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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 292643, 9 pages
http://dx.doi.org/10.1155/2013/292643
Research Article

Extremal Solutions and Relaxation Problems for Fractional Differential Inclusions

1Departamento de Análisis Matematico, Facultad de Matemáticas, Universidad de Santiago de Compostela, 15782 Santiago de Compostela, Spain
2Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
3Laboratory of Mathematics, Sidi-Bel-Abbès University, P.O. Box 89, 22000 Sidi-Bel-Abbès, Algeria
4Department of Mathematics, Periyar University, Salem 636 011, India

Received 10 May 2013; Accepted 31 July 2013

Academic Editor: Daniel C. Biles

Copyright © 2013 Juan J. Nieto et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We present the existence of extremal solution and relaxation problem for fractional differential inclusion with initial conditions.

1. Introduction

Differential equations with fractional order have recently proved to be valuable tools in the modeling of many physical phenomena [19]. There has also been a significant theoretical development in fractional differential equations in recent years; see the monographs of Kilbas et al. [10], Miller and Ross [11], Podlubny [12], and Samko et al. [13] and the papers of Kilbas and Trujillo [14], Nahušev [15], Podlubny et al. [16], and Yu and Gao [17].

Recently, some basic theory for initial value problems for fractional differential equations and inclusions involving the Riemann-Liouville differential operator was discussed, for example, by Lakshmikantham [18] and Chalco-Cano et al. [19].

Applied problems requiring definitions of fractional derivatives are those that are physically interpretable for initial conditions containing , , and so forth. The same requirements are true for boundary conditions. Caputo’s fractional derivative satisfies these demands. For more details on the geometric and physical interpretation for fractional derivatives of both Riemann-Liouville and Caputo types, see Podlubny [12].

Fractional calculus has a long history. We refer the reader to [20].

Recently fractional functional differential equations and inclusions and impulsive fractional differential equations and inclusions with standard Riemann-Liouville and Caputo derivatives with differences conditions were studied by Abbas et al. [21, 22], Benchohra et al. [23], Henderson and Ouahab [24, 25], Jiao and Zhou [26], and Ouahab [2729] and in the references therein.

In this paper, we will be concerned with the existence of solutions, Filippov’s theorem, and the relaxation theorem of abstract fractional differential inclusions. More precisely, we will consider the following problem: where is the Caputo fractional derivatives, , is a multifunction, and represents the set of extreme points of . is the family of all nonempty subsets of .

During the last couple of years, the existence of extremal solutions and relaxation problem for ordinary differential inclusions was studied by many authors, for example, see [3034] and the references therein.

The paper is organized as follows. We first collect some background material and basic results from multivalued analysis and give some results on fractional calculus in Sections 2 and 3, respectively. Then, we will be concerned with the existence of solution for extremal problem. This is the aim of Section 4. In Section 5, we prove the relaxation problem.

2. Preliminaries

The reader is assumed to be familiar with the theory of multivalued analysis and differential inclusions in Banach spaces, as presented in Aubin et al. [35, 36], Hu and Papageorgiou [37], Kisielewicz [38], and Tolstonogov [32].

Let be a real Banach space, an interval in , and the Banach space of all continuous functions from into with the norm

A measurable function is Bochner integrable if is Lebesgue integrable. In what follows, denotes the Banach space of functions , which are Bochner integrable with norm Denote by the space of equivalence classes of Bochner integrable function with the norm The norm is weaker than the usual norm , and for a broad class of subsets of , the topology defined by the weak norm coincides with the usual weak topology (see [37, Proposition 4.14, page 195]). Denote by A multivalued map has convex (closed) values if is convex (closed) for all . We say that is bounded on bounded sets if is bounded in for each bounded set of (i.e., .

Definition 1. A multifunction is said to be upper semicontinuous at the point , if, for every open such that , there exists a neighborhood of such that .

A multifunction is called upper semicontinuous (u.s.c. for short) on if for each it is u.s.c. at .

Definition 2. A multifunction is said to be lower continuous at the point , if, for every open such that , there exists a neighborhood of with property that for all .

A multifunction is called lower semicontinuous (l.s.c. for short) provided that it is lower semicontinuous at every point .

Lemma 3 (see [39, Lemma 3.2]). Let be a measurable multivalued map and a measurable function. Then for any measurable , there exists a measurable selection of such that for a.e. ,

First, consider the Hausdorff pseudometric defined by where and . is a metric space and is a generalized metric space.

Definition 4. A multifunction is called Hausdorff lower semicontinuous at the point , if for any there exists a neighbourhood of the point such that where is the unite ball in .

Definition 5. A multifunction is called Hausdorff upper semicontinuous at the point , if for any there exists a neighbourhood of the point such that

is called continuous, if it is Hausdorff lower and upper semicontinuous.

Definition 6. Let be a Banach space; a subset is decomposable if, for all and for every Lebesgue measurable set , one has where stands for the characteristic function of the set . We denote by the family of decomposable sets.

Let be a multivalued map with nonempty closed values. Assign to the multivalued operator defined by The operator is called the Nemyts’kiĭ operator associated to .

Definition 7. Let be a multivalued map with nonempty compact values. We say that is of lower semicontinuous type (l.s.c. type) if its associated Nemyts’kiĭ operator is lower semicontinuous and has nonempty closed and decomposable values.

Next, we state a classical selection theorem due to Bressan and Colombo.

Lemma 8 (see [40]). Let be a separable metric space and let be a Banach space. Then every l.s.c. multivalued operator with closed decomposable values has a continuous selection; that is, there exists a continuous single-valued function such that for every .

Let us introduce the following hypothesis. is a nonempty compact valued multivalued map such that(a)the mapping is measurable;(b)the mapping is lower semicontinuous for a.e. .

Lemma 9 (see, e.g., [41]). Let be an integrably bounded multivalued map satisfying . Then is of lower semicontinuous type.

Define where is a Banach space.

Lemma 10 (see [37]). Let be a weakly compact subset of . Then is relatively weakly compact subset of . Moreover if is convex, then is weakly compact in .

Definition 11. A multifunction possesses the Scorza-Dragoni property (- property) if for each , there exists a closed set whose Lebesgue measure and such that is continuous with respect to the metric .

Remark 12. It is well known that if the map is continuous with respect to for almost every and is measurable with respect to for every , then it possesses the S-D property.

In what follows, we present some definitions and properties of extreme points.

Definition 13. Let be a nonempty subset of a real or complex linear vector space. An extreme point of a convex set is a point with the property that if with and , then and/or .   will denote the set of extreme points of .

In other words, an extreme point is a point that is not an interior point of any line segment lying entirely in .

Lemma 14 (see [42]). A nonempty compact set in a locally convex linear topological space has extremal points.

Let be a denumerable, dense (in topology) subset of the set . For any and define the function

Lemma 15 (see [33]). if and only if for all .

In accordance with Krein-Milman and Trojansky theorem [43], the set is nonempty and .

Lemma 16 (see [33]). Let be a measurable, integrably bounded map. Then where is the closure of set in the topology of the space .

Theorem 17 (see [33]). Let be a multivalued map that has the - property and let it be integrable bounded on compacts from . Consider a compact subset and define the multivalued map , by Then for every compact in ,   and any continuous selection , there exists a continuous selector of the map such that for all one has

For a background of extreme point of see Dunford-Schwartz [42, Chapter 5, Section 8] and Florenzano and Le Van [44, Chapter 3].

3. Fractional Calculus

According to the Riemann-Liouville approach to fractional calculus, the notation of fractional integral of order () is a natural consequence of the well known formula (usually attributed to Cauchy) that reduces the calculation of the -fold primitive of a function to a single integral of convolution type. In our notation the Cauchy formula reads

Definition 18 (see [13, 45]). The fractional integral of order of a function is defined by where is the gamma function. When , we write , where for , and we write for and as , where is the delta function and is the Euler gamma function defined by For consistency, (identity operator), that is, . Furthermore, by we mean the limit (if it exists) of for ; this limit may be infinite.

After the notion of fractional integral, that of fractional derivative of order becomes a natural requirement and one is attempted to substitute with in the above formulas. However, this generalization needs some care in order to guarantee the convergence of the integral and preserve the well known properties of the ordinary derivative of integer order. Denoting by , with , the operator of the derivative of order , we first note that that is, is the left inverse (and not the right inverse) to the corresponding integral operator . We can easily prove that As a consequence, we expect that is defined as the left inverse to . For this purpose, introducing the positive integer such that , one defines the fractional derivative of order .

Definition 19. For a function given on interval , the Riemann-Liouville fractional-order derivative of is defined by where and is the integer part of .

Defining for consistency, , then we easily recognize that Of course, properties (25) and (26) are a natural generalization of those known when the order is a positive integer.

Note the remarkable fact that the fractional derivative is not zero for the constant function , if . In fact, (26) with illustrates that It is clear that , for , due to the poles of the gamma function at the points .

We now observe an alternative definition of fractional derivative, originally introduced by Caputo [46, 47] in the late sixties and adopted by Caputo and Mainardi [48] in the framework of the theory of Linear Viscoelasticity (see a review in [4]).

Definition 20. Let . The Caputo fractional-order derivative of is defined by

This definition is of course more restrictive than Riemann-Liouville definition, in that it requires the absolute integrability of the derivative of order . Whenever we use the operator we (tacitly) assume that this condition is met. We easily recognize that in general unless the function , along with its first derivatives, vanishes at . In fact, assuming that the passage of the -derivative under the integral is legitimate, we recognize that, for and , and therefore, recalling the fractional derivative of the power function (26), one has The alternative definition, that is, Definition 20, for the fractional derivative thus incorporates the initial values of the function and of lower order. The subtraction of the Taylor polynomial of degree at from means a sort of regularization of the fractional derivative. In particular, according to this definition, the relevant property for which the fractional derivative of a constant is still zero: We now explore the most relevant differences between the two fractional derivatives given in Definitions 19 and 20. From Riemann-Liouville fractional derivatives, we have From (32) and (33) we thus recognize the following statements about functions which, for , admit the same fractional derivative of order , with ,  : In these formulas, the coefficients are arbitrary constants. For proving all main results we present the following auxiliary lemmas.

Lemma 21 (see [10]). Let and let or . Then

Lemma 22 (see [10]). Let and . If or , then

For further readings and details on fractional calculus, we refer to the books and papers by Kilbas [10], Podlubny [12], Samko [13], and Caputo [4648].

4. Existence Result

Definition 23. A function is called mild solution of problem (1) if there exist such that where .

We will impose the following conditions on .The function such that(a)for all , the map is measurable,(b)for every , the multivalued map is continuous There exist and a continuous nondecreasing function such that with

Theorem 24. Assume that the conditions - and then the problem (2) have at least one solution.

Proof. From there exists such that for each .
Let We consider It is clear that all the solutions of (41) are solutions of (2).
Set It is clear that is weakly compact in . Remark that for every , there exists a unique solution of the following problem: this solution is defined by We claim that is continuous. Indeed, let converge in , as , set , . It is clear that is relatively compact in and converge to . Let Then Hence is compact and convex subset of . Let be the multivalued Nemitsky operator defined by It is clear that is continuous and and is integrably bounded, then by Theorem 17 (see also Theorem 6.5 in [32] or Theorem 1.1 in [34]), we can find a continuous function such that From Benamara [49] we know that Setting and letting , then Now, we prove that is continuous. Indeed, let converge to in .
Then Since and , then From Lemma 10, converge weakly to in as . By the definition of , we have Since , then there exists subsequence of converge in . Then This proves that is continuous. Hence by Schauder’s fixed point there exists such that .

5. The Relaxed Problem

In this section, we examine whether the solutions of the extremal problem are dense in those of the convexified one. Such a result is important in optimal control theory whether the relaxed optimal state can be approximated by original states; the relaxed problems are generally much simpler to build. For the problem for first-order differential inclusions, we refer, for example, to [35, Theorem 2, page 124] or [36, Theorem , page 402]. For the relaxation of extremal problems we see the following recent references [30, 50].

Now we present our main result of this section.

Theorem 25. Let be a multifunction satisfying the following hypotheses.The function such that, for all , the map is measurable.There exists such that Then .

Proof. By Coviz and Nadlar fixed point theorem, we can easily prove that , and since has compact and convex valued, then is compact in . For more information we see [25, 2729, 51, 52].
Let ; then there exists such that Let be a compact and convex set in such that . Given that and , we define the following multifunction by The multivalued map is measurable and is continuous. In addition, if has compact values, then is graph measurable, and the mapping is a measurable multivalued map for fixed . Then by Lemma 3, there exists a measurable selection such that this implies that . We consider defined by Since the measurable multifunction is integrable bounded, Lemma 9 implies that the Nemyts’kiĭ operator has decomposable values. Hence is l.s.c. with decomposable values. By Lemma 8, there exists a continuous selection such that From Theorem 17, there exists function such that From we can prove that there exists such that Consider the sequence ,  as  , and set , . Set Let be the map such that each assigns the unique solution of the problem As in Theorem 24, we can prove that is compact in and the operator is compact; then by Schauder’s fixed point there exists such that and Hence Let be a limit point of the sequence . Then, it follows that from the above inequality, one has which implies . Consequently, is a unique limit point of .

Example 26. Let with where are Carathéodory functions and bounded.
Then (2) is solvable.

Example 27. If, in addition to the conditions on of Example 26, and are Lipschitz functions, then .

Acknowledgments

This work is partially supported by the Ministerio de Economia y Competitividad, Spain, project MTM2010-15314, and cofinanced by the European Community Fund FEDER.

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