EXISTENCE AND STABILITY ANALYSIS OF IMPULSIVE

DELAY DYNAMIC SYSTEMS ON TIME SCALES

A Thesis Submitted in Partial Fulfillment for the Requirement of the

Degree of Doctor of Philosophy in Mathematics

By

Syed Omar Shah

Supervised by

Dr. Akbar Zada

DEPARTMENT OF MATHEMATICS

UNIVERSITY OF PESHAWAR

JULY, 2019

DEDICATED TO MY BELOVED PARENTS AND MY LOVELY

WIFE

iv

Table of Contents

Table of Contents v

Acknowledgements vii

List of Publications x

Notations and Abbreviations xi

Abstract xiv

1 Introduction 1

2 Preliminaries 52.1 Introduction to Time Scale . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Differentiation and Integration on Time Scale . . . . . . . . . . . . . . 102.3 Dynamic Equations and System of Dynamic Equations on Time Scale 142.4 Impulsive Dynamic Equations on Time Scale . . . . . . . . . . . . . . 192.5 Delay Dynamic Equations on Time Scale . . . . . . . . . . . . . . . . . 21

3 Non–Linear Impulsive Delay Differential Equations 243.1 Basic Concepts and Remarks . . . . . . . . . . . . . . . . . . . . . . . 253.2 Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . . . . 283.3 Hyers–Ulam Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

4 First Order Non–Linear Impulsive Time Varying Delay Dynamic System 344.1 First Order Non–Linear Time Varying Delay Dynamic System with

Instantaneous Impulses . . . . . . . . . . . . . . . . . . . . . . . . . . 344.1.1 Basic Concepts and Remarks . . . . . . . . . . . . . . . . . . . 354.1.2 Existence and Uniqueness of Solutions of Equation (4.1.1) . . 39

v

4.1.3 Hyers–Ulam Stability of Equation (4.1.1) . . . . . . . . . . . . 404.1.4 Existence and Uniqueness of Solutions of Equation (4.1.2) . . 434.1.5 Hyers–Ulam Stability of Equation (4.1.2) . . . . . . . . . . . . 45

4.2 Non–Linear Delay Dynamic System withNon–Instantaneous Impulses . . . . . . . . . . . . . . . . . . . . . . . 504.2.1 Basic Concepts and Remarks . . . . . . . . . . . . . . . . . . . 504.2.2 Existence and Uniqueness of Solutions . . . . . . . . . . . . . 534.2.3 Hyers–Ulam Stability . . . . . . . . . . . . . . . . . . . . . . . 55

5 Non–linear Impulsive Volterra Integro–Delay Dynamic System 605.1 Non–Linear Volterra Integro–Delay Dynamic System with Instanta-

neous Impulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.1.1 Basic Concepts and Remarks . . . . . . . . . . . . . . . . . . . 615.1.2 Existence and Uniqueness of Solutions . . . . . . . . . . . . . 635.1.3 Hyers–Ulam Stability . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Non–Linear Volterra Integro–Delay Dynamic System with FractionalIntegrable Impulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.2.1 Basic Concepts and Remarks . . . . . . . . . . . . . . . . . . . 725.2.2 Existence and Uniqueness of Solutions . . . . . . . . . . . . . 745.2.3 Hyers–Ulam Stability . . . . . . . . . . . . . . . . . . . . . . . 77

6 Applications 836.1 Hammerstein Integro–Delay Dynamic System . . . . . . . . . . . . . 83

6.1.1 Basic Concepts and Remarks . . . . . . . . . . . . . . . . . . . 846.1.2 Existence and Uniqueness of Solutions . . . . . . . . . . . . . 866.1.3 Hyers–Ulam Stability . . . . . . . . . . . . . . . . . . . . . . . 89

6.2 Non–linear Mixed Integro–Dynamic System . . . . . . . . . . . . . . 936.2.1 Basic Concepts and Remarks . . . . . . . . . . . . . . . . . . . 946.2.2 Existence and Uniqueness of Solutions . . . . . . . . . . . . . 966.2.3 Hyers–Ulam Stability . . . . . . . . . . . . . . . . . . . . . . . 98

7 Conclusion and Future Research Work 103

Bibliography 105

vi

Acknowledgements

First of all, I am thankful to Allah Almighty, the most gracious, compassionate

and ever merciful, who gave me the power to complete this assignment, the sight

to observe and the mind to think, judge and analyze. I also offer Darood and

Salam to Holy Prophet Hazrat Mohammad (S.A.W.W) who is ideal human being

and ocean of sympathy for humanity.

My deep appreciation to my research supervisor Dr. Akbar Zada, Assistant

Professor, Department of Mathematics, University of Peshawar for providing me

encouragement, support, help and continued backing throughout this thesis writ-

ing and research work. He always inspired me to set a good example for the others.

As my master’s thesis supervisor he also provided support and endeavor in set-

ting up this thesis, too. My PhD has been an astonishing experience and I thank

Dr. Akbar Zada enthusiastically, not only for his marvelous academic support, but

also for giving me so many pleasing occasions.

I am very thankful to Dr. Imran Aziz, Chairman, Department of Mathematics,

University of Peshawar for creative guidance and valuable suggestions. He taught

me everything I know about mathematics, have often had to bear the burden of

my aggravation and rages against the world.

I acknowledge the people who mean a lot to me, my father, Prof. Dr. Akram

Shah, Department of Zoology, University of Peshawar and my Mother for giving

me liberty to choose what I desired. I salute them for the selfless love, care, pain

and sacrifices they did to shape my life. Although you hardly understood what I

researched on, you were willing to support any decision I made. I would never be

vii

viii

able to pay back the love and affection showered upon me by my parents.

I have no words to express my gratitude to my beloved wife who has been

extremely supportive to me throughout this entire process and has made countless

sacrifices to help me getting to this point. She was always around at times I thought

that it is impossible to continue and helped me to keep things in perception.

Special thanks are due to my brothers, Syed Ibrahim Shah and Syed Haris Shah,

my Uncles Syed Azam Shah Advocate, Syed Maroof Shah and Syed Asif Shah and

to all my family for their selfless love, care and dedicated efforts which contributed

a lot for completion of this thesis.

I would like to communicate my sincere thankfulness to my teachers Dr. Muham-

mad Farooq, Dr. Suhail Ahmed, Dr. Adil Khan, Dr. Tahir Hussain, Dr. Tahir Saeed,

Dr. Abdul Samad, Dr. Rohul Amin, Dr. Imran Khan, Dr. Fawad Khan, Dr. Haider

Ali, Dr. Muhammad Asif and Mr. Rashid Ali Jan for the unremitting support

during my PhD study. I am thankful to them for their endurance, inspiration, ea-

gerness, and conferring knowledge. I could not have anticipated having a better

advisors and mentors for my education.

I am very grateful to Prof. Dr. Yongjin Li, Department of Mathematics, Sun

Yat-sen University, Guangzhou, 510275, P. R. China and Dr. Alaa E. Hamza, De-

partment of Mathematics, Faculty of Science, Cairo University, Giza, Egypt, for

their collaboration and support.

I am very indebted to Prof. Dr. Martin Bohner, Curators’ Distinguished Profes-

sor of Mathematics and Statistics, Department of Mathematics and Statistics Mis-

souri University of Science and Technology Rolla, Missouri 65409–0020, USA, who

always guided me about time scales calculus.

I am very thankful to my friends Mr. Nisar Ahmed Afridi, Mr. Inamullah. Mr.

Waqas Ahmed, Mr. Muhammad Amir Zia, Mr. Muhammad Arif, Mr. Rizwan and

Mr. Hassan Shah for their continuous encouragement.

I would like to thank Dr. Nisar Ahmed Khattak, Mr. Wajid Ali, Miss Asia

Mashal, Miss Shaleena, Miss Hira, Mr. Zeeshan Ali, Mr. Ihsanullah Khan and Mr.

ix

Bakht Zada for helping me in checking this thesis.

In every day work I have been blessed with a gracious and smiling group of

fellow students, thanks all guys for providing company.

Finally, but by no means least, thanks go to all the clerical staff of Department

of Mathematics for their help.

Syed Omar Shah,

July, 2019.

List of Publications

• Zada, A.; Shah, S. O. Hyers–Ulam stability of first–order non–linear delay

differential equations with fractional integrable impulses, Hacettepe J. Math.

Stat. 2018, 47(5), 1196–1205 (Impact Factor: 0.558).

• Zada, A.; Shah, S. O.; Li, Y. Hyers–Ulam stability of nonlinear impulsive

Volterra integro–delay dynamic system on time scales, J. Nonlinear Sci. Appl.

2017, 10(11), 5701–5711 (2016 Impact Factor: 1.34).

• Shah, S. O.; Zada, A.; Hamza, A. E. Stability analysis of the first order non-

linear impulsive time varying delay dynamic system on time scales, Qual.

Theory Dyn. Syst. 2019, https://doi.org/10.1007/s12346-019-00315-x (Impact

Factor: 1.019).

• Shah, S. O.; Zada, A. Hyers–Ulam stability of non–linear Volterra integro–

delay dynamic system with fractional integrable impulses on time scales, Ira-

nian Journal of Mathematical Sciences and Informatics,

http://ijmsi.ir/browse accepted.php(in press) [ESCI].

• Shah, S. O.; Zada, A. On the stability analysis of non–linear Hammerstein

impulsive integro–dynamic system on time scales with delay, Punjab Univ. J.

Math. 2019, 51(7), 89–98 (HEC Recognized Journal, X Category).

• Shah, S. O.; Zada, A. Existence, uniqueness and stability of solution to mixed

integral dynamic systems with instantaneous and noninstantaneous impulses

on time scales, Appl. Math. Comput. 2019, 359, 202–213 (Impact Factor: 2.300).

x

Notations and Abbreviations

• N= Set of natural numbers

• Z= Set of integers

• R= Set of real numbers

• R+= Set of all non–negative real numbers

• Rn= Set of n–tuples of real numbers

• FP= Fixed point

• RL= Riemann–Liouville

• IVP= Initial Value Problems

• DEs= Differential equations

xi

xii

• DyEs= Dynamic equations

• DDEs= Delay differential equations

• DDyEs= Delay dynamic equations

• DDyS= Delay dynamic system

• IDyS= Integro–dynamic system

• IDDyS= Integro–delay dynamic system

• EU= Existence and uniqueness

• EUS= Existence and uniqueness of solutions

• HU= Hyers–Ulam

• HUR= Hyers–Ulam–Rassias

• 0, 1, 2, . . . , m = 0, m and 1, 2, . . . , m = 1, m

• C([t0 − τ, t f ]TS ,Rn) (respectively PC([t0 − τ, t f ]TS ,Rn)) is a Banach space of

continuous functions (respectively the Banach space of piecewise continuous

functions) with the norm ‖x‖ = supt∈[t0−τ,t f ]TS‖x(t)‖. P1

C(TS0,Rn) = x ∈

xiii

PC([t0 − τ, t f ]TS ,Rn) : x∆ ∈ PC([t0 − τ, t f ]TS ,Rn), the Banach space with

norm ‖x‖1 = max‖x‖, ‖x∆‖

• C(J,Rn) (respectively PC(J,Rn)) is a Banach space of all continuous func-

tions (respectively the Banach space of all piecewise continuous functions)

with the norm ‖x‖ = supt∈J ‖x(t)‖, J = [v0 − τ, u f ] ∩ TS. P1C(J,Rn) =

x ∈ PC(J,Rn) : x∆ ∈ PC(J,Rn), the Banach space with norm ‖x‖1 =

sup‖x‖, ‖x∆‖

Abstract

The main purpose of this thesis is to study the existence, uniqueness and Ulam’s

type stability of solutions of impulsive delay dynamic systems on time scale. First,

we study the existence, uniqueness and Ulam’s type stability of solutions of impul-

sive delay differential equations and then we extended it to impulsive delay dy-

namic systems on time scale. More specifically, the integral impulses of fractional

order are utilized for the study of first order non–linear delay differential equations

and then the results are extended to system of non–linear delay differential equa-

tions with instantaneous and fractional integrable impulses on time scale. Also,

the results are then generalized to non–linear impulsive Volterra integro–delay dy-

namic system and non–linear impulsive Hammerstein integro–delay dynamic sys-

tem on time scale. Furthermore, the mixed impulsive integro–dynamic system on

time scale is also studied. The main tools for obtaining the existence, uniqueness

and Ulam’s type stability results are Gronwall’s inequality, abstract Gronwall’s

lemma, Banach contraction principle and Picard operator. For our results, some

assumptions are made along with Lipschitz conditions. Some examples are also

presented to support our results.

xiv

Chapter 1

Introduction

The qualitative theory of DEs studies the behavior of properties of different phe-

nomena. From the literature of the study area, it can be seen that the most im-

portant aspect of the qualitative theory of DEs is stability analysis [1]. There are

different types of stability but recently the concept of Ulam’s type stability is a

central topic for researchers because it is very important for the error analysis in

approximation theory. In 1940, Ulam [2, 3] asked a famous question related to the

stability of homomorphisms: “With which requirements does an additive mapping near

an approximate additive mapping exists?”. Hyers [4], partially answered this ques-

tion, for the case of Banach spaces. This concept proposed by Ulam and Hyers,

was named as HU stability or stability in terms of HU. In 1978, Rassias [5] gen-

eralized HU stability concept and named it as HUR stability or stability in terms

of HUR. At the end of 19th century, many researchers contributed to the stabil-

ity idea of Ulam’s type, for various types of functional and DEs. There are many

advantages of Ulam’s type stability in tackling problems, related to optimization

techniques, numerical analysis, control theory and many more. For more details

on stability in terms of HU, see [6–45].

1

2

There are numerous implications of DEs. However, depending upon the cir-

cumstances, a bit change in the real world action endure DEs with abrupt devi-

ation, e.g., in control theory, changes in population, blood flows, blood pressure,

heart beats etc. [46–48]. To deal with such circumstances, impulsive DEs [49,50] can

be utilized to overcome such abruptness. One of the mathematical models about

such circumstances can be described by the following impulsive DEsψ′(u) = f (u, ψ(u)), u ∈ [u0, u f ]\uk,

zψ(uk) = ψ(u+k )− ψ(u−k ) = χk(ψ(u−k )), k = 1, m,

where the impulsive conditions are the combination of the traditional IVPs and the

short-term perturbations whose duration can be negligible in comparison with the

duration of such a process.

It is to be noted that the pioneers of the Ulam’s type stability for impulsive or-

dinary DEs are Wang et al. [30]. Following their own work, in 2014, they proved

the stability in terms of HUR and generalized stability in terms of HUR of impul-

sive evolution equations on a compact interval [31] which then they extended for

infinite impulses in the same paper. Wang and Zhang [34] proved Bielecki–Ulam’s

type stabilities of non–linear DEs having integral impulses of fractional order. The

work of Wang and Zhang [34] was extended by Zada et al. in [36], in which they

discussed stability in terms of HU of higher–order non–linear DEs with fractional

integrable impulses. They established Bielecki stability in terms of HUR, gener-

alized Bielecki stability in terms of HUR and Bielecki stability in terms of HU for

this class of DEs on a compact interval. Wang et al. [33] obtained very interesting

results about the Ulam’s type stability of first order impulsive DEs with variable

delay in quasi–Banach spaces.

3

Despite the situations where only impulsive factor is involved or delay effects

happen, we have a wide variety of evolutionary processes together where delay

and impulsive effects exist in their states. To model such phenomena, which are

subject to impulsive perturbations as the time delays, an impulsive DDE is used,

see [33, 39, 41].

The theory of dynamic equations on time scale has been rising fast and has ac-

knowledged a lot of interest in recent years. This theory was introduced by Hilger

[51] in 1988, with the inspiration to provide a unification of continuous and discrete

calculus. The time scales calculus is a significant research field for the researchers

that are dealing with DyEs from the last two decades. In fact, the major features of

time scales calculus are “unification” and “extension”. Bohner and Peterson pro-

vided the basics of time scales calculus in the excellent monographs [52, 53]. For

more details on time scale, see [6–8, 14, 18, 23–27, 40, 42, 43, 49, 50, 54–74].

In this thesis, a detailed study is presented regarding EUS and stability analysis

of impulsive DDEs and impulsive DDySs on time scale with supporting examples

and applications.

In Chapter 2, we give the time scales calculus i.e., the definition of time scale,

forward jump operator, backward jump operator, graininess function and their

examples. We study about the differentiation and integration for the function

g : TS → R on time scale. We state the fundamental results e.g., sum rule, product

rule etc. of differentiation and integration for the function g on time scale. We dis-

cuss about the DyEs, system of DyEs and solution of system of DyEs on time scale.

In order to define the generalized exponential function on time scale, we have used

4

cylinder transformation. We recall the properties of exponential function and fun-

damental matrix on time scale. Also the impulsive DyEs, DDyEs, integro–dynamic

equations and some important types of integro–dynamic equations on time scale

along with their solutions and examples are defined.

In Chapter 3, we prove the EUS and stability in terms of HU and HUR of first

order non–linear DDEs with fractional integrable impulses. We state Gronwall’s

inequality, abstract Gronwall’s lemma and Banach contraction principle. Also the

Picard operator is defined.

In Chapter 4, we obtain the EUS and stability in terms of HU and HUR of first

order non–linear DDyS on time scale with instantaneous and non–instantaneous

impulses. We state Gronwall’s inequalities on time scale in order to obtain the

stability in terms of HU and HUR of delay dynamic systems.

In Chapter 5, we establish the EUS and stability in terms of HU and HUR

results of non–linear Volterra IDDyS on time scale with instantaneous and non–

instantaneous impulses.

In Chapter 6, the applications of our results are discussed. The first section of

Chapter 6 is about the EUS and stability in terms of HU and HUR of impulsive

Hammerstein IDDyS (special class of Volterra IDDyS) on time scale. While the

second section of Chapter 6 is about the EUS and stability in terms of HU and

HUR of non–linear impulsive mixed IDyS on time scale.

In Chapter 7, the whole work is concluded and future research work is dis-

cussed.

Chapter 2

Preliminaries

In this chapter, the fundamental concepts and results of time scales calculus are

stated, that are applied throughout in this thesis. It comprises the idea of differ-

entiation, integration, dynamic equations, integro–dynamic equations, solution of

system of dynamic equations, impulsive dynamic equations and delay dynamic

equations on time scale. We start our work from the following definitions of time

scales calculus.

2.1 Introduction to Time Scale

In this section, we define time scale and give some examples. Also, we present

some basic definitions.

Definition 2.1.1. [52] Any non-empty arbitrary closed subset of real numbers is

called a time scale and is denoted byTS. Some examples of time scale are [1, 10], set

of natural numbers, set of whole numbers, [1, 20]∪ 30, 31, 32 . . . , [6, 15]∪ [31, 40]

etc.

Definition 2.1.2. [52] Let TS be any time scale and t ∈ TS. The forward jump

5

6

operator Θ : TS → TS is defined as:

Θ(t) = infs ∈ TS : s > t.

The forward jump operator gives the next point in time scale if it exists and if it

does not exist, then it gives that point i.e., inf φ = supTS, so either Θ(t) > t or

Θ(t) = t. For any t ∈ TS, if t < Θ(t), then point t is said to be right–scattered and

if Θ(t) = t, then point t is called right–dense.

Definition 2.1.3. [52] The backward jump operator $ : TS → TS is defined as:

$(t) = sups ∈ TS : s < t.

The backward jump operator gives the previous point in time scale if it exists and

if it does not exist, then it gives that point i.e., sup φ = infTS, so either $(t) < t

or $(t) = t. For any t ∈ TS, if t > $(t), then point t is said to be left–scattered and

if t = $(t), then t is called left–dense.

Definition 2.1.4. [52] The graininess function µ : TS → [0, ∞), which is used to

find the distance between two consecutive points in time scale, is defined as:

µ(t) = Θ(t)− t, t ∈ TS.

Example 2.1.1. Find the corresponding Θ(t), $(t) and µ(t) at t = 7.5 for the time scale

TS = [−15, 15]?

Solution: By definition, we can find Θ(7.5), $(7.5) and µ(7.5) as:

Θ(7.5) = infs ∈ TS : s > 7.5

= inf(7.5, 15]

= 7.5,

7

similarly

$(7.5) = sups ∈ TS : s < 7.5

= sup[−15, 7.5)

= 7.5

and

µ(7.5) = Θ(7.5)− 7.5

= 0.

Since in this example Θ(7.5) = 7.5 and $(7.5) = 7.5, so 7.5 is right–dense and left–

dense respectively and so 7.5 is dense. As time scale consist of real numbers in this

example, so whenever TS = R, then Θ(t) = t = $(t) and µ(t) = 0, ∀ t ∈ TS.

Example 2.1.2. Find the corresponding Θ(t), $(t) and µ(t) for the time scale TS =

[1, 5] ∪ 7, 8, 9 at t = 5?

Solution: By definition, we can find Θ(5), $(5) and µ(5) as:

Θ(5) = infs ∈ TS : s > 5

= inf7, 8, 9

= 7,

similarly

$(5) = sups ∈ TS : s < 5

= sup[1, 5)

= 5

8

and

µ(5) = Θ(5)− 5

= 2.

Since Θ(5) > 5, so 5 is right–scattered and also $(5) = 5, so 5 is left–dense.

Example 2.1.3. Find the corresponding Θ(t) at t = 25 and $(t) at t = 1 for the time

scale TS = [1, 25]?

Solution: By definition

Θ(25) = infs ∈ TS : s > 25

= infφ

= supTS

= 25

and

$(1) = sups ∈ TS : s < 1

= supφ

= infTS

= 1.

From this example, we can see that when we consider the maximum point in time

scale, then the forward jump operator gives that point in time scale and whenever

we take the smallest point in time scale, then the backward jump operator gives

that point in time scale.

9

Example 2.1.4. Find the corresponding Θ(t), $(t) and µ(t) at t = −2 for the time scale

TS = −6,−5− 4,−3,−2,−1, 0, 1, 2, 3, 4, 5, 6?

Solution: By definition, we can find Θ(−2), $(−2) and µ(−2) as:

Θ(−2) = infs ∈ TS : s > −2

= inf−1, 0, 1, 2, 3, 4, 5, 6

= −1,

similarly

$(−2) = sups ∈ TS : s < −2

= sup−6,−5,−4,−3

= −3

and

µ(−2) = Θ(−2)− (−2)

= −1 + 2

= 1.

Since in this example, Θ(−2) > −2 and $(−2) < −2, so −2 is right–scattered

and left–scattered, respectively and hence −2 is scattered. As time scale consist

of integers in this example, so we can see that Θ(t) = t + 1, $(t) = t − 1 and

µ(t) = 1, ∀ t ∈ TS.

Definition 2.1.5. [52] Let f : TS → R be a real valued function, then f Θ : TS → R

is defined as f Θ(t) = f (Θ(t)), ∀ t ∈ TS.

10

Definition 2.1.6. [52] The derived form of a time scale TS, denoted by TSz, is de-

fined as:

TSz =

TS\($(supTS), supTS], if supTS < ∞,

TS, if supTS = ∞.

If the time scale TS has left–scattered upper bound, then the time scale derivative

of f : TS → R is not defined on all ofTS but only onTSz e.g., for the time scaleTS =

1, 2, 3, 4, 5, the forward derivative of a function is only defined on 1, 2, 3, 4, 5z =

1, 2, 3, 4.

Definition 2.1.7. The time scales intervals are defined as

[a, b]TS = [a, b] ∩TS = t ∈ TS : a ≤ t ≤ b,

(a, b]TS = (a, b] ∩TS = t ∈ TS : a < t ≤ b.

2.2 Differentiation and Integration on Time Scale

This section is devoted to differentiation, integration and some properties of dif-

ferentiation and integration on time scale.

Definition 2.2.1. [52] The delta derivative of the function W : TS → R at t ∈ TSz

is defined by

W∆(t) = lims→t, s 6=Θ(t)

W(Θ(t))−W(s)Θ(t)− s

.

Remark 2.2.1. Note that when TS = R, then g∆ = g′

is the usual derivative and when

TS = Z, then g∆ = ∆g is the usual forward difference operator.

11

Theorem 2.2.1. (Properties of Differentiation) [52] Let g, h : TS → R be the differen-

tiable functions at t ∈ TSz. Then

1. (g + h)∆(t) = g∆(t) + h∆(t).

2. For any constant c,

(ch)∆(t) = ch∆(t).

3. (gh)∆(t) = g∆(t)h(t) + g(Θ(t))h∆(t) = g(t)h∆(t) + g∆(t)h(Θ(t)).

4. If g(Θ(t))g(t) 6= 0, then (1g

)∆

(t) = − g∆(t)g(t)g(Θ(t))

.

5. If h(t)h(Θ(t)) 6= 0, then(gh

)∆

(t) =g∆(t)h(t)− h∆(t)g(t)

h(t)h(Θ(t)).

Example 2.2.1. Find the delta derivative of the function h(t) = t2, t ∈ TS?

Solution: We have

h∆(t) = limu→t

h(Θ(t))− h(u)Θ(t)− u

= limu→t

Θ2(t)− u2

Θ(t)− u

= limu→t

(Θ(t)− u)(Θ(t) + u)Θ(t)− u

= Θ(t) + t.

Now if TS = R, then Θ(t) = t and

h∆(t) = t + t

= 2t

= h′(t).

12

But when TS = Z, then Θ(t) = t + 1 and

h∆(t) = t + 1 + t

= 2t + 1

= ∆h(t).

Definition 2.2.2. [52] The function W : TS → R is called right–dense (rd) con-

tinuous if it is continuous at every right–dense point on TS and its left–sided limit

exists at every left–dense point on TS.

Definition 2.2.3. [52] The function W : TS → R is called regressive (respec-

tively positively regressive ) if 1 + µ(t)W(t) 6= 0, (respectively 1 + µ(t)W(t) > 0)

∀ t ∈ TSz. The set of all right–dense continuous regressive functions (respec-

tively right–dense continuous positively regressive functions) will be denoted by

RG(TS) (respectivelyRG(TS)+).

Example 2.2.2. For the time scale TS = [1, 5] ∪ 6, 7, 8, 9, 10, the functionW(t) = t2

is rd continuous and regressive.

Definition 2.2.4. [52] The indefinite integral of the function W : TS → R is given

by: ∫W(t)∆t = w(t) + C,

where C is any constant. Now the Cauchy integral of W is defined as:∫ b

aW(t)∆t = w(b)−w(a), ∀ a, b ∈ TS,

where w∆ = W on TSz.

Theorem 2.2.2. (Properties of Integration) [52] If a0, b0, c0 ∈ TS, β ∈ R and g, h :

TS → R be the rd continuous functions, then

13

1.∫ b0

a0[g(s) + h(s)]∆s =

∫ b0a0

g(s)∆s +∫ b0

a0h(s)∆s.

2.∫ b0

a0βg(s)∆s = β

∫ b0a0

g(s)∆s.

3.∫ b0

a0g(s)∆s = −

∫ a0b0

g(s)∆s.

4.∫ b0

a0g(s)∆s =

∫ c0a0

g(s)∆s +∫ b0

c0g(s)∆s.

5.∫ b0

a0g(Θ(s))h∆(s)∆s = (gh)(b0)− (gh)(a0)−

∫ b0a0

g∆(s)h(s)∆s.

6.∫ b0

a0g(s)h∆(s)∆s = (gh)(b0)− (gh)(a0)−

∫ b0a0

g∆(s)h(Θ(s))∆s.

7.∫ a0

a0g(s)∆s = 0.

Theorem 2.2.3. [52] Let s, s0 ∈ TS and g : TS → R be the rd continuous function.

1. If TS = R, then

∫ s0

sg(t)∆t =

∫ s0

sg(t)dt.

2. If TS = Z, then

∫ s0

sg(t)∆t =

s0−1

∑t=s

g(t), if s < s0,

0, if s = s0,

−s−1

∑t=s0

g(t), if s > s0.

Example 2.2.3. If TS = Z, find the indefinite integral∫

ct∆t, where c 6= 1 is some

constant?

14

Solution: Since (ct

c− 1

)∆

= ∆(

ct

c− 1

)=

ct+1 − ct

c− 1

= ct.

So we get ∫ct∆t =

ct

c− 1+ C,

where C is any arbitrary constant.

2.3 Dynamic Equations and System of Dynamic Equa-

tions on Time Scale

This section deals with DyEs, solution of DyEs and system of DyEs on time scale.

Also the generalized exponential function and properties of fundamental matrix

on time scale are discussed.

Definition 2.3.1. [52] The generalized exponential function eW(a, b) for W ∈ RG(TS)

on TS is defined by

eW(a, b) = exp(∫ b

aφµ(s)W(s)∆s

)∀ a, b ∈ TS,

where

φµ(t)W(t) =

Log(1 + µ(t)W(t))

µ(t), if µ(t) 6= 0,

W(t), if µ(t) = 0,

is the cylindrical transformation.

15

Lemma 2.3.1. (Semi–group Property) [52] Let W ∈ RG(TS) and t, s, r ∈ TS, then

eW(t, s)eW(s, r) = eW(t, r).

Lemma 2.3.2. [42] If λ ∈ RG(TS)+, then for DyE ψ∆(t) = λψ(t) the following in-

equality holds

eλ(t, s) ≤ eλ(t−s), ∀ t, s ∈ TS.

Definition 2.3.2. [52] If W ∈ RG(TS), then the DyE

ψ∆(t) = W(t)ψ(t), (2.3.1)

is regressive.

Theorem 2.3.1. [52] If (2.3.1) is regressive and t0 ∈ TS. Then eW(., t0) satisfies the IVP

ψ∆(t) = W(t)ψ(t), ψ(t0) = 1.

Definition 2.3.3. [52] The non–homogeneous linear DyE

ψ∆(t) = W(t)ψ(t) + g(t), (2.3.2)

is called regressive if W ∈ RG(TS) and g is the rd continuous function.

Theorem 2.3.2. [52] Suppose that (2.3.2) is regressive. Let t0 ∈ TS and ψ0 ∈ R. The

only one solution of IVP

ψ∆(t) = W(t)ψ(t) + g(t), ψ(t0) = ψ0,

is

ψ(t) = eW(t, t0)ψ0 +∫ t

t0

eW(t, Θ(s))g(s)∆s.

16

Definition 2.3.4. [58] The fundamental matrix is defined to be the general solution

to the matrix dynamic system ψ∆(t) = M(t)ψ(t), ψ(t0) = ψ0, t ∈ TS and is

denoted by ΦM(t, t0).

Lemma 2.3.3. [58] If M ∈ RG(TS) be matrix valued function, then the family M =

ΦM(t, s) : t, s ∈ TS has the following properties:

1. Φ0(t, s) = 1 and ΦM(t, t) = 1.

2. ΦM(Θ(t), s) = (1 + µ(t)M(t))ΦM(t, s).

3. ΦM(t, s)ΦM(s, u) = ΦM(t, u).

4. Φ∆M(t, s) = M(t)ΦM(t, s).

Definition 2.3.5. [58] The system

ψ∆(t) = M(t)ψ(t) + g(t), t ∈ TS,

is regressive provided M ∈ RG(TS) and g : TS → Rn is rd continuous.

Theorem 2.3.3. [58]. The only one solution of regressive IVP

ψ∆(t) = M(t)ψ(t) + g(t), ψ(t0) = ψ0, t ∈ TS,

is

ψ(t) = ΦM(t, t0)ψ0 +∫ t

t0

ΦM(t, Θ(s))g(s)∆s.

Definition 2.3.6. An integro–dynamic equation is a type of DyE that involves both

delta derivative and integral of some unknown function say ψ(t). In general, the

first order integro–dynamic equation is of the form

ψ∆(t) = p(t)ψ(t) +∫ t

0f (s, ψ(s))∆s, t ∈ TS, ψ(0) = 1,

17

where p ∈ RG(TS) and f : TS ×R → R is continuous function. The solution of

above equation is

ψ(t) = ep(t, 0) +∫ t

0ep(t, Θ(s))

∫ s

0f (u, ψ(u))∆u∆s.

Example 2.3.1. The equation

ψ∆(t) = ψ(t) +∫ t

0s∆s, ψ(0) = 1, t ∈ TS,

is an integro–dynamic equation with solution

ψ(t) = e1(t, 0) +∫ t

0e1(t, Θ(s))

∫ s

0u∆u∆s.

Definition 2.3.7. If at least one limit of integration is a variable, then such integro–

dynamic equation is called Volterra integro–dynamic equation. The linear form of

Volterra integro–dynamic equation is

ψ∆(t) = p(t)ψ(t) +∫ t

0K(t, s)ψ(s)∆s, ψ(0) = 1, t ∈ [t0, ∞)TS ,

with solution

ψ(t) = ep(t, 0) +∫ t

0ep(t, Θ(s))

∫ s

0K(s, u)ψ(u)∆u∆s,

while the non–linear form of Volterra integro–dynamic equation is

ψ∆(t) = p(t)ψ(t) +∫ t

0K(t, s, ψ(s))∆s, t ∈ [t0, ∞)TS , ψ(0) = 1,

and its solution is

ψ(t) = ep(t, 0) +∫ t

0ep(t, Θ(s))

∫ s

0K(s, u, ψ(u))∆u∆s,

where p ∈ RG(TS), t0 ≥ 0, K(t, s) and K(t, s, ψ(s)) are continuous functions on

Ω = (t, s, ψ) : t0 ≤ s ≤ t < ∞.

18

Example 2.3.2. The integro–dynamic equation

ψ∆(t) = t2ψ(t) +∫ t

0(1 + s)ψ(s)∆s, ψ(0) = 1, t ∈ [0, ∞)TS ,

is a linear Volterra integro–dynamic equation and its solution is

ψ(t) = et2(t, 0) +∫ t

0et2(t, Θ(s))

∫ s

0(1 + u)ψ(u)∆u∆s.

Example 2.3.3. The integro–dynamic equation

ψ∆(t) =t2

2ψ(t) +

∫ t

0ep(t, ψ(s))∆s, ψ(0) = 1, t ∈ [0, ∞)TS ,

is a non–linear Volterra integro–dynamic equation, where p = p(t) = t2

2 and its solution

is

ψ(t) = ep(t, 0) +∫ t

0ep(t, Θ(s))

∫ s

0ep(s, ψ(u))∆u∆s.

Definition 2.3.8. The special class of non–linear Volterra integro–dynamic equa-

tion which is called Hammerstein integro–dynamic equation and it is of the form

ψ∆(t) = p(t)ψ(t) + f (t, s, ψ(s))∫ t

0K(t, s)g(t, s, ψ(s))∆s, ψ(0) = 1, t ∈ [t0, ∞)TS ,

p ∈ RG(TS), t0 ≥ 0, K(t, s), g(t, s, ψ(s)) and f (t, s, ψ(s)) are continuous functions

on Ω = (t, s, ψ) : t0 ≤ s ≤ t < ∞. The general solution of Hammerstein integro–

dynamic equation is

ψ(t) = ep(t, 0) +∫ t

0ep(t, Θ(s)) f (s, u, ψ(u))

∫ s

0K(s, u)g(s, u, ψ(u))∆u∆s.

Example 2.3.4. The integro–dynamic equation

ψ∆(t) = (1 + t)ψ(t) + e1(t, ψ(s))∫ t

0K(t, s)ep(t, ψ(s))∆s, t ∈ [0, ∞)TS , ψ(0) = 1,

is a non–linear Hammerstein integro–dynamic equation, where p(t) = 1 + t and its solu-

tion is

ψ(t) = ep(t, 0) +∫ t

0ep(t, Θ(s))e1(s, ψ(u))

∫ s

0K(s, u)ep(s, ψ(u))∆u∆s.

19

2.4 Impulsive Dynamic Equations on Time Scale

This section comprises of impulsive DyEs on time scale.

An impulsive DyE is a DyE which is subjected to abrupt changes in state of

dynamical processes. Mainly, there are two types of DyE with impulse impact:

• An instantaneous impulsive DyE is one in which the impulse action is de-

fined at certain discrete points. The general form of first order non–linear

DyE with instantaneous impulses isψ∆(u) = f (u, ψ(u)), u ∈ [u0, u f ]TS\uk,

zψ(uk) = ψ(u+k )− ψ(u−k ) = χk(ψ(u−k )), k = 1, m,

where u f > u0 ≥ 0, TS0 = [u0, u f ]TS , f : TS0 × R → R and χk : R → R

are continuous functions. Also ψ(u+k ) = limυ→0+ ψ(uk + υ) and ψ(u−k ) =

limυ→0− ψ(uk − υ) are respectively the right and left side limits of ψ(u) at uk,

where uk satisfies u0 < u1 < u2 < · · · < um < um+1 = u f < +∞.

• A non–instantaneous impulsive DyE is a DyE which establishes the effect of

impulse on an interval. The general form of first order non–linear DyE with

non–instantaneous impulses is

ψ∆(u) = f (u, ψ(u)), u ∈ (vi, ui+1] ∩TS, i = 0, m,

ψ(u) = gi(u, ψ(u)), u ∈ (ui, vi] ∩TS, i = 1, m,

where 0 = u0 = v0 < u1 ≤ v1 ≤ u2 < . . . um ≤ vm ≤ um+1 = u f are pre–

fixed numbers, f : (vi, ui+1] ∩ TS ×R → R, i = 0, m and gi : (ui, vi] ∩ TS ×

R→ R, i = 1, m are continuous functions. The fractional integrable impulses

are the special case of non–instantaneous impulses, the general form of first

20

order non–linear DyE with integral impulses of fractional order isψ∆(u) = F(u, ψ(u)), u ∈ (vi, ui+1] ∩TS, i = 0, m,

ψ(u) = Iui,uα gi(u, ψ(u)), u ∈ (ui, vi] ∩TS, i = 1, m,

where Iui,uα gi are called RL (left) integrals having fractional order α ∈ (0, 1)

( [74], Definition 2.7) on time scale, with the representation:

Iui,uα gi(u, ψ(u)) =

1Γ(α)

∫ u

ui

(u− v)α−1gi(v, ψ(v))∆v.

Example 2.4.1. The equationψ∆(u) =

1u− 1

ψ(u), ψ(0) = 1, u ∈ [0, 2]TS\1,

zψ(uk) = ψ(u+k )− ψ(u−k ) = χk(ψ(u−k )), k = 1,

is an instantaneous impulsive DyE with solution

ψ(u) = χk(ψ(u−k )) + ep(u, 0),

where p = p(u) = 1u−1 .

Example 2.4.2. For TS = R, an equationψ′(u) =

|ψ(u)|(1 + 9eu)(1 + |ψ(u)|) , u ∈ (0, 1] ∪ (2, 3],

ψ(u) =|ψ(u)|

(5− e + eu)(2 + |ψ(u)|) , u ∈ (1, 2].

is non–instantaneous impulsive DyE with solution

ψ(u) =|ψ(u)|

(5− e + eu)(2 + |ψ(u)|) +∫ u

0

|ψ(v)|(1 + 9ev)(1 + |ψ(v)|)dv.

21

2.5 Delay Dynamic Equations on Time Scale

This section deals with the DDyEs on time scale.

From the common observation, it can be seen that some physical processes take

more time than the usual time for completion. Such processes can be modeled in

the form of DDyEs. In fact, the DDyEs are those DyEs which involves the deriva-

tives of some unknown functions at present time and these derivatives are depen-

dent on the values of the functions at previous times [59, 63, 72]. The first order

non–linear DDyE is of the form

ψ∆(t) = f (t, ψ(t), ψ(t− ς)), t ∈ [t0, t f ]TS ,

where ς > 0 is called delay constant, the physical quantity ψ(t) changes between

initial time t0 and final time t f on the time scale. The delta derivative ψ∆(t) does

not depends only on t and ψ(t) but also on the state of the quantity at previous

time t− ς. DDyEs are different from DyEs due to the reason that the delta deriva-

tive at any time also depends on prior times. DDyEs involves history function that

describes the behavior of ψ(t) on the initial interval instead of an initial value as in

the case of ordinary DyEs. Moreover, the solution of DDyE can become unstable

due to the delay factor. In fact, DDyEs are the generalization of ordinary DyEs be-

cause ordinary DyEs can be considered as the DDyEs with zero delay or very small

delay constant ς. DDyEs with more than one constant delay are called DDyEs with

multiple constant delays and they are of the form

ψ∆(t) = f (t, ψ(t), ψ(t− ς1), ψ(t− ς2), . . . , ψ(t− ςn)),

22

where ς1, ς2, . . . , ςn are positive constants. The IVP in case of DDyE is of the formψ∆(t) = f (t, ψ(t), ψ(t− ς)), t ∈ [t0, t f ]TS ,

ψ(t) = ρ(t), t ∈ [t0 − ς, t0]TS ,

where ρ(t) is the history function that describes the history of ψ(t) at initial time

interval [t0 − ς, t0]TS .

Example 2.5.1. Let TS = R, consider the IVP,ψ′(t) = ψ(t− 1), t ∈ [0, 3],

ψ(t) = 1, t ∈ [−1, 0].(2.5.1)

For t ∈ [0, 1], integrating from 0 to t, we get

ψ(t) = ψ(0) +∫ t

0ψ′(s)ds, t ∈ [0, 1]

= 1 +∫ t

0ψ(s− 1)ds

= 1 +∫ t

01ds

= 1 + t− 0

= 1 + t.

Hence ψ(t) = 1 + t, t ∈ [0, 1]. Now consider the interval [1, 2], using ψ(t) = 1 + t, t ∈

[0, 1] and integrating from 1 to t, we get

ψ(t) = ψ(1) +∫ t

1ψ′(s)ds, t ∈ [1, 2],

= 2 +∫ t

1ψ(s− 1)ds

= 2 +∫ t

1(1 + s− 1)ds

= 2 +s2

2|t1

=t2

2+

32

.

23

So ψ(t) = t2

2 + 32 . In the similar way, we get

ψ(t) =t3

6− t2

2+ 2t +

16

, t ∈ [2, 3].

Thus, the solution of (2.5.1) on [−1, 3] is

ψ(t) =

1, t ∈ [−1, 0],

t2

2+

32

, t ∈ [1, 2],

t3

6− t2

2+ 2t +

16

, t ∈ [2, 3].

Chapter 3

Non–Linear Impulsive DelayDifferential Equations

This chapter deals with the existence and uniqueness of solutions, Hyers–Ulam

stability and Hyers–Ulam–Rassias stability of first–order non–linear delay differ-

ential equations with integral impulses having fractional order,ξ′(u) = F(u, ξ(u), ξ(h(u))), u ∈ (vi, ui+1], i = 0, m,

ξ(u) = Iui,uα gi(u, ξ(u), ξ(h(u))), u ∈ (ui, vi], i = 1, m, α ∈ (0, 1),

ξ(u) = ρ(u), u ∈ [v0 − τ, v0],

(3.0.1)

where τ > 0, 0 = u0 = v0 < u1 ≤ v1 ≤ u2 < · · · ≤ um ≤ vm ≤ um+1 = u f are pre–

fixed numbers, F : (vi, ui+1]×R2 → R, i = 0, m and gi : (ui, vi]×R2 → R, i = 1, m

are continuous functions, ρ : [v0 − τ, v0]→ R is history function and Iui,uα gi are RL

fractional integrals of order α and are defined as:

Iui,uα gi(u, ξ(u), ξ(h(u))) =

1Γ(α)

∫ u

ui

(u− v)α−1gi(v, ξ(v), ξ(h(v)))dv.

Furthermore, the delay function h : [v0− τ, u f ]→ (vi, ui+1] is continuous such that

h(u) ≤ u. These results can be seen in [41].

24

25

3.1 Basic Concepts and Remarks

Let C(J,R) and PC(J,R) be the Banach spaces of all continuous and piecewise

continuous functions, respectively, with norm ‖ξ‖ = max|ξ(u)| : u ∈ J, where

J = [v0 − τ, u f ]. We denote P1C(J,R) =ξ ∈ PC(J,R) : ξ

′ ∈ PC(J,R) the Banach

space with norm ‖ξ‖P1C= max‖ξ‖PC , ‖ξ ′‖PC.

Consider the following inequalities,∣∣ζ ′(u)− F(u, ζ(u), ζ(h(u)))

∣∣ ≤ ε, u ∈ (vi, ui+1], i = 0, m,∣∣ζ(u)− Iui,uα gi(u, ζ(u), ζ(h(u)))

∣∣ ≤ ε, u ∈ (ui, vi], i = 1, m, α ∈ (0, 1),(3.1.1)

∣∣ζ ′(u)− F(u, ζ(u), ζ(h(u)))

∣∣ ≤ ϕ(u), u ∈ (vi, ui+1], i = 0, m,∣∣ζ(u)− Iui,uα gi(u, ζ(u), ζ(h(u)))

∣∣ ≤ κ, u ∈ (ui, vi], i = 1, m, α ∈ (0, 1),(3.1.2)

where ε > 0, κ ≥ 0 and ϕ ∈ PC(J,R+) is an increasing function.

Definition 3.1.1. Eq. (3.0.1) is stable in terms of HU on J if for every ζ ∈ P1C(J,R)

satisfying (3.1.1), there exists a solution ζ0 ∈ P1C(J,R) of (3.0.1) such that |ζ0(u)−

ζ(u)| ≤ Kε, K > 0, ∀ u ∈ J.

Definition 3.1.2. Eq. (3.0.1) is stable in terms of HUR on J with respect to (ϕ, κ),

if for every ζ ∈ P1C(J,R) satisfying (3.1.2), there exists a solution ζ0 ∈ P1

C(J,R) of

(3.0.1) such that |ζ0(u)− ζ(u)| ≤ Mϕ(u), M > 0, ∀ u ∈ J.

Definition 3.1.3. Let (X, d) be a metric space. An operator Λ : X → X is called a

Picard operator if it has a unique FP x∗ ∈ X such that for every x ∈ X, the sequence

Λ(n)(x)n∈N converges to x∗.

Definition 3.1.4. Let (X, d) be a complete metric space. A mapping T : X → X is

said to be contraction if there exists k < 1 such that for any u, v ∈ X, the inequality

d(T(u), T(v)) ≤ kd(u, v),

26

holds.

Theorem 3.1.1. (Banach Contraction Principle [75]): Let (X, d) be a complete metric

space and T : X → X be a contraction. Then T has a unique FP in X.

Lemma 3.1.1. (Abstract Gronwall’s Lemma [76]): Let (X, d,≤) be an ordered metric

space and Λ : X → X be an increasing Picard operator with FP x∗. Then for any x ∈ X,

x ≤ Λ(x) implies x ≤ x∗ and x ≥ Λ(x) implies x ≥ x∗, where x∗ is the FP of Λ in X.

Lemma 3.1.2. (Gronwall’s Inequality [77]): If for u ≥ u0 ≥ 0, we have,

x(u) ≤ ξ(u) +∫ u

u0

ζ(v)x(v)dv + ∑u0<uk<u

ηkx(u−k ),

where x, ξ, ζ ∈ PC[[u0, ∞),R+], ξ is non–decreasing and ζ(u), ηk > 0, then for u ≥ u0

the following inequality holds

x(u) ≤ ξ(u) ∏u0<uk<u

(1 + ηk) exp( ∫ u

u0

ζ(v)dv)

.

Remark 3.1.3. If (3.1.1) holds for ζ ∈ P1C(J,R), then there exists f ∈ PC(J,R) with

sequence fk bounded by positive real number ε such thatζ′(u) = F(u, ζ(u), ζ(h(u))) + f (u), u ∈ (vi, ui+1], i = 0, m,

ζ(u) = Iui,uα gi(u, ζ(u), ζ(h(u))) + fi, u ∈ (ui, vi], i = 1, m, α ∈ (0, 1).

Lemma 3.1.4. Every ζ ∈ P1C(J,R) that satisfies (3.1.1) also comes out perfect on the

following inequality;

∣∣∣∣ζ(u)− ρ(u0)−∫ u

vi

F(v, ζ(v), ζ(h(v)))dv− Iui,uα gi(u, ζ(u), ζ(h(u)))

∣∣∣∣ ≤ (m

+ u f − vi)ε, u ∈ (vi, ui+1], i = 1, m,∣∣ζ(u)− Iui,uα gi(u, ζ(u), ζ(h(u)))

∣∣ ≤ mε, u ∈ (ui, vi], i = 1, m, α ∈ (0, 1).

27

Proof. If ζ ∈ P1C(J,R) satisfies (3.1.1), then by Remark 3.1.3, we have

ζ′(u) = F(u, ζ(u), ζ(h(u))) + f (u), u ∈ (vi, ui+1], i = 0, m,

ζ(u) = Iui,uα gi(u, ζ(u), ζ(h(u))) + fi, u ∈ (ui, vi], i = 1, m, α ∈ (0, 1),

ζ(u) = ρ(u), u ∈ [v0 − τ, v0].

(3.1.3)

Clearly the solution of (3.1.3) is given as

ζ(u) =

ρ(u0) +∫ u

vi

(F(v, ζ(v), ζ(h(v))) + f (v)

)dv

+ Iui,uα gi(u, ζ(u), ζ(h(u))) +

m

∑i=1

fi, u ∈ (vi, ui+1], i = 1, m,

Iui,uα gi(u, ζ(u), ζ(h(u))) +

m

∑i=1

fi, u ∈ (ui, vi], i = 1, m, α ∈ (0, 1).

For u ∈ (vi, ui+1], i = 1, m, α ∈ (0, 1), we get∣∣∣∣ζ(u)− ρ(u0)−∫ u

vi

F(v, ζ(v), ζ(h(v)))dv

−Iui,uα gi(u, ζ(u), ζ(h(u)))

∣∣∣∣ ≤ ∫ u

vi

| f (v)|dv +m

∑i=1| fi|

≤ (u− vi + m)ε

≤ (u f − vi + m)ε.

Similarly, we can derive

∣∣ζ(u)− Iui,uα gi(u, ζ(u), ζ(h(u)))

∣∣ ≤ mε, u ∈ (ui, vi], i = 1, m, α ∈ (0, 1).

We have similar remarks for (3.1.2).

28

3.2 Existence and Uniqueness of Solutions

This section deals with the EUS of Eq. (3.0.1). First, lets assume the following con-

ditions:

(A′1) F : (vi, ui+1]×R2 → R satisfies the Lipschitz condition∣∣F(u, u1, u2)− F(u, v1, v2)∣∣ ≤ ∑2

k=1 L|uk − vk|, L > 0, for all u ∈ (vi, ui+1], i = 0, m

and uk, vk ∈ R, k ∈ 1, 2;

(A′2) gi : (ui, vi]×R2 → R satisfies the Lipschitz condition

|gi(u, u1, u2)− gi(u, v1, v2)| ≤ ∑2k=1 Lgi |uk− vk|, Lgi > 0, for all u ∈ (ui, vi], i = 1, m

and u1, u2, v1, v2 ∈ R;

(A′3)(

2LgiΓ(α)

∫ viui(vi − v)α−1dv + 2L(u f − vi)

)< 1;

(A′4) ϕ ∈ PC(J,R+) is increasing such that for some ρ > 0,

∫ u

u0

ϕ(r)dr ≤ ρϕ(u),

holds.

Theorem 3.2.1. If conditions (A′1)− (A′3) hold, then Eq. (3.0.1) has a unique solution in

P1C(J,R).

29

Proof. i) Define Λ : PC(J,R)→ PC(J,R) as

(Λξ)(u) =

ρ(u), u ∈ [v0 − τ, v0],

Iui,uα gi(u, ξ(u), ξ(h(u))), u ∈ (ui, vi], i = 1, m, α ∈ (0, 1),

ρ(u0) + Iui,viα gi(vi, ξ(vi), ξ(h(vi))) +

∫ u

vi

F(v, ξ(v), ξ(h(v)))dv,

u ∈ (vi, ui+1], i = 1, m, α ∈ (0, 1).

For any ξ1, ξ2 ∈ PC(J,R), u ∈ (vi, ui+1], i = 1, m, we have

∣∣(Λξ1)(u)− (Λξ2)(u)∣∣ ≤ ∣∣∣∣Iui,vi

α gi(vi, ξ1(vi), ξ1(h(vi)))

−Iui,viα gi(vi, ξ2(vi), ξ2(h(vi)))

∣∣∣∣+∫ u

vi

∣∣∣∣F(v, ξ1(v), ξ1(h(v)))− F(v, ξ2(v), ξ2(h(v)))∣∣∣∣dv

≤ 1Γ(α)

∫ vi

ui

(vi − v)α−1∣∣gi(v, ξ1(v), ξ1(h(v)))

−gi(v, ξ2(v), ξ2(h(v)))∣∣dv + L

∫ u

vi

∣∣ξ1(v)− ξ2(v)∣∣dv

+L∫ u

vi

∣∣ξ1(h(v))− ξ2(h(v))∣∣dv

≤Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1∣∣ξ1(v)− ξ2(v)∣∣dv

+Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1∣∣ξ1(h(v))− ξ2(h(v))∣∣dv

+2L∫ u

vi

maxvi≤v≤ui+1

|ξ1(v)− ξ2(v)|dv

≤2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1 maxui≤v≤vi

|ξ1(v)− ξ2(v)|dv +

2L∫ u

vi

maxvi≤v≤ui+1

|ξ1(v)− ξ2(v)|dv

≤ ‖ξ1 − ξ2‖2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1dv + 2L‖ξ1 − ξ2‖∫ u

vi

dv

≤(

2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1dv + 2L(u− vi)

)‖ξ1 − ξ2‖

30

≤(

2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1dv + 2L(u f − vi)

)‖ξ1 − ξ2‖.

From (A′3), Λ is a contraction on (vi, ui+1], i = 1, m, and so it is a Picard operator

having unique FP which is actually the unique solution of (3.0.1) in P1C(J,R).

3.3 Hyers–Ulam Stability

In this section, we establish our results concerning stability in terms of HU and

HUR of Eq. (3.0.1).

Theorem 3.3.1. If conditions (A′1)− (A′3) hold, then Eq. (3.0.1) has stability in terms of

HU on J.

Proof. Let ζ ∈ P1C(J,R) be a solution to (3.1.1). The unique solution ξ ∈ P1

C(J,R)

of the DEξ′(u) = F(u, ξ(u), ξ(h(u))), u ∈ (vi, ui+1], i = 0, m,

ξ(u) = Iui,uα gi(u, ξ(u), ξ(h(u))), u ∈ (ui, vi], i = 1, m, α ∈ (0, 1),

ξ(u) = ζ(u), u ∈ [v0 − τ, v0],

is given by

ξ(u) =

ζ(u), u ∈ [v0 − τ, v0],

Iui,uα gi(u, ξ(u), ξ(h(u))), u ∈ (ui, vi], i = 1, m, α ∈ (0, 1),

ζ(u0) + Iui,viα gi(vi, ξ(vi), ξ(h(vi))) +

∫ u

vi

F(v, ξ(v), ξ(h(v)))dv,

u ∈ (vi, ui+1], i = 1, m, α ∈ (0, 1).

31

We observe that for all u ∈ [v0 − τ, v0], |ζ(u)− ξ(u)| = 0. For u ∈ (vi, ui+1], i =

1, m, using Lemma 3.1.4, we have

∣∣ζ(u)− ξ(u)∣∣ ≤ ∣∣ζ(u)− ζ(u0)−

∫ u

vi

F(v, ζ(v), ζ(h(v)))dv

−Iui,uα gi(v, ζ(v), ζ(h(v)))

∣∣+ ∣∣Iui,viα gi(vi, ζ(vi), ζ(h(vi)))

−Iui,viα gi(vi, ξ(vi), ξ(h(vi)))

∣∣+ ∫ u

vi

∣∣F(v, ζ(v), ζ(h(v)))

−F(v, ξ(v), ξ(h(v)))∣∣dv

≤ (m + u f − vi)ε +Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1∣∣ζ(v)− ξ(v)∣∣dv

+Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1∣∣ζ(h(v))− ξ(h(v))∣∣dv + L

∫ u

vi

∣∣ζ(v)− ξ(v)∣∣dv

+L∫ u

vi

∣∣ζ(h(v))− ξ(h(v))∣∣dv.

Now we define the operator T : PC(J,R+)→ PC(J,R+) as:

(Tg)(u) = (m + u f − vi)ε +Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1g(v)dv

+Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1g(h(v))dv + L∫ u

vi

g(v)dv + L∫ u

vi

g(h(v))dv.

For any g1, g2 ∈ PC(J,R+), u ∈ (vi, ui+1], i = 1, m, we get

∣∣(Tg1)(u)− (Tg2)(u)∣∣ ≤ Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1∣∣g1(v)− g2(v)∣∣dv

+Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1∣∣g1(h(v))− g2(h(v))∣∣dv

+L∫ u

vi

∣∣g1(v)− g2(v)∣∣dv + L

∫ u

vi

∣∣g1(h(v))− g2(h(v))∣∣dv

≤2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1 maxui≤v≤vi

|g1(v)− g2(v)|dv

+2L∫ u

vi

maxvi≤v≤ui+1

|g1(v)− g2(v)|dv

≤ ‖g1 − g2‖2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1dv + 2L‖g1 − g2‖∫ u

vi

dv

32

≤(

2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1dv + 2L(u− vi)

)‖g1 − g2‖

≤(

2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1dv + 2L(u f − vi)

)‖g1 − g2‖.

Again from (A′3), T is strictly contractive on (vi, ui+1], i = 1, m. Applying Banach

contraction principle, T is Picard operator with unique FP g∗ ∈ PC(J,R+) i.e.,

g∗(u) = (m + u f − vi)ε +Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1g∗(v)dv

+Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1g∗(h(v))dv + L∫ u

vi

g∗(v)dv + L∫ u

vi

g∗(h(v))dv.

Since g∗ is increasing, so g∗(h(u)) ≤ g∗(u) and for i = 1, m, we can write,

g∗(u) ≤ (m + u f − vi)ε +2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1g∗(v)dv + 2L∫ u

vi

g∗(v)dv

≤ (m + u f − vi)ε + ∑0<vi<u

(2Lgi

mΓ(α)

∫ vi

ui

(vi − v)α−1g∗(v)dv)

+2L∫ u

v0

g∗(v)dv.

Using Lemma 3.1.2, we get

g∗(u) ≤ (m + u f − vi)ε ∏0<vi<u

(1 +

2Lgi

mΓ(α)

∫ vi

ui

(vi − v)α−1dv)

exp(2L(u f − v0)

).

If we set g = |ζ− ξ|, then g(u) ≤ (Tg)(u) from which by using abstract Gronwall’s

lemma, it follows that g(u) ≤ g∗(u), thus

|ζ(u)− ξ(u)| ≤ Kε,

where K = (m + u f − vi)∏0<vi<u

(1+

2LgimΓ(α)

∫ viui(vi − v)α−1dv

)exp

(2L(u f − v0)

),

hence Eq. (3.0.1) has stability in terms of HU on J.

In the following theorem, the stability in terms of HUR of (3.0.1) on J is stated.

Its proof is identical to above theorem. The remarked Lemma 3.1.4 for inequality

(3.1.2) is consumed in the proof.

33

Theorem 3.3.1. If conditions (A′1)− (A′4) hold, then Eq. (3.0.1) has stability in terms of

HUR on J.

Chapter 4

First Order Non–Linear ImpulsiveTime Varying Delay Dynamic System

In this chapter, we prove the existence and uniqueness of solutions, Hyers–Ulam

stability and Hyers–Ulam–Rassias stability of the first order non–linear time vary-

ing delay dynamic system with instantaneous and non–instantaneous impulses on

time scale. We claim that the results obtained in this chapter are interesting. Some

results of this chapter can be found in [24].

4.1 First Order Non–Linear Time Varying Delay Dy-

namic System with Instantaneous Impulses

This section focuses on the EUS and stability in terms of HU and HUR results of

first order non–linear time varying DDyS of the formx∆(t) = A(t)x(t) + F(t, x(t), x(h(t))), t ∈ TS0,

x(t) = ρ(t), t ∈ [t0 − τ, t0]TS ,

x(t0) = ρ(t0) = x0,

(4.1.1)

34

35

also of first order non–linear impulsive time varying DDyS of the form

x∆(t) = M(t)x(t) + F(t, x(t), x(h(t))), t ∈ TS′ = TS0\tk,

zx(tk) = x(t+k )− x(t−k ) = χk(x(t−k )), k = 1, m,

x(t) = ρ(t), t ∈ [t0 − τ, t0]TS ,

x(t0) = ρ(t0) = x0,

(4.1.2)

where τ > 0, the regressive matrices A(t) and M(t) of order n are continuous and

piecewise continuous on TS0 = [t0, t f ]TS , respectively, t f > t0 ≥ 0 and F : TS0 ×

Rn × Rn → Rn, χk : Rn → Rn, ρ : [t0 − τ, t0]TS → R are continuous functions.

Also x(t+k ) = limυ→0+ x(tk + υ) and x(t−k ) = limυ→0− x(tk − υ) are respectively the

right and left side limits of x(t) at tk, where tk satisfies t0 < t1 < t2 < · · · < tm <

tm+1 = t f < +∞.

4.1.1 Basic Concepts and Remarks

Consider the following inequalities:∣∣∣∣∣∣∣∣y∆(s)− A(s)y(s)− F(s, y(s), y(h(s)))∣∣∣∣∣∣∣∣ ≤ ε, s ∈ TS0, (4.1.3)

∣∣∣∣∣∣∣∣y∆(s)− A(s)y(s)− F(s, y(s), y(h(s)))∣∣∣∣∣∣∣∣ ≤ ϕ(s), s ∈ TS0, (4.1.4)

∣∣∣∣∣∣∣∣y∆(s)−M(s)y(s)− F(s, y(s), y(h(s)))∣∣∣∣∣∣∣∣ ≤ ε, s ∈ TS′,

‖zy(sk)− χk(y(s−k ))‖ ≤ ε, k = 1, m,(4.1.5)

∣∣∣∣∣∣∣∣y∆(s)−M(s)y(s)− F(s, y(s), y(h(s)))

∣∣∣∣∣∣∣∣ ≤ ϕ(s), s ∈ TS′,

‖zy(sk)− χk(y(s−k ))‖ ≤ κ, k = 1, m,(4.1.6)

where ε > 0, κ ≥ 0 and ϕ : [t0 − τ, t f ]TS → R+ is rd continuous function.

36

Definition 4.1.1. Eq. (4.1.1) is said to be stable in terms of HU on [t0 − τ, t f ]TS if

∀ y ∈ C([t0 − τ, t f ]TS ,Rn) satisfying (4.1.3), there exists a solution y0 ∈ C([t0 −

τ, t f ]TS ,Rn) of (4.1.1) such that ‖y0(t)− y(t)‖ ≤ Kε, K > 0, ∀ t ∈ [t0 − τ, t f ]TS .

Definition 4.1.2. Eq. (4.1.1) is said to be stable in terms of HUR on [t0 − τ, t f ]TS

if ∀ y ∈ C([t0 − τ, t f ]TS ,Rn) satisfying (4.1.4), there exists a solution y0 ∈ C([t0 −

τ, t f ]TS ,Rn) of (4.1.1) such that ‖y0(t)− y(t)‖ ≤ Kϕ(t), K > 0, ∀ t ∈ [t0 − τ, t f ]TS .

Definition 4.1.3. Eq. (4.1.2) is said to be stable in terms of HU on [t0 − τ, t f ]TS if

∀ y ∈ P1C(TS

0,Rn) satisfying (4.1.5), there exists a solution y0 ∈ P1C(TS

0,Rn) of

(4.1.2) such that ‖y0(t)− y(t)‖ ≤ Kε, K > 0, ∀ t ∈ [t0 − τ, t f ]TS .

Definition 4.1.4. Eq. (4.1.2) is said to be stable in terms of HUR on [t0 − τ, t f ]TS

if ∀ y ∈ P1C(TS

0,Rn) satisfying (4.1.6), there exists a solution y0 ∈ P1C(TS

0,Rn) of

(4.1.2) such that ‖y0(t)− y(t)‖ ≤ Kϕ(t), K > 0, ∀ t ∈ [t0 − τ, t f ]TS .

Lemma 4.1.1. (Gronwall’s Inequality [52], Corollary 6.7): Let y be a rd–continuous

function, p ∈ RG(TS)+ and β ∈ R. Then

y(t) ≤ β +∫ t

t0

y(u)p(u)∆u, ∀ t ∈ TS,

implies

y(t) ≤ βep(t, t0), ∀ t ∈ TS.

Lemma 4.1.2. [66] Let υ ∈ T+S , y, b ∈ RG(TS+), p ∈ RG(TS+)+ and c, bk ∈

R+, k = 1, 2, . . . , then

y(t) ≤ c +∫ t

υp(s)y(s)∆s + ∑

υ<tk<tbky(tk),

implies

y(t) ≤ c ∏υ<tk<t

(1 + bk)ep(t, υ), t ≥ υ.

37

Remark 4.1.3. If (4.1.3) holds for y ∈ C([t0− τ, t f ]TS ,Rn) then there exists f ∈ C([t0−

τ, t f ]TS ,Rn), such that ‖ f (t)‖ ≤ ε, ∀ t ∈ [t0 − τ, t f ]TS and

y∆(t) = A(t)y(t) + F(t, y(t), y(h(t))) + f (t), y(t0) = y0.

A similar remark can be given for inequality (4.1.4).

Lemma 4.1.4. Every solution y ∈ C([t0− τ, t f ]TS ,Rn) of inequality (4.1.3) also satisfies

the following inequality:∣∣∣∣∣∣∣∣y(t)− y(t0)−ΦA(t, t0)y0 −∫ t

t0

ΦA(t, Θ(s))F(s, y(s), y(h(s)))∆s∣∣∣∣∣∣∣∣ ≤ C(t f − t0)ε,

for t ∈ [t0 − τ, t f ]TS , where ‖ΦA(t, Θ(s))‖ ≤ C.

Proof. If y ∈ C([t0 − τ, t f ]TS ,Rn) satisfies (4.1.3), then by Remark 4.1.3, we get

y∆(t) = A(t)y(t) + F(t, y(t), y(h(t))) + f (t), y(t0) = y0.

Therefore,

y(t) = y(t0) + ΦA(t, t0)y0 +∫ t

t0

ΦA(t, Θ(s))F(s, y(s), y(h(s)))∆s

+∫ t

t0

ΦA(t, Θ(s)) f (s)∆s.

So, ∣∣∣∣∣∣∣∣y(t)− y(t0)−ΦA(t, t0)y0 −∫ t

t0

ΦA(t, Θ(s))F(s, y(s), y(h(s)))∆s∣∣∣∣∣∣∣∣

≤∫ t

t0

‖ΦA(t, Θ(s))‖‖ f (s)‖∆s

≤ C(t f − t0)ε.

38

Remark 4.1.5. If (4.1.5) holds for y ∈ P1C(TS

0,Rn), then there is a function f ∈

PC([t0 − τ, t f ]TS ,Rn) with sequence fk bounded by positive real number ε such thaty∆(t) = M(t)y(t) + F(t, y(t), y(h(t))) + f (t), y(t0) = y0, t ∈ TS′,

zy(tk) = χk(y(t−k )) + fk, k = 1, m.

A similar remark for inequality (4.1.6) can also be stated.

Lemma 4.1.6. Every solution y ∈ P1C(TS

0,Rn) of inequality (4.1.5) also satisfies the

following inequality:∣∣∣∣∣∣∣∣y(t)− y(t0)−ΦM(t, t0)y0 −m

∑j=1

χ(y(t−j )) −∫ t

t0

ΦM(t, Θ(s))F(s, y(s), y(h(s)))∆s∣∣∣∣∣∣∣∣

≤ δε,

for t ∈ (tk, tk+1] ⊂ TS0, where δ = (m + C(t f − t0)) and ‖ΦM(t, Θ(s))‖ ≤ C.

Proof. If y ∈ P1C(TS

0,Rn) satisfies (4.1.5), then by Remark 4.1.5, we havey∆(t) = M(t)y(t) + F(t, y(t), y(h(t))) + f (t), y(t0) = y0, t ∈ T′S,

zy(tk) = χk(y(t−k )) + fk, k = 1, m.

Hence

y(t) = y(t0) + ΦM(t, t0)y0 +m

∑j=1

χ(y(t−j )) +m

∑j=1

f j

+∫ t

t0

ΦM(t, Θ(s))(F(s, y(s), y(h(s))) + f (s)

)∆s.

So,∣∣∣∣∣∣∣∣y(t)− y(t0)−ΦM(t, t0)y0 −m

∑j=1

χ(y(t−j ))−∫ t

t0

ΦM(t, Θ(s))F(s, y(s), y(h(s)))∆s∣∣∣∣∣∣∣∣

≤∫ t

t0

‖ΦM(t, Θ(s))‖‖ f (s)‖∆s +m

∑j=1‖ f j‖

≤ δε.

39

4.1.2 Existence and Uniqueness of Solutions of Equation (4.1.1)

Here, we prove the EUS of Eq. (4.1.1). First, we assume the following conditions:

(A1) The Lipschitz condition holds for the continuous function F : TS0 × Rn ×

Rn → Rn, i.e., ‖F(t, µ1, µ2) − F(t, ν1, ν2)‖ ≤ ∑2i=1 L‖µi − νi‖, L > 0, t ∈ TS

0,

µj, νj ∈ Rn, j ∈ 1, 2;

(A2) For some C ≥ 0, we have ‖ΦA(t, Θ(s))‖ ≤ C; t, s ∈ TS0;

(A3) 2CL(t f − t0) < 1;

(A4) For rd continuous and increasing function ϕ : [t0 − τ, t f ]TS → R+, we have

∫ t

t0

ϕ(u)∆u ≤ βϕ(t), β > 0.

Theorem 4.1.1. If conditions (A1)− (A3) hold, then Eq. (4.1.1) has a unique solution in

C([t0 − τ, t f ]TS ,Rn).

Proof. Define Λ : C([t0 − τ, t f ]TS ,Rn)→ C([t0 − τ, t f ]TS ,Rn) as

(Λx)(t) =

ρ(t), t ∈ [t0 − τ, t0]TS ,

ρ(t0) + ΦA(t, t0)x0

+∫ t

t0

ΦA(t, Θ(s))F(s, x(s), x(h(s)))∆s, t ∈ [t0, t f ]TS .

Now for x1, x2 ∈ C([t0 − τ, t f ]TS ,Rn) and for every t ∈ [t0 − τ, t0]TS , we have

‖(Λx1)(t)− (Λx2)(t)‖ = 0.

40

For t ∈ [t0, t f ]TS , we proceed as follows

‖(Λx1)(t)− (Λx2)(t)‖ =

∣∣∣∣∣∣∣∣ ∫ t

t0

ΦA(t, Θ(s))(

F(s, x1(s), x1(h(s)))

−F(s, x2(s), x2(h(s))))

∆s∣∣∣∣∣∣∣∣

≤∫ t

t0

‖ΦA(t, Θ(s))‖∣∣∣∣∣∣∣∣(F(s, x1(s), x1(h(s)))

−F(s, x2(s), x2(h(s))))∣∣∣∣∣∣∣∣∆s

≤∫ t

t0

CL‖x1(s)− x2(s)‖∆s

+∫ t

t0

CL‖x1(h(s))− x2(h(s))‖∆s

≤∫ t

t0

CL supt∈[t0−τ,t f ]TS

‖x1(t)− x2(t)‖∆s

+∫ t

t0

CL supt∈[t0−τ,t f ]TS

‖x1(t)− x2(t)‖∆s

≤ 2‖x1 − x2‖∫ t

t0

CL∆s

≤ ‖x1 − x2‖(

2CL(t− t0)

)≤ ‖x1 − x2‖

(2CL(t f − t0)

).

From (A3), Λ is strictly contractive on C([t0− τ, t f ]TS ,Rn). Therefore, the operator

Λ is a Picard operator having unique FP which is the unique solution of (4.1.1) in

C([t0 − τ, t f ]TS ,Rn).

4.1.3 Hyers–Ulam Stability of Equation (4.1.1)

In this subsection, we establish our results concerning stability in terms ofHU and

HUR of Eq. (4.1.1).

41

Theorem 4.1.2. If conditions (A1)− (A3) are satisfied, then Eq. (4.1.1) is stable in terms

of HU on [t0 − τ, t f ]TS .

Proof. If y ∈ C([t0 − τ, t f ]TS ,Rn) be a solution to (4.1.3), then the unique solution

x ∈ C([t0 − τ, t f ]TS ,Rn) of DyEx∆(t) = A(t)x(t) + F(t, x(t), x(h(t))), t ∈ TS0,

x(t) = y(t), t ∈ [t0 − τ, t0]TS ,

x(t0) = y(t0) = y0,

is

x(t) =

y(t), t ∈ [t0 − τ, t0]TS ,

y(t0) + ΦA(t, t0)y0 +∫ t

t0

ΦA(t, Θ(s))F(s, x(s), x(h(s)))∆s,

t ∈ [t0 − τ, t f ]TS .

We have ‖y(t) − x(t)‖ = 0 for t ∈ [t0 − τ, t0]TS and for t ∈ [t0 − τ, t f ]TS , using

Lemma 4.1.4, we get

‖y(t)− x(t)‖ ≤∣∣∣∣∣∣∣∣y(t)− y(t0)−ΦA(t, t0)y0 −

∫ t

t0

ΦA(t, Θ(s))F(s, y(s), y(h(s)))∆s∣∣∣∣∣∣∣∣

+

∣∣∣∣∣∣∣∣ ∫ t

t0

ΦA(t, Θ(s))(

F(s, y(s), y(h(s)))− F(s, x(s), x(h(s))))

∆s∣∣∣∣∣∣∣∣

≤ C(t f − t0)ε +∫ t

t0

‖ΦA(t, Θ(s))‖∣∣∣∣∣∣∣∣F(s, y(s), y(h(s)))

−F(s, x(s), x(h(s)))∣∣∣∣∣∣∣∣∆s

≤ C(t f − t0)ε +∫ t

t0

‖ΦA(t, Θ(s))‖L‖y(s)− x(s)‖∆s

+∫ t

t0

‖ΦA(t, Θ(s))‖L‖y(h(s))− x(h(s))‖∆s.

42

Let us define T : C([t0 − τ, t f ]TS ,R+)→ C([t0 − τ, t f ]TS ,R+) as

(Tw)(t) = C(t f − t0)ε+∫ t

t0

‖ΦA(t, Θ(s))‖Lw(s)∆s+∫ t

t0

‖ΦA(t, Θ(s))‖Lw(h(s))∆s.

(4.1.7)

For any w1, w2 ∈ C([t0 − τ, t f ]TS ,R+), we derive

‖(Tw1)(t)− (Tw2)(t)‖ =

∣∣∣∣∣∣∣∣ ∫ t

t0

‖ΦA(t, Θ(s))‖L(w1(s)−w2(s)

)∆s∣∣∣∣∣∣∣∣

+

∣∣∣∣∣∣∣∣ ∫ t

t0

‖ΦA(t, Θ(s))‖L(w1(h(s))−w2(h(s))

)∆s∣∣∣∣∣∣∣∣

≤∫ t

t0

‖ΦA(t, Θ(s))‖L‖w1(s)−w2(s)‖∆s

+∫ t

t0

‖ΦA(t, Θ(s))‖L‖w1(h(s))−w2(h(s))‖∆s

≤ 2∫ t

t0

‖ΦA(t, Θ(s))‖L supt∈[t0−τ,t f ]TS

‖w1(t)−w2(t)‖∆s

≤ ‖w1 −w2‖∫ t

t0

2CL∆s

≤ ‖w1 −w2‖(

2CL(t f − t0)

).

Since(

2CL(t f − t0)

)< 1, so the operator is contractive on C([t0 − τ, t f ]TS ,R+).

Applying Theorem 3.1.1, T is an increasing Picard operator having unique FP w∗

i.e.,

w∗(t) = C(t f − t0)ε+∫ t

t0

‖ΦA(t, Θ(s))‖Lw∗(s)∆s+∫ t

t0

‖ΦA(t, Θ(s))‖Lw∗(h(s))∆s.

Since w∗ is increasing, then w∗(h(s)) ≤ w∗(s) and in view of (A2), we get

w∗(t) ≤ C(t f − t0)ε + 2∫ t

t0

CLw∗(s)∆s.

By Lemma 4.1.1,

w∗(t) ≤ C(t f − t0)εeP(t, t0),

43

where P = 2CL is a positively regressive constant function. Setting w(t) = ‖y(t)−

x(t)‖ and applying (4.1.7), w(t) ≤ (Tw)(t), thus by abstract Gronwall’s lemma, we

get w(t) ≤ w∗(t), so

‖y(t)− x(t)‖ ≤ C(t f − t0)εeP(t, t0)

≤ C(t f − t0)εeP(t−t0)

≤ C(t f − t0)εeP(t f−t0).

The desired assertion follows by choosing K = C(t f − t0)eP(t f−t0). Hence the Eq.

(4.1.1) is stable in terms of HU on [t0 − τ, t f ]TS .

The following theorem can be proved by repeating the same process of above

theorem.

Theorem 4.1.7. If conditions (A1)− (A4) hold, then Eq. (4.1.1) is stable in terms of

HUR on [t0 − τ, t f ]TS .

4.1.4 Existence and Uniqueness of Solutions of Equation (4.1.2)

Here, we prove the EUS of Eq. (4.1.2). Consider the following additional assump-

tions:

(A5) χk : Rn → Rn satisfies ‖χk(µ1)− χk(µ2)‖ ≤ Mk‖µ1 − µ2‖, Mk > 0, k = 1, m,

µ1, µ2 ∈ Rn;

(A6)

(∑m

j=1Mj + 2CL(t f − t0)

)< 1;

44

Theorem 4.1.3. If conditions (A1), (A5) and (A6) are satisfied, then Eq. (4.1.2) has a

unique solution in P1C(TS

0,Rn).

Proof. Define Λ : PC([t0 − τ, t f ]TS ,Rn)→ PC([t0 − τ, t f ]TS ,Rn) as

(Λx)(t) =

ρ(t), t ∈ [t0 − τ, t0]TS ,

ρ(t0) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), x(h(s)))∆s, t ∈ (t0, t1],

ρ(t0) +i

∑j=1

χj(x(t−j )) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), x(h(s)))∆s,

t ∈ (ti, ti+1], i = 1, m.

We see that for any x1, x2 ∈ PC([t0 − τ, t f ]TS ,Rn) and t ∈ [t0 − τ, t0]TS , we have

‖(Λx1)(t)− (Λx2)(t)‖ = 0. For t ∈ (ti, ti+1], we get

‖(Λx1)(t)− (Λx2)(t)‖ ≤m

∑j=1‖χj(x1(t−j ))− χj(x2(t−j ))‖

+∫ t

t0

∣∣∣∣∣∣∣∣ΦM(t, Θ(s))(

F(s, x1(s), x1(h(s)))

−F(s, x2(s), x2(h(s))))∣∣∣∣∣∣∣∣∆s

≤m

∑j=1‖χj(x1(t−j ))− χj(x2(t−j ))‖

+∫ t

t0

‖ΦM(t, Θ(s))‖∣∣∣∣∣∣∣∣(F(s, x1(s), x1(h(s)))

−F(s, x2(s), x2(h(s))))∣∣∣∣∣∣∣∣∆s

45

≤m

∑j=1Mj‖x1(t−j )− x2(t−j )‖+

∫ t

t0

CL‖x1(s)− x2(s)‖∆s

+∫ t

t0

CL‖x1(h(s))− x2(h(s))‖∆s

≤m

∑j=1Mj sup

t∈[t0−τ,t f ]TS

‖x1(t)− x2(t)‖

+2∫ t

t0

CL supt∈[t0−τ,t f ]TS

‖x1(t)− x2(t)‖∆s

≤m

∑j=1Mj‖x1 − x2‖+ 2‖x1 − x2‖

∫ t

t0

CL∆s

≤ ‖x1 − x2‖( m

∑j=1Mj + 2CL(t− t0)

)

≤ ‖x1 − x2‖( m

∑j=1Mj + 2CL(t f − t0)

).

From (A6), Λ is a contraction on PC([t0 − τ, t f ]TS ,Rn). So Λ is a Picard operator

having unique FP which is the unique solution of (4.1.2) in P1C(TS

0,Rn).

4.1.5 Hyers–Ulam Stability of Equation (4.1.2)

In this subsection, we establish our results concerning stability in terms ofHU and

HUR of Eq. (4.1.2).

Theorem 4.1.4. If conditions (A1), (A5) and (A6) are satisfied, then Eq. (4.1.2) is stable

in terms of HU on [t0 − τ, t f ]TS .

Proof. If y ∈ P1C(TS

0,Rn) is a solution to (4.1.5), then the unique solution x ∈

46

P1C(TS

0,Rn) of the DyE

x∆(t) = M(t)x(t) + F(t, x(t), x(h(t))), t ∈ TS′ = TS0\tk,

zx(tk) = χk(x(t−k )), k = 1, m,

x(t) = y(t), t ∈ [t0 − τ, t0]TS ,

x(t0) = y(t0) = y0,

is

x(t) =

y(t), t ∈ [t0 − τ, t0]TS ,

y(t0) + ΦM(t, t0)y0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), x(h(s)))∆s, t ∈ (t0, t1],

y(t0) +i

∑j=1

χj(x(t−j )) + ΦM(t, t0)y0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), x(h(s)))∆s,

t ∈ (ti, ti+1], i = 1, m.

We see that for every t ∈ [t0− τ, t0]TS , ‖y(t)− x(t)‖ = 0. For t ∈ (ti, ti+1], utilizing

Lemma 4.1.6, we get

‖y(t)− x(t)‖ ≤∣∣∣∣∣∣∣∣y(t)− y(t0)−ΦM(t, t0)y0 −

m

∑j=1

χ(y(t−j ))

−∫ t

t0

ΦM(t, Θ(s))F(s, y(s), y(h(s)))∆s∣∣∣∣∣∣∣∣

+m

∑j=1‖χj(y(t−j ))− χj(x(t−j ))‖

+

∣∣∣∣∣∣∣∣ ∫ t

t0

ΦM(t, Θ(s))(

F(s, y(s), y(h(s)))− F(s, x(s), x(h(s))))

∆s∣∣∣∣∣∣∣∣

47

≤ δε +m

∑j=1Mj‖y(t−j )− x(t−j )‖

+∫ t

t0

‖ΦM(t, Θ(s))‖L‖y(s)− x(s)‖∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖L‖y(h(s))− x(h(s))‖∆s.

Consider the operator T : PC([t0 − τ, t f ]TS ,R+)→ PC([t0 − τ, t f ]TS ,R+) as:

(Tw)(t) =

0, t ∈ [t0 − τ, t0]TS ,

C(t f − t0)ε +∫ t

t0

‖ΦM(t, Θ(s))‖Lw(s)∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖Lw(h(s))∆s, t ∈ (t0, t1],

δε +i

∑j=1Mjw(t−j ) +

∫ t

t0

‖ΦM(t, Θ(s))‖Lw(s)∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖Lw(h(s))∆s, t ∈ (ti, ti+1], i = 1, m.

(4.1.8)

For any w1, w2 ∈ PC([t0 − τ, t f ]TS ,R+), ‖(Tw1)(t) − (Tw2)(t)‖ = 0 for all t ∈

[t0 − τ, t0]TS . For t ∈ (ti, ti+1], we have

‖(Tw1)(t)− (Tw2)(t)‖ ≤m

∑j=1Mj‖w1(t−j )−w2(t−j )‖

+∫ t

t0

‖ΦM(t, Θ(s))‖L‖w1(s)−w2(s)‖∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖L‖w1(h(s))−w2(h(s))‖∆s

≤m

∑j=1Mj sup

t∈[t0−τ,t f ]TS

‖w1(t)−w2(t)‖

+2CL∫ t

t0

supt∈[t0−τ,t f ]TS

‖w1(t)−w2(t)‖∆s

≤m

∑j=1Mj‖w1 −w2‖+ 2CL‖w1 −w2‖

∫ t

t0

∆s

48

≤ ‖w1 −w2‖( m

∑j=1Mj + 2CL(t f − t0)

).

From (A6), T is a contraction on PC([t0− τ, t f ]TS ,R+). By utilizing Theorem 3.1.1,

T is Picard operator having unique FP w∗ i.e.,

w∗(t) = δε +m

∑j=1Mjw∗(t−j ) +

∫ t

t0

‖ΦM(t, Θ(s))‖Lw∗(s)∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖Lw∗(h(s))∆s.

Since w∗ is increasing, so w∗(h(s)) ≤ w∗(s), hence

w∗(t) ≤ δε +m

∑j=1Mjw∗(t−j ) + 2

∫ t

t0

CLw∗(s)∆s.

By Lemma 4.1.2,

w∗(t) ≤ δε ∏t0<tj<t

(1 +Mj)eP(t, t0),

where P = 2CL is a positively regressive constant function. By setting w(t) =

‖y(t)− x(t)‖, w(t) ≤ (Tw)(t) (from (4.1.8)), thus by abstract Gronwall’s lemma,

we get w(t) ≤ w∗(t), so

‖y(t)− x(t)‖ ≤ δε ∏t0<tj<t

(1 +Mj)eP(t, t0)

≤ Kε,

where K = δ ∏t0<tj<t(1 +Mj)eP(t f−t0). Hence Eq. (4.1.2) is stable in terms of HU

on [t0 − τ, t f ]TS .

Similarly, we can establish the stability in terms of HUR of (4.1.2) on [t0 −

τ, t f ]TS .

Theorem 4.1.8. If conditions (A1), (A4), (A5) and (A6) are satisfied, then Eq. (4.1.2) is

stable in terms of HUR on [t0 − τ, t f ]TS .

49

Example 4.1.1. Consider the following semilinear impulsive dynamic equation:

x∆(t) =

1t− 1

x(t) + ep(t, x(t)), x(0) = 1, t ∈ [0, 2]TS\1,

zx(tk) = x(t+k )− x(t−k ) = χk(x(t−k )), k = 1,(4.1.9)

and its associated inequality∣∣y∆(t)− 1

t− 1y(t)− ep(t, y(t))

∣∣ ≤ 1, t ∈ [0, 2]TS\1,∣∣zy(tk)− χk(y(t−k ))∣∣ ≤ 1, k = 1.

(4.1.10)

By setting TS′ = [0, 2]TS\1, t1 = 1 and p(t) = 1t−1 . Here F(t, x(t)) = ep(t, x(t)) for

t ∈ TS′ and put ε = 1. If y ∈ P1C([0, 2]TS ,R) satisfies the inequality (4.1.10), then there

exist f ∈ P1C([0, 2]TS ,R) and f0 ∈ R such that | f (t)| ≤ 1 for t ∈ TS′ and | f0| ≤ 1. So

we have y∆(t) =

1t− 1

y(t) + ep(t, y(t)) + f (t), t ∈ TS′,

zy(tk) = χk(y(t−k )) + f0, k = 1,

and the solution of Eq. (4.1.9) is

x(t) = χ1(x(t−1 )) + ep(t, t0) +∫ t

0ep(t, Θ(s))ep(s, x(s))∆s.

Based on our theoretical results, Eq. (4.1.9) has a unique solution in P1C([0, 2]TS ,R) and

is stable in terms of HU on TS′.

50

4.2 Non–Linear Delay Dynamic System with

Non–Instantaneous Impulses

This section comprises of the EUS and stability in terms of HU and HUR of first

order non–linear DDyS with integral impulses having fractional order α ∈ (0, 1),

x∆(u) = M(u)x(u) + F(u, x(u), x(h(u))), u ∈ (vi, ui+1] ∩TS, i = 0, m,

x(u) = Iui,uα gi(u, x(u), x(h(u))), u ∈ (ui, vi] ∩TS, i = 1, m,

x(u) = ρ(u), u ∈ [v0 − τ, v0] ∩TS,

x(u0) = ρ(u0) = x0,

(4.2.1)

where τ > 0, M(u) is a regressive piecewise continuous square matrix, 0 = u0 =

v0 < u1 ≤ v1 ≤ u2 < · · · ≤ um ≤ vm ≤ um+1 = u f , F : (vi, ui+1]∩TS×Rn×Rn →

Rn, i = 0, m, gi : (ui, vi] ∩TS ×Rn ×Rn → Rn, i = 1, m are continuous functions,

ρ : [v0 − τ, v0] ∩TS → Rn is history function and Iui,uα gi are the so called RL (left)

integrals having fractional order α ∈ (0, 1) on time scale, with the representation:

Iui,uα gi(u, x(u), x(h(u))) =

1Γ(α)

∫ u

ui

(u− v)α−1gi(v, x(v), x(h(v)))∆v.

Moreover, (vi, ui+1] ∩ TS, (ui, vi] ∩ TS, [v0 − τ, v0] ∩ TS are non–empty sets and

the delay function h : [v0 − τ, u f ] ∩ TS → (vi, ui+1] ∩ TS is continuous such that

h(u) ≤ u.

4.2.1 Basic Concepts and Remarks

Consider the following inequalities,∣∣∣∣ψ∆(u)−M(u)ψ(u)− F(u, ψ(u), ψ(h(u)))‖ ≤ ε, u ∈ (vi, ui+1] ∩TS,

‖ψ(u)− Iui,uα gi(u, ψ(u), ψ(h(u)))‖ ≤ ε, u ∈ (ui, vi] ∩TS,

(4.2.2)

51

‖ψ∆(u)−M(u)ψ(u)− F(u, ψ(u), ψ(h(u)))‖ ≤ ϕ(u), u ∈ (vi, ui+1] ∩TS,

‖ψ(u)− Iui,uα gi(u, ψ(u), ψ(h(u)))‖ ≤ κ, u ∈ (ui, vi] ∩TS,

(4.2.3)

where ε > 0, κ ≥ 0 and ϕ ∈ C(J,R+) is an increasing function.

Definition 4.2.1. Eq. (4.2.1) is said to be stable in terms of HU, if for every ε > 0

and ψ ∈ P1C(J,Rn) satisfying (4.2.2), there exists a solution x ∈ P1

C(J,Rn) of (4.2.1)

such that ‖x(u) − ψ(u)‖ ≤ Kε for all u ∈ J. Here K is a positive number that

depends on ε.

Definition 4.2.2. Eq. (4.2.1) is said to be stable in terms ofHUR, provided for every

(ϕ, κ) ∈ C(J,R+)×R+ and for each ψ ∈ P1C(J,Rn) satisfying (4.2.3), there exists a

solution x ∈ P1C(J,Rn) of (4.2.1) such that the inequality ‖x(u)− ψ(u)‖ ≤ Mϕ(u)

is true for all u ∈ J. Here M > 0 depends on (ϕ, κ).

Remark 4.2.1. A function ψ ∈ P1C(J,Rn) satisfies inequality (4.2.2) (respectively in-

equality (4.2.3)) if and only if there exist f ∈ P1C(J,Rn) with sequence fk (dependent on

ψ) such that ‖ f (u)‖ ≤ ε for all u ∈ J and ‖ fk‖ ≤ ε (respectively ‖ fk‖ ≤ κ) for every

k = 1, m andψ∆(u) = M(u)ψ(u) + F(u, ψ(u), ψ(h(u))) + f (u),

u ∈ (vi, ui+1] ∩TS, i = 0, m,

ψ(u) = Iui,uα gi(u, ψ(u), ψ(h(u))) + fi, u ∈ (ui, vi] ∩TS, i = 1, m.

Lemma 4.2.2. If ψ ∈ P1C(J,Rn) satisfies inequality (4.2.2) (respectively inequality (4.2.3)),

52

then the following inequalities

‖ψ(u)− ψ(u0)−ΦM(u, u0)ψ(u0)−∫ u

vi

ΦM(u, Θ(v))F(v, ψ(v), ψ(h(v)))∆v

− Iui,uα gi(u, ψ(u), ψ(h(u)))‖ ≤ (Cu f − Cvi + m)ε,

u ∈ (vi, ui+1] ∩TS, i = 1, m,

‖ψ(u)− Iui,uα gi(u, ψ(u), ψ(h(u)))‖ ≤ mε, (respectively mκ),

u ∈ (ui, vi] ∩TS, i = 1, m,

are true. Here ‖ΦM(u, Θ(v))‖ ≤ C.

Proof. If ψ ∈ P1C(J,Rn) satisfies (4.2.2), then by Remark 4.2.1, we have

ψ∆(u) = M(u)ψ(u) + F(u, ψ(u), ψ(h(u))) + f (u),

u ∈ (vi, ui+1] ∩TS, i = 0, m,

ψ(u) = Iui,uα gi(u, ψ(u), ψ(h(u))) + fi, u ∈ (ui, vi] ∩TS, i = 1, m.

(4.2.4)

Clearly the solution of (4.2.4) is given as

ψ(u) =

ψ(u0) + ΦM(u, u0)ψ(u0) +

∫ u

vi

ΦM(u, Θ(v))(

F(v, ψ(v), ψ(h(v)))

+ f (v))∆v + Iui,u

α gi(u, ψ(u), ψ(h(u))), u ∈ (vi, ui+1] ∩TS,

Iui,uα gi(u, ψ(u), ψ(h(u))) + fi, u ∈ (ui, vi] ∩TS.

For u ∈ (vi, ui+1] ∩TS, i = 1, m, we get

‖ψ(u)− ψ(u0)−ΦM(u, u0)ψ(u0)−∫ u

vi

ΦM(u, Θ(v))F(v, ψ(v), ψ(h(v)))∆v

−Iui,uα gi(u, ψ(u), ψ(h(u)))‖

≤∫ u

vi

‖ΦM(u, Θ(v))‖‖ f (v)‖∆v +m

∑i=1‖ fi‖

≤ (Cu− Cvi + m)ε

≤ (Cu f − Cvi + m)ε.

53

As above, we see

‖ψ(u)− Iui,uα gi(u, ψ(u), ψ(h(u)))‖ ≤ mε, u ∈ (ui, vi] ∩TS, i = 1, m.

Similar remarks are for inequality (4.2.3).

4.2.2 Existence and Uniqueness of Solutions

Here, we prove the EUS of Eq. (4.2.1). Consider the following assumptions:

(A1) The Lipschitz condition holds for F : (vi, ui+1] ∩ TS × Rn × Rn → Rn i.e,

‖F(u, µ1, µ2) − F(u, ν1, ν2)‖ ≤ ∑2k=1 L‖µk − νk‖, for some L > 0, u ∈ (vi, ui+1] ∩

TS, i = 0, m and µk, νk ∈ Rn, k ∈ 1, 2;

(A2) gi : (ui, vi]∩TS×Rn×Rn → Rn satisfies the Lipschitz condition ‖gi(u, µ1, µ2)−

gi(u, ν1, ν2)‖ ≤ Li ∑2k=1 ‖µk − νk‖, for some Li > 0, u ∈ (ui, vi] ∩ TS, i = 1, m and

µ1, µ2, ν1, ν2 ∈ Rn;

(A3)

(2Li

Γ(α)

∫ viui(vi − v)α−1∆v + 2CL(u f − vi)

)< 1, i = 1, m;

(A4) ϕ ∈ C(J,R+) is rd continuous and increasing, so that for some γ > 0,

∫ u

u0

ϕ(r)∆r ≤ γϕ(u).

Theorem 4.2.1. If conditions (A1)− (A3) are satisfied, then Eq. (4.2.1) has precisely

only one solution in P1C(J,Rn).

54

Proof. Define an operator Λ : P1C(J,Rn)→ P1

C(J,Rn) by

(Λx)(u) =

ρ(u), u ∈ [v0 − τ, v0] ∩TS,

Iui,uα gi(u, x(u), x(h(u))), u ∈ (ui, vi] ∩TS, i = 1, m,

x0 + ΦM(u, u0)x0 + Iui,viα gi(vi, x(vi), x(h(vi)))

+∫ u

vi

ΦM(u, Θ(v))F(v, x(v), x(h(v)))∆v,

u ∈ (vi, ui+1] ∩TS, i = 1, m, α ∈ (0, 1).

For any x1, x2 ∈ P1C(J,Rn), u ∈ (vi, ui+1] ∩TS, i = 1, m, we have

‖(Λx1)(u)− (Λx2)(u)‖ ≤ ‖Iui,viα gi(vi, x1(vi), x1(h(vi)))

−Iui,viα gi(vi, x2(vi), x2(h(vi)))‖

+∫ u

vi

‖ΦM(u, Θ(v))‖ ‖F(v, x1(v), x1(h(v)))

−F(v, x2(v), x2(h(v)))‖∆v

≤ 1Γ(α)

∫ vi

ui

(vi − v)α−1‖gi(v, x1(v), x1(h(v)))

−gi(v, x2(v), x2(h(v)))‖∆v

+L∫ u

vi

‖ΦM(u, Θ(v))‖ ‖x1(v)− x2(v)‖∆v

+L∫ u

vi

‖ΦM(u, Θ(v))‖ ‖x1(h(v))− x2(h(v))‖∆v

≤ Li

Γ(α)

∫ vi

ui

(vi − v)α−1‖x1(v)− x2(v)‖∆v

+Li

Γ(α)

∫ vi

ui

(vi − v)α−1‖x1(h(v))− x2(h(v))‖∆v

+2CL∫ u

vi

supvi≤v≤ui+1

‖x1(u)− x2(u)‖∆v

≤ 2Li

Γ(α)

∫ vi

ui

(vi − v)α−1 supui≤v≤vi

‖x1(v)− x2(v)‖∆v

+2CL∫ u

vi

supvi≤v≤ui+1

‖x1(v)− x2(v)‖∆v

55

≤ ‖x1 − x2‖2Li

Γ(α)

∫ vi

ui

(vi − v)α−1∆v + 2CL‖x1 − x2‖∫ u

vi

∆v

≤(

2Li

Γ(α)

∫ vi

ui

(vi − v)α−1∆v + 2CL(u− vi)

)‖x1 − x2‖

≤(

2Li

Γ(α)

∫ vi

ui

(vi − v)α−1∆v + 2CL(u f − vi)

)‖x1 − x2‖.

In view of condition (A3), Λ is strictly contractive operator and hence a Picard

operator on P1C(J,Rn). Therefore, it has only one FP which is actually the only one

solution of Eq. (4.2.1).

4.2.3 Hyers–Ulam Stability

In this subsection, we establish our results concerning stability in terms ofHU and

HUR of Eq. (4.2.1).

Theorem 4.2.2. If conditions (A1)− (A3) hold, then Eq. (4.2.1) has stability in terms of

HU on J.

Proof. Assume that (4.2.2) has a solution ψ ∈ P1C(J,Rn). Then for DyE

x∆(u) = M(u)x(u) + F(u, x(u), x(h(u))), u ∈ (vi, ui+1] ∩TS, i = 0, m,

x(u) = Iui,uα gi(u, x(u), x(h(u))), u ∈ (ui, vi] ∩TS, i = 1, m,

x(u) = ψ(u), u ∈ [v0 − τ, v0] ∩TS,

x(u0) = ψ(u0) = x0,

56

we have the unique solution

x(u) =

ψ(u), u ∈ [v0 − τ, v0] ∩TS,

Iui,uα gi(u, x(u), x(h(u))), u ∈ (ui, vi] ∩TS, i = 1, m,

ψ(u0) + ΦM(u, u0)x0 + Iui,viα gi(vi, x(vi), x(h(vi)))

+∫ u

vi

ΦM(u, Θ(v))F(v, x(v), x(h(v)))∆v,

u ∈ (vi, ui+1] ∩TS, i = 1, m, α ∈ (0, 1).

We observe that for all u ∈ (vi, ui+1] ∩TS, i = 1, m, using Lemma 4.2.2, we obtain

‖ψ(u)− x(u)‖ ≤∣∣∣∣∣∣∣∣ψ(u)− ψ(u0)−ΦM(u, u0)ψ0

−∫ u

vi

ΦM(u, Θ(v))F(v, ψ(v), ψ(h(v)))∆v

−Iui,uα gi(v, ψ(v), ψ(h(v)))

∣∣∣∣∣∣∣∣+‖Iui,vi

α gi(vi, ψ(vi), ψ(h(vi)))− Iui,viα gi(vi, x(vi), x(h(vi)))‖

+∫ u

vi

‖ΦM(u, Θ(v))‖‖F(v, ψ(v), ψ(h(v)))− F(v, x(v), x(h(v)))‖∆v

≤ (m + Cu f − Cvi)ε +Li

Γ(α)

∫ vi

ui

(vi − v)α−1‖ψ(v)− x(v)‖∆v

+Li

Γ(α)

∫ vi

ui

(vi − v)α−1‖ψ(h(v))− x(h(v))‖∆v

+CL∫ u

vi

‖ψ(v)− x(v)‖∆v + CL∫ u

vi

‖ψ(h(v))− x(h(v))‖∆v.

Consider the operator T : PC(J,R+)→ PC(J,R+) as

(Tg)(u) = (m + Cu f − Cvi)ε +Li

Γ(α)

∫ vi

ui

(vi − v)α−1g(v)∆v

+Li

Γ(α)

∫ vi

ui

(vi − v)α−1g(h(v))∆v + CL∫ u

vi

g(v)∆v

+CL∫ u

vi

g(h(v))∆v.

57

For any g1, g2 ∈ PC(J,R+), u ∈ (vi, ui+1] ∩TS, i = 1, m, we have

‖(Tg1)(u)− (Tg2)(u)‖ ≤Li

Γ(α)

∫ vi

ui

(vi − v)α−1‖g1(v)− g2(v)‖∆v

+Li

Γ(α)

∫ vi

ui

(vi − v)α−1‖g1(h(v))− g2(h(v))‖∆v

+CL∫ u

vi

‖g1(v)− g2(v)‖∆v + CL∫ u

vi

‖g1(h(v))− g2(h(v))‖∆v

≤ 2Li

Γ(α)

∫ vi

ui

(vi − v)α−1 supui≤v≤vi

‖g1 − g2‖∆v

+2CL∫ u

vi

supvi≤v≤ui+1

‖g1 − g2‖∆v

≤ 2Li

Γ(α)

∫ vi

ui

(vi − v)α−1‖g1 − g2‖∆v + 2CL∫ u

vi

‖g1 − g2‖∆v

≤(

2Li

Γ(α)

∫ vi

ui

(vi − v)α−1∆v + 2CL(u− vi)

)‖g1 − g2‖

≤(

2Li

Γ(α)

∫ vi

ui

(vi − v)α−1∆v + 2CL(u f − vi)

)‖g1 − g2‖.

Again according to (A3), we are dealing here with a strictly contractive operator

on (vi, ui+1]∩TS and so it is Picard operator on PC(J,R+). So from Theorem 3.1.1,

T is a Picard operator having unique FP g∗ ∈ PC(J,R+) i.e.,

g∗(u) = (m + Cu f − Cvi)ε +Li

Γ(α)

∫ vi

ui

(vi − v)α−1g∗(v)∆v

+Li

Γ(α)

∫ vi

ui

(vi − v)α−1g∗(h(v))∆v + CL∫ u

vi

g∗(v)∆v

+CL∫ u

vi

g∗(h(v))∆v.

As, g∗ is increasing, therefore g∗(h(u)) ≤ g∗(u), (m + Cu f − Cvi) ≤ δ for some

δ > 0 and for i = 1, m, we can write

g∗(u) ≤ δε + ∑0<vi<u

(2Li

mΓ(α)

∫ vi

ui

(vi − v)α−1g∗(v)∆v)+ 2CL

∫ u

v0

g∗(v)∆v.

58

Using Lemma 4.1.2, we conclude that

g∗(u) ≤ δε ∏0<vi<u

(1 +

2Li

mΓ(α)

∫ vi

ui

(vi − v)α−1∆v)

eq(u, vi),

where q = 2CL is a positively regressive constant function. If we put g = ‖ψ −

x‖, then g(u) ≤ (Tg)(u). It follows, by utilizing abstract Gronwall’s lemma that

g(u) ≤ g∗(u), u ∈ J. Therefore,

‖ψ(u)− x(u)‖ ≤ δε ∏0<vi<u

(1 +

2Li

mΓ(α)

∫ vi

ui

(vi − v)α−1∆v)

eq(u, vi)

≤ δε ∏0<vi<u

(1 +

2Li

mΓ(α)

∫ vi

ui

(vi − v)α−1∆v)

eq(u−vi)

≤ δε ∏0<vi<u

(1 +

2Li

mΓ(α)

∫ vi

ui

(vi − v)α−1∆v)

eq(u f−vi).

By choosing K = δ ∏0<vi<u

(1 + 2Li

mΓ(α)

∫ viui(vi − v)α−1∆v

)eq(u f−vi), the Eq. (4.2.1)

has stability in terms of HU on J.

In the same way, we can prove the following theorem.

Theorem 4.2.3. If conditions (A1)− (A4) hold, then Eq. (4.2.1) has stability in terms of

HUR on J.

Example 4.2.1. Consider the semilinear dynamic equation with fractional integrable im-

pulses:

x∆(u) = arctan(u2 + x(u)), u ∈ (0, 1] ∩TS,

x(u) =1

Γ(23)

∫ u

1(u− v)

−13

|x(v)|16(1 + |x(v)|)∆v, u ∈ (1, 2] ∩TS,

(4.2.5)

59

and its associated inequality∣∣ψ∆(u)− arctan(u2 + ψ(u))

∣∣ ≤ 0.5, u ∈ (0, 1] ∩TS,∣∣ψ(u)− 1Γ(2

3)

∫ u

1(u− v)

−13

|ψ(v)|16(1 + |ψ(v)|)∆v

∣∣ ≤ 0.5, u ∈ (1, 2] ∩TS.(4.2.6)

By setting J = [0, 2] ∩ TS, 0 = u0 = v0 < u1 = 1 < v1 = 2 and α = 23 . De-

note F(u, x(u)) = arctan(u2 + x(u)) for u ∈ (0, 1] ∩ TS and Iu1,u23

g1(u, x(u)) =

1Γ( 2

3 )

∫ u1 (u− v)

−13

|x(v)|16(1+|x(v)|)∆v for u ∈ (1, 2] ∩TS. We put ε = 0.5. If y ∈ P1

C([0, 2] ∩

TS,R) satisfies the inequality (4.2.6), then there exist f ∈ P1C([0, 2]∩TS,R) and f0 ∈ R

such that | f (u)| ≤ 0.5 for u ∈ (0, 1] ∩TS and | f0| ≤ 0.5. So we haveψ∆(u) = arctan(u2 + ψ(u)) + f (u), u ∈ (0, 1] ∩TS,

ψ(u) =1

Γ(23)

∫ u

1(u− v)

−13

|ψ(v)|16(1 + |ψ(v)|)∆v + f0, u ∈ (1, 2] ∩TS,

and the solution of Eq. (4.2.5) is

x(u) =1

Γ(23)

∫ u

1(u− v)

−13

|x(v)|16(1 + |x(v)|)∆v +

∫ u

0arctan(v2 + x(v))∆v.

Based on our theoretical results, Eq. (4.2.5) has a unique solution in P1C([0, 2] ∩ TS,R)

and is stable in terms of HU on J = [0, 2] ∩TS.

Chapter 5

Non–linear Impulsive VolterraIntegro–Delay Dynamic System

This chapter focuses on the existence and uniqueness of solutions, Hyers–Ulam

stability and Hyers–Ulam–Rassias stability of non–linear impulsive Volterra integro–

delay dynamic system on time scale. The results of this chapter can be found

in [23, 43].

5.1 Non–Linear Volterra Integro–Delay Dynamic Sys-

tem with Instantaneous Impulses

This section deals with the EUS and stability in terms of HU and HUR of non–

linear Volterra IDDyS with instantaneous impulses

x∆(t) = M(t)x(t) +∫ t

t0

K(t, s, x(s), x(h(s)))∆s, t ∈ TS′ = TS0\tk,

zx(tk) = x(t+k )− x(t−k ) = χk(x(t−k )), k = 1, m,

x(t) = ρ(t), t ∈ [t0 − τ, t0]TS ,

x(t0) = ρ(t0) = x0,

(5.1.1)

60

61

where τ > 0, M(t) is piecewise continuous and a regressive square matrix of order

n on TS0 = [t0, t f ]TS , t f > t0 ≥ 0 and K(t, s, x(s), x(h(s))) is piecewise continuous

operator on Ω = (t, s, x(s), x(h(s))) : t0 ≤ s ≤ t ≤ t f , x ∈ Rn. Also χk : Rn →

Rn, ρ : [t0 − τ, t0]TS → R are continuous functions, x(t+k ) = limυ→0+ x(tk + υ) and

x(t−k ) = limυ→0− x(tk − υ) are respectively the right and left side limits of x(t) at

tk, where tk satisfies t0 < t1 < t2 < · · · < tm < tm+1 = t f < +∞. Moreover,

h : TS0 → [t0 − τ, t f ]TS is a continuous delay function such that h(t) ≤ t.

5.1.1 Basic Concepts and Remarks

Consider the following inequalities,∣∣∣∣∣∣∣∣y∆(t)−M(t)y(t)−

∫ t

t0

K(t, s, y(s), y(h(s)))∆s∣∣∣∣∣∣∣∣ ≤ ε; t ∈ TS′,

‖zy(tk)− χk(y(t−k ))‖ ≤ ε, k = 1, m,(5.1.2)

∣∣∣∣∣∣∣∣y∆(t)−M(t)y(t)−

∫ t

t0

K(t, s, y(s), y(h(s)))∆s∣∣∣∣∣∣∣∣ ≤ ϕ(t); t ∈ TS′,

‖zy(tk)− χk(y(t−k ))‖ ≤ κ, k = 1, m,(5.1.3)

where ε > 0, κ ≥ 0 and ϕ ∈ C([t0 − τ, t f ]TS ,R+) is an increasing function.

Definition 5.1.1. Eq. (5.1.1) is stable in terms of HU on [t0 − τ, t f ]TS if for each

y ∈ P1C(TS

0,Rn) satisfying (5.1.2), there exists a solution y0 ∈ P1C(TS

0,Rn) of

(5.1.1) with ‖y0(t)− y(t)‖ ≤ Kε, K > 0, ∀ t ∈ [t0 − τ, t f ]TS .

Definition 5.1.2. Eq. (5.1.1) is stable in terms of HUR on [t0 − τ, t f ]TS if for each

y ∈ P1C(TS

0,Rn) satisfying (5.1.3), there exists a solution y0 ∈ P1C(TS

0,Rn) of

(5.1.1) with ‖y0(t)− y(t)‖ ≤ Kϕ(t), K > 0, ∀ t ∈ [t0 − τ, t f ]TS .

62

Remark 5.1.1. If (5.1.2) holds for y ∈ P1C(TS

0,Rn), then there is a function f ∈

PC([t0 − τ, t f ]TS ,Rn) with sequence fk (which depends on y) bounded by positive real

number ε such thaty∆(t) = M(t)y(t) +

∫ t

t0

K(t, s, y(s), y(h(s)))∆s + f (t), y(t0) = y0, t ∈ TS′,

zy(tk) = χk(y(t−k )) + fk, k = 1, m.

Lemma 5.1.2. Every y ∈ P1C(TS

0,Rn) that satisfies (5.1.2) also comes out perfect on the

following inequality;∣∣∣∣∣∣∣∣y(t) − y(t0)−ΦM(t, t0)y0 −m

∑j=1

χ(y(t−j ))

−∫ t

t0

ΦM(t, Θ(s))∫ s

t0

K(s, u, y(u), y(h(u)))∆u∆s∣∣∣∣∣∣∣∣

≤ δε, t ∈ (tk, tk+1] ⊂ TS0,

where ‖ΦM(t, Θ(s))‖ ≤ C and δ = (m + C(t f − t0)).

Proof. If y ∈ P1C(TS

0,Rn) satisfies (5.1.2), then by Remark 5.1.1, we havey∆(t) = M(t)y(t) +

∫ t

t0

K(t, s, y(s), y(h(s)))∆s + f (t), t ∈ T′S

zy(tk) = χk(y(t−k )) + fk, k = 1, m.

Then

y(t) = y(t0) + ΦM(t, t0)y0 +m

∑j=1

χ(y(t−j )) +m

∑i=1

fi

+∫ t

t0

ΦM(t, Θ(s))( ∫ s

t0

K(s, u, y(u), y(h(u)))∆u + f (s))

∆s.

63

So, ∣∣∣∣∣∣∣∣y(t)− y(t0)−ΦM(t, t0)y0 −m

∑j=1

χ(y(t−j ))

−∫ t

t0

ΦM(t, Θ(s))∫ s

t0

K(s, u, y(u), y(h(u)))∆u∆s∣∣∣∣∣∣∣∣

≤∫ t

t0

‖ΦM(t, Θ(s))‖‖ f (s)‖∆s +m

∑i=1‖ fi‖

≤ δε.

We have similar remarks for (5.1.3).

5.1.2 Existence and Uniqueness of Solutions

Here, we prove the EUS of Eq. (5.1.1). First, we consider the following assump-

tions:

(C1) The Lipschitz condition holds for piecewise function K, i.e., ‖K(t, s, µ1, µ2)−

K(t, s, ν1, ν2)‖ ≤ ∑2i=1 L‖µi − νi‖, L > 0 and µ1, µ2, ν1, ν2 ∈ Rn;

(C2) χk : Rn → Rn satisfies ‖χk(µ1)− χk(µ2)‖ ≤ Mk‖µ1 − µ2‖, Mk > 0, k = 1, m;

(C3)

(∑m

j=1 Mj + 2∫ t

t0‖ΦM(t, Θ(s))‖

∫ st0

L∆u∆s)< 1;

(C4) ϕ ∈ C([t0 − τ, t f ]TS ,R+) is increasing function such that for some β > 0,∫ t

t0

ϕ(r)∆r ≤ βϕ(t).

Theorem 5.1.1. If conditions (C1) − (C3) are satisfied, then Eq. (5.1.1) has a unique

solution in P1C(TS

0,Rn).

64

Proof. Define Λ : PC([t0 − τ, t f ]TS ,Rn)→ PC([t0 − τ, t f ]TS ,Rn) as

(Λx)(t) =

ρ(t), t ∈ [t0 − τ, t0]TS ,

ρ(t0) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))∫ s

t0

K(s, u, x(u), x(h(u)))∆u∆s, t ∈ (t0, t1],

ρ(t0) + χ1(x(t−1 )) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))∫ s

t0

K(s, u, x(u), x(h(u)))∆u∆s, t ∈ (t1, t2],

ρ(t0) +2

∑j=1

χj(x(t−j )) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))∫ s

t0

K(s, u, x(u), x(h(u)))∆u∆s, t ∈ (t2, t3],

.

.

.

ρ(t0) +m

∑j=1

χj(x(t−j )) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))∫ s

t0

K(s, u, x(u), x(h(u)))∆u∆s, t ∈ (tm, tm+1].

(5.1.4)

We see that for x1, x2 ∈ PC([t0 − τ, t f ]TS ,Rn) and t ∈ [t0 − τ, t0]TS , we have

‖(Λx1)(t)− (Λx2)(t)‖ = 0. For t ∈ (tm, tm+1], consider

‖(Λx1)(t)− (Λx2)(t)‖ =m

∑j=1‖χj(x1(t−j ))− χj(x2(t−j ))‖

+

∣∣∣∣∣∣∣∣ ∫ t

t0

ΦM(t, Θ(s))∫ s

t0

(K(s, u, x1(u), x1(h(u)))

−K(s, u, x2(u), x2(h(u))))

∆u∆s∣∣∣∣∣∣∣∣

65

≤m

∑j=1

Mj‖x1(t−j )− x2(t−j )‖

+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

∣∣∣∣∣∣∣∣(K(s, u, x1(u), x1(h(u)))

−K(s, u, x2(u), x2(h(u))))∣∣∣∣∣∣∣∣∆u∆s

≤m

∑j=1

Mj supt∈[t0−τ,t f ]TS

‖x1(t)− x2(t)‖

+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

L‖x1(u)− x2(u)‖∆u∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

L‖x1(h(u))− x2(h(u))‖∆u∆s

≤m

∑j=1

Mj‖x1 − x2‖

+2∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

L supt∈[t0−τ,t f ]TS

‖x1(t)− x2(t)‖∆u∆s

≤ ‖x1 − x2‖( m

∑j=1

Mj + 2∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

L∆u∆s)

.

Following from (C3), the operator Λ is strictly contractive and hence a Picard op-

erator on PC([t0 − τ, t f ]TS ,Rn). Regarding to (5.1.4), we can see that unique FP of

operator Λ is actually the only one solution of (5.1.1) in P1C(TS

0,Rn).

5.1.3 Hyers–Ulam Stability

In this subsection, we prove the results concerning stability in terms of HU and

HUR of Eq. (5.1.1).

Theorem 5.1.2. If conditions (C1)− (C3) are satisfied, then Eq. (5.1.1) is stable in terms

of HU on [t0 − τ, t f ]TS .

66

Proof. Let y ∈ P1C(TS

0,Rn) be a solution to (5.1.2). The unique solution x ∈

P1C(TS

0,Rn) of the DyE

x∆(t) = M(t)x(t) +∫ t

t0

K(t, s, x(s), x(h(s)))∆s, t ∈ TS′ = TS0\tk,

zx(tk) = x(t+k )− x(t−k ) = χk(x(t−k )), k = 1, m,

x(t) = y(t), t ∈ [t0 − τ, t0]TS ,

x(t0) = y(t0) = x0,

is

x(t) =

y(t), t ∈ [t0 − τ, t0]TS ,

y(t0) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))∫ s

t0

K(s, u, x(u), x(h(u)))∆u∆s, t ∈ (t0, t1],

y(t0) + χ1(x(t−1 )) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))∫ s

t0

K(s, u, x(u), x(h(u)))∆u∆s, t ∈ (t1, t2],

y(t0) +2

∑j=1

χj(x(t−j )) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))∫ s

t0

K(s, u, x(u), x(h(u)))∆u∆s, t ∈ (t2, t3],

.

.

.

y(t0) +m

∑j=1

χj(x(t−j )) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))∫ s

t0

K(s, u, x(u), x(h(u)))∆u∆s, t ∈ (tm, tm+1].

We observe that for each t ∈ [t0 − τ, t0]TS , we have ‖y(t) − x(t)‖ = 0. Utilizing

67

Lemma 5.1.2 for t ∈ (tm, tm+1], we have

‖y(t)− x(t)‖ ≤∣∣∣∣∣∣∣∣y(t)− y(t0)−ΦM(t, t0)y0 −

m

∑j=1

χ(y(t−j ))

−∫ t

t0

ΦM(t, Θ(s))∫ s

t0

K(s, u, y(u), y(h(u)))∆u∆s∣∣∣∣∣∣∣∣

+m

∑j=1‖χj(y(t−j ))− χj(x(t−j ))‖

+

∣∣∣∣∣∣∣∣ ∫ t

t0

ΦM(t, Θ(s))∫ s

t0

(K(s, u, y(u), y(h(u)))

−K(s, u, x(u), x(h(u))))

∆u∆s∣∣∣∣∣∣∣∣

≤ δε +m

∑j=1

Mj‖y(t−j )− x(t−j )‖

+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

L‖y(u)− x(u)‖∆u∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

L‖y(h(u))− x(h(u))‖∆u∆s.

68

Next, we define the operator T : PC([t0− τ, t f ]TS ,R+)→ PC([t0− τ, t f ]TS ,R+) as

(Tg)(t) =

0, t ∈ [t0 − τ, t0]TS ,

(t f − t0)ε +∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

Lg(u)∆u∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

Lg(h(u))∆u∆s, t ∈ (t0, t1],

(1 + t f − t0)ε + M1g(t−1 ) +∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

Lg(u)∆u∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

Lg(h(u))∆u∆s, t ∈ (t1, t2],

(2 + t f − t0)ε +2

∑j=1

Mjg(t−j ) +∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

Lg(u)∆u∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

Lg(h(u))∆u∆s, t ∈ (t2, t3],

.

.

.

δε +m

∑j=1

Mjg(t−j ) +∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

Lg(u)∆u∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

Lg(h(u))∆u∆s, t ∈ (tm, tm+1].

(5.1.5)

For any g1, g2 ∈ PC([t0 − τ, t f ]TS ,R+), ‖(Tg1)(t) − (Tg2)(t)‖ = 0 for all t ∈

69

[t0 − τ, t0]TS . Now for t ∈ (tm, tm+1], consider

‖(Tg1)(t)− (Tg2)(t)‖

≤m

∑j=1

Mj‖g1(t−j )− g2(t−j )‖+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

L‖g1(u)− g2(u)‖∆u∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

L‖g1(h(u))− g2(h(u))‖∆u∆s

≤m

∑j=1

Mj supt∈[t0−τ,t f ]TS

‖g1(t)− g2(t)‖

+2∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

L supt∈[t0−τ,t f ]TS

‖g1(t)− g2(t)‖∆u∆s

≤m

∑j=1

Mj‖g1 − g2‖+ 2‖g1 − g2‖∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

L∆u∆s

≤ ‖g1 − g2‖( m

∑j=1

Mj + 2∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

L∆u∆s)

.

Since(

∑mj=1 Mj + 2

∫ tt0‖ΦM(t, Θ(s))‖

∫ st0

L∆u∆s)

< 1, so the operator is contrac-

tive onPC([t0− τ, t f ]TS ,R+). By Theorem 3.1.1, T is Picard operator having unique

FP g∗ i.e.,

g∗(t) = δε +m

∑j=1

Mjg∗(t−j ) +∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

Lg∗(u)∆u∆s

+∫ t

t0

‖ΦM(t, Θ(s))‖∫ s

t0

Lg∗(h(u))∆u∆s.

Since g∗ is increasing, so g∗(h(u)) ≤ g∗(u) , also ‖ΦM(t, Θ(s))‖ ≤ C, so we have

g∗(t) ≤ δε +m

∑j=1

Mjg∗(t−j ) + 2∫ t

t0

∫ s

t0

CLg∗(u)∆u∆s.

Lemma 4.1.2 implies

g∗(t) ≤ δε ∏t0<tj<t

(1 + Mj)eP(t, t0),

70

where P(s) = 2∫ s

t0CL∆u is a positively regressive function. If we set g(t) =

‖y(t) − x(t)‖, then from (5.1.5), g(t) ≤ (Tg)(t), so by using abstract Gronwall’s

lemma, it follows that g(t) ≤ g∗(t). Thus

‖y(t)− x(t)‖ ≤ δε ∏t0<tj<t

(1 + Mj)eP(t, t0)

≤ Kε,

where K = δ ∏t0<tj<t(1 + Mj)eP(t f−t0). So Eq. (5.1.1) is stable in terms of HU on

[t0 − τ, t f ]TS .

Similarly, we can establish the stability in terms of HUR of (5.1.1) on [t0 −

τ, t f ]TS .

Theorem 5.1.3. If conditions (C1) − (C4) hold, then Eq. (5.1.1) is stable in terms of

HUR on [t0 − τ, t f ]TS .

Example 5.1.1. Consider the following semilinear impulsive Volterra integro–dynamic

equation:x∆(t) =

1t− 2

x(t) +∫ t

t0

ep(t, x(s))∆s, x(0) = 1, t ∈ [0, 3]TS\2,

zx(tk) = x(t+k )− x(t−k ) = χk(x(t−k )), k = 1,(5.1.6)

and its associated inequality∣∣y∆(t)− 1

t− 2y(t)−

∫ t

t0

ep(t, y(s))∆s∣∣ ≤ 1.5, t ∈ [0, 3]TS\2,∣∣zy(tk)− χk(y(t−k ))

∣∣ ≤ 1.5, k = 1.(5.1.7)

By setting TS′ = [0, 3]TS\2, t0 = 0, t1 = 2 and p(t) = 1t−2 . Denote K(t, s, x(s)) =

ep(t, x(s)) = ep(t, s)ep(s, x(s)) for t ∈ TS′ and put ε = 1.5. If y ∈ P1C([0, 3]TS ,R)

71

satisfies the inequality (5.1.7), then there exist f ∈ P1C([0, 3]TS ,R) and f0 ∈ R such that

| f (t)| ≤ 1.5 for t ∈ TS′ and | f0| ≤ 1.5. So we havey∆(t) =

1t− 2

y(t) +∫ t

t0

ep(t, y(s))∆s + f (t), t ∈ TS′,

zy(tk) = χk(y(t−k )) + f0, k = 1,

and the solution of Eq. (5.1.6) is given as

x(t) = χ1(x(t−1 )) + ep(t, 0) +∫ t

0ep(t, Θ(s))

∫ s

0ep(s, x(u))∆u∆s.

Based on our theoretical results, Eq. (5.1.6) has a unique solution in P1C([0, 3]TS ,R) and

is stable in terms of HU on TS′.

5.2 Non–Linear Volterra Integro–Delay Dynamic Sys-

tem with Fractional Integrable Impulses

This section comprises of the EUS and stability in terms of HU and HUR for the

following non–linear Volterra IDDyS with integral impulses having fractional or-

der α ∈ (0, 1),

x∆(u) = M(u)x(u) +∫ u

u0

K(u, v, x(v), x(h(v)))∆v,

u ∈ (vi, ui+1] ∩TS, i = 0, m,

x(u) = Iui,uα gi(v, x(v), x(h(v))), u ∈ (ui, vi] ∩TS, i = 1, m,

x(u) = ρ(u), u ∈ [v0 − τ, v0] ∩TS,

x(u0) = ρ(u0) = x0,

(5.2.1)

where τ > 0, M(u) is a regressive piecewise continuous square matrix, 0 = u0 =

v0 < u1 ≤ v1 ≤ u2 < · · · ≤ um ≤ vm ≤ um+1 = u f are pre–fixed numbers,

72

K(u, v, x(v), x(h(v))) is piecewise continuous operator on Ω = (u, v, x(v), x(h(v))) :

u0 ≤ v ≤ u ≤ u f , x ∈ Rn, gi : (ui, vi] ∩TS ×Rn ×Rn → Rn, i = 1, m are contin-

uous functions, ρ : [v0 − τ, v0] ∩TS → Rn is history function and Iui,uα gi are the so

called RL (left) integrals having fractional order α ∈ (0, 1) on time scale, with the

representation:

Iui,uα gi(u, x(u), x(h(u))) =

1Γ(α)

∫ u

ui

(u− v)α−1gi(v, x(v), x(h(v)))∆v.

Moreover, (vi, ui+1]∩TS, (ui, vi]∩TS, [v0− τ, v0]∩TS are non-empty sets and the

delay function h : [v0 − τ, u f ] ∩TS → (vi, ui+1] ∩TS is continuous with h(u) ≤ u.

5.2.1 Basic Concepts and Remarks

Consider the following inequalities,‖y∆(u)−M(u)y(u)−

∫ u

u0

K(u, v, y(v), y(h(v)))∆v‖ ≤ ε,

u ∈ (vi, ui+1] ∩TS, i = 0, m,

‖y(u)− Iui,uα gi(u, y(u), y(h(u)))‖ ≤ ε, u ∈ (ui, vi] ∩TS, i = 1, m,

(5.2.2)

‖y∆(u)−M(u)y(u)−

∫ u

u0

K(u, v, y(v), y(h(v)))∆v‖ ≤ ϕ(u),

u ∈ (vi, ui+1] ∩TS, i = 0, m,

‖y(u)− Iui,uα gi(u, y(u), y(h(u)))‖ ≤ κ, u ∈ (ui, vi] ∩TS, i = 1, m,

(5.2.3)

where ε > 0, κ ≥ 0 and ϕ ∈ C(J,R+) is an increasing function.

Definition 5.2.1. Eq. (5.2.1) is stable in terms ofHU, if for every ε > 0 there exists a

positive number K such that for every y ∈ P1C(J,Rn) satisfying (5.2.2), there exists

a solution y0 ∈ P1C(J,Rn) of (5.2.1) such that ‖y0(u)− y(u)‖ ≤ Kε ∀ u ∈ J. Here K

is a positive number that depends on ε and do not depend on f i.

73

Definition 5.2.2. Eq. (5.2.1) is stable in terms ofHUR, provided ∀ (ϕ, κ) ∈ C(J,R+)×

R+ there exists M > 0 such that ∀ y ∈ P1C(J,Rn) satisfying (5.2.3), there exists a so-

lution y0 ∈ P1C(J,Rn) of (5.2.1) such that the inequality ‖y0(u)− y(u)‖ ≤ Mϕ(u)

is true ∀ u ∈ J. Here M > 0 depends on (ϕ, κ).

Remark 5.2.1. A function y ∈ P1C(J,Rn) satisfies inequality (5.2.2) (respectively in-

equality (5.2.3)) if and only if there exist f ∈ P1C(J,Rn) and a finite sequence fi such that

‖ f (u)‖ ≤ ε ∀ u ∈ J and ‖ fi‖ ≤ ε (respectively ‖ fi‖ ≤ κ) for every i = 1, m andy∆(u) = M(u)y(u) +

∫ u

u0

K(u, v, y(v), y(h(v)))∆v + f (u),

u ∈ (vi, ui+1] ∩TS, i = 0, m,

y(u) = Iui,uα gi(u, y(u), y(h(u))) + fi, u ∈ (ui, vi] ∩TS, i = 1, m.

Lemma 5.2.2. If y ∈ P1C(J,Rn) satisfies inequality (5.2.2) (respectively inequality (5.2.3)),

then the following inequalities

∣∣∣∣∣∣∣∣y(u)− y(u0)−ΦM(u, u0)y0 −∫ u

vi

ΦM(u, Θ(v))∫ v

v0

K(v, s, y(s), y(h(s)))∆s∆v

− Iui,uα gi(u, y(u), y(h(u)))

∣∣∣∣∣∣∣∣ ≤ (Cu f − Cvi + m)ε,

u ∈ (vi, ui+1] ∩TS, i = 1, m,

‖y(u)− Iui,uα gi(u, y(u), y(h(u)))‖ ≤ mε, (respectively mκ),

u ∈ (ui, vi] ∩TS, i = 1, m,

are true. Here C is the bound of fundamental matrix ΦM(u, Θ(v)).

Proof. If y ∈ P1C(J,Rn) satisfies (5.2.2), then by Remark 5.2.1, we have

y∆(u) = M(u)y(u) +∫ u

u0

K(u, v, y(v), y(h(v)))∆v + f (u),

u ∈ (vi, ui+1] ∩TS, i = 0, m,

y(u) = Iui,uα gi(u, y(u), y(h(u))) + fi, u ∈ (ui, vi] ∩TS, i = 1, m.

(5.2.4)

74

Clearly the solution of (5.2.4) is given as

y(u) =

y(u0) + ΦM(u, u0)y0

+∫ u

vi

ΦM(u, Θ(v))( ∫ v

v0

K(v, s, y(s), y(h(s)))∆s + f (v))

∆v

+ Iui,uα gi(u, y(u), y(h(u))), u ∈ (vi, ui+1] ∩TS, i = 1, m,

Iui,uα gi(u, y(u), y(h(u))) + fi, u ∈ (ui, vi] ∩TS, i = 1, m.

For u ∈ (vi, ui+1] ∩TS, i = 1, m, we get∣∣∣∣∣∣∣∣y(u)− y(u0)−ΦM(u, u0)y0 −∫ u

vi

ΦM(u, Θ(v))∫ v

v0

K(v, s, y(s), y(h(s)))∆s∆v

−Iui,uα gi(u, y(u), y(h(u)))

∣∣∣∣∣∣∣∣≤

∫ u

vi

‖ΦM(u, Θ(v))‖‖ f (v)‖∆v +m

∑i=1‖ fi‖

≤ (Cu− Cvi + m)ε

≤ (Cu f − Cvi + m)ε.

Proceeding as above, we derive

‖y(u)− Iui,uα gi(u, y(u), y(h(u)))‖ ≤ mε, u ∈ (ui, vi] ∩TS, i = 1, m.

We have similar processions for (5.2.3).

5.2.2 Existence and Uniqueness of Solutions

Here, we prove the EUS of Eq. (5.2.1). First, we consider the following assump-

tions:

(A′1) The Lipschitz condition holds for piecewise continuous function K, i.e.,

75

‖K(u, v, µ1, µ2)−K(u, v, ν1, ν2)‖ ≤ ∑2k=1 L‖µk− νk‖, L > 0, u ∈ (vi, ui+1]∩TS, i =

0, m;

(A′2) gi : (ui, vi]∩TS×Rn×Rn → Rn satisfies the Lipschitz condition ‖gi(u, µ1, µ2)−

gi(u, ν1, ν2)‖ ≤ ∑2k=1 Lgi‖µk − νk‖, Lgi > 0, ∀ u ∈ (ui, vi] ∩ TS, i = 1, m and

µ1, µ2, ν1, ν2 ∈ Rn ;

(A′3)(

2LgiΓ(α)

∫ viui(vi − v)α−1∆v + 2CL

∫ uvi

∫ vv0

∆s∆v)< 1, i = 1, m;

(A′4) ϕ ∈ C(J,R+) is increasing so that for some β > 0,

∫ u

u0

ϕ(r)∆r ≤ βϕ(u).

Theorem 5.2.1. If conditions (A′1)− (A′3) hold, then Eq. (5.2.1) has precisely unique

solution in P1C(J,Rn).

Proof. Consider the operator Λ : PC(J,Rn)→ PC(J,Rn),

(Λx)(u) =

ρ(u), u ∈ [v0 − τ, v0] ∩TS,

Iui,uα gi(u, x(u), x(h(u))), u ∈ (ui, vi] ∩TS, i = 1, m, α ∈ (0, 1),

ρ(u0) + ΦM(u, u0)x0 + Iui,viα gi(vi, x(vi), x(h(vi)))

+∫ u

vi

ΦM(u, Θ(v))∫ v

v0

K(v, s, x(s), x(h(s)))∆s∆v,

u ∈ (vi, ui+1] ∩TS, i = 1, m, α ∈ (0, 1).

76

For any x1, x2 ∈ PC(J,Rn), u ∈ (vi, ui+1] ∩TS, i = 1, m, we have

‖(Λx1)(u)− (Λx2)(u)‖

≤ ‖Iui,viα gi(vi, x1(vi), x1(h(vi)))− Iui,vi

α gi(vi, x2(vi), x2(h(vi)))‖

+

∣∣∣∣∣∣∣∣ ∫ u

vi

ΦM(u, Θ(v))∫ v

v0

K(v, s, x1(s), x1(h(s)))

−K(v, s, x2(s), x2(h(s)))∆s∆v∣∣∣∣∣∣∣∣

≤ ‖Iui,viα gi(vi, x1(vi), x1(h(vi)))− Iui,vi

α gi(vi, x2(vi), x2(h(vi)))‖

+∫ u

vi

‖ΦM(u, Θ(v))‖∫ v

v0

∣∣∣∣∣∣∣∣K(v, s, x1(s), x1(h(s)))

−K(v, s, x2(s), x2(h(s)))∣∣∣∣∣∣∣∣∆s∆v

≤ 1Γ(α)

∫ vi

ui

(vi − v)α−1‖gi(v, x1(v), x1(h(v)))

−gi(v, x2(v), x2(h(v)))‖∆v

+L∫ u

vi

‖ΦM(u, Θ(v))‖∫ v

v0

‖x1(s)− x2(s)‖∆s∆v

+L∫ u

vi

‖ΦM(u, Θ(v))‖∫ v

v0

‖x1(h(s))− x2(h(s))‖∆s∆v

≤ Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1‖x1(v)− x2(v)‖∆v

+Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1‖x1(h(v))− x2(h(v))‖∆v

+2CL∫ u

vi

∫ v

v0

supvi≤v≤ui+1

‖x1(v)− x2(v)‖∆s∆v

≤ 2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1 supui≤v≤vi

‖x1(v)− x2(v)‖∆v

+2CL∫ u

vi

∫ v

v0

supvi≤v≤ui+1

‖x1(v)− x2(v)‖∆s∆v

≤ ‖x1 − x2‖2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1∆v

+2‖x1 − x2‖CL∫ u

vi

∫ v

v0

∆s∆v

77

≤(

2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1∆v + 2CL∫ u

vi

∫ v

v0

∆s∆v)‖x1 − x2‖.

According to (A′3), we are dealing here with the strictly contractive operator on

(vi, ui+1] ∩ TS, i = 1, m and so it is Picard operator on PC(J,Rn). The unique

solution of Eq. (5.2.1) in P1C(J,Rn) is in fact the unique FP of this operator.

5.2.3 Hyers–Ulam Stability

In this subsection, we establish our results concerning stability in terms ofHU and

HUR of Eq. (5.2.1).

Theorem 5.2.2. If conditions (A′1)− (A′3) are satisfied, then Eq. (5.2.1) has stability in

terms of HU on J.

Proof. Assume that (5.2.2) has a solution y ∈ P1C(J,Rn). Then for DyE

x∆(u) = M(u)x(u) +∫ u

u0

K(u, v, x(v), x(h(v)))∆v,

u ∈ (vi, ui+1] ∩TS, i = 0, m,

x(u) = Iui,uα gi(u, x(u), x(h(u))), u ∈ (ui, vi] ∩TS, i = 1, m,

x(u) = y(u), u ∈ [v0 − τ, v0] ∩TS,

x(u0) = y(u0) = x0,

78

we have the unique solution

x(u) =

y(u), u ∈ [v0 − τ, v0] ∩TS,

Iui,uα gi(u, x(u), x(h(u))), u ∈ (ui, vi] ∩TS, i = 1, m, α ∈ (0, 1),

y(u0) + ΦM(u, u0)x0 + Iui,viα gi(vi, x(vi), x(h(vi)))

+∫ u

vi

ΦM(u, Θ(v))∫ v

v0

K(v, s, x(s), x(h(s)))∆s∆v,

u ∈ (vi, ui+1] ∩TS, i = 1, m.

We observe that ∀ u ∈ (vi, ui+1] ∩TS, i = 1, m, using Lemma 5.2.2, we have

‖y(u)− x(u)‖ ≤ ‖y(u)− y(u0)−ΦM(u, u0)y0

−∫ u

vi

ΦM(u, Θ(v))∫ v

v0

K(v, s, y(s), y(h(s)))∆s∆v

−Iui,uα gi(v, y(v), y(h(v)))‖+ ‖Iui,vi

α gi(vi, y(vi), y(h(vi)))

−Iui,viα gi(vi, x(vi), x(h(vi)))‖

+∫ u

vi

‖ΦM(u, Θ(v))‖∫ v

v0

‖K(v, s, y(s), y(h(s)))

−∫ v

v0

K(v, s, x(s), x(h(s)))‖∆s∆v

≤ (m + Cu f − Cvi)ε +Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1‖y(v)− x(v)‖∆v

+Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1‖y(h(v))− x(h(v))‖∆v

+CL∫ u

vi

∫ v

v0

‖y(s)− x(s)‖∆s∆v

+CL∫ u

vi

∫ v

v0

‖y(h(s))− x(h(s))‖∆s∆v.

Consider the operator T : PC(J,R+)→ PC(J,R+) as:

(Tg)(u) = (m + Cu f − Cvi)ε +Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1g(v)∆v

79

+Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1g(h(v))∆v + CL∫ u

vi

∫ v

v0

g(u)∆s∆v

+CL∫ u

vi

∫ v

v0

g(h(u))∆s∆v.

For any g1, g2 ∈ PC(J,R+), u ∈ (vi, ui+1] ∩TS, i = 1, m, we have

‖(Tg1)(u)− (Tg2)(u)‖ ≤Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1‖g1(v)− g2(v)‖∆v

+Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1‖g1(h(v))− g2(h(v))‖∆v

+CL∫ u

vi

∫ v

v0

‖g1(s)− g2(s)‖∆s∆v

+CL∫ u

vi

∫ v

v0

‖g1(h(s))− g2(h(s))‖∆s∆v

≤ 2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1 supui≤v≤vi

‖g1(v)− g2(v)‖∆v

+2CL∫ u

vi

∫ v

v0

supvi≤v≤ui+1

‖g1(v)− g2(v)‖∆s∆v

≤ ‖g1 − g2‖2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1∆v

+2‖g1 − g2‖CL∫ u

vi

∫ v

v0

∆s∆v

≤(

2Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1∆v

+2CL∫ u

vi

∫ v

v0

∆s∆v)‖g1 − g2‖.

Again according to (A′3), we are dealing here with the strictly contractive oper-

ator on (vi, ui+1] ∩ TS, i = 1, m and so a Picard operator on PC(J,R+). Banach

contraction theorem imply, T is Picard operator having unique FP g∗ ∈ PC(J,R+)

80

i.e.,

g∗(u) = (m + Cu f − Cvi)ε +Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1g∗(v)∆v

+Lgi

Γ(α)

∫ vi

ui

(vi − v)α−1g∗(h(v))∆v + CL∫ u

vi

∫ v

v0

g∗(s)∆s∆v

+CL∫ u

vi

∫ v

v0

g∗(h(s))∆s∆v.

As, g∗ is increasing, therefore g∗(h(u)) ≤ g∗(u), (m + Cu f − Cvi) ≤ δ for some

δ > 0 and for i = 1, m, we can write

g∗(u) ≤ δε + ∑0<vi<u

2Lgi

mΓ(α)

∫ vi

ui

(vi − v)α−1g∗(v)∆v + 2CL∫ u

v0

∫ v

v0

g∗(s)∆s∆v.

Using Lemma 4.1.2, we have

g∗(u) ≤ δε ∏0<vi<u

(1 +

2Lgi

mΓ(α)

∫ vi

ui

(vi − v)α−1∆v)

eq(u, vi).

where q = 2CL∫ v

v0∆s is a positively regressive function. If we determine g = ‖y−

x‖, then g(u) ≤ (Tg)(u), which follows by utilizing abstract Gronwall’s lemma

that g(u) ≤ g∗(u), hence

‖y(u)− x(u)‖ ≤ δε ∏0<vi<u

(1 +

2Lgi

mΓ(α)

∫ vi

ui

(vi − v)α−1∆v)

eq(u, vi)

≤ δε ∏0<vi<u

(1 +

2Lgi

mΓ(α)

∫ vi

ui

(vi − v)α−1∆v)

eq(u−vi)

≤ δε ∏0<vi<u

(1 +

2Lgi

mΓ(α)

∫ vi

ui

(vi − v)α−1∆v)

eq(u f−vi).

By choosing K = δ ∏0<vi<u

(1 +

2LgimΓ(α)

∫ viui(vi − v)α−1∆v

)eq(u f−vi), the Eq. (5.2.1)

has stability in terms of HU on J.

Similarly, we can establish the stability in terms of HUR of (5.2.1) on J.

81

Theorem 5.2.3. If conditions (A′1)− (A′4) hold, then Eq. (5.2.1) has stability in terms of

HUR on J.

Example 5.2.1. Consider the following semilinear impulsive Volterra integro–dynamic

equation with integral impulses:

x∆(u) = arctan(u)x(u) +

∫ u

0ep(u, x(v))∆v, x(0) = 1, u ∈ (0, 1] ∩TS,

x(u) =1

Γ(23)

∫ u

1(u− v)

−13

|x(v)|16(1 + |x(v)|)∆v, u ∈ (1, 2] ∩TS,

(5.2.5)

and its associated inequality∣∣y∆(u)− arctan(u)y(u)−

∫ u

0ep(u, y(v))∆v

∣∣ ≤ 0.5, u ∈ (0, 1] ∩TS,∣∣y(u)− 1Γ(2

3)

∫ u

1(u− v)

−13

|y(v)|16(1 + |y(v)|)∆v

∣∣ ≤ 0.5, u ∈ (1, 2] ∩TS.(5.2.6)

By setting J = [0, 2] ∩ TS, 0 = u0 = v0 < u1 = 1 < v1 = 2 and α = 23 . Denote

p(u) = arctan(u), K(u, v, x(v)) = ep(u, x(v)) = ep(u, v)ep(v, x(v)) for u ∈ (0, 1] ∩

TS and Iu1,u23

g1(u, x(u)) = 1Γ( 2

3 )

∫ u1 (u − v)

−13

|x(v)|16(1+|x(v)|)∆v for u ∈ (1, 2] ∩ TS. We

put ε = 0.5. If y ∈ P1C([0, 2] ∩ TS,R) satisfies the inequality (5.2.6), then there exist

f ∈ P1C([0, 2] ∩ TS,R) and f0 ∈ R such that | f (u)| ≤ 0.5 for u ∈ (0, 1] ∩ TS and

| f0| ≤ 0.5. So we havey∆(u) = arctan(u)y(u) +

∫ u

0ep(u, y(v))∆v + f (u), u ∈ (0, 1] ∩TS,

y(u) =1

Γ(23)

∫ u

1(u− v)

−13

|y(v)|16(1 + |y(v)|)∆v + f0, u ∈ (1, 2] ∩TS,

and the solution of Eq. (5.2.5) is

x(u) =1

Γ(23)

∫ u

1(u− v)

−13

|x(v)|16(1 + |x(v)|)∆v + ep(u, 0)

+∫ u

0ep(u, Θ(v))

∫ v

0ep(v, x(s))∆s∆v.

82

Based on our theoretical results, Eq. (5.2.5) has a unique solution in P1C([0, 2] ∩ TS,R)

and is stable in terms of HU on J = [0, 2] ∩TS.

Chapter 6

Applications

This chapter contains the applications of our results. We prove the existence and

uniqueness of solutions, Hyers–Ulam stability and Hyers–Ulam–Rassias stabil-

ity of the non–linear impulsive Hammerstein integro–delay dynamic system and

non–linear impulsive mixed integro–dynamic system on time scale. The results of

this chapter can be seen in [25, 26].

6.1 Hammerstein Integro–Delay Dynamic System

This section deals with the EUS and stability in terms of HU and HUR of the fol-

lowing non–linear impulsive Hammerstein IDDyS,

x∆(t) = M(t)x(t) + F(t, x(t), x(h(t)))∫ t

t0

g(t, s)H(s, x(s), x(h(s)))∆s,

t ∈ TS′ = TS0\tk,

zx(tk) = x(t+k )− x(t−k ) = χk(x(t−k )), k = 1, m,

x(t) = ρ(t), t ∈ [t0 − τ, t0]TS ,

x(t0) = ρ(t0) = x0,

(6.1.1)

83

84

where τ > 0, the M(t) is piecewise continuous regressive square matrix on TS0 =

[t0, t f ]TS , t f > s > t0 ≥ 0 and F : TS0 ×Rn ×Rn → Rn, H : TS0 ×Rn ×Rn →

Rn, χk : Rn → Rn, ρ : [t0 − τ, t0]TS → R, the kernal g : TS0 × TS0 → Rn are

continuous functions. Also x(t+k ) = limυ→0+ x(tk + υ) and x(t−k ) = limυ→0− x(tk −

υ) are respectively the right and left side limits of x(t) at tk, where tk satisfies t0 <

t1 < t2 < · · · < tm < tm+1 = t f < +∞. Moreover, h : TS0 → [t0 − τ, t f ]TS is a

continuous delay function such that h(t) ≤ t.

6.1.1 Basic Concepts and Remarks

Consider the following inequalities:∣∣∣∣∣∣∣∣ψ∆(s)−M(s)ψ(s)−F(s, ψ(s), ψ(h(s)))

∫ s

s0

g(s, t)H(t, ψ(t), ψ(h(t)))∆t∣∣∣∣∣∣∣∣ ≤ ε,

‖zψ(sk)− χk(ψ(s−k ))‖ ≤ ε, k = 1, m,(6.1.2)

∣∣∣∣∣∣∣∣ψ∆(s)−M(s)ψ(s)−F(s, ψ(s), ψ(h(s)))∫ s

s0

g(s, t)H(t, ψ(t), ψ(h(t)))∆t∣∣∣∣∣∣∣∣ ≤ ϕ(s),

‖zψ(sk)− χk(ψ(s−k ))‖ ≤ κ, k = 1, m,(6.1.3)

where s ∈ TS′, ε > 0, κ ≥ 0 and ϕ : [t0 − τ, t f ]TS → R+ is rd continuous and

increasing.

Definition 6.1.1. Eq. (6.1.1) is said to be stable in terms ofHU on [t0− τ, t f ]TS if for

each ψ ∈ P1C(TS

0,Rn) satisfying (6.1.2), there exists a solution ψ0 ∈ P1C(TS

0,Rn)

of (6.1.1) with ‖ψ0(s)− ψ(s)‖ ≤ Cε, C > 0, ∀ s ∈ [t0 − τ, t f ]TS .

Definition 6.1.2. Eq. (6.1.1) is said to be stable in terms of HUR on [t0 − τ, t f ]TS if

for each ψ ∈ P1C(TS

0,Rn) satisfying (6.1.3), there exists a solution ψ0 ∈ P1C(TS

0,Rn)

of (6.1.1) with ‖ψ0(s)− ψ(s)‖ ≤ Cϕ(s), C > 0, ∀ s ∈ [t0 − τ, t f ]TS .

85

Remark 6.1.1. If (6.1.2) holds for ψ ∈ P1C(TS

0,Rn), then there exist f ∈ PC([t0 −

τ, t f ]TS ,Rn) along with sequence fk bounded by positive real number ε such thatψ∆(t) = M(t)ψ(t) + F(t, ψ(t), ψ(h(t)))

∫ t

t0

g(t, s)H(s, ψ(s), ψ(h(s)))∆s + f (t),

ψ(t0) = ψ0, t ∈ TS′,

zψ(tk) = χk(ψ(t−k )) + fk, k = 1, m.

Lemma 6.1.2. Every solution ψ ∈ P1C(TS

0,Rn) of inequality (6.1.2) also satisfies the

following inequality:∣∣∣∣∣∣∣∣ψ(t)− ψ(t0)−ΦM(t, t0)ψ0 −

m

∑j=1

χ(ψ(t−j ))

−∫ t

t0

ΦM(t, Θ(s))F(s, ψ(s), ψ(h(s)))∫ s

s0

g(s, u)H(u, ψ(u), ψ(h(u)))∆u∆s∣∣∣∣∣∣∣∣ ≤ ηε,

for t ∈ (tk, tk+1] ⊂ TS0, where η = (m + C(t f − t0)) and ‖ΦM(t, Θ(s))‖ ≤ C.

Proof. If ψ ∈ P1C(TS

0,Rn) satisfies (6.1.2), so by Remark 6.1.1, we getψ∆(t) = M(t)ψ(t) + F(t, ψ(t), ψ(h(t)))

∫ t

t0

g(t, s)H(s, ψ(s), ψ(h(s)))∆s + f (t),

ψ(t0) = ψ0, t ∈ T′S,

zψ(tk) = χk(ψ(t−k )) + fk, k = 1, m.

Then

ψ(t) = ψ(t0) + ΦM(t, t0)ψ0 +m

∑j=1

χ(ψ(t−j )) +m

∑i=1

fi

+∫ t

t0

ΦM(t, Θ(s))F(s, ψ(s), ψ(h(s)))∫ s

s0

g(s, u)H(u, ψ(u), ψ(h(u)))∆u∆s

+∫ t

t0

ΦM(t, Θ(s)) f (s)∆s.

86

So ∣∣∣∣∣∣∣∣ψ(t)− ψ(t0)−ΦM(t, t0)ψ0 −m

∑j=1

χ(ψ(t−j ))

−∫ t

t0

ΦM(t, Θ(s))F(s, ψ(s), ψ(h(s)))∫ s

s0

g(s, u)H(u, ψ(u), ψ(h(u)))∆u∆s∣∣∣∣∣∣∣∣

≤∫ t

t0

‖ΦM(t, Θ(s))‖‖ f (s)‖∆s +m

∑i=1‖ fi‖

≤ ηε.

A similar remark for (6.1.3) can be derived.

6.1.2 Existence and Uniqueness of Solutions

Here, we prove the EUS of Eq. (6.1.1). First, consider the following conditions:

(C′1) H : TS0 ×Rn ×Rn → Rn satisfies Lipschitz condition

‖H(t, µ1, µ2)−H(t, ν1, ν2)‖ ≤ ∑2i=1 L‖µi − νi‖, L > 0, ∀ t ∈ TS0 and µ1, µ2, ν1, ν2 ∈

Rn;

(C′2) χk : Rn → Rn satisfies ‖χk(µ1)−χk(µ2)‖ ≤ Mk‖µ1− µ2‖, Mk > 0, ∀ k = 1, m

and µ1, µ2 ∈ Rn;

(C′3) ‖F(s, x1(s), x1(h(s)))−F(s, x2(s), x2(h(s)))‖ ≤ δ, ‖g(t, s)‖ ≤ ε, δ > 0, ε > 0,

∀ t, s ∈ TS0 ;

(C′4)(

∑mj=1 Mj + 2

∫ tt0

∫ ss0

CδεL∆u∆s)< 1;

87

(C′5) For ϕ : [t0 − τ, t f ]TS → R+, the following inequality

∫ t

t0

ϕ(u)∆u ≤ γϕ(t), γ > 0,

is satisfied.

Theorem 6.1.1. If conditions (C′1)− (C′4) hold, then Eq. (6.1.1) has only one solution

in P1C(TS

0,Rn).

Proof. Define Λ : PC([t0 − τ, t f ]TS ,Rn)→ PC([t0 − τ, t f ]TS ,Rn) as

(Λx)(t) =

ρ(t), t ∈ [t0 − τ, t0]TS ,

ρ(t0) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), x(h(s)))∫ s

s0

g(s, u)H(u, x(u), x(h(u)))∆u∆s,

t ∈ (t0, t1],

ρ(t0) +i

∑j=1

χj(x(t−j )) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), x(h(s)))∫ s

s0

g(s, u)H(u, x(u), x(h(u)))∆u∆s,

t ∈ (ti, ti+1], i = 1, m.

For any x1, x2 ∈ PC([t0 − τ, t f ]TS ,Rn), t ∈ [t0 − τ, t0]TS , we have ‖(Λx1)(t) −

88

(Λx2)(t)‖ = 0. For t ∈ (ti, ti+1], we have

‖(Λx1)(t)− (Λx2)(t)‖

≤m

∑j=1‖χj(x1(t−j ))− χj(x2(t−j ))‖+

∣∣∣∣∣∣∣∣ ∫ t

t0

ΦM(t, Θ(s))(

F(s, x1(s), x1(h(s)))

−F(s, x2(s), x2(h(s)))) ∫ s

s0

g(s, u)(

H(u, x1(u), x1(h(u)))

−H(u, x2(u), x2(h(u))))

∆u∆s∣∣∣∣∣∣∣∣

≤m

∑j=1‖χj(x1(t−j ))− χj(x2(t−j ))‖

+∫ t

t0

‖ΦM(t, Θ(s))‖∣∣∣∣∣∣∣∣(F(s, x1(s), x1(h(s)))

−F(s, x2(s), x2(h(s))))∣∣∣∣∣∣∣∣ ∫ s

s0

‖g(s, u)‖∣∣∣∣∣∣∣∣H(u, x1(u), x1(h(u)))

−H(u, x2(u), x2(h(u)))∣∣∣∣∣∣∣∣∆u∆s

≤m

∑j=1

Mj‖x1(t−j )− x2(t−j )‖+∫ t

t0

Cδ∫ s

s0

εL‖x1(u)− x2(u)‖∆u∆s

+∫ t

t0

Cδ∫ s

s0

εL‖x1(h(u))− x2(h(u))‖∆u∆s

≤m

∑j=1

Mj supt∈[t0−τ,t f ]TS

‖x1(t)− x2(t)‖

+2∫ t

t0

Cδ∫ s

s0

εL supt∈[t0−τ,t f ]TS

‖x1(t)− x2(t)‖∆u∆s

≤m

∑j=1

Mj‖x1 − x2‖+ 2‖x1 − x2‖∫ t

t0

∫ s

s0

CδεL∆u∆s

≤ ‖x1 − x2‖( m

∑j=1

Mj + 2∫ t

t0

∫ s

s0

CδεL∆u∆s)

.

From (C′4), Λ is a contraction on PC([t0 − τ, t f ]TS ,Rn). So Λ is Picard operator

with unique FP, which is the unique solution of (6.1.1) in P1C(TS

0,Rn).

89

6.1.3 Hyers–Ulam Stability

In this subsection, we establish our results concerning stability in terms ofHU and

HUR of Eq. (6.1.1).

Theorem 6.1.2. If conditions (C′1)− (C′4) hold, then Eq. (6.1.1) has stability in terms

of HU on [t0 − τ, t f ]TS .

Proof. If ψ ∈ P1C(TS

0,Rn) satisfies (6.1.2). The only one solution x ∈ P1C(TS

0,Rn)

of the DyE

x∆(t) = M(t)x(t) + F(t, x(t), x(h(t)))∫ t

t0

g(t, s)H(s, x(s), x(h(s)))∆s,

t ∈ TS′ = TS0\tk,

zx(tk) = x(t+k )− x(t−k ) = χk(x(t−k )), k = 1, m,

x(t) = ψ(t), t ∈ [t0 − τ, t0]TS ,

x(t0) = ψ(t0) = x0,

is

x(t) =

ψ(t), t ∈ [t0 − τ, t0]TS ,

ψ(t0) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), x(h(s)))∫ s

s0

g(s, u)H(u, x(u), x(h(u)))∆u∆s,

t ∈ (t0, t1],

ψ(t0) +i

∑j=1

χj(x(t−j )) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), x(h(s)))∫ s

s0

g(s, u)H(u, x(u), x(h(u)))∆u∆s,

t ∈ (ti, ti+1], i = 1, m.

90

We observe that for each t ∈ [t0 − τ, t0]TS , we have ‖ψ(t) − x(t)‖ = 0. For t ∈

(ti, ti+1], utilizing Lemma 6.1.2, we get

‖ψ(t)− x(t)‖ ≤∣∣∣∣∣∣∣∣ψ(t)− ψ(t0)−ΦM(t, t0)ψ0 −

m

∑j=1

χ(ψ(t−j ))

−∫ t

t0

ΦM(t, Θ(s))F(s, ψ(s), ψ(h(s)))∫ s

s0

g(s, u)H(u, ψ(u), ψ(h(u)))∆u∆s∣∣∣∣∣∣∣∣

+m

∑j=1‖χj(ψ(t−j ))− χj(x(t−j ))‖+

∣∣∣∣∣∣∣∣ ∫ t

t0

ΦM(t, Θ(s))(

F(s, ψ(s), ψ(h(s)))

−F(s, x(s), x(h(s)))) ∫ s

s0

g(s, u)(

H(u, ψ(u), ψ(h(u)))

−H(u, x(u), x(h(u))))

∆u∆s∣∣∣∣∣∣∣∣

≤ ηε +m

∑j=1

Mj‖ψ(t−j )− x(t−j )‖+∫ t

t0

‖ΦM(t, Θ(s))‖∣∣∣∣∣∣∣∣(F(s, ψ(s), ψ(h(s)))

−F(s, x(s), x(h(s))))∣∣∣∣∣∣∣∣ ∫ s

s0

‖g(s, u)‖∣∣∣∣∣∣∣∣(H(u, ψ(u), ψ(h(u)))

−H(u, x(u), x(h(u))))∣∣∣∣∣∣∣∣∆u∆s

≤ ηε +m

∑j=1

Mj‖ψ(t−j )− x(t−j )‖+∫ t

t0

Cδ∫ s

s0

εL‖ψ(u)− x(u)‖∆u∆s

+∫ t

t0

Cδ∫ s

s0

εL‖ψ(h(u))− x(h(u))‖∆u∆s.

91

Consider T : PC([t0 − τ, t f ]TS ,R+)→ PC([t0 − τ, t f ]TS ,R+) as:

(Tw)(t) =

0, t ∈ [t0 − τ, t0]TS ,

(t f − t0)ε +∫ t

t0

Cδ∫ s

s0

εLw(u)∆u∆s +∫ t

t0

Cδ∫ s

s0

εLw(h(u))∆u∆s,

t ∈ (t0, t1],

ηε +i

∑j=1

Mjw(t−j ) +∫ t

t0

Cδ∫ s

s0

εLw(u)∆u∆s

+∫ t

t0

Cδ∫ s

s0

εLw(h(u))∆u∆s, t ∈ (ti, ti+1], i = 1, m.

(6.1.4)

For any w1, w2 ∈ PC([t0 − τ, t f ]TS ,R+), ‖(Tw1)(t) − (Tw2)(t)‖ = 0 for all t ∈

[t0 − τ, t0]TS . For t ∈ (ti, ti+1], we can see that,

‖(Tw1)(t)− (Tw2)(t)‖ ≤m

∑j=1

Mj‖w1(t−j )−w2(t−j )‖

+∫ t

t0

Cδ∫ s

s0

εL‖w1(u)−w2(u)‖∆u∆s

+∫ t

t0

Cδ∫ s

s0

εL‖w1(h(u))−w2(h(u))‖∆u∆s

≤m

∑j=1

Mj‖w1 −w2‖+ 2‖w1 −w2‖∫ t

t0

∫ s

s0

CδεL∆u∆s

≤ ‖w1 −w2‖( m

∑j=1

Mj + 2∫ t

t0

∫ s

s0

CδεL∆u∆s)

.

From (C′4), the operator T is a contraction on PC([t0 − τ, t f ]TS ,R+). By using

Theorem 3.1.1, T is Picard operator having unique FP w∗ i.e.,

w∗(t) = ηε +m

∑j=1

Mjw∗(t−j ) +∫ t

t0

Cδ∫ s

s0

εLw∗(u)∆u∆s

+∫ t

t0

Cδ∫ s

s0

εLw∗(h(u))∆u∆s.

92

Since w∗(h(s)) ≤ w∗(s), so

w∗(t) ≤ ηε +m

∑j=1

Mjw∗(t−j ) + 2∫ t

t0

Cδ∫ s

s0

εLw∗(u)∆u∆s.

By Lemma 4.1.2,

w∗(t) ≤ ηε ∏t0<tj<t

(1 + Mj)eP(t, t0),

where P =∫ s

s02δεCL∆u. Taking w(t) = ‖ψ(t)− x(t)‖ and using (6.1.4), w(t) ≤

(Tw)(t). Utilizing abstract Gronwall’s lemma, w(t) ≤ w∗(t), thus

‖ψ(t)− x(t)‖ ≤ ηε ∏t0<tj<t

(1 + Mj)eP(t, t0)

≤ Kε,

where K = η ∏t0<tj<t(1 + Mj)eP(t f−t0). Hence Eq. (6.1.1) is stable in terms of HU

on [t0 − τ, t f ]TS .

Repeating the same steps, we can prove the following theorem.

Theorem 6.1.3. If conditions (C′1)− (C′5) hold, then Eq. (6.1.1) is stable in terms of

HUR on [t0 − τ, t f ]TS .

Example 6.1.1. Consider the following semilinear impulsive Hammerstein integro–dynamic

equation:x∆(t) =

tt− 2

x(t) + ep(t, Θ(x(t)))∫ t

t0

g(t, s)ep(s, x(s))∆s, x(0) = 1, t ∈ [0, 3]TS\2,

zx(tk) = x(t+k )− x(t−k ) = χk(x(t−k )), k = 1,(6.1.5)

and its associated inequality∣∣ψ∆(t)− t

t− 2ψ(t)− ep(t, Θ(ψ(t)))

∫ t

t0

g(t, s)ep(s, ψ(s))∆s∣∣ ≤ 1.5, t ∈ [0, 3]TS\2,∣∣zψ(tk)− χk(ψ(t−k ))

∣∣ ≤ 1.5, k = 1.(6.1.6)

93

By setting TS′ = [0, 3]TS\2, t0 = 0, t1 = 2 and p(t) = tt−2 . Denote F(t, x(t)) =

ep(t, Θ(x(t))) and H(t, x(t)) = ep(t, x(t)) for t ∈ TS′ and put ε = 1.5. If ψ ∈

P1C([0, 3]TS ,R) satisfies the inequality (6.1.6), then there exist f ∈ P1

C([0, 3]TS ,R) and

f0 ∈ R such that | f (t)| ≤ 1.5 for t ∈ TS′ and | f0| ≤ 1.5. So we haveψ∆(t) =

tt− 2

ψ(t) + ep(t, Θ(ψ(t)))∫ t

t0

g(t, s)ep(s, ψ(s))∆s + f (t), ψ(0) = 1, t ∈ TS′,

zψ(tk) = χk(ψ(t−k )) + f0, k = 1,

and the solution of Eq. (6.1.5) is

x(t) = χ1(x(t−1 ))+ ep(t, 0)+∫ t

t0

ep(t, Θ(x(s)))ep(s, Θ(x(s)))∫ s

s0

g(s, u)ep(u, x(u))∆u∆s.

Based on our theoretical results, Eq. (6.1.5) has a unique solution in P1C([0, 3]TS ,R) and

is stable in terms of HU on TS′.

6.2 Non–linear Mixed Integro–Dynamic System

This section comprises of the EUS and stability in terms of HU and HUR of non–

linear impulsive mixed IDySx∆(t) = M(t)x(t) + F(t, x(t), Ix(t), Jx(t)), t ∈ TS′ = TS

0\tk,

zx(tk) = x(t+k )− x(t−k ) = χk(x(t−k )), k = 1, m,

x(t0) = x0,

(6.2.1)

where Ix(t) =∫ t f

t0a(t, s, x(s))∆s, Jx(t) =

∫ t ft0

b(t, s, x(s))∆s, the n × n regressive

matrix M(t) is piecewise continuous, TS0 = [t0, t f ]TS , 0 ≤ t0 = s0 < s f ≤ t f

and F : TS0 × Rn × Rn × Rn → Rn, χk : Rn → Rn, a : TS0 × TS0 × Rn → Rn,

b : TS0×TS0×Rn → Rn are continuous functions. Also x(t−k ) = limυ→0− x(tk− υ)

94

and x(t+k ) = limυ→0+ x(tk + υ) are left and right side limits of x(t) at tk respectively

and tk satisfies t0 < t1 < t2 < · · · < tm < tm+1 = t f < +∞.

6.2.1 Basic Concepts and Remarks

Consider,∣∣∣∣∣∣∣∣ψ∆(s)−M(s)ψ(s)−F(s, ψ(s), Iψ(s), Jψ(s))

∣∣∣∣∣∣∣∣ ≤ ε; s ∈ TS′,

‖zψ(sk)− χk(ψ(s−k ))‖ ≤ ε, k = 1, m,(6.2.2)

∣∣∣∣∣∣∣∣ψ∆(s)−M(s)ψ(s)−F(s, ψ(s), Iψ(s), Jψ(s))

∣∣∣∣∣∣∣∣ ≤ ϕ(s); s ∈ TS′,

‖zψ(sk)− χk(ψ(s−k ))‖ ≤ κ, k = 1, m,(6.2.3)

where ε > 0, κ ≥ 0 and ϕ : TS0 → R+ is rd continuous and increasing.

Definition 6.2.1. Eq. (6.2.1) is said to be stable in terms of HU on TS0 if for every

ψ ∈ P1C(TS

0,Rn) satisfying (6.2.2), there exists a solution ψ0 ∈ P1C(TS

0,Rn) of

(6.2.1) with ‖ψ0(s)− ψ(s)‖ ≤ Cε, C > 0, ∀ s ∈ TS0.

Definition 6.2.2. Eq. (6.2.1) is said to be stable in terms ofHUR on TS0 if for every

ψ ∈ P1C(TS

0,Rn) satisfying (6.2.3), there exists a solution ψ0 ∈ P1C(TS

0,Rn) of

(6.2.1) with ‖ψ0(s)− ψ(s)‖ ≤ Cϕ(s), C > 0, ∀ s ∈ TS0.

Remark 6.2.1. A function ψ ∈ P1C(TS

0,Rn) satisfies (6.2.2) if and only if there exists

f ∈ PC(TS0,Rn) and sequence fk with ‖ f (t)‖ ≤ ε, ∀ t ∈ TS0, ‖ fk‖ ≤ ε, ∀ k = 1, m,

and ψ∆(t) = M(t)ψ(t) + F(t, ψ(t), Iψ(t), Jψ(t)) + f (t), ψ(t0) = ψ0, t ∈ TS′,

zψ(tk) = χk(ψ(t−k )) + fk.

95

Lemma 6.2.2. Each solution ψ ∈ P1C(TS

0,Rn) of inequality (6.2.2) also satisfies:∣∣∣∣∣∣∣∣ψ(t)−ΦM(t, t0)ψ0 −

m

∑j=1

χ(ψ(t−j ))

−∫ t

t0

ΦM(t, Θ(s))F(s, ψ(s), Iψ(s), Jψ(s))∆s∣∣∣∣∣∣∣∣ ≤ ηε,

for t ∈ (tk, tk+1] ⊂ TS0, where ‖ΦM(t, Θ(s))‖ ≤ C and η = (m + C(t f − t0)).

Proof. If ψ ∈ P1C(TS

0,Rn) satisfies (6.2.2), so by Remark 6.2.1, we haveψ∆(t) = M(t)ψ(t) + F(t, ψ(t), Iψ(t), Jψ(t)) + f (t), ψ(t0) = ψ0, t ∈ TS′,

zψ(tk) = χk(ψ(t−k )) + fk, k = 1, m.

Then,

ψ(t) = ΦM(t, t0)ψ0 +m

∑j=1

χ(ψ(t−j )) +m

∑i=1

fi

+∫ t

t0

ΦM(t, Θ(s))F(s, ψ(s), Iψ(s), Jψ(s))∆s +∫ t

t0

ΦM(t, Θ(s)) f (s)∆s.

So, ∣∣∣∣∣∣∣∣ψ(t)−ΦM(t, t0)ψ0 −m

∑j=1

χ(ψ(t−j ))

−∫ t

t0

ΦM(t, Θ(s))F(s, ψ(s), Iψ(s), Jψ(s))∆s∣∣∣∣∣∣∣∣

≤∫ t

t0

‖ΦM(t, Θ(s))‖‖ f (s)‖∆s +m

∑i=1‖ fi‖

≤ ηε.

Similar remarks are for inequality (6.2.3).

96

6.2.2 Existence and Uniqueness of Solutions

Here, we prove the EUS of Eq. (6.2.1). Consider the following assumptions:

(C1) F : TS0×Rn×Rn×Rn → Rn satisfies the Lipschitz condition ‖F(t, µ1, µ2, µ3)−

F(t, ν1, ν2, ν3)‖ ≤ ∑3i=1 L‖µi − νi‖, L > 0, ∀ t ∈ TS0 and µi, νi ∈ Rn, i ∈ 1, 2, 3;

(C2) χk : Rn → Rn satisfies ‖χk(µ1)− χk(µ2)‖ ≤ Mk‖µ1− µ2‖, Mk > 0, ∀ k = 1, m

and µ1, µ2 ∈ Rn;

(C3) For some positive constant L f ≥ 0, the functions a : TS0 × TS0 ×Rn → Rn

and b : TS0 ×TS0 ×Rn → Rn satisfy the following condition,

‖a(t, s, x1)− a(t, s, x2)‖ ≤ L f ‖x1 − x2‖,

‖b(t, s, x1)− b(t, s, x2)‖ ≤ L f ‖x1 − x2‖,

for all t, s ∈ TS0 and x1, x2 ∈ Rn;

(C4)(

∑mj=1 Mj + CL(t f − t0) + 2CLL f (t f − t0)

2)< 1;

(C5) ϕ : TS0 → R+ is increasing such that

∫ t

t0

ϕ(s)∆s ≤ βϕ(t), β > 0.

Theorem 6.2.1. If conditions (C1)− (C4) hold, then Eq. (6.2.1) has a unique solution in

P1C(TS

0,Rn).

97

Proof. Define an operator Λ : PC(TS0,Rn)→ PC(TS0,Rn) by

(Λx)(t) =

ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), Ix(s), Jx(s))∆s, t ∈ (t0, t1],

i

∑j=1

χj(x(t−j )) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), Ix(s), Jx(s))∆s,

t ∈ (ti, ti+1], i = 1, m.

For t ∈ (ti, ti+1] and x1, x2 ∈ PC(TS0,Rn), simple calculation shows that

‖(Λx1)(t)− (Λx2)(t)‖ ≤m

∑j=1‖χj(x1(t−j ))− χj(x2(t−j ))‖

+∫ t

t0

‖ΦM(t, Θ(s))‖∣∣∣∣∣∣∣∣(F(s, x1(s), Ix1(s), Jx1(s))

−F(s, x2(s), Ix2(s), Jx2(s)))∣∣∣∣∣∣∣∣∆s

≤m

∑j=1

Mj‖x1(t−j )− x2(t−j )‖+∫ t

t0

CL‖x1(s)− x2(s)‖∆s

+∫ t

t0

CL‖Ix1(s)− Ix2(s)‖∆s +∫ t

t0

CL‖Jx1(s)− Jx2(s)‖∆s

≤m

∑j=1

Mj supt∈TS0

‖x1(t)− x2(t)‖+∫ t

t0

CL supt∈TS0

‖x1(t)− x2(t)‖∆s

+∫ t

t0

CL∣∣∣∣∣∣∣∣ ∫ s f

s0

a(s, u, x1(u))∆u−∫ s f

s0

a(s, u, x2(u))∆u∣∣∣∣∣∣∣∣∆s

+∫ t

t0

CL∣∣∣∣∣∣∣∣ ∫ s f

s0

b(s, u, x1(u))∆u−∫ s f

s0

b(s, u, x2(u))∆u∣∣∣∣∣∣∣∣∆s

98

≤m

∑j=1

Mj‖x1 − x2‖+ ‖x1 − x2‖∫ t

t0

CL∆s

+∫ t

t0

CL∫ s f

s0

L f ‖x1(u)− x2(u)‖∆u∆s

+∫ t

t0

CL∫ s f

s0

L f ‖x1(u)− x2(u)‖∆u∆s

≤m

∑j=1

Mj‖x1 − x2‖+ ‖x1 − x2‖CL(t− t0)

+2∫ t

t0

CL∫ s f

s0

L f supt∈TS0

‖x1(t)− x2(t)‖∆u∆s

≤m

∑j=1

Mj‖x1 − x2‖+ ‖x1 − x2‖CL(t f − t0)

+2‖x1 − x2‖∫ t

t0

CLL f (s f − s0)∆s

≤m

∑j=1

Mj‖x1 − x2‖+ ‖x1 − x2‖CL(t f − t0)

+2‖x1 − x2‖CLL f (t f − t0)2

≤ ‖x1 − x2‖( m

∑j=1

Mj + CL(t f − t0) + 2CLL f (t f − t0)2)

.

From (C4), Λ is a contraction on PC(TS0,Rn). So Λ is a Picard operator such that

it has a unique FP which is the unique solution of (6.2.1) in P1C(TS

0,Rn).

6.2.3 Hyers–Ulam Stability

In this subsection, we establish our results concerning stability in terms ofHU and

HUR of Eq. (6.2.1).

Theorem 6.2.2. If conditions (C1)− (C4) are satisfied, then Eq. (6.2.1) has stability in

terms of HU on TS0.

99

Proof. Let ψ ∈ P1C(TS

0,Rn) satisfies (6.2.2). The solution x ∈ P1C(TS

0,Rn) of Eq.

(6.2.1) is:

x(t) =

ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), Ix(s), Jx(s))∆s, t ∈ (t0, t1],

i

∑j=1

χj(x(t−j )) + ΦM(t, t0)x0

+∫ t

t0

ΦM(t, Θ(s))F(s, x(s), Ix(s), Jx(s))∆s,

t ∈ (ti, ti+1], i = 1, m.

For t ∈ (ti, ti+1], with the help of Lemma 6.2.2, we get

‖ψ(t)− x(t)‖ ≤∣∣∣∣∣∣∣∣ψ(t)−ΦM(t, t0)ψ0 −

m

∑j=1

χ(ψ(t−j ))

−∫ t

t0

ΦM(t, Θ(s))F(s, ψ(s), Iψ(s), Jψ(s))∆s∣∣∣∣∣∣∣∣

+m

∑j=1‖χj(ψ(t−j ))− χj(x(t−j ))‖+

∣∣∣∣∣∣∣∣ ∫ t

t0

ΦM(t, Θ(s))(

F(s, ψ(s), Iψ(s), Jψ(s))

−F(s, x(s), Ix(s), Jx(s)))

∆s∣∣∣∣∣∣∣∣

≤ ηε +m

∑j=1

Mj‖ψ(t−j )− x(t−j )‖+∫ t

t0

‖ΦM(t, Θ(s))‖∣∣∣∣∣∣∣∣(F(s, ψ(s), Iψ(s), Jψ(s))

−F(s, x(s), Ix(s), Jx(s)))∣∣∣∣∣∣∣∣∆s

≤ ηε +m

∑j=1

Mj‖ψ(t−j )− x(t−j )‖+∫ t

t0

CL‖ψ(s)− x(s)‖∆s

+∫ t

t0

CL‖Iψ(s)− Ix(s)‖∆s +∫ t

t0

CL‖Jψ(s)− Jx(s)‖∆s

100

≤ ηε +m

∑j=1

Mj‖ψ(t−j )− x(t−j )‖+∫ t

t0

CL‖ψ(s)− x(s)‖∆s

+∫ t

t0

CL∣∣∣∣∣∣∣∣ ∫ s f

s0

a(s, u, ψ(s))∆u−∫ s f

s0

a(s, u, x(u))∆u∣∣∣∣∣∣∣∣∆s

+∫ t

t0

CL∣∣∣∣∣∣∣∣ ∫ s f

s0

b(s, u, ψ(u))∆u−∫ s f

s0

b(s, u, x(u))∆u∣∣∣∣∣∣∣∣∆s

≤ ηε +m

∑j=1

Mj‖ψ(t−j )− x(t−j )‖+∫ t

t0

CL‖ψ(s)− x(s)‖∆s

+2∫ t

t0

CL∫ s f

s0

L f ‖ψ(u)− x(u)‖∆u∆s.

Consider T : PC(TS0,R+)→ PC(TS0,R+) as:

(Tw)(t) =

ηε +

i

∑j=1

Mjw(t−j ) +∫ t

t0

CLw(s)∆s

+ 2∫ t

t0

CL∫ s f

s0

L f w(u)∆u∆s, t ∈ (ti, ti+1], i = 1, m.

(6.2.4)

For t ∈ (ti, ti+1] and w1, w2 ∈ PC(TS0,R+), we can see that,

‖(Tw1)(t)− (Tw2)(t)‖ ≤m

∑j=1

Mj‖w1(t−j )−w2(t−j )‖

+∫ t

t0

CL‖w1(s)−w2(s)‖∆s + 2∫ t

t0

CL∫ s f

s0

L f ‖w1(u)−w2(u)‖∆u∆s

≤m

∑j=1

Mj‖w1 −w2‖+ ‖w1 −w2‖CL(t f − t0) + 2‖w1 −w2‖CLL f (t f − t0)2

≤ ‖w1 −w2‖( m

∑j=1

Mj + CL(t f − t0) + 2CLL f (t f − t0)2)

.

From (C4), the operator is contractive on PC(TS0,R+). Utilizing Theorem 3.1.1, T

is Picard operator having unique FP w∗ i.e.,

w∗(t) = ηε +m

∑j=1

Mjw∗(t−j ) +∫ t

t0

CLw∗(s)∆s + 2∫ t

t0

CL∫ s f

s0

L f w∗(u)∆u∆s.

101

By simple calculation, we get

w∗(t) = ηε +m

∑j=1

Mjw∗(t−j ) +∫ t

t0

CLw∗(s)∆s + 2∫ s f

s0

L f w∗(u)∆u∫ t

t0

CL∆s

= ηε +m

∑j=1

Mjw∗(t−j ) +∫ t

t0

CLw∗(s)∆s + 2∫ s f

s0

L f w∗(u)∆u(CL(t− t0)

)≤

(η +

2ε(CL(t f − t0)

) ∫ s f

s0

L f w∗(u)∆u)

ε +m

∑j=1

Mjw∗(t−j )

+∫ t

t0

CLw∗(s)∆s.

By Lemma 4.1.2,

w∗(t) ≤(

η +2ε(CL(t f − t0)

) ∫ s f

s0

L f w∗(u)∆u)

ε ∏t0<tj<t

(1 + Mj)eP(t, t0),

where P = CL. Taking w(t) = ‖ψ(t)− x(t)‖ and using (6.2.4), w(t) ≤ (Tw)(t).

So by Lemma 3.1.1,

‖ψ(t)− x(t)‖ ≤(

η +2ε(CL(t f − t0)

) ∫ s f

s0

L f w∗(u)∆u)

ε ∏t0<tj<t

(1 + Mj)eP(t, t0)

≤ Kε,

where K =

(η + 2

ε (CL(t f − t0)) ∫ s f

s0L f w∗(u)∆u

)∏t0<tj<t(1 + Mj)e

P(t f−t0). So Eq.

(6.2.1) is stable in terms of HU on TS0.

The proof of the following theorem is similar to above theorem. So its proof

will be omitted.

Theorem 6.2.3. If conditions (C1)− (C5) are satisfied, then Eq. (6.2.1) has stability in

terms of HUR on TS0.

102

Example 6.2.1. Consider the following mixed impulsive DyE,x∆(t) =

1t− 1

x(t) + ep(t, Θ(x(t))) +∫ t

0ep(s, x(s))∆s, x(0) = 1, t ∈ [0, 2]TS\1,

zx(tk) = x(t+k )− x(t−k ) = χk(x(t−k )), k = 1,(6.2.5)

and its associated inequality∣∣ψ∆(t)− 1

t− 1ψ(t)− ep(t, Θ(ψ(t)))−

∫ t

0ep(s, ψ(s))∆s

∣∣ ≤ 1, t ∈ [0, 2]TS\1,∣∣zψ(tk)− χk(ψ(t−k ))∣∣ ≤ 1, k = 1.

(6.2.6)

By setting TS′ = [0, 2]TS\1, t1 = 1 and p(t) = 1t−1 . Denote F(t, x(t), Ix(t)) =

ep(t, Θ(x(t))) + Ix(t), where Ix(t) =∫ t

0 ep(s, x(s))∆s for t ∈ TS′ and put ε = 1. If

ψ ∈ P1C([0, 2]TS ,R) satisfies the inequality (6.2.6), then there exist f ∈ P1

C([0, 2]TS ,R)

and f0 ∈ R such that | f (t)| ≤ 1 for t ∈ TS′ and | f0| ≤ 1. So we haveψ∆(t) =

1t− 1

ψ(t) + ep(t, Θ(ψ(t))) +∫ t

0ep(s, ψ(s))∆s + f (t), ψ(0) = 1, t ∈ TS′,

zψ(tk) = χk(ψ(t−k )) + f0, k = 1,

and the solution of Eq. (6.2.5) is

x(t) = χ1(x(t−1 )) + ep(t, 0) +∫ t

0ep(t, Θ(s))

(ep(s, Θ(x(s))) +

∫ s

0ep(u, x(u))∆u

)∆s.

Based on our theoretical results, Eq. (6.2.5) has a unique solution in P1C([0, 2]TS ,R) and

is stable in terms of HU on TS′.

Remark 6.2.4. Similarly, the results of (6.2.1) can be proved for non–linear mixed IDyS

with fractional integrable impulses.

Chapter 7

Conclusion and Future ResearchWork

Conclusion

In this thesis, we proved the EUS, stability in terms of HU and HUR of first order

non–linear DDEs having fractional integrable impulses, first order non–linear im-

pulsive DDyS, non–linear impulsive Volterra IDDyS, non–linear impulsive Ham-

merstein IDDyS and mixed impulsive IDyS on time scale, by using FP methods.

The EU results are proved with the help of Banach contraction principle, while the

stability in terms ofHU andHUR results are obtained with the help of Gronwall’s

inequality, abstract Gronwall’s lemma, Banach contraction principle and Picard

operator. Also some particular assumptions are made to establish EU and stability

results. When finding the exact solution is difficult, then the concept of stability in

terms ofHU is very important i.e., our results are fruitful in approximation theory.

Stability in terms of HU helps us to construct a neighborhood inside which there

exists a solution of the equation near to the solution of inequality (approximate

solution). Moreover, from the obtained results, we conclude that the FP methods

103

104

are very powerful, effectual and suitable for the solutions of impulsive DDEs and

DDySs on time scale.

Future Research Work

Here, we list some open problems that can be achieved in future.

Open Problem 1

Most of our results used Gronwall’s inequality. Now a natural question arises that

can we achieve these results without using Gronwall’s inequality?

Open Problem 2

A good analysis will be to give a relationship between exponential stability and

Ulam’s type stability of system of DyEs on time scale.

Open Problem 3

An interesting problem will be to extend our results to fractional DEs on time scale.

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