapping T 7→ Rwith respect to the Borel σ− algebras B(Rk) and B(R). Such mappings aremeasurable if they are continuous. For set valued mappings we have

Proposition 1.5.2 F : T → Kc(Rn) is measurable if it is upper semicontinu-ous or lower semicontinuous, and hence if it is continuous.

For set valued mappings, measurability is also preserved on taking limits.

Proposition 1.5.3 Let Fi : T → Kc(Rn) be measurable for i = 1, 2, 3, · · · andsuppose that limt→∞D[Fi(t), F (t)] = 0 for almost all t ∈ T. Then F : T →Kc(Rn) is measurable.

A selector of a set valued mapping F from T into Rn is a single valuedmapping f : T → Rn such that

f(t) ∈ F (t) for all t ∈ T. (1.5.13)

Proposition 1.5.4 If F : T → Kc(Rn) is measurable then it has a measurableselector f : T → Rn.

The following result, known as the Castaing Representation Theorem, givesan additional characterization of measurability of a set valued mapping.

Theorem 1.5.1 F : T → Kc(Rn) is measurable if and only if there exists asequence fi of measurable selectors of F such that

F (t) =⋃

fi(t) : i = 1, 2, · · ·, (1.5.14)

for each t ∈ T.

If the set valued mapping is at least lower semicontinuous, then it has contin-uous selectors. In fact, any point in F (t) is attainable by a continuous selector.

Proposition 1.5.5 Let F : T → Kc(Rn) be lower semicontinuous. Then foreach x ∈ F (t) and t ∈ T there is a continuous selector f of F such that f(t) = x.

On the other hand, a set valued mapping need not have a continuous selectorif it is only upper semicontinuous.


Example 1.5.3 The set valued mapping F from R to R defined by

F (t) =

−1, if t < 0,[−1, 1], if t = 0,+1, if t > 0,

is upper semicontinuous, but has no continuous selectors. Note that F is notlower semicontinuous at t = 0.

When the set valued mappingF is Lipschitz continuous, then it has Lipschitzcontinuous selectors satisfying the attainability property of Proposition 1.5.4.

Let ≤ be the partial ordering on Kc(Rn) induced by set inclusion, that is,A ≤ B if and only if A ⊆ B. We say that a mapping F : T → Kc(Rn) has a≤ − maximum at t0 ∈ T if

F (t) ≤ F (t0) for all t ∈ T, (1.5.15)

and a ≤ − minimum at t0 ∈ T if

F (t0) ≤ F (t) for all t ∈ T. (1.5.16)

1.6 Differentiation

We begin with the definition of the Hukuhara derivative.

Definition 1.6.1 Let I be an interval of real numbers. Let a multifunctionU : I → Kc(Rn) be given. U is Hukuhara differentiable at a point t0 ∈ I, ifthere exists DHU (t0) ∈ Kc(Rn) such that the limits

lim∆t→0+

U (t0 + ∆t) − U (t0)∆t

(1.6.1)

and

lim∆t→0+

U (t0) − U (t0 − ∆t)∆t

(1.6.2)

both exist and are equal to DHU (t0).

Clearly, implicit in the definition of DHU (t0) is the existence of the differencesU (t0 + ∆t) − U (t0) and U (t0) − U (t0 − ∆t), for all ∆t > 0 sufficiently small.Using the difference quotient in (1.6.2) is not equivalent to using the differencequotient in

lim∆t→0−

U (t0 + ∆t) − U (t0)∆t

, (1.6.2′)

contrary to the situation for ordinary functions from I into a topological vectorspace. In general the existence of A−B, A,B,∈ Kc(Rn) implies nothing aboutthe existence of B −A.

The integral of a continuous multifunction F : [a, b] → Kc(Rn) is defined inHukuhara [1,2], and is shown that DH

∫ t

a F (s)ds = F (t). In order that such aresult holds, one must use (1.6.2) instead of (1.6.2′), since the difference quotientin (1.6.2′) may not exist, as shown in the following example.

1.6 DIFFERENTIATION 19

Example 1.6.1 Let A ∈ Kc(Rn) and define F (t) = A, t ∈ R; then for anyt > 0, we have

∫ t

0F (s)ds = tA. Taking U (t) = tA, t > 0 we see that the

difference quotient (1.6.2 ′) does not exist.

The following proposition illustrates an important property of Hukuhara deriva-tive.

Proposition 1.6.1 If the multifunction U : I → Kc(Rn) is Hukuhara differen-tiable on I, then the real valued function t → diam(U (t)), t ∈ I is nondecreasingon I.

Proof If U is Hukuhara differentiable at a point t0 ∈ I, then there is a δ(t0) > 0,such that U (t0 + ∆t)− U (t0) and U (t0) − U (t0 − ∆t) are defined for 0 < ∆t <δ(t0). Since A − B, A,B ∈ Kc(Rn) is defined only if some translate of B iscontained in A, thus A − B exists only if diam(A) ≥ diam(B). Let t1, t2 ∈ Ibe fixed with t1 < t2. Then for each τ ∈ [t1, t2] there is a δ(τ ) > 0 such thatdiam(U (s)) ≤ diam(U (τ )), for s ∈ [τ−δ(τ ), τ ], and diam(U (s)) ≥ diam(U (τ )),for s ∈ [τ, τ + δ(τ )]. The collection

Iτ : τ ∈ [t1, t2], Iτ = (τ − δ(τ ), τ + δ(τ )),

forms an open covering of [t1, t2]. Choose a finite subcover Iτ1 , · · · , IτN withτi < τi+1. We then arrive at diam(U (t1)) ≤ diam(U (τ1)) and diam(U (τN )) ≤diam(U (t2)). There is no loss in generality to assume Iτi ∩ Iτi+1 6= ∅, i =1, · · · , N − 1. Thus for each i = 1, · · · , N − 1, there exists an si ∈ Iτi ∩ Iτi+1

with τi < si < τi+1, and hence

diam(U (τi)) ≤ diam(U (si)) ≤ diam(U (τi+1)).

Therefore we have diam(U (t1)) ≤ diam(U (t2)), which proves the proposition.The example given below, utilizes the above proposition.

Example 1.6.2 Let U (t) = (2 + sin t)Sn1 .(Sn

1 is the closed unit ball in Rn). Uis not Hukuhara differentiable on (0, 2π), since diam(U (t)) = 2(2 + sin t) is notnon-decreasing on (0, 2π).

Remark 1.6.1 Note that the existence of the limits in (1.6.1) and (1.6.2) isnot used in the proof of proposition 1.6.1. In fact, instead of the hypothesis thatU (t) is Hukuhara differentiable on I, one could substitute the assumption thatfor each t ∈ I, the differences U (t+ ∆t)−U (t) and U (t)−U (t−∆t) both existfor all sufficiently small ∆t > 0.

Also, we observe that a multifunctionU : I → Kc(Rn) is Hukuhara differentiableon I, and diam(U (t)) > 0 for t ∈ I need not imply U is monotone with respectto set inclusion. The following example illustrates this fact.

Example 1.6.3 If U (t) = [t, 2t], 0 < t < 1, then DH (t) = [1, 2], 0 < t < 1 andyet U (t1) 6⊂ U (t2) and U (t2) 6⊂ U (t1) for any t1, t2, 0 < t1 < t2 < 1.


We now proceed to prove the following standard result on the real line forthe Hukuhara derivative.

Proposition 1.6.2 The set valued mapping U is constant if and only if we have

DHU = 0 (1.6.3)

identically on I.

Proof If U is a constant, the result follows. Conversely, suppose (1.6.3) holds.For a fixed t0 ∈ (0, 1), if t > t0, using (1.3.15) we get

D[U (t), U (t0)] = D[U (t) − U (t0), θ],

which gives

limt→t+0

D[U (t), U (t0)]t− t0

= 0.

Similarly, if t < t0, we obtain

limt→t−0

D[U (t), U (t0)]t− t0

= 0

and hence,

limt→t0

∣∣∣∣D[U (t), U (t0)]

t − t0

∣∣∣∣ = 0 (1.6.4)

Fixing t1 ∈ (0, 1), from the inequality

|D[U (t1), U (t)] −D[U (t1), U (t0)]| ≤ D[U (t), U (t0)],

upon dividing by |t− t0| and considering (1.6.4), we obtain that the real valuedfunction t → D[U (t1), U (t)] is a constant, and since it is zero at t1, it must beidentically zero.

1.7 Integration

Let F : [0, 1] → Kc(Rn) and let S(F ) denote the set of integrable selectors of Fover [0, 1]. Then the Aumann integral of F over [0, 1] is defined as

∫ 1

0

F (t)dt =∫ 1

0

f(t)dt : f ∈ S(F ). (1.7.1)

If S(F ) 6= ∅, then the Aumann integral exists and F is said to be Aumannintegrable.

We shall say that F is integrally bounded on [0, 1] if there exists an integrablefunction g : [0, 1] → R such that

‖F (t)‖ ≤ g(t), for almost all t ∈ [0, 1]. (1.7.2)

If such an F has measurable selectors, then they are also integrable and S(F )is nonempty.

1.7 INTEGRATION 21

Theorem 1.7.1 If F : [0, 1] → Kc(Rn) is measurable and integrally bounded,then it is Aumann integrable over each [a, s] ⊂ [0, 1] with

∫ s

aF (t)dt ∈ Kc(Rn)

for all s ∈ [a, 1].

For a set valued mapping F as in Theorem 1.7.1, the Castaing RepresentationTheorem 1.5.1 applies and provides a sequence fi of integrable selectors whichare pointwise dense in F. Moreover,

∫ 1

0

F (t)dt =∫ 1

0

fi(t)dt : i = 1, 2, · · ·

(1.7.3)

and so we need only consider these selectors to evaluate∫ 1

0F (t)dt.

The Aumann integrability of a mapping F : [0, 1] → Kc(Rn) is fundamen-tally related to the Bochner integrability of its support function.

Theorem 1.7.2 Suppose that F : [0, 1] → Kc(Rn) is measurable. Then F isAumann integrable if and only if s(·, F (·)) : Sn−1× [0, 1] → C(Sn−1) is Bochnerintegrable, in which case

s

(·,∫ 1

0

F (t)dt)

=∫ 1

0

s(·, F (t))dt, (1.7.4)

where the integral on the right is Bochner integral.

From (1.7.4), we obtain the pointwise equality

s

(p,

∫ 1

0

F (t)dt)

=∫ 1

0

s(p, F (t))dt (1.7.5)

for all p ∈ Sn−1, where the integral on the right is now the Lebesgue integral.Using Theorem 1.7.2, we find that the Aumann integral satisfies

∫ 1

0

(F (t) +G(t))dt =∫ 1

0

F (t)dt+∫ 1

0

G(t)dt, (1.7.6)

∫ c

a

F (t)dt =∫ b

a

F (t)dt+∫ c

b

F (t)dt, 0 ≤ a ≤ b ≤ c ≤ 1, (1.7.7)

and ∫ 1

0

λF (t)dt = λ

∫ 1

0

F (t)dt, λ ∈ R, (1.7.8)

for all Aumann integrable F,G : [0, 1] → Kc(Rn), with

∫ 1

0

F (t)dt ⊆∫ 1

0

G(t)dt if F (t) ⊆ G(t) for all t ∈ [0, 1]. (1.7.9)

In addition, the Aumann integral uniquely determines its integrand.


Proposition 1.7.1 If F,G : [0, 1] → Kc(Rn) are Aumann integrable with∫ 1

0F (t)dt =

∫ 1

0G(t)dt, then F (t) = G(t) for almost all t ∈ [0, 1].

Similarly the following convergence properties can be established.

Theorem 1.7.3 Let Fi, F : [0, 1] → Kc(Rn), i = 1, 2, · · · , be measurable anduniformly integrally bounded. If Fi(t) → F (t) for all t ∈ [0, 1] as i→ ∞, then

Ai =∫ 1

0

Fi(t)dt→ A =∫ 1

0

F (t)dt as i → ∞. (1.7.10)

Theorem 1.7.4 Let Fi : [0, 1] → Kc(Rn), i = 1, 2, · · · , be measurable anduniformly integrally bounded, and suppose that Ai =

∫ 1

0Fi(t)dt → A ∈ Kc(Rn)

as i → ∞. Then there exists a measurable mapping F : [0, 1] → Kc(Rn) suchthat A =

∫ 1

0F (t)dt.

Theorem 1.7.5 If F,G : [0, 1] → Kc(Rn) are integrable, then so also is

D[F (·), G(·)] : [0, 1] → R

and

D

[∫ 1

0

F (t)dt,∫ 1

0

G(t)dt]≤∫ 1

0

D[F (t), G(t)]dt. (1.7.11)

Integration and differentiation of set valued mappings F : [0, 1] → Kc(Rn) areessentially inverse operations.

Proposition 1.7.2 Let F : [0, 1] → Kc(Rn) be measurable and integrally bounded.Then

lim∆t→0+

1∆t

∫ t0+∆t

t0

F (t)dt = F (t0), (1.7.12)

for almost all t0 ∈ [0, 1]. In particular, (1.7.12) holds at all t0 ∈ [0, 1) when Fis continuous.

Theorem 1.7.6 Let F : [0, 1] → Kc(Rn) be measurable and integrally bounded.Then A : [0, 1] → Kc(Rn) defined by

A(t) =∫ t

0

F (s)ds, (1.7.13)

for all t ∈ [0, 1] is Hukuhara differentiable for almost all t0 ∈ (0, 1), with theHukuhara derivative DHA(t0) = F (t0).

A counterpart of the first Fundamental Theorem of Calculus

F (t1) = F (t0) +∫ t1

t0

DHF (t)dt, 0 ≤ t0 ≤ t1 ≤ 1, (1.7.14)

holds for a Hukuhara differentiable F : [0, 1] → Kc(Rn) with continuous Hukuharaderivative DHF on [0, 1].

1.8 SUBSETS OF BANACH SPACES 23

1.8 Subsets of Banach Spaces

Let E be a real Banach space with the norm ‖·‖ and the metric generated by it.Let (2E)b be a collection of all nonempty bounded subsets of E with Hausdorffpseudometric,

D[A,B] = max[supx∈B

d(x,A), supy∈A

d(y,B)], (1.8.1)

where d(x,A) = inf[d(x, y) : y ∈ A], A,B ∈ (2E)b. Denote by K(E) (Kc(E))the collection of all nonempty compact (compact convex) subsets of E, whichare considered as subspaces of (2E)b. We note that on K(E), (Kc(E)) thetopology of the space (2E)b induces the Hausdorff metric. Also K(E) (Kc(E))is a complete metric space and K(E) (Kc(E)) is separable if E is separablespace.

It is known that if the space Kc(E) is equipped with the natural algebraicoperations of addition and nonnegative scalar multiplication, then Kc(E) be-comes a semilinear metric space which can be embedded as a complete cone ina corresponding Banach space. See Tolstonogov [1], and Brandao et. el [1].

Let T = [0, a], a > 0. Then the mapping F : T → K(E) is said to bestrongly measurable, if it is almost everywhere (a.e.) in T a point-wise limit ofthe sequence Fn : T → K(E), n ≥ 1, of step mappings.

If D(F (t),Θ) ≤ λ(t), a.e. on T, where λ(t) is summable on T and Θ is thezero element of E, which is regarded as a one-point set, then F is said to beintegrally bounded on T. For a set-valued mapping F : T → E, we shall denoteby (A)

∫T0F (s)ds the integral in the sense of Aumann on the measurable set

T0 ⊂ T, that is,

(A)∫

T0

F (s)ds =[∫

T0

f(s)ds : f is a Bochner integrable selector of F].

For a strongly measurable mapping F : T → Kc(E), the integral∫T0F (s)ds in

the sense of Bochner is introduced in a natural way, since as pointed out earlier,Kc(E) can be embedded as a complete cone into a corresponding Banach Space.

If a multifunction F : T → E with compact convex values is strongly mea-surable and integrally bounded then

∫

T0

F (s)ds = (A)∫

T0

F (s)ds, (1.8.2)

on the measurable set T0 ⊂ T.Let A,B ∈ Kc(E). The set C ∈ Kc(E) satisfying A = B+C is known as the

Hukuhara difference of the sets A and B and is denoted by the symbol A −B.We say that the mapping F : T → Kc(E) has a Hukuhara derivative DHF (t0)at a point t0 ∈ T if there exists an element DHF (t0) ∈ Kc(E) such that

limh→0+

F (t0 + h) − F (t0)h

, and limh→0+

F (t0) − F (t0 − h)h


exist in the topology of Kc(E) and are equal DHF (t0).By embedding Kc(E) as a complete cone in a corresponding Banach space

and taking into account the result on differentiation of Bochner integral, we findthat if

F (t) = X0 +∫ t

0

Φ(s) ds, X0 ∈ Kc(E), (1.8.3)

where Φ : T → Kc(E) is integrable in the sense of Bochner, then DHF (t) existsa.e. on T and the equality

DHF (t) = Φ(t) a.e. on T, (1.8.4)

holds.Let µ be Lebesgue measure, L be a σ− algebra of Lebesgue measurable

subsets of J = [t0, b], t0 ≥ 0, b ∈ (t0,∞), and E be a metrizable space.A multiplication F : J → E with closed values is measurable if the set

F−1(B) = t ∈ J ;F (t)∩B 6= ∅ is measurable for each closed subset B of E. IfF : Y → E, where Y is a topological space, then F is measurable implies thatF is measurable when Y is assigned the σ−algebra BY of Borel subsets of Y.Similarly, if F : J × Y → E, then the measurability of F is defined in terms ofthe product of σ−algebras L ⊗ BY generated by the sets A× B, where A ∈ L

and B ∈ BY . If E is separable then for multifunction F : J → E with compactvalues the definitions of strong measurability and measurability are equivalent.

A multifunction from a topological space Y into space E is upper semicon-tinuous (usc) at a point y0 ∈ Y if, for any ε > 0, there exists a neighborhoodU (y0) of the point y0 such that F (y) ⊂ F (y0) + ε ·B for all y ∈ U (y0), where Bis unit ball of E.

A multifunction F : Y → E is said to be usc if it is usc at any point y0 ∈ E.For a multifunction F : Y → E with compact values the definition of usc is

equivalent to the following: the set F−1(U ) is closed for each closed subset Uof E.

Let us recall that the Hausdorff metric (1.8.1) satisfies the following proper-ties:

D[U +W,V +W ] = D[U, V ], (1.8.5)

D[λU, λV ] = λD[U, V ], (1.8.6)

D[U, V ] ≤ D[U,W ] +D[W,V ], (1.8.7)

for all U, V,W ∈ Kc(E) and λ ∈ R+. Also for U ∈ Kc(E), we set

D[U,Θ] = ‖U‖ = sup[‖u‖ : u ∈ U ]. (1.8.8)

1.9 Notes and Comments

The preliminary material including calculus of the set valued maps assembledin this chapter is taken from Arstein [1], Banks and Jacobs [1], De Blasi and

1.9 NOTES AND COMMENTS 25

Iervolino [1], Diamond and Kloeden [1], Hukuhara [1,2], Radstrom [1] and Tol-stonogov [1]. See also for further details Aubin and Frankowska [1], Aumann[1], Castaing and Valadier [1], Debreu [1], Hermes [1], Hausdorff [1], Lay [1] andRockafeller [1].


Chapter 2

Basic Theory

2.1 Introduction

This chapter is devoted to the basic theory of set differential equations (SDEs).We begin Section 2.2 with the formulation of the initial value problem of SDEsin the metric space (Kc(Rn), D). Utilizing the properties of the Hausdorff metricD[·, ·] and employing the known theory of differential and integral inequalities,we establish a variety of comparison results, that are required for later discus-sion. Section 2.3 deals with the convergence of successive approximations of theinitial value problem (IVP) under the general uniqueness assumption of Perrontype, using the comparison function that is rather instructive. The continuousdependence of solutions relative to the initial conditions is also studied underthe same conditions. In Section 2.4, we investigate an existence result of Peano’stype and then consider the existence of extremal solutions of SDE. For this pur-pose, one needs to introduce a partial order in (Kc(Rn), D), prove the requiredcomparison result for strict inequalities, and then, utilizing it, discuss the ex-istence of extremal solutions. Having the notion of maximal solution for SDE,we then prove the comparison result analogous to the well known comparisontheorem in the ordinary differential system.

The monotone iterative technique is considered for SDE in Section 2.5, em-ploying the method of upper and lower solutions. The results considered are sogeneral that they contain several special cases of interest. Section 2.6 containsa global existence result, and Section 2.7 considers the error estimate betweenthe solutions and approximate solutions of SDEs.

In Section 2.8, we discuss the IVP of SDE without assuming any continuity ofthe function involved and obtain the existence of an Euler solution which reducesto the actual solution when the continuity assumption is made. Using theproximal normal aiming condition, we study in Section 2.9, the flow invarianceof solutions relative to a closed set. Here weak and strong flow invarianceresults are considered in terms of nonsmooth analysis. Section 2.10 establishesan existence result for SDE when the function involved is upper semicontinuous.

27

28 CHAPTER 2. BASIC THEORY

The conditions are also provided to obtain the function involved in SDE fromthe multifunction with compact values only by suitably utilizing convexification.Notes and comments are provided in Section 2.11.

2.2 Comparison Principles

Let us consider the initial value problem (IVP) for the set differential equation

DHU = F (t, U ), U (t0) = U0 ∈ Kc(Rn), t0 ≥ 0, (2.2.1)

where F ∈ C[R+ ×Kc(Rn),Kc(Rn)] and DHU is the Hukuhara derivative of U.The mapping U ∈ C1[J,Kc(Rn)] where J = [t0, t0 + a], a > 0, is said to be

a solution of (2.2.1) on J , if it satisfies (2.2.1) on J .Since U (t) is continuously differentiable, we have

U (t) = U0 +∫ t

t0

DHU (s)ds, t ∈ J. (2.2.2)

We therefore associate with the IVP(2.2.1) the following integral equation

U (t) = U0 +∫ t

t0

F (s, U (s))ds, t ∈ J, (2.2.3)

where the integral in (2.2.3) is the Hukuhara integral. Observe also that U (t)is a solution of (2.2.1) if and only if it satisfies (2.2.3) on J.

Utilizing the properties of the Hausdorff metric D[·, ·] and the integral, andemploying the known theory of differential and integral inequalities for ordinarydifferential equations, we shall first establish the following comparison principles,which we need for later discussion.

Theorem 2.2.1 Assume that F ∈ C[J × Kc(Rn),Kc(Rn)] and t ∈ J, U, V ∈Kc(Rn),

D[F (t, U ), F (t, V )] ≤ g(t,D[U, V ]), (2.2.4)

where g ∈ C[J×R+,R+] and g(t, w) is monotone nondecreasing in w for each t ∈J. Suppose further that the maximal solution r(t, t0, w0) of the scalar differentialequation

w′ = g(t, w), w(t0) = w0 ≥ 0, (2.2.5)

exists on J. Then, if U (t), V (t) are any two solutions through (t0, U0), (t0, V0)respectively on J, it follows that

D[U (t), V (t)] ≤ r(t, t0, w0), t ∈ J, (2.2.6)

provided D[U0, V0] ≤ w0.

2.2 COMPARISON PRINCIPLES 29

Proof Set m(t) = D[U (t), V (t)], so that m(t0) = D[U0, V0] ≤ w0. Then, inview of the properties of the metric D, we get

m(t) = D

[U0 +

∫ t

t0

F (s, U (s))ds, V0 +∫ t

t0

F (s, V (s))ds]

≤ D

[U0 +

∫ t

t0

F (s, U (s))ds, U0 +∫ t

t0

F (s, V (s))ds]

+D[U0 +

∫ t

t0

F (s, V (s))ds, V0 +∫ t

t0

F (s, V (s))ds]

= D

[∫ t

t0

F (s, U (s))ds,∫ t

t0

F (s, V (s))ds]

+D[U0, V0].

Now using the properties of the integrals and condition (2.2.4), we observe that,

m(t) ≤ m(t0) +∫ t

t0

D [F (s, U (s)), F (s, V (s))] ds

≤ m(t0) +∫ t

t0

g(s,D[U (s), V (s)])ds

= m(t0) +∫ t

t0

g(s,m(s))ds, t ∈ J.

Now applying Theorem 1.9.2 given in Lakshmikantham and Leela [1], we con-clude that

m(t) ≤ r(t, t0, w0), t ∈ J.

This establishes Theorem 2.2.1.

Remark 2.2.1 If we employ the theory of differential inequalities instead ofintegral inequalities, we can dispense with the monotone character of g(t, w)assumed in Theorem 2.2.1. This is the content of the next comparison principle.

Theorem 2.2.2 Let the assumptions of Theorem 2.2.1 hold except the nonde-creasing property of g(t, w) in w. Then the conclusion (2.2.6) is valid.

Proof For small h > 0, the Hukuhara differences U (t+h)−U (t), V (t+h)−V (t)exist, and we have for t ∈ J,

m(t+ h) −m(t) = D[U (t+ h), V (t+ h)] −D[U (t), V (t)].

Using the property (1.3.8) for D, we get

D[U (t + h), V (t + h)] ≤ D[U (t+ h), U (t) + hF (t, U (t))]+D[U (t) + hF (t, U (t)), V (t + h)],


and

D[U (t) + hF (t, U (t)), V (t+ h)]≤ D[V (t) + hF (t, V (t)), V (t + h)]

+D[U (t) + hF (t, U (t)), V (t) + hF (t, V (t))].

Also, we observe that

D[U (t) + hF (t, U (t)), V (t) + hF (t, V (t))]≤ D[U (t) + hF (t, U (t)), U (t) + hF (t, V (t))]

+D[U (t) + hF (t, V (t)), V (t) + hF (t, V (t))]= D[hF (t, U (t)), hF (t, V (t))] +D[U (t), V (t)].

Hence, it follows that

m(t + h) −m(t)h

≤ 1hD[U (t+ h), U (t) + hF (t, U (t))]

+1hD[V (t) + hF (t, V (t)), V (t+ h)]

+1hD[hF (t, U (t)), hF (t, V (t))]

and consequently, in view of the properties of D and the fact that U (t), V (t)are solutions of (2.2.1), we find that

D+m(t) ≤ lim suph→0+

1h

[m(t + h) −m(t)]

≤ lim suph→0+

D

[U (t+ h) − U (t)

h, F (t, U (t))

]

+ lim suph→0+

D

[F (t, V (t)),

V (t+ h) − V (t)h

]

+D[F (t, U (t)), F (t, V (t))].

Here, we have used the fact that

D[U (t + h), U (t) + hF (t, U (t))] = D[U (t) + Z(t, h), U (t) + hF (t, U (t))]= D[Z(t, h) + U (t), U (t) + hF (t, U (t))]= D[Z(t, h), hF (t, U (t))]= D[U (t+ h) − U (t), hF (t, U (t))].

The conclusion (2.2.6) follows from Theorem 1.4.1 in Lakshmikantham and Leela[1].

The next comparison result provides an estimate under weaker assumptions.

Theorem 2.2.3 Assume that F ∈ C[J ×Kc(Rn),Kc(Rn)] and

lim suph→0

1h

[D[U + hF (t, U ), V + hF (t, V )] −D[U, V ]] ≤ g(t,D[U, V ]), t ∈ J,

2.2 COMPARISON PRINCIPLES 31

where U, V ∈ Kc(Rn), g ∈ C[J × R+,R]. The maximal solution r(t, t0, w0) of(2.2.5) exists on J. Then the conclusion of Theorem 2.2.1 is valid.

Proof Proceeding as in the proof of Theorem 2.2.2, we see that

m(t + h) −m(t)= D[U (t+ h), V (t + h)] −D[U (t), V (t)]≤ D[U (t+ h), U (t) + hF (t, U (t))]

+D[V (t) + hF (t, V (t)), V (t+ h)]+D[U (t) + hF (t, U (t)), V (t) + hF (t, V (t))] −D[U (t), V (t)].

D+m(t) = lim suph→0+

1h

[m(t + h) −m(t)]

≤ lim suph→0+

1h

[D[U (t) + hF (t, U (t)),

V (t) + hF (t, V (t))] −D[U (t), V (t)]]

+ lim suph→0+

D

[U (t+ h) − U (t)

h, F (t, U (t))

]

+ lim suph→0+

D

[F (t, V (t)),

V (t+ h) − V (t)h

]

≤ g(t,D[U (t), V (t)]) = g(t,m(t)), t ∈ J.

The conclusion follows as before by Theorem 1.4.1 in Lakshmikantham andLeela [1] and the proof is complete.

We wish to remark that in Theorem 2.2.3, g(t, w) need not be nonnegativeand therefore the estimate in Theorem 2.2.3 would be finer than the estimatesin Theorems 2.2.1 and 2.2.2.

As special cases of Theorems 2.2.1, 2.2.2 and 2.2.3, we have the followingimportant corollaries.

Corollary 2.2.1 Assume that F ∈ C[J ×Kc(Rn),Kc(Rn)] and either

(a) D[F (t, U ), θ] ≤ g(t,D[U, θ]) or

(b) lim suph→0

1h

[D[U + hF (t, U ), θ] − D[U, θ]] ≤ g(t,D[U, θ]), where g ∈ C[J ×

R+,R].

Then, if D[U0, θ] ≤ w0, we have

D[U (t), θ] ≤ r(t, t0, w0), t ∈ J,

where r(t, t0, w0) is the maximal solution of (2.2.5) on J.


Corollary 2.2.2 The function g(t, w) = λ(t)w, λ(t) ≥ 0 and continuous isadmissible in Theorem 2.2.1 to give

m(t) ≤ m(t0) +∫ t

t0

λ(s)m(s)ds, t ∈ J.

Then the Gronwall inequality implies

m(t) ≤ m(t0) exp[∫ t

t0

λ(s)ds], t ∈ J,

which shows that (2.2.6) reduces to

D[U (t), V (t)] ≤ D[U0, V0] exp[∫ t

t0

λ(s)ds], t ∈ J.

Corollary 2.2.3 The function g(t, w) = −λ(t)w, with λ(t) as in Corollary2.2.2, is also admissible in Theorem 2.2.3, and we get,

D[U (t), V (t)] ≤ D[U0, V0] exp[−∫ t

t0

λ(s)ds], t ∈ J.

If λ(t) = λ > 0, we find that

D[U (t), V (t)] ≤ D[U0, V0] e−λ(t−t0), t ∈ J.

If J = [t0,∞), we see that limt→∞D[U (t), V (t)] = 0, showing the advantage ofTheorem 2.2.3.

2.3 Local Existence and Uniqueness

We shall begin by proving the existence and uniqueness result under assumptionsmore general than the Lipschitz type condition, which exhibits the idea of thecomparison principle.

Theorem 2.3.1 Assume that

(a) F ∈ C[R0,Kc(Rn)] and D[F (t, U ), θ] ≤ M0 where R0 = J × B(U0, b),B(U0, b) = [U ∈ Kc(Rn) : D[U,U0] ≤ b] on R0;

(b) g ∈ C [J × [0, 2b],R+] , g(t, w) ≤ M1 on J × [0, 2b], g(t, 0) ≡ 0, g(t, w) isnondecreasing in w for each t ∈ J and w(t) ≡ 0 is the only solution of

w′ = g(t, w), w(t0) = 0, (2.3.1)

on J ;

(c) D[F (t, U ), F (t, V )] ≤ g(t,D[U, V ]) on R0.

2.3 LOCAL EXISTENCE AND UNIQUENESS 33

Then the successive approximations defined by

Un+1(t) = U0 +∫ t

t0

F (s, Un(s))ds, n = 0, 1, 2, . . ., (2.3.2)

exist on J0 = [t0, t + η), where η = mina, bM , M = maxM0,M1, as

continuous functions and converge uniformly to the unique solution U (t) of theIVP (2.2.1) on J0.

Proof Using the properties of Hausdorff metric, we get by induction,

D[Un+1(t), U0] = D

[U0 +

∫ t

t0

F (s, Un(s)) ds, U0

]

= D

[∫ t

t0

F (s, Un(s)) ds, θ]

≤∫ t

t0

D [F (s, Un(s)), θ] ds

≤ M0(t − t0) ≤M0a ≤ b,

and consequently, the successive approximations Un(t) are well defined on J0.We shall next define the successive approximations of (2.3.1) as follows:

w0(t) = M (t − t0),

wn+1(t) =∫ t

t0

g(s, wn(s)) ds, t ∈ J0, n = 0, 1, 2, . . . (2.3.3)

An easy induction, in view of the monotone character of g(t, w) in w, provesthat wn(t) are well defined and

0 ≤ wn+1(t) ≤ wn(t), t ∈ J0. (2.3.4)

Since |w′n(t)| ≤ g(t, wn−1(t)) ≤ M1, we conclude by Ascoli-Arzela Theorem and

the monotonicity of the sequence wn(t) that

limn→∞

wn(t) = w(t),

uniformly on J0. It is also clear that w(t) satisfies (2.3.1) and therefore bycondition (b), w(t) ≡ 0 on J0.

We observe that

D[U1(t), U0] ≤∫ t

t0

D[F (s, U0), θ] ds ≤ M (t− t0) ≡ w0(t).

Assume that for some k > 1, we have

D[Uk(t), Uk−1(t)] ≤ wk−1(t), on J0.


Since

D[Uk+1(t), Uk(t)] ≤∫ t

t0

D[F (s, Uk(s)), F (s, Uk−1(s))] ds,

using condition (c) and the monotone character of g(t, w), we get

D[Uk+1(t), Uk(t)] ≤∫ t

t0

g(s,D[Uk(s), Uk−1(s)]) ds

≤∫ t

t0

g(s, wk−1(s)) ds = wk(t).

Thus by induction, the estimate

D[Un+1(t), Un(t)) ≤ wn(t), t ∈ J0, (2.3.5)

is true for all n.Letting u(t) = D[Un+1(t), Un(t)], t ∈ J0, the proof of Theorem of 2.2.2 shows

thatD+u(t) ≤ g (t,D[Un(t), Un−1(t)]) ≤ g(t, wn−1(t)), t ∈ J0.

Now let n ≤ m. Setting v(t) = D[Un(t), Um(t)], we obtain from (2.3.2)

D+v(t) ≤ D[DHUn(t), DHUm(t)] = D[F (t, Un−1(t)) , F (t, Um−1(t))]≤ D [F (t, Un(t)) , F (t, Un−1(t))] +D [F (t, Un(t)) , F (t, Um(t))]

+D[F (t, Um(t)) , F (t, Um−1(t))]≤ g (t, wn−1(t)) + g (t, wm−1(t)) + g (t,D[Un(t), Um(t)])≤ g (t, v(t)) + 2g (t, wn−1(t)) , t ∈ J0.

Here we have used the arguments of the proof of Theorem 2.2.2, the mono-tone character of g(t, w) and the fact that wm−1 ≤ wn−1, since n ≤ m and wn(t)is a decreasing sequence. The comparison Theorem 1.4.1 in Lakshmikanthamand Leela [1] yields the estimate

v(t) ≤ rn(t), t ∈ J0,

where rn(t) is the maximal solution of

r′n = g(t, rn) + 2g(t, wn−1(t)), rn(t0) = 0, (2.3.6)

for each n. Since as n → ∞, 2g (t, wn−1(t)) → 0 uniformly on J0, it followsby Lemma 1.3.1 in Lakshmikantham and Leela [1] that rn(t) → 0 as n → ∞uniformly, on J0. This implies from (2.2.5) and the definition of v(t) that Un(t)converges uniformly to U (t), and clearly U (t) is a solution of (2.2.1).

To show uniqueness, let V (t) be another solution of (2.2.1), on J0. Thensetting m(t) = D[U (t), V (t)] and noting that m(t0) = 0, we get, as before,

D+m(t) ≤ g (t,m(t)) , t ∈ J0,

2.3 LOCAL EXISTENCE AND UNIQUENESS 35

and m(t) ≤ r(t, t0, 0), t ∈ J0, by Theorem 2.2.1. By assumption r(t, t0, 0) ≡ 0,we get U (t) ≡ V (t) on J0, proving the theorem.

We shall discuss, in the next result, the continuous dependence of solutionswith initial values. We need the following lemma before we proceed.

Lemma 2.3.1 Let F ∈ C[J ×Kc(Rn),Kc(Rn)] and let

G(t, r) = max[D [F (t, U ), θ] : D[U,U0] ≤ r].

Assume that r∗(t, t0, 0) is the maximal solution of

w′ = G(t, w), w(t0) = 0, on J.

Let U (t) = U (t, t0, U0) be the solution of (2.2.1). Then

D[U (t), U0] ≤ r∗(t, t0, 0), t ∈ J.

Proof Define m(t) = D[U (t), U0], t ∈ J . Then Corollary 2.2.1 shows that

D+m(t) ≤ D[DHU (t), θ]= D[F (t, U (t)) , θ]≤ max

D[U,U0]≤m(t)D[F (t, U ), θ]

= G(t,m(t)).

This implies by Theorem 1.4.1 in Lakshmikantham and Leela [1] that

D[U (t), U0] ≤ r∗(t, t0, 0), t ∈ J,

proving the lemma.

Theorem 2.3.2 Suppose that the assumptions (a), (b), (c) of Theorem 2.3.1hold. Assume further that the solutions w(t, t0, w0) of (2.2.5) through everypoint (t0, w0) are continuous with respect to (t0, w0). Then the solutions U (t) =U (t, t0, U0) of (2.2.1) are continuous relative to (t0, U0).

Proof Let U (t) = U (t, t0, U0), V (t) = V (t, t0, V0), U0, V0 ∈ Kc(Rn), be the twosolutions of (2.2.1). Then defining m(t) = D[U (t), V (t)], we get from Theorem2.2.1, the estimate

D[U (t), V (t)] ≤ r (t, t0, D[U0, V0]) , t ∈ J.

Since limU0→V0 r (t, t0, D[U0, V0]) = r(t, t0, 0) uniformly on J and by hypothesisr(t, t0, 0) ≡ 0, it follows that limU0→V0 U (t, t0, U0) = V (t, t0, V0) uniformly andhence continuity of U (t, t0, U0) relative to U0 is valid.

To prove the continuity relative to t0, we let U (t) = U (t, t0, U0), V (t) =V (t, τ0, U0) be the two solutions of (2.2.1) and let τ0 > t0. As before, settingm(t) = D[U (t), V (t)], noting thatm(τ0) = D[U (τ0), U0], we obtain from Lemma2.3.1,

m(τ0) ≤ r∗(τ0, t0, 0),


and consequently, by Theorem 2.2.1, we arrive at

m(t) ≤ r(t), t ≥ τ0,

where r(t) = r(t, τ0, r∗(τ0, t0, 0)) is the maximal solution of (2.2.5) through(τ0, r∗(τ0, t0, 0)). Since r∗(t0, t0, 0) = 0, we have

limτ0→t0

r(t, τ0, r∗(τ0, t0, 0)) = r(t, t0, 0),

uniformly on J . By hypothesis r(t, t0,0) ≡ 0 which proves the continuity ofU (t, t0,U0), with respect to t0 and the proof is complete.

2.4 Local Existence and Extremal Solutions

We begin by proving the local existence result corresponding to Peano’s theoremfor the IVP (2.2.1). For this purpose, we need the Ascoli-Arzela theorem suitablygeneralized in the present set up, which we state below, see Morales[1].

Since Kc(Rn) is a closed subset of K(Rn), and the family of compact setsincluded in a closed ball of Rn is compact, the following Ascoli-Arzela theoremholds.

Theorem 2.4.1 If Un(t) is a sequence of equicontinuous and equiboundedmultimappings defined on an interval J, we can extract a subsequence that con-verges uniformly to a continuous multimapping U (t) on J.

Using Theorem 2.4.1, we can prove the local existence result for the IVP(2.2.1).

Theorem 2.4.2 Assume that F ∈ C[R0,Kc(Rn)] where R0 = J × B[U0, b],B[U0, b] = U ∈ Kc(Rn) : D[U,U0] ≤ b and D[F (t, U ), θ] ≤ M on R0. Thenthere exists at least one solution U (t) for the IVP (2.2.1) on J0 = [t0, t0 + α],where α = mina, b

M . Here, as before, J = [t0, t0 + a], a > 0.

Proof Let U0 ∈ C1[[t0−δ, t0],Kc(Rn)], δ > 0 such that U0(t0) = U0, D[U0(t), U0] ≤b and D[DHU0(t), θ] ≤ M. Consider 0 < ε ≤ δ and define

[Uε(t) = U0(t), t0 − δ ≤ t ≤ t0,

Uε(t) = U0 +∫ t

t0F (s, Uε(s − ε))ds, t0 ≤ t ≤ t0 + α1,

(2.4.1)

where α1 = minα, ε. We then have, using the properties (1.3.8), and (1.7.11)of D,

D[Uε(t), U0] = D

[U0 +

∫ t

t0

F (s, Uε(s− ε))ds, U0

]

= D

[∫ t

t0

F (s, Uε(s − ε))ds, θ]

≤∫ t

t0

D[F (s, Uε(s− ε)), θ]ds

≤ M (t− t0) ≤Mα1 ≤ b.

2.4 LOCAL EXISTENCE AND EXTREMAL SOLUTIONS 37

Further more,DHUε(t) = F (t, Uε(t− ε)) on [t0, t0 + α1], (2.4.2)

and therefore,D[DHUε(t), θ] ≤ M. (2.4.3)

If α1 < α, we can use (2.4.1) to extend Uε(t) satisfying the relations (2.4.2)and (2.4.3) on [t0 − δ, t0 +α2] where α2 = minα1, 2ε. Continuing this processwe arrive at an a.e. differentiable function Uε(t) such that (2.4.1), (2.4.2) and(2.4.3) hold on [t0 − δ, t0 + α]. Thus Uε(t) forms a family of equicontinuousand uniformly bounded functions. By Theorem 2.4.1, we obtain a decreasingsequence εn such that εn → 0 as n → ∞ and U (t) = limn→∞Uεn (t) existsuniformly for t0 − δ ≤ t ≤ t0 +α. Since F is uniformly continuous, we show thatF (t, Uεn(t− εn)) → F (t, U (t)) uniformly, as n → ∞ on J0. This allows for termby term integration in (2.4.1) with ε = εn and α1 = α which yields

U (t) = U0 +∫ t

t0

F (s, U (s))ds, t ∈ J0.

Hence U (t) is a solution of (2.2.1) on J0 and the proof is complete.In order to discuss the existence of extremal solutions for the IVP (2.2.1),

we require a comparison result which demands introducing a partial order inthe metric space (Kc(Rn), D).

We denote byK(K0) the subfamily ofKc(Rn) consisting of sets U ∈ Kc(Rn)such that any u ∈ U is a nonnegative (positive) vector of n−components satis-fying ui ≥ 0 (ui > 0) for i = 1, 2, · · ·n. Thus K is a cone in Kc(Rn) and K0

is the nonempty interior of K. We can therefore induce a partial ordering inKc(Rn) as follows.

Definition 2.4.1 For any U and V ∈ Kc(Rn), if there exists a Z ∈ Kc(Rn)such that Z ∈ K(K0) and

U = V + Z, (2.4.4)

then, we write U ≥ V (U > V ). Similarly, one can define U ≤ V (U < V ).

We are now in a position to define the maximal and minimal solutions of(2.2.1).

Definition 2.4.2 Let R(t) be a solution of the set differential equation (2.2.1).Then we say that R(t) is the maximal solution of (2.2.1) if, for every solutionU (t) of (2.2.1) existing on J0, we have

U (t) ≤ R(t), t ∈ J0. (2.4.5)

We define the minimal solution of (2.2.1) similarly by reversing the inequalityin (2.4.5).

We now state and prove the basic comparison result.



(i) F ∈ C[R+×Kc(Rn),Kc(Rn)], F (t, U ) is monotone nondecreasing in U foreach t ∈ R+; that is, whenever U ≤ V, we haveF (t, U ) ≤ F (t, V ), t ∈ R+;

(ii) V,W ∈ C1[R+,Kc(Rn)],

DHV < F (t, V ) and DHW ≥ F (t,W ), t ∈ R+; (2.4.6)

(iii) V (t0) < W (t0).

Then,V (t) < W (t), t ≥ t0. (2.4.7)

Proof Let t1 > 0 be the supremum of all positive numbers δ > 0 such thatV (t0) < W (t0) implies V (t) < W (t) on [t0, δ].

Clearly t1 > t0 and V (t1) ≤ W (t1). Now using the nondecreasing nature ofF (t, U ) in U and the assumption (ii), we arrive at

DHV (t1) < F (t1, V (t1)) ≤ F (t1,W (t1)) ≤ DHW (t1).

It therefore follows that there exists an η > 0 satisfying

V (t) −W (t) > V (t1) −W (t1), t1 − η < t < t1.

This implies that t1 > t0 cannot be the supremum due to the continuity of thefunctions involved, and hence the relation (2.4.7) holds, completing the proof.

Remark 2.4.1 It is clear that the inequalities 2.4.6 can be replaced by

DHV ≤ F (t, V ) and DHW > F (t,W ), t ∈ R+,

respectively, to get the conclusion of Theorem 2.4.3.

We are now ready to prove the existence of extremal solutions of (2.2.1).

Theorem 2.4.4 Let the assumptions of Theorem 2.4.2 hold and suppose furtherF (t, U ) is nondecreasing in U for each t ∈ J. Then the IVP (2.2.1) possessesthe extremal solutions on J0 = [t0, t0 + α0], where α0 = mina, b

2M+b.

Proof Let ε = (ε1, ε2, · · · , εn) > 0 be such that ‖ε‖ ≤ b2 . Then consider for each

positive integer N, the following IVP for t ∈ J,

DHU = F (t, U ) +ε

N, U (t0) = U0 +

ε

N. (2.4.8)

We observe that FN (t, U ) = F (t, U ) + εN

is defined and is continuous on Rε =J × B[U0,

b2], where B[U0,

b2] = U ∈ Kc(Rn) : D[U,U0 + ε

N] ≤ b

2. Also

2.4 LOCAL EXISTENCE AND EXTREMAL SOLUTIONS 39

D[FN (t, U ), θ] ≤ M + b2 on Rε. Hence we deduce from Theorem 2.4.2 that

(2.4.8) has a solution UN (t, ε) ∈ Kc(Rn) on J0. For 0 < ε2 < ε1 ≤ ε, we see that

UN (t0, ε2) < UN (t0, ε1),

DHUN (t, ε2) ≤ F (t, UN (t, ε2)) +ε2N,

DHUN (t, ε1) > F (t, UN (t, ε1)) +ε2N, on J0.

We can apply the Theorem 2.4.3 (in fact Remark 2.4.1) to get

UN (t, ε2) < UN (t, ε1), on J0.

Since the family of functions UN (t, ε) is equicontinuous and uniformly boundedon J0, it follows by Theorem 2.4.1 that there exists a decreasing sequence ε

Nk

such that εNk

→ 0 as k → ∞ and the uniform limit

R(t) = limk→∞

UNk (t, ε), (2.4.9)

exists on J0. Obviously R(t0) = U0. The uniform continuity of F implies thatF (t, UNk(t, ε)) tends uniformly to F (t, R(t)) as k → ∞, and thus term by termintegration is applicable to

UNk (t, ε) = U0 +ε

Nk+∫ t

t0

F (s, UNk(t, ε))ds,

which in turn yields that the limit R(t) is a solution of (2.2.1) on J0.We shall next show that R(t) is the required maximal solution of IVP (2.2.1)

on J0. For this purpose, we observe that

U (t0) = U0 < U0 +ε

N= UN (t0, ε),

DHU (t) < F (t, U (t)) +ε

N,

DHUN (t, ε) ≥ F (t, UN (t, ε)) +ε

N, on J0.

We then obtain from Theorem 2.4.3 (or Remark 2.4.1), that

U (t) < UN (t, ε), on J0.

The uniqueness of maximal solution R(t) shows that UN (t, ε) tends uniformlyto R(t) on J0 as N → ∞. This proves that R(t) is the maximal solution of IVP(2.2.1). Similarly, one can prove the existence of the minimal solution ρ(t) ofIVP (2.2.1) by considering the IVP

DHU = F (t, U )− ε

N, U (t0) = U0 −

ε

N,

and proceeding with suitable change of arguments. Hence the proof is complete.Having the notion of maximal solution and how to obtain it, one can prove

the following comparison result analogous to the well known comparison resultin ordinary differential system.


Theorem 2.4.5 Assume that the conditions of Theorem 2.4.4 are satisfied.Suppose that M ∈ C1[J,Kc(Rn

+)] and

DHM (t) ≤ F (t,M (t)), M (t0) ≤ U0. (2.4.10)

Then M (t) ≤ R(t) on J0.

Proof Let UN (t, ε) be a solution of (2.4.8) on J0. Then

M (t0) < U0 +ε

N,

DHUN (t, ε) > F (t, UN (t, ε)) on J0.

This together with (2.4.10), yields, by Theorem 2.4.3,

M (t) < UN (t, ε) on J0.

The last inequality, in view of (2.4.9) proves the assertion of the Theorem.

2.5 Monotone Iterative Technique

The method of lower and upper solutions coupled with the monotone iterativetechnique offers an effective and flexible mechanism to provide constructive ex-istence results for nonlinear problems. In the development of this technique,one uses the fact that when the right hand side is not monotone, it can be mademonotone by adding a suitable function. A generalization of this idea has beenrecently developed where one considers the situation when the right hand sidecan be split into the difference of two monotone functions. This unified settingprovides very general results which cover several known cases of importance inaddition to providing new results.

In this section, we develop the monotone iterative technique, in the samegeneral set up. See Ladde, Lakshmikantham and Vatsala [1], Pao [1], and Koksaland Lakshmikantham [1] for details.

In the previous section we introduced a partial ordering in the metric space(Kc(Rn), D) and proved the comparison Theorem 2.4.3 for strict inequalities,which was essential for discussing the existence of extremal solutions for IVPsof set differential equations. We now prove first the following basic result onnonstrict set differential inequalities.


(i) V,W ∈ C1[R+,Kc(Rn)], F ∈ C[R+ ×Kc(Rn),Kc(Rn)], F (t,X) is mono-tone nondecreasing in X for each t ∈ R+ and

DHV ≤ F (t, V ), DHW ≥ F (t,W ), t ∈ R+;

(ii) for any X,Y ∈ Kc(Rn) such that X ≥ Y, t ∈ R+,

F (t,X) ≤ F (t, Y ) + L(X − Y )

for some L > 0.

2.5 MONOTONE ITERATIVE TECHNIQUE 41

Then V (t0) ≤ W (t0) implies

V (t) ≤ W (t), t ≥ t0. (2.5.1)

Proof Let ε = (ε1, ε2, . . . , εn) > 0 and define W = W + εe2Lt. Since V (t0) ≤W (t0) < W (t0), it is enough to prove that

V (t) < W (t), t ≥ t0, (2.5.2)

to arrive at the conclusion (2.5.1) in view of the fact ε > 0 is arbitrary.Let t1 > 0 be the supremum of all positive numbers δ > 0 such that V (t0) <

W (t0) implies V (t) < W (t) on [t0, δ]. It is clear that t1 > t0 and V (t1) ≤ W (t1).From this follows, using the nondecreasing nature of F and condition (ii), that

DHV (t1) ≤ F (t1, V (t1))≤ F (t1, W (t1))≤ F (t1,W (t1)) + L(W −W )≤ DHW (t1) + Lεe2Lt1

< DHW (t1) + 2Lεe2Lt1

= DHW (t1).

Consequently, it follows that there exists an η > 0 satisfying

V (t) − W (t) > V (t1) − W (t1), t1 − η < t < t1.

This implies that t1 > t0 cannot be the supremum in view of the continuity ofthe functions involved and therefore the relation (2.5.2) is true, which, in turn,leads to the conclusion (2.5.1). The proof is complete.

The following corollary is useful.

Corollary 2.5.1 Let V,W ∈ C1[R+,Kc(Rn)], σ ∈ C[R+,Kc(Rn)]. Supposethat

DHV ≤ σ, DHW ≥ σ, for t ≥ t0.

Then V (t) ≤ W (t), t ≥ t0, provided V (t0) ≤ W (t0).

In order to develop the monotone iterative technique, we shall consider thefollowing set differential equation,

DHU = F (t, U ) + G(t, U ), U (0) = U0 ∈ Kc(Rn), (2.5.3)

where F,G ∈ C[J ×Kc(Rn),Kc(Rn)] and J = [0, T ].We need the following definition which gives various possible notions of lower

and upper solutions relative to (2.5.3).

Definition 2.5.1 Let V,W ∈ C1[J,Kc(Rn)]. Then V,W are said to be

(a) natural lower and upper solutions of (2.5.3) if

DHV ≤ F (t, V ) + G(t, V ), DHW ≥ F (t,W ) +G(t,W ), t ∈ J ; (2.5.4)


(b) coupled lower and upper solutions of type I of (2.5.3) if

DHV ≤ F (t, V ) +G(t,W ), DHW ≥ F (t,W ) + G(t, V ), t ∈ J ; (2.5.5)

(c) coupled lower and upper solutions of type II of (2.5.3) if

DHV ≤ F (t,W ) + G(t, V ), DHW ≥ F (t, V ) +G(t,W ), t ∈ J ; (2.5.6)

(d) coupled lower and upper solutions of type III of (2.5.3) if

DHV ≤ F (t,W ) + G(t,W ), DHW ≥ F (t, V ) +G(t, V ), t ∈ J. (2.5.7)

We observe that whenever we have V (t) ≤ W (t), t ∈ J , if F (t,X) is nonde-creasing in X for each t ∈ J and G(t, Y ) is nonincreasing in Y for each t ∈ J ,the lower and upper solutions defined by (2.5.4) and (2.5.7) reduce to (2.5.6)and consequently, it is sufficient to investigate the cases (2.5.5) and (2.5.6).

We are now in a position to prove the following result.


(A1) V,W ∈ C1[J,Kc(Rn)] are coupled lower and upper solutions of type Irelative to (2.5.3) with V (t) ≤ W (t), t ∈ J ;

(A2) F,G ∈ C[J×Kc(Rn),Kc(Rn)], F (t,X) is nondecreasing in X and G(t, Y )is nonincreasing in Y , for each t ∈ J ;

(A3) F and G map bounded sets into bounded sets in Kc(Rn).

Then there exist monotone sequences Vn(t), Wn(t) in Kc(Rn) such thatVn(t) → ρ(t), Wn(t) → R(t) in Kc(Rn) and (ρ,R) are the coupled minimal andmaximal solutions of (2.5.3) respectively, that is, they satisfy

DHρ = F (t, ρ) + G(t, R), ρ(0) = U0,

DHR = F (t, R) + G(t, ρ), R(0) = U0, on J.

Proof For each n ≥ 0, define the unique solutions Vn+1(t), Wn+1(t) by

DHVn+1 = F (t, Vn) +G(t,Wn), Vn+1(0) = U0, (2.5.8)DHWn+1 = F (t,Wn) +G(t, Vn), Wn+1(0) = U0, t ∈ J, (2.5.9)

where V (0) ≤ U0 ≤ W (0). We set V0 = V, W0 = W .Our aim is to prove

V0 ≤ V1 ≤ V2 ≤ ... ≤ Vn ≤ Wn ≤ ... ≤ W2 ≤ W1 ≤W0, t ∈ J. (2.5.10)

Since V0 is the coupled lower solution of type I of (2.5.3), we have using thefact V0 ≤ W0 and the nondecreasing character of F ,

DHV0 ≤ F (t, V0) + G(t,W0).


Also from (2.5.8), we get for n = 0,

DHV1 = F (t, V0) +G(t,W0).

Consequently, following the proof of Theorem 2.5.1, we arrive at V0 ≤ V1 onJ . A similar argument shows that W1 ≤ W0 on J . We next prove V1 ≤W1 onJ . For this purpose consider

DHV1 = F (t, V0) +G(t,W0)DHW1 = F (t,W0) +G(t, V0),V1(0) = W1(0) = U0.

Then, the monotone nature of F and G respectively yield

DHV1 ≤ F (t,W0) + G(t,W0), DHW1 ≥ F (t,W0) + G(t,W0), t ∈ J.

We therefore have, by Corollary 2.5.1, V1 ≤W1 on J . As a result, we obtain

V0 ≤ V1 ≤ W1 ≤ W0 on J. (2.5.11)

Assume that for some j > 1, we have

Vj−1 ≤ Vj ≤Wj ≤Wj−1 on J. (2.5.12)

Then we show thatVj ≤ Vj+1 ≤Wj+1 ≤Wj on J. (2.5.13)

To do this, consider

DHVj = F (t, Vj−1) +G(t,Wj−1), Vj(0) = U0,

DHVj+1 = F (t, Vj) + G(t,Wj) ≥ F (t, Vj−1) +G(t,Wj−1), t ∈ J.

Here we have employed (2.5.12) and the monotone nature of F and G. Corol-lary 2.5.1 now gives Vj ≤ Vj+1 on J . Similarly, we can get Wj+1 ≤ Wj onJ .

Next we show that Vj+1 ≤ Wj+1, t ∈ J . We have from (2.5.8) and (2.5.9)

DHVj+1 = F (t, Vj) + G(t,Wj), Vj+1(0) = U0,

DHWj+1 = F (t,Wj) + G(t, Vj), Wj+1(0) = U0, t ∈ J.

Using (2.5.12) and the monotone character of F and G, we arrive at

DHVj+1 ≤ F (t,Wj) + G(t,Wj),DHWj+1 ≥ F (t,Wj) + G(t,Wj), t ∈ J,

and therefore Corollary 2.5.1 implies that Vj+1 ≤ Wj+1, t ∈ J . Hence (2.5.13)follows and consequently, by induction (2.5.10) is valid for all n. Clearly thesequences Vn, Wn are uniformly bounded on J .


To show that they are equicontinuous, consider for any s < t, where t, s ∈ J,

D[Vn(t), Vn(s)] = D

[U0 +

∫ t

0

F (ξ, Vn−1(ξ)) + G(ξ,Wn−1(ξ))dξ,

U0 +∫ s

0

F (ξ, Vn−1(ξ)) +G(ξ,Wn−1(ξ))dξ]

= D

[∫ t

0

F (ξ, Vn−1(ξ)) + G(ξ,Wn−1(ξ)dξ,∫ s

0

F (ξ, Vn−1(ξ) +G(ξ,Wn−1(ξ))dξ]

≤∫ t

s

D[F (ξ, Vn−1(ξ)) + G(ξ,Wn−1(ξ)), θ]dξ ≤M |t− s|.

Here we have utilized the properties of integral and the metric D, together withthe fact F + G are bounded since Vn, Wn are uniformly bounded. HenceVn(t) is equicontinuous on J . The corresponding Ascoli’s Theorem 2.4.1 nowgives a subsequence Vnk(t) which converges uniformly to ρ(t) ∈ Kc(Rn), t ∈J , and since Vn(t) is a monotone nondecreasing sequence, the entire sequenceVn(t) converges uniformly to ρ(t) on J .

Similar arguments apply to the sequence Wn(t) and Wn(t) → R(t) uni-formly on J . It therefore follows, using the integral representation of (2.5.8)and (2.5.9) that ρ(t), R(t) satisfy

DHρ(t) = F (t, ρ(t)) +G(t, R(t)), ρ(0) = U0,

DHR(t) = F (t, R(t)) +G(t, ρ(t)), R(0) = U0, t ∈ J,(2.5.14)

and thatV0 ≤ ρ ≤ R ≤ W0, t ∈ J. (2.5.15)

Next we claim that (ρ,R) are coupled minimal and maximal solutions of(2.5.3), that is, if U (t) is any solution of (2.5.3) such that

V0 ≤ U ≤ W0, t ∈ J, (2.5.16)

thenV0 ≤ ρ ≤ U ≤ R ≤ V0, t ∈ J. (2.5.17)

Suppose that for some n,

Vn ≤ U ≤ Wn on J. (2.5.18)

Then we have, using the monotone nature of F, G and (2.5.18),

DHU = F (t, U ) +G(t, U ) ≥ F (t, Vn) + G(t,Wn), U (0) = U0,

DHVn+1 = F (t, Vn) +G(t,Wn), Vn+1(0) = U0.

Corollary 2.5.1 yields Vn+1 ≤ U on J . Similarly Wn+1 ≥ U on J . Hence byinduction (2.5.18) is true for all n ≥ 1. Now taking the limit as n → ∞, we get(2.5.17), proving the claim. The proof is therefore complete.


Corollary 2.5.2 If, in addition to the assumptions of Theorem 2.5.2, F andG satisfy, whenever X ≥ Y, X, Y ∈ Kc(Rn),

F (t,X) ≤ F (t, Y ) +N1(X − Y )

andG(t,X) +N2(X − Y ) ≥ G(t, Y ),

where N1, N2 > 0. Then ρ = R = U is the unique solution of (2.5.3).

Proof Since ρ ≤ R, we have R = ρ +m or m = R− ρ. Now

DHρ+DHm = DHR = F (t, R) +G(t, ρ),≤ F (t, ρ) + N1m +G(t, R) + N2m,

= DHρ + (N1 +N2)m,

which meansDHm ≤ (N1 + N2)m, m(0) = 0,

which by Theorem 2.5.1 leads to R ≤ ρ on J , proving the uniqueness of ρ =R = U , completing the proof.

Several remarks are now in order.

Remark 2.5.1 (1) In Theorem 2.5.2, if G(t, Y ) ≡ 0, then we get a resultwhen F is nondecreasing.

(2) In (1) above, suppose that F is not nondecreasing, but F (t,X) = F (t,X)+MX is nondecreasing in X for each t ∈ J , for some M > 0, then one canconsider the IVP

DHU +MU = F (t, U ), U (0) = U0,

where F (t,X) = F (t,X) +MX to obtain the same conclusion as in (1).To see this, use the transformation U (t) = U (t)eMt so that

DH U = [DHU +MU ]eMt = F (t, U e−Mt)eMt ≡ F0(t, U ), U (0) = U0.(2.5.19)

Clearly (2.5.19) has V (t) = V (t)eMt as a lower solution and W (t) =W (t)eMt as an upper solution, and therefore we have the same conclusionas in (1). Here we assume that DH U exists.

(3) If f(t,X) ≡ 0 in Theorem 2.5.2, then we obtain the result for G nonin-creasing.

(4) If in (3) above, G is not monotone but there exists a function G(t, U ) thatis nonincreasing in U for each t ∈ J , and a constant M > 0 such that

G(t, U ) = MU + G(t, U )and

G(t, U ) = G(t, U ) −MU.


Then setting U (t) = U(t)eMt, we obtain

DH U = G0(t, U),

U(0) = U0,(2.5.20)

where G0(t, U ) = G(t, UeMt)e−Mt. In this case, we need to assume that(2.5.20) has coupled lower and upper solutions to get the same conclusionas in (3).

(5) Suppose that in Theorem 2.5.2, G(t, Y ) is nonincreasing in Y and F (t,X)is not monotone but F (t,X) = F (t,X) +MX,M > 0 is nondecreasing inX. Then we consider the IVP

DH U +MU = F (t, U ) +G(t, U ), U (0) = U0. (2.5.21)

The transformation in (2) yields the conclusion by Theorem 2.5.2 in thiscase as well.

(6) If in Theorem 2.5.2, F is nondecreasing and G is not monotone then wesuppose that there exists a function G(t, U ) and a constant M > 0 as in(4) and consider the IVP

DH U = F0(t, U ) + G(t, U ), U (0) = U0, (2.5.22)

where F0(t, U) = F (t, UeMt)e−Mt and G0(t, U) = G(t, U eMt)e−Mt.

(7) If both F and G are not monotone in Theorem 2.5.2, then suppose thatthere are functions F (t, U ), G(t, U ), and a constant M > 0 such thatF (t, U ) + G(t, U ) + MU = F (t, U ) + G(t, U ), where F (t, U ) is nonde-creasing in U and G(t, U ) is nonincreasing in U . Now the transformationU (t) = U (t)eMt gives,

DHU = F0(t, U) +G0(t, U ), U (0) = U0 (2.5.22∗)

where F0(t, U) = F (t, U eMt)e−Mt, G0(t, U ) = G(t, UeMt)e−Mt. Assum-ing that (2.5.22∗) has coupled lower and upper solutions of type I, one getsthe same conclusion by Theorem 2.5.2.

Let us next consider utilizing the coupled lower and upper solutions of typeII. In this case, we don’t need to assume the existence of coupled lower andupper solutions of type II of (2.5.3) since one can construct them under thegiven assumptions. However, we have to pay a price to get monotone flows,by assuming certain conditions on the second iterates. Also, we get alternativesequences which are monotone but complicated.

Theorem 2.5.3 Assume that (A2) and (A3) of Theorem 2.5.2 hold. Then forany solution U (t) of (2.5.3) with V0 ≤ U ≤ W0 on J , we have the iteratesVn, Wn satisfying

V0 ≤ V2 ≤ ... ≤ V2n ≤ U ≤ V2n+1 ≤ ... ≤ V3 ≤ V1 on J, (2.5.23)


W1 ≤W3 ≤ ... ≤W2n+1 ≤ U ≤W2n ≤ ... ≤ W2 ≤ W0 on J, (2.5.24)

provided V0 ≤ V2,W2 ≤ W0 on J , where the iterative schemes are given by

DHVn+1 = F (t,Wn) + G(t, Vn), Vn+1(0) = U0, (2.5.25)

DHWn+1 = F (t, Vn) + G(t,Wn), Wn+1(0) = U0, on J. (2.5.26)

Moreover, the monotone sequences V2n, V2n+1, W2n, W2n+1 in Kc(Rn)converge to ρ,R, ρ∗, R∗ in Kc(Rn) respectively and verify

DHR = F (t, R∗) + G(t, ρ), R(0) = U0,

DHρ = F (t, ρ∗) + G(t, R), ρ(0) = U0,

DHR∗ = F (t, R) + G(t, ρ∗), R∗(0) = U0,

DHρ∗ = F (t, ρ) + G(t, R∗), ρ∗(0) = U0, on J.

Proof We shall first show that coupled lower and upper solutions V0,W0 oftype II of (2.5.3) exist on J satisfying V0 ≤ W0 on J . For this purpose, considerthe IVP

DHZ = F (t, θ) + G(t, θ), Z(0) = U0. (2.5.27)

Let Z(t) be the unique solution of (2.5.27) which exists on J . Define V0,W0 by

R0 + V0 = Z and W0 = Z + R0,

where the positive vector R0 = (R01, R02, ..., R0n) is chosen sufficiently large sothat we have V0 ≤ θ ≤ W0 on J . Then using the monotone character of F andG, we arrive at

DHV0 = DHZ = F (t, θ) +G(t, θ) ≤ F (t,W0) + G(t, V0),

V0(0) = Z(0) − R0 ≤ Z(0) = U0.

Similarly, DHW0 ≥ F (t, V0) + G(t,W0), W0(0) ≥ U0. Thus V0,W0 are thecoupled lower and upper solutions of type II of (2.5.3).

Let U (t) be any solution of (2.5.3) such that V0 ≤ U ≤ W0 on J . We shallshow that

V0 ≤ V2 ≤ U ≤ V3 ≤ V1, W1 ≤ W3 ≤ U ≤ W2 ≤W0 on J. (2.5.28)

Since U is a solution of (2.5.3), we have, using the monotone character of F andG, and the fact V0 ≤ U ≤ W0,

DHU = F (t, U ) + G(t, U ) ≤ F (t,W0) +G(t, V0), U (0) = U0,

and V1 satisfies

DHV1 = F (t,W0) + G(t, V0), V1(0) = U0, on J. (2.5.29)

Hence Corollary (2.5.1) yields U ≤ V1 on J . Similarly, W1 ≤ U on J .


Next we show that V2 ≤ U on J . Note that

DHV2 = F (t,W1) +G(t, V1), V2(0) = U0,

and because of monotonicity of F and G, we get

DHU = F (t, U ) +G(t, U ) ≥ F (t,W1) + G(t, V1), U (0) = U0 on J.

Corollary 2.5.1 therefore gives V2 ≤ U on J . A similar argument shows thatU ≤ W2 on J . Next we find utilizing the assumption V0 ≤ V2,W2 ≤ W0 on Jand monotonicity of F and G,

DHV3 = F (t,W2) + G(t, V2) ≤ F (t,W0) + G(t, V0), V3(0) = U0 on J.

This together with (2.5.29) shows by Corollary 2.5.1 that V3 ≤ V1, on J . In thesame way one can show that W1 ≤W3 on J . Also, employing similar reasoning,one can prove that U ≤ V3 and W3 ≤ U on J , proving the relations (2.5.28).

Now assuming for some n > 2, the inequalities

V2n−4 ≤ V2n−2 ≤ U ≤ V2n−1 ≤ V2n−3,

W2n−3 ≤ W2n−1 ≤ U ≤W2n−2 ≤ W2n−4, on J,

hold, it can be shown, employing similar arguments that

V2n−2 ≤ V2n ≤ U ≤ V2n+1 ≤ V2n−1,

W2n−1 ≤W2n+1 ≤ U ≤W2n ≤ W2n−2, on J.

Thus by induction (2.5.23) and (2.5.24) are valid for all n = 0, 1, 2, · · · .Since Vn,Wn ∈ Kc(Rn) for all n, employing a similar reasoning as in The-

orem 2.5.2, we conclude that the limits limn→∞ V2n = ρ, limn→∞ V2n+1 =R, limn→∞Wn+1 = ρ∗, and limn→∞W2n = R∗, exist, inKc(Rn), uniformly onJ . It therefore follows by suitable use of the integral representation (2.5.25)and(2.5.26) that ρ, ρ∗, R, R∗ satisfy corresponding set differential equations givenin Theorem 2.5.3 on J . Also, from (2.5.23)and 2.5.24), we arrive at

ρ ≤ U ≤ R, ρ∗ ≤ U ≤ R∗ on J.

The proof is therefore complete.

Corollary 2.5.3 Under the assumptions of Theorem 2.5.3 if F and G satisfythe assumptions of Corollary 2.5.2, then ρ = ρ∗ = R = R∗ = U is the uniquesolution of (2.5.3).

Proof Let q1 + ρ = R, q2 + ρ∗ = R∗ so that q1, q2 ≥ 0 on J , since ρ ≤ R andρ∗ ≤ R∗ on J . It then follows using the assumptions, that

DH(q1 + q2) ≤ (N1 +N2)(q1 + q2), q1(0) + q2(0) = 0 on J.

2.6 GLOBAL EXISTENCE 49

This implies that q1 + q2 ≤ 0 on J and consequently, we get

U = ρ = R and ρ∗ = R∗ = U on J,

and this proves the claim of Corollary 2.5.2.Theorem 2.5.3 also has several remarks which correspond to the remarks of

Theorem 2.5.2. To repetition we do not list them again. For similar resultswhich unify monotone iterative technique refer to Lakshmikantham and Koksal[1].

2.6 Global Existence

We consider the set differential equation

DHU = F (t, U ), U (t0) = U0 ∈ Kc(Rn), (2.6.1)

where F ∈ C[R+ × Kc(Rn),Kc(Rn)]. In this section, we shall investigate theglobal existence of solutions for t ≥ t0. Assuming local existence, we shall provethe following global existence result.

Theorem 2.6.1 Assume that F ∈ C[R+ ×Kc(Rn),Kc(Rn)] and

D[F (t, U ), θ] ≤ g(t,D[U, θ]), (t, U ) ∈ R+ ×Kc(Rn),

where g ∈ C[R2+,R+], g(t, w) is nondecreasing in w for each t ∈ R+ and the

maximal solution r(t, t0, w0) of (2.2.5) exists on [t0,∞). Suppose further thatF is smooth enough to guarantee local existence of solutions of (2.6.1) for any(t0, U0) ∈ R+ × Kc(Rn). Then the largest interval of existence of any solutionU (t, t0, U0) of (2.6.1) such that D[U0, θ] ≤ w0 is [t0,∞).

Proof Let U (t) = U (t, t0, U0) be any solution of (2.6.1) with D[U0, θ] = w0,which exists on [t0, β), t0 < β < ∞ and the value of β cannot be increased.Define m(t) = D[U (t), θ]. Then Corollary 2.2.1 shows that

m(t) ≤ r(t, t0, D[U0, θ]) t0 ≤ t < β. (2.6.2)

For any t1, t2 such that t0 < t1 < t2 < β, we have

D[U (t1), U (t2)] = D

[U0 +

∫ t1

t0

F (s, U (s))ds, U0 +∫ t2

t0

F (s, U (s))ds]

= D

[∫ t2

t1

F (s, U (s))ds, θ]

≤∫ t2

t1

D[F (s, U (s)), θ]ds

≤∫ t2

t1

g(s,D[U (s), θ])ds.


The relation (2.6.2) and the nondecreasing nature of g(t, w) now yields

D[U (t1), U (t2)] ≤∫ t2

t1

g(s, r(s, t0, w0))ds (2.6.3)

= r(t2, t0, w0) − r(t1, t0, w0).

Since limt→β− r(t, t0, w0) exists and is finite by hypothesis, taking the limit ast1, t2 → β− and using the Cauchy criterion for convergence, it follows from(2.6.3) that limt→β− U (t, t0, U0) exists.

We defineU (β, t0, U0) = lim

t→β−U (t, t0, U0)

and consider the initial value problem

DHU = F (t, U ), U (β) = U (β, t0, U0).

By the assumed local existence, we see that U (t, t0, U0) can be continued be-yond β, contradicting our assumption that β cannot be continued. Hence everysolution U (t, t0, U0) of (2.6.1) such that D[U0, θ] ≤ w0 exists globally on [t0,∞)and the proof is complete.

Remark 2.6.1 Since r(t, t0, w0) is nondecreasing because of the fact that g(t, w) ≥0, if we assume that r(t, t0, w0) is bounded on [t0,∞) it follows thatlimt→∞ r(t, t0, w0) exists and is finite. This, together with (2.6.2) which nowholds for t ∈ [t0,∞), implies that limt→∞ U (t, t0, U0) = Y ∈ Kc(Rn) exists.

2.7 Approximate Solutions

We shall obtain an error estimate between the solutions and approximate solu-tions of IVP (2.6.1). Let us define the notion of approximate solutions.

Definition 2.7.1 A function V (t) = V (t, t0, V0, ε), ε > 0, is said to be anε−approximate solutions of IVP (2.6.1) ifV ∈ C[R+,Kc(Rn)], V (t0, t0, V0, ε) = V0 and

D[DHV (t), F (t, V (t))] ≤ ε, t ≥ t0.

In case ε = 0, V (t) is a solution of (2.6.1).

Theorem 2.7.1 Assume that F ∈ C[R+ ×Kc(Rn),Kc(Rn)] and fort ≥ t0, U, V ∈ Kc(Rn),

D[F (t, U ), F (t, V )] ≤ g(t,D[U, V ]), (2.7.1)

where g ∈ C[R2+,R+]. Suppose that r(t) = r(t, t0, w0, ε) is the maximal solution

ofw′ = g(t, w) + ε, w(t0) = w0 ≥ 0, (2.7.2)

2.8 EXISTENCE OF EULER SOLUTIONS 51

existing for t ≥ t0. Let U (t) = U (t, t0, U0) be any solution of (2.6.1) and V (t) =V (t, t0, V0, ε) be an ε−approximate solution of IVP 2.6.1 existing for t ≥ t0.Then

D[U (t), V (t)] ≤ r(t, t0, w0, ε), t ≥ t0, (2.7.3)


Proof We proceed, as in the proof of Theorem 2.2.3, withm(t) = D[U (t), V (t)],until we arrive at

D+m(t) ≤ lim suph→0+

D

[U (t + h) − U (t)

h, F (t, U (t))

]

+ lim suph→0+

D

[F (t, V (t)),

V (t + h) − V (t)h

]

+D[F (t, U (t)), F (t, V (t))], t ≥ t0.

This implies, using the definition of approximate solution and (2.7.2), the dif-ferential inequality

D+m(t) ≤ g(t,m(t)) + ε, t ≥ t0,

and m(t0) ≤ w0. The stated estimate follows from Theorem 1.4.1 in Laksh-mikantham and Leela [1].

The following corollary provides the well-known error estimate between thesolution and an ε−approximate solution of (2.6.1).

Corollary 2.7.1 The function g(t, w) = Lw, L > 0, is admissible in Theorem2.7.1 to yield

D[U (t, t0, U0), V (t, t0, V0, ε)]

≤ D[U0, V0]eL(t−t0) +ε

L

(eL(t−t0) − 1

), t ≥ t0. (2.7.4)

Proof Since (2.7.2) in this case reduces to

w′ = Lw + ε, w(t0) = D[U0, V0] (2.7.5)

it is easy to obtain the estimate (2.7.4) by solving the linear differential equation(2.7.5).

2.8 Existence of Euler Solutions

We consider the initial value problem (IVP) for set differential equation

DHU = F (t, U ), U (t0) = U0 ∈ Kc(Rn), (2.8.1)

where F is any function from [t0, T ]×Kc(Rn) → Kc(Rn). Let

π = [t0, t1, ...., tN = T ] (2.8.2)


be a partition of [t0, T ].Consider the interval [t0, t1]. Observe that the right hand side of the set

differential equation

DHU (t) = F (t0, U0), U (t0) = U0,

on [t0, t1] is a constant. Therefore, this IVP clearly has a unique solution U (t) =U (t, t0, U0) on [t0, t1].

Define the node U1 = U (t1) and iterate next by considering on [t1, t2] theIVP

DHU = F (t1, U1) U (t1) = U1 ∈ Kc(Rn).

The next node is U2 = U (t2) and we proceed this way till an arc Uπ = Uπ(t)has been defined on all [t0, T ]. We employ the notation Uπ to emphasize therole played by the particular partition π in determining Uπ which is the Eulerpolygonal arc corresponding to the partition π. The diameter µπ of the partitionπ is given by

µπ = max ti − ti−1 : 1 ≤ i ≤ N. (2.8.3)

Definition 2.8.1 By an Euler solution of (2.8.1) we mean any arc U = U (t)which is the uniform limit of Euler polygonal arcs UπJ , corresponding to somesequence πJ such that πJ → 0, where this means the convergence of the diame-ters µπJ → 0 as J → ∞.

Clearly the corresponding number NJ of the partition points in πJ must thengo to infinity. Obviously, the Euler arc satisfies the initial condition U (t0) = U0.

We can now prove the following result on existence of an Euler solution for(2.8.1).


(i) D[F (t, A), θ] ≤ g(t,D[A, θ]), (t, A) ∈ [t0, T ] × Kc(Rn), where g ∈C[[t0, T ]× R+,R+] g(t, u) is nondecreasing in (t, u);

(ii) the maximal solution r(t) = r(t, t0, u0) of the scalar differential equation

u′ = g(t, u), u(t0) = u0 ≥ 0, (2.8.4)

exists on [t0, T ].

Then,

(a) there exists at least one Euler solution U (t) = U (t, t0, U0) to the IVP(2.8.1), which satisfies a Lipschitz condition;

(b) any Euler solution U (t) of (2.8.1) satisfies the relation

D[U (t), U0] ≤ r(t, t0, u0) − u0, t ∈ [t0, T ], (2.8.5)

where u0 = D[U0, θ].


Proof Let π be the partition of [t0, T ] defined by (2.8.2) and let Uπ = Uπ(t)denote the corresponding arc with nodes of Uπ represented by U0, U1, ...., UN.

Let us set Uπ(t) = Ui(t) on ti ≤ t ≤ ti+1, i = 0, 1.....,N − 1, and observethat Ui(ti) = Ui, i = 0, 1, ...., N − 1.

On the interval (ti, ti+1) we have

D[DHUπ(t), θ] = D[F (ti, Ui), θ] ≤ g(ti, D[Ui, θ]). (2.8.6)

On the interval [t0, t1], we obtain

D[U1(t), U0] = D

[U0 +

∫ t

t0

F (t0, U0) ds, U0

]

= D

[∫ t

t0

F (t0, U0) ds, θ]

≤∫ t

t0

D[F (t0, U0), θ] ds

≤∫ t

t0

g(t0, D[U0, θ]) ds

≤∫ t

t0

g(s, r(s)) ds

= r(t, t0, D[U0, θ]) −D[U0, θ]≤ r(T, t0, D[U0, θ]) −D[U0, θ] = M (say).

Here we have employed the properties of the metric D and the integral, mono-tone character of g(t, u) in (t, u) and the fact that r(t, t0, U0) ≥ 0 is nondecreas-ing in t.

Similarly on [t1, t2], we get

D[U2(t), U0] = D

[U1 +

∫ t

t1

F (t1, U1) ds, U0

]

= D

[U0 +

∫ t1

t0

F (t0, U0) ds+∫ t

t1

F (t1, U1) ds, U0)]

= D

[∫ t1

t0

F (t0, U0) ds+∫ t

t1

F (t1, U1) ds, θ]

≤∫ t1

t0

D[F (t0, U0), θ] ds+∫ t

t1

D[F (t1, U1), θ] ds

≤∫ t1

t0

g(s, r(s)) ds+∫ t

t1

g(s, r(s) ds =∫ t

t0

g(s, r(s)) ds

≤ r(T, t0, D[U0, θ]) −D[U0, θ] = M (say).

Proceeding in this way, we arrive at

D[Ui(t), U0] ≤ r(T, t0, D[U0, θ]) −D[U0, θ] = M, on [ti, ti+1].


Hence it follows that

D[ Uπ(t), U0] ≤M, on [t0, T ].

Also, from (2.8.6) we obtain,

D[DHUπ(t), θ] ≤ g(T, r(T )) = r′(T, t0, D[U0, θ]) = k, say.

Consequently, using similar arguments, we can find for t0 ≤ s ≤ t ≤ T,

D[Uπ(t), Uπ(s)] ≤∫ t

t0

D[F (τ, Uπ(τ ), θ] dτ +∫ s

t0

D[F (τ, Uπ(τ )), θ] dτ

≤∫ t

t0

g(τ, r(τ )) dτ +∫ s

t0

g(τ, r(τ )) dτ

=∫ t

s

g(τ, r(τ )) dτ

= r(t) − r(s) = r′(σ) | t− s |≤ k | t− s |

for some σ such that s ≤ σ ≤ t, proving Uπ(t) is Lipschitz of rank k on[t0, T ].

Now, let πJ be a sequence of partitions of [t0, T ] such that πJ → 0, that issuch that µπJ → 0 and therefore NJ → ∞. Then the corresponding polygonalarcs UπJ on [t0, T ] all satisfy

UπJ (t0) = U0, D[UπJ (t), U0] ≤ M and D[DHUπJ (t), θ] ≤ k on [t0, T ].

Hence the family UπJ is equicontinuous and uniformly bounded, and, as aconsequence, Ascoli-Arzela Theorem 2.4.1 guarantees the existence of a subse-quence which converges uniformly to a continuous function U (t) on [t0, T ] andthus absolutely continuous on [t0, T ]. Thus, by definition, U (t) is an Euler so-lution of the IVP (2.8.1) on [t0, T ] and the claim of the theorem follows. Theinequality (2.8.5) in part (b) is inherited by U (t) from the sequence of polygonalarcs generating it when we identify T with t. Hence the proof is complete.

If F (t, U ) in (2.8.1) is assumed to be continuous, then one can show thatU (t) actually satisfies the IVP (2.8.1).

Theorem 2.8.2 Under the assumptions of Theorem 2.8.1, if we suppose inaddition that F ∈ C[[t0, T ] × Kc(Rn), Kc(Rn)], then U (t) is a solution of(2.8.1).

Proof Let UπJ (t) denote a sequence of polygonal arcs for IVP (2.8.1) converg-ing uniformly to an Euler solution U (t) on [t0, T ]. Clearly, the arcs UπJ (t) all liein B(U0,M ) and satisfy a Lipschitz condition of the same rank k. Since a con-tinuous function is uniformly continuous on compact sets, for any given ε > 0,one can find a δ > 0 such that

t, t∗ ∈ [t0, T ], U, U∗ ∈ UπJ , | t− t∗ |< δ, D[U,U∗] < δ


implies D[F (t, U ), F (t∗, U∗)] < ε.

Let J be sufficiently large so that the partition diameter µπJ satisfies µπJ < δand kµπJ < δ. For any t, which is not one of the finitely many points at whichUπJ (t) is a node, we have DHUπJ (t) = F (t, UπJ (t)) for some t within µπJ < δof t.

Since

D[UπJ (t), UπJ (t)] ≤ kµπJ < δ, we get

D[DHUπJ (t), F (t, UπJ (t))] = D[F (t, UπJ (t)), F (t, UπJ (t))] < ε.

It follows that for any t ∈ [t0, T ], we obtain,

D

[UπJ (t), UπJ (t0) +

∫ t

t0

F (τ, UπJ (τ ))dτ]

= D

[UπJ (t0) +

∫ t

t0

DHUπJ (τ ) dτ, UπJ (t0) +∫ t

t0

F (τ, UπJ (τ )) dτ]

= D

[∫ t

t0

DHUπJ (τ ) dτ,∫ t

t0

F (τ, UπJ (τ ))dτ]

≤∫ t

t0

D [DHUπJ (τ ), F (τ, UπJ (τ ))] dτ

≤ ε(t− t0) ≤ ε(T − t0).

Letting J → ∞, we have from this,

D[U (t), U0 +∫ t

t0

F (τ, U (τ )) dτ ] < ε(T − t0).

Since ε is arbitrary, it follows that

U (t) = U0 +∫ t

t0

F (τ, U (τ )) dτ, t ∈ [t0, T ],

which implies that U (t) is C1 and therefore

DHU (t) = F (t, U (t)), U (t0) = U0, t ∈ [t0, T ].

The proof is therefore complete.

Remark 2.8.1 We can extend the notion of an Euler solution of (2.8.1) fromthe interval [t0, T ] to [t0,∞), if we define F and g on [t0,∞) instead of [t0, T ]and assume that the maximal solution r(t) exists on [t0,∞) and show that anEuler solution exists on every [t0, T ] where T ∈ (t0,∞).


2.9 Proximal Normal and Flow Invariance

Let Ω ⊂ Kc(Rn) be a nonempty, closed set. Assume that for any U ∈ Kc(Rn)such that U and Ω are disjoint, and for any S ∈ Ω, there exists a Z ∈ Kc(Rn)such that U = S + Z. Then U − S is called the Hukuhara difference. Supposenow that, for any U ∈ Kc(Rn) there is an element S ∈ Ω whose distance to Uis minimal, that is,

D0[U,Ω] = ‖U − S‖ = infS0∈Ω

‖U − S0‖. (2.9.1)

Then S is called a projection of U onto Ω. The set of all such elements isdenoted by projΩ(U ). The element U − S will be called the proximal normaldirection to Ω at S. Any nonnegative multiple ξ = t(U − S), t ≥ 0, is calledproximal normal to Ω at S. The set of all ξ obtained in this way is said to beproximal normal cone to Ω at S and is denoted by NP

Ω (S).

Definition 2.9.1 The system (Ω, F ) is said to be weakly invariant provided thatfor all U0 ∈ Ω, there exists an Euler solution U (t) of (2.8.1) on [t0,∞) suchthat U (t0) = U0 and U (t) ∈ Ω, t ≥ t0.

Before proceeding further we introduce the following notation.For any A ∈ Kc(Rn), we get D[A, θ] = ‖A‖ = supa∈A ‖a‖ and we define for

any A,B ∈ Kc(Rn),

< A,B >= sup(a · b) : a ∈ A, b ∈ B

so that we obtain the relation

‖A+B‖2 ≤ ‖A‖2 + ‖B‖2 + 2 < A,B > .

We can now prove the following result which offers sufficient conditions in termsof proximal normal for the weak invariance of the system (Ω, F ).

Theorem 2.9.1 Let F and g satisfy the conditions of Theorem 2.8.1 on [t0,∞),t0 ≥ 0. Suppose that U (t) = U (t, t0, U0) is an Euler solution of (2.8.1) on[t0,∞), which lies in an open set Q ⊂ Kc(Rn). Suppose also that for every(t, Z) ∈ [t0,∞) × Q, the proximal aiming condition is satisfied: namely, thereexists an S ∈ projΩ(Z) such that

2 < F (t, Z), Z − S > ≤ q(t,D20[Z,Ω]), (2.9.2)

where q ∈ C[[t0,∞)× R+, R]. Assume that r(t) = r(t, t0, u0) is the maximalsolution of the scalar differential equation

u′ = q(t, u), u(t0) = u0 ≥ 0,

existing on [t0,∞). Then we have

D20 [U (t),Ω] ≤ r(t, t0, D2

0[U0,Ω]), t0 ≤ t < ∞. (2.9.3)

If, in addition r(t, t0, 0) ≡ 0 then U0 ∈ Ω implies U (t) ∈ Ω, t ≥ t0, that is, thesystem (Ω, F ) is weakly invariant.

2.9 PROXIMAL NORMAL AND FLOW INVARIANCE 57

Proof Let Uπ be one polygonal arc in the sequence converging uniformly to Uas per the definition of Euler solution of (2.8.1). We denote as before, its nodesat ti by Ui, i = 0, 1, ....., N and hence U0 = U (t0). Let Uπ(t) be in Q for allt0 ≤ t ≤ T , where T ∈ (t0,∞). Accordingly, there exists for each i, an elementSi ∈ projΩ(Ui) such that

2 < F (ti, Ui), Ui − Si > ≤ q(ti, D20[Ui,Ω]).

As in Theorem 2.8.1, letting D[DHUπ(t), θ] ≤ k, we find

D20 [U1,Ω] ≤ ‖U1 − S0‖2, since S0 ∈ Ω.

We note that U1 = U0 +Z1, where Z1 = F (t0, U0)(t1 − t0) and U0 = S0 +Z0

and therefore, we get successively

D20[U1,Ω] ≤ ‖Z1 + Z0‖2 ≤ ‖Z1‖2 + ‖Z0‖2 + 2 < Z1, Z0 >

≤ k2(t1 − t0)2 +D20[U0,Ω] + 2

∫ t1

t0

< DHUπ(t0), Z0 > dt

≤ k2(t1 − t0)2 +D20[U0,Ω] + 2

∫ t1

t0

< F (t0, U0), U0 − S0 > dt

≤ k2(t1 − t0)2 +D20[U0,Ω] + q(t0, D2

0[U0,Ω])(t1 − t0).

Since similar estimates hold at any node, we obtain,

D20[Ui,Ω] ≤ k2(ti − ti−1)2 +D2

0[Ui−1,Ω] + q(ti−1, D20[Ui−1,Ω])(ti − ti−1).

and therefore it follows that

D20[Ui,Ω] ≤ D2

0 [U0,Ω] + k2i∑

J=1

(tJ − tJ−1)2

+i∑

J=1

q(tJ−1, D20[UJ−1,Ω])(tJ − tJ−1)

≤ D20 [U0,Ω] + k2µπ

i∑

J=1

(tJ − tJ−1)

+i∑

J=1

q(tJ−1, D20[UJ−1,Ω])(tJ − tJ−1)

≤ D20 [U0,Ω] + k2µπ(T − t0) +

i∑

J=1

q(tJ−1, D20[UJ−1,Ω])(tJ − tJ−1).

Consider now, the sequence UπJ (t) of polynomial arcs converging to U (t).Since the last estimate is true at every node, µπJ → 0, as J → ∞, and the samek applies to each UπJ , we deduce in the limit the integral inequality

D20 [U (t),Ω] ≤ D2

0[U0,Ω] +∫ t

t0

q(s,D20[U (s),Ω]) ds, t0 ≤ t ≤ T, (2.9.4)


for every T ∈ (t0,∞). If we know that q(t, u) is nondecreasing in u, then wecan apply the theory of integral inequalities (see Theorem 1.9.2 in Lakshmikan-tham and Leela [1]), to arrive at

D20[U (t),Ω] ≤ r(t, t0, D2

0[U0,Ω]), t ≥ t0. (2.9.5)

If, on the other hand, q(t, u) is not nondecreasing in u, we can obtain insteadof (2.9.4), the following integral inequality for any t0 ≤ t ≤ t + h ≤ T, h > 0,employing similar reasoning,

D20[U (t+ h),Ω] ≤ D2

0[U (t),Ω] +∫ t+h

t

q(s,D20[U (s),Ω]) ds, (2.9.6)

from which we obtain, setting m(t) = D20[U (t),Ω], the differential inequality

D+m(t) ≤ q(t,m(t)), m(t0) = D20[U0,Ω], (2.9.7)

where D+m(t) is a Dini derivative.Applying now the theory of Differential inequalities (see Theorem 1.4.1

Lakshmikantham and Leela [1]), we arrive at the same estimate (2.9.5). Ifr(t, t0, 0) ≡ 0, then, supposing that U0 ∈ Ω implies that U (t) ∈ Ω for t ≥ t0 andtherefore the system (Ω, F ) is weakly invariant as claimed. The proof is hencecomplete.

We shall next discuss the strong invariance of the system (Ω, F ).

Definition 2.9.2 The system (Ω, F ) is said to be strongly invariant if everyEuler solution U (t) of (2.8.1) existing on [t0,∞) for which U (t0) = U0 ∈ Ω,satisfies U (t) ∈ Ω, t ≥ t0.

We can now prove the following result for strong invariance of Euler solutionsof (2.8.1).

Theorem 2.9.2 Let the assumptions of Theorem 2.8.1 hold. Suppose that Fsatisfies the generalized Lipschitz condition

< F (t, A), C > ≤ < F (t, B), C > + L ‖C‖2, (2.9.8)

where there exists a C ∈ Kc(Rn) for those A,B ∈ Kc(Rn) such that A = B +Cand L > 0. Then, if the proximal normal condition

< F (t, B), C > ≤ 0, (2.9.9)

holds, we have the strong invariance of the system (Ω, F ).

Proof Let V (t) be any Euler solution of (2.8.1) on [t0,∞) with V (t0) = U0 ∈ Ω.By Theorem 2.8.1, there exists a M > 0 such that

D[V (t), U0] ≤ M on [t0, T ], for any T ∈ (t0,∞).

If U lies in B[U0,M ] and S ∈ projΩ(U ) then we have

2.10 EXISTENCE, UPPER SEMICONTINUOUS CASE 59

D[S, U0] ≤ D[S, U ] +D[U,U0] ≤ 2D[U,U0] ≤ 2M,

which implies that S ∈ B[U0, 2M ]. Let L be the Lipschitz constant for F inB[U0, 2M ] and consider any U ∈ B[U0,M ] and S ∈ projΩ(U ). Then U − S ∈NP

Ω (S).Consequently, using (2.9.8), we get

< F (t, U ), U − S > ≤ 12D2

0[U,Ω]. (2.9.10)

The relation (2.9.10) is a special case of Theorem 2.9.1, with q(t, w) = L2w and

therefore we obtain the conclusion from Theorem 2.9.1,

D20[V (t),Ω] ≤ D2

0[U0,Ω] eL2 (t−t0), t ≥ t0. (2.9.11)

Since U0 ∈ Ω is assumed, we get readily from (2.9.11), V (t) ∈ Ω, t ≥ t0 andthe proof is complete.

2.10 Existence, Upper Semicontinuous Case

We shall consider the IVP for set differential equation

DHU = F (t, U ), U (t0) = U0 ∈ Kc(Rn), (2.10.1)

where F is a function from J × Kc(Rn) to Kc(Rn). By a solution of (2.10.1)we mean an absolutely continuous function U : J → Kc(Rn), U (t0) = U0,whose derivative DHU (t), in the sense of Hukuhara, satisfies (2.10.1) almosteverywhere (a.e.) on J = [t0, b], t0 ≥ 0, b ∈ (t0,∞).

In what follows, by F : J ×Kc(Rn) → Kc(Rn), we mean that F is a single-valued function and when we write F : J ×Kc(Rn) → Rn, it means that F is amultifunction defined on metric space J ×Kc(Rn) with values in Rn. From thecontext it would be clear when we consider F as a single-valued function or amultifunction.

Let F : J ×Kc(Rn) → Rn be a multifunction with compact, convex valuesand V be a compact convex subset of C(J,Rn). Then a function V (t) = x(t) :x(·) ∈ V , t ∈ J, is continuous from J to Kc(Rn). If the multifunction t →(t, V (t)) is measurable then there exists a measurable selector of F (t, V (t)). Fora compact convex subset V ⊂ C(T,Rn) we denote by T (V0, F, V ), V0 = V (t0)the collection of all functions x : J → Rn representable as

x(t) = x0 +∫ t

t0

v(s)ds, t ∈ J, x0 ∈ V (2.10.2)

where v(s) is a Bochner integrable selector of F (s, V (s)). Let us list the followingassumptions:

(i) F : J ×Kc(Rn) → Rn is a multifunction with compact, convex values andF (t, A) is monotone nondecreasing with respect to A ∈ Kc(Rn);


(ii) the map (t, U ) → F (t, U ) is L ⊕ B(Kc(Rn)) is measurable;

(iii) the map U → F (t, U ) is usc for almost all t ∈ J ;

(iv) ‖F (t, U )‖ ≤ g(t, ‖U‖), where g : J×R+ → R+ is a Caratheodory functionintegrally bounded on bounded subsets of J × R+, g(t, r) is monotonenondecreasing in r, a.e. in t ∈ J and r(t) = r(t, t0, r0) is the maximalsolution of the scalar differential equation

r′ = g(t, r), r(t0) = r0 ≥ 0, (2.10.3)

existing on J .

We are now in a position to prove the following existence result.

Theorem 2.10.1 Assume that conditions (i) to (iv) hold. Then for any U0 ∈Kc(Rn), there exists a solution U : J → Kc(Rn) of the IVP (2.10.1) on J .

Proof According to (ii), (iii) and Theorem 2.2 and Remark 2.1 in Tolstonogov[2] for any ε > 0, there exists a compact set Tε ⊂ J, µ(J\Tε) ≤ ε such thatthe restriction of F (t, A) on Tε × Kc(Rn) is usc. Then for any continuousfunction V : J → Kc(Rn) the restriction of F (t, U (t)) on Tε is usc. Hence themultifunction t→ F (t, U (t)) is measurable.

Let V ⊂ C(T,Rn) be a compact set of C(J,Rn) then we can define themultivalued operator T (V0, F, V ), V0 = V (0) by using (2.10.2). We note alsothat T (V0, F, V ) is monotone relative to V in view of the monotonicity of F (t, A)with respect to A assumed in (i).

Now let r0 = max‖x‖ : x ∈ V0 and r(t) = r(t, t0, r0) be the maximalsolution of (2.10.3) on J . We denote by U0 a set of all absolutely continuousfunctions x : J → Rn, x(t0) ∈ V0, whose derivatives x′(t) satisfy the estimate‖x(t)‖ ≤ r′(t) a.e. on J .

This implies that ‖x(t)‖ ≤ r(t), t ∈ J, for any x(·) ∈ U0 and therefore U0 isa convex, bounded and equicontinuous subset of C(T,Rn).

Using Theorem 1.5 in Tolstonogov [1] we obtain that U0 is a closed subset ofC(T,Rn). Hence U0 is a convex compact subset of C(T,Rn), U0(t0) = V0 anda multifunction U0(t) is continuous from J to Kc(Rn).

Set U1 = T (V0, F, U0). By assumption (iv) we have

‖F (t, U0(t))‖ ≤ g(t, ‖U0(t)‖) ≤ g(t, r(t)) = r′(t) a.e. (2.10.4)

and for any x(·) ∈ T (V0, F, U0), it follows, using (2.10.4) that

‖x′(t)‖ = ‖v(t)‖ ≤ ‖F (t, U0(t))‖ ≤ r′(t), a.e.

Hence U1 = T (V0, F, U0) ⊂ U0. By analogy with U0 we can prove that U1 iscompact convex subset of C(J,Rn) and U1(t0) = V0.

We now define U2 = T (V0, F, U1), and, since U1 ⊂ U0, it follows becauseof the monotone nature of T (V0, F, V ) in V , that T (V0, F, U1) ⊂ T (V0, F, U0).Thus U2 ⊂ U1.

2.10 EXISTENCE, UPPER SEMICONTINUOUS CASE 61

Continuing this process, we obtain a sequence Uk, k ≥ 1, of compact convexsets of C(J,Rn) decreasing relative to the inclusion.

Hence U =⋂∞

k=0 Uk is a nonempty compact convex subset of C(J,Rn)and the sequence Uk converges to U in Hausdorff metric on the space ofall nonempty, closed, bounded sets of C(J,Rn). It is clear that U (t0) = V0.Since U ⊂ Uk, k ≥ 0, we have

T (V0, F, U ) ⊂ T (V0, F, Uk−1) ⊂ Uk−1, k ≥ 1,

and therefore

T (V0, F, U ) ⊂∞⋂

k=0

Vk = U. (2.10.5)

It is easy to prove that U (t) =⋂∞

k=0Uk(t), t ∈ J, and the sequence Uk(t), k ≥ 1,converges pointwise on J to U (t).

Let x(·) ∈ U. Then x(·) ∈ T (V0, F, Uk−1), k ≥ 1. Hence x(t) is absolutelycontinuous and

x′(t) ∈ F (t, Uk−1(t)) a.e., k ≥ 1.

Thenx′(t) ∈ F (t, U (t)) a.e. (2.10.6)

due to (iii) and the convergence of Uk(t), k ≥ 1, to U (t) in Kc(Rn).As a consequence of (2.10.6) we have

x(·) ∈ T (V0, F, U ). (2.10.7)

Combining (2.10.5), (2.10.7) we obtain

T (V0, F, U ) = U. (2.10.8)

It is easy to see from (2.10.8) and (1.8.2) that

U (t) = V0 +∫ t

0

F (s, U (s))ds, t ∈ J. (2.10.9)

Taking into consideration (1.8.3), (1.8.4) and (2.10.9) we obtain

DHU (t) = F (t, U (t)) a.e. on J

and U (0) = V0 and the proof is complete.Let Γ : J ×Rn → Rn be a multifunction. We need the following hypotheses

H(Γ) : Γ : J × Rn → Rn is a multifunction with compact values such that

(i) (t, x) → Γ(t, x) is L ⊕ BRn measurable;

(ii) x→ Γ(t, x) is usc a.e. on J ;

(iii)‖Γ(t, x)‖ = sup‖y‖ : y ∈ Γ(t, x) ≤ g(t, ‖x‖). (2.10.10)


Consider a multifunction Φ : J ×Kc(Rn) → Rn defined by

Φ(t, U ) = co Γ(t, U ), U ∈ Kc(Rn), (2.10.11)

where co denotes closed convex hull.

Lemma 2.10.1 Suppose that hypotheses H(Γ) hold. Then there exists a mul-tifunction F : J ×Kc(Rn) → Rn such that assumptions (i) to (iv) hold and

F (t, U ) = Φ(t, U ), U ∈ Kc(Rn) a.e. on J,

where Φ(t, U ) is defined by (2.10.11).

Proof Let Tk, k ≥ 1, be a sequence of closed subsets of J increasing withrespect to inclusion such that µ(J\

⋃∞k=1 Tk) = 0. For every t ∈ Tk the multi-

function x→ Γ(t, x) is usc. Fix k ≥ 1. Then the restriction of Γ(t, x) on Tk ×Eis L ⊕ BRn measurable.

From Theorem 2.2 in Tolstonogov [2] we know that for every ε > 0 thereexists a closed set Tε ⊂ Tn, µ(Tk\Tε) ≤ ε such that the restriction of multifunc-tion on Tε × Rn is usc with respect to (t, x) ∈ Tε × Rn. Since the multifunctionx → Γ(t, x) is usc for every t ∈ Tε, the set Γ(t, A), A ∈ Kc(Rn), t ∈ Tε iscompact subset of Rn.

Let us show that the restriction of Γ(t, A) to Tε×Kc(Rn) is usc. To this endwe have to prove that restriction of Γ(t, U ) on Tε × M is usc for any compactset M ⊂ Kc(Rn), Tolstonogov [2].

Let M ⊂ Kc(Rn) be a compact set. Then the set M = ⋃U ; U ∈ M is

compact set of Rn. Since Γ(t, x) is usc on Tε ×M , there exists a compact setQ ⊂ Rn such that

Γ(t, x) ⊂ Q, t ∈ Tε, x ∈M. (2.10.12)

From (2.10.12) it follows that upper semicontinuity of restriction Γ(t, U ) onTε × M is equivalent to closedness of graph of restriction Γ(t, U ) on Tε × M.

Let tm → t, tm ∈ Tε, Um → U in Kc(Rn), Um ∈ M, and ym → y, ym ∈Γ(t, Um). Then there exists xm ∈ Um such that ym ∈ Γ(t, xm). Since xm ∈M, m ≥ 1, without loss of generality we can suppose that xm → x. It is clearthat x ∈ U.

From the upper semicontinuity of Γ(t, x) on Tε × M it follows that y ∈Γ(t, x) ⊂ Γ(t, U ). It means that the restriction of Γ(t, U ) on Tε × M has closedgraph in Tε ×Kc(Rn) × Rn. Hence the restriction of Γ(t, U ) on Tε × M is usc.Then, restriction Φ(t, U ) = coΓ(t, U ) on Tε × M. Therefore the restriction ofΦ(t, U ) on Tε ×Kc(Rn) is usc.

By using similar arguments we can prove that for every t ∈ Tk, k ≥ 1, themultifunction Φ(t, U ) is usc.

In this case Theorem 2.2 in Tolstonogov [2] tells us that the restriction of themultifunction Φ(t, U ) on Tk×Kc(Rn), k ≥ 1, is L⊕BKc(Rn) measurable. Hencefor every t ∈

⋃∞k=1 Tk the multifunction U → Φ(t, U ) is usc and the restriction

of the multifunction Φ(t, U ) on⋃∞

k=1 Tk ×Kc(Rn) is L⊕ BKc(Rn) measurable.


Set

F (t, U ) = Φ(t, U ), t ∈∞⋃

k=1

Tk, U ∈ Kc(Rn),

F (t, U ) = Θ, t ∈ J\∞⋃

k=1

Tk, U ∈ Kc(Rn),

where Θ is the zero element of E, which is regarded as an one-point set. It isclear that the multifunction U → F (t, U ) is usc for every t ∈ J and is monotonenondecreasing with respect to U ∈ Kc(Rn).

Since (T\⋃∞

k=1 Tk) × Kc(Rn) is a Borel subset of T × Kc(Rn), the mul-tifunction F (t, A) is L ⊕ BKc(Rn) measurable. From (2.10.10) it follows thatmultifunction F (t, U ) satisfies the assumption (iv). The theorem is proved.

Remark 2.10.1 If multifunction Γ(t, x) is L⊕BRn measurable and the multi-function x → Γ(t, x) is usc for every t ∈ J, then the multifunction co Γ(t, A) isL ⊕ BKc(R) measurable and the multifunction U → co Γ(t, U ) is usc for everyt ∈ J.

Corollary 2.10.1 Assume that the multifunction Γ satisfies hypotheses H(Γ).Then there exists, for any U0 ∈ Kc(Rn), a solution U (t) = U (t, t0, U0) ∈ Kc(Rn)of the IVP (2.10.1) with the multifunction Φ(t, U ) defined by (2.10.11).

By Lemma 2.10.1 without loss of generality we can consider that multifunctionΦ(t, U ) satisfies condition (i) to (iv). Then the Corollary follows from Theorem2.10.1.


For the formulation of SDEs in the metric space (Kc(Rn), D) and the initiationof preliminary results of existence, uniqueness and extremal solutions with asuitable partial order, see Brandao Lopes Pinto, De Blasi, and Iervolino [1], andDe Blasi and Iervolino [1]. For the case of SDEs in the metric space (Kc(E), D),E being a Banach space, as a tool to prove existence results of multivalueddifferential inclusions without compactness and convexity and for several generalresults refer Tolstonogov [1]. The results of Section 2.2 and 2.3 are taken fromLakshmikantham, Leela and Vatsala [2]. For the contents of Section 2.4, seeLakshmikantham and Vasundhara Devi[1], which are analogous to the resultsof Brandao Lopes Pinto, De Blasi, and Iervolino [1] in the present framework.

Monotone Iterative Technique of Section 2.5 is from Lakshmikantham andVatsala [1]. For earlier results on monotone iterative technique for ordinary andpartial differential equations see Ladde, Lakshmikantham and Vatsala [1] andmore recent general results Lakshmikantham and Koksal [1]. Sections 2.6 and2.7 contain new results parallel to the corresponding theorems in ODE. For theexistence of Euler solutions and flow invariance in Sections 2.8 and 2.9 in termsof nonsmooth analysis, see Gnana Bhaskar and Lakshmikantham [1]. For more


information on nonsmooth analysis see Clarke, Ledyaev, Stern, and Wolenski[1]. For some generalizations refer to Gnana Bhaskar and Lakshmikantham [2,4, 5]. The existence of solutions in USC case covered in 2.9 are adopted fromLakshmikantham and Tolstonogov [1].

Chapter 3

Stability Theory

3.1 Introduction

In this chapter, we investigate stability theory via Lyapunov-like functions. Weshall also initiate the development of set differential systems using generalizedmetric spaces.

In Section 3.2, we prove a comparison theorem in terms of Lyapunov-likefunctions which serves as a vehicle for the discussion of the stability theory ofLyapunov. Some special cases of the comparison result are given which are use-ful later. Section 3.3 considers a global existence result for solutions of SDE, interms of Lyapunov-like functions using Zorn’s lemma. Simple stability resultsare established in Section 3.4. Here an example is worked out to demonstratethe problems encountered in the study of stability theory of SDE, in view of thefact diam ‖U (t)‖ is nondecreasing as t increases. A way to avoid the problemsgenerated is suggested, which leads in certain cases to choosing an appropri-ate subset of the solution. The stability criteria is obtained in the suggestedformat throughout. Section 3.5 discusses non-uniform stability criteria underless restrictive conditions employing perturbing Lyapunov-like functions. Thecriteria for boundedness of solutions is dealt with in Section 3.6, where variousdefinitions of boundedness are also given. The results proved also include themethod of perturbing Lyapunov functions.

In Section 3.7, we embark on initiating the study of set differential systems,the consideration of which leads to generalized metric spaces, in terms of whichare proved comparison results utilizing vector Lyapunov-like functions. Section3.8 develops the method of vector Lyapunov-like functions and stability criteriain this set up. Since we get a comparison differential system in this situation,the study of which is sometimes difficult, we provide certain results to reducethe study of comparison systems to a single comparison equation.

We begin to utilize, in Section 3.9, the tools of nonsmooth analysis to in-vestigate the stability results via lower semicontinuous Lyapunov-like functions.Employing the connection between proximal normal theory and subdifferentials

65

66 CHAPTER 3. STABILITY THEORY

of lower semicontinuous functions, we provide the necessary framework in thissection. In Section 3.10, we prove stability criteria for Euler solutions of SDE,utilizing the tools provided in Section 3.9. Notes and comments are given insection 3.10.

3.2 Lyapunov-like Functions

We consider the IVP for set differential equations

DHU = F (t, U ), U (t0) = U0 ∈ Kc(Rn), (3.2.1)

where F ∈ C[R+ ×Kc(Rn),Kc(Rn)]. To investigate the qualitative behaviourof solutions of (3.2.1), the following comparison result in terms of a Lyapunov-like function is very important and can be proved via the theory of ordinarydifferential inequalities. Here the Lyapunov -like function serves as a vehicleto transform the set differential equation into a scalar comparison differentialequation. Therefore, it is enough to consider the qualitative properties of thesimpler comparison equation, under suitable conditions for the Lyapunov-likefunction.

We also require the IVP for scalar differential equation

w′ = g(t, w), w(t0) = w0 ≥ 0, (3.2.2)

where g ∈ C[R2+,R].


(i) V ∈ C[R+ ×Kc(Rn),R+] and | V (t, A) − V (t, B) | ≤ L D[A,B], where Lis the local Lipschitz constant , for A, B ∈ Kc(Rn), t ∈ R+;

(ii) g ∈ C[R2+,R] and for t ∈ R+, A ∈ Kc(Rn),

D+V (t, A) ≡ lim suph→0+

1h

[V (t+ h,A+ hF (t, A)) − V (t, A)] ≤ g(t, V (t, A)).

(3.2.3)

Then, if U (t) = U (t, t0, U0) is any solution of (3.2.1) existing on [t0, T ) suchthat V (t0, U0) ≤ w0, we have

V (t, U (t)) ≤ r(t, t0, w0), t ∈ [t0, T ), (3.2.4)

where r(t, t0, w0) is the maximal solution of (3.2.2) existing on [t0, T ).

Proof Let U (t) = U (t, t0, U0) be any solution of (3.2.1) existing on [t0, T ).Define m(t) = V (t, U (t)) so that m(t0) = V (t0, U0) ≤ w0. Now for small h > 0,

m(t + h) −m(t) = V (t+ h, U (t+ h)) − V (t, U (t))= V (t+ h, U (t+ h)) − V (t + h, U (t) + hF (t, U (t)))

+V (t + h, U (t) + hF (t, U (t))) − V (t, U (t))≤ LD[U (t + h), U (t) + hF (t, U (t))]

+V (t + h, U (t) + hF (t, U (t))) − V (t, U (t)),

3.2 LYAPUNOV-LIKE FUNCTIONS 67

using the Lipschitz condition given in (i). Thus


1h

[m(t + h) −m(t)]

≤ D+V (t, U (t)) + L lim suph→0+

1h

[D[U (t+ h), U (t) + hF (t, U (t))]].

Since

1hD[U (t+ h), U (t) + hF (t, U (t))] =

[U (t+ h) − U (t)

h, F (t, U (t))

],

we find that,

lim suph→0+

1h

[D[U (t+ h), U (t) + hF (t, U (t))]]

= lim suph→0+

D

[U (t + h) − U (t)

h, F (t, U (t))

],

= D[DHU (t), F (t, U (t))] ≡ 0,

since U (t) is a solution of (3.2.1). We therefore have the scalar differentialinequality

D+m(t) ≤ g(t,m(t)), m(t0) ≤ w0,

which yields, as before, the estimate

m(t) ≤ r(t, t0, w0), t ∈ [t0, T ).

This proves the claimed estimate of the Theorem.The following Corollaries are useful.

Corollary 3.2.1 The function g(t, w) = 0 is admissible in Theorem 3.2.1 toyield the estimate

V (t, U (t)) ≤ V (t0, U0), t ∈ [t0, T ).

Corollary 3.2.2 If, in Theorem 3.2.1, we assume that g(t, w) = −αw,α > 0,we get the relation

V (t, U (t)) ≤ V (t0, U0) exp(−α(t − t0)), t ∈ [t0, T ).

Corollary 3.2.3 If, in Theorem 3.2.1, we strengthen the assumption on D+V (t, U )to

D+V (t, U ) + c[w(t, U )] ≤ g(t, V (t, U )),

where w ∈ C[R+ × Kc(Rn),R+], c ∈ K and g(t, w) is nondecreasing in w foreach t ∈ [t0, T ), then we get

V (t, U (t)) +∫ t

t0

c[w(s, U (s))] ds ≤ r(t, t0, V (t0, U0)), t ∈ [t0, T ), (3.2.5)

whenever w0 = V (t0, U0).


Proof Set L(t, U (t)) = V (t, U (t)) +∫ t

t0c[w(s, U (s))] ds and note that

D+L(t, U (t)) ≤ D+V (t, U (t)) + c[w(t, U (t))]≤ g(t, V (t, U (t))) ≤ g(t, L(t, U (t))),

using the monotone character of g(t, w). We then get immediately by Theorem3.2.1 the desired estimate (3.2.4).

3.3 Global Existence

Employing the comparison Theorem 3.2.1, we shall prove the following globalexistence result.


(i) F ∈ C[R+ × Kc(Rn),Kc(Rn)], F maps bounded sets onto bounded sets,and there exists a local solution of (3.2.1) for every (t0, U0), t0 ≥ 0 andU0 ∈ Kc(Rn);

(ii) V ∈ C[R+ ×Kc(Rn),R+]; | V (t, A) − V (t, B) | ≤ L D[A,B] where L isthe local Lipschitz constant, for A,B ∈ Kc(Rn), t ∈ R+, V (t, A) → ∞ asD[A, θ] → ∞ uniformly for [0, T ] for every T > 0 and for t ∈ R+, A ∈Kc(Rn),

D+V (t, A) ≤ g(t, V (t, A)),

where g ∈ C[R2+,R], D+V (t, A) is as defined in Theorem 3.2.1.;

(iii) The maximal solution r(t) = r(t, t0, w0) of (3.2.2) exists on [t0,∞), andis positive whenever w0 > 0.

Then, for every U0 ∈ Kc(Rn) such that V (t0, U0) ≤ w0, the IVP (3.2.1) has asolution U (t) on [t0,∞) which satisfies

V (t, U (t)) ≤ r(t), t ≥ t0.

Proof Let S denote the set of all functions U defined on IU = [t0, cU) withvalues in Kc(Rn) such that U (t) is a solution of (3.2.1) on IU and

V (t, U (t)) ≤ r(t)on IU .

Define a partial order ≤ on S as follows:the relation U ≤ V implies IU ≤ IV and V (t) = U (t) on IU .We shall first show that S is nonempty. By (i) there exists a solution U (t)

of (3.2.1) defined on IU = [t0, cU ). Setting m(t) = V (t, U (t)), t ∈ IU and usingassumption (ii), we get by Theorem 3.2.1, that

V (t, U (t)) ≤ r(t), t ∈ IU ,

where r(t) is the maximal solution of (3.2.2). This proves that U ∈ S and henceS is not empty.

3.4 STABILITY CRITERIA 69

If (Uβ)β is a chain in (S,≤), then there is uniquely defined map V on IV =[t0, supβ cUβ ) that coincides with Uβ on IUβ . Clearly, V ∈ S and therefore Vis an upperbound of (Uβ)β in (S,≤). Then Zorn’s lemma assures the existenceof a maximal element Z in (S,≤). The proof of the Theorem is complete if weshow that cZ = ∞.

Suppose that it is not true, so that cZ < ∞. Since r(t) is assumed to exist on[t0,∞), r(t) is bounded on IZ . Since V (t, A) → ∞ as D[A, θ] → ∞ uniformlyin t on [t0, cZ] , the relation V (t, Z(t)) ≤ r(t) on IZ implies that D[Z(t), θ] isbounded on IZ . By (i), this shows that there is a constant M > 0 such that

D[F (t, Z(t)), θ] ≤M on IZ .

We then have, for all t1, t2 ∈ IZ , t1 ≤ t2,

D[Z(t2), Z(t1)] ≤∫ t2

t1

D[F (s, Z(s)), θ] ds ≤ M (t2 − t1),

which proves that Z is Lipschitizian on IZ and consequently has a continuousextension Z0(t) on [t0, cZ].

By continuity, we get

Z0(cZ) = U0 +∫ cZ

t0

F (s, Z0(s)) ds.

This implies that Z0(t) is a solution of (3.2.1) on [t0, cZ] and obviously V (t, Z0(t)) ≤r(t), t ∈ [t0, cZ].

Consider the IVP

DHU = F (t, U ), U (cZ) = Z0(cZ).

By the assumed local existence there is a solution U0(t) on [cZ , cZ + δ), δ > 0.Define

Z1(t) =[Z0(t) for t0 ≤ t ≤ cZ ,U0(t) for cZ ≤ t ≤ cZ + δ.

Clearly, Z1(t) is a solution of (3.2.1) on [t0, cZ + δ) and repeating the argument,we get

V (t, Z1(t)) ≤ r(t), t ∈ [t0, cZ + δ).

This contradicts the maximality of Z and hence cZ = ∞. The proof is complete.

3.4 Stability Criteria

Having necessary comparison results in terms of Lyapunov-like functions, it iseasy to establish the stability results for the set differential equations (SDE)(3.2.1) analogous to the original Lyapunov results and their extensions. How-ever, in order to investigate the stability criteria for the trivial solution of(3.2.1), we need to employ, in a natural way, the measure D[U (t), θ] = ‖U (t)‖ =


diamU (t) for t ≥ t0, where U (t) = U (t, t0, U0) is the solution of (3.2.1). Thisimplies by Proposition 1.6.1 that the diam U (t) is nondecreasing in t ≥ t0. Thisis due to the fact that, in the generation of the SDE from ordinary differentialequations(ODEs), certain undesirable elements may enter the solution U (t) andthe measure to be employed, namely, ‖U (t)‖ is therefore unsuitable to developstability theory without some adjustment. Recall that SDE (3.2.1) reduces toODE when U (t) is single valued and SDE (3.2.1) can be generated from ODEas well. The latter is done as follows.

We let F (t, A) = co f(t, A), A ∈ Kc(Rn), where f ∈ C[R+ ×Rn,Rn] arisingfrom the ODE

u′ = f(t, u), u(t0) = u0 ∈ Rn. (3.4.1)

Consequently, the solutions of u(t) of ODE (3.4.1) are imbedded in the solutionU (t) of the SDE (3.2.1).

Since the cause of the problem in SDE is the requirement of the existence ofHukuhara differences in formulating SDE, we need to incorporate the Hukuharadifference in the initial conditions also, in order to match the behavior of solu-tions of SDE with the corresponding ODE. This is precisely what we plan todo.

Suppose that the Hukuhara difference exists for any given initial valuesU0, V0 ∈ Kc(Rn) so that U0 − V0 = W0 is defined. Then we consider thestability of the solution U (t, t0, U0 − V0) = U (t, t0,W0) of (3.2.1). Before pre-senting the stability results, in this new set up, let us present some examples toillustrate our approach.

Consider the ODEu′ = −u, u(0) = u0 ∈ R, (3.4.2)

and the corresponding SDE

DHU = −U, U (0) = U0 ∈ Kc(R). (3.4.3)

Since the values of the solution (3.4.3) are interval functions, the equation (3.4.3)can be written as,

[u′1, u′2] = (−1)U = [−u2,−u1], (3.4.4)

where U (t) = [u1(t), u2(t)] and U (0) = [u10, u20]. The relation (3.4.4) is equiv-alent to the system of equations

u′1 = −u2, u1(0) = u10,

u′2 = −u1, u2(0) = u20,

whose solution for t ≥ 0, is

u1(t) = 12[u10 + u20]e−t + 1

2[u10 − u20]et,

u2(t) = 12 [u20 + u10]e−t + 1

2 [u20 − u10]et.(3.4.5)

Given U0 ∈ Kc(R), if there exists V0,W0 ∈ Kc(R) such that U0 = V0 +W0, thenthe Hukuhara difference U0 − V0 = W0, exists. Let us choose

U0 = [u10, u20], V0 =12[(u10 − u20), (u20 − u10)],


so thatW0 =

12[(u10 + u20), (u20 + u10)].

If u10 6= −u20, then we have for t ≥ 0,

U (t, U0) =12[−(u20 − u10), (u20 − u10)]et +

12[(u10 + u20), (u10 + u20)]e−t

U (t, V0) =12[(u10 − u20), (u20 − u10]et, and

U (t,W0) =12[(u10 + u20), (u10 + u20)]e−t.

If on the other hand, u10 = −u20, implies we choose U0 = [−d, d] with d = u20.Then U0 = V0 and W0 = [0, 0]. This choice eliminates the term with e−t andwe have only undesirable part of the solution. The other situation is to chooseU0 = [c, c] for some c, which eliminates the term with et and retains only thedesirable part of the solution compared with the ODE. Even when U0 is chosenas U0 = [−d, d], we can find V0 = [c − d, c + d] for some c so that we haveV0 = U0 +W0 where W0 = [c, c].

We note that for any general initial value U0, the solution of SDE (3.4.2)contains both desired and undesired parts compared to the solution of the ODE(3.4.2). In order to isolate the desired part of the solution U (t, U0) of (3.4.2)that matches the solution of the ODE (3.4.2), we need to use the initial valuessatisfying the desired Hukuhara difference of the given two initial values.

If, on the other hand, we have the SDE as

DHU = λ(t)U, U (0) = U0, (3.4.6)

which is generated byu′ = λ(t)u, u(0) = u0 (3.4.7)

where λ(t) > 0 is a real valued function from R+ → R+ such that λ ∈ L1(R+),then we see that, with similar computation,

U (t, U0) = U0 exp

[∫ t

0

λ(s) ds], t ≥ 0,

for any U0 ∈ Kc(Rn).Hence we get the stability of the trivial solution of (3.4.6). In this case, we

note that the solutions of both equations (3.4.6) and (3.4.7) match, providingthe same stability results. There is no necessity, therefore, to choose the initialvalues as before, since the undesirable part of the solution does not exist amongsolutions of (3.4.6). Consequently, it does not matter, whether we use theHukuhara difference or not, we get the same conclusion. In order to be consistentand to take care of all the situations, most of the results in this chapter areformulated in terms of Hukuhara differences of initial values.

In order to consider the stability of the trivial solution of (3.2.1), let usassume that F (t, θ) = θ, the solutions are unique and exist for all t ≥ t0. Also,


we assume, as a standard hypothesis, that the Hukuhara difference U0−V0 = W0

exists, since we suppose that U0 = V0 + W0. Consequently, we utilize thesolutions U (t) = U (t, t0, U0 − V0) = U (t, t0,W0), throughout.

Let us start with the following result on equi-stability.

Theorem 3.4.1 Assume that the following hold:

(i) V ∈ C[R+ × S(ρ),R+ ], |V (t, U1) − V (t, U2)| ≤ L D[U1, U2], L > 0 andfor (t, U ) ∈ R+ × S(ρ), where S(ρ) = [U ∈ Kc(Rn) : D[U, θ] < ρ],

D+V (t, U ) ≡ lim suph→0+

1h

[V (t+ h, U + hF (t, U ))− V (t, U )] ≤ 0; (3.4.8)

(ii) b(‖U‖) ≤ V (t, U ) ≤ a(t, ‖U‖), for (t, U ) ∈ R+ × S(ρ) where

b, a(t, .) ∈ K = σ ∈ C[[0, ρ),R+] : σ(0) = 0 and σ(ω) is increasing inω.

Then, the trivial solution of (3.2.1) is equi-stable.

Proof Let 0 < ε < ρ and t0 ∈ R+, be given. Choose a δ = δ(t0, ε) such that

a(t0, δ) < b(ε). (3.4.9)

We claim that with this δ, equi-stability holds. If not, there would exist asolution U (t) = U (t, t0,W0) of (3.2.1) and t1 > t0 such that

‖U (t1)‖ = ε and ‖U (t)‖ ≤ ε < ρ, t0 ≤ t ≤ t1, (3.4.10)

whenever ‖W0‖ < δ. By Corollary 3.2.1, we then have

V (t, U (t)) ≤ V (t0,W0), t0 ≤ t ≤ t1.

Consequently, using (ii), (3.4.9) and (3.4.10), we arrive at the following con-tradiction:

b(ε) = b(‖U (t1)‖) ≤ V (t1, U (t1)) ≤ V (t0,W0) ≤ a(t0, ‖W0‖) ≤ a(t0, δ) < b(ε).

Hence equi-stability holds, completing the proof.The next result provides sufficient conditions for equi-asymptotic stability.

In fact, it gives exponential asymptotic stability.

Theorem 3.4.2 Let the assumptions of Theorem 3.4.1 hold, except that theestimate (3.4.8) be strengthened to

D+V (t, U ) ≤ −βV (t, U ), (t, U ) ∈ R+ × S(ρ). (3.4.11)

Then the trivial solution of (3.2.1) is equi-asymptotically stable.


Proof Clearly, the trivial solution of (3.2.1) is equi-stable. Hence taking ε = ρand designating δ0 = δ(t0, ρ) , we have by Theorem 3.4.1,

‖W0‖ < δ0 implies ‖U (t)‖ < ρ, t ≥ t0,

where U (t) = U (t, t0,W0) as before.Consequently, we get from the assumption (3.4.11), the estimate

V (t, U (t)) ≤ V (t0,W0) exp[−β(t − t0)], t ≥ t0.

Given ε > 0, we choose T = T (t0, ε) = 1β lna(t0,δ0)

b(ε) + 1. Then it is easy to seethat,

b(‖U (t)‖) ≤ V (t, U (t)) ≤ a(t0, δ) e−β(t−t0) < b(ε), t ≥ t0 + T.

The proof is complete.We shall next consider the uniform stability criteria.

Theorem 3.4.3 Assume that, for (t, U ) ∈ R+ × S(ρ) ∩ Sc(η) for each 0 < η <ρ, V ∈ C[R+ × S(ρ) ∩ Sc(η),R+], we have,

|V (t, U1) − V (t, U2)| ≤ LD[U1, U2], L > 0,

D+V (t, U ) ≤ 0, (3.4.12)

andb(‖U‖) ≤ V (t, U ) ≤ a(‖U‖), a, b ∈ K. (3.4.13)

Then the trivial solution of (3.2.1) is uniformly stable.

Proof Let 0 < ε < ρ and t0 ∈ R+ be given. Choose δ = δ(ε) > 0 such thata(δ) < b(ε). Then we claim that with this δ, uniform stability follows. If not,there would exist a solution U (t) of (3.2.1), and a t2 > t1 > t0 satisfying

‖U (t1)‖ = δ, ‖U (t2)‖ = ε and δ ≤ ‖U (t)‖ ≤ ε < ρ, t1 ≤ t ≤ t2. (3.4.14)

Taking η = δ, we get from (3.4.12), the estimate

V (t2, U (t2)) ≤ V (t1, U (t1)),

and therefore, (3.4.13) and (3.4.14), together with the definition of δ, yield

b(ε) = b(‖U (t2)‖) ≤ V (t2, U (t2)) ≤ V (t1, U (t1))≤ a(‖U (t1)‖) = a(δ) < b(ε).

This contradiction proves uniform stability, completing the proof.Finally, we shall prove uniform asymptotic stability.

Theorem 3.4.4 Let the assumptions of Theorem 3.4.3 hold except that (3.4.12)is strengthened to

D+V (t, U ) ≤ −c(‖U‖), c ∈ K. (3.4.15)

Then the trivial solution of (3.2.1) is uniformly asymptotically stable.


Proof By Theorem 3.4.3, uniform stability follows. Now, for ε = ρ, we desig-nate δ0 = δ0(ρ). This means,

‖W0‖ < δ0 implies ‖U (t)‖ < ρ, t ≥ t0.

In view of the uniform stability, it is enough to show that there exists a t∗ suchthat for t0 ≤ t∗ ≤ t0 + T , where T = 1 + a(δ0)

c(δ),

‖U (t∗)‖ < δ. (3.4.16)

If this is not true, δ ≤ ‖U (t)‖, for t0 ≤ t ≤ t0 + T . Then, (3.4.15) gives,

V (t, U (t)) ≤ V (t0,W0) −∫ t

t0

c(‖U (s)‖) ds, t0 ≤ t ≤ t0 + T.

As a result, we have, in view of the choice of T ,

0 ≤ V (t0 + T, U (t0 + T )) ≤ a(δ0) − c(δ)T < 0

a contradiction. Hence there exists a t∗ satisfying (3.4.16) and uniform stabilitythen shows that

‖W0‖ < δ0 implies ‖U (t)‖ < ε, t ≥ t0 + T,

and the proof is complete.

3.5 Nonuniform Stability Criteria

In section 3.4, we discussed stability results parallel to Lyapunov’s original the-orems for set differential equations. We note that in proving nonuniform stabil-ity concepts, one needs to impose assumptions throughout R+ × S(ρ), whereasto investigate uniform stability notions it is enough to assume conditions inR+ × S(ρ) ∩ Sc(η) for 0 < η < ρ, where Sc(η) denotes the complement of S(η).The question therefore arises whether one can prove nonuniform stability no-tions under less restrictive assumptions. The answer is yes and one needs toemploy the method of perturbing Lyapunov functions to achieve this. This iswhat we plan to do in this section.

We begin with the following result which provides nonuniform stability cri-teria under weaker assumptions.


(i) V1 ∈ C[R+ × S(ρ),R+ ], | V1(t, U1) − V1(t, U2) |≤ LD[U1, U2], L1 > 0,V1(t, U ) ≤ a0(t, ‖U‖), where a0 ∈ C[R+ × [0, ρ),R+] and a0(t, .) ∈ K foreach t ∈ R+.

(ii) D+V1(t, U ) ≤ g1(t, V1(t, U )), (t, U ) ∈ R+ ×S(ρ), where g1 ∈ C[R2+,R] and

g1(t, 0) ≡ 0;

3.5 NONUNIFORM STABILITY CRITERIA 75

(iii) for every η > 0, there exists a Vη ∈ C[R+ × S(ρ) ∩ Sc(η),R+],

| Vη(t, U1) − Vη(t, U2) |≤ LηD[U1, U2]

b(‖U‖) ≤ Vη(t, U ) ≤ a(‖U‖), a, b ∈ K;

andD+V1(t, U ) +D+Vη(t, U ) ≤ g2(t, V1(t, U ) + Vη(t, U ))

for (t, U ) ∈ R+ × S(ρ) ∩ Sc(η), where g2 ∈ C[R2+,R] and g2(t, 0) ≡ 0;

(iv) the trivial solution w1 ≡ 0 of

w′1 = g1(t, w1), w1(t0) = w10 ≥ 0, (3.5.1)

is equistable.

(v) the trivial solution w2 ≡ 0 of

w′2 = g2(t, w2), w2(t0) = w20 ≥ 0, (3.5.2)

is uniformly stable.


Proof Let 0 < ε < ρ and t0 ∈ R+, be given. Since the trivial solution of (3.5.2)is uniformly stable, given b(ε) > 0 and t0 ∈ R+, there exists a δ0 = δ0(ε) > 0satisfying

0 < w20 < δ0 implying w2(t, t0, w20) < b(ε), t ≥ t0, (3.5.3)

where w2(t, t0, w20) is any solution of (3.5.2). In view of the hypothesis on a(w),there is a δ2 = δ2(ε) > 0 such that

a(δ2) <δ02. (3.5.4)

Since the trivial solution of (3.5.1) is equi-stable, given δ02> 0 and t0 ∈ R+, we

can find a δ∗ = δ∗(t0, ε) > 0 such that

0 < w10 < δ∗ implies w1(t, t0, w10) <δ02, t ≥ t0, (3.5.5)

where w1(t, t0, w10) is any solution of (3.5.1). Choose w10 = V1(t0,W0). SinceV1(t, U ) ≤ a0(t, ‖U‖), we see that there exists δ1 = δ1(t0, ε) > 0 satisfying

‖W0‖ < δ1 and a0(t0, ‖W0‖) < δ∗, (3.5.6)

simultaneously. Define δ = min(δ1, δ2).Then, we claim that

‖W0‖ < δ implies ‖U (t)‖ < ε, t ≥ t0, (3.5.7)


for any solution U (t) = U (t, t0,W0) of (3.2.1). If this is false, there would exista solution U (t) of (3.2.1) with ‖W0‖ < δ and t1, t2 > t0 such that

‖U (t1)‖ = δ2, ‖U (t2)‖ = ε,

andδ2 ≤ ‖U (t)‖ ≤ ε < ρ, (3.5.8)

for t1 ≤ t ≤ t2. We let η = δ2 so that the existence of a Vη satisfying hypothesis(iii) is assured. Hence setting

m(t) = V1(t, U (t)) + Vη(t, U (t)), t ∈ [t1, t2],

we obtain the differential inequality

D+m(t) ≤ g2(t,m(t)), t1 ≤ t ≤ t2,

which yieldsV1(t2, U (t2)) + Vη(t2, U (t2)) ≤ r2(t2, t1, w20), (3.5.9)

where w20 = V1(t1, U (t1)) + Vη(t1, U (t1)), and r2(t, t1, w20) is the maximalsolution of (3.5.2). We also have, because of assumptions (i) and (ii),

V1(t1, U (t1)) ≤ r1(t1, t0, w10),

with w10 = V1(t0,W0) where r1(t, t0, w10) is the maximal solution of (3.5.1).By (3.5.5) and (3.5.6), we get

V1(t1, U (t1)) <δ02. (3.5.10)

Also, by (3.5.4), (3.5.8) and assumption (iii) , we arrive at

Vη(t1, U (t1)) ≤ a(δ2) <δ02. (3.5.11)

Thus, (3.5.10) and (3.5.11) and the definition of w20 shows that w20 < δ0 which,in view of (3.5.3), shows that w2(t2, t1, w20) < b(ε). It then follows from (3.5.9),V1(t, U ) ≥ 0 and assumption (iii),

b(ε) = b(‖U (t2)‖) ≤ V1(t2, U (t2)) ≤ r2(t2, t1, w20) < b(ε).

This contradiction proves equi-stability of the trivial solution of (3.2.1) since(3.5.7) is then true. The proof is complete.

The next result offers conditions for equi-asymptotic stability.

Theorem 3.5.2 Let the assumptions of Theorem 3.5.1 hold except that condi-tion (ii) is strengthened to

D+V1(t, U ) ≤ −c(w(t, U )) + g1(t, V1(t, U )), c ∈ K, (ii∗)

3.5 NONUNIFORM STABILITY CRITERIA 77

w ∈ C[R+ × S(ρ),R+ ],

| w(t, U1) − w(t, U2) | ≤ N D[U1, U2], N > 0,

and D+w(t, U ) is bounded above or below. Then, the trivial solution of (3.2.1)is equi-asymptotically stable, if g1(t, w) is monotone nondecreasing in w and

w(t, U ) ≥ b0(‖U‖), b0 ∈ K. (3.5.12)

Proof By Theorem 3.5.1, the trivial solution of (3.2.1) is equi-stable. Henceletting ε = ρ so that δ0 = δ(ρ, t0), we get, by equi-stability

‖W0‖ < δ0 implies ‖U (t0)‖ < ρ, t ≥ t0.

We shall show that, for any solution U (t) of (3.2.1) with ‖W0‖ < δ0, we havelimt→∞w(t, U (t)) = 0. This and (3.5.12) imply limt→∞ ‖U (t)‖ = 0, completingthe proof.

Suppose that limt→∞ supw(t, U (t)) 6= 0. Then there would exist two diver-gent sequences t′i, t′′i and a σ > 0 satisfying

(a) w(t′i, U (t′i)) =σ

2, w(t′′i , U (t′′i )) = σ and w(t, U (t)) ≥ σ

2, t ∈ (t′i.t

′′i ),

or

(b) w(t′i, U (t′i)) = σ, w(t′′i , U (t′′i )) =σ

2and w(t, U (t)) ≥ σ

2, t ∈ (t′i, t

′′i ).

Suppose that D+w(t, U (t)) ≤ M. Then using (a) we obtain

σ

2= σ − σ

2= w(t′′i , U (t′′i )) −w(t′i, U (t′i)) ≤ M (t′′i − t′i),

which shows that t′′i − t′i ≥ σ2M

for each i. Hence by (ii∗) and Corollary 3.2.3we have

V1(t, U (t)) ≤ r1(t, t0, w10) −n∑

i=1

∫ t′′i

t′i

c[w(s, U (s))] ds, t ≥ t0.

Since w10 = V1(t0,W0) ≤ a0(t0, ‖W0‖) ≤ a0(t0, δ0) < δ∗(ρ), we get from(3.5.5), w1(t, t0, w10) <

δ0(ρ)2, t ≥ t0. we thus obtain

0 ≤ V1(t, U (t)) ≤ δ0(ρ)2

− c(σ

2)σ

2Mn.

For sufficiently large n, we get a contradiction and therefore lim supt→∞w(t, U (t)) =0. Since w(t, U ) ≥ b0(‖U (t)‖) by assumption, it follows that limt→∞ ‖U (t)‖ = 0and the proof is complete.

The following remarks are relevant.


Remark 3.5.1 The functions g1(t, w) = g2(t, w) ≡ 0 are admissible in The-orem 3.5.1, and so the same conclusion can be reached. If V1(t, U ) ≡ 0 andg1(t, w) ≡ 0, then we get uniform stability from Theorem 3.5.1. If, on the otherhand, Vη(t, U ) ≡ 0, g2(t, w) ≡ 0 and V1(t, U ) ≥ b(‖U‖), b ∈ K, then Theorem3.5.1 yields equi-stability. We note that known results on equi-stability requirethe assumption to hold everywhere in S(ρ) and Theorem 3.5.1 relaxes such arequirement considerably by the method of perturbing Lyapunov functions.

Remark 3.5.2 The functions g1(t, w) ≡ g2(t, w) ≡ 0 are admissible in Theo-rem 3.5.2 to yield equi-asymptotic stability. Similarly, if Vη(t, U ) ≡ 0, g2(t, w) ≡0 with V1(t, U ) ≥ b(‖U‖), b ∈ K, implies the same conclusion. If V1(t, U ) ≡ 0and g1(t, w) ≡ 0 in Theorem 3.5.1, to get uniform asymptotic stability, oneneeds to strengthen the estimate on D+Vη(t, U ). This we state as a corollary.

Corollary 3.5.1 Suppose that the assumptions of Theorem 3.5.1 hold withV1(t, U ) ≡ 0, g(t, w) ≡ 0. Suppose further that

D+Vη(t, U ) ≤ −c[w(t, U )]+g2(t, Vη(t, U )), (t, U ) ∈ R+×S(ρ)∩Sc (η), (3.5.13)

where w ∈ C[R+ ×S(ρ),R+ ], w(t, U ) ≥ b(‖U‖), c, b ∈ K and g2(t, w) is nonde-creasing in w. Then, the trivial solution of (3.2.1) is uniformly asymptoticallystable.

Proof The trivial solution of (3.2.1) is uniformly stable by Remark 3.5.1 in thepresent case. Hence taking ε = ρ and designating δ0 = δ(ρ), we have

‖W0‖ < δ0 implies ‖U (t)‖ < ρ, t ≥ t0.

To prove uniform attractivity, let 0 < ε < ρ be given. Let δ = δ(ε) > 0 be thenumber relative to ε in uniform stability. Choose T = b(ρ)

c(δ)+ 1. Then we shall

show that there exists a t∗ ∈ [t0, t0 + T ] such that w(t∗, U (t∗)) < b(δ) for anysolution U (t) of (3.2.1) with ‖W0‖ < δ0.

If this is not true, w(t, U (t)) ≥ b(δ), t ∈ [t0, t0+T ].Now using the assumption(3.5.13) and arguing as in Corollary 3.2.3, we get

0 ≤ Vη(t0 + T, U (t0 + T )) ≤ r2(t0 + T, t0, w20) −∫ t0+T

t0

w(s, U (s)) ds.

This yields, since r2(t, t0, w20) < b(ρ) and the choice of T ,

0 ≤ Vη(t0 + T, U (t0 + T )) ≤ b(ρ) − c(δ)T < 0,

which is a contradiction. Hence there exists a t∗ ∈ [t0, t0 + T ] satisfyingw(t∗, U (t∗)) < b(δ), which implies ‖U (t)‖ < δ. Consequently, it follows, byuniform stability that

‖W0‖ < δ0 implies ‖U (t)‖ < ε, t ≥ t0 + T,


3.6 CRITERIA FOR BOUNDEDNESS 79

3.6 Criteria for Boundedness

We shall, in this section, investigate the boundedness of solutions of the setdifferential equation

DHU = F (t, U ), U (t0) = U0 ∈ Kc(Rn), (3.6.1)

where F ∈ C[R+×Kc(Rn),Kc(Rn)]. Corresponding to the definitions of variousstability notions given in section 3.4, we also have boundedness concepts, whichwe define below.

Definition 3.6.1 The solution of (3.6.1) is said to be

(B1) equi-bounded, if for any α > 0 and t0 ∈ R+, there exists a β = β(t0, α) >0 such that

‖W0‖ < α implies ‖U (t)‖ < β, t ≥ t0;

(B2) uniform-bounded, if β in (B1)does not depend on t0;

(B3) quasi-equi-ultimately bounded for a bound B if for each α ≥ 0, t0 ∈ R+,there exists a B > 0 and a T = T (t0, α) > 0 such that

‖W0‖ < α implies ‖U (t)‖ < B, t ≥ t0 + T.

(B4) quasi-uniform ultimately bounded if T in (B3) is independent of t0;

(B5) equi-ultimately bounded, if (B1) and (B3) hold simultaneously;

(B6) uniform ultimately bounded if (B2) and (B4) hold simultaneously;

(B7) equi-Lagrange stable if (B1) and (S3) hold;

(B8) uniformly Lagrange stable if (B2) and (S4) hold.

Using the comparison results of section 3.2, we shall prove simple bounded-ness results.


(i) V ∈ C[R+ × Kc(Rn),R+], | V (t, U1) − V (t, U2) |≤ LD[U1, U2], L > 0,and for (t, U ) ∈ R+ ×Kc(Rn), D+V (t, U ) ≤ 0;

(ii) b(‖U‖) ≤ V (t, U ) ≤ a(t, ‖U‖) , for (t, U ) ∈ R+ ×Kc(Rn) where b, a(t, .) ∈K = [σ ∈ C[R+,R+] : σ(ω) is increasing in ω and σ(w) → ∞ as w → ∞].

Then, (B1) holds.

Proof Let 0 < α and t0 ∈ R+, be given. Choose β = β(t0, α) such that

a(t0, α) < b(β). (3.6.2)


With this β, (B1) holds. If this is not true, there would exist a solutionU (t) = U (t, t0,W0) of (3.6.1) and a t1 > t0 such that

‖U (t1)‖ = β and ‖U (t)‖ ≤ β, t0 ≤ t ≤ t1.

Assumption (i) and Corollary 3.2.1 show that

V (t, U (t)) ≤ V (t0,W0), t0 ≤ t ≤ t1.

As a result, condition (ii) and (3.6.2) yield

b(β) = b(‖U (t1)‖) ≤ V (t1, U (t1)) ≤ V (t0,W0)≤ a(t0, ‖W0‖) < a(t0, α) < b(β).

This contradiction proves (B1) and we are done.For uniform boundedness, the following result is obtained under weaker as-

sumptions.


(i) V ∈ C[R+ × Sc(ρ),R+], where ρ may be large; |V (t, U1) − V (t, U2)| ≤LD[U1, U2], and for (t, U ) ∈ R+ × Sc(ρ),D+V (t, U ) ≤ 0;

(ii) b(‖U‖) ≤ V (t, U ) ≤ a(‖U‖) , for (t, U ) ∈ R+×Sc(ρ) where a, b ∈ K, whichare defined only on [ρ,∞).

Then, (B2) holds.

Proof The proof is similar to the proof of Theorem 3.6.1 except that the choiceof β is now made so that a(α) < b(β) and consequently β is independent of t0.Also, α > ρ for the proof since the assumptions are only for Sc(ρ). However, if0 < α ≤ ρ, we can take β = β(ρ) and again the proof follows.

We shall give a typical result that offers conditions for equi-ultimate bound-edness, that is for (B5).

Theorem 3.6.3 Let the assumptions of Theorem 3.6.1 hold except that westrengthen the estimate on D+V (t, U ) as

D+V (t, U ) ≤ −ηV (t, U ), η > 0, (t, U ) ∈ R+ ×Kc(Rn), (3.6.3)

and suppose that condition (ii) holds for ‖U‖ ≥ B. Then (B5) holds.

Proof Clearly (B1) is obtained from Theorem 3.6.1. Hence

‖W0‖ < α implies ‖U (t)‖ < β, t ≥ t0.

Now (3.6.3) yields the estimate

V (t, U (t)) ≤ V (t0,W0)e−η(t−t0), t ≥ t0. (3.6.4)

3.6 CRITERIA FOR BOUNDEDNESS 81

Let T = 1η ln a(t0,α)

b(B) and suppose that for t ≥ t0 + T, ‖U (t)‖ ≥ B. Then, we getfrom (3.6.4)

b(B) ≤ b(‖U (t)‖) ≤ V (t, U (t)) < a(t0, α)e−ηT = b(B).

This contradiction proves (B5) and the proof is complete.Finally, we shall provide a result on nonuniform boundedness property using

the method of perturbing Lyapunov functions.


(i) ρ > 0, V1 ∈ C[R+ × S(ρ),R+ ], V1 is bounded for (t, U ) ∈ R+ × ∂S(ρ), and

|V1(t, U1) − V (t, U2)| ≤ L1D[U1, U2], L1 > 0,

D+V1(t, U ) = lim suph→0+

1h

[V1(t+ h, U + hF (t, U ))− V1(t, U )]

≤ g1(t, V1), (t, U ) ∈ R+ × Sc(ρ),

where g1 ∈ C[R2+,R];

(ii) V2 ∈ C[R+ × Sc(ρ),R+],

b(‖U‖) ≤ V2(t, U ) ≤ a(‖U‖), a, b ∈ K,

D+V1(t, U ) +D+V2(t, U ) ≤ g2(t, V1(t, U ) + V2(t, U )), g2 ∈ C[R2+,R],

(iii) the scalar differential equations

w′1 = g1(t, w1), w1(t0) = w10 ≥ 0, (3.6.5)

andw′

2 = g2(t, w2), w2(t0) = w20 ≥ 0, (3.6.6)

are equi-bounded and uniformly bounded respectively.

Then the system (3.6.1) is equi-bounded.

Proof Let B1 > ρ and t0 ∈ R+, be given. Let

α1 = α1(t0, B1) = maxα0, α∗,

where α0 = maxV1(t0,W0) : W0 ∈ clS(B1) ∩ Sc(ρ)andα∗ ≥ V1(t, U ) for (t, U ) ∈ R+ × ∂S(ρ).

Since equation (3.6.5) is equi-bounded, given α1 > 0, and t0 ∈ R+, thereexist a β0 = β(t0, α1) such that

w1(t, t0, w10) < β0, t ≥ t0, (3.6.7)


provided w10 < α1,where w1(t, t0, w10) is any solution (3.6.5). Let α2 = a(B1)+β0, the uniform boundedness of equation (3.6.6) yields that

w2(t, t0, w20) < β1(α2), t ≥ t0, (3.6.8)

provided w20 < α2, where w2(t, t0, w20) is any solution of (3.6.6). Choose B2

satisfyingb(B2) > β1(α2). (3.6.9)

We now claim that W0 ∈ S(B1) implies that

U (t, t0,W0) ∈ S(B2),

for t ≥ t0, where U (t, t0,W0) is any solution of (3.6.1).If it is not true, there exists a solution U (t, t0,W0) of (3.6.7) with W0 ∈

S(B1), such that for some t∗ > t0, ‖U (t∗, t0,W0)‖ = B2. Since B1 > ρ, thereare two possibilities to consider:

(1) U (t, t0,W0) ∈ Sc(ρ) for t ∈ [t0, t∗];

(2) there exists a t ≥ t0 such that U (t, t0,W0) ∈ ∂S(ρ) and U (t, t0,W0) ∈Sc(ρ) for t ∈ [t, t∗].

If (1) holds, we can find t1 > t0 such that

U (t1, t0,W0) ∈ ∂S(B1),

U (t∗, t0,W0) ∈ ∂S(B2), and (3.6.10)

U (t, t0,W0) ∈ Sc(B1), t ∈ [t1, t∗].

Setting m(t) = V1(t, U (t, t0,W0)) + V2(t, U (t, t0,W0)) for t ∈ [t1, t∗], and thenusing Theorem 3.2.1, we can obtain the differential inequality

D+m(t) ≤ g2(t,m(t)), t ∈ [t1, t∗],

and so,m(t) ≤ γ2(t, t1,m(t1)), t ∈ [t1, t∗],

where γ2(t, t1, v0) is the maximal solution of (3.6.6) with γ2(t1, t1, v0) = v0.Thus,

V1(t∗, U (t∗, t0,W0)) + V2(t∗, U (t∗, t0,W0))

≤ γ2(t∗, t1, V1(t1, U (t1, t0,W0)) + V2(t1, U (t1, t0,W0))). (3.6.11)

Similarly, we also have

V1(t1, U (t1, t0,W0)) ≤ γ1(t1, t0, V1(t0,W0)), (3.6.12)

where γ1(t, t0, u0) is the maximal solution of (3.6.5).

3.7 SET DIFFERENTIAL SYSTEMS 83

Set w10 = V1(t0,W0) < α1. Then

V1(t1, U (t1, t0,W0)) ≤ γ1(t1, t0, V (t0,W0)) ≤ β0,

since (3.6.7) holds.Furthermore, V2(t1, U (t1, t0,W0)) ≤ a(B1). Consequently, we have

w20 = V1(t1, U (t1, t0,W0)) + V2(t1, U (t1, t0,W0)) ≤ β0 + a(B1) = α2. (3.6.13)

Combining (3.6.8),(3.6.9),(3.6.10) and (3.6.13), we obtain

b(B2) ≤ m(t∗) ≤ γ(t∗) ≤ β1(α2) < b(B2), (3.6.14)

which is a contradiction.If case (2) holds, we also arrive at the inequality (3.6.11), where t1 > t

satisfies (3.6.10). We then have, in place of (3.6.12), the relation

V1(t1, U (t1, t0,W0)) ≤ γ1(t1, t, V1(t, U (t, t0,W0))).

Since U (t, t0,W0) ∈ ∂S(ρ) and V1(t, U (t, t0,W0)) ≤ α∗ ≤ α1, arguing as before,we get the contradiction (3.6.14). This proves that for any given B1 > ρ, t0 > 0,there exists a B2 such that

W0 ∈ S(B1) impliesU (t, t0,W0) ∈ S(B2), t ≥ t0.

For B1 < ρ, we set B2(t0, B1) = B2(t0, ρ) and hence the proof is complete.

3.7 Set Differential Systems

In this section we shall attempt to study the set differential system, given by

DHU = F (t, U ), U (t0) = U0, (3.7.1)

where F ∈ C[R+ ×Kc(Rn)N ,Kc(Rn)N ], U ∈ Kc(Rn)N ,Kc(Rn)N = (Kc(Rn)×Kc(Rn) × .....× Kc(Rn), N times), U = (U1, ...., UN) such that for each i, 1 ≤i ≤ N, Ui ∈ Kc(Rn). Note also U0 ∈ Kc(Rn)N .

We have the following two possibilities to measure the new variables U,U0, F .

(1) Define D0[U, V ] =∑N

i=1D[Ui, Vi], where U, V ∈ Kc(Rn)N and employ themetric space (Kc(Rn)N , D0).

(2) Define D : Kc(Rn)N ×Kc(Rn)N → RN+ such that

D[U, V ] = (D[U1, V1], D[U2, V2], ......D[UN, VN ]),

and employ the generalized metric space (Kc(Rn)N , D).


If we utilize option (1) above, the assumption (2.2.4) of Theorem 2.2.1 ap-pears as

D0[F (t, U ), F (t, V )] =N∑

i=1

D(fi(t, U ), fi(t, V ) ≤ g(t,D0[U, V ]). (3.7.2)

On the other hand, if we choose option (2) then the assumption (2.2.4) takesthe form

D[F (t, U ), F (t, V )] ≤ G(t, D[U, V ]), (3.7.3)

where G ∈ C[R+ × RN ,RN ].In this case, condition (2.2.4) reduces to

D[F (t, U ), F (t, V )] ≤ SD[U, V ]), (3.7.4)

where S = (Sij) is an N ×N matrix with Sij ≥ 0, for all i, j, which correspondsto the generalized contractive condition. Of course, the matrix S needs to satisfya suitable condition, that is , for some k > 1, Sk must be an A−matrix, whichmeans I − Sk is positive definite, where I is the identity matrix. For details ofgeneralized spaces and contraction mapping theorem in this set up, see Bernfeldand Lakshmikantham [1].

Moreover, in order to arrive at the corresponding estimate (3.2.4) of Theorem3.2.1, for example , one is required to utilize the corresponding theory of systemsof differential inequalities, which demands that G(t, w) have the quasi-monotoneproperty, which is defined as follows:

w1 ≤ w2 and w1i = w2i for some i, 1 ≤ i ≤ N,

impliesGi(t, w1) ≤ Gi(t, w2), w1, w2 ∈ RN .

If G(t, w) = Aw, where A is an N×N matrix then the quasi-monotone propertyreduces to requiring aij ≥ 0, i 6= j.

The method of vector Lyapunov-like functions has been very effective in theinvestigation of the qualitative properties of large-scale differential systems.

We shall extend this technique to set differential systems (3.7.1) where, as weshall see, both metrics described above are very useful. For this purpose, let usprove the following comparison result in terms of vector Lyapunov-like functionsrelative to the set differential system (3.7.1). We note that the inequalitiesbetween vectors in RN are to be understood as componentwise.

Theorem 3.7.1 Assume that V ∈ C[R+ ×Kc(Rn)N ,RN+ ],

|V (t, U1) − V (t, U2)| ≤ A D[U1, U2],

where A is on N ×N matrix with nonnegative elements, and for (t, U ) ∈ R+ ×Kc(Rn)N ,

D+V (t, U ) ≤ G(t, V (t, U )), (3.7.5)

3.7 SET DIFFERENTIAL SYSTEMS 85

where G ∈ C[R+ × RN+ ,RN ]. Suppose further that G(t, w) is quasi-monotone in

w for each t ∈ R+ and r(t) = r(t, t0, w0) is the maximal solution of

w′ = G(t, w), w(t0) = w0 ≥ 0, (3.7.6)

existing for t ≥ t0. Then

V (t, U (t)) ≤ r(t), t ≥ t0, (3.7.7)

where U (t) = U (t, t0,W0) is any solution of (3.7.1) existing for t ≥ t0.

Proof Let U (t) be any solution of (3.7.1) existing for t ≥ t0.Define m(t) = V (t, U (t)) so that m(t0) = V (t0,W0) ≤ w0. Now for small

h > 0, we have, in view of Lipschitz conditions,

m(t + h) −m(t) = V (t + h, U (t+ h)) − V (t, U (t))≤ AD[U (t+ h), U (t) + hF (t, U (t))]

+V (t+ h, U (t) + hF (t, U (t)))− V (t, U (t)).

It follows therefore that


1h

[m(t+ h) −m(t)] ≤ D+V (t, U (t))

+A lim suph→0+

1h

[D[U (t+ h), U (t) + hF (t, U (t)))].

Since DHU is assumed to exist, we see that U (t + h) = U (t) + Z(t) whereZ(t) = Z(t, h) is the Hukuhara difference for small h > 0. Hence utilizing theproperties of D[U, V ] we obtain,

D[U (t+ h), U (t) + hF (t, U (t))] = D[U (t) + Z(t), U (t) + hF (t, U (t))]= D[Z(t), hF (t, U (t)))]= D[U (t+ h) − U (t), hF (t, U (t)))].

As a result, we get

1hD[U (t+ h), U (t) + hF (t, U (t))] = D

[U (t + h) − U (t)

h, F (t, U (t))

]

and consequently

lim suph→0+

1hD[U (t+ h), U (t) + hF (t, U (t))]

= lim suph→0+

1h

[D

[U (t+ h) − U (t)

h, F (t, U (t))

]]= D[DHU (t), F (t, U (t))] = 0,

since U (t) is a solution of (3.7.1). We therefore have the vectorial differentialinequality,

D+m(t) ≤ G(t,m(t)), m(t0) ≤ w0, t ≥ t0,


which by the theory of differential inequalities for systems (Lakshmikanthamand Leela [1]) yields

m(t) ≤ r(t), t ≥ t0,

proving the claimed estimate (3.7.7).The following corollary of Theorem 3.7.1 is interesting.

Corollary 3.7.1 The function G(t, w) = Aw, where A is an N × N matrixsatisfying aij ≥ 0, i 6= j, is admissible in Theorem 3.7.1 and yields the estimate

V (t, U (t)) ≤ V (t0,W0)eA(t−t0), t ≥ t0.

3.8 The Method of Vector Lyapunov Functions

We shall prove a typical result that gives sufficient conditions in terms of vectorLyapunov-like functions for the stability properties of the trivial solution of theset differential system (3.7.1).


(i) G ∈ C[R+ × RN+ ,RN ], G(t, 0) ≡ 0 and G(t, w) is quasi-monotone nonde-

creasing in w for each t ∈ R+;

(ii) V ∈ C[R+ × S(ρ),RN+ ], |V (t, U1) − V (t, U2)| ≤ A D[U1, U2], where A is a

nonnegative N × N matrix and the function

V0(t, U ) =N∑

i=1

Vi(t, U ) (3.8.1)

satisfiesb(D0[U, θ]) ≤ V0(t, U ) ≤ a(D0[U, θ]), a, b ∈ K;

(iii) F ∈ C[R+ × S(ρ),Kc(Rn)N ], F (t, θ) ≡ θ and

D+V (t, U ) ≤ G(t, V (t, U )), (t, U ) ∈ R+ × S(ρ),

where S(ρ) = [U ∈ Kc(Rn)N : D0[U, θ] < ρ].

Then, the stability properties of the trivial solution of (3.7.6) imply the corre-sponding stability properties of the trivial solution of (3.7.1).

Proof We shall prove only equi-asymptotic stability of the trivial solution of(3.7.1). For this purpose, let us first prove equi-stability . Let 0 < ε < ρand t0 ∈ R+, be given. Assume that the trivial solution of (3.7.6) is equi-asymptotically stable. Then, it is equi-stable. Hence given b(ε) > 0 and t0 ∈ R+,there exists a δ1 = δ1(t0, ε) > 0 such that

N∑

i=1

wi0 < δ1 impliesN∑

i=1

wi(t, t0, w0) < b(ε), t ≥ t0, (3.8.2)

3.8 THE METHOD OF VECTOR LYAPUNOV FUNCTIONS 87

where w(t, t0, w0) is any solution of (3.7.6). Choose w0 = V (t0,W0) and aδ = δ(t0, ε) > 0 satisfying

a(δ) < δ1. (3.8.3)

Let D0[W0, θ] < δ. Then, we claim that D0[U (t), θ] < ε, t ≥ t0, for any solutionU (t) = U (t, t0,W0) of (3.7.1). If this is not true, there would exist a solutionU (t) of (3.7.1) with D0[W0, θ] < δ and a t1 > t0 such that

D0[U (t1), θ] = ε and D0[U (t), θ] ≤ ε < ρ, t0 ≤ t ≤ t1. (3.8.4)

Hence we have by Theorem 3.7.1,

V (t, U (t)) ≤ r(t, t0, w0), t0 ≤ t ≤ t1, (3.8.5)

where r(t, t0, w0) is the maximal solution of (3.7.6). Since

V0(t0,W0) ≤ a(D0[W0, θ]) < a(δ) < δ1,

the relations (3.8.2), (3.8.3), (3.8.4) and (3.8.5) yield

b(ε) ≤ V0(t1, U (t1)) ≤ r0(t1, t0, w0) < b(ε),

where r0(t, t0, w0) =∑N

i=1 ri(t, t0, w0). This contradiction proves that the trivialsolution of (3.7.1) is equi-stable.

Suppose next that the trivial solution of (3.7.6) is quasi-equi-asymptoticallystable . Set ε = ρ and δ0 = δ(t0, ρ). Let 0 < η < ρ. Then given b(η) andt0 ∈ R+, there exist δ∗1 = δ1(t0) > 0 and T = T (t0, η) > 0 satisfying

N∑

i=1

wi0 < δ∗1 impliesN∑

i=1

wi(t, t0, w0) < b(η), t ≥ t0 + T. (3.8.6)

Choosing w0 = V (t0,W0) as before, we find δ∗0 = δ0(t0) > 0 such that a(δ∗0) <δ∗1 . Let δ0 = min(δ∗1 , δ∗0) and D0[W0, θ] < δ0. This implies D0[U (t), θ] < ρ,t ≥ t0 and therefore the estimate (3.8.5) holds for all t ≥ t0.

Suppose now that there is a sequence tk, tk ≥ t0+T, tk → ∞ as k → ∞,and η ≤ D0[U (tk), θ], where U (t) is any solution of (3.7.1) with D0[W0, θ] < δ0.

In view of (3.8.6), this leads to the contradiction

b(η) ≤ V0(tk, U (tk)) ≤ r0(tk, t0, w0) < b(η).

Hence the trivial solution of (3.7.1) is equi-asymptotically stable and the proofis complete.

In order to apply the method of vector Lyapunov functions to concrete prob-lems, it is necessary to know the properties of solutions of the comparison system(3.7.6), which is difficult in general, except when G(t, w) = Aw, where A is aquasi-monotone N × N stability matrix. Hence we shall present some simpleand useful techniques to deal with this problem.

We shall first prove a result which reduces the study of the properties ofsolutions of (3.7.6) to that of a scalar differential equation

v′ = G0(t, v), v(t0) = v0 ≥ 0 (3.8.7)

where G0 ∈ C[R2+,R]. Specifically, we have the following result.


Lemma 3.8.1 Assume that L ∈ C1[R+,RN+ ], G ∈ C[R+ × RN

+ ,RN ], G0 ∈C[R2

+,R] and G,G0 are smooth enough to assure the existence and uniquenessof solutions for t ≥ t0 of (3.7.6) and (3.8.7) respectively. Suppose further thatfor (t, v) ∈ R2

+,

G(t, L(v)) ≤ dL(v)dv

G0(t, v).

Then w0 ≤ L(v0) implies

w(t, t0, w0) ≤ L(v(t, t0, v0)), t ≥ t0, (3.8.8)

where w(t, t0, w0), v(t, t0, w0) are the solutions of (3.7.6) and (3.8.7) respectively.

Proof Set m(t) = L(v(t, t0, v0), so that m(t0) = L(v0)) ≥ w0 and

m′(t) =dL(v(t, t0, v0))

dvG0(t, v(t, t0, v0))

≥ G(t, L(v(t, t0, v0))) = G(t,m(t)).

hence by the comparison Theorem 1.4.1 in Lakshmikantham and Leela[1] , weget the stated result in view of uniqueness of solutions.

Let us give an example to illustrate Lemma 3.8.1.Suppose that G1 = −2w2

1, G2 = −2w32 + 2w1w

322 , so that

w′1 = −2w2

1,

w′2 = −2w3

2 + 2w1w322 . (3.8.9)

Choosing L1(v) = 35v

32 , L2(v) = v and

G0(t, v) =

−25v

3, 0 ≤ v < 1,−2

5v

52 , 1 ≤ v,

the assumptions of Lemma 3.8.1 are satisfied. Clearly, the trivial solution of(3.8.7) is uniformly asymptotically stable and therefore the trivial solution of(3.8.9) is also uniformly asymptotically stable.Lemma 3.8.2. Assume that Q ∈ C1[RN

+ ,R+], G ∈ C[R+ × RN+ ,RN ], G0 ∈

C[R2+,R] and for (t, w) ∈ R+ × RN

+ ,

dQ(w)dw

G(t, w) ≤ G0(t, Q(w)). (3.8.10)

Then, any solution w(t) = w(t, t0, w0) of (3.7.6) existing for t ≥ t0, satisfies

Q(w(t)) ≤ v(t), t ≥ t0,

where v(t) = v(t, t0, v0) is the maximal solution of (3.8.7) existing for t ≥ t0,provided Q(w0) ≤ v0.Proof Let w(t) = w(t, t0, w0) be any solution of (3.7.6) existing for t ≥ t0.

3.9 NONSMOOTH ANALYSIS 89

Set p(t) = Q(w(t)), Then, we have

p′(t) =dQ(w(t))

dwG(t, w(t)) ≤ G0(t, Q(w(t))) = G0(t, p(t))),

and p(t0) ≤ v0. Hence by the Theorem 1.4.1 in Lakshmikantham and Leela [1],it follows that p(t) ≤ v(t), t ≥ t0, where v(t) is the maximal solution of (3.8.7).Hence the proof is complete.

As an example, consider the case G(t, w) = Aw where A is an N ×N matrixwith aij ≥ 0, i 6= j, and A is quasi-diagonally dominant, that is, for some di > 0,

di|aii| >n∑

j=1,i 6=j

dj|aij|. (3.8.11)

Choosing Q(w) =∑N

i=1 diwi for some di > 0, we see that (3.8.10) is satisfiedby G0(t, v) = −γv for some γ > 0 in view of (3.8.11). Consequently, the trivialsolution of (3.8.7) is exponentially asymptotically stable which implies that thetrivial solution of (3.7.6) does have the same property.

3.9 Nonsmooth Analysis

In the previous sections of this chapter, we developed several results in stabilitytheory by utilizing continuous Lyapunov-like functions and investigated analo-gous results parallel to standard Lyapunov stability theory for set differentialequations.

In this and the next section, we concern ourselves with Lyapunov stabilitytheory employing Lyapunov-like functions, which are only lower semi continu-ous (lsc) and this requires introducing the concepts and results of nonsmoothanalysis extending suitably to the present set up. We have already sketched inSections 2.7 and 2.8, some results related to the existence of Euler solutions andflow invariance in terms of proximal normals. There is an intimate connectionbetween proximal normal theory and subdifferentials of lower semicontinuousfunctions that we develop before proceeding to build Lyapunov theory in theframework of lsc functions and set differential equations. We shall embark onthis aspect in this section and consider Lyapunov’s theory in the following sec-tion.

Let us start with proximal normals again, since we did not provide in Section2.8, all the necessary tools needed for our purpose.

Recall that D[A, θ] = ‖A‖ = supa∈A‖a‖, and

‖A+ B‖2 ≤ ‖A‖2 + ‖B‖2 + 2 < A,B >, (3.9.1)

where for A,B ∈ Kc(Rn),

< A,B > = sup(a · b) : a ∈ A, b ∈ B. (3.9.2)


Let f : Kc(Rn) → R. Let U, V ∈ Kc(Rn) be such that there exists a Z ∈ Kc(Rn)satisfying V = U + Z. So V − U the Hukuhara difference of V, U , exists.

If there exists an element A(U ) ∈ Kc(Rn) such that

|f(V ) − f(U )− < A(U ), Z > | ≤ ε ‖Z‖, ε > 0,

where < A(U ), Z > is defined as in (3.9.2), then we say that fU (U ) = A(U ) isthe derivative of f at U . We note that fU : Kc(Rn) → Kc(Rn).

Consider next, F (U ) = fU (U ). For A,B ∈ Kc(Rn), we first define (A,B) asan element of Rn whose ith component is given by

(A,B)i = sup(aibi) : a ∈ A, b ∈ B, 1 ≤ i ≤ n.

Suppose there exists a map f defined for each U ∈ Kc(Rn), mapping Kc(Rn)into Kc(Rn) = the set of compact, convex subsets in Kc(Rn).

Then, we define,

D[F (V ), F (U ) + << f(U ), Z >>] ≤ ε ‖Z‖, ε > 0,

where

<< f (U ), Z >> = (B,Z) : B ∈ f (U ) and f : Kc(Rn) → Kc(Rn).

Then, we say that FU (U ) = f (U ) is the derivative of F at U .With these preliminaries, we consider, as in Section 2.8, the set differential

equationDHU = F (t, U ), U (t0) = U0 ∈ Kc(Rn), (3.9.3)

where F : [t0,∞)×Kc(Rn) → Kc(Rn) is any function, and the scalar differentialequation

u′ = g(t, u), u(t0) = u0 ≥ 0, (3.9.4)

where g ∈ C[R2+,R+], as in (2.8.4)

Let Ω ⊂ Kc(Rn) be a nonempty, closed set. Let U ∈ Kc(Rn) be not lyingin Ω. Suppose that the Hukuhara difference U − S exists for every S ∈ Ω. Thatis, for each S ∈ Ω, there exists a Z ∈ Kc(Rn) such that U = S + Z. Supposefurther that, there exists an element S ∈ Ω whose distance to U is minimal,that is,

D0[U,Ω] = ‖U − S‖ = infS0∈Ω

‖U − S0‖. (3.9.5)

We call such an S ∈ Ω a projection of U onto Ω and denote the set of all suchelements by projΩ(U ). The element U − S will be called the proximal normaldirection to Ω at S. Any nonnegative multiple ξ = t(U − S), t ≥ 0, we call aproximal normal to Ω at S. The set of all ξ obtained in this manner is said tobe a proximal normal cone to Ω at S and is denoted by NP

Ω (S). Suppose thatS ∈ Ω is such that S 6∈ projΩ(U ), for any U ∈ Kc(Rn), not in Ω, then we setNP

Ω (S) = θ. When S 6∈ Ω, NPΩ (S) is not defined.

3.9 NONSMOOTH ANALYSIS 91

Suppose that S ∈ projΩ(U ). Then, ‖U − S‖ ≥ ‖U − S‖ for all S ∈ Ω. As aresult, we have using (3.9.1),

‖U − S‖2 = ‖U − S + S − S‖2

≤ ‖U − S‖2 + ‖S − S‖2 + 2 < U − S, S − S >,

which implies

< U − S, S − S > ≤ 12‖S − S‖2 for all S ∈ Ω. (3.9.6)

However, any element C = U − S, satisfying (3.9.6) need not be such thatS ∈ projΩ(U ). Consequently, we set NP

Ω (S) = θ, for any ξ = t(U − S),satisfying

< ξ, S − S > ≤ σ ‖S − S‖2 for all S ∈ Ω, (3.9.7)

where σ = σ(ξ, S) > 0, but S 6∈ projΩ(U ). Thus, we assume as an axiom, thefollowing proposition.

Proposition 3.9.1 For any given δ > 0, ξ ∈ NPΩ (S), if and only if there

exists a σ = σ(ξ, S) > 0, such that

< ξ, S − S > ≤ σ‖S − S‖2, for all S ∈ Ω ∩B(S, δ). (3.9.8.)

It can be verified that S ∈ projΩ(U ) is equivalent to S ∈ projΩ(S + δ(U − S)).

We shall next consider the subgradients of lower semicontinuous(lsc) func-tions.

Let f : Kc(Rn) → (−∞,∞] be a lsc function with dom(f) = X ∈ Kc(Rn) :f(X) < ∞. Suppose that (ζ,−λ) ∈ Kc(Rn) × R belongs to NP

epi(f)(X, r). Itcan be verified that (i) λ ≥ 0. (ii) r = f(X) ifλ > 0, and (iii) λ =0 if r > f(X).

An element ζ ∈ Kc(Rn) is said to be the proximal subgradient of f at x ∈dom (f) provided that (ζ,−1) ∈ Nepi(f)(X, f(X)), where epi(f) = (X, r) ∈Kc(Rn) × R : f(X) ≤ r.

The set of all such ζ is denoted by ∂P f(X) and is referred to as proximalsubdifferential or P-subdifferential. Let us note that because a cone is involved,if λ > 0, and (ζ,−λ) ∈ NP

epi(f)(X, f(X)), then ζλ∈ ∂P f(X).

Also, the following result concerns the approximation of horizontal proximalnormals to epigraphs, by nonhorizontal proximal normals, and is needed for ourlater use.

Theorem 3.9.1 Let f : Kc(Rn) → R be a lsc function and let (θ, 0) ∈Nepi(f)(X, f(X)). Then, for every ε > 0 there exists X ′ ∈ X+εB and (ζ,−λ) ∈Nepi(f) (X ′, f(X ′)) such that

λ > 0, | f(X ′) − f(X) |< ε, ‖(θ, 0) − (ζ,−λ)‖ < ε (3.9.9)

We are now in a position to prove the following proximal subgradient in-equality.


Theorem 3.9.2 Let f : Kc(Rn) → (−∞,∞] be a lsc function and let X ∈dom(f). Then, ζ ∈ ∂P f(X) if and only if there exists positive numbers σ, andη such that

f(Y ) ≥ f(X)+ < ζ, Y −X > −σ‖Y −X‖2 (3.9.10)for all Y ∈ B(X, η).

Proof Suppose that (3.9.10) holds. we then have by adding σ(α− f(X))2,

α− f(X) + σ[‖Y −X‖2 + (α− f(X))2 ] ≥ < ζ, Y −X >

for all α ≥ f(Y ). This implies,

< (ζ,−1), [(Y, α)− (X, f(X))] > ≤ σ‖(Y, α) − (X, f(X))‖2

for all points (Y, α) ∈ epi(f) near (X, f(X)).In view of Proposition 3.9.1, this implies,

(ζ,−1) ∈ NPepi(f)(X, f(X)).

Conversely, suppose that (ζ,−1) ∈ NPepi(f)(X, f(X)). Then there exists a δ > 0

such that(X, f(X)) ∈ projepi(f)((X, f(X)) + δ(ζ,−1)).

This implies,

‖δ(ζ,−1)‖2 ≤ ‖[(X, f(X)) + δ(ζ,−1)]− (Y, α)‖2,

for all (Y, α) ∈ epi(f).Upon taking α = f(Y ), we get from the last inequality

δ2 + δ2‖ζ‖2 ≤ ‖X − Y + δζ‖2 + (f(X) − f(Y ) − δ)2,

which can be rewritten as

(f(Y ) − f(X) + δ)2 ≥ δ2 + 2δ < ζ, Y −X > −‖X − Y ‖2. (3.9.11)

Clearly, the left hand side of the expression (3.9.11) is positive for all Y , suffi-ciently near X, that is Y ∈ B(X, η) for some η > 0. Since f is lsc, by shrinkingη if necessary, we can make sure that

f(Y ) − f(X) + δ > 0, for all Y ∈ B(X, η).

Consequently, taking square roots of (3.9.11) we get,

f(Y ) ≥ h(Y ) = f(X) − δ + [δ2 + 2δ < ζ, Y −X > −‖Y −X‖2]12 (3.9.12)

for all Y ∈ B(X, η).Considering the function h(Y ), we can calculate and show that H(X) =

hX (X) = ζ, and HX(X) is bounded by say 2σ > 0 in a neighbourhood of X.Hence the function h satisfies the inequality, for some η > 0,

h(Y ) ≥ h(X)+ < ζ, Y −X > −σ‖Y −X‖2, for all Y ∈ B(X, η). (3.9.13)

But, then noting that f(X) = h(X) , the relations (3.9.12) and (3.9.13) yield

f(Y ) ≥ f(X)+ < ζ, Y −X > −σ‖Y −X‖2 (3.9.14)

for all Y ∈ B(X, η) which is (3.9.10) as desired. The proof is complete.

3.10 LYAPUNOV STABILITY CRITERIA 93

3.10 Lyapunov Stability Criteria

In this section, we plan to provide the Lyapunov stability criteria for Euler solu-tions of set differential equation (3.9.3). We begin with the following definitionof the Lyapunov function.

Let V : R+ ×Kc(Rn) → (−∞,∞] be an lsc function with domain dom(V ) =[(t,X) ∈ R+ ×Kc(Rn) : V (t,X) < ∞].

Definition 3.10.1 The pair (V, F ) is said to be weakly decreasing if for any U0

there exists an Euler solution U (t) = U (t, t0, U0) of (3.9.3) satisfying

V (t, U (t)) ≤ V (t0, U0), t ≥ t0 ≥ 0.

One can easily verify that, (V, F ) is weakly decreasing if and only if (epiV, 1×F (t, U ) × 0) is weakly invariant.

In order to deduce the Lyapunov theory of stability from the present set up,we need a sufficient condition, which assures the weakly decreasing nature ofthe system (V, F ).

Theorem 3.10.1 (V, F ) is weakly decreasing if for all (θ, ζ) ∈ ∂PV (t, Z), (t, Z) ∈dom(V ), we have

θ + < F (t, Z), ζ > ≤ 0. (3.10.1)

Proof Suppose that for (t, Z) ∈ dom(V ), (θ, ζ) ∈ ∂PV (t, Z), and (3.10.1)holds. (V, F ) is weakly decreasing if and only if (epiV, 1× F × 0) is weaklyinvariant.

To show that (epiV, 1 × F × 0) is weakly invariant, it suffices to showthat for any (θ, ζ, λ) ∈ NP

epiV (t, Z, r)

< (1, F (t, Z), 0), (θ, ζ, λ) > ≤ 0.

Let (θ, ζ, λ) ∈ NPepiV (t, Z, r). Then, λ ≤ 0.

If λ < 0, then (θ, ζ, λ) ∈ NPepiV (t, Z, r).

This implies ( θ−λ ,

ζ−λ ,−1) ∈ NP

epiV (t, Z, r),which in turn leads to ( θ−λ ,

ζ−λ) ∈

∂PV (t, Z).By hypothesis, we get θ

−λ+ < F (t, Z), ζ−λ > ≤ 0. This implies, θ + <

F (t, Z), ζ > ≤ 0, (t, Z) ∈ dom(V ), which is the required condition.In case, λ = 0, then we have (θ, ζ, 0) ∈ NP

epiV (t, Z, V (t, Z)). We invoke Theo-

rem 3.9.1 to deduce the existence of sequences (θi, ζi,−εi) ∈ NPepiV (ti, Zi, V (ti, Zi))

with εi > 0, and (θi, Zi, V (ti, Zi)) such that (θi, ζi,−εi) → (θ, ζ, 0),

(ti, Zi, V (ti, Zi)) → (t, Z, V (t, Z)).

Then, as in case λ < 0 above, θi + < F (ti, Zi), ζi > ≤ 0. Since F is locallybounded, the sequence F (ti, Zi) is bounded. Passing to a subsequence, we maysuppose that F (ti, Zi) converges to F (t, Z).


This in turn implies θ + < F (t, Z), ζ > ≤ 0 and hence the proof is complete.We shall now consider the Lyapunov theory of stability, employing the lsc

functions V (t,X) so that we can utilize the set of proximal subdifferentials ofV , namely ∂PV (t,X), for providing sufficient conditions for Lyapunov stabilityin the weak sense. We shall derive the weak stability results from Theorem3.10.1. We assume that F (t, θ) ≡ θ so that we can discuss the weak stability ofX = θ. As in section 3.4, we assume that, given U0, V0 ∈ Kc(Rn), the Hukuharadifference U0 − V0 ≡ X0 exists, and we consider solutions

U (t) = U (t, t0, X0) or X(t) = X(t, t0, X0) for stability purposes.We list the following conditions concerning V :

(i) V : R+ × Kc(Rn) → [0,∞] is lsc with dom V = (t,X) ∈ R+ × Kc(Rn) :V (t,X) <∞ and V (t,X) is positive definite.

(ii) the sets [V ]q = (t,X) ∈ R+ × Kc(Rn) : V (t,X) ≤ q are compact forevery q > 0.

(iii) θ + < F (t,X), ζ > −G(t, V (t,X)) ≤ 0, for (t,X) ∈ R+×Kc(Rn), (θ, ζ) ∈∂PV (t,X)) where G ∈ C[R2

+,R], G(t, 0) ≡ 0 and satisfies a nonlineargrowth condition similar to F . (See Theorem 2.8.1).

(iv) θ + < F (t,X), ζ > +W (t,X) ≤ 0, (t,X) ∈ Kc(Rn), where W ∈ C[R+×Kc(Rn),R+], and F is bounded on bounded sets. Here W (t,X) is positivedefinite and satisfies a nonlinear growth condition similar to F .

We can now prove the following result on weak stability.

Theorem 3.10.2 Assume that (i), (ii) and (iii) hold. Then the weak stabilityproperties of the scalar differential equation

w′ −G(t, w) = 0, w(0) = w0 = V (t0, X0), (3.10.2)

imply that the corresponding weak stability properties of X = θ of (3.9.3).

Proof Let us define Q : R+×Kc(Rn)×R → (−∞,∞] byQ(t,X,w) = V (t,X)−w and the function F (t,X,w) = F (t,X) × −G(t, w). We observe that F isalso an usc function satisfying the nonlinear growth condition similar to the onesatisfied by F .

Let us claim that the system (Q, F ) is weakly decreasing. We wish to applyTheorem 3.10.1 and so let (θ, ζ, η) ∈ ∂PQ(t,X,w).

Then, (θ, ζ) ∈ ∂PV (t,X) and η = −1.The condition θ + < F (t,X), ζ > −G(t, V (t,X)) ≤ 0, provides the condition

thatθ+ < (F (t,X),−G(t, V (t,X)), (ζ,−1) > ≤ 0,

which verifies the inequality of the Theorem 3.10.1.Hence, we deduce the existence of a solution (t,X,w) of F with (t0, X0, w0)

satisfying

Q(t,X(t), w(t)) ≤ Q(t0, X0, w0) = V (t0, X0) −w0 = 0, t ≥ t0,


which reduces to the comparison principle

V (t,X(t)) ≤ w(t), t ≥ t0, (3.10.3)

where X(t) = X(t.t0, X0) is a solution of (3.9.3), and w(t) is a solution of(3.10.2).

Once we have the comparison principle given by (3.10.3), if we suppose thatw = 0, of (3.10.2) possesses any weak stability property, it follows employing thestandard arguments, see Lakshmikantham and Leela [1] and Lakshmikantham,Leela and Martynyuk [1] that the corresponding weak stability property of X =θ of (3.9.3) holds. Hence the proof is complete.

In a similar manner, one can prove the following result relative to the con-dition (iv).

Theorem 3.10.3 Assume that (i), (ii) and (iv) hold. then, X = θ of (3.9.3)is weakly asymptotically stable.

Proof In this case we set, Q(t,X, y) = V (t,X) + y and F (t,X, y) = F (t,X)×W (t,X) and, similar to the proof of Theorem 3.10.2, we can deduce fromTheorem 2.8.1, the existence of a solution (X, y) of F at (X0, 0) such that

Q(t,X(t), y(t)) ≤ Q(t0, X0, 0) = V (t0, X0), t ≥ 0.

This in turn reduces to

V (t,X(t)) +∫ t

0

W (s,X(s)) ds ≤ V (t0, X0)), t ≥ 0, (3.10.4)

where X(t) is a solution of (3.9.3).The weak stability of X = θ of (3.9.3) follows immediately from (3.10.4)

using (i) and (ii) in a straightforward way.To prove ‖X(t)‖ → 0 as t → ∞, we observe that (3.10.4) implies that

V (t,X(t)) and∫ t

0W (s,X(s)) ds are bounded on R+, as well as X(t). Since F is

bounded on bounded sets, we also have DHX(t) bounded, and as a consequenceX(t) is globally Lipschitz. With this information, it is now standard to showthat ‖X(t)‖ → 0 as t → ∞, for otherwise the divergence of

∫∞0 W (s,X(s)) ds

results. Hence the proof is complete.


Lyapunov-like functions and needed comparison theorems including global ex-istence described in Sections 3.2 and 3.3 are from Lakshmikantham, Leela andVatsala [2]. The stability criteria provided in Section 3.4, parallel to the orig-inal Lyapunov results for ODE, is from Gnana Bhaskar and Vasundhara Devi[1]. The idea of utilizing the concept of Hukuhara difference in choosing theinitial values suitably to delete any undesirable part of the solutions that maybe present in certain cases, is from Lakshmikantham, Leela and Vasundhara


Devi [1]. The example worked to demonstrate this idea is also from the abovementioned paper. The example uses interval analysis, for which, see Moore [1]and Markov [1].

The contents of Sections 3.5 and 3.6, where the method of perturbing Lya-punov functions is employed to prove nonuniform stability and boundednessresults, are from Gnana Bhaskar and Vasundhara Devi [2].For the idea of per-turbing Lyapunov functions pertaining to ODE, refer to Lakshmikantham andLeela [4]. See also Lakshmikantham, Leela and Martynyuk [1].The introductionof set differential systems and extension of the method of vector Lyapunov func-tions for such systems reported in Sections 3.7 and 3.8, is adapted from GnanaBhaskar and Vasundhara Devi [3]. For more details on the method of vec-tor Lyapunov functions, see Lakshmikantham, Matrosov and Sivasundaram [1].The criteria described in Sections 3.9 and 3.10 is taken from Gnana Bhaskarand Lakshmikantham [3], where the ideas of nonsmooth analysis and weakerlower semicontinuous functions are employed.

Chapter 4

Connection to FuzzyDifferential Equations

4.1 Introduction

When a real world problem is transferred into a deterministic IVP of ordinarydifferential equations, namely,

x′ = f(t, x), x(t0) = x0,

we cannot usually be sure that the model is perfect. If the underlying structureof the model depends upon subjective choices, one way to incorporate these intothe model, is to utilize the aspect of fuzziness, which leads to the considerationof fuzzy differential equations(FDE). There exists sufficient literature on thetheory of FDEs. The intricacies involved in incorporating fuzziness into thetheory of ordinary differential equations pose a certain disadvantage and otherpossibilities are being explored to address this problem. One of the approachesis to transform FDEs into multivalued differential inclusions so as to employthe existing theory of differential inclusions. Another approach is to connectFDEs to SDEs and examine the interconnection between them. In this chapterwe shall be concerned with the latter approach, since the former framework isalready known, see Lakshmikantham and Mohapatra [1].

In Section 4.2, we provide a short account of the necessary preliminary ma-terial on fuzzy set theory, formulate FDEs and list the required known resultsconcerning local and global existence, uniqueness, continuous dependence of so-lutions and comparison results. Section 4.3 is devoted to the investigation ofLyapunov stability theory through Lyapunov-like functions. For this purpose,we develop a comparison result in terms of Lyapunov-like functions and thengive some simple stability results. We sketch an example to expose the difficul-ties involved in general and suggest a way out in those cases where there is aproblem, by suitably choosing the initial value in order to weed out the unde-sirable part of the solution. This may be considered parallel to partial stability

97

98 CHAPTER 4. CONNECTION TO FDES

or conditional stability in ODEs. The interconnection of FDEs with SDEs isexplored in Section 4.4. We indicate also the alternative formulation of fuzzyIVPs into a sequence of multivalued differential inclusions and the advantage ofthe approach.

In Section 4.5, we continue to study the interconnection for the case whenthe function involved in the SDE is only upper semicontinuous and prove someresults parallel to the continuous case considered in Section 4.4. Section 4.6introduces impulses into FDEs and shows how the impulses help to overcomethe disadvantage that exists in the study of FDEs. Some important resultsrelative to impulsive fuzzy differential equations are proved in this section anda familiar example is discussed to point out the advantage gained by addingimpulses. Hybrid fuzzy differential equations are considered in Section 4.7 andthe required structure is developed for the investigation of the stability of suchsystems. Section 4.8 introduces another concept of differential equations in ametric space which can be applied to study FDEs. Notes and comments aregiven in Section 4.9.

4.2 Preliminaries

Let Kc(Rn) denote the family of all nonempty, compact, convex subsets of R.If α, β ∈ R and A,B ∈ Kc(Rn), then

α(A+B) = αA+ αB, α(βA) = (αβ)A, 1A = A

and if α, β ≥ 0, then (α+ β)A = αA+ βA.Let I = [t0, t0 + a], t0 ≥ 0 and a > 0 and En be the set of all functions

u : Rn → [0, 1] such that u satisfies (i)-(iv) mentioned below :

(i) u is normal, that is , there exists an x0 ∈ Rn such that u(x0) = 1;

(ii) u is fuzzy convex , that is , for x, y ∈ Rn and 0 ≤ λ ≤ 1,

u(λx+ (1 − λ)y) ≥ minu(x), u(y);

(iii) u is upper semicontinuous;

(iv) [u]0 = closure of x ∈ Rn : u(x) > 0 is compact.

For 0 < α ≤ 1, we denote [u]α = x ∈ Rn : u(x) ≥ α. Then from (i)-(iv), itfollows that the α− level sets [u]α ∈ Kc(Rn), for 0 ≤ α ≤ 1.

Let D[A,B] be the Hausdorff distance between the sets A,B ∈ Kc(Rn). Wedefine,

D0[u, v] = sup0≤α≤1D[[u]α, [v]α], (4.2.1)

which is a metric in En, and (En, D0) is a complete metric space.We list the following properties of D0[u, v], which are similar to D[A,B]

where A,B ∈ Kc(Rn) :

D0[u+ w, v +w] = D0[u, v], (4.2.2)

4.2 PRELIMINARIES 99

D0[λu, λv] = |λ|D0[u, v], (4.2.3)

D0[u, v] ≤ D0[u,w] +D0[w, v], (4.2.4)

for all u, v, w ∈ En and λ ∈ R.For x, y ∈ En if there exists a z ∈ En such that x = y+z, then z is called the

Hukuhara difference of x and y and is denoted by x−y. A mapping F : I → En

is differentiable at t ∈ I if there exists a DHF (t) ∈ En such that the limits

limh→0+

F (t+ h) − F (t)h

, and limh→0+

F (t) − F (t− h)h

(4.2.5)

exist and are equal to DHF (t). Here the limits are taken in the metric space(En, D0).

Moreover, if F : I → En is continuous , then it is integrable and∫ t2

t0

F =∫ t1

t0

F +∫ t2

t1

F, t0 ≤ t1 ≤ t2 ≤ t0 + a.

Further, If F,G : I → En are integrable, λ ∈ R, then the following propertiesof the integral hold: ∫

(F + G) =∫F +

∫G;

∫λF = λ

∫F, λ ∈ R;

D0[F,G] is integrable;

D0

[∫F,

∫G

]≤∫D0[F,G].

Finally, let F : I → En be continuous. Then the integralG(t) =

∫ t

t0F (s) ds is differentiable and DHG(t) = F (t). Furthermore,

F (t) − F (t0) =∫ t

t0

DHF (s) ds.

See for details Lakshmikantham and Mohapatra[1].Consider the fuzzy differential system

DHu = f(t, u), u(t0) = u0, (4.2.6)

where f ∈ C[I ×En, En] and I = [t0, t0 + a], t0 ≥ 0, a > 0.Before proceeding further, we note that a mapping u : I → En is a solution

of the initial value problem (4.2.6) if and only if it is continuous and satisfiesthe integral equation

u(t) = u0 +∫ t

t0

f(s, u(s)), ds for t ∈ I.


Using the properties of D0[u, v] and of the integral listed above, and the theoryof differential and integral inequalities, one can establish the following resultsconcerning comparison principles, existence and uniqueness, continuous depen-dence and global existence of solutions of (4.2.6). We merely state such resultswhose proofs are almost similar to the corresponding results for set differentialequations discussed in Chapter 2. One can also see the proofs in Lakshmikan-tham and Mohapatra [1].

Theorem 4.2.1 Assume that f ∈ C[I × En, En] and for t ∈ I, u, v ∈ En,

D0[f(t, u), f(t, v)] ≤ g(t,D0[u, v]),

where g ∈ C[I × R+,R+] and g(t, w) is nondecreasing in w for each t. Supposefurther that the maximal solution r(t) = r(t, t0, w0) of the scalar differentialequation

w′ = g(t, w), w(t0) = w0 ≥ 0, (4.2.7)

exists on I. Then, if u(t), v(t) are any two solutions of (4.2.6) through (t0, u0),(t0, v0) respectively on I, and if D0[u0, v0] ≤ w0, we have

D0[u(t), v(t)] ≤ r(t, t0, w0), t ∈ I. (4.2.8)

Remark 4.2.1 If we employ the theory of differential inequalities instead ofintegral inequalities, we can dispense with the monotone character of g(t, w)assumed in Theorem 4.2.1. This is the next comparison principle.

Theorem 4.2.2 Let the assumptions of Theorem 4.2.1 hold except for the non-decreasing property of g(t, w) in w. Then the conclusion (4.2.8) is valid.

The next comparison result provides an estimate under weaker assumptions.

Theorem 4.2.3 Assume that f ∈ C[I × En, En] and

lim suph→0+

D0[u+ hf(t, u), v + hf(t, v)] −D0[u, v]h

≤ g(t,D0[u, v]), t ∈ I,

where g ∈ C[I ×R+,R], u, v ∈ En. The maximal solution r(t, t0, w0) of (4.2.7)exists on I. Then, the conclusion of the Theorem 4.2.1 is valid.

We wish to remark that in Theorem 4.2.3, g(t, w) need not be non-negative, andtherefore the estimate (4.2.8) would be finer than the estimates in Theorems4.2.1 and 4.2.2.

As a special case of Theorems 4.2.1, 4.2.2, 4.2.3 we have the following im-portant corollary.

Corollary 4.2.1 Assume that f ∈ C[I ×En, En] and either

(a) D0[f(t, u), 0 ] ≤ g(t,D0[u, 0 ]) or

(b) lim suph→0+

D0[u+ hf(t, u), 0 ] −D0[u, 0 ]h

≤ g(t,D0[u, 0 ]), t ∈ I,


where g ∈ C[I × R+,R]. Then, if D0[u0, 0 ] ≤ w0, we have

D0[u(t), 0 ] ≤ r(t, t0, w0), t ∈ I,

where r(t, t0, w0) is the maximal solution of (4.2.7) on I and 0 ∈ En is definedas

0 (x) =

1 if x = 0,0 if x 6= 0.


(a) f ∈ C[R0, En], D0[f(t, x), 0 ] ≤ M0, on R0; where R0 = I×B(u0, b), B(u0, b) =

x ∈ En : D0[u, u0] ≤ b and

(b) g ∈ C[I × [0, 2b],R+], 0 ≤ g(t, w) ≤ M1 on I × [0, 2b], g(t, 0) = 0, g(t, w)is nondecreasing in w for each t ∈ I and w(t) ≡ 0 is the unique solutionof (4.2.7) on I;

(c) D0[f(t, u), f(t, v)] ≤ g(t,D0[u, v]) on R0.


un+1(t) = u0 +∫ t

t0

f(s, un(s)) ds, n = 0, 1, 2, · · · ,

exist on [t0, t0 + η], where η = mina, bM , M = maxM0,M1, as continuous

functions and converge uniformly to the unique solution u(t) of the IVP (4.2.6)on [t0, t0 + η].

Theorem 4.2.5 Suppose that the assumptions of Theorem 4.2.4. hold. Also,further that the solutions w(t, t0, w0) of (4.2.7) through every point (t0, w0) arecontinuous with respect to (t0, w0). Then the solutions u(t, t0, u0) of (4.2.6) arecontinuous relative to (t0, u0).

Theorem 4.2.6 Assume that f ∈ C[R+ × En, En] and

D0[f(t, u), 0 ] ≤ g(t,D0[u, 0 ] ), (t, u) ∈ R+ ×En,

where g ∈ C[R2+,R+], g(t, w) is nondecreasing in w for each t ∈ R+ and

the maximal solution r(t, t0, w0) of (4.2.7) exists on [t0,∞). Suppose furtherthat f is smooth enough to guarantee local existence of solutions of (4.2.6) forany (t0, u0) ∈ R+ × En. Then the largest interval of existence of any solutionu(t, t0, u0) of (4.2.6) such that D0[u0, 0 ] ≤ w0 is [t0,∞).

4.3 Lyapunov-like functions

Consider the fuzzy differential equation

DHu = f(t, u), u(t0) = u0, (4.3.1)


where f ∈ C[R+ × S(ρ), En] and S(ρ) = u ∈ En : D0[u, 0 < ρ].We assume that f(t, 0) = 0, so that we have the trivial solution for (4.3.1).To investigate the stability criteria, the following comparison result in terms

of the Lyapunov-like function is very important and can be proved via the the-ory of ordinary differential inequalities. Here the Lyapunov-like function servesas a vehicle to transform the fuzzy differential equation into scalar compari-son differential equation, and therefore it is enough to consider the qualitativeproperties of the simpler comparison equation.


(i) V ∈ C[R+ × S(ρ),R+ ] and | V (t, u1) − V (t, u2) | ≤ L D0[u1, u2], whereL > 0,

(ii) D+V (t, u) ≡ lim suph→0+

1h

[V (t+h, u+hf(t, u))−V (t, u)] ≤ g(t, V (t, u)), where

g ∈ C[R2+,R].

Then, if u(t) is any solution of (4.3.1) existing on (t0,∞) such that V (t0, u0) ≤w0, we have

V (t, u(t)) ≤ r(t, t0, w0), t ≥ t0,

where r(t, t0, w0) is the maximal solution of the scalar differential equation

w′ = g(t, w), w(t0) = w0 ≥ 0, (4.3.2)

existing on [t0,∞).

Proof Let u(t) be any solution of (4.3.1) existing on [t0,∞). Define m(t) =V (t, u(t)) so that m(t0) = V (t0, u0) ≤ w0. For small h > 0,

m(t + h) −m(t) = V (t + h, u(t+ h)) − V (t, u(t))= V (t + h, u(t+ h)) − V (t+ h, u(t) + hf(t, u(t)))

+V (t+ h, u(t) + hf(t, u(t))) − V (t, u(t))≤ LD0[u(t+ h), u(t) + hf(t, u(t))]

+V (t+ h, u(t) + hf(t, u(t))) − V (t, u(t)),

using the Lipschitz condition given in (i). Thus


1h

[m(t+ h) −m(t)]

≤ D+V (t, u(t)) + L lim suph→0+

1h

[D0[u(t+ h), u(t) + hf(t, u(t))]].

Let u(t+ h) = u(t) + z(t, h), where z(t, h) is the Hukuhara difference for smallh > 0, which is assumed to exist. Hence employing the properties of D0[u, v],we see that

D0[u(t+ h), u(t) + hf(t, u(t))] = D0[u(t) + z(t, h), u(t) + hf(t, u(t))]= D0[z(t, h), hf(t, u(t))]= D0[u(t+ h) − u(t), hf(t, u(t))].


Consequently,

1hD0[u(t+ h), u(t) + hf(t, u(t))] = D0[

u(t+ h) − u(t)h

, f(t, u(t))],

and therefore,

lim suph→0+

1h

[D0[u(t+ h), u(t) + hf(t, u(t))]]

= lim suph→0+

1hD0

[u(t+ h) − u(t)

h, f(t, u(t))

],

= D0[DHu(t), f(t, u(t))] ≡ 0,

since u(t) is a solution of (4.3.1). We therefore have the scalar differentialinequality

D+m(t) ≤ g(t,m(t)), m(t0) ≤ w0, t ≥ t0,

which yields by the theory of differential inequalities (see Lakshmikantham andLeela [1])

m(t) ≤ r(t, t0, w0), t ≥ t0.

This proves the claimed estimate of the theorem.The following corollaries are useful.

Corollary 4.3.1 The function g(t, w) ≡ 0 is admissible in Theorem 4.3.1 toyield the estimate

V (t, u(t)) ≤ V (t0, u0), t ≥ t0.

Corollary 4.3.2 If, in Theorem 4.3.1, we strengthen the assumption on D+V (t, u)to

D+V (t, u) ≤ −c[w(t, u)] + g(t, V (t, u)),

where w ∈ C[R+ × S(ρ),R+ ], c ∈ K = a ∈ C[[0, ρ),R+] : a(w) is increasing inw and a(0) = 0, and g(t, w) is nondecreasing in w for each t ∈ R+, then weget the estimate

V (t, u(t)) +∫ t

t0

c[w(s, u(s))] ds ≤ r(t, t0, w0), t ≥ t0,

whenever V (t0, u0) ≤ w0.

Proof Set L(t, u(t)) = V (t, u(t)) +∫ t

t0c[w(s, u(s))] ds, and note that

D+L(t, u(t)) ≤ D+V (t, u(t)) + c[w(t, u(t))]≤ g(t, V (t, u(t))) ≤ g(t, L(t, u(t))),

using the monotone character of g(t, w). We then get immediately by Theorem4.3.1, the estimate

L(t, u(t)) ≤ r(t, t0, w0), t ≥ t0,


where u(t) is any solution of (4.3.1). This implies the stated estimate.A simple example of V (t, u) is D0[u, 0] so that

D+V (t, u) = lim suph→0+

1h

[D0[u+ hf(t, u), 0 ] −D0[u, 0 ]].

Having the necessary comparison results in terms of Lyapunov-like functions, itis easy to establish stability results analogous to original Lyapunov results forfuzzy differential equations.

We shall assume that (4.3.1) possesses the trivial solution, and the solutionsexist and are unique for t ≥ 0.

Recall that the approach in the formulation of fuzzy differential equation(FDE) (4.3.1) is based on the fuzzification of the differential operator, whosevalues are in En and therefore suffers from the disadvantage ( in view of Corol-lary 2.5.1 in Lakshmikantham and Mohapatra[1]) that the solution u(t) of (4.3.1)has the property that diam[u(t)]α is nondecreasing as t increases. Consequently,it is concluded that this formulation of FDE is not suitable for reflecting the richbehaviour of the solutions of the corresponding ordinary differential equation(ODE). As a result, following the suggestion of Hullermeier[1], an alternativeformulation leading to multivalued differential inclusions has been investigated,which we shall discuss in the next section. Here we shall utilize the Hukuharadifference in the initial values in such a way that a subset of the solutions of(4.3.1) matches the behaviour of the solutions of the corresponding ODE.

For this purpose, we suppose that for any u0, v0 ∈ En, there exists a z0 ∈ En

such that Hukuhara difference u0 − v0 = z0 exists. Then, we consider thestability of the solutions u(t, t0, u0 − v0) = u(t, t0, z0) of (4.3.1) with respect tothe trivial solution of (4.3.1). This approach helps to delete an undesirable partof the solution generated.

On the other hand, if the FDE is given by

DHu = λ(t)u, λ ∈ C[R+,R+], λ ∈ L1(R+),

the foregoing situation does not exist, and therefore it matches in its behaviourwith corresponding ODE. Hence there is no need to use the Hukuhara differencein the initial values to obtain the desirable part of the solution.

Let us consider the following standard example. Let a ∈ E1 have level sets[a]α = [aα

1 , aα2 ] for α ∈ I = [0, 1] and suppose that a solution u : [0, T ] → E1 of

the fuzzy differential equation

DHu = au, u(0) = u0 ∈ E1, (4.3.3)

on E1 has level sets [u(t)]α = [uα1 (t), uα

2 (t)], for α ∈ I and t ∈ [0, T ].The Hukuhara derivative DHu also has level sets

[du

dt(t)]α

=[duα

1

dt(t),

duα2

dt(t)],

for α ∈ I and t ∈ [0, T ].


By the extension principle, the fuzzy set f(t, u(t)) = au(t) has level sets,

[au(t)]α = [minaα1u

α1 (t), aα

2uα1 (t), aα

1uα2 (t), aα

2uα2 (t),

maxaα1u

α1 (t), aα

2uα1 (t), aα

1uα2 (t), aα

2uα2 (t)],

for all α ∈ I and t ∈ [0, T ].Thus the fuzzy differential equation (4.3.3) is equivalent to the coupled sys-

tem of ordinary differential equations

duα1

dt= minaα

1uα1 , a

α2u

α1 , a

α1u

α2 , a

α2u

α2,

duα2

dt= maxaα

1uα1 , a

α2u

α1 , a

α1u

α2 , a

α2u

α2 ,

(4.3.4)

for α ∈ I.For a = χ−1 ∈ E1, the fuzzy differential equation (4.3.3) becomes

DHu = −u, u(0) = u0, (4.3.5)

and the system of ordinary differential equations (4.3.4) reduces to

duα1

dt= −uα

2 ,duα

2

dt= −uα

1 ,

for α ∈ I.If the level sets of the initial value u0 ∈ E1 are [u0]α = [uα

01, uα02] for α ∈ I,

then the level sets of the solution u of (4.3.5)are given by, [u(t)]α = [uα1 (t), uα

2 (t)]where

uα1 (t) =

12(uα

01 − uα02)e

t +12(uα

01 + uα02)e

−t, (4.3.6)

uα2 (t) =

12(uα

02 − uα01)e

t +12(uα

01 + uα02)e

−t, (4.3.7)

for 0 ≤ α ≤ 1 and t ≥ 0.Let us suppose that for v0, z0 ∈ E1, such that the Hukuhara difference

u0 − v0 = z0 exists, we have

[u0]α = [v0 + z0]α = [v0]α + [z0]α.

Let us suppose that for v0, z0 ∈ E1, such that the Hukuhara difference u0−v0 =z0 exists, we have

[u0]α = [v0 + z0]α = [v0]α + [z0]α.

Since [u0]α = [uα01, u

α02], let us choose [v0]α =

[uα

01 − uα02

2,uα

02 − uα01

2

], so that

[z0]α =[uα

01 + uα02

2,uα

02 + uα01

2

].

Then it follows, assuming uα01 6= −uα

02, that

[u(t, u0)]α =[−uα

02 − uα01

2,uα

02 − uα01

2

]et +

[uα

01 + uα02

2,uα

02 + uα01

2

]e−t,


[u(t, v0)]α =[12(uα

01 − uα02),

12(uα

02 − uα01)]et,

and

[u(t, z0)]α =[12(uα

01 + uα02),

12(uα

02 + uα01)]e−t, t ≥ 0.

If on the other hand, uα01 = −uα

02, that is [u0]α = [−dα, dα] with dα = uα02. Then

the choice of [v0]α = [cα − dα, cα + dα] for some cα so that [z0]α = [cα, cα] andwe find [v0]α = [u0]α + [z0]α, by changing the roles of u0, v0.

We note that, in general, for any initial value [u0]α for which uα01 6= uα

02,the solution of (4.3.5) contains both desired and undesired parts of solutioncompared with the solution of the corresponding ODE. In order to isolate thedesired part of the solution u(t, u0) of (4.3.5) that matches the solution of ODE,we need to utilize the initial values satisfying the desired Hukuhara difference.

We are now ready to prove the following stability results by means of Lyapunov-like functions utilizing the solutions u(t, t0, z0) = u(t, t0, u0 − v0) of (4.3.1). Forthis purpose, we list a typical definition of stability so that others can be for-mulated on the basis of this and standard definitions in stability theory.

Definition 4.3.1 The trivial solution of (4.3.1) is said to be equi-stable if foreach ε > 0 and t0 ≥ 0, there exists a δ = δ(t0, ε) > 0 such that D0[z0, 0 ] < δimplies D0[u(t, t0, z0), 0 ] < ε, t ≥ t0.

Theorem 4.3.2 Assume that the following hold:

(i) V ∈ C[R+ × S(ρ),R+ ], |V (t, u1) − V (t, u2)| ≤ LD0[u1, u2], L > 0 and for(t, u) ∈ R+ × S(ρ), where S(ρ) = u ∈ En : D0[u, 0 < ρ,

D+V (t, u) ≡ lim suph→0+

1h

[V (t+ h, u+ hf(t, u)) − V (t, u)] ≤ 0; (4.3.8)

(ii) b(D0[u, 0 ]) ≤ V (t, u) ≤ a(t,D0[u, 0 ]) , for (t, u) ∈ R+ × S(ρ) whereb, a(t, .) ∈ K= σ ∈ C[[0, ρ),R+] : σ(0) = 0 and σ(ω) is increasing in ω.


Proof Let 0 < ε < ρ and t0 ∈ R+ be given. Choose a δ = δ(t0, ε) such that

a(t0, δ) < b(ε). (4.3.9)

We claim that with this δ, equi-stability holds. If not, there would exist asolution u(t) = u(t, t0, z0) of (4.3.1) with D0[z0, 0] < δ and t1 > t0 such that

D0[u(t1), 0 ] = ε and D0[u(t), 0 ] ≤ ε < ρ, t0 ≤ t ≤ t1. (4.3.10)

By Corollary 4.3.1, we then have

V (t, u(t)) ≤ V (t0, z0), t0 ≤ t ≤ t1.


Consequently, using (ii), (4.3.9) and (4.3.10), we arrive at the following contra-diction:

b(ε) = b(D0[u(t1), 0 ]) ≤ V (t1, u(t1)) ≤ V (t0, z0) ≤ a(t0, D0[z0, 0 ]) < b(ε).

Hence equi-stability holds, completing the proof.The next result provides sufficient conditions for equi-asymptotic stability.

In fact, it gives exponential asymptotic stability.

Theorem 4.3.3 Let the assumptions of Theorem 4.3.2 hold except that theestimate (4.3.8) be strengthened to

D+V (t, u) ≤ −βV (t, u), (t, u) ∈ R+ × S(ρ). (4.3.11)

Then the trivial solution of (4.3.1) is equi-asymptotically stable.

Proof Clearly, the trivial solution of (4.3.1) is equi-stable. Hence taking ε = ρand designating δ0 = δ(t0, ρ) , we have by Theorem 4.3.2.

D0[z0, 0 ] < δ0 implies D0[u(t), 0 ] < ρ, t ≥ t0.

Consequently, we get from the assumption (4.3.11), the estimate

V (t, u(t)) ≤ V (t0, z0) exp[−β(t − t0)], t ≥ t0.

Given ε > 0, we choose T = T (t0, ε) =1βlna(t0, δ0)b(ε)

+ 1. Then, it is easy to see

that,

b(D0[u(t), 0 ]) ≤ V (t, u(t)) ≤ a(t0, δ) e−β(t−t0) < b(ε), t ≥ t0 + T.

The proof is complete.We shall next consider the uniform stability criteria.

Theorem 4.3.4 Assume that, for (t, u) ∈ R+×S(ρ)∩Sc(η) for each 0 < η < ρ,V ∈ C[R+ × S(ρ) ∩ Sc(η),R+], |V (t, u1) − V (t, u2)| ≤ LD0[u1, u2], L > 0,

D+V (t, u) ≤ 0, (4.3.12)

andb(D0[u, 0 ]) ≤ V (t, u) ≤ a(D0[u, 0 ]), a, b ∈ K. (4.3.13)

Then the trivial solution of (4.3.1) is uniformly stable.

Proof Let 0 < ε < ρ and t0 ∈ R+ be given. Choose δ = δ(ε) > 0 such thata(δ) < b(ε). Then we claim that with this δ, uniform stability follows. If not,there would exist a solution u(t) = u(t, t0, z0) of (4.3.1), and a t2 > t1 > t0satisfying

D0[u(t1), 0 ] = δ, D0[u(t2), 0 ] = ε and δ ≤ D0[u(t), 0 ] ≤ ε < ρ. (4.3.14)


Taking η = δ, we get from (4.3.12), the estimate

V (t2, u(t2)) ≤ V (t1, u(t1)).

This, together with (4.3.13), (4.3.14), and the definition of δ, yield

b(ε) = b(D0[u(t2), 0 ])≤ V (t2, u(t2)) ≤ V (t1, u(t1))≤ a(D0[u(t1), 0 ]) = a(δ) < b(ε).

This contradiction proves uniform stability, completing the proof.Finally, we shall prove uniform asymptotic stability.

Theorem 4.3.5 Let the assumptions of Theorem 4.3.4 hold except that (4.3.12)is strengthened to

D+V (t, u) ≤ −c(D0[u, 0 ]), c ∈ K. (4.3.15)

Then the trivial solution of (4.3.1) is uniformly asymptotically stable.

Proof By Theorem 4.3.4, uniform stability follows. And, for ε = ρ, we desig-nate δ0 = δ0(ρ). This means,

D0[z0, 0 ] < δ0 implies D0[u(t), 0 ] < ρ, t ≥ t0,

where u(t) = u(t, t0, z0) is the solution of (4.3.1).In view of the uniform stability, it is enough to show that there exists a t∗

such that for t0 ≤ t∗ ≤ t0 + T , where T = 1 +a(δ0)c(δ)

,

D0[u(t∗), 0 ] < δ. (4.3.16)

If this is not true, δ ≤ D0[u(t), 0 ], for t0 ≤ t ≤ t0 + T . Then, (4.3.15) gives,

V (t, u(t)) ≤ V (t0, z0) −∫ t

t0

c(D0[u(s), 0 ]) ds, t0 ≤ t ≤ t0 + T.

As a result, we have, in view of the choice of T ,

0 ≤ V (t0 + T, u(t0 + T )) ≤ a(δ0) − c(δ)T < 0,

a contradiction. Hence there exists a t∗ satisfying (4.3.16) and from uniformstability we conclude that

D0[z0, 0 ] < δ0 implies D0[u(t), 0 ] < ε, t ≥ t0 + T,


4.4 CONNECTION WITH SDES 109

4.4 Connection with Set Differential Equations

Recall that the IVP for fuzzy differential equations proposed in Section 4.2, isof the type

DHu = f(t, u), u(t0) = u0 ∈ En, (4.4.1)

where f ∈ C[R+ ×En, En], for which basic results are listed there. As noted inSection 4.3., this approach is based on the fuzzification of the differential opera-tor, whose values are in En and therefore suffers from the disadvantage that thesolution u(t) of (4.4.1) has the property that diam[u(t)]α is nondecreasing as tincreases. Consequently, this formulation cannot fully reflect the rich behaviourof solutions of corresponding ordinary differential equations.

Recently, Hullermeier [1] has suggested an alternative formulation of fuzzyIVPs by replacing the RHS of a system of ordinary differential equations by afuzzy function

f : R+ × Rn → En,

with the initial fuzzy set x0 ∈ En, so that one can consider the fuzzy multivalueddifferential inclusion

x′ ∈ f(t, x), x(t0) = x0 ∈ En, (4.4.2)

on J = [t0, T ], where now f is defined from R+ × Rn → En, rather thanR+×En → En, as in (4.4.1). However, instead of (4.4.2), a family of multivalueddifferential inclusions,

x′β ∈ F (t, xβ; β), xβ(t0) ∈ [x0]β, 0 ≤ β ≤ 1, (4.4.3)

is investigated on J where F (t, x, β) ≡ [f(t, x)]β and F (t, x, 0) = supp(f(t, x)).The idea is that the set of all solutions Sβ(x0, T ), t0 ≤ t ≤ T , would be β− levelof a fuzzy set S(x0, T ), in the sense that all attainable sets Aβ(x0, t), t0 < t ≤ T,are levels of a fuzzy set on Rn. Considering S(x0, T ) to be the solution of (4.4.1)thus captures both uncertainty and the rich behaviour of differential inclusionin one and the same technique.

For this purpose, the standard results of multivalued differential inclusions,under the usual conditions on F in (4.4.3) yield that the attainable set Aβ(x0, t)is compact subset of Rn. If F is assumed to be quasiconcave in addition, one canconclude, under reasonable assumptions, utilizing the representation theorem,the existence of a fuzzy set u(t) such that [u(t)]β = Aβ(x0, t) with a similarrelation for the solution set Sβ(x0, T ). See for details Lakshmikantham andMohapatra [1].

In the literature, the following example is often quoted to demonstrate theadvantage gained by the alternative approach when compared to the originalone.

Consider the crisp initial value problem of ODE with unknown initial valuex0, that is ,

x′ = −x, x(0) = x0 ∈ [−1, 1]. (4.4.4)


The solution of (4.4.4) when restricted to the interval [−1, 1] is

x(t) = [−e−t, e−t], t ≥ 0.

The fuzzy differential equation corresponding to (4.4.4) in E1 is

DHx = −x, x(0) = x0 = [−1, 1], x0 ∈ E1. (4.4.5)

Suppose that [x]β = [xβ1 , x

β2 ], [DHx]β =

[dxβ

1

dt,dxβ

2

dt

]are the β−level sets for

0 ≤ β ≤ 1. By extension principle, (4.4.5) becomes

dxβ1

dt= −xβ

2 ,dxβ

2

dt= −xβ

1 , 0 ≤ β ≤ 1. (4.4.6)

The solution of (4.4.6) is given by xβ1 (t) = −et, xβ

2 (t) = et and therefore thefuzzy function x(t) solving (4.4.5) is [x(t)]β = [−et, et], t ≥ 0, which shows thatthe diam[x(t)]β → ∞ as t→ ∞.

In the framework of Hullermeier, on the other hand, fuzzy differential equa-tion (4.4.5) is replaced by the family of inclusions

x′β ∈ F (t, xβ; β) = [−xβ2 ,−x

β1 ], xβ(0) = [−1, 1]; (4.4.7)

which has a fuzzy solution set S([−1, 1], T ), and fuzzy attainable set Aβ([−1, 1], t),0 ≤ t ≤ T respectively which are defined by β− level sets

Sβ([−1, 1], t) = x(.) : x(t) ∈ [−e−t, e−t], 0 ≤ t ≤ T, (4.4.8)

Aβ [[−1, 1], t) = [−e−t, e−t], (4.4.9)

which matches the kind of behaviour a fuzzification of the ordinary differentialequation (4.4.4) should have.

The new approach shows that a fuzzification of ODE has no effect on the be-haviour of solutions. Then, a question arises, why should we bother to introducefuzziness in the originally modelled ODE without fuzziness? It is natural, infact, when one incorporates in the ODE other phenomena such as randomness,delay, uncertainties such as fuzziness or even transform ODE into a differenceequation or generate a set differential equation (SDE), to name a few, the cor-responding dynamic system should exhibit the effect of such phenomena. It isnot natural to expect always to have the same behaviour as that of the solu-tions of ODE from which the new dynamic systems are generated. Generallyspeaking, the theory of the corresponding dynamic systems is a lot richer thanthe theory of ODEs and therefore it would be interesting to investigate it as anindependent discipline.

Seen from this point of view, the original formulation of FDEs, does meet thecriteria. Let us give some examples, including the often quoted one describedearlier, to show other possibilities. Let us start with the ODEs, that is, crispIVPs:


(1a) u′ = − u, or (1b) u′ + u = 0, u(0) = u0,

(2a) u′ = u, or (2b) u′ − u = 0, u(0) = u0.

Clearly the solutions of (1) and (2) are u(t) = u0e−t and u(t) = u0e

t, t ≥ 0,respectively.

The corresponding IVPs for FDEs are, respectively ,

(i) DHu = (−1)u, u(0) = u0 ∈ E1,(ii) DHu+ u = 0, u(0) = u0 ∈ E1,(iii) DHu = u, u(0) = u0 ∈ E1,(iv) DHu + (−1)u = 0, u(0) = u0 ∈ E1.

Since (ii) and (iv) are not essentially FDE’s, we introduce as a forcing terma σ ∈ E1 and consider the following FDE’s:

(ii∗) DHu+ u = σ(t), u(0) = u0 ∈ E1,(iv∗) DHu + (−1)u = σ(t), u(0) = u0 ∈ E1.

Suppose that the solutions of FDEs have level sets

[u(t)]α = [uα1 (t), uα

2 (t)], [u0]α = [α− 1, 1 − α],

and [DHu]α = [duα

1

dt,duα

2

dt] and σα(t) = [(α− 1), (1− α)]e−t, for 0 ≤ α ≤ 1.

The FDE’s (i), (ii∗), (iii) and (iv∗) reduce to the following systems of ODEs

(I) [duα

1

dt,duα

2

dt] = [−uα

2 ,−uα1 ],

(II∗) [duα

1

dt,duα

2

dt] + [uα

1 , uα2 ] = [(α− 1), (1− α)]e−t,

(III) [duα

1

dt,duα

2

dt] = [uα

1 , uα2 ],

(IV ∗) [duα

1

dt,duα

2

dt] + [−uα

2 ,−uα1 ] = [(α− 1), (1 − α)]e−t.

with the same initial condition [u0]α = [α − 1, 1 − α]. The solutions of theseequations, using the methods of interval analysis, are

(Ia) [u(t)]α = [α− 1, 1− α]et,(II∗a) [u(t)]α = [α− 1, 1− α](e−t(1 + t)),(IIIa) [u(t)]α = [α− 1, 1− α]et,(IV ∗a) [u(t)]α = [α− 1, 1− α](e−t(1 + t)).

The solutions (Ia) and (II∗a) represent typical behaviours corresponding toODE from which they are generated. They show that introducing fuzziness into


ODEs, some times destroys the good behaviour of solutions and helps at someother times. For a discussion that solution behavior depends on the choice ofthe forcing term, see Kaleva [3].

On the other hand, if we consider the family of multivalued differential in-clusions (MDI), for (IV ∗), we obtain,

0 ∈ u′α + [−uα2 ,−uα

1 ] + [(α− 1), (1− α)]e−t, u(0) = [α− 1, 1 − α],

which has as its attainable set

Aα[[α− 1, 1− α], t] = [α− 1, 1− α] (32et − e−t), t ≥ 0.

The foregoing analysis of examples indicates a variety of behaviors of solutionsof FDEs compared to that of ODEs, from which they are generated if the initialvalues are chosen appropriately. Thus, it appears that study of FDEs as origi-nally formulated is much richer than expected and needs further investigation.

We next state a known result that relates the solution of set differentialequation to the attainable set of multivalued differential inclusion is the following(see Tolstonogov [1]).

Theorem 4.4.1 Assume that F ∈ C[R+ × Rn,Kc(Rn)];

D[F (t, x), F (t, y)] ≤ g(t, ‖x− y‖), (t, x) (t, y) ∈ R+ × Rn,

and D[F (t, x), θ] ≤ q(t, ‖x‖), (t, x) ∈ R+ × Rn, where g and q satisfy thefollowing assumptions. The functions g, q ∈ C[R2

+,R+], g(t, w), q(t, w) arenondecreasing in w for each t ∈ R+, w(t) ≡ 0 is the only solution of the scalardifferential equation

w′ = g(t, w), w(t0) = 0,

on [t0,∞) and r(t, t0, w0) is the maximal solution of the scalar differential equa-tion

w′ = q(t, w), w(t0) = w0 > 0,

existing on [t0,∞). Then, there exists a unique solution U (t) = U (t, t0, U0) on[t0,∞) of IVP (3.2.1) and the attainable set A(U0, t) of differential inclusion

x′ ∈ F (t, x), x(t0) ∈ U0,

satisfying A(U0, t) ⊂ U (t), t0 ≤ t < ∞.

Finally, we need the following representation result (see Lakshmikanthamand Mohapatra [1]).

Theorem 4.4.2 Let Yβ ⊂ Rn, 0 ≤ β ≤ 1 be a family of compact subsetssatisfying

(i) Yβ ∈ K(Rn) for all 0 ≤ β ≤ 1;

(ii) Yβ ⊆ Yα whenever α ≤ β, α, β ∈ [0, 1];


(iii) Yβ =∏∞

i=1 Yβi , for any nondecreasing sequence βi → β in [0, 1].

Then, there is a fuzzy set u ∈ Dn, such that [u]β = Yβ. If Yβ are also convex,then u ∈ En. (Here Dn denotes the set of usc normal fuzzy sets with compactsupport and thus En ⊂ Dn). Conversely, the level sets [u]β, of any u ∈ En, areconvex and satisfy these conditions.

It should be noted that Theorem 4.4.2 can be easily generalized from Rn toa Banach space.

We propose here another formulation of fuzzy differential equation (4.4.1)by a set differential equation which is generated by β− level set of the R.H.S.of (4.4.1) where f : R+ × Rn → En, as before.

For this purpose, consider the level set for each β, 0 ≤ β ≤ 1, and write

F (t, x; β) = [f(t, x)]β ∈ Kc(Rn).

Next generate the mapping H : R+ × Kc(Rn) × I → Kc(Rn), I = [0, 1] bydefining

H(t, A; β) = coF (t, A; β) (4.4.10)

for each A ∈ Kc(Rn). Then consider the family of set differential equations givenby

DHUβ = H(t, Uβ; β) Uβ(t0) = U0β ∈ Kc(Rn), (4.4.11)

on [t0, T ], T > t0, where DHUβ is the Hukuhara derivative for each β.Let us list the following conditions.

(i) F (t, x; β) is quasi-concave, that is,

1. for (t, x) ∈ R+ × Rn, α, β ∈ I, F (t, x;α) ⊇ F (t, x; β), wheneverα ≤ β;

2. if βn is nondecreasing sequence in I, converging to β, then for (t, x) ∈R+ × Rn, ∩∞

n=1F (t, x; βn) = F (t, x; β);

(ii) D[H(t, A; β), H(t, B; β)] ≤ g(t,D[A,B]), for t ∈ R+, A,B ∈ Kc(Rn),β ∈ I;

(iii) g ∈ C[R2+,R+], g(t, 0) ≡ 0, g(t, w) is nondecreasing in w for each t ∈ R+

and w(t) ≡ 0 is the only solution of

w′ = g(t, w), w(t0) = 0, for t ≥ t0;

(iv) D[H(t, A;α), H(t, A; β)] ≤ L|α− β|, α, β ∈ I,(t, A) ∈ R+ ×Kc(Rn), L > 0.

We are now in a position to prove the following result.

Theorem 4.4.3 Suppose that the conditions (i) to (iv) are satisfied. Then thereexists a unique solution Uβ(t) = Uβ(t, t0, U0β) ∈ Kc(Rn), β ∈ I of (4.4.11) andUβ(t) is quasiconcave in β for t ≥ t0.Moreover, there exists a fuzzy set u(t) ∈ En

such that[u(t)]β = Uβ(t), t ≥ t0.


Proof Since f is continuous on R+ × Rn, F (t, x; β) is also continuous for(t, x, β) ∈ R+ × Rn × I. This implies that H(t, A; β) is continuous map for(t, A; β) ∈ R+ × Kc(Rn) × I. Consequently, by Theorems 2.3.1 and 2.6.1 itfollows that there exists a unique solution Uβ(t) = U (t, t0, U0β) ∈ Kc(Rn), fort ≥ t0 of (4.4.11).

We first show that if α ≤ β, then Uβ(t) ⊆ Uα(t) for t ≥ t0. From thedefinition of quasi-concavity of F (t, x; β), it follows that H(t, A; β) is also quasi-concave in β. Let Uα(t), Uβ(t) be the solution of

DHUα = H(t, Uα;α), Uα(t0) = U0 ∈ Kc(Rn),

DHUβ = H(t, Uβ; β) Uβ(t0) = U0 ∈ Kc(Rn), α ≤ β.

Then we find, using quasi-concavity

DHUα = H(t, Uα;α) ⊇ H(t, Uα; β), α ≤ β.

But H(t, Uα; β) = H(t, Uβ; β) because of Uα(t0) = U0 = Uβ(t0) and thereforeUα(t) ≡ Uβ(t) by uniqueness of solutions of (4.4.11).

Thus it is clear that Uβ(t) ⊆ Uα(t), α ≤ β, t ≥ t0.We shall next prove that if βn is a nondecreasing sequence, βn ∈ I, converg-

ing to β, then Uβn(t) → Uβ(t), uniformly on compact subsets of [t0,∞). Forthis purpose, set m(t) = D[Uβn (t), Uβ(t)] and note that D[U0βn , U0β] = m(t0).We shall assume that U0βn → U0β, as n → ∞. Then employing the propertiesof the metric D, the definition of Hukuhara derivative and the conditions (ii)and (iv), we arrive at the scalar differential inequality,

D+m(t) ≤ g(t,m(t)) + L|βn − β|, m(t0) = D[U0βn , U0β], t ≥ t0.

Hence by Lemma 1.3.1 in (Lakshmikantham and Leela [1]), we obtain

m(t) ≤ rn(t, t0, ηn),

on any compact set J ⊂ [t0,∞), where ηn = D[U0βn , U0β] and rn(t, t0, ηn) isthe maximal solution of

w′ = g(t, w) + L|βn − β|, w(t0) = ηn, on J.

By assumption (iii), rn(t, t0, ηn) → r(t, t0, 0) ≡ 0, uniformly on J as n → ∞.Since βn → β as n → ∞, it follows that m(t) ≡ 0 on J , which in turn impliesthat D[Uβn (t), Uβ(t)] → 0 as n→ ∞. Thus, Uβ(t) is quasi-concave in β ∈ I, fort ≥ t0. Consequently, by Theorem 4.4.2, there exists a fuzzy set u(t) ∈ En suchthat [u(t)]β = Uβ(t), t ≥ t0, and this completes the proof.

To find the connection between the solution Uβ(t) of (4.4.11) and the attain-ability set Aβ(U0, t) of (4.4.3) we have the following result.

Theorem 4.4.4 Let F ∈ C[R+ × Rn × I,Kc(Rn)] satisfy the assumptions ofTheorem 4.4.1 for each β ∈ I = [0, 1] and assume that it is also quasiconcave inβ as well. Then there exists a unique solution Uβ(t) = Uβ(t, t0, U0) of (4.4.1)for t ≥ t0 and the attainable set Aβ(U0, t) of the inclusion (4.4.3) such that

Aβ(U0, t) ⊂ Uβ(t, t0, U0), t ≥ t0. (4.4.12)


Proof It is easy to verify that when F (t, x; β) satisfies the assumptions re-quired in Theorem 4.4.1, the desired conditions in Theorems 2.3.1 and 2.6.1are also satisfied, in view of the monotone nondecreasing nature of the func-tions g(t, w), q(t, w), the definition of D and the fact H(t, A; β) is generated byF (t, x; β). We have assumed conditions in terms of H(t, A; β) since the set dif-ferential equations are treated as an independent subject. Thus for each β ∈ I,Theorem 4.4.1 yields the relation (4.4.12). Also, both Uβ(t) and Aβ(U0, t) sat-isfy the assumptions of Theorem 4.4.2., because one can prove similarly thequasi-concavity of Aβ(U0, t). Therefore, there exist fuzzy sets u(t), v(t) ∈ En

such that[v(t)]β = Aβ(U0, t) and [u(t)]β = Uβ(t), t ≥ t0.

The proof is complete.We note that, in general, since Aβ(U0, t) is only compact and not convex,

only (4.4.12) holds. Equality in (4.4.12) is valid only in some special cases.Recalling (4.4.7) in the example, let us generate the set differential equation

from F in (4.4.7), that is

H(t, U, β) = coF (t, A; β), for A ∈ Kc(R),

and thus,DHUβ = −Uβ , Uβ(0) = U0β ∈ Kc(R). (4.4.13)

Since the values of the solution (4.4.13) are intervals, the equation (4.4.13) canbe written as,

[u′1β, u′2β] = [−u2β,−u1β], (4.4.14)

where U = [u1β, u2β]. The relation (4.4.14) is equivalent to, taking U0β =[u10β, u20β], the system of equations,

u′1β = −u2β, u1β(0) = u10β,

u′2β = −u1β, u2β(0) = u20β,

whose solution corresponds to (4.3.6) and (4.3.7), duly altered to the presentframework, for 0 ≤ β ≤ 1 and t ≥ 0,

u1β(t) =12[u10β + u20β]e−t +

12[u10β − u20β]et,

u2β(t) =12[u20β + u10β]e−t +

12[u20β − u10β]et.

Given U0β ∈ Kc(R), there exists V0β,W0β ∈ Kc(R) such that U0β = V0β +W0β ,and hence the Hukuhara difference U0β − V0β = W0β exists.

ChooseV0β = [

12[(u10β − u20β), (u20β − u10β)],

so that W0β =12[(u10β + u20β), (u20β + u10β)].


It then follows, assuming that u10β 6= −u20β, that

Uβ(t, U0β) =12[−(u20β − u10β), (u20β − u10β)]et

+12[(u10β + u20β), (u10β + u20β)]e−t

Uβ(t, V0β) =12[(u10β − u20β), (u20β − u10β)]et, and

Uβ(t,W0β) =12[(u10β + u20β), (u10β + u20β)]e−t, t ≥ 0.

It therefore follows that

Aβ(W0β, t) = Uβ(t,W0β) ⊂ Uβ(t, U0β), t ≥ 0. (4.4.16)

4.5 Upper Semicontinuous Case Continued

Recall that from fuzzy differential equation (4.4.1), we did generate the sequenceof set differential equations given by

DHUβ = H(t, Uβ; β), Uβ(t0) = U0β ∈ Kc(Rn), (4.5.1)

where H(t, A; β) = coF (t, A; β) and F (t,X; β) = [f(t,X)]β, 0 ≤ β ≤ 1.( seeequations (4.4.10) and (4.4.11).)

In this section, we continue to investigate the upper semicontinuous (usc)case discussed in Section 2.9 and prove some results parallel to the continuouscase considered in Section 4.4.

Let us list the following hypotheses, H(F).F : J × Rn × I → Rn, I = [0, 1], J = [t0, b], t0 ≥ 0 and b ∈ [t0,∞), is a

multifunction with compact convex values such that

(a) (t,X) → F (t,X; β) is L⊕

B(Rn) measurable for every β ∈ I;

(b) for every (x, β) ∈ Rn × I and ε > 0 for almost every t ∈ J there existsδ > 0 such that for all α, β − δ < α ≤ β, y ∈ x+ ε.B, the inclusion

F (t, y;α) ⊂ F (t, x; β) + ε.B (4.5.2)

holds;

(c) F (t, x; β) is quasiconcave, that is for x ∈ Rn, α, β ∈ I, α ≤ β,

F (t, x;α) ⊃ F (t, x; β) a.e. (4.5.3)

(d) the inequality (2.9.10) holds for every β ∈ I.

Lemma 4.5.1 Assume that hypotheses H(F) hold. Then for every β ∈ I themultifunction H(t, A; β) has all properties listed in Lemma 2.9.1. and

4.5 UPPER SEMICONTINUOUS CASE CONTINUED 117

(i) H(t, A; β) is quasiconcave a.e.;

(ii) if Uk ∈ Kc(Rn), k ≥ 1, is a nonincreasing sequence with respect toinclusion, and βk ∈ I, k ≥ 1, is a nondecreasing sequence converging toβ.

Then the sequence H(t, Uk; βk), k ≥ 1, converges a.e. to H(t,∩∞k=1 Uk; β) in

space Kc(Rn).

Proof By (4.5.2) for fixed β ∈ I the multifunction x → F (t, x; β) is usc foralmost every t ∈ J . Hence for fixed β ∈ I the multifunction (t, x) → H(t, x; β)for almost every t ∈ J has compact convex values and possesses all propertieslisted in Lemma 2.9.1.

According to (4.5.3) for every A ∈ Kc(Rn),

F (t, A;α) ⊃ F (t, A; β), α ≤ β, a.e. (4.5.4)

Hence (i) is true.Let a sequence Uk ∈ Kc(Rn), k ≥ 1 be nonincreasing with respect to inclu-

sion, and βk ∈ I, k ≥ 1, be a nondecreasing sequence converging to β. ThenV = ∩∞

k=1Uk is a nonempty compact convex set and sequence Uk, k ≥ 1, con-verges to V in Kc(Rn). By using (4.5.4) and the monotonicity of F (t, U ; β) withrespect to U we obtain

F (t, V ; β) ⊂∞⋂

k=1

F (t, V ; βk) ⊂∞⋂

k=1

F (t, Uk; βk). (4.5.5)

Let y ∈ ∩∞k=1F (t, Uk; βk). Then there exists a sequence xk ∈ Uk ⊂ U1 such that

y ∈ F (t, xk; βk), k ≥ 1. Since xk ∈ U1, k ≥ 1, without loss of generality we canassume that xk, k ≥ 1, converges to x. It is clear that x ∈ V. Take any ε > 0.According to (4.5.2) there exists a number N ≥ 1 such that

y ∈ F (t, xk; βk) ⊂ F (t, x; β) + ε.B (4.5.6)

for all k ≥ N. Since ε > 0 is arbitrary then y ∈ F (t, x; β) ⊂ F (t, V ; β). Henceby (4.5.5)

F (t, V ; β) ⊂ ∩∞k=1F (t, Uk; βk) ⊂ F (t, V ; β). (4.5.7)

The inclusion (4.5.7) tells us that the sequence F (t, Uk; βk), k ≥ 1 converges toF (t, V ; β) in Kc(Rn). Hence the sequence H(t, Uk; βk) = coF (t, Uk; βk), k ≥ 1,converges to H(t,∩∞

k=1Uk; β) in Kc(Rn). Thus the Lemma is proved.We can now prove the following result which provides the connection be-

tween the fuzzy differential equation (4.4.1) and the sequence of set differentialequations (4.5.1).

Theorem 4.5.1 Assume that the multifunction F (t, x; β) satisfies hypothesesH(F ). Then there exists a solution Uβ(t) = Uβ(t, t0, U0β) ∈ Kc(Rn), β ∈ I, t ∈ Jof the equation (4.5.1). If the solution Uβ(t), β ∈ I, is unique then Uβ(t) isquasiconcave for t ∈ J . Moreover, there exists a fuzzy set u(t) ∈ En such that[u(t)]β = Uβ(t), β ∈ I, t ∈ J and fuzzy set t → u(t) is continuous from J to En.


Proof The existence of solution Uβ(t), β ∈ I follows from Lemma 4.5.1. andCorollary 2.9.1. Let us show that Uβ(t), t ∈ J, is quasiconcave if the solutionUβ(t), β ∈ I, is unique.

Let α < β and Uα(t) be a solution of equation (4.5.1). Then

Uα(t) = U0α +∫ t

t0

H(s, Uα(s);α) ds, t ∈ J. (4.5.8)

Denote by Vα the collection of all functions x→ Rn representable as

x(t) = x0 +∫ t

t0

v(s) ds, t ∈ J, x0 ∈ U0α, (4.5.9)

where v(s) is a Bochner integrable selector ofH(s, Uα(s);α).Then Vα is compactconvex set of C(J,Rn) and Vα(t) = Uα(t), t ∈ J .

Using (1.8.2) , (4.5.8), (4.5.9) and the definition of the operator T (V0, F, V )we obtain

Vα = T (U0α,Hα, Vα). (4.5.10)

Let V 0β be a collection of all functions x : J → Rn representable as (4.5.9) with

x0 ∈ U0β and v(s) being a Bochner integrable selector of H(s, Uα(s);α).As has been shown in the proof of Theorem 2.9.1, V 0

β is a compact convexset of C(J,Rn),

V 0β = T (U0β ,Hα, Vα), (4.5.11)

andV 0

β ⊂ Vα, (4.5.12)

because U0β ⊂ U0α.From quasiconcavity and monotonicity of H(t, A;α) and (4.5.11) , (4.5.12)

it follows

V 1β = T (U0β ,Hβ, V

0β ) ⊂ T (U0β ,Hα, V

0β ) ⊂ T (U0β),Hα, Vα) = V 0

β , (4.5.13)

and V 1β is compact convex subset of C(J,Rn).

We now define V 2β = T (U0β,Hβ, V

1β ). Because V 1

β ⊂ V 0β we have

V 2β = T (U0β,Hβ, V

1β ) ⊂ T (U0β ,Hβ, V

0β ) = V 1

β .

Continuing this process we obtain a sequence V kβ , k ≥ 1, of compact convex

subsets of C(J,Rn) decreasing relative to the inclusion. Repeating the proofof Theorem 2.9.1 we obtain that Uβ = ∩∞

k=1Vkβ = T (U0β,Hβ, Uβ) and Uβ(t) =

x(t) : x(.) ∈ Uβ is a solution of equation (4.5.1). Since Uβ ⊂ V kβ ⊂ Vα then

Uβ(t) ⊂ Vα(t) = Uα(t), t ∈ J .For the construction of Uβ(t) we use the solution Uα(t) of equation (4.5.1).

Since the equation (4.5.1) has a unique solution, the solution Uβ(t) does notdepend on α < β. Hence Uβ(t) ⊂ Uα(t), t ∈ J, for any α, β ∈ I, α < β.

4.5 UPPER SEMICONTINUOUS CASE CONTINUED 119

Let r0 = max‖x‖;x ∈ U00 and r(t) = r(t, t0, r0) be a maximal solution of(2.9.3) on J . By hypothesis H(F )(d) we have

‖Uβ(t)‖ ≤ ‖U0β‖ +∫ t

0

‖H(s, Uβ(s); β)‖ ds ≤∫ t

0

g(s, ‖Uβ(s)‖) ds, t ∈ J.

Hence by Lemma 1.3.1. in Lakshmikantham and Leela[1] we obtain

‖H(t, Uβ(t); β)‖ ≤ g(t, r(t)) = r′(t). (4.5.14)

If t∗ ≤ t, then

Uβ(t) = Uβ(t∗) +∫ t

t∗

H(s, Uβ(s); β) ds. (4.5.15)

If t ≤ t∗, then

Uβ(t∗) = Uβ(t) +∫ t∗

t

H(s, Uβ(s); β) ds. (4.5.16)

Taking into consideration (1.3.9),(4.5.14),(4.5.15),(4.5.16) we obtain

D(Uβ(t), Uβ(t∗)) ≤ |∫ t

t∗

r′(s) ds|, β ∈ J.

Hence the family of functions Uβ(.), β ∈ I is equicontinuous from J to Kc(Rn).Let βk ∈ I, k ≥ 1, be any nondecreasing sequence converging to β. We claimthat Uβ(t) = ∩∞

k=1Uβk (t), t ∈ J. Since for every t ∈ J the sequence Uβk(t)is nondecreasing with respect to inclusion, the sequence Uβk(t) is convergingpointwise to a function V (t) = ∩∞

k=1Uβk(t) in Kc(Rn). Moreover, the sequenceUβk (t), k ≥ 1, converges uniformly to V (t) because the sequence Uβk(t) isequicontinuous. Hence V : J → Kc(Rn) is continuous.

Taking into consideration the statement (ii) of Lemma 4.5.1 we obtain

H(t, Uβk(t); βk) → H(t, V (t); β) in Kc(Rn). (4.5.17)

From (4.5.14) it follows

D(H(t, V (t); β),H(t, Uβk(t); βk)) ≤ 2r′(t). (4.5.18)

Let

U (t) = U0β +∫ t

0

H(s, V (s); β) ds.

Then the inequality

D(U (t), Uβk (t)) ≤ D(U0β , U0βk)+∫ t

0

D(H(s, V (s); β),H(t, Uβk(t), Uβk)) ds, t ∈ J,

(4.5.19)holds. Since U0β = [x0]β, then

limk→∞

D(U0β, U0βk) = 0. (4.5.20)


Now from (4.5.18) to (4.5.20) and Lebesgue bounded convergence theorem weobtain limk→∞D(U (t), Uβk(t)) = 0.

Since Uβk(t) → V (t) then U (t) = V (t), t ∈ J and the equality

V (t) = U0β +∫ t

0

H(s, V (s)) ds, t ∈ J

is true.Hence V (t) is a solution of equation (4.5.1). Due to the uniqueness of the

solution of equation (4.5.1)

Uβ(t) = V (t) =∞⋂

k=1

Uβk(t), t ∈ J.

Consequently, by Theorem 4.4.2, there exists a fuzzy set u(t) ∈ En such that[u(t)]β = Uβ(t), t ∈ J. Since the family Uβ(t), β ∈ J, is equicontinuous , thenthe fuzzy set u(t) is continuous from J to En and this completes the proof.

Remark 4.5.1 In the original formulation of fuzzy differential equations (FDEs),the function f in (4.4.1)is assumed to be continuous to prove several basic re-sults. This implies that the function F (t, x; β) is continuous for each β. InSection 4.4, under the assumption of continuity several results are investigated.The function f is assumed to be usc, F (t, x; β) is usc for each β, and, conse-quently , the standard results of multivalued inclusions can be utilized to capturethe rich behaviour of solutions of inclusions. See Lakshmikantham and Mohap-atra [1] for further details. In this section, we took a step further to study setdifferential equations (SDEs) which are generated by FDEs, since SDEs haveseveral advantages.

Remark 4.5.2 If u ∈ Dn, that is u is not assumed fuzzy convex , then thelevel set [u]β need not be convex. Hence, when the fuzzy convexity is discarded,[f(t, x)]β = F (t, x; β) need not be convex. Nonetheless, the generated functionH(t, A; β) is convex and therefore, we can still apply our results.

4.6 Impulsive Fuzzy Differential Equations

Let PC denote the class of piecewise continuous functions from R+ to R withdiscontinuities of first kind only at t = tk, k = 1, 2, ... . We need the followingknown result (see Lakshmikantham, Bainov and Simeonov [1]).


(A0) The sequence tk satisfies 0 ≤ t0 < t1 < t2, ..., with limk→∞ tk = ∞;

(A1) m ∈ PC1[R+,R] and m(t) is left continuous at tk, k = 1, 2, ...;

4.6 IMPULSIVE FDES 121

(A2) for k = 1, 2, ..., t ≥ t0, and m′(t) ≤ g(t,m(t)), t 6= tk, m(t0) ≤ w0,

m(t+k ) ≤ ψk(m(tk)), (4.6.1)

where g ∈ C[R+ × R,R], ψk : R → R, ψk(w) is nondecreasing in w;

(A3) r(t) = r(t, t0, w0) is the maximal solution of

w′ = g(t, w), t 6= tk, w(t0) = w0,

w(t+k ) = ψk(w(tk)), tk > t0 ≥ 0, (4.6.2)

existing on [t0,∞). Then m(t) ≤ r(t), t ≥ t0.

Proof For t ∈ [t0, t1], we have by the classical comparison theorem m(t) ≤ r(t).Hence using the facts that ψ1(w) is nondecreasing in w and m(t1) ≤ r(t1) weobtain

m(t+1 ) ≤ ψ1(m(t1)) ≤ ψ1(r(t1)) = w+1 .

Now, for t1 < t ≤ t2, it follows, using again the classical comparison theoremm(t) ≤ r(t), where r(t) = r(t, t1, w+

1 ) is the maximal solution of (4.6.2) on theinterval t1 ≤ t ≤ t2. Moreover, as before, we get

m(t+2 ) ≤ ψ2(m(t2)) ≤ ψ2(r(t2)) = w+2 .

Repeating the arguments, we finally arrive at the desired result, and the proofis complete. Repeating the arguments, we finally arrive at the desired result,and the proof is complete.

Let us consider now the impulsive fuzzy differential equation

u′ = f(t, u), t 6= tk,

u(t+k ) = u(tk) + Ik(u(tk)), u(t0) = u0,(4.6.3)

where (A0) holds and f : R+ × En → En, Ik : En → En, f is continuous in(tk−1, tk] × En and for each u ∈ En, lim f(t, v) = f(t+k , u) exists as (t, v) →(t+k , u).

We assume that, for each (tk−1, tk]×En, there exists a unique solution u(t)of (4.6.3) in each interval [tk−1, tk]. As a result, employing impulsive conditionin (4.6.3) at each tk, we can define the solution u(t) on the entire interval [t0,∞).

Theorem 4.6.2 Assume that f ∈ C[R+ × En, En] and

lim suph→0+

1h

[D0[u+ hf(t, u), v + hf(t, v)] −D0[u, v]]]

≤ g(t,D0[u, v]), t ∈ R+, u, v ∈ En, t 6= tk,

where g ∈ C[R+ × R+,R]. Suppose that

D0[u+ Ik(u), v + Ik(v)] ≤ ψk(D0[u, v])


where ψk : R+ → R+, ψk(w) is nondecreasing in w. The maximal solutionr(t) = r(t, t0, w0) of (4.6.2) exists for t ≥ t0. Then

D0[u(t), v(t)] ≤ r(t), t ≥ t0,

where u(t), v(t) are the solutions of (4.6.3) existing on [t0,∞).

Proof Proceeding as in the proof of Theorem 4.3.1, we find that for t 6= tk,

m(t + h) −m(t) = D0[u(t+ h), v(t + h)] −D0[u(t), v(t)]

≤ D0[u(t+ h), u(t) + hf(t, u(t)] +D0[v(t) + hf(t, v(t), v(t + h)]

+ D0[u(t) + hf(t, u(t)), v(t) + hf(t, v(t))] −D0[u(t), v(t)].

Hence


1h

[m(t+ h) −m(t)]

+ ≤ lim suph→0+

1h

[D0(u(t) + hf(t, u(t)), v(t) + hf(t, v(t))]

−D0[u(t), v(t)]

+ lim suph→0+

D0[u(t+ h) − u(t)

h, f(t, u(t))]

+ lim suph→0+

D0[f(t, v(t)),v(t + h) − v(t)

h], t 6= tk,

≤ g(t,D0[u(t), v(t)]) = g(t,m(t)), t 6= tk.

Also, for t = tk,

m(t+k ) = D0[u(t+k ), v(t+k )]= D0[u(tk) + Ik(u(tk)), v(tk) + Ik(v(tk))]≤ ψk(D0[u(tk), v(tk)] = ψk(m(tk)).

We therefore obtain from Theorem 4.6.1, the stated result, namely,

D0[u(t), v(t)] ≤ r(t), t ≥ t0,

where r(t) = r(t, t0, w0) is the maximal solution of (4.6.2) provided D0[u0, v0] ≤w0, completing the proof.

Let V : R+ ×En → R+. Then V is said to belong to class V0 if

(i) V is continuous in (tk−1, tk] × En and for each u ∈ En, k = 1, 2, ...,lim(t,v)→(t+k ,u) V (t, v) = V (t+k , u) exists;

(ii) V satisfies |V (t, u) − V (t, v)| ≤ LD0[u, v], L ≥ 0.

4.6 IMPULSIVE FDES 123

For (t, u) ∈ (tk−1, tk] ×En, we define

D+V (t, u) = lim suph→0+

1h

[V (t+ h, u+ hf(t, u)) − V (t, u)],

then we can prove the following comparison theorem.

Theorem 4.6.3 Let V : R+ × En → R+ and V ∈ V0. Suppose that

D+V (t, u) ≤ g(t, V (t, u)), t 6= tk, (4.6.4)

V (t, u+ Ik(u)) ≤ ψk(V (t, u)), t = tk, (4.6.5)

where g : R2+ → R is continuous in (tk−1, tk] × R+ and for each w ∈ R+,

lim(t,z)→(t+k ,w) g(t, z) = g(t+k , w) exists, ψk : R+ → R is nondecreasing. Letr(t) be the maximal solution of the scalar impulsive differential equation (4.6.2)existing for t ≥ t0. Then V (t+0 , u0) ≤ w0 implies

V (t, u(t)) ≤ r(t), t ≥ t0.

Proof Let u(t) = u(t, t0, u0) be any solution of (4.6.3) existing on t ≥ t0, suchthat V (t+0 , u0) ≤ w0. Define m(t) = V (t, u(t)) for t 6= tk. Then using standardarguments, we arrive at the differential inequality

D+m(t) ≤ g(t,m(t)), t 6= tk.

From (4.6.5), we get for t = tk,

m(t+k ) = V (t+k , u(t+k )) = V (t+k , u(tk)+Ik(u(tk))) ≤ ψk(V (tk, u(tk))) = ψk(m(tk)).

Hence by Theorem 4.6.1,m(t) ≤ r(t), t ≥ t0, which proves the claim of Theorem4.6.3.

Some special cases of g(t, w) and ψk(w) which are instructive and useful aregiven below as a corollary.

Corollary 4.6.1 In Theorem 4.6.3, suppose that

(i) g(t, w) = 0, ψk(w) = w for all k, then V (t, u(t)) is nondecreasing in tand V (t, u(t)) ≤ V (t+0 , u0), t ≥ t0;

(ii) g(t, w) ≡ 0, ψk(w) = dkw, dk ≥ 0 for all k, then

V (t, u(t)) ≤ V (t+0 , u0)∏

t0<tk<t

dk, t ≥ t0;

(iii) g(t, w) = −αw,α > 0, ψk(w) = dkw, dk ≥ 0 for all k, then

V (t, u(t)) ≤ V (t+0 , u0)

( ∏

t0<tk<t

dk

)e−α(t−t0), t ≥ t0;


(iv) g(t, w) = λ′(t)w, ψk(w) = dkw, dk ≥ 0 for all k, λ ∈ C1[R+,R+].

Then

V (t, u(t)) ≤ V (t+0 , u0)

( ∏

t0<tk<t

dk

)exp[λ(t) − λ(t0)], t ≥ t0;

Recall the example considered in Section 4.3 and note that when we choose[x0]α = [α− 1, 1− α], 0 ≤ α ≤ 1, we get

[x(t)]α = [(α− 1), (1 − α)]et = [−1, 1](1− α)et, t ≥ 0.

In particular, diam [x(t)]α = 2(1 − α)et, t ≥ 0.In order to show the effect of impulses, we now introduce the impulsive

condition as in (4.6.3), that is, [xk]+α = [x+α,1k , x+α

2k ] and [xk]α = [xα1k, x

α2k] for

each k , where x+k = x(t+k ) and xk = x(tk). Because of impulse condition [x(t)]α

reduces to

[x(t)]α =

[(α− 1)

∏

0<tk<t

dk, (1 − α)∏

0<tk<t

dk

]et, t ≥ 0.

It follows that, if dk, tk satisfy the condition

tk+1 + ln dk ≤ tk,

then x = 0 is stable and if tk+1 + β ln dk ≤ tk, β > 0, then x = 0 is asymptoti-cally stable.

This demonstrates that the impulsive action helps to obtain stability of FDEwithout utilizing Hukuhara difference for the initial values, as we have proposedin Section 4.3.

4.7 Hybrid Fuzzy Differential Equations

The problem of stabilizing a continuous plant governed by differential equationthrough the interaction with a discrete time controller has recently been investi-gated. This study leads to the consideration of hybrid systems. In this section,we shall extend this approach to fuzzy differential equations.

Consider the hybrid fuzzy differential system

u′(t) = f(t, u(t), λk(z)), u(tk) = z, (4.7.1)

on [tk, tk+1] for any fixed z ∈ En, k = 0, 1, 2, ...,where f ∈ C[R+×En×En, En],and λk ∈ C[En, En]. Here we assume that 0 ≤ t0 < t1 < t2 < ..., are such thattk → ∞ as k → ∞ and the existence and uniqueness of solutions of the hybrid

4.7 HYBRID FDES 125

system hold on each [tk, tk+1]. To be specific, the system is of the form

u′(t) =

u′0(t) = f(t, u0(t), λ0(u0)), u0(t0) = u0, t0 ≤ t ≤ t1,

u′1(t) = f(t, u1(t), λ1(u1)), u1(t1) = u1, t1 ≤ t ≤ t2,...

......

u′k(t) = f(t, uk(t), λk(uk)), uk(tk) = uk, tk ≤ t ≤ tk+1,

......

...

where uk = uk−1(tk) for each k. By the solution of (4.7.1), we therefore meanthe following function

u(t) = u(t, t0, u0) =

u0(t), t0 ≤ t ≤ t1,

u1(t), t1 ≤ t ≤ t2,...

...uk(t), tk ≤ t ≤ tk+1,

......

We note that the solutions of (4.7.1) are differentiable in each interval for t ∈(tk, tk+1) for any fixed uk ∈ En and k = 0, 1, 2, ....

Let V ∈ C[En,R+]. For t ∈ [tk, tk+1], u, z ∈ En, we define

D+V (u; z) = lim suph→0+

1h

[V (u+ hf(t, u, λk(z))) − V (u)].

We can then prove the following comparison theorem in terms of Lyapunov-likefunction V .


(i) V ∈ C[En,R+], V (u) satisfies |V (u) − V (v)| ≤ LD0[u, v], L > 0 foru, v ∈ En;

(ii) D+V (u; z) ≤ g(t, V (u), σk(V (z))), t ∈ (tk, tk+1], where g ∈ C[R3+, R],

σk ∈ C[R+,R+], u, z ∈ En, k = 0, 1, 2, ...;

(iii) the maximal solution r(t) = r(t, t0, w0) of the hybrid scalar differentialequation.

w′ = g(t, w(t), σk(wk)), t ∈ (tk, tk+1],

w(tk) = wk, k = 0, 1, 2, ...,(4.7.2)

exists on [t0,∞).

Then any solution u(t) = u(t, t0, u0) of (4.7.1) such that V (u0) ≤ w0 satisfiesthe estimate

V (u(t)) ≤ r(t), t ≥ t0.


Proof Let u(t) be any solution of (4.7.1) existing on [t0,∞) and set m(t) =V (u(t)). Then using (i) and (ii), and proceeding as in the proof of Theorem4.3.1 and Theorem 4.6.2, we get the differential inequality

D+m(t) ≤ g(t,m(t), σk(mk)) for tk < t ≤ tk+1,

where mk = V (u(tk)). For t ∈ [t0, t1], since m(t0) = V (u0) ≤ w0, we obtain

V (u0(t)) ≤ r0(t, t0, w0), t0 ≤ t ≤ t1,

where r0(t) = r0(t, t0, w0) is the maximal solution of

w′0 = g(t, w0, σ0(w0)), w0(t0) = w0 ≥ 0, t0 ≤ t ≤ t1,

and u0(t) is the solution of

u′0 = f(t, u0(t), λ0(u0)), u(t0) = u0 ≥ 0, t0 ≤ t ≤ t1.

Similarly, for t ∈ [t1, t2], it follows that

V (u1(t)) ≤ r1(t, t1, w1), t1 ≤ t ≤ t2,

where w1 = r0(t1, t0, w0), r1(t, t1, w1) is the maximal solution of

w′1 = g(t, w1, σ1(w1)), w1(t1) = w1 ≥ 0, t1 ≤ t ≤ t2,

and u1(t) is the solution of

u′1 = f(t, u1(t), λ1(u1)), u1(t1) = u1, t1 ≤ t ≤ t2.

Proceeding similarly, we can obtain

V (uk(t)) ≤ rk(t, tk, wk), tk ≤ t ≤ tk+1,

where uk(t) is the solution of

u′k(t) = f(t, uk(t), λk(uk)), uk(tk) = uk, tk ≤ t ≤ tk+1,

and rk(t, tk, wk) is the maximal solution of

w′k = g(t, wk(t), σk(wk)), wk(tk) = wk, tk ≤ t ≤ tk+1,

where wk = rk−1(tk, tk−1, rk−2(tk−1, tk−2, wk−1)).Thus defining r(t, t0, w0)as the maximal solution of the comparison hybrid

system (4.7.2) as

r(t, t0, w0) =

r0(t, t0, w0), t0 ≤ t ≤ t1,r1(t, t1, w1), t1 ≤ t ≤ t2,...

...rk(t, tk, wk), tk ≤ t ≤ tk+1,

......

4.7 HYBRID FDES 127

and taking w0 = V (u0), we obtain the desired estimate

V (u(t)) ≤ r(t), t ≥ t0.

The proof is therefore complete.Consider now the hybrid impulsive fuzzy differential system given by

u′ = f(t, u, λ(tk, uk)), t ∈ [tk, tk+1],

u(t+k ) = u(tk) + Ik(u(tk)), t = tk, (4.7.3)

u(t+0 ) = u0,

where f ∈ C[R+ × En × En, En], Ik : En → En, λk ∈ C[R+,×En, En], andk = 0, 1, 2, ... .

We assume that I0(u0) = 0, and the existence of solution of the system

u′ = f(t, u, λ(tk, z)), t ∈ (tk, tk+1],

u(t+k ) = z + Ik(z), t 6= tk, (4.7.4)

u(t+0 ) = u0

on (tk, tk+1] for any fixed z ∈ En and all k = 0, 1, 2, ... .Note that the solution of (4.7.4) is a piecewise continuous function with

points of discontinuity of the first type at t = tk at which they are assumed tobe left continuous.

Let V : R+ ×En → R+. Then V is said to belong to class V0, if

(i) V is continuous in (tk, tk+1] × En and for each u ∈ En, k = 1, 2, ....lim(t,v)→(t+k ,u) V (t, v) = V (t+k , u) exists;

(ii) V is locally Lipschitzian in u. Then we define, as before,

D+V (t, u, z) ≡ lim suph→0+

1h

[V (t+ h, u+ hf(t, u, λk(tk, z)) − V (t, u)].

We need the following comparison result.


(i) V ∈ C[R+ ×En,R+], V (t, u) is locally Lipschitzian in u that is |V (t, u)−V (t, v)| ≤ L D0[u, v], L > 0, and

D+V (t, u, z) ≤ g(t, V (t, u), σk(tk, z)), t ∈ (tk, tk+1],

u, z ∈ En, where σk ∈ C[R+ × En,R], g ∈ C[R3+,R];

(ii) there exist a ψk ∈ C[R+,R+], ψk(w) is nondecreasing in w and

V (t, u+ Ik(u)) ≤ ψk(V (t, u)), k = 1, 2, · · · ; u ∈ En;


(iii) the maximal solution r(t) = r(t, t0, w0) of the scalar hybrid impulsive dif-ferential equation

w′ = g(t, w, η(tk, wk)), t ∈ (tk, tk+1],

w(t+k ) = ψ(w(tk)), t = tk, (4.7.5)

w(t0) = w0 ≥ 0,

existing on [t0,∞), where η ∈ C[R2+,R] and wk = V (tk, uk). Then any solution

u(t) = u(t, t0, u0) of (4.7.3) satisfies

V (t, u(t)) ≤ r(t, t0, w0), t ≥ t0,

provided w0 ≥ V (t0, u0).

The proof of this comparison theorem follows on similar lines to Theorem 4.7.2defining u(t) and r(t), piece by piece suitably. We omit the proof to avoidmonotony. Having the foregoing comparison result at our disposal, we canformulate stability criteria of the solutions of (4.7.3) relative to any kind ofstability such as Lyapunov stability , practical stability, stability in terms oftwo different measures, which includes several known stability concepts or thenew concept of stability, which includes Lyapunov and orbital stability as specialcases. We simply state a typical result whose proof can be constructed basedon stability criteria of impulsive differential equations.


(i) V ∈ V0 and V (t, u) is positive definite and decrescent;

(ii) D+V (t, u, z) ≤ g(t, V (t, u), σk(tk, z)), t ∈ (tk, tk+1], u, z ∈ En, σk, g areas defined in Theorem 4.7.2 ;

(iii) V (t, u+ Ik(u)) ≤ ψk(V (t, u)), t = tk, u ∈ En, where ψk(w) is nondecreas-ing in w, as in Theorem 4.7.2.

Then stability properties of the trivial solution w = 0 of (4.7.5) imply the cor-responding stability properties of (4.7.3) respectively.

All that is needed to get any kind of stability properties of (4.7.3) is to requirepositive definiteness and decresence of V (t, u) suitably relative to that particularstability demands. For example if we want stability criteria in terms of twodifferent measures say, (h0, h), then V (t, u) need to satisfy positive definitenessrelative to h and decrescence with respect to h0, where h0 is finer than h. Seefor details Lakshmikantham and Liu [1].

4.8 ANOTHER FORMULATION 129

4.8 Another Formulation

Consider a differential equation in a given space X

u′(t) = f(t, u(t)), t ∈ [0, T ], (4.8.1)

with a specific initial condition

u(0) = u0, (4.8.2)

where T > 0, and f : [0, T ]×X → X.Now, suppose that X is a Banach space with norm ‖.‖ inducing a distance

d. If f is continuous, then equation (4.8.1) is equivalent to

limh→0

‖u(t+ h) − u(t) − f(t, u(t))h‖h

= 0, t ∈ [0, T ).

Therefore, any solution of (4.8.1) satisfies

limh→0+

d(u(t+ h), F (t, h, u(t)))h

= 0, (4.8.3)

whereF : [0, T ]× R+ ×X → X, F (t, h, u) = u+ hf(t, u). (4.8.4)

With the notation of Kloeden, Sadovsky and Vasiyeva [1], (4.8.1) is equivalentto

u(t+ dt) − u(t) −Dt,u(t)(dt) = o(dt), (4.8.5)

withDt,u(dt) = f(t, u)dt = F (t, dt, u)− u. (4.8.6)

There are three possible definitions for continuous processes in a Banach space.Formulation (4.8.5) allows the study of nonsmooth systems such as “stop

nonlinearities” and is called an equation with a nonlinear differential. Withan adequate choice of nonlinear differential D it is possible to obtain existenceresults for classical ordinary differential equations, Caratheodory differentialequations, and differential inclusions with maximal monotone operators.

However, (4.8.1) and (4.8.5) have both the same shortcoming: One needs analgebraic structure in the underlying space. On the other hand, (4.8.3) seemsadequate to study the evolution of a process in a metric space making it possibleto obtain results of calculus and differential equations without employing anyconcept of derivative or requiring that the underlying metric space be linear.

This motivates the following definition.Let (X, d) be a complete metric space and F : [0, T ] × R+ × X → X. We

shall consider (X,F ) as a metric differential equation in the following sense: Afunction u : [0, T ] → X is a solution of the metric differential equation (X,F )with initial condition (4.8.2) if u satisfies (4.8.3) and u(0) = u0.

This conception is related to the concept of quasi-differential equations andto mutations in a metric space. See Melnik [1], Panasyuk [1], Plotnikov [1],Aubin [1,2].


Of course, if X = Rn and F is given by (4.8.4) with f continuous, thenwe have a classical ordinary differential equation, i.e., a continuous dynamicalsystem.

If X = En, note that for a function f : [0, T ] × En → En, (4.8.4) makessense, and we can reconsider a fuzzy differential equation as a metric differentialequation in the metric space En. As we know that in En the difference of twoelements is not always well defined which precludes us from using (4.8.5) tostudy fuzzy differential equations.

We recall that a fuzzy subset of Rn is just a map

u : Rn → [0, 1]

where u(x) is the grade of membership of x ∈ Rn to the fuzzy set. For eachα ∈ (0, 1] the α− level set [u]α = x ∈ Rn : u(x) ≥ α. The support of u ,denoted by [u]0 is the closure of the union of all its level sets. Of course, anyclassical subset A ⊂ Rn is identified with its characteristic function χA. The setof normal, fuzzy convex, upper semicontinuous functions, with compact supportis denoted by En.

It is possible to define addition in En levelwise:

u, v ∈ En, [u+ v]α = [u]α + [v]α, α ∈ [0, 1].

For c 6= 0, scalar multiplication is defined also levelwise:

u ∈ En, [cu]α = c[u]α.

Note that it is not possible to define 0u levelwise since 0u = χφ 6∈ En. Wedefine u − v = u + (−1)v. Observe that u + v = χ0 implies u = −v, butu = −v does not imply, in general, that u + v = χ0. Indeed, for example, foru = χ[0,1], −u = χ[−1,0], and u− u = χ[−1,1].

The distance between elements of En is given by the supremum of the Haus-dorff distance between the level sets:

u, v ∈ En, D0[u, v] = supα∈[0,1]D[[u]α, [v]α].

Thus, (En, D0) is a complete metric space (see Lakshmikantham and Mohapatra[1]). Moreover, for u, v, w, z ∈ En and c, c′ 6= 0, we have

D0[cu, cv] = |c|D0[u, v],D0[u+ w, v +w] = D0[u, v],D0[u+w, v + z] = D0[u, v] +D0[w, z],

D0[cu, c′u] = |c− c′|D0[u, χ0].

If a ∈ Rn, then χa ∈ En, and for a, b ∈ Rn. D0[χa, χb] = |a− b|.Now, consider a map f : En → En.


Definition 4.8.1 For fixed u, v ∈ En we say that f is differentiable at u in thedirection v if there exists w ∈ En such that

limh→0+

D0[f(u + hv), f(u) + hw]h

= 0.

We say that w is the derivative of f at u in the direction v and write w = f ′(u)v.

Definition 4.8.2 For a function u : [0, T ] → En and t ∈ [0, T ) we say that uis differentiable at t if there exists w ∈ En such that

limh→0+

D0[u(t+ h), u(t) + hw]h

= 0,

and we write DHu(t) = w.

Example 4.8.1 Let u0 ∈ En and take u : [0, T ] → En, u(t) = u0. Then,u′(t) = χ0 for every t ∈ [0, T ). Indeed,

limh→0+

D0[u(t+ h), u(t) + hχ0]h

= limh→0+

D0[u0, u0 + hχ0]h

= limh→0+

D0[u0, u0]h

= 0.

Example 4.8.2 For u0, u1 ∈ En let u(t) = u0 + tu1. Then, DHu(t) = u1 since

limh→0+

D0[u(t+ h), u(t) + hu1]h

= limh→0+

D0[u0 + (t+ h)u1, u0 + tu1 + hu1]h

= 0.

Example 4.8.3 Let f : [0, T ] → Rn be differentiable. Define

f : [0, T ] → En, f (t) = χf(t).

Then f is differentiable and f ′(t) = χf ′(t).

Example 4.8.4 For any λ > 0, u0 ∈ En, the function u(t) = eλtu0 satisfiesDHu(t) = λu(t). Indeed,

limh→0+

1h

(D0[u(t+ h), u(t) + hλeλtu0])

= limh→0+

1h

(D0[eλ(t+h)u0, eλtu0 + hλeλtu0])

= limh→0+

eλt

hD0[eλhu0, (1 + λh)u0]

= limh→0+

eλt

h(eλh − (1 + λh))D0[u0, χ0] = 0.


Definition 4.8.3 Given v : [0, T ] → En, a primitive of v is a function u :[0, T ] → En such that DHu(t) = v(t) a.e. [0, T ], i.e. for almost all t ∈ [0, T ] wehave

limh→0+

D0[u(t+ h), u(t) + hv(t)]h

= 0.

If u is a primitive of v satisfying the initial condition 4.8.2., we say that u is aprimitive starting at u0.

For example, a primitive of v(t) = etu0 is itself.

Lemma 4.8.1 If u : [0, T ] → En is differentiable at t, then u is right continuousat t.

Proof Let DHu(t) = v(t). For every ε > 0, there exists δ > 0 such that forh ∈ (0, δ), we have

D0[u(t+ h), u(t)] ≤ D0[u(t+ h), u(t) + hv(t)] +D0[u(t) + hv(t), u(t)]

≤ εh+D0[hv(t), χ0] = εh+ hD0[v(t), χ0].

This shows the right continuity of u at t.

Lemma 4.8.2 Suppose that v ∈ [0, T ] → En is piecewise constant, then vhas a primitive starting at u0. Moreover, if t0 = 0 < t1 < t2 < · · · < tk =T, and v(t) = vi ∈ En for t ∈ (ti, ti+1), i = 0, 1, ......., k − 1, it is possibleto construct a Lipschitz continuous primitive with Lipschitz constant equal tomax0≤i≤k−1D0[vi, χ0].

Proof Let u(0) = u0 and for t ∈ (0, t1), define u(t) = u0+ tv0. Thus, DHu(t) =v0 = v(t) for every t ∈ [0, t1). For i ≥ 1, set u(ti) = u(ti−1) + (ti − ti−1)vi−1,and for t ∈ (ti, ti+1), u(t) = u(ti) + (t− ti)vi. It is clear that DHu(t) = v(t) forevery t ∈ (ti, ti+1), i = 0, 1....., k− 1. Now, if t, τ ∈ [ti, ti+1], then

D0[u(t), u(τ )] = D0[u(ti) + (t− ti)vi, u(ti) + (τ − ti)vi]= D0[(t− ti)vi, (τ − ti)vi]= |t− τ |D0[vi, χ0].

For arbitrary t, τ ∈ [0, T ], suppose that t ∈ [ti, ti+1], and τ ∈ [tj, tj+1], i < j.Hence,

D0[u(t), u(τ )] ≤ D0[u(t), u(ti+1)] +D0[u(ti+1), u(ti+2)] + · · ·· · ·+D0[u(tj−1), u(tj)] +D0[u(tj), u(τ )]

≤ |t− ti+1|D0[vi, χ0] + |ti+2 − ti+1|D0[vi+1, χ0] + · · ·· · ·+ |tj − tj−1|D0[vj−1, χ0] + |τ − tj|D0[vj, χ0]

≤ max0≤i≤(k−1)

D0[vi, χ0]|τ − t|.


Lemma 4.8.3 Let v1, v2 : [0, T ] → En with primitives u1, u2. Suppose that thefunction s ∈ [0, T ] → D0[v1(s), v2(s)] is integrable on [0, T ] (for example if v1and v2 are piecewise continuous). Then,

D0[u1(t), u2(t)] ≤ D0[u1(0), u2(0)] +∫ t

0

D0[v1(s), v2(s)] ds. (4.8.7)

Proof Define ξ(t) = D0[u1(t), u2(t)], t ∈ [0, T ]. We have

ξ(t + h) − ξ(t)≤ D0[u1(t + h), u1(t) + hv1(t)] +D0[u1(t) + hv1(t), u2(t) + hv1(t)]

+ D0[u2(t) + hv1(t), u2(t) + hv2(t)] +D0[u2(t) + hv2(t), u2(t+ h)]− D0[u1(t), u2(t)]= D0[u1(t + h), u1(t) + hv1(t)] +D0[u1(t), u2(t)] + hD0[v1(t), v2(t)]

+ D0[u2(t) + hv2(t), u2(t+ h)] −D0[u1(t), u2(t)].

Hence,ξ(t + h) − ξ(t)

h≤ 1hD0[u1(t + h), u1(t) + hv1(t)]

+D0[v1(t), v2(t)] +1hD0[u2(t) + hv2(t), u2(t+ h)].

Therefore, D+ξ(t) ≤ D0[v1(t), v2(t)], t ∈ [0, T ], and

ξ(t) ≤ ξ(0) +∫ t

0

D0[v1(s), v2(s)] ds.

Corollary 4.8.1 (Uniqueness) : For a given initial state we have at most onecontinuous primitive.

Lemma 4.8.4 Let u be a continuous primitive of v. Then it satisfies the fol-lowing inequality:

1hD0[u(t+ h), u(t) + hv(t)] ≤ 1

h

∫ h

0

D0[v(t + s), v(t)] ds. (4.8.8)

Moreover, assume that supτ∈[0,T ] D0[v(τ ), χ0] = k < +∞, then u is Lipschitzcontinuous with Lipschitz constant k.

Proof Fix t0 ∈ [0, T ], and note that

µ : [0, T ] → En, µ(t) = u(t0) + tv(t0)

is a primitive of the constant function v(t0) with µ(0) = u(t0). Also the function

ut0 : [0, T − t0] → En, ut0(t) = u(t+ t0)


is a primitive of vt0 : [0, T − t0] → En, vt0(t) = v(t + t0) with ut0(0) = u(t0).Using (4.8.7) we get

D0[µ(h), ut0(h)] ≤ D0[µ(0), ut0(0)] +∫ h

0

D0[v(t0), v(s + t0)] ds

and

D0[u(t0) + hv(t0), u(h+ t0)] ≤∫ h

0

D0[v(t0 + s), v(t0)] ds

Dividing by h we obtain (4.8.8).Now, the constant function u(t) is a primitive of the constant function χ0

starting at u(t). Let t′ > t. hence,

D0[u(t′), u(t)] = D0[ut(t′ − t), u(t)] ≤∫ t′−t

0

D0[vt(s), χ0] ds

=∫ t′−t

0

D0[v(t+ s), χ0] ds ≤ k.(t′ − t).

We now prove the main result of the Section: Any continuous function has aprimitive.

Theorem 4.8.1 Let v : [0, T ] → En be continuous. Then there exists a uniqueprimitive of v starting at a given u0.

Proof For ε > 0 there exists δ > 0 such that D0[v(t), v(s)] < ε whenever|t − s| < δ since v is uniformly continuous on [0, T ]. Take m > T

δ, h = T

m, and

for i = 0, 1, 2....,m− 1 define the piecewise continuous function

vm : [0, T ] → En, vm(t) = v(ih), t ∈ (ih, (i+ 1)h).

For any t ∈ [0, T ], let t ∈ (ih, (i + 1)h), then

D0[vm(t), v(t)] = D0[v(ih), v(t)] < ε,

since |t− ih| < h = Tm< δ. Also, for every m = 1, 2, ..... and t ∈ [0, T ] we have

D0[vm(t), χ0] ≤ supτ∈[0,T ]D0[v(τ ), χ0] = k < ∞.

In view of Lemma 4.8.2., let um be the primitive of vm starting at u0. ByLemma 4.8.4, um is Lipschitz continuous with Lipschitz constant k. Now, usingLemma 4.8.3., for l,m > T

δand t ∈ [0, T ] we have

D0[um(t), ul(t)] ≤∫ t

0

D0[vm(s), vl(s)] ds ≤ 2εt ≤ 2Tε. (4.8.9)

In consequence, for every t ∈ [0, T ] we see that the sequence um(t)∞m=1 is aCauchy sequence in En. Therefore, there exists limm→∞ um(t) = u(t). Passing


to the limit when l → ∞ in (4.8.9) we see that um∞m=1 converges uniformlyto u.

Now, for m = 1, 2..... consider the functions

µm : [0, T ] → En, µm(t) = um(t) + hvm(t),

andµ : [0, T ] → En, µ(t) = u(t) + hv(t).

Using (4.8.8) we can write

D0[um(t + h), um(t) + hvm(t)] ≤∫ h

0

D0[vm(t+ s), vm(t)] ds.

On the other hand,

limm→∞

D0[um(t+ h), um(t) + hvm(t)] = D0[u(t+ h), u(t) + hv(t)],

and

limm→∞

∫ h

0

D0[vm(t + s), vm(t)] ds =∫ h

0

D0[v(t+ s), v(t)] ds.

Hence,1hD0[u(t+ h), u(t) + hv(t)] ≤ 1

h

∫ h

0

D0[v(t + s), v(t)] ds.

Now, v is uniformly continuous on [0, T ] and hence

limh→0+

1h

∫ h

0

D0[v(t + s), v(t)] ds = 0,

uniformly on t ∈ [0, T ]. Therefore,

limh→0+

1hD0[u(t+ h), u(t) + hv(t)] = 0,

uniformly on t ∈ [0, T ], and u is a primitive of v. For f : En → En, considerthe fuzzy differential equation

DHu(t) = f(u(t)), t ∈ [0, T ], (4.8.10)

with the fuzzy initial condition

u(0) = u0 ∈ En. (4.8.11)

Definition 4.8.4 We say that u : [0, T ] → En is a solution of (4.8.10)-(4.8.11)if u is a primitive of f(u) starting at u0.

For u ∈ C([0, T ], En), define the function

Fu : [0, T ] → En, [Fu](t) = f(u(t)),

and denote by Gu the unique continuous primitive of Fu starting at u0. Hence,a function u ∈ C([0, T ], En) is a solution of the initial value problem (4.8.10)-(4.8.11) if and only if u coincides with Gu.


Theorem 4.8.2 Suppose that f : En → En is such that there exists k ≥ 0 with

D0[f(x), f(y)] ≤ kD0[x, y], x, y ∈ En. (4.8.12)

Then the initial fuzzy problem (4.8.10),(4.8.11) has a unique solution.

Proof . In the space C([0, T ], En), consider the metric

D[u1, u2] = supt∈[0,T ]D0[u1(t), u2(t)]e−kt.

Thus, using Lemma 4.8.3, we have for any t ∈ [0, T ],

D0[[Gu1](t), [Gu2](t)] ≤∫ t

0

D0[[Fu1])(s), [Fu2](s)] ds

=∫ t

0

D0[f(u1(s)), f(u2(s))] ds

≤ k

∫ t

0

D0[u1(s), u2(s)] ds

= k

∫ t

0

ekse−ksD0[u1(s), u2(s)] ds

≤ k

∫ t

0

eksD[u1, u2] ds = (ekt − 1)D[u1, u2].

HenceD[Gu1, Gu2] ≤ [1− e−kT ] D[u1, u2],

and G is a contraction and has a unique fixed point.


The preliminaries introduced for the formulation of fuzzy differential equationsand the basic results reported for such equations in Section 4.2, are adaptedfrom Lakshmikantham and Mohapatra [1]. For further results on basic fuzzyset theory see Kaleva [1, 2], Seikkala [1], Vorobiev and Seikkala [1], and O’Regan,Lakshmikantham and Nieto [1]. For the results of Section 4.3 concerning sta-bility criteria in terms of Lyapunov-like functions with necessary comparisonprinciples see Gnana Bhaskar, Lakshmikantham, and Vasundhara Devi [1]. Toeliminate the possible undesirable part of the solutions, the Hukuhara differencein initial conditions is employed suitably extending the ideas given Lakshmikan-tham, Leela and Vasundhara Devi [1]. For the suggestion to reduce FDEs toa sequence of multivalued differential equations, see Hullermeier [1]. See Di-amond[1], Diamond and Watson [1] and Lakshmikantham and Mohapatra [1]where Hulermeier’s approach is exploited fruitfully. For recent results in thisconnection see Agarwal, O’Regan, and Lakshmikantham [1]. For results in mul-tivalued differential equations see Deimling [1]. The interconnection between


FDEs and SDEs sketched in Section 4.4, dealing with the continuous case, istaken from Lakshmikantham, Leela and Vatsala [1]. See Lakshmikantham andTolstonogov [1] for similar results in USC case described in Section 4.5 and alsoTolstonogov [1]. The introduction to the theory of impulsive FDEs in Section4.6 and hybrid FDEs in Section 4.7 and the corresponding results reported areadapted from Lakshmikantham and Vatsala [2]. See Lakshmikantham and Ni-eto [1] for the formulation of differential equations in metric space and relatedresults for FDEs studied in Section 4.8. For other kinds of formulation of dif-ferential equations in metric spaces, see also Aubin [1], Kloeden, Sadovsky andVasiyeva [1], Melnik [1], Panasyuk [1] and Plotnikov [1].


Chapter 5

Miscellaneous Topics

5.1 Introduction

This chapter is devoted to the introduction of several topics in the frameworkof set differential equations, that need further investigation.

We begin Section 5.2 with set differential equations involving impulsive ef-fects and prove certain basic results. In Section 5.3, we consider impulsive setdifferential equations and develop the fruitful monotone iterative technique ina general set up so as to include several special important results.

Section 5.4 is dedicated to the investigation of set differential equations withdelay. Here we provide some fundamental results for such equations. In Sec-tion 5.5, we introduce impulses into the study of SDEs with delay and considersuitable interesting results. The discussion of set difference equations forms thecontent of Section 5.6. Employing Causal or nonanticipative maps of Volterratype, we discuss, in Section 5.7, set differential equations involving such mapsand extend appropriate basic results to such equations. Lyapunov-like functionswhose values are in the metric space (Kc(Rd), D) are introduced in Section 5.8.A necessary comparison theorem in terms of such Lyapunov-like functions isproved using suitable partial order, to discuss qualitative properties of solutionsof set differential systems. The general set up considered for Lyapunov-like func-tions covers not only existing single, vector, matrix and cone-valued Lyapunovfunction theory, but also provides a very general framework for further progress.

Since all through this book, we did employ the metric space (Kc(Rn), D) forthe investigation of several situations, in Section 5.9 we indicate how one canextend most of the results discussed to the metric space (Kc(E), D), where Eis any real Banach space with suitable modifications demanded by the infinitedimensional framework. Finally, we provide notes and contents in Section 5.10.

139

140 CHAPTER 5. MISCELLANEOUS TOPICS

5.2 Impulsive Set Differential Equations

Many evolution processes are characterized by the fact that at certain momentsof time they experience a change of state abruptly. These processes are subjectto short term perturbations whose duration is negligible in comparison to theduration of the process. Consequently, it is natural to suppose that these per-turbations act instantaneously, that is, in the form of impulses. Thus impulsivedifferential equations have become a natural description of observed evolutionphenomena of several real world problems. The study of impulsive differen-tial equations has been growing as a well deserved discipline, and a systematictreatment of the theory is available. There has been much progress in the in-vestigation of impulsive dynamic systems of other kinds.

In this section, we shall extend the ideas of impulsive ordinary differentialequations, to set differential equations and investigate some basic properties.

We first introduce the following notation.

(i) Let tk be a sequence such that 0 ≤ t1 < t2 < . . . < tk < . . . withlimk→∞ tk = ∞.

(ii) F ∈ PC[R+ × Kc(Rn), Kc(Rn)], implies F : R+ × Kc(Rn) → Kc(Rn)is continuous in (tk−1, tk] × Kc(Rn), for each k = 1, 2, ..., and for eachU ∈ Kc(Rn), k = 1, 2, . . . , lim(t,Y )→(t+k ,U) F (t, Y ) = F (t+k , U ) exists.

(iii) g ∈ PC[R+ × R+ , R ] if g : (tk−1, tk] × R+ → R is continuous and foreach w ∈ R+, lim(t,z)→(t+k ,w) g(t,z) = g(t+k , w) exists.

(iv) h ∈ PC1[R+,Kc(Rn)] means that h ∈ PC[R+, Kc(Rn)] and is differen-tiable in each interval (tk−1, tk).

Now, consider the impulsive set differential equation (ISDE) given by

DHU = F (t, U ), t 6= tk,U (t+k ) = Ik(U (tk)), t = tk,U (t0) = U0 ∈ Kc(Rn),

(5.2.1)

where F ∈ PC[R+ × Kc(Rn), Kc(Rn)], Ik : Kc(Rn) → Kc(Rn) and tkis a sequence of points such that 0 ≤ t0 < t1 < t2 < . . . < tk < . . . withlimk→∞ tk = ∞.

Definition 5.2.1 By a solution U (t, t0, U0) of the impulsive set differentialequation (5.2.1), we mean a piecewise continuous function on [t0,∞) whichis left continuous in each subinterval (tk, tk+1] and is given by

U (t, t0, U0) =

U0(t, t0, U0), t0 ≤ t ≤ t1,U1(t, t1, U+

1 ), t1 < t ≤ t2,...

...Uk(t, tk, U+

k ), tk < t ≤ tk+1,...

...

(5.2.2)

5.2 IMPULSIVE SET DIFFERENTIAL EQUATIONS (SDES) 141

where Uk(t, tk, U+k ) is the solution of the set differential equation

DHU = F (t, U ), U (t+k ) = U+k = Ik(U (tk)).

We begin with a basic differential inequality result, which is a useful tool instudying monotone method in the impulsive setup later.


(i) V,W ∈ PC1[R+,Kc(Rn)], F ∈ PC[R+ × Kc(Rn),Kc(Rn)]. F (t,X) ismonotone nondecreasing in X for each t ∈ R+ and

DHV ≤ F (t, V ), t 6= tk,

V (t+k ) ≤ Ik(V (tk)), t = tk,

and DHW ≥ F (t,W ), t 6= tk,

W (t+k ) ≥ Ik(W (tk)), t = tk, k = 1, 2, · · · ;

(ii) Ik : Kc(Rn) → Kc(Rn), Ik(U ) is nondecreasing in U for each k;

(iii) for any X,Y ∈ Kc(Rn) such that X ≥ Y, t ∈ R+,

F (t,X) ≤ F (t, Y ) + L(X − Y ) for some L > 0.

Then V (0) ≤W (0) implies V (t) ≤W (t), t ≥ 0.

Proof Consider J = [0, t1]. Let V (0) ≤ W (0). Then applying Theorem 2.5.1,we have V (t) ≤ W (t) on J . This implies V (t1) ≤ W (t1) and since I1(U ) isnondecreasing,

V (t+1 ) ≤ I1(V (t1)) ≤ I1(W (t1)) ≤ W (t+1 ).

Thus V (t+1 ) ≤ W (t+1 ). Next set J = (t1, t2], and apply Theorem 2.5.1, to get

V (t) ≤ W (t) on (t1, t2].

Proceeding as before, we can obtain the conclusion of the theorem.

Corollary 5.2.1 Let V,W ∈ PC1[R+,Kc(Rn)], p, σ ∈ C[R+,Kc(Rn)]

DHV ≤ σ, t 6= tk ,

V (t+k ) ≤ p(tk), t = tk ,

and

DHW ≥ σ, t 6= tk ,

W (t+k ) ≥ p(tk), t = tk .

Then V (t0) ≤ W (t0) implies

V (t) ≤ W (t), t ≥ t0.


We first prove an existence theorem for impulsive set differential equationswith fixed moments of impulse.

Theorem 5.2.2 Suppose that

(i) F ∈ PC[R+ ×Kc(Rn),Kc(Rn)],

(ii) D[F (t, U ), θ] ≤ g(t,D(U, θ)], t 6= tk, where g ∈ PC[R2+,R+], g(t, w) is

nondecreasing in (t, w),

(iii) D[U (t+k ), θ] ≤ ψk(D[U (tk), θ]), t = tk,

(iv) ψk(w) is a nondecreasing function of w,

(v) r(t, t0, w0) is the maximal solution of the impulsive scalar differential equa-tion

w′ = g(t, w), t 6= tk,w(t+k ) = ψk(w(tk)), t = tk,w(t0) = w0,

(5.2.3)

existing on [0,∞),

(vi) tk is a sequence of points of impulse with 0 ≤ t0 < t1 < t2 < . . .< tk < . . . and limk→∞ tk = ∞.

Then there exists a solution for the ISDE (5.2.1).

Before we proceed with the proof, let us define the notion of a maximalsolution of (5.2.3).

Definition 5.2.2 By a maximal solution r(t, t0, w0) of the impulsive differentialequation (5.2.3), we mean the solution r(t, t0, w0) defined by By a maximalsolution r(t, t0, w0) of the impulsive differential equation (5.2.3), we mean thesolution r(t, t0, w0) defined by

r(t, t0, w0) =

r0(t, t0, r0), t0 ≤ t ≤ t1,r1(t, t1, r+1 ), t1 < t ≤ t2,

......

rk(t, tk, r+k ), tk < t ≤ tk+1,...

...

satisfying the relation

w(t, t0, w0) ≤ r(t, t0, w0), t ∈ R+,

for every solution w(t, t0, w0) of (5.2.3), where r+k = ψk(rk−1(tk)).


We now present the proof of Theorem 5.2.2.Proof Set J0 = [t0, t1] and restrict F to J0 ×Kc(Rn).

Consider the set differential equation given by

DHU = F (t, U )

U (t0) = U0 on J0.

Then the hypothesis of Theorem 2.8.2 is satisfied with D[U0, θ] = w0. Hencethere exists a solution U0(t, t0, U0), for the set differential equation such thatD[U0(t), θ] ≤ r(t, t0, w0) on J0.

For t = t1, U0(t1) = U0(t1, t0, U0). Set U+1 = U0(t+1 ) = I1(U0(t1)). From

hypothesis,

D[U+1 , θ] = D[I1(U0(t1)), θ]

≤ ψ(D[U0(t1), θ])≤ ψ(r(t1)) = r(t+1 ).

Put J1 = [t1, t2] and consider the set differential equation

DHU = F (t, U ), t ∈ J1,

U (t+1 ) = U+1 .

Then again, restricting F to the domain J1×Kc(Rn), the hypothesis of Theorem2.8.2 is satisfied and thus there exists a solution U1(t, t1, U+

1 ) for t ∈ J1 satisfyingthe set differential equation restricted to J1. We have

U1(t2) = U1(t2, t1, U+1 ) and U1(t+2 ) = I2(U (t2)).

Set U+2 = U1(t+2 )and J2 = [t2, t3]

Now repeating the above process, we obtain the existence of a solution ofthe impulsive set differential equation.

We next give a basic comparison theorem in the impulsive set differentialequation set up.


(i) F ∈ PC[R+ ×Kc(Rn), Kc(Rn)];

(ii) for t ∈ R+, t 6= tk, U, V ∈ Kc(Rn),

D[F (t, U ), F (t, V )] ≤ g(t,D(U, V )], (5.2.4)

where g ∈ PC[R2+,R+];

(iii) D[U (t+k ), V (t+k )] ≤ ψk(D[U (tk), V (tk)]), where ψk(w) is a nondecreasingfunction of w.


Further, suppose that the maximal solution r(t, t0, w0) of the impulsive scalardifferential equation (5.2.3) exists on R+.

If U (t), V (t) are any two solutions of ISDE (5.2.1) through U (t0) = U0 andV (t0) = V0, U0, V0 ∈ Kc(Rn) on J respectively, then we have

D[U (t), V (t)] ≤ r(t, t0, w0), t ∈ R+,


Proof We set J0 = [t0, t1] and restrict the domain of F to J0 ×Kc(Rn). ThenF is continuous on this domain and further the hypothesis of Theorem 2.2.1 issatisfied. Hence we can conclude that

D[U (t), V (t)] ≤ r(t, t0, w0], t ∈ J0,

which implies D[U (t1), V (t1)] ≤ r(t1, t0, w0].Now using the hypothesis for t = t+1 , we have

D[U (t+1 ), V (t+1 )] ≤ ψ1(D[U (t1), V (t1)])≤ ψ1[r(t1, t0, w0)]= r(t+1 ),

since ψ1 is a nondecreasing function. Thus

D[U (t+1 ), V (t+1 )] ≤ r(t+1 ). (5.2.5)

Next, set J1 = [t1, t2], domF = J1×Kc(Rn). Then using the inequalities (5.2.4),(5.2.5) and applying Theorem 2.2.1, we conclude that

D[U (t), V (t)] ≤ r(t, t0, w0), t ∈ J1.

Repeating the above process, the conclusion of the theorem is obtained.We now state a corollary which will be a useful tool in our work.

Corollary 5.2.2 Assume that,

(i) F ∈ PC[R+ ×Kc(Rn),Kc(Rn)];

(ii) D[F (t, U ), θ] ≤ g(t,D[U, θ]), t 6= tk, where g ∈ PC[R2+, R];

(iii) D[U (t+k ), θ] ≤ ψk(D[U (tk), θ]), t = tk, and ψk(w) is nondecreasing in w;

(iv) r(t, t0, w0) is the maximal solution of the impulsive scalar differential equa-tion (5.2.3).

Then, if D[U0, θ] ≤ w0, we have

D[U (t), θ] ≤ r(t, t0, w0), t ∈ J.


We next prove the comparison theorem using Lyapunov-like functions. Thiswill help us to study stability criteria for ISDE (5.2.1). Before proceeding fur-ther, we make the following assumption.

Let V : R+ ×Kc(Rn) → R+. We say that V belongs to the class V0 if

(i) V is continuous in (tk−1, tk] × Kc(Rn) and for each U ∈ Kc(Rn), k =1, 2, . . . , lim(t,Y )→(t+k ,U) V (t, Y ) = V (t+k , U ) exists,

(ii) |V (t, A) − V (t, B)| ≤ LD[A,B] for A,B ∈ Kc(Rn), t ∈ R+, where L isthe local Lipschitz constant.

Theorem 5.2.4 Assume that,

(i) V ∈ V0;

(ii) for t ∈ R+, U ∈ Kc(Rn),

D+V (t, U ) ≤ g(t, V (t, U )), t 6= tk,

where g ∈ C[R2+, R];

(iii) V (t+k , U (t+k )] ≤ ψk(V (tk, U (tk))), t = tk, where ψk(w) is nondecreasing inw.

Further, suppose that r(t, t0, w0) is the maximal solution of the impulsive scalardifferential equation (5.2.3) existing on R+. Then, if U (t) = U (t, t0, U0) isany solution of (5.2.1) existing on R+ such that V (t0, U0) ≤ w0, we have

V (t, U (t)) ≤ r(t, t0, w0), t ∈ R+.

Proof Set J0 = [t0, t1]. Applying Theorem 3.2.1 on J0 ×Kc(Rn), we obtain

V (t, U (t)) ≤ r(t, t0, w0), t ∈ J0.

At t = t1, V (t1, U (t1)) ≤ r(t1, t0, w0). Since ψ1 is a nondecreasing function

ψ1(V (t1, U (t1))) ≤ ψ1(r(t1, t0, w0) = r(t+1 ).

Hence proceeding as in the earlier theorems, step by step, we can prove theconclusion of the theorem.

We now define stability properties of the null solution of an impulsive setdifferential equation (5.2.1).

Definition 5.2.3 Let U (t) = U (t, t0, U0) be any solution of ISDE (5.2.1). Thenthe trivial solution U (t) ≡ θ is said to be

(S1) stable, if for each ε > 0 and t0 ∈ R+, there exists a δ = δ(t0, ε) > 0 suchthat D[U0, θ] < δ implies D[U (t), θ] < ε, for t ≥ t0.


(S2) attractive, if for each ε > 0 and t0 ∈ R+ there exist δ0 = δ0(t0) > 0 and aT = T (t0, ε) > 0 such that D[U0, θ] < δ0 implies

D[U (t), θ] < ε, for t ≥ t0 + T.

The other definitions can be formulated similarly.We denote B(U0, b) = U ∈ Kc(Rn) : D[U,U0] ≤ b,

K = σ ∈ C[R+,R+] : σ(0) = 0 and σ(t) is strictly increasing in t.

The following theorem connects the stability properties of the trivial solutionof ISDE (5.2.1) with the stability properties of the trivial solution of the impul-sive scalar differential equation (5.2.3) through the Lyapunov-like function.

In order to obtain the trivial solution for ISDE (5.2.1) we assume thatF (t, θ) ≡ θ and Ik(θ) = θ for all k.



(i) V : R+ ×B(θ, b) → R+, V ∈ V0,

D+V (t, U ) ≤ g(t, V (t, U )), t 6= tk,

where g : R+ × R+ → R+, g(t, 0) ≡ 0 and g satisfies A0(ii).

(ii) there exists b0 > 0 such that U ∈ B(θ, b0) implies that Ik(U ) ∈ B(θ, b) forall k, and

V (tk, Ik(U (tk))) ≤ ψk(V (tk, U (tk))), U ∈ B(θ, b0)

and ψk : R+ → R+ is nondecreasing, ψk(0) = 0;

(iii) b(D[U, θ]) ≤ V (t, U ) ≤ a(D[U, θ]), where a, b ∈ K.

Then the stability properties of the trivial solution of the impulsive scalardifferential equation (5.2.3) imply the corresponding stability properties of thetrivial solution of ISDE (5.2.1).

Proof Let 0 < ε < b∗ = min(b0, b) and t0 ∈ R+ be given. Suppose the trivialsolution of (5.2.3) is stable. Then given b(ε) > 0 there exists a δ1(t0, ε) > 0 suchthat

0 ≤ w0 < δ1 implies w(t, t0, u0) < b(ε), t ≥ t0,

where w(t, t0, u0) is any solution of (5.2.3).Let w0 = a(D[U0, θ]) and choose δ2 = δ2(ε) such that a(δ2) < δ1.Define δ = min(δ1, δ2). With this δ, we claim that if D[U0, θ] < δ then

D[U (t), θ] < ε, t ≥ t0, where U (t) = U (t, t0, U0) is any solution of ISDE (5.2.1).Suppose this does not hold.

Then there exists a solution U (t) = U (t, t0, U0) of ISDE (5.2.1) withD[U0, θ] < δ and a t∗ > t0 such that tk < t∗ ≤ tk+1 for some k satisfyingε ≤ D[U (t∗), θ] and D[U (t), θ] < ε for t0 ≤ t ≤ tk. Since 0 < ε < b0 fromcondition (ii) we have

D[U (t+k ), θ] = D[Ik(U (tk)), θ] < b,

and D[U (tk), θ] < ε.Hence, we can find a t0 such that tk < t0 ≤ t∗ satisfying

ε ≤ D[U (t0), θ] < b.

Setting m(t) = V (t, U (t)) for t0 ≤ t ≤ t0, and using the hypothesis (i) and (ii),we get from Theorem 5.2.4, the estimate

V (t, U (t)) ≤ r(t, t0, a(D[U0, θ])), t0 ≤ t ≤ t0,

where r(t, t0, w0) is the maximal solution of impulsive scalar differential equation(5.2.3).


Now consider

b(ε) ≤ b(D[U (t0), θ]) ≤ V (t0, U (t0))≤ r(t0, t0, a(D[U0, θ])) < b(ε),

which is a contradiction. This proves that U (t) ≡ θ of ISDE (5.2.1) is stable.Next, if we suppose that w ≡ 0 of (5.2.3) is uniformly stable. Then clearly

δ is independent of t0 and this gives the uniform stability of U ≡ θ of the ISDE(5.2.1).

Let us suppose that w ≡ 0 of (5.2.3) is asymptotically stable. This impliesthat U ≡ θ of ISDE (5.2.1) is stable. Hence, set ε = b∗ and δ∗0 = δ(t0, b∗), wehave

D[U0, θ] < δ∗0 implies D[U (t), θ] < b∗, t ≥ t0. (5.2.6)

To prove attractivity, we let 0 < ε < b∗ and t0 ∈ R+. Since w ≡ 0 of (5.2.3)is attractive, given b(ε) > 0 and t0 ∈ R+, there exists a δ10 = δ10(t0) > 0 and aT = T (t0, ε) > 0 such that 0 ≤ w0 < δ10 implies

w(t, t0, w0) < b(ε) for t ≥ t0 + T.

Then using (5.2.6) and reasoning as in the earlier case, we get

V (t, U (t)) ≤ r(t, t0, a(D[U0, θ])).

Thus we get

b(D[U (t), θ]) ≤ V (t, U (t)) ≤ r(t, t0, a(D[U0, θ]))

< b(ε), t ≥ t0 + T,

which impliesD[U (t), θ] < ε, t ≥ t0 + T.

Thus U ≡ θ is attractive and hence asymptotically stable.We now consider the example given in 3.4.3 and illustrate how impulses

control the behavior of the solutions and as such the notion of using Hukuharadifference in initial values becomes redundant in this case.

Example 5.2.1 Consider the set differential equation,

DHU = (−1)U, U (0) = U0 ∈ Kc(R). (5.2.7)

Since the values of the solution U (t) of (5.2.7) are intervals, the equation (5.2.7)can be written as

[u′1, u′2] = (−1)U = [−u2,−u1], (5.2.8)

where U = [u1, u2] and U0 = [u10, u20]. Recall that the solution is given by

u1(t) = 12 [u10 + u20]e−t + 1

2 [u10 − u20]et,

u2(t) = 12 [u20 + u10]e−t + 1

2 [u20 − u10]et.

(5.2.9)


If U0 = [u0, u0], that is U0 is a singleton, we get from (5.2.9),

U (t) = [u1(t), u2(t)] = u0e−t, t ≥ 0.

In this situation, the impulses have no role to play and hence we can take

u(t+k ) = u(tk), k = 1, 2, . . . .

If, on the other hand, we take U0 = [−u0, u0], then (5.2.9)reduces to

U (t) = [−u0, u0]et, t ≥ 0.

Suppose that we choose the impulses as

U (t+k ) = dkU (tk), for t = tk, (5.2.10)

where the dk’s satisfy 0 < dk < 1 and

tk+1 + ln dk ≤ tk for all k, (5.2.11)

then the solution of the corresponding ISDE (5.2.7),(5.2.10) is given by

U (t) = U0 Π0<tk<t dk et, t ≥ 0, (5.2.12)

we know that D[U (t), θ] = ‖U (t)‖, therefore

‖U (t)‖ ≤ ‖U0‖Π0<tk<t dk et, t ≥ 0. (5.2.13)

Choosing δ = ε2 e−t1 and using (5.2.11) it follows that ‖U (t)‖ < ε, t ≥ 0

provided that ‖U0|| < δ. Hence the stability of the trivial solution of (5.2.7),(5.2.10) follows.

To prove asymptotic stability, we strengthen the assumption (5.2.11) to

tk+1 + ln α dk ≤ tk for all k, where α > 1.

Then dk ≤ 1α exp[tk − tk+1]. Using this estimate on dk in (5.2.13), we see from

the relationlim

k→∞‖U (t)‖ = 0.

Thus, the trivial solution U ≡ θ of the ISDE (5.2.7), (5.2.10) is asymptoticallystable.

Remark 5.2.1 If U , F and I in (5.2.1) are single-valued mappings then theHukuhara derivative and integral reduce to the ordinary derivative and integral.Consequently, the impulsive set differential equation (5.2.1) reduces to the cor-responding ordinary impulsive differential system. Thus the results obtained inthis section include the corresponding results of such equations as a very specialcase.


5.3 Monotone Iterative Technique

We develop the monotone technique for ISDE corresponding to the various no-tions of upper and lower solutions of SDE 2.5.3. We can define similar conceptsfor the ISDE

DHU = F (t, U ) + G(t, U ), t 6= tk,

U (t+k ) = Ik(U (tk)) + Jk(U (tk)), t = tk,

U (0) = U0 ∈ Kc(Rn),

(5.3.1)

where F,G ∈ PC[J × Kc(Rn),Kc(Rn)], Ik, Jk : Kc(Rn) → Kc(Rn) for each kand 0 < t1 < t2 < · · · < tk < · · · , with limk→∞ tk = T with J = [0, T ].

Definition 5.3.1 Let V,W ∈ PC1[J,Kc(Rn)]. Then V,W are said to be

(a) coupled lower and upper solutions of type I of (5.3.1) if

DHV ≤ F (t, V ) +G(t,W ), t 6= tk,

V (t+k ) ≤ Ik(V (tk)) + Jk(W (tk)), t = tk,

V (0) ≤ U0,

(5.3.2)

and

DHW ≥ F (t,W ) + G(t, V ), t 6= tk,

W (t+k ) ≥ Ik(W (tk)) + Jk(V (tk)), t = tk,

W (0) ≥ U0,

(5.3.3)

(b) coupled lower and upper solutions of type II of (5.3.1) if

DHV ≤ F (t,W ) + G(t, V ), t 6= tk,

V (t+k ) ≤ Ik(W (tk)) + Jk(V (tk)), t = tk,

V (0) ≤ U0,

(5.3.4)

and

DHW ≥ F (t, V ) +G(t,W ), t 6= tk,

W (t+k ) ≥ Ik(V (tk)) + Jk(W (tk)), t = tk,

W (0) ≥ U0.

(5.3.5)



(A1) V,W ∈ PC1[J,Kc(Rn)] are coupled lower and upper solutions of type Irelative to (5.3.1) with V (t) ≤ W (t), t ∈ J ;

(A2) F,G ∈ C[J×Kc(Rn),Kc(Rn)], F (t,X) is nondecreasing inX and G(t, Y )is nonincreasing in Y , for each t ∈ J, F,G map bounded sets into bounded sets;

(A3) Ik(U ) is continuous and nondecreasing in U , Jk(U ) is continuous andnonincreasing in U , for each k = 1, 2, · · · .

Then there exist monotone sequences Vn(t), Wn(t) in Kc(Rn) such thatVn(t) → ρ(t),Wn(t) → R(t) in Kc(Rn) and (ρ,R) are the coupled minimaland maximal solutions of type I of (5.3.1) respectively, that is, they satisfy therelations

DHρ = F (t, ρ) + G(t, R), t 6= tk,

ρ(t+k ) = Ik(ρ(tk)) + Jk(R(tk)), t = tk,

ρ(0) = U0,

(5.3.6)

and

DHR = F (t, R) +G(t, ρ), t 6= tk,

R(t+k ) = Ik(R(tk)) + Jk(ρ(tk)), t = tk,

R(0) = U0,

(5.3.7)

for t ∈ J .

Proof Consider, for each n ≥ 0, the ISDEs given by

DHVn+1 = F (t, Vn) +G(t,Wn), t 6= tk,

Vn+1(t+k ) = Ik(Vn(tk)) + Jk(Wn(tk)), t = tk,

Vn+1(0) = U0,

(5.3.8)

and

DHWn+1 = F (t,Wn) + G(t, Vn), t 6= tk,

Wn+1(t+k ) = Ik(Wn(tk)) + Jk(Vn(tk)), t = tk,

Wn+1(0) = U0,

(5.3.9)

where V (0) ≤ U0 ≤W (0).It is clear that the equations (5.3.8) and (5.3.9) have unique solutions say

Vn+1(t) and Wn+1(t), t ∈ J . We set V0(t) = V (t) and W0(t) = W (t), t ∈ J .Our aim is to prove

V0 ≤ V1 ≤ V2 ≤ · · · ≤ Vn ≤ Wn ≤ · · · ≤ W2 ≤ W1 ≤W0, t ∈ J . (5.3.10)

From the hypothesis, we have that V0 and W0 are coupled lower and uppersolutions of type I of (5.3.1). Setting, Vn = V0 and Wn = W0 in (5.3.8) and(5.3.9), we get, V1(t) and W1(t), t ∈ J , which are unique solutions of (5.3.8)and (5.3.9). We now claim


(i) V0 ≤ V1,

(ii) V1 ≤W1, and

(iii) W1 ≤W0 for t ∈ J .

To prove (i), consider the equation (5.3.8) with n = 0,then

DHV1 = F (t, V0) +G(t,W0), t 6= tk,

V1(t+k ) = Ik(V0(tk)) + Jk(W0(tk)), t = tk,

V1(0) = U0,

(5.3.11)

and from hypothesis (A1), we have,

DHV0 ≤ F (t, V0) +G(t,W0), t 6= tk,

V0(t+k ) ≤ Ik(V0(tk)) + Jk(W0(tk)), t = tk,

V0(0) ≤ U0.

(5.3.12)

Now arguing as in Theorem 2.5.1, we get

V0(t) ≤ V1(t), t ∈ (tk−1, tk].

Now using the fact that

V0(t+k ) ≤ V1(t+k ), at each t = tk,

we get V0(t) ≤ V1(t), t ∈ J.Next, to prove (ii), we consider the relations (5.3.8), (5.3.9) with n = 0. We

use the monotone properties of F and G, and Ik and Jk for each k = 1, 2, · · · .Then we arrive at the following equations:

DHV1 ≤ F (t,W0) + G(t,W0), t 6= tk,

V1(t+k ) ≤ Ik(W0(tk)) + Jk(W0(tk)), t = tk,

V1(0) = U0,

and

DHW1 ≥ F (t,W0) +G(t,W0), t 6= tk,

W1(t+k ) ≥ Ik(W0(tk)) + Jk(W0(tk)), t = tk,

W1(0) = U0,

which yield from Corollary 5.2.1,

V1(t) ≤ W1(t), t ∈ J.

Proceeding as in the proof of (i), we obtain, W1(t) ≤ W0(t), t ∈ J , which is(iii).


Thus, we have

V0(t) ≤ V1(t) ≤ W1(t) ≤ W0(t) for t ∈ J.

Assume that for some j > 1, we have

Vj−1(t) ≤ Vj(t) ≤ Wj(t) ≤ Wj−1(t) for t ∈ J. (5.3.13)

Then we prove Vj(t) ≤ Vj+1(t) ≤Wj+1(t) ≤ Wj(t) for t ∈ J .Consider

DHVj = F (t, Vj−1) + G(t,Wj−1), t 6= tk,

Vj(t+k ) = Ik(Vj−1(tk)) + Jk(Wj−1(tk)), t = tk,

Vj(0) = U0,

(5.3.14)

and

DHVj+1 = F (t, Vj) + G(t,Wj), t 6= tk,

Vj+1(t+k ) = Ik(Vj(tk)) + Jk(Wj(tk)), t = tk,

Vj+1(0) = U0.

(5.3.15)

Using the nondecreasing nature of F , for each t ∈ J and the nonincreasingnature of G, for each t ∈ J , and also using the nondecreasing nature of Ik andthe nonincreasing nature of Jk, for each k, along with the relation (5.3.13) wearrive at

DHVj+1 ≥ F (t, Vj−1) + G(t,Wj−1), t 6= tk,

Vj+1(t+k ) ≥ Ik(Vj−1(tk)) + Jk(Wj−1(tk)), t = tk,

Vj+1(0) ≥ U0.

(5.3.16)

By applying Corollary 5.2.1, to the equations (5.3.14) and (5.3.16) we get,

Vj(t) ≤ Vj+1(t), t ∈ J.

Similarly, we can show that Wj+1(t) ≤ Wj(t), t ∈ J .We next prove that Vj+1(t) ≤ Wj+1(t), t ∈ J . Taking n = j in (5.3.8) and

(5.3.9), we have

DHVj+1 = F (t, Vj) + G(t,Wj), t 6= tk,

Vj+1(t+k ) = Ik(Vj(tk)) + Jk(Wj(tk)), t = tk,

Vj+1(0) = U0,

(5.3.17)

and

DHWj+1 = F (t,Wj) +G(t, Vj), t 6= tk,

Wj+1(t+k ) = Ik(Wj(tk)) + Jk(Vj(tk)), t = tk,

Wj+1(0) = U0.

(5.3.18)


Again, using the fact that G is nonincreasing in Y for each t, and Jk(U ) isnonincreasing in U in the relation (5.3.18), and F is nondecreasing in X foreach t and Ik(U ) is nondecreasing in U for each k = 1, 2, · · · , in the relations(5.3.17),(5.3.18) we have,

DHVj+1 ≤ F (t,Wj) +G(t,Wj), t 6= tk,

Vj+1(t+k ) ≤ Ik(Wj(tk)) + Jk(Wj(tk)), t = tk,

Vj+1(0) ≤ U0,

and

DHWj+1 ≥ F (t,Wj) + G(t,Wj), t 6= tk,

Wj+1(t+k ) ≥ Ik(Wj(tk)) + Jk(Wj(tk)), t = tk,

Wj+1(0) ≥ U0, t ∈ J,

which on using Corollary 5.2.1, yields

Vj+1 ≤ Wj+1 for t ∈ J.

Thus we have the sequences of functions Vn, Wn which are piecewise con-tinuous functions and also satisfy the relation (5.3.10). Clearly these sequencesare uniformly bounded on J . In each subinterval [tk, tk+1], the sequence of func-tions Vn and Wn are equi-continuous; hence using Arzela–Ascoli Theoremon each subinterval, we show that the entire sequence Vn(t) converges uni-formly to ρ(t) on [tk, tk+1] and Wn converges uniformly to R(t) on [tk, tk+1].Since Ik, Jk are continuous functions for each k = 1, 2, · · · , we obtain from

limn→∞

Vn(t+k ) = limn→∞

[Ik(Vn−1(tk)) + Jk(Wn−1(tk))]

that ρ(t+k ) = Ik(ρ(tk)) + Jk(R(tk)), similarly R(t+k ) = Ik(R(tk)) + Jk(ρ(tk)).We now consider the integral equations,

Vn+1(t) = U0 +∫ t

0

[F (s, Vn(s)) +G(s,Wn(s))] ds

Wn+1(t) = U0 +∫ t

0

[F (s,Wn(s)) +G(s, Vn(s))] ds

Taking limits as n → ∞, using the uniform continuity of F and G on eachsubinterval [tk, tk+1], we get (5.3.6) and (5.3.7). Further, V0 ≤ ρ ≤ R ≤ W0 fort ∈ J .

Next, we claim that (ρ,R) are coupled minimal and maximal solutions ofISDE(5.3.1). For proof, we show that if U (t) is any solution of (5.3.1) such thatV0 ≤ U ≤ W0 for t ∈ J then,

V0 ≤ ρ ≤ U ≤ R ≤ W0, for t ∈ J. (5.3.19)


Suppose for some n,Vn ≤ U ≤ Wn, t ∈ J. (5.3.20)

Using (5.3.20) along with the monotone properties of F,G for each t, and Ik, Jk

for each k, we arrive at

DHU ≥ F (t, Vn) +G(t,Wn), t 6= tk,

U (t+k ) ≥ Ik(Vn(tk)) + Jk(Wn(tk)), t = tk,

U (0) ≥ U0,

and

DHVn+1 = F (t, Vn) +G(t,Wn), t 6= tk,

Vn+1(t+k ) = Ik(Vn(tk)) + Jk(Wn(tk)), t = tk,

Vn(0) = U0,

which yields, on using Corollary 5.2.1, Vn+1(t) ≤ U (t), t ∈ J .SimilarlyWn+1(t) ≥ U (t) for t ∈ J . This holds for all n. Hence taking limits

as n → ∞, we come up with the relation (5.3.19), thus proving our claim.

Corollary 5.3.1 If in addition to the assumptions of Theorem 5.3.1, supposethat the following hold.

(i) F and G satisfy the relations, for X,Y ∈ Kc(Rn), whenever X ≥ Y ,

F (t,X) ≤ F (t, Y ) + N1(X − Y ), N1 ≥ 0and G(t,X) +N2(X − Y ) ≥ G(t, Y ), N2 ≥ 0;

(ii) for each k, Ik, Jk satisfy the relations

Ik(X) ≤ Ik(Y ) +M1k(X − Y ), M1k ≥ 0and Jk(X) +M2k(X − Y ) ≥ Jk(Y ), M2k ≥ 0,

such that M1k +M2k < 1.Then ρ = U = R is the unique solution of the ISDE (5.3.1).

Proof Since ρ ≤ R, we have R = ρ +m or m = R− ρ. Now

DHρ+DHm = DHR = F (t, R) +G(t, ρ), t 6= tk,

≤ F (t, ρ) + N1m +G(t, R) + N2m, t 6= tk,

= DHρ + (N1 +N2)m, t 6= tk,

and for t = tk,

m(t+k ) + ρ(t+k ) = R(t+k )= Ik(R(tk)) + Jk(ρ(tk))≤ Ik(ρ(tk)) + Jk(R(tk)) +M1km(tk) +M2km(tk)

= ρ(t+k ) + (M1k +M2k)m(tk).


Thus we have,

DHm ≤ Nm, t 6= tk,

m(t+k ) ≤ Mkm(tk), t = tk,

m(0) = 0,

with N = N1 + N2 > 0 and Mk = M1k +M2k with 0 < Mk < 1 for each k.Using a special case of Theorem 1.4.1 in Lakshmikantham, Bainov, Simeonov

[1], we obtain m(t) ≤ 0, that is R ≤ ρ. Hence ρ = U = R is the unique solutionof the ISDE (5.3.1).

Remark 5.3.1

(1) In Theorem 5.3.1, if G(t, Y ) ≡ 0, Jk(U ) ≡ 0 for every k, then we get theresult when F is nondecreasing in X for each t and Ik(U ) is nondecreasing inU for every k.

(2) In (1) above, suppose that F is not nondecreasing in X for every t and forevery k, Ik(X) is not nondecreasing in X, but F (t,X) = F (t,X)+MX, M > 0is nondecreasing in X and I(X) = Ik(X) + NkX is nondecreasing in X, forNk > 0.

Now we consider the IVP of ISDE

DHU +MU = F (t, U ), t 6= tk,

U (t+k ) +NkU (tk) = Ik(U (tk)), t = tk,

U (0) = U0.

Then we obtain the same conclusion as in (1). To see this, consider the trans-formation

U (t) =

U (t)eMt, t 6= tk,

11+Nk

U (t), t = tk,

then

DH U = F (t, U e−Mt)eMt = F0(t, U), t 6= tk,

U (t+k ) + Nk U (tk) = [1 + Nk]Ik

(U (tk)1 +Nk

)=

Ik(U (tk)), t = tk,

U (0) = U0.

For this system

V (t) =

V (t)eMt, t 6= tk,

(1 + Nk)V (t), t = tk,

and

W (t) =

W (t)eMt, t 6= tk,

(1 + Nk)W (t), t = tk,

are lower and upper solutions. Here we have assumed that DH U exists.


(3) If F (t,X) ≡ 0, Ik(X) ≡ 0, k = 1, 2, · · · in Theorem 5.3.1, then we obtainthe result for G(t, Y ) nonincreasing in Y for each t and Jk(Y ) nonincreasing inY , for each k = 1, 2, · · · .

(4) If in (3) above, G and Jk, for each k, are not monotone but

(i) there exists a function G(t, Y ) which is nonincreasing in Y for eacht ∈ J , and a constant M > 0 such that G(t, Y ) = MY + G(t, Y ), that isG(t, Y ) = G(t, Y ) −MY, and

(ii) there exists a function Jk(Y ) which is nonincreasing in Y for each k anda constant Nk > 0, with 0 < Nk < 1, for each k such thatJk(Y ) = NkY + Jk(Y )

Then using the transformation

U (t) =

U (t)eMt, t 6= tk,

11−Nk

U (t), t = tk,

we obtain

DH U = G0(t, U), t 6= tk,

U (t+k ) =Jk[U(tk)], t = tk,

U (0) = U0,

(5.3.21)

where G0[t, U ] = G(t, UeMt)e−Mt andJk[U(tk)] = [1 − Nk][Jk[ 1

1−NkU (tk)] +

NkU (tk)].In this case we need to assume that (5.3.21) has coupled lower and upper

solutions of type I, to get the same conclusion as in (3).

(5) Suppose that in Theorem 5.3.1, G(t, Y ) is nonincreasing in Y and F (t,X)is not monotone but F (t,X) = F (t,X) + MX, M > 0 is nondecreasing in X.Further, suppose that Jk(Y ) is nonincreasing in Y , for each k and Ik(U ) is notmonotone but Ik(U ) = Ik(U ) + NkU , Nk > 0, is nondecreasing in U. Then,consider the IVP of ISDE

DHU +MU = F (t, U ) +G(t, U ), t 6= tk,

U (t+k ) + NkU (tk) = Ik(U (tk)) + Jk(U (tk)), t = tk,

U (0) = U0,

(5.3.22)

In this case also, we obtain the same conclusion of Theorem 5.3.1, by utilizingthe transformation used in (2).

(6) If F (t,X) is nondecreasing in X, and Ik is nondecreasing for each k. ButG(t, Y ) is not monotone in Y for each t ∈ J , and Jk is not nonincreasing for


each k. Then we assume there exist functions G(t, Y ) and Jk(Y ), and constantsM,Nk > 0 as in (4). Now, we consider the IV P

DHU = F (t, U ) + G(t, U ) +MU, t 6= tk,

U (t+k ) = Ik(U ) + Jk(U ) + NkU, t = tk,

U (0) = U0.

(5.3.23)

Then using the transformation

U (t) =

U (t)eMt, t 6= tk,

11−Nk

U (t), t = tk,

we get

DHU = F0(t, U) +G0(t, U), t 6= tk,

U(t+k ) =Ik(U (tk)) +

Jk(U (tk)), t = tk,

U (0) = U0,

(5.3.24)

where for t 6= tk, F0[t, U] = F (t, UeMt)e−Mt

and G0[t, U ] = G(t, U eMt)e−Mt

Jk[U (tk)] = [1 −Nk][Jk[ 1

1−NkU (tk)] + NkU (tk),

andIk[U (tk)] = Ik[ 1

1−NkU (tk)].

If we assume that the system (5.3.24) has coupled lower and upper solutionsof type I, then we get by Theorem 5.3.1 the same conclusion.

(7) If both F and G are not monotone and also Ik, Jk for each k are not mono-tone in Theorem 5.3.1, then we suppose

(i) there exist functions F (t, U ), G(t, U ) and a constant M > 0 such that

F (t, U ) + G(t, U ) = F (t, U ) +G(t, U ) +MU,

exists, and F (t, U ) is nondecreasing in U and G(t, U ) is nonincreasing in U .(ii) there exist functions Ik and Jk and NkU , with 0 < Nk < 1 for each k

such thatIk(U ) + Jk(U ) = Ik(U ) + Jk(U ) + NkU,

where Ik is nondecreasing in U and Jk is nonincreasing in U for each k.Now using the transformation

U (t) =

U (t)eMt, t 6= tk,

11−Nk

U (t), t = tk,


we get,

DH U = F0(t, U) + G0(t, U ), t 6= tk,

U (t+k ) =Jk(U (tk)), t = tk,

U (0) = U0.

Assuming that the above ISDE has coupled lower and upper solutions of type I,we conclude Theorem 5.3.1.

Next, we try to utilize the coupled lower and upper solution of type II inour study. In this case, we need not assume the existence of coupled lower andupper solutions of type II of (5.3.1) as we can construct them under the givenassumptions. But this leads to assumptions on second iterates. Further, we getcomplicated alternative sequences which are monotone.

Theorem 5.3.2 Assume that (A2) and (A3) of Theorem 5.3.1 hold. Then forany solution U (t) of (5.3.1) with V0 ≤ U ≤ W0, t ≥ 0, we have the iteratesVn, Wn satisfying

V0 ≤ V2 ≤ · · · ≤ V2n ≤ U ≤ V2n+1 ≤ · · · ≤ V3 ≤ V1 on R+ (5.3.25)

and W1 ≤ W3 ≤ · · · ≤ W2n+1 ≤ U ≤ W2n ≤ · · · ≤ W2 ≤ W0 on R+,(5.3.26)

provided V0 ≤ V2, W2 ≤ W0 on J , where the iterative schemes are given by

DHVn+1 = F (t,Wn) +G(t, Vn), t 6= tk,

Vn+1(t+k ) = Ik(Wn(tk)) + Jk(Vn(tk)), t = tk,

Vn+1(0) = 0,

(5.3.27)

and

DHWn+1 = F (t, Vn) + G(t,Wn), t 6= tk,

Wn+1(t+k ) = Ik(Vn(tk)) + Jk(Wn(tk)), t = tk,

Wn+1(0) = U0 on J.

(5.3.28)

Moreover, the monotone sequences V2n, V2n+1, W2n, W2n+1inKc(Rn)converge to ρ,R, ρ∗, R∗ in Kc(Rn) respectively and verify,

DHR = F (t, R∗) + G(t, ρ), t 6= tk,

R(t+k ) = Ik(R∗(tk)) + Jk(ρ(tk)), t = tk,

R(0) = U0;

DHρ = F (t, ρ∗) +G(t, R), t 6= tk,

ρ(t+k ) = Ik(ρ∗(tk)) + Jk(R(tk)), t = tk,

ρ(0) = U0;


DHR∗ = F (t, R) + G(t, ρ∗), t 6= tk,

R∗(t+k ) = Ik(R(tk)) + Jk(ρ∗(tk)), t = tk,

R∗(0) = U0;

and

DHρ∗ = F (t, ρ) +G(t, R∗), t 6= tk,

ρ∗(t+k ) = Ik(ρ(tk)) + Jk(R∗(tk)), t = tk,

ρ∗(0) = U0,

respectively on J .

Proof First, we prove the existence of coupled lower and upper solutions V0,W0

of type II, satisfying V0(t) ≤ W0(t), t ∈ J .To achieve this, consider,

DHZ = F (t, θ) + G(t, θ), t 6= tk,

Z(t+k ) = Ik(θ) + Jk(θ), t = tk,

Z(0) = U0;

Let Z(t) be the unique solution which exists on J . Define V0 and W0 by

R0 + V0 = Z and W0 = Z + R0,

where the positive vector R0 = (R01, · · · , R0n) is chosen sufficiently large sothat we have V0 ≤ θ ≤W0 on J .

Next, using the monotone character of F,G, Ik and Jk, for each k, we getfor t 6= tk

DHV0 = DHZ

= F (t, θ) + G(t, θ)≤ F (t,W0) + G(t, V0),

andV0(t+k ) ≤ Z(t+k ),= Ik(θ) + Jk(θ),≤ Ik(W0(tk)) + Jk(V0(tk)),

and V0(0) ≤ U0.Similarly,

DHW0 ≥ F (t, V0) +G(t,W0), t 6= tk,

W0(t+k ) ≥ Ik(V0(tk)) + Jk(W0(tk)), t = tk,

W0(0) ≥ U0,

Thus V0 and W0 are coupled lower and upper solutions of type II of (5.3.1).Let U (t) be any solution of (5.3.1) such that V0 ≤ U ≤ W0 on J . We prove

that,

V0 ≤ V2 ≤ U ≤ V3 ≤ V1 and W1 ≤ W3 ≤ U ≤W2 ≤ W0 on J. (5.3.29)


The monotonicity of F and G, Ik and Jk along with the facts V0 ≤ U ≤ W0

and U is a solution of (5.3.1) gives,

DHU = F (t, U ) + G(t, U ) ≤ F (t,W0) + G(t, V0), t 6= tk

U (t+k ) = Ik(U (tk)) + Jk(U (tk)) ≤ Ik(W0(tk)) + Jk(V0(tk)), t = tk

U (0) ≤ U0.

The relations (5.3.27) for n = 0 are

DHV1 = F (t,W0) +G(t, V0), t 6= tk,

V1(t+k ) = Ik(W0(tk)) + Jk(V0(tk)), t = tk,

V1(0) = U0.

On using Corollary 5.2.1, for U and V1, we get,

U ≤ V1 on J.

Again, setting n = 1 in (5.3.27), we get

DHV2 = F (t,W1) +G(t, V1),≤ F (t, U ) +G(t, U ), t 6= tk,

V2(t+k ) = Ik(W1(tk)) + Jk(V1(tk)),≤ Ik(U (tk)) + Jk(U (tk)), t = tk,

V2(0) = U0.

Now, since U is a solution of (5.3.1), the above differential inequalities of V2,along with Corollary 5.2.1, imply V2 ≤ U on J . Thus we have V2 ≤ U ≤ V1 onJ .

Similarly , we can show that U ≤ W2 on J .Further, using the fact, V0 ≤ V2 andW2 ≤ W0 on J along with the properties

of F,G, Ik and Jk for each k, we have,

DHV3 = F (t,W2) +G(t, V2) ≤ F (t,W0) + G(t, V0), t 6= tk,

V3(t+k ) = Ik(W2(tk)) + Jk(V2(tk)) ≤ Ik(W0(tk)) + Jk(V0(tk)), t = tk,

V3(0) = U0.

Now considering equations (5.3.27) with n = 0 and using the Corollary 5.2.1,we get V3 ≤ V1 on J . In a similar fashion, we can show that W1 ≤ W3 on J .Also, we get U ≤ V3 and W3 ≤ U on J , thus proving the relations (5.3.29).

Now assume for some n > 2, the inequalities,

V2n−4 ≤ V2n−2 ≤ U ≤ V2n−1 ≤ V2n−3, (5.3.30)

W2n−3 ≤ W2n−1 ≤ U ≤W2n−2 ≤ W2n−4, (5.3.31)

hold on J . We claim that

V2n−2 ≤ V2n ≤ U ≤ V2n+1 ≤ V2n−1, (5.3.32)


W2n−1 ≤ W2n+1 ≤ U ≤ W2n ≤ W2n−2, on J. (5.3.33)

Taking n = 2n − 1 in the equations (5.3.27) and using the relations (5.3.30),(5.3.31) and monotone character of F,G, Ik and Jk for each k, yields

DHV2n = F (t,W2n−1) +G(t, V2n−1)≤ F (t, U ) +G(t, U ), t 6= tk,

V2n(t+k ) ≤ Ik(W2n−1(tk)) + Jk(V2n−1(tk))≤ Ik(U (tk)) + Jk(U (tk)), t = tk,

V2n(0) ≤ U0.

Since U is a solution of (5.3.1), U satisfies the reverse inequalities.Now arguing as in the proof of Theorem 2.5.1, we get V2n ≤ U on J .We now show W2n ≤ W2n−2. Setting n = 2n − 1 in relations (5.3.28), and

using the hypothesis,

DHW2n = F (t, V2n−1) + G(t,W2n−1)≤ F (t, V2n−3) + G(t,W2n−3), t 6= tk,

W2n(t+k ) = Ik(V2n−1(tk)) + Jk(W2n−1(tk))≤ Ik(V2n−3(tk)) + Jk(W2n−3(tk)), t = tk,

W2n(0) = U0.

Observing that by setting n = 2n − 3 in (5.3.28), we get the equalities(reverse inequalities) of the above relations with W2n replaced by W2n−2. Nowusing Corollary 5.2.1, gives W2n ≤ W2n−2 on J .

The proofs of the remaining relations are an exact repetition of the abovediscussion. Hence we avoid them. Thus we conclude the relations (5.3.32) and(5.3.33). By induction on n, we have the conclusion of the relations (5.3.25) and(5.3.26).

Since Vn,Wn ∈ PC1[J,Kc(Rn)] for all n, reasoning as in Theorem 5.3.1, weget

limn→∞

V2n = ρ, and limn→∞

V2n+1 = R,

limn→∞

W2n+1 = ρ∗, and limn→∞

W2n = R∗

exist over each subinterval [tk, tk+1], and the convergence is uniform in eachsubinterval.

Further, for each t = tk,

limn→∞

V2n(t+k ) = Ik( limn→∞

W2n−1(tk)) + Jk( limn→∞

V2n−1(tk))

which implies, ρ(t+k ) = Ik(ρ∗(tk)) + Jk(R(tk)).With a similar reasoning, we observe that the functions ρ∗, R,R∗, satisfy

their corresponding impulse conditions, at t = tk. Also, using the integralrepresentation for the differential equations in (5.3.27) and (5.3.28) suitably,

5.4 SET DIFFERENTIAL EQUATIONS WITH DELAY 163

we obtain that ρ, ρ∗, R,R∗ satisfy their corresponding impulsive set differentialequations, given in the statement of the theorem.

Also from (5.3.25) and (5.3.26), we get

ρ ≤ U ≤ R and ρ∗ ≤ U ≤ R∗, on J.

Thus the proof is complete.

Corollary 5.3.2 Assume that the hypothesis of the Theorem 5.3.2 hold. Fur-ther, suppose F,G, Ik and Jk satisfy the hypothesis in the corollary 5.3.1. Thenρ = ρ∗ = R = R∗ = U is the unique solution of the ISDE (5.3.1).

Proof Since ρ ≤ R and ρ∗ ≤ R∗, let q1 + ρ = R and q2 + ρ∗ = R∗. Thenconsidering DH (q1 + q2) and using the hypothesis, we get

DH (q1 + q2) ≤ (N1 + N2)(q1 + q2), t 6= tk,

q1(t+k ) + q2(t+k ) ≤ (M1k +M2k)(q1 + q2), t = tk,

and (q1 + q2)(0) = 0.

Using Theorem 1.4.1 in Lakshmikantham, Bainov, Simeonov [1] in this context,since N1 + N2 ≥ 0 and 0 < M1k +M2k < 1, we get

(q1 + q2)(t) ≤ 0, t ∈ J,

which means R + R∗ ≤ ρ + ρ∗ ≤ R + R∗,. This gives U = ρ = R = ρ∗ = R∗,and hence the solution is unique.

Remark 5.3.2 Corresponding to the Remark 5.3.1, we can make similar re-mark following from Theorem 5.3.2. To avoid monotony we do not list them.

Remark 5.3.3 The impulsive set differential equation (5.3.1) reduces to an or-dinary impulsive differential equation if F,G, Ik, Jk are all single valued map-pings. In this case, Theorem 5.3.1 and 5.3.2 along with the remarks give rise tomany new results in the theory of impulsive differential equations.

5.4 Set Differential Equations with Delay

In ordinary differential difference equations or, more generally, in differentialequations with delay, the history exerts its influence in a significant way onthe future of solutions. There are several applications in which future dependson the past history (finite or infinite). This area of differential equations withdelay or usually known as functional differential equations is investigated as anindependent subject and is very interesting.

In this section, we shall incorporate delay in the formulation of set differentialequations and provide some basic results of interest. We start by describing theset differential equation with delay.


Given any τ > 0, consider C0 = C[[−τ, 0],Kc(Rn)]. For any Φ, Ψ ∈ C0,define the metric

D0[Φ,Ψ] = max−τ≤s≤0

D[Φ(s),Ψ(s)]

. Also, we write ‖Φ‖0 = D0[Φ, θ].Suppose that J0 = [t0 − τ, t0 + a], a > 0. Let U ∈ C[J0,Kc(Rn)]. For any

t ≥ t0, t ∈ J0, let Ut denote a translation of the restriction of U to the interval[t− τ, t]. That is, Ut ∈ C0 is defined by Ut(s) = U (t+ s), −τ ≤ s ≤ 0.

Consider the set differential equation with finite delay given by

DHU = F (t, Ut), Ut0 = Φ0 ∈ C0 (5.4.1)

where F ∈ C[J × C0,Kc(Rn)] and J = [t0, t0 + a].The following existence result is obtained using the contraction principle.


D[F (t,Φ), F (t,Ψ)] ≤ KD0[Φ,Ψ], K > 0 (5.4.2)

for t ∈ J, Φ,Ψ ∈ C0. Then the IVP (5.4.1) possesses a unique solution U (t)on J0.

Proof Consider the set of functions U ∈ C[J0,Kc(Rn)] such that U (t) =Φ0(t), t0 − τ ≤ t ≤ t0 and U ∈ C[J,Kc(Rn)] with U (t0) = Φ0(0) withΦ0(t) ∈ Kc(Rn), − τ ≤ t ≤ 0.

Define the metric on C[J0,Kc(Rn)] by

D1(U, V ) = maxt0−τ≤t≤t0+a

D[U (t), V (t)]e−λt, λ > 0, is chosen suitably later.

(5.4.3)Next, define the operator T on C[J0,Kc(Rn)] by

TU (t) = Φ0(t), t0 − τ ≤ t ≤ t0

TU (t) = Φ0(0) +∫ t

t0

F (s, Us) ds, t ∈ J.(5.4.4)

Then, for −τ ≤ s ≤ 0,

D[TU (t0 + s), TV (t0 + s)],= D[Φ0(t0 + s),Φ0(t0 + s)],= 0 .

We get, using the properties of Hausdorff metric (1.3.9) and (1.7.11), for


t ∈ J,

D[TU (t), TV (t)]

= D[Φ0(0) +∫ t

t0

F (ξ, Uξ) dξ,Φ0(0) +∫ t

t0

F (ξ, Vξ) dξ]

= D[∫ t

t0

F (ξ, Uξ) dξ,∫ t

t0

F (ξ, Vξ) dξ]

≤∫ t

t0

D[F (ξ, Uξ), F (ξ, Vξ)] dξ

≤ K

∫ t

t0

D0[(Uξ , Vξ)] dξ.

Consider∫ t

t0

D[U (ξ + s), V (ξ + s)] dξ ≤∫ t

t0−τ

D[U (σ), V (σ)] dσ

≤∫ t

t0−τ

maxt0−τ≤σ≤t0+a

[D[U (σ), V (σ)]e−λσ] eλσ dσ

= D1[U, V ]∫ t

t0−τ

eλσdσ

= D1[U, V ]1λ

[eλt − eλ(t0−τ)]

≤ 1λD1[U, V ]eλt.

Thus we obtain

K

∫ t

t0

D0[Uξ, Vξ] dξ ≤K

λD1[U, V ] eλt,

which implies that, on J0,

e−λt D[TU (t), TV (t)] ≤ K

λD1[U, V ].

Choosing λ = 2K and taking maximum over t, on J0, we have

D1[TU, TV ] ≤ 12D1[U, V ],

which means that the operator T on C[J0,Kc(Rn)] is a contraction. Thus thereexists a unique fixed point U ∈ C[J0,Kc(Rn)] of T by the contraction principle.Hence U (t) = U (t0,Φ0)(t) is the unique solution of the IVP (5.4.1).

We now prove a comparison theorem in this context, which is a useful toolin proving the global existence theorem.


Theorem 5.4.2 Assume that F ∈ C[R+×C0,Kc(Rn)] and D[F (t,Φ), F (t,Ψ)] ≤g(t,D0[Φ,Ψ]) for t ∈ R+, Φ0,Ψ0 ∈ C0, where g ∈ C[R2

+,R+]. Let r(t) =r(t, t0, w0) be the maximal solution of

w′ = g(t, w)w(t0) = w0 ≥ 0 existing for t ≥ t0.

Then D0[Φ0,Ψ0] ≤ w0 implies

D[U (t), V (t)] ≤ r(t), t ≥ t0,

where U (t) = U (t0,Φ0)(t) and V (t) = V (t0,Ψ0)(t) are the solutions of (5.4.1).

Proof Since U (t), V (t) are solutions of (5.4.1) for small h > 0, the differencesU (t+h)−U (t), V (t+h)−V (t) exist. Now for t ∈ R+, set m(t) = D[U (t), V (t)].Then using the properties of Hausdorff metric, (1.3.8) and (1.3.9), we have

m(t + h) −m(t) = D [U (t+ h), V (t+ h)]−D[U (t), V (t)]≤ D[U (t + h), U (t) + hF (t, Ut)] +D[U (t) + hF (t, Ut), V (t) + hF (t, Vt)]

+D[V (t) + hF (t, Vt), V (t+ h)] −D[U (t), V (t)]≤ D[U (t + h), U (t) + hF (t, Ut)] +D[V (t) + hF (t, Vt), V (t+ h)]

+hD[F (t, Ut), F (t, Vt)] ,

from which we get, using (1.3.8) and (1.3.9) again,

m(t + h) −m(t)h

≤ D

[U (t + h) − U (t)

h, F (t, Ut)

]

+D

[F (t, Vt),

V (t + h) − V (t)h

]+D[F (t, Ut), F (t, Vt)].

Taking limit supremum as h → 0+, gives,


m(t+ h) −m(t)h

≤ D[F (t, Ut), F (t, Vt)]≤ g(t,D0[Ut, Vt]) = g(t, |mt|0).

The above inequality, along with the fact that |mt0 |0 = D0[Φ0,Ψ0] ≤ w0, impliesfrom the comparison theorem for ordinary delay differential equations (Laksh-mikantham and Leela [1]), that

D[U (t), V (t)] ≤ r(t), t ≥ t0.

We are now ready to prove the global existence theorem.


Theorem 5.4.3 Let F ∈ C[R+ × C0,Kc(Rn)] and for (t,Φ) ∈ R+ × C0,

D[F (t,Φ), θ] ≤ g(t,D0[Φ, θ]),

where g ∈ C[R2+,R+], g(t, w) is nondecreasing in w for each t ∈ R+. Assume

that the solutions w(t, t0, w0) of w′ = g(t, w), w(t0) = w0 exist for t ≥ t0,and F is smooth enough to assure local existence. Then the largest interval ofexistence of any solution U (t0,Φ0)(t) of (5.4.1) is [t0,∞).

Proof Suppose that U (t0,Φ0)(t) is a solution of (5.4.1) existing on some in-terval [t0−τ, β), where t0 < β < ∞. Assume that β cannot be increased. Definefor t ∈ [t0 − τ, β),

m(t) = D[U (t0,Φ0)(t), θ]

mt = D[Ut(t0,Φ0), θ] and |mt|0 = D0[Ut(t0,Φ0), θ]

Then reasoning and proceeding as in the comparison theorem, we get the dif-ferential inequality

D+m(t) ≤ g(t, |mt|0), t0 ≤ t < β.

Choosing |mt0|0 = D0[Φ0, θ] ≤ w0, we arrive at

D[U (t0,Φ0)(t), θ] ≤ r(t, t0, w0), t0 ≤ t < β.

Now g(t, w) ≥ 0 implies that r(t, t0, w0) is nondecreasing in t, which furtheryields

D[Ut(t0,Φ0), θ] ≤ r(t, t0, w0), t0 ≤ t < β. (5.4.5)

Consider t1, t2 such that t0 < t1 < t2 < β, then using (1.3.8) and (1.7.11), weget

D[U (t0,Φ0)(t1), U (t0,Φ0)(t2)]

= D[U (t0,Φ0)(t1), U (t0,Φ0)(t1) +∫ t2

t1

F (s, Us) ds]

= D[θ,∫ t2

t1

F (s, Us) ds]

≤∫ t2

t1

D[F (s, Us), θ] ds

≤∫ t2

t1

g(s,D0[Us(t0,Φ0), θ]) ds.

Next, using the fact that g is monotonically nondecreasing in w and the relation(5.4.5) in the above inequality, we obtain

D[U (t0,Φ0)(t1), U (t0,Φ0)(t2)] ≤∫ t2

t1

g(s, r(s, t0, w0)) ds

= r(t2, t0, w0) − r(t1, t0, w0).(5.4.6)


If we let t1, t2 → β in the above relation (5.4.6), then limt→β− U (t0,Φ0)(t)exists, because of Cauchy’s criterion for convergence.

We now define U (t0,Φ0)(β) = limt→β− U (t0,Φ0)(t) and consider Ψ0 =Uβ(t0,Φ0)

as the new initial function at t = β. The assumption of local existenceimplies that there exists a solution U (β,Ψ0)(t) of (5.4.1) on [β−τ, β+α], α > 0.This means that the solution U (t0,Φ0)(t) can be continued beyond β, which iscontrary to our assumption that the value of β cannot be increased. Hence thetheorem.

Next, we present a result on nonuniform practical stability of (5.4.1) usingperturbing Lyapunov functions. See (Lakshmikantham Leela and Martynyuk[1]) for details.

Before proceeding further, we need the following classes of functions

K = a ∈ C[[0, A],R+], a(0) = 0 and a(u) is strictly increasing

CK = σ ∈ C[R+ × [0, A],R+] : σ(t, .) ∈ K for each t ∈ R+We now define practical stability in this context.

Definition 5.4.1 The system (5.4.1) is practically stable, given (λ,A) with 0 <λ < A, we have D0[Φ0, θ] < λ implies D[U (t), θ] < A, t ≥ t0 for some t0 ∈ R+.

We setS(A) = U ∈ Kc(Rn) : D[U, θ] < A

andΩ(A) = Φ ∈ C0 : D0[Φ, θ] < A.

We are now in a position to prove the following result on practical stability.

Theorem 5.4.4 Assume that (i) 0 < λ < A ;(ii) V1 ∈ C[ R+ × S(A) × Ω(A),R+],for (t, U1,Φ), (t, U2,Φ) ∈ R+ × S(A) × Ω(A)

|V1(t, U1,Φ) − V1(t, U2,Φ)| ≤ L1 D[U1, U2], L1 > 0;

for each (t, U,Φ) ∈ R+ × S(A) × Ω(A),

V1(t, U,Φ) ≤ a1(t,D0[Φ, θ]), a1 ∈ CK,

and

D+V1(t, U,Φ) ≡ lim suph→0+

1h

[V1(t+ h, U + hF (t, Ut), Ut+h) − V1(t, U, Ut)]

≤ g1(t, V1(t, U,Φ)),


(iii) V2 ∈ C[ R+ × S(A) × Ω(A),R+],for (t, U1,Φ), (t, U2,Φ) ∈ R+ × S(A) × Ω(A),

|V2(t, U1,Φ) − V2(t, U2,Φ)| ≤ L2 D[U1, U2], L2 > 0;


for each (t, U,Φ) ∈ R+ × S(A) × Ω(A),

b(D[U, θ]) ≤ V2(t, U,Φ) ≤ a2(D0[Φ, θ]),

D+V1(t, U,Φ) +D+V2(t, U,Φ) ≤ g2(t, V1(t, U,Φ) + V2(t, U,Φ)),

where a2, b ∈ K and g2 ∈ C[R2+,R];

(iv) a1(t0, λ) + a2(λ) < b(A) for some t0 ∈ R+;(v) u0 < a1(t0, λ) implies u(t, t0, u0) < a1(t0, λ) for t ≥ t0 where u(t, t0, u0) isany solution of

u′ = g1(t, u), u(t0) = u0, (5.4.7)

and v0 < a1(t0, λ) + a2(λ) implies

v(t, t0, v0) < b(A), t ≥ t0,

for every t0 ∈ R+, where v(t, t0, v0) is any solution of

v′ = g2(t, v), v(t0) = v0 ≥ 0. (5.4.8)

Then the system (5.4.1) is practically stable.

Proof We have to prove that given 0 < λ < A, D0[Φ, θ] < λ thenD[U (t), θ] <A where U (t) = U (t0,Φ0)(t) is any solution of (5.4.1), for t ≥ t0. Suppose it isnot true, then there exists t2 > t1 > t0 and a solution U (t0,Φ0)(t) of (5.4.1)such that

D[U (t1), θ] = λ and D[U (t2), θ] = A (5.4.9)

and λ ≤ D[U (t), θ] ≤ A, for t1 ≤ t ≤ t2. Now using the hypothesis (iii) and (v)and the standard arguments, we get

V1(t, U (t0,Φ0)(t), Ut(t0,Φ0)) + V2(t, U (t0,Φ0)(t), Ut(t0,Φ0))≤ r2(t, t1, V1(t, U (t0,Φ0)(t1), Ut1(t0,Φ0))

+ V2(t1, U (t0,Φ0)(t1), Ut1(t0,Φ0)))(5.4.10)

for t1 ≤ t ≤ t2, where r2(t, t1, v0) is the maximal solution of (5.4.8) through(t1, v0), and v0 = V1(t1, U (t0,Φ0)(t1), Ut1(t0,Φ0))+V2(t1, U (t0,Φ0)(t1), Ut1(t0,Φ0)).

Similarly, condition (ii) gives the estimate

V1(t, U (t0,Φ0)(t), Ut(t0,Φ0)) ≤ r1(t, t0, V1(t0,Φ0(0),Φ0)), t0 ≤ t ≤ t1,

where r1(t, t0, u0) is the maximal solution of (5.4.7) with u0 = V1(t0,Φ0(0),Φ0).Since D0[Φ0, θ] < λ, using hypothesis (ii)

V1(t0,Φ0(0),Φ0) ≤ a1(t0, D0[Φ0, θ]) ≤ a1(t0, λ).

Also, we have

V2(t1, U (t0,Φ0)(t1), Ut1(t0,Φ0)) ≤ a2(D0[Ut1 , θ]) ≤ a2(λ).


Thus we get

V1(t1, U (t0,Φ0)(t1), Ut1(t0,Φ0)) + V2(t1, U (t0,Φ0)(t1), Ut1(t0,Φ0))≤ a1(t0, λ) + a2(λ).

Now using the relation (5.4.10) and the hypothesis (v), we obtain

V1(t2, U (t0,Φ0)(t2), Ut2(t0,Φ0)) + V2(t2, U (t0,Φ0)(t2), Ut2(t0,Φ0))≤ r2(t2, t1, a1(t0, λ) + a2(λ)) < b(A).

(5.4.11)

However, using the relation (5.4.9) and the hypothesis (ii) and (iii), we get

V1(t2, U (t0,Φ0)(t2), Ut2(t0,Φ0)) + V2(t2, U (t0,Φ0)(t2), Ut2(t0,Φ0))≥ V2(t2, U (t0,Φ0)(t2), Ut2(t0,Φ0))≥ b(D[U (t2), θ]) = b(A),

which contradicts (5.4.11). Thus the proof of our claim.We next study the nonuniform boundedness property for the system (5.4.1).

We define the concept of boundedness as follows.

Definition 5.4.2 The differential system (5.4.1) is said to be

(1) Equibounded, if for any α > 0 and t0 ∈ R+, there exists a β where β =β(t0, α) > 0 such that

D0[Φ0, θ] < α implies D[U (t), θ] < β, t ≥ t0,

where U (t) = U (t0,Φ0)(t) is any solution of (5.4.1).

(2) Uniform Bounded, if β in (1) does not depend on t0.

The following theorem uses the method of perturbing Lyapunov functionsto obtain nonuniform boundedness property.

For that purpose, setS(ρ) = U ∈ Kc(Rn) : D[U, θ] < ρ and S(ρ) = Φ ∈ C : D0[Φ, θ] < ρ.


(i) ρ > 0, V1 ∈ C[R+ × S(ρ) × S(ρ),R+], V1 is bounded for(t, U,Φ) ∈ R+ × ∂S(ρ) × ∂S(ρ);

|V1(t, U1,Φ) − V1(t, U2,Φ)| ≤ L1D[U1, U2], L1 > 0,

and for (t, U,Φ) ∈ R+ × Sc(ρ) × Sc(ρ),

D+V1(t, U,Φ) ≡ lim suph→0+

1h

[V1(t + h, U + hF (t, Ut), Ut+h) − V1(t, U, Ut)]

≤ g1(t, V1(t, U,Φ)),



(ii) V2 ∈ C[R+ × Sc(ρ) × Sc(ρ),R+],

b(D[U, θ]) ≤ V2(t, U,Φ) ≤ a(D0[Φ, θ]),

D+V1(t, U,Φ) +D+V2(t, U,Φ) ≤ g2(t, V1(t, U,Φ) + V2(t, U,Φ)),

where a, b ∈ K and g2 ∈ C[R2+,R];

(iii) The scalar differential equations

w′1 = g1(t, w1), w1(t0) = w10 ≥ 0, (5.4.12)

w′2 = g2(t, w2), w2(t0) = w20 ≥ 0, (5.4.13)

are equibounded and uniformly bounded respectively.

Then the system (5.4.1) is equibounded.

Proof Let B1 > ρ and t0 ∈ R+ be given. Let

α0 = maxV1(t0, U0,Φ0) : U0 = Φ0(0) ∈ clS(B1) ∩ Sc(ρ),Φ0 ∈ clS(B1) ∩ Sc(ρ)

α∗ ≥ V1(t, U,Φ) for (t, U,Φ) ∈ R+ × ∂S(ρ) × ∂S(ρ),

and set α1 = α1(t0, B1) = max(α0, α∗).

Since the scalar differential equation (5.4.12) is equibounded, given α1 > 0and t0 ∈ R+, there exists a β0 = β0(t0, α1) such that

w1(t, t0, w0) < β0, t ≥ t0 , (5.4.14)

provided w10 < α1, where w1(t, t0, w10) is any solution of (5.4.12).Let α2 = a(B1)+β0. Then the uniform boundedness of the equation (5.4.13)

yieldsw2(t, t0, w20) < β1(α2), t ≥ t0 (5.4.15)

provided w20 < α2, where w2(t, t0, w20) is any solution of (5.4.13).Choose B2 satisfying

b(B2) > β1(α2). (5.4.16)

We now claim that Φ0 ∈ S(B1) implies that U (t) ∈ S(B2) for t ≥ t0, whereU (t) = U (t0,Φ0)(t) is any solution of (5.4.1).

If it is not true, there exists a solution U (t0,Φ0)(t) of (5.4.1) with Φ0 ∈S(B1), such that, for some t∗ > t0, D[U (t0,Φ0)(t∗), θ] = B2. Since B1 > ρ,there are two possibilities to consider,

(i) U (t0,Φ0)(t) ∈ Sc(ρ) for t ∈ [t0, t∗],

(ii) there exists a t ≥ t0 such that U (t0,Φ0)(t) ∈ ∂S(ρ) andU (t0,Φ0)(t) ∈ Sc(ρ) for t ∈ [t, t∗].


If (i) holds, we can find a t1 > t0 such that:

U (t0,Φ0)(t1) ∈ ∂S(B1),U (t0,Φ0)(t∗) ∈ ∂S(B2),U (t0,Φ0)(t) ∈ Sc(B1), t ∈ [t1, t∗].

(5.4.17)

Setting m(t) = V1(t, U (t0,Φ0)(t), Ut(t0,Φ0)) + V2(t, U (t0,Φ0)(t), Ut(t0,Φ0))for t ∈ [t1, t∗] and using the standard arguments, we obtain the differentialinequality

D+m(t) ≤ g2(t,m(t)), t ∈ [t, t∗] .

It then follows from the Comparison Theorem 1.4.1 of Lakshmikantham andLeela [1]

m(t) ≤ r2(t, t1,m(t1)), t ∈ [t1, t∗]

where r2(t, t1, w20) is the maximal solution of (5.4.13) with

r2(t1, t1, w20) = w20 =V1(t1, U (t0,Φ0)(t1), Ut1(t0,Φ0))+ V2(t1, U (t0,Φ0)(t1), Ut1(t0,Φ0)).

Thus,

V1(t∗, U (t0,Φ0)(t∗), Ut∗(t,Φ0)) + V2(t∗, U (t0,Φ0)(t∗), Ut∗(t,Φ0))≤ r2(t∗, t1, w20).

(5.4.18)

Similarly, we get

V1(t, U (t0,Φ0)(t), Ut(t0,Φ0)) ≤ r1(t1, t0, V1(t0, U (t0,Φ0)(t0), Ut0(t0,Φ0))(5.4.19)

where r1(t, t0, u0) is the maximal solution of (5.4.12) with

u0 = V1(t0, U (t0,Φ0)(t0), Ut0(t0,Φ0)) = V1(t0,Φ0(0),Φ0).

Setting w10 = V1(t0,Φ0(0),Φ0) < α1, and using the relation (5.4.14) we get,

V1(t1, U (t0,Φ0)(t1), Ut1(t0,Φ0)) ≤ r1(t1, t0, w10) ≤ β0.

Furthermore, V2(t1, U (t0,Φ0)(t1), Ut1(t0,Φ0)) ≤ a(B1) and we have w20 ≤ β0 +a(B1) = α2.

Now combining (5.4.15), (5.4.16),(5.4.17) we have

b(B2) ≤ m(t∗) ≤ r(t∗) ≤ β1(α2) < b(B2), (5.4.20)

which is a contradiction.If case (ii) holds, we also come up with the inequality (5.4.18), where t1 > t

satisfies (5.4.17). We then obtain, in place of (5.4.19) the relation

V1(t1, U (t0,Φ0)(t1),Φt1) ≤ r1(t1, t, V1(t, U (t0,Φ0)(t)),Φt) ,

5.5 IMPULSIVE SET DIFFERENTIAL EQUATIONS WITH DELAY 173

since U (t0,Φ0)(t) ∈ ∂S(ρ) and

V1(t, U (t0,Φ0)(t),Φt) ≤ α∗ ≤ α1 ,

arguing as before, we get a contradiction.This proves that, for any given B1 > ρ, t0 > 0 there exists a B2 such that

Φ0 ∈ S(B1) implies U (t0,Φ0)(t) ∈ S(B2), t ≥ 0. For the case B1 < ρ, we getB2(t0, B1) = B2(t0, ρ) and hence the proof.

5.5 Impulsive Set Differential Equations with De-

lay

In this section we establish basic results in the theory of impulsive set differentialequations with delay.

Consider the impulsive set differential equation with delay

DHU = F (t, Ut), t 6= tk,Ut+k

= Ik(Utk), t = tk,

Ut0 = Φ0 ∈ Kc(Rn),

(5.5.1)

where F ∈ PC[R+ × C,Kc(Rn)], Ik : C −→ C with C = C[[−τ, 0],Kc(Rn)]and tk is a sequence of points such that 0 ≤ t0 < t1 < · · · < tk < . . . withlimk→∞ tk = ∞.

By a solution of (5.5.1) we mean a piecewise continuous function U (t0,Φ0)(t)on [t0,∞) which is left continuous on (tk, tk+1] and defined by

U (t0,Φ0)(t) =

Φ0, t0 − τ ≤ t ≤ t0,U0(t0,Φ0)(t), t0 ≤ t ≤ t1,U1(t1,Φ1)(t), t1 < t ≤ t2,

......

Uk(tk,Φk)(t), tk < t ≤ tk+1,...

...

(5.5.2)

where Uk(tk,Φk)(t) is the solution of the set differential equation with delay

DHU = F (t, Ut), Ut+k

= Φk, k = 0, 1, 2, . . . .

We will first prove an existence theorem for impulsive set differential equationswith delay.


(i) F ∈ PC[R+ × C, Kc(Rn],

(ii) D[F (t,Φ), θ] ≤ g(t,D0[Φ, θ)]), t 6= tk, where g ∈ PC[R+2,R+], g(t, w) is

nondecreasing in w for each t ∈ R+,


(iii) D0[I(Utk), θ] ≤ Jk(D0[Utk , θ]), t = tk, where Jk(w) is a nondecreasingfunction of w,

(iv) r(t, t0, w0) is the maximal solution of the impulsive scalar differential equa-tion

w ′ = g(t, w), t 6= tk,w(t+k ) = Jk(w(tk)), t = tk,w(t0) = w0,

(5.5.3)

existing on [t0,∞] and F is smooth enough to assure local existence.

Then there exists a solution for (5.5.1) on [t0,∞).

Proof Set J0 = [t0, t1] and restrict F to J0 × C. Note that F is continuous onJ0 × C.

Consider on J0 the set differential equationDHU = F (t, Ut),Ut0 = Φ0.

Then, the hypothesis of Theorem 5.4.3 is satisfied, and hence there exists asolution U0(t0,Φ0)(t), t ∈ J0, for the set differential equation with delay on J0.

Now, for t = t1, U0(t1) = U0(t0,Φ0)(t1) and U0,t+1= I1(U0,t1). Set Φ1 =

U0,t+1. Let J1 = (t1, t2] and consider the set differential equation with delay

DHU = F (t, Ut), t ∈ J1,Ut+1

= Φ1.

Once again, restricting the domain of F to J1 × C and employing the im-pulsive condition in (iii), the hypothesis of Theorem 5.4.3 is satisfied and thusthere exists a solution U1(t1,Φ1)(t),t ∈ J1, satisfying the set differential equation with delay restricted to J1.We have U1(t2) = U1(t1,Φ1)(t2), U1,t+2

= I2(U1,t2). Set Φ2 = U1,t+2and let

J2 = (t2, t3]. Repeating the above process, we get the existence of a solution ofthe impulsive set differential equation with delay on [t0,∞).Next, we give a basic comparison theorem for impulsive set differential equationswith delay.


(i) F ∈ PC[R+ × C,Kc(Rn)];

(ii) D[F (t,Φ), F (t,Ψ)] ≤ g(t,D0[Φ,Ψ]) for t ∈ R+, t 6= tk, Φ,Ψ ∈ C, andg ∈ PC[R+

2 ,R+];

(iii) D0[Ik(U tk), Ik(V tk)] ≤ Jk(D0[Utk , Vtk]), t = tk, where Jk(w) is a nonde-creasing function of w;

(iv) r(t) = r(t, t0, w0) is the maximal solution of the scalar impulsive differen-tial equation (5.5.3) existing on [t0,∞).


Then, if U (t) = U (t0,Φ0)(t) and V (t) = V (t0,Ψ0)(t) are any two solutions of(5.5.1) on [t0,∞), we have

D[U (t), V (t)] ≤ r(t), t ≥ t0,

provided D0[Φ0,Ψ0] ≤ w0.

Proof We set J0 = [t0, t1] and restrict the domain of F to J0 × C. Then Fis continuous on this domain and the hypothesis of Theorem 5.4.2 is satisfied.Hence we have that

D[U (t), V (t)] ≤ r(t), t ∈ J0,

which implies that D[U (t1), V (t1)] ≤ r(t1). Using hypothesis (iii) for t = t+1 , weget

D0[Ut+1, Vt+1

] = D0[I1(Ut1 ), I1(Vt1)]≤ J1(D0[Ut1 , Vt1])≤ J1(r(t1)) ≡ r(t+1 ).

ThusD0[U+

t1 , V+t1 ] ≤ r(t+1 ). (5.5.4)

Next, restrict the domain of F to J1 × C, where J1 = (t1, t2]. Then using(ii), (5.5.4) and Theorem 5.4.2 we can conclude that

D[U (t), V (t)] ≤ r(t, t0, w0), t ∈ J1.

Repeating the above process, the conclusion of the theorem is obtained.We shall next extend a typical result in Lyapunov-like theory.Let V : R+ ×Kc(Rn) × C → R+. Then V is said to belong to class V0 if

(A1) V (t, U,Φ) is continuous in (tk−1, tk]×Kc(Rn) × C and for eachU ∈ Kc(Rn), Φ ∈ C, k = 1, 2, · · · ,

lim(t,W,Φ)→(t+k ,U,Φ)

V (t,W,Φ) = V (t+k , U,Φ)

exists;

(A2) V ∈ C[(tk−1, tk] ×Kc(Rn) × C,Rn+] satisfies

|V (t, U,Φ)− V (t,W,Φ| ≤ LD[U,W ], L > 0.

For (t, U, φ) ∈ (tk−1, tk] ×Kc(Rn) × C, we define

D+V (t, U,Φ) = lim suph→0+

1h

[V (t+ h, U + hF (t, Ut), Ut+h) − V (t, U, Ut)].

To investigate stability criteria the following comparison result in terms of aLyapunov function on product spaces is needed. (See Lakshmikantham, Leelaand Sivasundaram [1]).


Theorem 5.5.3 Suppose that

(i) V : R+ ×Kc(Rn) × C → R+ and V ∈ V0.

(ii) D+V (t, U,Φ) ≤ g(t, V (t, U,Φ)), t 6= tk, where g : (tk−1, tk] × R+ → R iscontinuous and for each w ∈ R+, lim(t,z)→(t+k ,w) g(t,z) = g(t+k , w) exists.

(iii) V (t+k , U (t0,Φ0)(t+k ), Ut+k(t0,Φ0)) ≤ Jk[V (tk, U (t0,Φ0)(tk), Utk(t0,Φ0)],

t = tk, and Jk(w) is nondecreasing in w.

Let r(t) = r(t, t0, w0) be the maximal solution of the scalar impulsive differ-ential equation (5.5.3) existing on t ≥ t0. Then

V (t, U (t0,Φ0)(t), Ut(t0,Φ0)) ≤ r(t), t ≥ t0,

where U (t0,Φ0)(t) is any solution of the impulsive set differential equation withdelay (5.5.1) existing on t ≥ t0.

Proof Let U (t0,Φ0)(t) be any solution of (5.5.1) existing on [t0,∞). Definem(t) = V (t, U (t0,Φ0)(t), Ut(t0,Φ0)), so thatm(t0) = V (t0, U (t0,Φ0)(t0),Φ0), and suppose that m(t0) ≤ w0.

Now for t ∈ (tk−1, tk], k = 1, 2, · · · ,

m(t + h) −m(t) = V (t+ h, U (t0,Φ0)(t+ h), Ut+h(t0,Φ0))−V (t, U (t0,Φ0)(t), Ut(t0,Φ0))

= V (t+ h, U (t0,Φ0)(t+ h), Ut+h(t0,Φ0))−V (t + h, U (t0,Φ0)(t) + hF (t, Ut), Ut+h(t0,Φ0))+V (t + h, U (t0,Φ0)(t) + hF (t, Ut), Ut+h(t0,Φ0))−V (t, U (t0,Φ0)(t), Ut(t0,Φ0))

≤ LD[U (t0,Φ0)(t) + hF (t, Ut), U (t0,Φ0)(t+ h))+V (t + h, U (t0,Φ0)(t) + hF (t, Ut), Ut+h(t0,Φ0))−V (t, U (t0,Φ0)(t), Ut(t0,Φ0)),

using (A2). Thus


1h

[m(t + h) −m(t)]

≤ D+V (t, U (t0,Φ0)(t), Ut(t0,Φ0))+L lim sup

h→0+D[U (t0,Φ0)(t + h), U (t0,Φ0)(t) + hF (t, Ut)].

Using the properties of the Hausdorff metric D and the fact that U (t0,Φ0)(t) isa solution (5.5.1), it is not difficult to show that

lim suph→0+

D[U (t0,Φ0)(t+ h), U (t0,Φ0)(t) + hF (t, Ut)] = 0.


Therefore, using (ii), we have

D+m(t) ≤ D+V (t, U (t0,Φ0)(t), Ut(t0,Φ0))≤ g(t, V (t, U (t0,Φ0)(t), Ut(t0,Φ0))= g(t,m(t)).

For t = tk, we get from (iii)

m(t+k ) = V (t+k , U (t0,Φ0)(t+k ), Ut+k(t0,Φ0))

≤ Jk[V (tk, U (t0,Φ0)(tk), Utk(t0,Φ0))]= Jk[m(tk)].

Hence by Theorem 4.6.1 we get

m(t) ≤ r(t), t ≥ t0.

We will now define stability of the null solution of an impulsive set differentialequation with delay.

Definition 5.5.1 Let U (t) = U (t0,Φ0)(t) be any solution of (5.5.1). Then thetrivial solution U (t) ≡ θ is said to be stable if for each ε > 0 and t0 ∈ R+, thereexists a δ = δ(t0, ε) > 0 such that D0[Φ0, θ] < δ implies D[U (t), θ] < ε, t ≥ t0.

The other definitions can be formulated similarly. (See Lakshmikanthamand Leela [1]).

We set, as before

S(ρ) = [U ∈ Kc(Rn) : D[U, θ] < ρ]S(ρ) = [Φ ∈ C : D0[Φ, θ] < ρ]

K = a ∈ C[R+,R+] : a(0) = 0 and a(u) is strictly increasing.

We shall now give a typical result on stability criteria.In order to obtain the trivial solution of (5.5.1) we assume that F (t, θ) ≡ θ

and Ik(θ) ≡ θ for all k.


(i) V : R+ × S(ρ) × S(ρ) → R+, V ∈ V0

and D+V (t, U, φ) ≤ g(t, V (t, U, φ)), t 6= tk, g : R2+ → R, g(t, 0) ≡ 0

and g satisfies the assumptions given in Theorem 5.5.3;

(ii) there exists ρ0 > 0 such that Utk ∈ S(ρ0) implies that Ik(Utk) ∈ S(ρ) forall k andV (t+k , U (t0,Φ0)(t+k ), Ut+k

(t0,Φ0)) ≤ Jk[V (tk, U (t0,Φ0)(tk), Utk(t0,Φ0))],

t = tk, Utk ∈ S(ρ0) and Jk : R+ → R+ is nondecreasing and Jk(0) = 0for all k;

(iii) b(D0[U, θ]) ≤ V (t, U,Φ) ≤ a(D0[Φ, θ]), where a, b ∈ K.


Then the stability properties of the trivial solution of (5.5.3)imply the corre-sponding stability properties of the trivial solution of (5.5.1).

Proof Let 0 < ε < min(ρ, ρ0), t0 ∈ R+ be given. Suppose that the trivialsolution of (5.5.3) is stable. Then, given b(ε) > 0 and t0 ∈ R+, there exists aδ1 = δ1(t0, ε) > 0 such that 0 ≤ w0 < δ1 implies w(t, t0, w0) < b(ε), t ≥ t0,where w(t, t0, w0) is any solution of (5.5.3). Let w0 = a(D0[Φ0, θ]) and choosea δ = δ(t0, ε) such that a(δ) < δ1.

We claim that with this δ we have D0[Φ0, θ] < δ implies D[U (t0,Φ0)(t), θ] <ε, t ≥ t0 for any solution U (t0,Φ0)(t) of (5.5.1). If this is not true therewould exist a solution U (t) = U (t0,Φ0)(t) of (5.5.1) with D0[Φ0, θ] < δ anda t∗ > t0 satisfying tk < t∗ ≤ tk+1, for some k, ε ≤ D[U (t0,Φ0)(t∗), θ] andD[U (t0,Φ0)(t), θ] < ε, t0 ≤ t ≤ tk.

Since 0 < ε < ρ0, condition (ii) shows that D[U (t0,Φ0)(tk), θ] < ε andD0[Ut+k

(t0,Φ0), θ] = D0[Ik(Utk (t0,Φ0)), θ] < ρ.

Hence we can find a t0 such that tk < t0 ≤ t∗ satisfying

ε ≤ D[U (t0, U (t0,Φ0)(t0), θ] < ρ.

Now setting m(t) = V (t, U (t0,Φ0)(t), Ut(t0,Φ0)), t0 ≤ t ≤ t0 and usinghypothesis (i), (ii), and Theorem 5.5.3, we get the estimate

V (t, U (t0,Φ0)(t), Ut(t0,Φ0)) ≤ r(t, t0, a(D0[Φ0, θ])), t0 ≤ t ≤ t0,

where r(t, t0, w0) is the maximal solution of (5.5.3).We are then led, because of(iii), to the contradiction

b(ε) ≤ b(D[U (t0,Φ0)(t0), θ)≤ V (t0, U (t0,Φ0)(t0), Ut0(t0,Φ0))≤ r(t0, t0, a(D0[Φ0, θ]))< r(t0, t0, a(δ)) < r(t0, t0, δ1) < b(ε),

which proves that the trivial solution of (5.5.1) is stable.

Example 5.5.1 Consider the set differential equation with delay on R

DHU = −U (t− τ ), U0 = [φ1, φ2], (5.5.5)

where U (t) = [u1(t), u2(t)].This can be written as

[u ′1, u

′2] = [−u2(t− τ ),−u1(t− τ )]

which is equivalent to the system of ordinary differential equations with delay

u ′1 = −u2(t − τ ), u1,0 = φ1,

u ′2 = −u1(t − τ ), u2,0 = φ2, (5.5.6)


which reduces to u ′′

1 = u1(t− 2τ )

u ′′2

= u2(t− 2τ )(5.5.7)

Suppose the initial functions are given by

φ1(s) =(u10 − u20

2

)eλ1s +

(u10 + u20

2

)e−λ2s,

φ2(s) =(u20 − u10

2

)eλ1s +

(u10 + u20

2

)e−λ2s,

− 2τ ≤ s ≤ 0. (5.5.8)

We choose λ1, λ2 > 0 such that

λ21 = e−2λ1τ , λ2

2 = e2λ2τ ,

so that eλ1t, e−λ2t satisfy (5.5.7).As a result , we have

u1(t) = c1e

λ1t + c2e−λ2t

u2(t) = c3eλ1t + c4e

−λ2t t ≥ 0. (5.5.9)

Using (5.5.6), (5.5.8) and (5.5.9) at t = 0, we compute the values of c1, c2, c3 and c4to find that c1 =

u10 − u20

2, c2 =

u10 + u20

2, c3 =

u20 − u10

2and c4 =

u10 + u20

2.

Hence, the solutions (5.5.9) are given by

u1(t) =(u10 − u20

2

)eλ1t +

(u10 + u20

2

)e−λ2t,

u2(t) =(u20 − u10

2

)eλ1t +

(u10 + u20

2

)e−λ2t, t ≥ 0.

(5.5.10)

Thus, the solution of (5.5.7) is given by (5.5.8) and (5.5.10), whereφ1(0) = u10 and φ2(0) = u20.Case 1: If u20 = u10 = u0, then (5.5.10) reduces to

u1(t) = u0e

−λ2t,

u2(t) = u0e−λ2t, t ≥ 0,

orU (t) = [u0, u0]e−λ2t, t ≥ 0.

In this case impulses have no role to play.Case 2: If u10 = −u20 = −u0, then (5.5.10) reduces to

u1(t) = −u0e

λ1t

u2(t) = u0eλ1t, t ≥ 0,


orU (t) = [−u0, u0]eλ1t, t ≥ 0.

Now suppose we introduce impulses to the set differential equation with delay,(5.5.5) at t = tk, k = 1, 2, . . . as

Ut+k= akUtk , ak > 0. (5.5.11)

Then the solution to the impulsive set differential equation with delay (5.5.5),(5.5.11), is given by

U (t) =∏

0<tk<t

ak[−u0, u0]eλ1t, t ≥ 0. (5.5.12)

If the ak’s satisfyλ1tk+1 + ln ak ≤ λ1tk for all k, (5.5.13)

then ak ≤ eλ1(tk−tk+1) and using this estimate in (5.5.12), we obtain

||U (t)|| ≤ ||U0|| eλ1t1

where ||U (t)|| = D[U (t), θ]. Choosing δ = ε2e

−λ1t1 , it follows that ||U (t)|| <ε, t ≥ 0, provided ||U0|| < δ. Hence the stability of the trivial solution of(5.5.5) and (5.5.11) follows.

For asymptotic stability, we strengthen the assumption (5.5.13) to

λ1tk+1 + ln(αak) ≤ λ1tk for all k, where α > 1. (5.5.14)

Then ak ≤ 1αe

λ1(tk−tk+1) and using this estimate in (5.5.12), we obtain

lim suph→∞

‖U (t)‖ = 0.

Thus the trivial solution of the impulsive differential equation with delay (5.5.5)and (5.5.11) is asymptotically stable.

5.6 Set Difference Equations

It is well known that difference equations appear as the natural description ofobserved evolution phenomenon, because most measurements of time evolvingvariables are discrete and as such these equations are, in their own right, impor-tant models. More importantly, difference equations also appear in the studyof discretization methods for differential equations. Consequently, initial valueproblems (IVPs) of difference equations should be formulated as,

∆un ≡ un+1 − un = F (n, un), un0 = u0, n ≥ n0 ≥ 0

to represent discretization of corresponding ODEs. Nonetheless, in the literatureof difference equations, we find usually the following type of formulation

un+1 = f(n, un), un0 = u0, n ≥ n0 ≥ 0,

5.6 SET DIFFERENCE EQUATIONS 181

where, of course , f(n, un) = un + F (n, un).In this section, we plan to discuss set difference equations parallel to set

differential equations, and develop the theory of such equations. We shall onlyprovide a few typical results so as to advance the investigation of set differenceequations further, since the theory of such equations is a lot richer than thecorresponding SDEs.

Let N denote the natural numbers and N+ = N∪0.We denote by N+n0

theset

N+n0

= n0, n0 + 1, · · · , n0 + l, · · · ,

with l ∈ N+ and n0 ∈ N+.We consider the set difference equation of the form

Un+1 = F (n, Un), Un0 = U0, (5.6.1)

where F : N+n0

× Kc(Rq) → Kc(Rq) is continuous in U for each n and Un ∈Kc(Rq) for each n ≥ n0.

Since we shall be using n in difference equations, we shall employ the metricspace (Kc(Rq), D) for (Kc(Rn), D) used earlier. This will avoid confusion. Thepossibility of obtaining the values of solutions of (5.6.1) recursively is very im-portant and does not have a counter part in other kinds of equations. For thisreason, we sometimes reduce continuous problems to approximate differenceproblems. For simple set difference equations we can find solutions in closedform. However, deducing information on the qualitative and quantitative be-havior of solutions of (5.6.1) by the comparison principle, is very effective asusual.

We need the following comparison principle for ordinary difference equations.See Lakshmikantham and Trigiante [1] for details.

Theorem 5.6.1 Let n ∈ N+n0, r ≥ 0 and g(n, r) be a nondecreasing function

in r for each n. Suppose that for each n ≥ n0, the inequalities

yn+1 ≤ g(n, yn), (5.6.2)

zn+1 ≥ g(n, zn), (5.6.3)

hold. If yn0 ≤ zn0 , then yn ≤ zn for all n ≥ n0.

Corollary 5.6.1 Let n ∈ N+n0, kn ≥ 0 and yn+1 ≤ knyn + pn. Then,

yn ≤ yn0

n−1∏

s=n0

ks +n−1∑

s=n0

ps

n−1∏

τ=s+1

kτ , n ≥ n0. (5.6.4)

Corollary 5.6.2 (Discrete Gronwall Inequality)Let n ∈ N+

n0, kn ≥ 0 and

yn+1 ≤ yn0 +n∑

s=n0

[ksys + ps].


Then,

yn ≤ yn0

n−1∏

s=n0

(1 + ks) +n−1∑

s=n0

ps

n−1∏

τ=s+1

(1 + kτ ) (5.6.5)

≤ yn0 exp

(n−1∑

s=n0

ks

)+

n−1∑

s=n0

ps exp

(n−1∑

τ=s+1

kτ

), n ≥ n0.

The following theorem estimates the solution of the set difference equationin terms of the solution of the scalar difference equation

zn+1 = g(n, zn), zn0 = z0, (5.6.6)

where g(n, r) is continuous in r for each n and nondecreasing in r for each n.We prove the following result.

Theorem 5.6.2 Assume that F (n, U ) is continuous in U for each n and

D[F (n, U ), θ] = ‖F (n, U )‖ ≤ g(n, ‖U‖) (5.6.7)

where g(n, r) is given in (5.6.6). Then, ‖Un0‖ ≤ zn0 implies,

‖Un+1‖ ≤ zn+1, for n ≥ n0. (5.6.8)

Proof Set yn+1 = ‖Un+1‖, so that we get

yn+1 = ‖F (n, Un)‖ ≤ g(n, ‖Un‖) = g(n, yn), n ≥ n0.

Let zn+1 be the solution of (5.6.6) with zn0 = yn0 . Then, Theorem 5.6.1 yieldsimmediately,

yn+1 ≤ zn+1, n ≥ n0,

which implies (5.6.8), completing the proof.The assumption (5.6.7) can be replaced by a weaker condition, namely

D[F (n, U ), θ] = ‖F (n, U )‖ ≤ ‖U‖+ w(n, ‖U‖), U ∈ Kc(Rq).

Now, set g(n, r) = r + w(n, r) and assume that g(n, r) is nondecreasing in r,for each n. This version of Theorem 5.6.2 is more suitable because w(n, r) neednot be positive, and hence the solutions of (5.6.6) could have better properties.This observation is useful in extending the Lyapunov-like method for (5.6.1).

Let V : N+n0

× Kc(Rq) → R+ be a given function. We have the followingcomparison result.

Theorem 5.6.3 Let V (n, U ) given above satisfy

V (n+ 1, Un+1) ≤ V (n, Un) + w(n, V (n, Un))≡ g(n, V (n, Un)), n ≥ n0.

Then, V (n, Un0) ≤ zn0 implies,

V (n + 1, Un+1) ≤ zn+1, n ≥ n0, (5.6.9)

where zn+1 = zn+1(n0, zn0) is the solution of (5.6.6).

5.6 SET DIFFERENCE EQUATIONS 183

Proof Set yn+1 = V (n+ 1, Un+1), so that yn0 = V (n0, Un0) ≤ zn0 and

yn+1 ≤ yn + w(n, yn), n ≥ n0.

Consequently, g(n, r) = r + w(n, r). Hence, by Theorem 5.6.1, we get

yn+1 ≤ zn+1, n ≥ n0,

where zn+1 is the solution of (5.6.7). This implies the stated estimate.Using Theorem 5.6.3., we can prove the stability results for the solutions of

set difference equation (5.6.1).

Theorem 5.6.4 Let the assumptions of Theorem 5.6.3 hold. Suppose furtherthat

b(‖U‖) ≤ V (n, U ) ≤ a(‖U‖),

where a, b ∈ K, n ∈ N+n0

and U ∈ Kc(Rq). The stability properties of thetrivial solution of (5.6.6) imply the corresponding stability properties of the trivialsolution of (5.6.1).

Proof Suppose that the trivial solution of (5.6.6) is asymptotically stable.Then, it is stable. Let, 0 < ε, n0 ∈ N be given. Then, b(ε) > 0, n0 ∈ N,there exists a δ1 = δ1(n0, ε) such that

0 ≤ zn0 < δ1 implies zn+1 < b(ε), n ≥ n0.

Choose δ = δ(n0, ε) satisfyinga(δ) < δ1.

Then Theorem 5.6.3 gives,

V (n+ 1, Un+1) ≤ zn+1, n ≥ n0,

which shows that

b(‖Un+1‖) ≤ V (n+ 1, Un+1) ≤ zn+1, n ≥ n0.

Let ‖Un0‖ < δ. Choose zn0 = V (n0, Un0) so that we have

zn0 ≤ a(‖Un0‖) ≤ a(δ) < δ1.

We then get,b(‖Un+1‖) < b(ε), n ≥ n0,

which implies the stability of the trivial solution of (5.6.1).For asymptotic stability, we observe that

b(‖Un+1‖) ≤ V (n+ 1, Un+1) ≤ zn+1, n ≥ n0.

Since zn+1 → 0 as n → ∞, we get ‖Un+1‖ → 0 as n → ∞. The proof iscomplete.


As an example, take g(n, r) = anr where an ∈ R. Then the solution of

zn+1 = anzn, zn0 = z0,

is given by

zn = z0

n−1∏

i=n0

ai.

We have the following two cases.(a) If

|n−1∏

i=n0

ai| ≤M (n0),

then |zn| ≤ |z0|M (n0) and therefore enough to take δ(ε, n0) =ε

M (n0), to get

stability.(b) If

limn→∞

|n−1∏

i=n0

ai| = 0,

then asymptotic stability results.Consequently, Theorem 5.6.3. yields the corresponding stability properties

of the trivial solution of (5.6.1).Corresponding to the familiar example in SDE, namely DHU = (−1)U, let

us consider the example in set difference equation

∆Un ≡ Un+1 − Un = (−1)Un, U0 = U0 ∈ Kc(R).

Of course, we need to assume that the Hukuhara difference Un+1 − Un existsfor all n. Letting Un = [un, vn], U0 = U0 = [u0, v0], using the interval methodsas before, we solve the ordinary difference equations,

un+1 = un − vn, vn+1 = vn − un,

which yieldun+1 = 2n[u0 − v0], vn+1 = 2n[v0 − u0],

and thereforeUn+1 = [u0 − v0, v0 − u0]2n.

Thus, diam ‖Un+1‖ → ∞ as n → ∞.If, on the other hand, we consider set difference equation

Un+1 = (−1)Un, U0 = U0 ∈ Kc(R),

we get , using the interval methods,

un+1 = −vn, vn+1 = −un,

5.7 SET DIFFERENTIAL EQUATIONS WITH CAUSAL OPERATORS 185

which give us

U2n = [u0, v0], U2n+1 = [−v0,−u0] = (−1)U0.

Hence diam ‖Un‖ = diam ‖U0‖, which is finite.As a last example, if we analyze

Un+1 =12Un, U0 = U0 = [u0, v0],

we find that Un =(

12

)nU0 and consequently, diam‖Un‖ → 0 as n → ∞.

5.7 Set Differential Equations with Causal Op-

erators

We shall devote this section to extend certain basic results to set differentialequations (SDEs) with causal or nonanticipative maps of Volterra type, sincesuch equations provide a unified treatment of the basic theory of SDEs, SDEswith delay and set integro differential equations, which in turn include ordinarydynamic systems of the corresponding types.

Let E = C[[t0, T ],Kc(Rn)] with norm

D0[U, θ] = supt0≤t≤T

D[U (t), θ].

Definition 5.7.1 Suppose that Q ∈ C[E,E], then Q is said to be a causal mapor a nonanticipative map if U (s) = V (s), t0 ≤ s ≤ t ≤ T, where U, V ∈ E,then (QU )(s) = (QV )(s), t0 ≤ s ≤ t.

We define the IVP for SDE with causal map, using the Hukuhara derivativeas follows:

DHU (t) = (QU )(t), U (t0) = U0 ∈ Kc(Rn). (5.7.1)

Before we proceed to prove an existence and uniqueness result for (5.7.1), weneed the following comparison results.

Theorem 5.7.1 Assume that m ∈ C[J,R+], g ∈ C[J × R+,R+] and for t ∈J = [t0, T ],

D−m(t) ≤ g(t, |m|0(t)),

where |m|0(t) = sup0≤s≤t |m(s)|. Suppose that r(t) = r(t, t0, w0) is the maximalsolution of the scalar differential equation

w′ = g(t, w), w(t0) = w0 ≥ 0, (5.7.2)

existing on J. Then m(t0) ≤ w0 implies m(t) ≤ r(t), t ∈ J.Proof To prove the stated inequality, it is enough to prove that

m(t) < w(t, t0, w0, ε), t ≥ t0, t ∈ J, (5.7.3)


where w(t, t0, w0, ε) is any solution of

w′ = g(t, w) + ε, w(t0) = w0 + ε, ε > 0,

since limε→0+ w(t, t0, w0, ε) = r(t, t0, w0).If (5.7.3) is not true, there exists a t1 > t0 such that m(t1) = w(t1, t0, w0, ε)

and m(t) < w(t, t0, w0, ε), t0 ≤ t < t1, in view of the fact m(t0) < w0 + ε.Hence,

D−m(t1) ≥ w′(t1, t0, w0, ε) = g(t1, w(t1, t0, w0, ε)) + ε. (5.7.4)

Now g(t, w) ≥ 0 implies that w(t, t0, w0, ε) is nondecreasing in t, and this gives

|m|0(t1) = w(t1, t0, w0, ε) = m(t1),

which in turn yields

D−m(t1) ≤ g(t1, |m|0(t1)) = g(t1, w(t1, t0, w0, ε))

which is a contradiction to (5.7.4). Hence the theorem follows.Next we consider an estimate of any two solutions of (5.7.1) in terms of the

maximal solution of (5.7.2) utilizing Theorem 5.7.1.We define D0[U, V ](t) = maxt0≤s≤tD[U (s), V (s)].

Theorem 5.7.2 Let Q ∈ C[E,E] be a causal map such that for t ∈ J,

D[(QU )(t), (QV )(t)] ≤ g(t,D0[U, V ](t)), (5.7.5)

where g ∈ C[J ×R+,R+]. Suppose further that the maximal solution r(t, t0, w0)of the differential equation (5.7.2) exists on J. Then, if U (t), V (t) are any twosolutions of (5.7.1) through U (t0) = U0, V (t0) = V0, U0, V0 ∈ Kc(Rn) on Jrespectively, we have

D[U (t), V (t)] ≤ r(t, t0, w0), t ∈ J, (5.7.6)


Proof Set m(t) = D[U (t), V (t)]. Then m(t0) = D[U0, V0] ≤ w0. Now for smallh > 0, t ∈ J, consider m(t + h) = D[U (t + h), V (t + h)]. Using the property(1.3.8) of the Hausdorff metric D, we get successively, the following relations:

m(t + h) ≤ D[U (t+ h), U (t) + h(QU )(t)] +D[U (t) + h(QU )(t), V (t + h)]≤ D[U (t+ h), U (t) + h(QU )(t)]

+D[U (t) + h(QU )(t), V (t) + h(QV )(t)]+D[V (t) + h(QV )(t), V (t + h)]

≤ D[U (t+ h), U (t) + h(QU )(t)]+D[U (t) + h(QU )(t), U (t) + h(QV )(t)]+D[U (t) + h(QV )(t), V (t) + h(QV )(t)]+D[V (t) + h(QV )(t), V (t + h)].


Next using the property (1.3.9) of the Hausdorff metric D and the fact that theHukuhara differences U (t+h)−U (t) and V (t+h)−V (t) exist for small h > 0,we arrive at,

m(t + h) ≤ D[U (t) + Z(t, h), U (t) + h(QU )(t)]+D[h(QU )(t), h(QV )(t)] +D[U (t), V (t)]+D[V (t) + h(QV )(t), V (t) + Y (t, h)],

where U (t + h) = U (t) + Z(t, h) and V (t + h) = V (t) + Y (t, h). Again theproperty (1.3.9) gives

m(t+ h) ≤ D[Z(t, h), h(QU )(t)] +D[h(QU )(t), h(QV )(t)]+D[U (t), V (t)] +D[h(QV )(t), Y (t, h)].

Since the Hukuhara differences exist, we can replace Z(t, h) and Y (t, h) withU (t + h) − U (t) and V (t + h) − V (t) respectively. This gives, on subtractingm(t) and dividing both sides with h > 0,

m(t + h) −m(t)h

≤ D

[U (t+ h) − U (t)

h, (QU )(t)

]

+D[(QU )(t), (QV )(t)]

+D[(QV )(t),

V (t+ h) − V (t)h

].

Now, taking limit supremum as h → 0+ and using the fact that U (t) and V (t)are solutions of (5.7.1) along with assumption (5.7.5), we obtain

D+m(t) ≤ D[(QU )(t), (QV )(t)]≤ g(t,D0[U, V ](t))= g(t, |m|0(t)), t ∈ J.

Now, Theorem 5.7.1 guarantees the stated conclusion and the proof is complete.

Corollary 5.7.1 Let Q ∈ C[E,E] be a causal map and

D[(QU )(t), θ] ≤ g(t,D0[U, θ](t)),

where g ∈ C[J ×R+,R+]. Also, suppose that r(t, t0, w0) is the maximal solutionof the scalar differential equation (5.7.2). If U (t, t0, U0) is any solution of (5.7.1)through (t0, U0) with U0 ∈ Kc(Rn), then

D[U0, θ] ≤ w0 implies D[U (t), θ] ≤ r(t, t0, w0), t ∈ J.

Let us begin by proving a local existence result using successive approxima-tions.



(a) Q ∈ C[B,E] where B = B(U0, b) = U ∈ E : D0[U,U0] ≤ b, is a causalmap and D0[(QU ), θ](t) ≤M1, on B;

(b) g ∈ C[J × [0, 2b],R+], g(t, w) ≤ M2 on J × [0, 2b], g(t, 0) ≡ 0, g(t, w) isnondecreasing in w for each t ∈ J and w(t) = 0 is the only solution of

w′ = g(t, w), w(t0) = 0 on J ; (5.7.7)

(c) D[(QU )(t), (QV )(t)] ≤ g(t,D0[U, V ](t)) on B.

Then, the successive approximations defined by

Un+1(t) = U0 +∫ t

t0

(QUn)(s) ds, n = 0, 1, 2, · · · , (5.7.8)

exist on J0 = [t0, t0 +η) where η = min[T − t0, bM ] and M = max(M1,M2), and

converge uniformly to the unique solution U (t) of (5.7.1).

Proof For t ∈ J0, we have, by induction, using property (1.3.9) and (1.7.11) ofthe Hausdorff metric D,

D[Un+1(t), U0] = D

[U0 +

∫ t

t0

(QUn)(s) ds, U0

]

= D

[∫ t

t0

(QUn)(s) ds, θ]

≤∫ t

t0

D[(QUn)(s), θ] ds

≤∫ t

t0

D0[QUn, θ](t) ds ≤M1(t − t0) ≤ M (t− t0) ≤ b,

which shows the successive approximations are well defined on J0.

Next, we define successive approximations for the problem (5.7.7) as follows:

w0(t) = M (t− t0)

wn+1(t) =∫ t

t0

g(s, wn(s)) ds, t ∈ J0, n = 0, 1, 2, · · · .

Then,

w1(t) =∫ t

t0

g(s, w0(s)) ds ≤ M2(t− t0) ≤ M (t− t0) = w0(t).

Assume for some k > 1, t ∈ J0, that

wk(t) ≤ wk−1(t).


Then, using the monotonicity of g, we get

wk+1(t) =∫ t

t0

g(s, wk(s)) ds ≤∫ t

t0

g(s, wk−1(s)) ds = wk(t).

Hence, the sequence wk(t) is monotone decreasing.Since w′

k(t) = g(t, wk−1(t)) ≤ M2, t ∈ J0, we conclude by the Ascoli-Arzelatheorem and the monotonicity of the sequence wk(t) that

limt→∞

wn(t) = w(t)

uniformly on J0. Since w(t) satisfies equation (5.7.7), we get w(t) ≡ 0 on J0

from condition (b) on J0. Observing that for each t ∈ J0, t0 ≤ s ≤ t,

D[U1(s), U0] = D

[U0 +

∫ s

t0

(QU0)(ξ) dξ, U0

]

= D

[∫ s

t0

(QU0)(ξ) dξ, θ

]

≤∫ s

t0

D [(QU0)(ξ), θ] dξ

≤ D0[(QU0), θ](ξ) (s − t0) ≤ D0[(QU0), θ](ξ) (t − t0)≤ M1(t− t0) ≤ M (t− t0) = w0(t),

which implies that D0[U1, U0](t) ≤ w0(t).We assume for some k > 1,

D0[Uk, Uk−1](t) ≤ wk−1(t), t ∈ J0.

Consider, for any t ∈ J0, t0 ≤ s ≤ t,

D[Uk+1(s), Uk(s)] ≤∫ s

t0

D[(QUk)(ξ), (QUk−1)(ξ)] dξ

≤∫ s

t0

g(ξ,D0[Uk, Uk−1](ξ)) dξ

≤∫ s

t0

g(ξ, wk−1(ξ)) dξ

≤∫ t

t0

g(ξ, wk−1(ξ)) dξ = wk(t),

which further gives,

D0[Uk+1, Uk](t) ≤ wk(t), t ∈ J0.

Thus we conclude that

D0[Un+1, Un](t) ≤ wn(t), (5.7.9)


for t ∈ J0 and for all n = 0, 1, 2, · · · .We claim that Un(t) is a Cauchy sequence. To show this, let n ≤ m.

Setting v(t) = D[Un(t), Um(t)] and using (5.7.8), we get

D+v(t) ≤ D[DHUn(t), DHUm(t)]= D[(QUn−1)(t), (QUm−1)(t)]≤ D[(QUn−1)(t), (QUn)(t)] +D[(QUn)(t), (QUm)(t)]

+D[(QUm)(t), (QUm−1)(t)]

≤ g(t,D0[Un−1, Un](t)) + g(t,D0[Un, Um](t))+g(t,D0[Um−1, Um](t))

≤ g(t, wn−1(t)) + g(t, |v|0(t)) + g(t, wn−1(t))= g(t, |v|0(t)) + 2g(t, wn−1(t)).

The above inequalities yield, on using the Theorem 5.7.1, the estimate

v(t) ≤ rn(t), t ∈ J0,

where rn(t) is the maximal solution of

r′n(t) = g(t, rn) + 2g(t, wn−1(t)),rn(t0) = 0,

for each n. Since as n → ∞, 2g(t, wn−1(t)) → 0 uniformly on J0, it followsby Lemma 1.3.1 in Lakshmikantham and Leela [1] that rn(t) → 0, as n → ∞uniformly on J0. This implies from (5.7.9) that Un(t) converges uniformly toU (t) on J0 and clearly U (t) is a solution of (5.7.1).

To prove uniqueness, let V (t) be another solution of (5.7.1) on J0. Set m(t) =D[U (t), V (t)]. Then m(t0) = 0 and

D+m(t) ≤ g(t, |m|0(t)), t ∈ J0.

Since m(t0) = 0, it follows from Theorem 5.7.1 that

m(t) ≤ r(t, t0, 0), t ∈ J0,

where r(t, t0, 0) is the maximal solution of (5.7.7). The assumption (b) nowshows that U (t) = V (t), t ∈ J0, proving uniqueness.

Assuming local existence, we next discuss a global existence result.

Theorem 5.7.4 Let Q ∈ C[E,E] be a causal map such that

D[(QU )(t), θ] ≤ g(t,D0[U, θ](t)), (5.7.10)

where g ∈ C[R2+,R+], g(t, w) is nondecreasing in w for each t ∈ R+ and the

maximal solution r(t) = r(t, t0, w0) of (5.7.2) exists on [t0,∞). Suppose furtherthat Q is smooth enough to guarantee the local existence of solutions of (5.7.1)for any (t0, U0) ∈ R+ ×Kc(Rn). Then, the largest interval of existence for anysolution U (t, t0, U0) of (5.7.1) is [t0,∞), whenever D[U0, θ] ≤ w0.


Proof Suppose that U (t) = U (t, t0, U0) is any solution of (5.7.1) existing on[t0, β), t0 < β <∞, with D[U0, θ] ≤ w0, and the value of β cannot be increased.Define m(t) = D[U (t), θ] and note that m(t0) ≤ w0. Then it follows that,

D+m(t) ≤ D[DHU (t), θ] ≤ D[(QU )(t), θ] ≤ g(t,D0[U, θ](t)).

Using Comparison Theorem 5.7.1, we obtain

m(t) ≤ r(t), t0 ≤ t < β. (5.7.11)

For any t1, t2 such that t0 < t1 < t2 < β, using (5.7.10) and the properties ofHausdorff metric D,

D[U (t1), U (t2)] = D

[∫ t1

t0

(QU )(s) ds,

∫ t2

t0

(QU )(s) ds]

≤∫ t2

t1

D[(QU )(s), θ] ds

≤∫ t2

t1

g(s,D0[U, θ](s)) ds.

Employing the estimate (5.7.11) and the monotonicity of g(t, w), we find,

D[U (t1), U (t2)] ≤∫ t2

t1

g(s, r(s)) ds = r(t2) − r(t1).

Since limt→β− r(t, t0, w0) exists, taking the limit as t1, t2 → β−, we get thatU (tn) is a Cauchy sequence and therefore limt→β− U (t, t0, U0) = Uβ exists.

We then consider the IVP

DHU (t) = (QU )(t), U (β) = Uβ.

As we have assumed the local existence, we note that U (t, t0, U0) can be contin-ued beyond β, contradicting our assumption that β cannot be increased. Thusevery solution U (t, t0, U0) of (5.7.1) such that D[U0, θ] ≤ w0 exists globally on[t0,∞) and hence the proof.

To prove a comparison result in terms of Lyapunov-like functions, we needthe following known result from Lakshmikantham and Rama Mohana Rao [1].

Lemma 5.7.1 Let g0, g ∈ C[R2+,R] be such that

g0(t, w) ≤ g(t, w), (t, w) ∈ R2+. (5.7.12)

Then the right maximal solution r(t, t0, w0) of (5.7.2) and the left maximalsolution η(t, T0, v0) of

v′ = g0(t, v), v(T0) = v0 ≥ 0, (5.7.13)

satisfy the relation

r(t, t0, w0) ≤ η(t, T0, v0), t ∈ [t0, T0],

whenever r(T0, t0, w0) ≤ v0.



(i) L ∈ C[R+ ×Kc(Rn),R+], L(t, U ) is locally Lipschitzian in U, i.e.

|L(t, U ) − L(t, V )| ≤ KD[U, V ], U, V ∈ Kc(Rn), t ∈ R+;

(ii) g0, g ∈ C[R+ × R+,R],g0(t, w) ≤ g(t, w),

η(t, T0, v0) is the left maximal solution of

v′ = g0(t, v), v(T0) = v0 ≥ 0,

existing on t0 ≤ t ≤ T0 and r(t, t0, w0) is the maximal solution of (5.7.2)existing on [t0,∞);

(iii) D−L(t, U (t)) ≤ g(t, L(t, U (t))) on Ω, where

Ω = [U ∈ E : L(s, U (s)) ≤ η(s, t, L(t, U (t))), t0 ≤ s ≤ t],

and

D−L(t, U (t)) = liminfh→0−1h

[L(t+ h, U (t) + h(QU )(t)) − L(t, U (t))] .

Then we have,L(t, U (t, t0, U0)) ≤ r(t, t0, w0), t ≥ t0. (5.7.14)

whenever L(t0, U0) ≤ w0.

Proof To prove (5.7.14), we setm(t) = L(t, U (t, t0, U0)), t ≥ t0 so thatm(t0) =L(t0, U0) ≤ w0. Let w(t, ε) be any solution of

w′ = g(t, w) + ε, w(t0) = w0 + ε,

for sufficiently small ε > 0.Then, since r(t, t0, w0) = limε→0+ w(t, ε), it is enoughto prove that

m(t) < w(t, ε) for t ≥ t0.

If this is not true, there exists a t1 > t0 such that m(t1) = w(t1, ε),m(t) < w(t, ε), t0 ≤ t < t1. This implies that

D−m(t1) ≥ w′(t, ε) = g(t1,m(t1)) + ε. (5.7.15)

Now consider the left maximal solution η(s, t1,m(t1)) of (5.7.13) withv(t1) = m(t1) on the interval t0 ≤ s ≤ t1. By Lemma 5.7.1, we have

r(s, t0, w0) ≤ η(s, t1,m(t1)), s ∈ [t0, t1].

Sincer(t1, t0, w0) = lim

ε→0+w(t1, ε) = m(t1) = η(t1, t1,m(t1))


and m(s) ≤ w(s, ε) for t0 ≤ s ≤ t1, it follows that

m(s) ≤ r(s, t0, w0) ≤ η(s, t1,m(t1)), t0 ≤ s ≤ t1.

This inequality implies that hypothesis (iii) holds for U (s, t0, U0) ont0 ≤ s ≤ t1, and hence, standard computation yields,

D−m(t1) ≤ g(t1,m(t1))

which contradicts the relation (5.7.15). Thus m(t) ≤ r(t, t0, w0), t ≥ t0 and theproof is complete.

The above comparison result in terms of Lyapunov like functions is a usefultool to establish some appropriate stability and boundedness results for setdifferential equations with causal maps (5.7.1) analogous to original Lyapunovresults and their extensions. In order to match the behavior of solutions of(5.7.1) with the corresponding ordinary differential equation with causal map,we need to use the existence of Hukuhara difference U0 − V0 = W0 in the initialcondition as in Section 3.3 and study the stability or boundedness criteria ofU (t, t0, U0 − V0) = U (t, t0,W0) of (5.7.1).

We present a simple example in Kc(R).Consider

DHU (t) = −∫ t

0

U (s) ds, U (0) = U0 ∈ Kc(R).

Then using interval methods, we get

u′1 = −∫ t

0

u2(s) ds,

u′2 = −∫ t

0

u1(s) ds,

where U = [u1, u2] and U0 = [u10, u20]. Clearly this yields

u(4)1 = u1, u1(0) = u10,

u(4)2 = u2, u2(0) = u20,

whose solutions are given by

u1(t) =(u10 − u20

2

)(et + e−t

2

)+(u10 + u20

2

)cos t

u2(t) =(u20 − u10

2

)(et + e−t

2

)+(u10 + u20

2

)cos t.


That is, t ≥ 0,

U (t, 0, U0) =[−1

2(u20 − u10),

12(u20 − u10)

](et + e−t

2

)

+[12

(u10 + u20

2

),12

(u10 + u20

2

)]cos t, t ≥ 0.

Thus choosing

V0 =[−1

2(u20 − u10),

12(u20 − u10)

],

we obtain

U (t, 0,W0) =[12

(u10 + u20

2

),12

(u10 + u20

2

)]cos t, t ≥ 0.

which implies the stability of the trivial solution of (5.7.1).We next give an example which illustrates that one can get asymptotic sta-

bility as well in SDE with causal maps.Consider the following differential equation,

DHU = −aU − b

∫ t

0

U (s) ds, U (0) = U0 ∈ Kc(Rn), (5.7.16)

a, b > 0.As before, we take U (t) = [u1(t), u2(t)], U0 = [u10, u20], which reduces to

u′1 = −au2 − b

∫ t

0

u2(s) ds,

u′2 = −au1 − b

∫ t

0

u1(s) ds,

u(4)1 = a2u′′1 + 2abu′1 + b2u1,

u(4)2 = a2u′′2 + 2abu′2 + b2u2,

from which we obtain by choosing a = 1 and b = 2. Using the initial conditions,

u1(t) =16

(u10 − u20) e−t +13

(u10 − u20) e2t

+e−12 t

[12

(u10 + u20) cos

(√7

2t

)− 1

2√

7(u10 + u20) sin

(√7

2t

)]

u2(t) =16

(u20 − u10) e−t +13

(u20 − u10) e2t

+e−12 t

[12

(u20 + u10) cos

(√7

2t

)− 1

2√

7(u20 + u10) sin

(√7

2t

)].

5.8 LYAPUNOV-LIKE FUNCTIONS IN KC(RD+ ) 195

Thus it follows that,

U (t) = (u20 − u10)[−1

6,16

]e−t + (u20 − u10)

[−1

3,13

]e2t

+(u10 + u20)[12,12

]e−

12 t cos

(√7

2t

)

−(u10 + u20)[

12√

7,

12√

7

]e−

12 t sin

(√7

2t

).

Now choosing u10 = u20, we eliminate the undesirable terms and therefore, weget asymptotic stability of the zero solution of (5.7.16).

5.8 Lyapunov-like Functions in Kc(Rd+)

Recall that in Section 2.3, a partial order in (Kc(Rn), D) is introduced and em-ploying it, the existence of extremal solutions and a suitable comparison resultwere discussed. In this section, using the comparison result, we shall considerLyapunov-like functions whose values are in (Kc(Rd

+), D). For this purpose, weshall also utilize the set differential systems developed in Section 3.7, and con-sequently, we consider the set differential system, namely,

DHU = F (t, U ), U (t0) = U0 ∈ Kc(Rn)N , (5.8.1)

where F ∈ C[R+ ×Kc(Rn)N ,Kc(Rn)N ], Kc(Rn)N = Kc(Rn)×Kc(Rn)× · · ·×Kc(Rn), N times. Since we need a partial order in (Kc(Rd), D), we shall assumethat we have introduced it in a similar way. We shall begin by proving a com-parison result in terms of Lyapunov-like functions whose values are in Kc(Rd

+).We also need a map,

ρ : Kc(Rn)N → Kc(Rd+). (5.8.2)


(i) L ∈ C[R+×Kc(Rn)N ,Kc(Rd+)] and whenever Hukuhara difference of A,B ∈

Kc(Rn)N exists,L(t, A) ≤ L(t, B) +Kρ[A− B], (5.8.3)

where K ≥ 0 is a local Lipschitz constant;

(ii) G ∈ C[R+ ×Kc(Rd+),Kc(Rd)], the Hukuhara difference

L(t + h, U (t+ h)) − L(t, U (t)) (5.8.4)

exists, and for t ∈ R+, U ∈ Kc(Rn),

D+L(t, U ) ≡ lim suph→0+

1h

[L(t+ h, U + hF (t, U )) − L(t, U )]

≤ G(t, L(t, U )). (5.8.5)


Then, if U (t) = U (t, t0, U0) is any solution of (5.8.1) existing for t ≥ t0, suchthat L(t0, U0) ≤ R0, we have

L(t, U (t)) ≤ R(t, t0, R0), t ≥ t0, (5.8.6)

where R(t, t0, R0) is the maximal solution of SDE

DHR = G(t, R), R(t0) = R0 ∈ Kc(Rd+), (5.8.7)

existing for t ≥ t0.

Proof Define m(t) = L(t, U (t)), where U (t) = U (t, t0, U0) is any solution of(5.8.1) existing on [t0,∞). Clearly, m(t) ∈ Kc(Rd

+) for each t ∈ [t0,∞) andm(t0) ≤ R0. Now for small h > 0,

m(t + h) = L(t+ h, U (t+ h)) = L(t + h, U (t) + hF (t, U (t)) + o(h))= L(t, U (t)) + Z(t, U (t);h)

because of (5.8.4). It therefore follows, using (5.8.3), that

m(t + h) −m(t) = Z(t, U (t);h)= L(t + h, U (t) + hF (t, U (t)) + o(h)) − L(t, U (t))≤ L(t + h, U (t) + hF (t, U (t))) − L(t, U (t)) +Kρ(o(h)).

Hence we arrive at

D+m(t) ≡ lim suph→0+

1h

[m(t+ h) −m(t)]

≤ lim suph→0+

Kρ(o(h))h

+ lim suph→0+

[L(t+ h, U (t) + hF (t, U (t))− L(t, U (t))]h

≤ D+V (t, U (t)) ≤ G(t, L(t, U (t))) = G(t,m(t)), t ≥ t0,

in view of (5.8.5). This implies by Theorem 2.4.5,

L(t, U (t)) = m(t) ≤ R(t, t0, R0), t ≥ t0,

R(t, t0, R0) being the maximal solution of (5.8.7). The proof is complete.Employing the estimate (5.8.6), we can now discuss the desired qualitative

properties of solutions U (t) of set differential system (5.8.1). For this purpose,we need suitable notions of stability for SDEs (5.8.1) and (5.8.7).

Definition 5.8.1 The trivial solution of (5.8.1) is said to be equi-stable if, givenε > 0 and t0 ∈ R+, there exists a δ = δ(t0, ε) > 0 such that

D0[W0, θ] < δ implies D0[U (t, t0,W0), θ] < ε, t ≥ t0,

where W0 is chosen in such a way that U0 = V0 + W0, that is, the Hukuharadifference U0 − V0 exists and is equal to W0.

5.9 SET DIFFERENTIAL EQUATIONS IN (KC(E), D), 197

Definition 5.8.2 The trivial solution of (5.8.7) is said to be equi-stable if givenε > 0 and t0 ∈ R+, there exists a δ = δ(t0, ε) > 0 such that

‖Q0‖ < δ implies ‖R(t, t0, Q0)‖ < ε, t ≥ t0,

where the Hukuhara difference R0 − S0 = Q0 is assumed to exist and‖R‖ = sup[‖r‖ : r ∈ R ∈ Kc(Rd

+)].

One can construct the other definitions of various stability and boundednessconcepts based on the foregoing definitions. We shall now state a typical resulton stability.

Theorem 5.8.2 Assume that conditions (i), (ii) of Theorem 5.8.1 are satisfied.Suppose further that

b(D0[U, θ]) ≤ ‖L(t, U )‖ ≤ a(D0[U, θ]), a, b ∈ K. (5.8.8)

Then the stability properties of the trivial solution of (5.8.7) imply the corre-sponding stability properties of the trivial solution of (5.8.1) respectively.

One can construct the proof of the theorem following the standard proofsemployed already, once we have the estimates (5.8.6) and (5.8.8).

Since the comparison system in the present situation is also a SDE andnot a scalar differential equation as before, it would be necessary to choose(when connecting the initial values of the two SDEs) a(D0[W0, θ]) = ‖Q0‖ sothat ‖L(t0,W0)‖ ≤ ‖Q0‖ holds, which is required when we utilize the estimate(5.8.6).

The use of comparison SDE in Kc(Rd+) provides a very general set up, which

includes several possibilities. For example, if R,G ∈ Kc(Rd+) in (5.8.7) are single

valued maps, then as observed earlier, (5.8.7) reduces to ordinary differentialsystem, and, consequently, there results the method of Vector Lyapunov-likefunctions.

Similarly, if d = 1, then the Lyapunov-like functions and the comparisonsystem reduce to interval valued maps, which, of course, include as a specialcase, the usual Lyapunov-like method. There is a need to further explore thisframework to get interesting results.

5.9 Set Differential Equations in (Kc(E), D),

In Section 1.8, we provided the necessary background for the metric space(Kc(E), D), where E is a Banach space. As we have seen, SDEs generated bythe differential inclusions can be used as a tool to prove existence results of dif-ferential inclusions. The results of Section 4.4 offer such results in (Kc(Rn), D).In this Section, we shall only indicate some basic results similar to the onesproved in Chapter 2.


Let Γ : J × Kc(E) → K(E) =all nonempty compact subsets of E, andassume Γ is continuous on J × Kc(E), where J = [t0, t0 + a]. Consider thedifferential inclusion

x′ ∈ Γ(t, x), x(t0) = x0 ∈ E. (5.9.1)

Then we can define a mapping F : J × Kc(E) → Kc(E), F (t, A) = coΓ(t, A)where A ∈ Kc(E). Since Γ is assumed to be continuous, we have F is continuous.We can now define SDE

DHU = F (t, U ), U (t0) = U0 ∈ Kc(E), (5.9.2)

where, as before, DHU is the Hukuhara derivative.One can prove several results parallel to the ones we have investigated for

SDEs in the metric space (Kc(Rn), D). In fact, Tolstonogov [11] presents asystematic study of differential inclusions in a Banach space as well as SDEsgenerated by the differential inclusions. This reference has several general resultsfor SDEs.

We shall therefore list a couple of results which can be proved almost parallelto the results considered already in (Kc(Rn), D). We begin with the followingcomparison result.

Theorem 5.9.1 Assume that F ∈ C[J ×Kc(E),K(E)] and for t ∈ J, U, V ∈Kc(E),

D[F (t, U ), F (t, V )] ≤ g(t,D[U, V ]) (5.9.3)

where g ∈ C[J × R+,R+]. Suppose further that the maximal solution r(t) =r(t, t0, w0) of the scalar differential equation

w′ = g(t, w), w(t0) = w0 ≥ 0, (5.9.4)

exists on J. Then, if U (t) = U (t, t0, U0), V (t) = V (t, t0, V0) are any two solu-tions of (5.9.2) existing on J, we have the estimate

D[U (t), V (t)] ≤ r(t, t0, w0), t ∈ J,


The proof follows exactly as in the proof of Theorem 2.2.1 with suitablemodifications and hence omitted. The following Corollary is immediate and isuseful later.

Corollary 5.9.1 Assume that F ∈ C[J ×Kc(E),Kc(E)] and

D[F (t, U ), θ] ≤ g(t,D[U, θ]), (5.9.5)

for t ∈ J, U ∈ Kc(E), where g satisfies the same assumptions as in Theo-rem 5.9.1. Then, if U (t) = U (t, t0, U0) is any solution of (5.9.2) existing onJ, D[U0, θ] ≤ w0 implies

D[U (t), θ] ≤ r(t, t0, w0), t ∈ J,

r(t, t0, w0) being the maximal solution of (5.9.4) existing on J.


We recall that D[U, θ] = ‖U‖ = sup[‖u‖ : u ∈ U ], U ∈ Kc(E).Next, we state an existence and uniqueness result under assumptions more

general than the Lipschitz type condition, which provides the idea inherent inthe comparison principle.


(i) F ∈ C[R0,Kc(E)] where R0 = J × B(U0, b), B(U0, b) = U ∈ Kc(E) :D[U,U0] ≤ b and D[F (t, U ), θ] ≤ M0 on R0;

(ii) g ∈ C[J × [0, 2b],R+], g(t, w) ≤ M1 on J × [0, 2b], g(t, 0) ≡ 0, g(t, w) ismonotone nondecreasing in w for each t ∈ J, and w(t) ≡ 0 is the onlysolution of

w′ = g(t, w), w(t0) = 0 on J ;

(iii) D[F (t, U ), F (t, V )] ≤ g(t,D[U, V ]) for t ∈ J, U, V ∈ R0.


Un+1(t) = U0 +∫ t

t0

F (s, Un(s)) ds, n = 0, 1, 2, · · · , (5.9.6)

exist on J0 = [t0, t0 + η] where η = min(a, bM ), M = max[M0,M1], as a contin-

uous function and converge to the unique solution U (t) = U (t, t0, U0) of (5.9.2)on J0.

The proof of Theorem 5.9.2 proceeds similarly to the proof of Theorem 2.3.1,and, therefore, we omit the proof. We thus find that several results proved in(Kc(Rn), D) can be adapted to (Kc(E), D), with additional conditions whenevernecessary to match the Banach Space set up. For example, to prove Peano’stheorem, we need to impose a suitable condition in terms of the measure ofnoncompactness and also employ the corresponding Ascoli-Arzela’s theorem.

As we indicated earlier, there are several results in Tolstonogov [1], in thisframework which connect differential inclusion in a Banach Space when themultifunction involved is not convex, as well as not continuous, and even whenit is not compact, convex. Then the proofs of corresponding results becomemore complicated but can be constructed.


Section 5.2 begins by introducing impulses to SDEs and obtains basic results.This material is from Vasundhara Devi [1]. Section 5.3 extends the monotoneiterative technique to impulsive SDEs and this is adapted from VasundharaDevi and Vatsala [1]. In Section 5.4, delay is incorporated in the SDEs, and thefundamental results described are from Vasundhara Devi and Vatsala [1]. Forordinary differential equations with delay, refer to Lakshmikantham and Leela[3] and Hale [1]. For practical stability for ODE see Lakshmikantham, Leela


and Martynyuk [1]. Monotone iterative technique for SDEs with delay wasdeveloped in McRae and Vasundhara Devi [2]. An interesting combination ofimpulses and delay was infused into the SDEs, and basic results were obtainedin McRae and Vasundhara Devi [1] which form Section 5.5.

Section 5.6 deals with set difference equations, the material of which is com-piled from Gnana Bhaskar and Shaw [1]. For the basic theory of differenceequations see Lakshmikantham and Trigiante [1]. The results on differentialequations with causal maps investigated in Section 5.7 are adapted from thepapers of Drici, Mc Rae and Vasundhara Devi [1,2]. For a good reference fordifferential equations with causal operators, see Corduneanu [1]. See also Lak-shmikantham and Rama Mohana Rao [1].

Also, for results in abstract spaces see Lakshmikantham and Leela [2]. Sec-tion 5.8 investigates the concept of Lyapunov-like functions, whose values are inmetric spaces, which includes single, vector, matrix and cone valued Lyapunovfunctions as a special case. For details, see Lakshmikantham and VasundharaDevi [1]. Finally, in Section 5.9 we indicate how one can extend most of the re-sults obtained in the metric space (Kc(Rn), D) to the metric space (Kc(E), D),where E is a Banach space. For details and more general results see Tolstonogov[1].

References

Ambrosio, L. and Cassno, F.S.[1] Lecture notes on “Analysis in Metric Spaces”, Scuola Normale Superiore,PISA, 2000.

Ambrosio, L. and Tilli, P.[1] Selected topics on ”Analysis in Metric Spaces”, Scuola Normale Superiore,PISA, 2000.

Agarwal, R.P., O’Regan, D. and Lakshmikantham, V.[1] Fuzzy Volterra Integral equations : a stacking theorem approach, NonlinearAnalysis 55, 2003, 299-312.

Arstein, Z.[1] On the calculus of closed set-valued functions, , Indiana Univ J. Math.,24(1974), 433-441.

Aubin, J.-P.[1] Mutational and Morphological Analysis, Birkhauser, Berlin, 1999.[2] Mutational equations in metric spaces, Set Valued Analysis, 1 (1993), 3-16.

Aubin, J.-P. and Frankowska, H.[1] Set Valued Analysis, Birkhauser, Basel, 1990.

Aumann, R.J.[1] Integrals of set-valued functions, JMAA 12(1965), 1-12.

Banks, H.T. and Jacobs, M.Q.[1] A differential calculus of multifunctions, JMAA 29 (1970), 246-272.

Brandao Lopes Pinto, A.J., De Blasi, F.S. and Iervolino, F.[1] Uniqueness and existence theorems for differential equations with convex-valued solutions, Boll. Unione. Mat. Italy 3 (1970), 47-54.

Castaing, C. and Valadier, M.[1] Convex analysis and measurable multifunctions, Springer-Verlag, Berlin 1977.

201

202 REFERENCES

Corduneanu, C.[1] Functional Equations with Causal Operators, Taylor & Francis, New York,2003.

Clarke, F.H., Yu.S.Ledyaev, Stern, R.J. and Wolenski, P.R.[1] Nonsmooth Analysis and Control Theory, Springer Verlag, New York, 1998.

De Blasi. F. S. and Iervolino, F.[1] Equazioni differenziali con soluzioni a valore compatio convesso, Boll. U.M.I.4, 2(1969), pp 491-501.

Debreu, G.[1] Integration of correspondences, in: Proc. Fifth Berkeley Symp.Math. Statist.Praobab., Vol 2, part#1, Univ California Press, Berkeley, CA(1967), 351-372.

Deimling, K.[1] Multivalued Differential Equations, Walter de Grutyer, New York, 1992.

Diamond, P. [1] Stability and periodicity in fuzzy differential equations, IEEETrans. Fuzzy Systems, 8, (2000), 583-590.

Diamond, P. and Kloeden, P.[1] Metric spaces of fuzzy sets, World Scientific, Singapore 1994.

Diamond, P. and Watson, P.[1] Regularity of solution sets for differential inclusions quasi-concave in param-eter, Appl. Math. Lett.13(2000), 31-35.

Drici, Z., McRae, F.A. and Vasundhara Devi, J.[1] Basic results for Set Differential Equations with Causal maps.(to appear)[2] Stability results for Set Differential Equations with Causal maps.(to appear)

Gnana Bhaskar, T., Lakshmikantham, V.[1] Set Differential Equations and Flow Invariance, Appl. Anal. 82(2003), 357-368.[2] Generalized flow invariance for differential inclusions, To appear in Int.J.Nonlinear Diff.Eq. TMA.[3] Lyapunov stability for set differential equations, Dyn. Sys. Appl. 13(1),2004, 1-10.[4] Generalized proximal normals and flow invariance for differential inclusions,Nonlinear Studies, 10(1), 2003, 29-37[5] Generalized proximal normals and Lyapunov stability, Nonlinear Studies,10(4), 2003, 335-341.

203

Gnana Bhaskar, T., Lakshmikantham, V. and Vasundhara Devi, J.[1] Revisiting fuzzy differential equations, Non.Anal. TMA 58(3-4), 2004, 351-358.

Gnana Bhaskar, T. and Shaw, M.[1] Stability results for Set Difference Equations, Dynamical Systems and Ap-plications, 13 (2004), 479-485.

Gnana Bhaskar, T. and Vasundhara Devi, J.[1] Stability Criteria in Set Differential Equations, to appear in Math.Comp.Mod.[2] Nonuniform stability and boundedness criteria for Set Differential Equations,Applicable Analysis, 84, No. 2 (2005), 131-142.[3] Set Differential Systems and Vector Lyapunov functions, to appear Appl.Math.Comp.

Hale, J.[1] Functional Differential Equations, Springer Verlag, New york, 1971.

Hausdorff, F.[1] Set Theory, Chelsea, New York 1957.

Hermes, H.[1] Calculus of set-valued functions and control, J. Math. Mech., 18(1968), 47-59.

Hukuhara, M.[1] Sul l’application semicontinue dont la valeur est un pact convexe, Functial.Ekvac. 10 (1967), pp.48-66[2] Integration des applications measurables dont la valeur est. un compact con-vexe, Funkcialaj, Ekavacioj 10(1967), 205-223.

Hullermeier, E.[1] An approach to modelling and simulation of uncertain dynamical systems,Int. J.Uncertainity, Fuzziness & Knowledge-based Systems, 5 (1997), 117-137.

Kaleva, O.[1] Fuzzy differential equations, Fuzzy Sets and Systems, 24(1987), 301-317.[2] The Cauchy problem for fuzzy differential equations, Fuzzy Sets and Sys-tems, 35(1990), 389-396.[3] A note on fuzzy differential equations, to appear in Nonlinear Analysis: The-ory and Methods (2005).

Kloeden, P.E., Sadovsky, B.N. and Vasilyeva, I.E.[1] Quasi-flows and equations with nonlinear differentials, Nonlinear Analysis51 (2002), 1143-1158.

204 REFERENCES

Ladde, G.S. , Lakshmikantham, V. and Vatsala, A.S.[1] Monotone Iterative Techniques for Nonlinear Differential Equations, Pitman,London, 1985.

Lakshmikantham, V.,[1] The connection between set and fuzzy differential equations, (to appear).

Lakshmikantham, V. and Bainov, D.D. and Simeonov, P.S.[1] Theory of Impulsive Differential Equations, World Scientific, Singapore,1989.

Lakshmikantham, V. and Koksal, S.[1] Monotone Flows and Rapid Convergence for Nonlinear Partial DifferentialEquations, Taylor & Francis, New York, 2003.

Lakshmikantham, V., and Leela, S.[1] Differential and Intergral Inequalities, Vol.I and II, Academic Press, NewYork, 1969.[2] On perturbing Lyapunov functions, Math.Sys.Theory 10(1976), 85-90.[3] Nonlinear Differential Equations in Abstract Spaces, Pergamon Press, Ox-ford, 1981.

Lakshmikantham, V., Leela, S. and Martynyuk, A.A.[1] Stability Analysis of Nonlinear Systems, Marcel Dekker, 1969.[2] Practical Stability of Nonlinear Systems, World Scientific, Singapore, 1990.

Lakshmikantham, V., Leela, S. and Vasundhara Devi, J.,[1] Stabiliy theory for set differential equations, Dynamics of Continuous, Dis-crete, and Impulsive Systems (DCDIS) Series A, 11 (2004), 181-190.

Lakshmikantham, V., Leela, S. and Vatsala, A.S.[1] Set valued hybrid differential equations and stability in terms of two mea-sures, J. Hybrid.Syst. 2(2002), 169-188.[2] Interconnection between set and fuzzy differential equations, Nonlinear Anal-ysis, 54(2), 2003, 351-360.

Lakshmikantham, V. and Liu, X.Z.[1] Stability analysis in terms of two measures, World Scientific, Singapore, 1993.

Lakshmikantham, V., Mohapatra, R.[1] Theory of Fuzzy Differential Equations and Inclusions, Taylor& Francis,2003.

Lakshmikantham, V., Matrosov, V. and Sivasundaram, S.[1] Vector Lyapunov Functions and Stability Analysis of Nonlinear Systems,Kluwer Academic Publishers, Dordrecht 1991.

205

Lakshmikantham, V., and Nieto, J.[1] Differential Equations in metric spaces: an introduction and application tofuzzy differential equations, DCDIS, 10 (2003), 991-1000.

Lakshmikantham, V. and Rama Mohana Rao, M.[1] Theory of Integro-Differential Equations, Gordon and Breach, Amsterdam,1995.

Lakshmikantham, V. and Tolstonogov, A.[1] Existence and interrrelation between set and fuzzy differential equations,Nonlinear Analysis TMA, 55(2003), 255-268.

Lakshmikantham, V. and Trigiante, D.[1] Theory of Difference Equations, Numerical Methods and Applications, 2ndEdition, Marcel Dekker, New york, 2002.

Lakshmikantham, V., and Vasundhara Devi, J.[1] Lyapunov-like functions in metric spaces, Dynamical Systems and Applica-tions, 13 (2004), 553-559.

Lakshmikantham, V., Vatsala, A.S.[1] Set differential equations and monotone flows, Nonlinear Dyn. Stability The-ory, 3(2), 2003, 33-43.[2] Hybrid Impulsive Fuzzy Differential Equations, (to appear).

Lay, S.R.[1] Convex sets and their applications, John Wiley and Sons, NY, 1982.

Markov, S.M.[1] Existence and uniqueness of solutions of the interval differential equationX ′ = F (t,X), comptes rendus de l’academie bulgare des sciences, 31(1978),1519-1522.

McRae, F.A. and Vasundhara Devi, J.[1] Impulsive Set Differential Equations with Delay, to appear.[2] Monotone Iterative Techniques for Set Differential Equations with Delay, toappear.

Mel’nik, T.A.[1] An iterative method for solving systems of quasi differential equations withslow and fast variables, Differential Equations, 33 (1997), 1335-1340.

Moore, R.E.[1] Methods and applications of interval analysis, SIAM, Philadelphia, 1979.

206 INDEX

Morales, P.[1] Non-Hausdorff Ascoli theory, Dissertation Math, Rozprawy Math. 119 (1974).

O’Regan, D., Lakshmikantham, V. and Nieto, J.[1] Initial and boundary value problems for fuzzy differential equations, Nonlin-ear Analysis, TMA 54 (2003), 405-415.

Panasyuk, A.I.[1] Quasi-differential equations in metric spaces, Differential Equations, 21(1985),914-921.

Plotnikov, V.A.[1] Controlled quasi-differential equations and some of their properties, Differ-ential Equations, 34, (1998), 1332-1336.

Radstrom, H.[1] An embedding theorem for spaces of convex sets, Proc. Amer.Math.Soc.3(1952), 165-169.

Rockafellar, R.T.[1] Convex analysis, Princeton University Press, Princeton, N.J., 1970.

Seikkala, S.[1] On the fuzzy initial value problem, Fuzzy Sets and Systems, 24 (1987), 319-330.

Tolstonogov, A.[1] Differential Inclusions in a Banach Space, Kluwer Academic Publishers,Dodrecht,2000.[2] On Scorza- Dragono theorem for multifunctions with variable domain of def-inition, Mathematicheskie zametki, 48, N 5 (1990), 109-120 (in Russian).

Vasundhara Devi, J. and Vatsala, A.S.[1] Monotone Iterative Technique for Impulsive and Set Differential Equations,Nonlinear Studies, 11, No. 4 (2004), 639-658.[2] A study of set differential equations with delay, Dynamics of Continuous,Discrete, and Impulsive Systems (DCDIS) Series A: Mathematical Analysis, 11(2004), 287-300.

Vasundhara Devi, J.[1] Basic Results in Set Differential Equations, Nonlinear Studies, 10, No.3(2003), 259-272.

Vorobiev, D., Seikkala, S.[1] Towards the theory of fuzzy differential equations, Fuzzy Sets and Systems,125(2), 2002, 231-237.

Index

Approximate Solutions, 50Ascoli-Arzela Theorem, 33

bounded solutionsequi, 79nonuniformly, 80uniformly, 80

Castaing Representation Theorem,17

causal map, 183comparison theorem for any two so-

lutions ofimpulsive FDE, 121impulsive SDE, 141SDE, 28, 29SDE with delay, 163

comparison theorem for Lyapunov-like functions for

fuzzy differential equations, 102hybrid FDE, 125hybrid impulsive FDE, 127impulsive FDE, 123impulsive SDE, 143impulsive SDE with delay, 174SDE, 66SDE with causal operator, 190set differential systems, 84

comparison theorem for set differen-tial

nonstrict inequalities, 40, 139nonstrict inequalities, with im-

pulses, 114strict inequalities, 37

continuous dependence, 35

epigraph, 91

equi-bounded, 79Euler solution

existence, 52weak asymptotic stability, 95

Existencefor impulsive SDE with delay,

171global, for SDE with delay, 165impulsive SDE, 140in a metric space, 134

Existence for SDEglobal, 49, 68Peano’s type, 36successive approximations, 32,

185USC case, 60

extension principle, 105extremal solutions

existence, 38for impulses, 149, 157via monotone iterates, 42, 46

Hausdorffmetric, 9separation, 9

Hukuharaderivative, 18difference, 7

integralAumann, 20Bochner, 21

integrally bounded, 20

Lipschitz continuous, 16lower and upper solutions

coupled, 42

207

208 INDEX

natural, 41lower semicontinuous, 15

maximal and minimal solutions, 37metric differential equation, 129Monotone iterative technique for SDE,

42

partial ordering in Kc(Rn), 37proximal

aiming condition, 56normal, 56

selector, 17set differential inequality, 40stability

equi, 72, 106equi, asymptotic, 72, 76, 107nonuniform, 74practical for SDE with delay,

166uniform, 73, 107uniform, asymptotic, 73, 78, 108

stabilty properties forimpulsive SDE, 145impulsive SDE with delay, 175set difference equations, 181

strongly invariant, 58strongly measurable, 23subdifferential, 91subgradient, 91support function, 13

uniform bounded solution, 79upper semicontinuous, 15

weakly decreasing, 93weakly invariant, 56

Date post:	08-Dec-2016
Category:	Documents
Upload:	lytuyen
View:	244 times
Download:	1 times

Theory of Set Differential Equations in Metric Spaces

Documents