Abstract and Applied Analysis

Volume 2012 (2012), Article ID 423796, 24 pages

doi:10.1155/2012/423796

Research Article

Existence Results for Fractional Differential Inclusions with Multivalued Term Depending on Lower-Order Derivative

Xiaoyou Liu¹ and Zhenhai Liu²

¹School of Mathematical Science and Computing Technology, Central South University, Changsha, Hunan 410075, China

²School of Science, Guangxi University for Nationalities, Nanning, Guangxi 530006, China

Received 24 October 2012; Accepted 14 December 2012

Academic Editor: Svatoslav Staněk

Copyright © 2012 Xiaoyou Liu and Zhenhai Liu. 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.

This paper is concerned with a class of fractional differential inclusions whose multivalued term depends on lower-order fractional derivative with fractional (non)separated boundary conditions. The cases of convex-valued and non-convex-valued right-hand sides are considered. Some existence results are obtained by using standard fixed point theorems. A possible generalization for the inclusion problem with integral boundary conditions is also discussed. Examples are given to illustrate the results.

1. Introduction

Recently, the subject of fractional differential equations has emerged as an important area of investigation. Indeed, we can find numerous applications of fractional-order derivatives in the mathematical modeling of physical and biological phenomena in various fields of science and engineering [1–3]. A variety of results on initial and boundary value problems of fractional differential equations and inclusions can easily be found in the literature on this topic. For some recent results, we can refer to, for instance, [4–20] (equations) [21–27] (inclusions) and the references therein.

Ahmad and Ntouyas [22] considered a boundary value problem of fractional differential inclusions with fractional separated boundary conditions given by

where denotes the Caputo fractional derivative of order , is a multivalued map, and , , () are real constants, with .

In Cernea [24], the following multipoint boundary value problem for a fractional-order differential inclusion was studied

where is the standard Riemann-Liouville fractional derivative, , , , , , , and is a multivalued map.

In Khan et al. [11], the authors studied the existence and uniqueness results of nonlinear fractional differential equation of the type

where , , , (or , ) and , are the Caputo fractional derivatives. The results in [11, 22, 24] are obtained by using appropriate standard fixed point theorems.

Motivated by the papers cited above, in this paper, we consider the existence results for a new class of fractional differential inclusions of the form

where denotes the Caputo fractional derivative of order , is a multivalued map, , , and . We study (1.4) subject to two families of boundary conditions:

separated boundary conditions

Nonseparated boundary conditions

 where , , , are real constants and .

The results of this paper can easily to be generalized to the boundary value problems of fractional differential inclusions (1.4) with the following integral boundary conditions:

where are given functions.

We remark that when the third variable of the multifunction in (1.4) vanishes, the problem (1.4), (1.5) reduces to the case considered in [22]. When , , and , the problem (1.4), (1.6) reduces to an antiperiodic fractional boundary value problem (the case of a given continuous function was studied in [4, 15]). Our results generalize some results from the literature cited above and constitute a contribution to this emerging field of research.

The rest of the paper is organized as follows: in Section 2 we present the notations and definitions and give some preliminary results that we need in the sequel, Section 3 is dedicated to the existence results of the fractional differential inclusion (1.4) with boundary conditions (1.5) and (1.6), in Section 4 we indicate a possible generalization for the inclusion problem (1.4) with integral boundary conditions (1.7) and (1.8), and two illustrative examples are given in Section 5.

2. Preliminaries

In this section, we introduce notations, definitions, and preliminary facts that will be used in the remainder of this paper.

Let be a normed space. We use the notations , , , , , , and so on.

Let ; the Pompeiu-Hausdorff distance of , is defined as

A multivalued map is convex (closed) valued if is convex (closed) for all . is said to be completely continuous if is relatively compact for every . is called upper semicontinuous on , if for every , the set is a nonempty closed subset of , and for every open set of containing , there exists an open neighborhood of such that . Equivalently, is upper semicontinuous if the set is open for any open set of . is called lower semicontinuous if the set is open for each open set in . If a multivalued map is completely continuous with nonempty compact values, then is upper semicontinuous if and only if has a closed graph, that is, if and , then implies [28].

A multivalued map is said to be measurable if, for every , the function is a measurable function.

Definition 2.1. A multivalued map is called

-Lipschitz if there exists such that

a contraction if it is -Lipschitz with .

Definition 2.2. A multivalued map is said to be Carathéodory if

is measurable for each ; is upper semicontinuous for a.e. . Further, a Carathéodory function is said to be - Carathéodory if for each , there exists such that

 for all , and a.e. .

Lemma 2.3 (see [29]). Let be a Banach space. Let be an - Carathéodory multivalued map and a linear continuous map from to , then the operator

is a closed graph operator in .

Here for a.e. .

Definition 2.4 (see [30]). The Riemann-Liouville fractional integral of order for a function is defined as

provided the integral exists.

Definition 2.5 (see [30]). For at least -times differentiable function , the Caputo derivative of order is defined as

where denotes the integer part of the real number .

Lemma 2.6 (see [20]). Let ; then the differential equation

has solutions and

here , , .

The following lemma obtained in [6] is useful in the rest of the paper.

Lemma 2.7 (see [6]). For a given , the unique solution of the fractional separated boundary value problem

is given by

where

We notice that the solution (2.10) of the problem (2.9) does not depend on the parameter , that is to say, the parameter is of arbitrary nature for this problem. And by (2.10), we should assume that and .

Lemma 2.8. For any , the unique solution of the fractional nonseparated boundary value problem

is given by

Proof. For , by Lemma 2.6, we know that the general solution of the equation can be written as

where , are arbitrary constants. Since ( is a constant), , (see [30]), from (2.14), we have

Using the boundary conditions, we obtain

Therefore, we have

Substituting the values of , in (2.14), we obtain (2.13). This completes the proof.

From the proof of the above lemma, we notice that the solution (2.13) of the problem (2.12) does not depend on the parameter , that is to say, the parameter is of arbitrary nature for this problem. In this situation, we need to assume that and .

Let us define what we mean by a solution of the problem (1.4), (1.5) and the problem (1.4), (1.6).

Definition 2.9. A function is a solution of the problem (1.4), (1.5) if it satisfies the boundary conditions (1.5) and there exists a function such that a.e. on and

Definition 2.10. A function is a solution of the problem (1.4), (1.6) if it satisfies the boundary conditions (1.6) and there exists a function such that a.e. on and

Let be the space of all continuous functions defined on . Define the space and () endowed with the norm . We know that is a Banach space (see [14]).

We end this section with two fixed point theorems, which will be used in the sequel.

Theorem 2.11 (nonlinear alternative of Leray-Schauder type [31]). Let be a Banach space, a closed convex subset of , an open subset of with . Suppose that is an upper semicontinuous compact map. Then either    has a fixed point in , or there is a and such that .

Theorem 2.12 (Covitz and Nadler Jr. [32]). Let be a complete metric space. If is a contraction, then has a fixed point.

3. Existence Results

In this section, we will give some existence results for the problems (1.4), (1.5) and (1.4), (1.6).

For each , define the set of selections of by

In view of Lemmas 2.7 and 2.8, we define operators as

with

It is clear that if is a fixed point of the operator (the operator ), then is a solution of the problem (1.4), (1.5) (the problem (1.4), (1.6)).

Now we are in a position to present our main results. The methods used to prove the existence results are standard; however, their exposition in the framework of problems (1.4), (1.5) and (1.4), (1.6) is new.

3.1. Convex Case

We consider first the case when is convex valued.

:   is a Carathéodory multivalued map; there exist and continuous, nondecreasing such that

 for and a.e. .

Theorem 3.1. Assume that (H1) is satisfied and there exists such that

where

Then the problem (1.4), (1.5) has at least one solution on .

Proof. Consider the operator defined by (3.2). From , we have for each , the set is nonempty [29]. For , let and , that is, , we have

where is a constant given by

Hence we know that the operator is well defined.

We put where

Here means that the constant defined by (3.9) is related to .

We will show that satisfies the requirements of the nonlinear alternative of Leray-Schauder type. The proof will be given in five steps.

Step 1 ( is convex valued). Since is convex valued, we know that is convex and therefore it is obvious that is convex for each .

Step 2 ( maps bounded sets into bounded sets in ). Let be a bounded subset of such that for any , , . We prove that there exists a constant such that for each , one has for each . Let and , then there exists such that

By simple calculations, we have

Hence we obtain

Step 3 ( maps bounded sets into equicontinuous sets in ). Let be a bounded set of as in Step 2. Let and . For each , then there is such that . Since

we obtain that (since , and )

and the limits are independent of and .

Step 4 ( has a closed graph). Let , , and ; we need to show . Now implies that there exists such that for . Let us consider the continuous linear operator given by

and denote . Then and

We apply Lemma 2.3 to find that has closed graph and from the definition of we get . Since , , it follows the existence of such that . This means that .

Step 5 (a priori bounds on solutions). Let for some . Then there exists such that for . With the same arguments as in Step 2 of our proof, for each , we obtain

Thus

Now we set

Clearly, is an open subset of and . As a consequence of Steps 1–4, together with the Arzela-Ascoli theorem, we can conclude that is upper semicontinuous and completely continuous. From the choice of the , there is no such that for some . Therefore, by the nonlinear alternative of Leary-Schauder type (Theorem 2.11), we deduce that has a fixed point , which is a solution of the problem (1.4), (1.5). This completes the proof.

Theorem 3.2. Assume that (H1) is satisfied and there exists such that

where

Then the problem (1.4), (1.6) has at least one solution on .

Proof. To obtain the result, the main aim is to study the properties of the operator defined in (3.3). The proof of them is similar to those of Theorem 3.1, so we omit the details. Here we just give some estimations, which are needed in the following theorems. Let and ; then there exists such that

We put and

here and are constants given by

By simple calculations, we have

Hence we obtain

This is the end of the proof.

3.2. Nonconvex Case

Now we study the case when is not necessarily convex valued.

A subset of is decomposable if for all and Lebesgue measurable, then , where stands for the characteristic function.

: is a multivalued map such that    is measurable;    is lower semicontinuous for a.e. .

Theorem 3.3. Let (H1)(1.5), (H2), and relation (3.6) hold; then the problem (1.4), (1.5) has at least one solution on .

Proof. From (H1)(1.5), (H2), and [33, Lemma 4.1], the map

is lower semicontinuous and has nonempty closed and decomposable values. Then from a selection theorem due to Bressan and Colombo [34], there exists a continuous function such that for all . That is to say, we have for a.e. . Now consider the problem

with the boundary conditions (1.5). Note that if is a solution of the problem (3.29), then is a solution to the problem (1.4), (1.5).

Problem (3.29) is then reformulated as a fixed point problem for the operator defined by

It can easily be shown that is continuous and completely continuous and satisfies all conditions of the Leray-Schauder nonlinear alternative for single-valued maps [31]. The remaining part of the proof is similar to that of Theorem 3.1, so we omit it. This completes the proof.

Theorem 3.4. Let (H1)(1.5), (H2), and relation (3.21) hold, then the problem (1.4), (1.6) has at least one solution on .

The proof of this theorem is similar to that of Theorem 3.3.

: is a multivalued map such that    is integrably bounded and the map is measurable for all ; there exists such that for a.e. and all , , , ,

Theorem 3.5. Let (H3) hold, if, in addition,

then the problem (1.4), (1.5) has at least one solution on .

Proof. From , we have that the multivalued map is measurable [28, Proposition ] and closed valued for each . Hence it has measurable selection [28, Theorem ] and the set is nonempty. Let be defined in (3.2). We will show that, under this situation, satisfies the requirements of Theorem 2.12.

Step 1. For each , . Let , such that in . Then and there exists , such that

By , the sequence is integrable bounded. Since has compact values, we may pass to a subsequence if necessary to get that converges to in . Thus and for each

This means that and is closed.

Step 2. There exists such that

Let , and ; then there exists such that

From , we deduce

Hence, for a.e. , there exists such that

Consider the multivalued map given by

Since , are measurable, [35, Theorem ] implies that is measurable. It follows from that the map is measurable. Hence by (3.38) and [28, Proposition  ], the multivalued map is measurable and nonempty closed valued. Therefore, we can find such that for a.e. ,

Let , that is, . Since

we obtain

Denote

By using an analogous relation obtained by interchanging the roles of and , we get

Therefore, from condition (3.32), Theorem 2.12 implies that has a fixed point, which is a solution of the problem (1.4), (1.5). This completes the proof.

Theorem 3.6. Let (H3) hold, if, in addition,

then the problem (1.4), (1.6) has at least one solution on .

Using the arguments employed in the proof of Theorem 3.5, we can prove this theorem similarly. Hence the details are omitted here.

4. Integral Boundary Conditions

In this section, the existence results of the problems (1.4), (1.5) and (1.4), (1.6) obtained in the previous section will be extended to the ones of the problems of fractional differential inclusions (1.4) subject to the integral boundary conditions (1.7) and (1.8).

Lemma 4.1. For any ,,, the unique solution of the fractional separated integral boundary value problem,

is given by

Lemma 4.2. For any , , , the unique solution of the fractional nonseparated integral boundary value problem,

is given by

To obtain the existence results of the problems (1.4), (1.7) and (1.4), (1.8), in view of Lemmas 4.1 and 4.2, we define two operators as

with

Observe that if is a fixed point of the operator (the operator ), that is, (), then is a solution of the problem (1.4), (1.7) (the problem (1.4), (1.8)).

From the definitions of the operators , (see (3.2), (4.5)), we know that the difference between them is very apparent, that is, , in (3.2) were replaced by and in (4.5). This fact is also true for the operators , (see (3.3), (4.6)).

In the following, we state some existence results for the problems (1.4), (1.7) and (1.4), (1.8). We omit the proofs as these are similar to the ones given in Section 3.

: The functions , are continuous. There exist functions , and , continuous, nondecreasing such that

 for all and a.e. .

Theorem 4.3. Assume that (H1) and (A1) hold. If there exists a constant such that

here is defined by (3.7) and

Then the boundary value problem (1.4), (1.7) has at least one solution on .

Theorem 4.4. Assume that (H1) and (A1) hold. If there exists a constant such that

here is defined by (3.22) and

Then the boundary value problem (1.4), (1.8) has at least one solution on .

Theorem 4.5. Assume that (H1)(1.5), (H2), (A1) and condition (4.9) hold. Then the boundary value problem (1.4), (1.7) has at least one solution on .

Theorem 4.6. Assume that (H1)(1.5), (H2), (A1) and condition (4.11) hold. Then the boundary value problem (1.4), (1.8) has at least one solution on .

: The functions are continuous and satisfy

 for all and a.e. ; here , .

Theorem 4.7. Assume that (H3) and (A2) hold. If, in addition,

here is defined by (3.7) and , are defined by (4.10), then the boundary value problem (1.4), (1.7) has at least one solution on .

Theorem 4.8. Assume that (H3) and (A2) hold. If, in addition,

here is defined by (3.22) and , are defined by (4.12), then the boundary value problem (1.4), (1.8) has at least one solution on .

5. Examples

In this section, we give two simple examples to show the applicability of our results.

Example 5.1. Consider the following fractional boundary value problem:

where , , , , , , , , , , and is a multivalued map given by

In the context of this problem, we have

It is clear that is convex compact valued and is of Carathéodory type. Let and , ; we get for , ,

As for the condition (3.6), since (, defined in (3.6)) is a constant, we can choose large enough so that

Thus, by the conclusion of Theorem 3.1, the boundary value problem (5.1) has at least one solution on .

Example 5.2. Consider the following fractional differential inclusion with integral boundary conditions:

where , , , , , , , , , ,

and , .

From the data given above, we have for , , ,

Then let , , and ; we have

Here is defined by (3.22) and , are defined by (4.12). Hence all the assumptions of Theorem 4.8 are satisfied, and by the conclusion of it, the boundary value problem (5.6) has at least one solution on .

Acknowledgment

This project was supported by NNSF of China Grants nos. 11271087 and 61263006.

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