Information Sciences 222 (2013) 544–558

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Information Sciences

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Periodic boundary value problems for first-order linear differentialequations with uncertainty under generalized differentiability

A. Khastan a, J.J. Nieto b, Rosana Rodríguez-López b,⇑a Department of Mathematics, Institute for Advanced Studies in Basic Sciences, Zanjan 45137-66731, Iranb Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain

a r t i c l e i n f o a b s t r a c t

Article history:Received 7 August 2010Received in revised form 22 April 2012Accepted 25 July 2012Available online 7 August 2012

Keywords:Fuzzy real numberFuzzy setFirst-order fuzzy differential equationPeriodic boundary value problemGeneralized differentiability

0020-0255/$ - see front matter � 2012 Elsevier Inchttp://dx.doi.org/10.1016/j.ins.2012.07.057

⇑ Corresponding author. Fax: +34 881 81 31 97.E-mail address: rosana.rodriguez.lopez@usc.es (R

We study the existence of solutions to a class of first-order linear fuzzy differential equa-tions subject to periodic boundary conditions from the point of view of generalized differ-entiability. The objective of this paper is to show that fuzzy differential equations undergeneralized differentiability can be used in the study of periodic phenomena, by consider-ing a combination of two types of derivatives with a switching point. We provide sufficientconditions which guarantee that the piecewise-defined solutions match adequately andillustrate, through some examples, the process of construction and calculation of solutions.

� 2012 Elsevier Inc. All rights reserved.

1. Introduction

One of the first approaches to study, by means of differential equations, the imprecise behavior which is inherent to manyreal processes takes into account the Hukuhara derivative (see [34] for details on H-derivative). We find some existence anduniqueness results for fuzzy differential equations in [9,23], for instance. It is also well-known [18] that the use of the Huku-hara derivative produces the nondecreasing length of the diameter of the level sets of the solutions and, therefore, the prop-erties of this derivative make it impossible to find periodic solutions to fuzzy differential equations except very restrictivesituations. To deal with periodicity in the context of fuzzy differential equations, in reference [35], it is considered a bound-ary value problem for a class of impulsive fuzzy differential equations, and the extremal solutions to this kind of problems ina fuzzy functional interval are approximated by using the monotone method. For some references on fuzzy equations andapplications of fuzzy dynamics, we cite [1,2,4,10,15,17–19,22–24,26,28,30–32] and other recent works such as the studyof some topological properties of the solutions to the Cauchy problem for fuzzy differential systems (see [14]), the extensionof the differential transformation method given in [3], the iterative method developed in [21] to get approximate-analyticalsolutions for first-order linear fuzzy differential systems with constant coefficients, the approach illustrated in [20] to solvethe Cauchy problem for fuzzy differential equations by using feed-forward neural networks, the study of the generalizedderivative and p-derivative for interval-valued functions [12], or the resolution of fuzzy relational equations [33]. Concern-ing two point boundary value problems for fuzzy differential equations, we cite [5,16], where the authors state the limita-tions of the difference in the sense of Hukuhara in the study of this kind of problems. In particular, in [16], relativederivatives are introduced to deal with two-point boundary value problems. See also [13] for the study of two-point bound-ary value problems of undamped uncertain dynamical systems, and a proof of the existence of periodic solutions from thepoint of view of differential inclusions.

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. Rodríguez-López).

A. Khastan et al. / Information Sciences 222 (2013) 544–558 545

The approach followed in this paper, strongly generalized differentiability, was introduced in [6], and further studies canbe found in Refs. [7,8,25,29,37]. Taking this concept of derivative as the basis of the analysis of fuzzy differential equations,new possibilities arise since the restriction consisting in the increasing character of the diameter of solution’s level sets isrelaxed. See also reference [36], which generalizes the difference of Hukuhara and the division to the context of real intervalsand fuzzy numbers, showing some applications to fuzzy equations, and [27], where fuzzy and fuzzy-delay differential equa-tions are considered for a general type of closed fuzzy sets by using reachable sets.

In this work, we provide explicitly some conditions which guarantee the existence of periodic solutions for a class of fuzzydifferential equations from the point of view of strongly generalized differentiability. We show different types of solutionsand give some examples in which our results are applied, deducing the existence of solution and also calculating its exactexpression. Our aim is to illustrate how this approach allows the effective modeling of periodic phenomena through fuzzydifferential equations, since it is a useful method to avoid the inconveniences of the Hukuhara derivative. To this purpose, weselect a combination of the two types of derivatives using a switching point (see [37]). Although this method is natural, it isnecessary to show that the solutions using different types of differentiability indeed match adequately.

In Section 2, we recall some preliminary concepts and results and then, in Section 3, we study the existence of solution tofirst-order linear fuzzy differential equations with periodic boundary value conditions. In Section 4, we show some applica-tions and, finally, in Section 5, we include some conclusions.

2. Basic concepts

We denote by RF the space of fuzzy sets in R, that is, functions u : R! ½0;1�, such that

(i) u is normal, i.e., there exists s0 2 R such that u(s0) = 1,(ii) u is fuzzy-convex, that is, uðtsþ ð1� tÞrÞP minfuðsÞ;uðrÞg; 8t 2 ½0;1�; s; r 2 R,

(iii) u is upper semicontinuous on R,(iv) clfs 2 RjuðsÞ > 0g is compact, where cl denotes the closure of a subset.

The space RF (also denoted by E1) is named as the space of fuzzy intervals. It is clear that the space of real numbers can beembedded in RF . Given 0 < a 6 1, we denote ½u�a ¼ fs 2 RjuðsÞP ag and ½u�0 ¼ clfs 2 RjuðsÞ > 0g. From properties (i)-(iv), itfollows that the a-level sets of u 2 RF are nonempty compact intervals. The notation ½u�a ¼ ½ua; �ua� represents explicitly the a-level set of u and its diameter is calculated as diam½u�a ¼ �ua � ua. We refer to u and �u as the lower and upper branches of u,respectively. Here, we use the notation ual and uar, instead of ua and �ua, respectively.

If we take u;v 2 RF and k 2 R, then we can define the addition, u + v, and the multiplication by an scalar, ku, in terms of theusual operations with real intervals, by the expressions [u + v]a = [u]a + [v]a, [ku]a = k[u]a, "a 2 [0,1].

The metric structure in RF is given by the Hausdorff distance

D : RF � RF ! Rþ [ f0g;

defined by

Dðu; vÞ ¼ supa2½0;1�

maxfjua � vaj; j�ua � �vajg; for u;v 2 RF ;

in such a way that ðRF ;DÞ has the structure of a complete metric space.

Definition 2.1. Given two elements x; y 2 RF , we say that the Hukuhara difference of x and y exists, if there exists z 2 RF withx = y + z. In this case, we denote z = x � y (the Hukuhara difference of x and y).

Note that x � y – x + (�1)y, in general. Usually, we denote x + (�1)y by x � y. In the sequel, we fix I = (a,b), wherea < b 2 R. In Refs. [6–8], the concept of strongly generalized differentiability is presented as follows.

Definition 2.2. Let F : I ! RF . Fix t0 2 I. We say that F is differentiable at t0, if there exists an element F 0ðt0Þ 2 RF such that

(i) for all h > 0 sufficiently close to 0, there exist F(t0 + h) � F(t0), F(t0) � F(t0 � h) and the limits (in the metric D)

limh!0þ

Fðt0 þ hÞ � Fðt0Þh

¼ limh!0þ

Fðt0Þ � Fðt0 � hÞh

¼ F 0ðt0Þ;

or(ii) for all h > 0 sufficiently close to 0, there exist F(t0) � F(t0 + h), F(t0 � h) � F(t0) and the limits (in the metric D)

limh!0þ

Fðt0Þ � Fðt0 þ hÞ�h

¼ limh!0þ

Fðt0 � hÞ � Fðt0Þ�h

¼ F 0ðt0Þ;

or

546 A. Khastan et al. / Information Sciences 222 (2013) 544–558

(iii) for all h > 0 sufficiently close to 0, there exist F(t0 + h) � F(t0), F(t0 � h) � F(t0) and the limits (in the metric D)

limh!0þ

Fðt0 þ hÞ � Fðt0Þh

¼ limh!0þ

Fðt0 � hÞ � Fðt0Þ�h

¼ F 0ðt0Þ;

or

(iv) for all h > 0 sufficiently close to 0, there exist F(t0) � F(t0 + h), F(t0) � F(t0 � h) and the limits (in the metric D)

limh!0þ

Fðt0Þ � Fðt0 þ hÞ�h

¼ limh!0þ

Fðt0Þ � Fðt0 � hÞh

¼ F 0ðt0Þ:

As stated in [8], case (i) corresponds to the H-derivative (see [34]). A function that is strongly generalized differentiable asin cases (i) and (ii), will be referred as (i)-differentiable or (ii)-differentiable, respectively. For cases (iii) and (iv), it is known(see [6]) that a function may be (iii) or (iv)-differentiable only at a discrete set of points, consisting exactly of the pointswhere differentiability switches between cases (i) and (ii).

Remark 2.1. It is shown in [6] that a fuzzy differential equation considered under strongly generalized differentiability has(locally) two solutions (see also [7]). In [8,10,25], the authors consider those solutions for which there are no changesbetween the different types of differentiability. Thus, they find solutions of a FDE which are (i) or (ii)-differentiable on aninterval. In this paper, we consider those solutions for which we have a switching point from one type of differentiability toanother.

3. Existence of periodic solutions

For the existence results included in this section, we consider the following set of functions.

Definition 3.1. Let J = [0,T] be a real interval and d 2 (0,T) fixed. We define the space F d, consisting on the functionsu 2 CðJ;RFÞ such that:

(I) u is differentiable in the sense of strong generalized differentiability on (0,T),(II) u has a switching point at t = d,

(III) there exist the limits limt!0þu0ðtÞ, limt!T�u0ðtÞ in RF .

Remark 3.1. The functions in F d might be:

� (i)-differentiable on (0,d), (ii)-differentiable on (d,T) and (iv)-differentiable at t = d; or� (ii)-differentiable on (0,d), (i)-differentiable on (d,T) and (iii)-differentiable at t = d.

For d 2 (0,T) fixed, we consider the periodic boundary value problem

y0ðtÞ ¼ aðtÞyðtÞ þ bðtÞ; t 2 J;yð0Þ ¼ yðTÞ;

�ð1Þ

where J = [0,T], and a : ½0; T� ! R, b : ½0; T� ! RF are continuous functions. This problem is related to Eq. (6) in [8].In this section, we first consider problem (1) where a(t) > 0, for t 2 (0,d) and a(t) < 0, for t 2 (d,T). Then, in Theorem 3.2, we

consider this problem with a(t) < 0, for t 2 (0,d) and a(t) > 0, for t 2 (d,T).In the following, we consider specific solutions of (1) which are functions in the space F d satisfying (1).

Theorem 3.1. Suppose that J = [0,T], a : ½0; T� ! R, b : ½0; T� ! RF are continuous functions satisfying a(t) > 0, for t 2 (0,d), anda(t) < 0, for t 2 (d,T). Suppose that the following conditions hold

Z T

ddiam½bðsÞ�a e�

R s

0aðuÞ duds 6

Z d

0diam½bðsÞ�a e�

R s

0aðuÞduds; 8a 2 ½0;1�; ð2ÞZ d

0bðsÞal e�

R s

0aðuÞdudsþ

Z t

dbðsÞar e�

R s

0aðuÞduds is nondecreasing in a; 8t 2 ðd; TÞ; ð3Þ

Z d

0bðsÞar e�

R s

0aðuÞ dudsþ

Z t

dbðsÞal e�

R s

0aðuÞduds is nonincreasing in a; 8t 2 ðd; TÞ ð4Þ

and

Z T

0aðuÞdu < 0:

A. Khastan et al. / Information Sciences 222 (2013) 544–558 547

Then there exists a solution to problem (1) in the set F d. Furthermore, this solution u 2 CðJ;RFÞ is (i)-differentiable on (0,d), (ii)-differentiable on (d, T) and (iv)-differentiable at t = d.

Proof. To find a solution to the periodic boundary value problem, we fix y0 2 RF and consider the initial value problem

y0ðtÞ ¼ aðtÞyðtÞ þ bðtÞ; t 2 J;

yð0Þ ¼ y0:

�ð5Þ

Hence, by the results in [8], since a(t) > 0, for t 2 (0,d), we have that

y1ðtÞ ¼ eR t

0aðuÞdu y0 þ

Z t

0bðsÞ e�

R s

0aðuÞduds

� �

is (i)-differentiable and it is the solution of problem (5) (as well as equation in (1)) on (0,d).Now, from the equation,

y01ðtÞ ¼ aðtÞy1ðtÞ þ bðtÞ; t 2 ð0; dÞ;

and we deduce that the limits

limt!0þ

y01ðtÞ ¼ limt!0þðaðtÞy1ðtÞ þ bðtÞÞ ¼ að0Þy0 þ bð0Þ;

limt!d�

y01ðtÞ ¼ limt!d�ðaðtÞy1ðtÞ þ bðtÞÞ ¼ aðdÞy1ðdÞ þ bðdÞ

exist in RF . Hence, for each y0 fixed, we take

y1ðdÞ ¼ eR d

0aðuÞdu y0 þ

Z d

0bðsÞ e�

R s

0aðuÞduds

� �¼ Cy0

:

Next, we solve the initial value problem

y0ðtÞ ¼ aðtÞyðtÞ þ bðtÞ; t 2 ðd; TÞ;yðdÞ ¼ Cy0

;

�ð6Þ

and look for (ii)-differentiable solutions since a(t) < 0, for t 2 (d,T). Thus, the solution is given (see again [8]) by the expression

y2ðtÞ ¼ eR t

daðuÞdu e

R d

0aðuÞdu y0 þ

Z d

0bðsÞ e�

R s

0aðuÞ duds

� ��Z t

dð�bðsÞÞ e�

R s

daðuÞduds

� �;

which is a (ii)-differentiable function and a solution to the equation in (1) on (d,T), provided that the Hukuhara difference

eR d

0aðuÞdu y0 þ

Z d

0bðsÞ e�

R s

0aðuÞduds

� ��Z t

dð�bðsÞÞ e�

R s

daðuÞduds ð7Þ

exists, for t 2 (d,T). Note that

limt!dþ

y02ðtÞ ¼ limt!dþðaðtÞy2ðtÞ þ bðtÞÞ ¼ aðdÞy2ðdÞ þ bðdÞ

exists in RF due to the properties of a and b. This value coincides with y01ðd�Þ and, by construction, the combination of y1 and

y2 is continuous at t = d. Besides, limt!T�y02ðtÞ exists in RF .We check, under the hypotheses of the Theorem, that the Hukuhara difference

eR d

0aðuÞdu y0 þ

Z d

0bðsÞ e�

R s

0aðuÞdu ds

� ��Z t

dð�bðsÞÞ e�

R s

daðuÞ duds

exists, for t 2 (d,T). Indeed, we get

diam eR d

0aðuÞdu y0 þ

Z d

0bðsÞ e�

R s

0aðuÞduds

� �� �a

P diamZ t

dð�bðsÞÞ e�

R s

daðuÞ duds

� �a

;

for every t 2 (d,T) and a 2 [0,1]. To this purpose, we calculate the level sets appearing in the previous expression and provethat it is equivalent to

eR d

0aðuÞdu diam½y0�

a þZ d

0diam½bðsÞ�a e�

R s

0aðuÞduds

� �PZ t

ddiam½ð�bðsÞÞ�a e�

R s

daðuÞduds;

or

diam½y0�a þ

Z d

0diam½bðsÞ�a e�

R s

0aðuÞduds P

Z t

ddiam½ð�bðsÞÞ�a e�

R s

0aðuÞduds;

for every t 2 (d,T) and a 2 [0,1].

548 A. Khastan et al. / Information Sciences 222 (2013) 544–558

Using that diam[(�b(s))]a = diam[b(s)]a, for every s and a 2 [0,1], the existence of the Hukuhara difference on every levelset is equivalent to the validity of the following inequality

diam½y0�a P

Z t

ddiam½bðsÞ�a e�

R s

0aðuÞduds�

Z d

0diam½bðsÞ�a e�

R s

0aðuÞduds; ð8Þ

for every t 2 (d,T) and a 2 [0,1].The validity of inequality (8) is equivalent to the validity of

diam½y0�a P

Z T

ddiam½bðsÞ�a e�

R s

0aðuÞduds�

Z d

0diam½bðsÞ�a e�

R s

0aðuÞduds; ð9Þ

for every a 2 [0,1], and this is trivially derived from hypothesis (2). On the other hand, by conditions (3) and (4), it is easy todeduce that

eR d

0aðuÞdu ðy0Þal þ

Z d

0bðsÞal e�

R s

0aðuÞdu ds

� �þZ t

dbðsÞar e�

R s

daðuÞduds

¼ eR d

0aðuÞdu ðy0Þal þ

Z d

0bðsÞal e�

R s

0aðuÞdu dsþ

Z t

dbðsÞar e�

R s

0aðuÞ duds

� �

is nondecreasing in a, for every t 2 (d,T), and, similarly,

eR d

0aðuÞdu ðy0Þar þ

Z d

0bðsÞar e�

R s

0aðuÞdu ds

� �þZ t

dbðsÞal e�

R s

daðuÞduds

¼ eR d

0aðuÞdu ðy0Þar þ

Z d

0bðsÞar e�

R s

0aðuÞdu dsþ

Z t

dbðsÞal e�

R s

0aðuÞ duds

� �

is nonincreasing in a, for every t 2 (d,T).Hence, the Hukuhara differences (7) exist for every y0 2 RF . Next, we define the operator

A : RF ! RF

given, for y0 2 RF , by

Aðy0Þ ¼ eR T

daðuÞdu e

R d

0aðuÞdu y0 þ

Z d

0bðsÞ e�

R s

0aðuÞduds

� ��Z T

dð�bðsÞÞ e�

R s

daðuÞdu ds

� �:

We check that the operator A is a contractive mapping. Indeed,

d1ðAðy0Þ;Aðw0ÞÞ 6 eR T

daðuÞdud1 e

R d

0aðuÞdu y0 þ

Z d

0bðsÞ e�

R s

0aðuÞdu ds

� ��

�Z T

dð�bðsÞÞ e�

R s

daðuÞ duds; e

R d

0aðuÞdu w0 þ

Z d

0bðsÞ e�

R s

0aðuÞdu ds

� ��Z T

dð�bðsÞÞ e�

R s

daðuÞ duds

�

¼ eR T

0aðuÞdud1 y0 þ

Z d

0bðsÞ e�

R s

0aðuÞduds;w0 þ

Z d

0bðsÞ e�

R s

0aðuÞduds

� �¼ eR T

0aðuÞdud1ðy0;w0Þ:

Since, by hypothesis, eR T

0aðuÞdu

< 1, we have proved that A is a contractive mapping and there exists a unique fixed point of A.For any y0 2 RF , limn!þ1A

ny0 ¼ y�0 is the unique fixed point of A. Using y�0 as initial condition for the equationy0(t) = a(t)y(t) + b(t), t 2 J, we obtain a solution to problem (1). h

Remark 3.2. Note that, following this approach, the diameter of the level sets of the solution obtained is nondecreasing in(0,d) and nonincreasing in (d,T).

Under the conditions of Theorem 3.1, the sought solution is obtained by taking the initial condition defined as

ðy0Þal ¼1

1� eR T

0aðuÞdu

eR T

0aðuÞdu

Z d

0bðsÞal e�

R s

0aðuÞdudsþ e

R T

daðuÞ du

Z T

dbðsÞar e�

R s

daðuÞ duds

� �;

ðy0Þar ¼1

1� eR T

0aðuÞdu

eR T

0aðuÞdu

Z d

0bðsÞar e�

R s

0aðuÞdudsþ e

R T

daðuÞ du

Z T

dbðsÞal e�

R s

daðuÞ duds

� �:

Now, consider again the periodic boundary value problem (1), where a : ½0; T� ! R is continuous and satisfies that a(t) < 0,for t 2 (0,d) and a(t) > 0, for t 2 (d,T), for a fixed d 2 (0,T).

Theorem 3.2. Let J = [0,T], a : ½0; T� ! R, b : ½0; T� ! RF be continuous functions satisfying that a(t) < 0, for t 2 (0,d) and a(t) > 0,for t 2 (d, T).

A. Khastan et al. / Information Sciences 222 (2013) 544–558 549

Suppose that

eR T

0aðuÞdu

Z T

ddiam½bðsÞ�a e�

R s

0aðuÞduds P

Z d

0diam½bðsÞ�ae�

R s

0aðuÞduds; 8a 2 ½0;1� ð10Þ

Z T

dbðsÞal e

R T

saðuÞ dudsþ

Z t

0bðsÞar e�

R s

0aðuÞ duds is nondecreasing in a; 8t 2 ð0; dÞ; ð11Þ

Z T

dbðsÞar e

R T

saðuÞ dudsþ

Z t

0bðsÞal e�

R s

0aðuÞ duds is nonincreasing in a; 8t 2 ð0; dÞ ð12Þ

and

Z T

0aðuÞdu < 0:

Then there exists a solution to problem (1) in F d. Furthermore, this solution u 2 CðJ;RFÞ is (ii)-differentiable on (0,d), (i)-differen-tiable on (d, T) and (iii)-differentiable at t = d.

Proof. For a fixed y0 2 RF , we consider the initial value problem (5) on the interval (0,d) from the point of view of (ii)-dif-ferentiability, which has a unique solution (see [8]) provided that the Hukuhara difference

y0 �Z t

0ð�bðsÞÞe�

R s

0aðuÞduds

exists, for each t 2 (0,d). In this case, the unique solution is given by

y2ðtÞ ¼ eR t

0aðuÞdu y0 �

Z t

0ð�bðsÞÞ e�

R s

0aðuÞduds

� �; t 2 ð0; dÞ:

Note that

y2ðdÞ ¼ eR d

0aðuÞdu y0 �

Z d

0ð�bðsÞÞ e�

R s

0aðuÞdu ds

� �¼ Dy0

:

For the existence of the Hukuhara differences y0 �R t

0ð�bðsÞÞe�R s

0aðuÞduds, for each t 2 (0,d), we have to check in first place that

diam½y0�a P

Z t

0diam½bðsÞ�ae�

R s

0aðuÞ duds; 8a 2 ½0;1�; and t 2 ð0; dÞ;

which is satisfied assuming that

diam½y0�a P

Z d

0diam½bðsÞ�ae�

R s

0aðuÞduds; 8a 2 ½0;1�:

Moreover, we have to guarantee that

ðy0Þal þZ t

0bðsÞare

�R s

0aðuÞ duds is nondecreasing in a; for all t 2 ð0; dÞ; ð13Þ

ðy0Þar þZ t

0bðsÞale

�R s

0aðuÞ duds is nonincreasing in a; for all t 2 ð0; dÞ; ð14Þ

conditions which imply its validity also for t = d. We work with initial values y0 satisfying these restrictions, and hence weconsider the initial value problem

y0ðtÞ ¼ aðtÞyðtÞ þ bðtÞ; t 2 ðd; TÞ;yðdÞ ¼ y2ðdÞ ¼ Dy0

;

�ð15Þ

from the point of view of (i)-differentiability. Hence the function

y1ðtÞ ¼ eR t

daðuÞdu e

R d

0aðuÞdu y0 �

Z d

0ð�bðsÞÞ e�

R s

0aðuÞduds

� �þZ t

dbðsÞ e�

R s

daðuÞduds

� �; t 2 ðd; TÞ;

is (i)-differentiable on (d,T) and it is the solution of problem (15) and the equation in (1) on (d,T). For this construction tomake sense, we set

C ¼ y0 2 RF : y0 �Z t

0ð�bðsÞÞe�

R s

0aðuÞduds exists for every t 2 ð0; dÞ

� �

¼ y0 2 RF : diam½y0�1 P

Z d

0diam½bðsÞ�1e�

R s

0aðuÞduds; and ð13Þ; ð14Þ hold

� �;

which is a closed subset in RF and, therefore, it is a complete metric space.

550 A. Khastan et al. / Information Sciences 222 (2013) 544–558

We define the mapping

B : C # RF ! RF

given, for y0 2 C, by

Bðy0Þ ¼ eR T

daðuÞdu e

R d

0aðuÞdu y0 �

Z d

0ð�bðsÞÞ e�

R s

0aðuÞdu ds

� �þZ T

dbðsÞ e�

R s

daðuÞduds

� �: ð16Þ

Note that, ifR d

0 diam½bðsÞ�ae�R s

0aðuÞ duds ¼ 0, for every a 2 [0,1], that is, b is real on (0,d), then diam[y0]a P 0, "a 2 [0,1],

8y0 2 RF , ðy0Þal þR t

0 bðsÞe�R s

0aðuÞduds is nondecreasing in a, for all t 2 (0,d) and ðy0Þar þ

R t0 bðsÞe�

R s

0aðuÞdu ds is nonincreasing in

a, for all t 2 (0,d), hence C ¼ RF and B maps C into itself. In the general case, we check that B maps C into itself. Indeed, takey0 2 C, and prove that By0 2 C. In consequence, we get

diam½By0�a ¼ e

R T

daðuÞdu e

R d

0aðuÞdu diam½y0�

a �Z d

0diam½bðsÞ�a e�

R s

0aðuÞduds

� �þZ T

ddiam½bðsÞ�a e�

R s

daðuÞduds

� �

P eR T

daðuÞdu

Z T

ddiam½bðsÞ�a e�

R s

daðuÞ duds ¼ e

R T

0aðuÞdu

Z T

ddiam½bðsÞ�a e�

R s

0aðuÞduds:

Therefore, by hypothesis (10), By0 satisfies the first restriction in C. Moreover, using y0 2 C, (11) and (12), we get

ðBy0Þal þZ t

0bðsÞare

�R s

0aðuÞ duds ¼ e

R T

0aðuÞdu ðy0Þal þ

Z d

0bðsÞar e�

R s

0aðuÞduds

� �þ eR T

daðuÞdu

Z T

dbðsÞal e�

R s

daðuÞduds

þZ t

0bðsÞare

�R s

0aðuÞduds

is nondecreasing in a; for all t 2 ð0; dÞ;

and

ðBy0Þar þZ t

0bðsÞale

�R s

0aðuÞ duds ¼ e

R T

0aðuÞdu ðy0Þar þ

Z d

0bðsÞal e�

R s

0aðuÞduds

� �þ eR T

daðuÞdu

Z T

dbðsÞar e�

R s

daðuÞduds

þZ t

0bðsÞale

�R s

0aðuÞ duds

is nonincreasing in a; for all t 2 ð0; dÞ:

Thus, we deduce that By0 2 C.Finally, we check that B is a contractive mapping. For y0; w0 2 C, we have

d1ðBðy0Þ;Bðw0ÞÞ ¼ eR T

daðuÞdue

R d

0aðuÞ dud1ðy0;w0Þ ¼ e

R T

0aðuÞdud1ðy0;w0Þ;

which joint to the hypothesis provides the contractive character of B, completing the proof. h

Remark 3.3. Using this approach, the diameter of the level sets of the solution obtained for (1) is nonincreasing on (0,d) andnondecreasing on (d,T). Under the hypotheses of Theorem 3.2, the periodic solution proposed is obtained by taking as initialcondition, y0, the element given by

ðy0Þal ¼1

1� eR T

0aðuÞdu

Z T

dbðsÞal e

R T

saðuÞdu dsþ

Z d

0bðsÞar e

R T

saðuÞ duds

� �; ð17Þ

ðy0Þar ¼1

1� eR T

0aðuÞdu

Z T

dbðsÞar e

R T

saðuÞdu dsþ

Z d

0bðsÞal e

R T

saðuÞ duds

� �: ð18Þ

In the following result, we illustrate how we can obtain the existence of (periodic) solutions to (1) which are differentia-ble on (0,T) under strongly generalized differentiability and real at t = d, that is, satisfying that diam[u(d)]a = 0, for everya 2 [0,1].

Theorem 3.3. Assume the hypotheses of Theorem 3.2 and, moreover, that the following condition holds:

eR T

0aðuÞdu

Z T

ddiam½bðsÞ�a e�

R s

0aðuÞduds ¼

Z d

0diam½bðsÞ�ae�

R s

0aðuÞduds; 8a 2 ½0;1�: ð19Þ

Then there exists a solution to problem (1) in F d, which is (ii)-differentiable on (0,d), (i)-differentiable on (d, T) and real at t = d.

Proof. The solution given by Theorem 3.2 corresponds to the expression y2, y1, with initial condition the fixed point of B, thatis, y0 given by (17) and (18). The solution is real at t = d if

A. Khastan et al. / Information Sciences 222 (2013) 544–558 551

y2ðdÞ ¼ eR d

0aðuÞdu y0 �

Z d

0ð�bðsÞÞ e�

R s

0aðuÞduds

� �

is real, that is,

diam½y2ðdÞ�a ¼ diam e

R d

0aðuÞdu y0 �

Z d

0ð�bðsÞÞ e�

R s

0aðuÞdu ds

� �� �a

¼ eR d

0aðuÞdu diam½y0�

a �Z d

0diam½bðsÞ�a e�

R s

0aðuÞdu ds

� �¼ 0;

for every a 2 [0,1]. Hence, we check that

diam½y0�a ¼

Z d

0diam½bðsÞ�ae�

R s

0aðuÞduds; 8a 2 ½0;1�:

Indeed, this condition is written as

1

1� eR T

0aðuÞdu

Z T

ddiam½bðsÞ�a e

R T

saðuÞduds�

Z d

0diam½bðsÞ�a e

R T

saðuÞ duds

� �¼Z d

0diam½bðsÞ�ae�

R s

0aðuÞduds;

which is equivalent to (19). h

Next, we present some examples where the conditions in Theorem 3.3 are satisfied. Since condition (19) implies (10), wejust check the validity of (11), (12), (19) and the contractivity condition

R T0 aðuÞdu < 0.

Example 3.1. If b is a fuzzy function defined on [0,T] with b(t) a (nonreal) fuzzy number for every t 2 [0,T], and assumingthat, for each fixed level a, the diameter of the a-level set of b(t), diam([b(t)]a), is the constant Da for every t 2 [0,T], then thecondition (19) is written, for Da > 0, as

eR T

0aðuÞdu

Z T

de�R s

0aðuÞ duds ¼

Z d

0e�R s

0aðuÞdu ds; ð20Þ

and it is trivially satisfied if Da = 0. Note that the choice of a is such that a < 0 on (0,d) and a > 0 on (d,T).On the other hand, if b is real, condition (19) is trivially satisfied.For instance, as a particular case of this situation, we can consider a ‘triangular’ fuzzy function of the type

½bðtÞ�a ¼ uðtÞ � Da

2;uðtÞ þ Da

2

� �; t 2 ½0; T�;

with u : ½0; T� ! R. For this choice of b, conditions (11) and (12) are deduced from (20), as we show in the following examplewhich considers a more general situation. Next example also illustrates the way to select an initial value for the iterativeprocess, assuming that

R T0 aðuÞdu < 0.

Example 3.2. Consider a fuzzy-valued function b defined on [0,T] such that, for each t 2 [0,T], b(t) is a symmetric fuzzy num-ber, that is, satisfies that the midpoints of its level sets remain constant in the variable a:

a! mpð½bðtÞ�aÞis a constant function; for each t 2 ½0; T� fixed:

We denote u(t) :¼mp([b(t)]a), t 2 [0,T].In this case, even if the diameter of the level sets of b(t) is nonconstant in t, or a, conditions (11) and (12) are derived from

(19). Denoting Da(t) :¼ diam([b(t)]a), for every t 2 [0,T] and a 2 [0,1], then the condition (19) is written as

Z T

dDaðsÞ e

R T

saðuÞduds ¼ e

R T

0aðuÞdu

Z T

dDaðsÞ e�

R s

0aðuÞduds ¼

Z d

0DaðsÞe�

R s

0aðuÞduds; ð21Þ

for every a 2 [0,1]. With the restrictions imposed over function b, we check that (11) and (12) hold, sincebðsÞal ¼ mpð½bðsÞ�aÞ � diamð½bðsÞ�aÞ

2 ¼ uðsÞ � DaðsÞ2 and bðsÞar ¼ mpð½bðsÞ�aÞ þ diamð½bðsÞ�aÞ

2 ¼ uðsÞ þ DaðsÞ2 , for s 2 [0,T]. Then,

Z T

duðsÞ � DaðsÞ

2

� �eR T

saðuÞdudsþ

Z t

0uðsÞ þ DaðsÞ

2

� �e�R s

0aðuÞ duds ¼

Z T

duðsÞ e

R T

saðuÞ dudsþ

Z t

0uðsÞ e�

R s

0aðuÞ duds

þ 12

Z t

0DaðsÞ e�

R s

0aðuÞduds�

Z T

dDaðsÞ e

R T

saðuÞduds

� �is nondecreasing in a; 8t 2 ð0; dÞ;

and

552 A. Khastan et al. / Information Sciences 222 (2013) 544–558

Z T

duðsÞ þ DaðsÞ

2

� �eR T

saðuÞdudsþ

Z t

0uðsÞ � DaðsÞ

2

� �e�R s

0aðuÞ duds ¼

Z T

duðsÞ e

R T

saðuÞ dudsþ

Z t

0uðsÞ e�

R s

0aðuÞ duds

þ 12

Z T

dDaðsÞ e

R T

saðuÞduds�

Z t

0DaðsÞ e�

R s

0aðuÞduds

� �is nonincreasing in a; 8t 2 ð0; dÞ;

due to the nonincreasing character of Da(s) in a, for each s 2 [0,T], and the fact that, using (21), we get

Z t

0DaðsÞ e�

R s

0aðuÞduds�

Z T

dDaðsÞ e

R T

saðuÞdu ds ¼ �

Z d

tDaðsÞ e�

R s

0aðuÞ duds; for t 2 ð0; dÞ:

If, moreover,R T

0 aðuÞdu < 0, we can take, in the iterative process leading to a solution of the problem, the initial elementy0 2 C given by

½y0�a ¼ p0 �

12

Z d

0DaðsÞ e�

R s

0aðuÞduds;p0 þ

12

Z d

0DaðsÞ e�

R s

0aðuÞduds

� �; 8a 2 ½0;1�;

where p0 2 R, with diameter diam½y0�a ¼

R d0 DaðsÞ e�

R s

0aðuÞdu ds, "a 2 [0,1], and satisfying (13) and (14), due to

p0 �12

Z d

0DaðsÞ e�

R s

0aðuÞ dudsþ

Z t

0uðsÞ þ DaðsÞ

2

� �e�R s

0aðuÞduds

¼ p0 þZ t

0uðsÞ e�

R s

0aðuÞduds� 1

2

Z d

tDaðsÞ e�

R s

0aðuÞduds is nondecreasing in a; 8t 2 ð0; dÞ;

and

p0 þZ t

0uðsÞ e�

R s

0aðuÞdudsþ 1

2

Z d

tDaðsÞ e�

R s

0aðuÞdu ds is nonincreasing in a; 8t 2 ð0; dÞ:

Remark 3.4. If b is a fuzzy function defined on [0,T] and, for every level a 2 [0,1] fixed, diam([b(t)]a) = g(t)Da, for everyt 2 [0,T], where g is a nonnegative continuous real function g : ½0; T� ! R, then the condition (19) is trivially fulfilled ifDa = 0, and written, for Da > 0, as

Z d

0gðsÞ e�

R s

0aðuÞduds�

Z T

dgðsÞ e

R T

saðuÞduds ¼ 0; ð22Þ

that is,

Z T

0gðsÞ HðsÞds ¼ 0; ð23Þ

where

HðsÞ ¼ e�R s

0aðuÞdu

; if s 6 d;

�eR T

saðuÞdu ¼ �e�

R s

0aðuÞdue

R T

0aðuÞdu

; if s > d:

8<:

Note that H(s) – 0, for every s 2 [0,T], but H > 0 on (0,d) and H < 0 on (d,T).Fixing the values of T, d 2 (0,T) and a continuous function a(t) with a < 0 on (0,d), a > 0 on (d,T) and

R T0 aðsÞds < 0, H is

completely determined and there always exists a nonnegative continuous function g: ½0; T� ! R satisfying (23). Indeed, thereare infinite possibilities to choose g. Just let h: ½0; T� ! R be a function which is continuous and such that h > 0 on (0,d),h(d) = 0, h < 0 on (d,T) and

R T0 hðsÞds ¼ 0. Then we obtain an admissible function g as

gðtÞ ¼ hðtÞHðtÞ ; t 2 ½0; T�;

which is nonnegative, continuous and satisfies (23). If, moreover, (11) and (12) are fulfilled, then Theorem 3.3 is applicable tothe periodic boundary value problem (1) for the continuous function b satisfying that diam([b(t)]a) = g(t)Da, for everyt 2 [0,T], and a 2 [0,1]. As particular cases, we can consider b(t) = g(t)C, or b(t) = �g(t)C, t 2 [0,T], with g satisfying the spec-ified requirements and C any fixed symmetric fuzzy number (i.e., with mp([C]a) constant in a). Once property (23) has shownto be true, the symmetry property is necessary (and sufficient) for the validity of (11) and (12) (the sufficient character wasalready shown in Example 3.2).

Example 3.3. Consider the function aðtÞ ¼ t � 23, t 2 J = [0,1], which satisfies that

R 10 aðtÞ dt ¼ � 1

6 < 0, and the function b givenby b(t) = g(t)C, t 2 [0,1], where C is the (symmetric) triangular fuzzy number defined as [C]a = [a � 1,1 � a], a 2 [0,1], and g isthe nonpositive real function given by

0 0.2 0.4 0.6 0.8 1−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Fig. 1. Endpoints of the support and core of b(t) = g(t)C.

A. Khastan et al. / Information Sciences 222 (2013) 544–558 553

gðtÞ ¼34 t � 1

2

� e�

23tþt2

2 ; if t 6 23 ;

�ð3t � 2Þe16�

23tþt2

2 ; if t > 23 :

8<:

Therefore, diam([b(t)]a) = jg(t)jdiam([C]a) = �g(t)(2 � 2a), for every t 2 [0,1] and a 2 [0,1]. In Fig. 1, we find three curveswhich represent the endpoints of the support and the core of b(t), for t 2 [0,1].

The periodic boundary value problem of interest is

y0ðtÞ ¼ t � 23

� yðtÞ þ gðtÞ C; t 2 J ¼ ½0;1�;

yð0Þ ¼ yð1Þ;

(ð24Þ

and we fix d ¼ 23.

In this example,

HðsÞ ¼e

23s�s2

2 ; if s 6 23 ;

�e�16þ

23s�s2

2 ; if s > 23 :

8<:

Note that function

hðtÞ ¼ jgðtÞjHðtÞ ¼� 3

4 t þ 12 ; if t 6 2

3 ;

�3t þ 2; if t > 23

(

satisfies thatR 1

0 hðsÞds ¼ 0, then condition (23) is satisfied, and hypotheses (11) and (12) are also valid by the symmetriccharacter of C. Therefore, by Remark 3.4 and Theorem 3.3, there exists a solution u to problem (24) such that uð23Þ is real.

Note that, in this example, if we set b(t) = jg(t)jC, for t 2 [0,1], we obtain the same results, since C is a symmetric fuzzynumber.

To complete this section, we analyze the connection between the solutions obtained and those of differential inclusions.First, consider the case where b is a real function. In the interval where a > 0, since g(t,u) = a(t)u + b(t) is continuous and non-decreasing with respect to the second argument, and the crisp initial value problem u0 = g(t,u), u(0) = c, has a unique solutionthen, by Theorem 3 in [24], the (i)-differentiable solution to the linear fuzzy differential equation coincides with the solutionobtained by using differential inclusions. Similarly, for the interval where a < 0, g is nonincreasing with respect to the secondargument and, hence, by Theorem 8 in [10], the (ii)-differentiable solution to the linear fuzzy differential equation coincideswith the solution by differential inclusions (see also [11]). However, for different choices of d and types of differentiabilitywith independence of the sign of a, different solutions may appear with respect to the differential inclusions approach, andan specific study should be made in order to determine if the (local) solutions are defined on the intervals we select. More-over, if b is a nonreal fuzzy function, solutions different from those corresponding to differential inclusions’ approach mayappear (and they might be less fuzzy than differential inclusions’ solutions).

4. Examples

In this section, we present two examples where the expression of the solution is obtained explicitly.

554 A. Khastan et al. / Information Sciences 222 (2013) 544–558

Example 4.1. Consider the problem

y0ðtÞ ¼ 13� t�

yðtÞ þ e�4tc; t 2 J ¼ ½0;1�;yð0Þ ¼ yð1Þ;

(ð25Þ

where the fuzzy number c is given by [c]a = [a,2 � a], a 2 [0,1], and take d ¼ 13. We check that conditions in Theorem 3.1 are

satisfied and then we have a periodic solution to problem (25) on [0,1]. Indeed,

Z 1

13

2e�4se�R s

013�uð Þduds�

Z 13

02e�4se�

R s

0ð13�uÞduds 6 0;

Z 13

0e�4sa e�

R s

013�uð Þdudsþ

Z t

13

e�4sð2� aÞ e�R s

013�uð Þduds is nondecreasing in a; 8t 2 1

3;1

� �;

Z 13

0e�4sð2� aÞ e�

R s

013�uð Þdudsþ

Z t

13

e�4sa e�R s

013�uð Þdu ds is nonincreasing in a; 8t 2 1

3;1

� �;

where we have used that

Z t

13

e�4s e�R s

013�uð Þ duds�

Z 13

0e�4s e�

R s

0ð13�uÞduds <

Z 1

13

e�4s e�R s

013�uð Þ duds�

Z 13

0e�4s e�

R s

013�uð Þduds 6 0; 8t 2 1

3;1

� �;

and it is also satisfied thatR 1

013� u�

du ¼ � 16 < 0. The operator A is defined as

Ay0 ¼ e

R 113

13�uð Þdu

eR 1

30

13�uð Þdu y0 þ

Z 13

0bðsÞe�

R s

013�uð Þduds

!�Z 1

13

ð�bðsÞÞe�R s

13

13�uð Þdu

ds

( );

for any y0 2 RF .Therefore, for any y0 2 RF , there exists limn!1A

ny0 ¼ y�0. Furthermore,

½Ay0�a ¼ e�

29 e

118ð½y0; y0� þ

Z 13

0½bðsÞ; �bðsÞ�e�

R s

013�uð ÞdudsÞ �

Z 1

13

½��bðsÞ;�bðsÞ�e�R s

13

13�uð Þdu

ds

( )

¼ e�29 e

118 y0 þ

Z 13

0bðsÞe�

R s

013�uð Þduds; y0 þ

Z 13

0

�bðsÞe�R s

013�uð Þduds

" # !(

�Z 1

13

ð��bðsÞÞe�R s

13

13�uð Þdu

ds;Z 1

13

ð�bðsÞÞe�R s

13

13�uð Þdu

ds

" #);

in which expression we have skipped the level, for simplicity. Hence,

Ay0 ¼ e�29 e

118ðy0 þ

Z 13

0bðsÞe�

R s

013�uð ÞdudsÞ þ

Z 1

13

�bðsÞe�R s

13

13�uð Þdu

ds

( )

¼ e�29 e

118y0 þ e

118

Z 13

0bðsÞe�

R s

013�uð Þdudsþ

Z 1

13

�bðsÞe�R s

13

13�uð Þdu

ds

( )

¼ e�29 e

118y0 þ e

118

Z 13

0ae�4se�

R s

013�uð Þdudsþ

Z 1

13

ð2� aÞe�4se�R s

13

13�uð Þdu

ds

( );

and it is easy to derive that

Ay0 ¼ e�29fe 1

18y0 þ Bg;

and, recursively,

Any0 ¼ e�29 e

118An�1y0 þ B

n o;

so that

Any0 ¼ e�3

18

�ny0 þ Be�

29Xn�1

i¼0

e�3

18

�i;

where B, depending on a, is given by

B ¼ e1

18

Z 13

0ae�4se�

R s

013�uð Þdudsþ

Z 1

13

ð2� aÞe�4se�R s

13

13�uð Þdu

ds:

A. Khastan et al. / Information Sciences 222 (2013) 544–558 555

Also, similarly, we have:

Any0 ¼ e�3

18

�ny0 þ B1e�

29

Xn�1

i¼0

e�3

18

�i;

where

B1 ¼ e1

18

Z 13

0ð2� aÞe�4se�

R s

013�uð Þdudsþ

Z 1

13

ae�4se�R s

13

13�uð Þdu

ds:

Note that limn!1Pn�1

i¼0 e�3

18

�i¼ e

16

e16�1

and limn!1 e�3

18

�n¼ 0. Then limn!1A

ny0 ¼ y�0, with1

y�0 ¼ Be�18

e16 � 1

; ð26Þ

and

y�0 ¼ B1e�

118

e16 � 1

: ð27Þ

We note that B ’ 0.125268a + 0.126958 and B1 ’ �0.125268a + 0.377494.This y�0 is the initial condition giving a periodic solution of (25).In consequence,

y1ðtÞ ¼ eR t

013�uð Þdu y�0 þ

Z t

0bðsÞe�

R s

013�uð Þduds

� �

is a (i)-differentiable solution of problem (25) on 0; 1

3

� . See Fig. 2, where we represent the lower and upper branches of u in

the interval 0; 13

� for the levels a = 0 and a = 1 (the central curve). We see that the diameter of the support of the solution is

increasing. Now, using the value of y1 at 13� we obtain

y2ðtÞ ¼ e

R t

13

13�uð Þdu

y113

� ��Z t

13

ð�bðsÞÞe�R s

13

13�uð Þdu

ds

!;

which is a (ii)-differentiable solution of (25) on 13 ;1�

(see Fig. 2). We see that its support has decreasing diameter. Finally, wecan check easily that, for the initial value y�0 indicated in (26) and (27), it is satisfied that y1ð0Þ ¼ y2ð1Þ ¼ Ay�0.

Example 4.2. Now, consider

y0ðtÞ ¼ t � 23

� yðtÞ þ tc; t 2 J ¼ ½0;1�;

yð0Þ ¼ yð1Þ;

(ð28Þ

where [c]a = [a,2 � a], a 2 [0,1], and fix d ¼ 23.

We check that conditions in Theorem 3.2 are satisfied, and thus we have a periodic solution to (28) on [0,1]. Indeed,

eR 1

0u�2

3ð ÞduZ 1

23

2se�R s

0u�2

3ð Þduds�Z 2

3

02se�

R s

0u�2

3ð Þduds P 0;

Z 1

23

s a eR 1

su�2

3ð Þ dudsþZ t

0s ð2� aÞ e�

R s

0u�2

3ð Þduds is nondecreasing in a; 8t 2 0;23

� �;

Z 1

23

s ð2� aÞ eR 1

su�2

3ð ÞdudsþZ t

0s a e�

R s

0u�2

3ð Þduds is nonincreasing in a; 8t 2 0;23

� �;

Fig. 2. (i)-Differentiable solution on 0; 13

� and (ii)-differentiable solution on 1

3 ;1�

for Example 4.1.

556 A. Khastan et al. / Information Sciences 222 (2013) 544–558

where we have used the inequality

Z t

0s e�

R s

0u�2

3ð Þduds�Z 1

23

s eR 1

su�2

3ð Þdu ds <Z 2

3

0s e�

R s

0u�2

3ð Þ duds�Z 1

23

s eR 1

su�2

3ð Þduds 6 0; 8t 2 0;23

� �;

and, finally,R 1

0 u� 23

� du < 0. The operator B is defined as

By0 ¼ e

R 123

u�23ð Þdu

eR 2

30

u�23ð Þdu y0 �

Z 23

0ð�bðsÞÞe�

R s

0u�2

3ð Þduds

!þZ 1

23

bðsÞe�R s

23

u�23ð Þdu

ds

( );

for every y0 2 RF .Hence, for every a 2 [0,1],

½By0�a ¼ e

118 e�

29ð½y0; y0� �

Z 23

0½��bðsÞ;�bðsÞ�e�

R s

0ðu�2

3ÞdudsÞ þZ 1

23

½bðsÞ; �bðsÞ�e�R s

23ðu�2

3Þduds

( );

which gives

By0 ¼ e1

18 e�29y0 þ e�

29

Z 23

0ð2� aÞse�

R s

0u�2

3ð ÞdudsþZ 1

23

ase�R s

23

u�23ð Þdu

ds

( );

thus

By0 ¼ e�3

18y0 þ e1

18B;

and, for Bny0, we have

Bny0 ¼ e�3

18

�ny0 þ Be

118Xn�1

i¼0

e�3

18

�i;

where

B ¼ e�29

Z 23

0ð2� aÞse�

R s

0ðu�2

3ÞdudsþZ 1

23

ase�R s

23

u�23ð Þdu

ds:

In consequence,

limn!1

Bny0 ¼ y�0 ¼ Be

29

e16 � 1

:

Similarly

y�0 ¼ B1e

29

e16 � 1

;

where

B1 ¼ e�29

Z 23

0ase�

R s

0u�2

3ð ÞdudsþZ 1

23

ð2� aÞse�R s

23

u�23ð Þdu

ds:

Then

y2ðtÞ ¼ eR t

0ðu�2

3Þdu y�0 �Z t

0ð�bðsÞÞe�

R s

0u�2

3ð Þduds� �

is a (ii)-differentiable solution to problem (28) on 0; 23

� . Besides,

y1ðtÞ ¼ e

R t23

u�23ð Þdu

y223

� �þZ t

23

bðsÞe�R s

23

u�23ð Þdu

ds

!

is a (i)-differentiable solution to (28) on 23 ;1�

. Finally, we check that, for y�0, we have y2(0) = y1(1). This solution to (28) isrepresented in Fig. 3.

The cases considered in this paper are those for which the equation a(t) = 0 has a unique solution in the interval (0,T). Anissue which deserves further research is to extend this study to more general situations, for instance, the case where theequation a(t) = 0 has no solution or finitely many solutions in (0,T). This could be applicable to study the existence of periodicsolutions to fuzzy differential equations with periodic coefficients.

Fig. 3. (ii)-Differentiable solution on 0; 23

� and (i)-differentiable solution on 2

3 ;1�

for Example 4.2.

A. Khastan et al. / Information Sciences 222 (2013) 544–558 557

5. Conclusions

The method followed shows that, in fact, there exist several possibilities to obtain solutions to the fuzzy periodic bound-ary value problem. For example, we can proceed by:

� starting with a (i)-differentiable solution and choosing a point t⁄ 2 (0,T) as the switching point where the solutionchanges from (i) to (ii) differentiability, or

� starting with a (ii)-differentiable solution and choosing a point t⁄ 2 (0,T) as the switching point where the solutionchanges from (ii) to (i) differentiability.

The type of (i) and (ii)-solutions considered are obtained by application of the variation-of-constants formula (see [8]),but the periodic boundary value problem of interest can be approached more generally, by using other solutions basedon generalized differentiability, as those in [7,10,37]. Thus, we could analyze the existence of solutions to the periodicboundary value problem corresponding to a finite number of switching points between intervals of (i)- and (ii)-differentia-bility, independently of the sign of function a on those intervals. This type of considerations allow to obtain, for instance,different (i)–(ii) differentiable solutions to the problem in Example 4.1, and even (ii)–(i) differentiable solutions.

The proposed solution is, therefore, not unique, but it serves to illustrate how this procedure can be useful to provide peri-odic solutions to fuzzy differential equations from the point of view of generalized differentiability. The particular solutionpresented is, in some sense, natural, since the switching point is located exactly at the point where function a changes itssign. On the other hand, if y is a solution to (1) and a(d) = 0, then yal and yar are differentiable at t = d if b(d)al = b(d)ar, thatis, yal and yar are differentiable at t = d, for every a 2 [0,1], if b(d) is real.

Moreover, as we have pointed out in Section 3, following this approach for equations with a nonreal fuzzy function b, wemay find solutions different from those corresponding to differential inclusions’ approach.

On the other hand, the study of two-point boundary value problems, consisting of fuzzy linear differential equations sub-ject to the conditions y(0) = y1 and y(T) = y2, where y1; y2 2 RF , can also be made under the approach of generalized differ-entiability similarly to the above mentioned procedure.

Acknowledgements

We thank the Editor in Chief and the anonymous referees for their interesting and helpful comments and suggestions.Research partially supported by Ministerio de Educación y Ciencia and FEDER, Projects MTM2007-61724 and MTM2010-

15314, and by Xunta de Galicia and FEDER, Project PGIDIT06PXIB207023PN.

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