Abstract and Applied Analysis

Volume 2013 (2013), Article ID 869837, 17 pages

http://dx.doi.org/10.1155/2013/869837

Research Article

Existence Results for Langevin Fractional Differential Inclusions Involving Two Fractional Orders with Four-Point Multiterm Fractional Integral Boundary Conditions

Ahmed Alsaedi,¹ Sotiris K. Ntouyas,² and Bashir Ahmad¹

¹Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

²Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

Received 2 January 2013; Accepted 22 March 2013

Academic Editor: Juan J. Nieto

Copyright © 2013 Ahmed Alsaedi 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 discuss the existence of solutions for Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions. Our study relies on standard fixed point theorems for multivalued maps and covers the cases when the right-hand side of the inclusion has convex as well as nonconvex values. Illustrative examples are also presented.

1. Introduction

We consider a boundary value problem of nonlinear Langevin fractional differential inclusions involving two fractional orders with four-point multiterm fractional integral boundary conditions given by

where , denotes the Caputo fractional derivative of order , is a real number, is the Riemann-Liouville fractional integral of order   , and are suitably chosen constants.

In recent years, the boundary value problems of fractional order differential equations have emerged as an important area of research, since these problems have applications in various disciplines of science and engineering such as mechanics, electricity, chemistry, biology, economics, control theory, signal and image processing, polymer rheology, regular variation in thermodynamics, biophysics, aerodynamics, viscoelasticity and damping, electrodynamics of complex medium, wave propagation, and blood flow phenomena [1–5]. Many researchers have studied the existence theory for nonlinear fractional differential equations with a variety of boundary conditions; for instance, see the papers [6–17] and the references therein.

The Langevin equation (first formulated by Langevin in 1908) is found to be an effective tool to describe the evolution of physical phenomena in fluctuating environments [18]. For some new developments on the fractional Langevin equation, see, for example, [19–26].

The main objective of this paper is to develop the existence theory for a class of problems of the type (1), when the right-hand side is convex as well as nonconvex valued. We establish three existence results: the first result is obtained by means of the nonlinear alternative of Leray-Schauder type; the second one relies on the nonlinear alternative of Leray-Schauder type for single-valued maps together with a selection theorem due to Bressan and Colombo for lower semicontinuous multivalued maps with nonempty closed and decomposable values; and a fixed point theorem due to Covitz and Nadler for contraction multivalued maps is applied to get the third result. The methods used are well known; however their exposition in the framework of problem (1) is new.

The paper is organized as follows: in Section 2, we recall some preliminary facts that we need in the sequel, and Section 3 contains our main results.

2. Preliminaries

2.1. Fractional Calculus

Let us recall some basic definitions of fractional calculus [1–3].

Definition 1. For at least -times differentiable function , the Caputo derivative of fractional order is defined as

where denotes the integer part of the real number .

Definition 2. The Riemann-Liouville fractional integral of order is defined as

provided the integral exists.

Lemma 3. For , the general solution of the fractional differential equation is given by

where   ().

In view of Lemma 3, it follows that

for some   ().

In the following, will denote the space of functions that are absolutely continuous and whose first derivative is absolutely continuous.

Definition 4. A function is called a solution of problem (1) if there exists a function with , a.e. such that , a.e. , and .

Lemma 5. Let . Then the boundary value problem

has a unique solution

where

Proof. As argued in [23], the solution of can be written as

Using the given conditions in (9) together with (8), we find that

where

Solving (10) for and , we find that

Substituting these values in (9), we find the desired solution.

In order to simplify the computations in the main results, we present a technical lemma, concerning the bounds of the operators and defined in the proof of the above lemma.

Lemma 6. One has

Proof. By using the following property of beta function

we have

By a similar way, we have

which completes the proof.

In the following, for convenience, we put

2.2. Background Material for Multivalued Analysis

Now we recall some basic definitions on multivalued maps [27–29].

Let denote a Banach space of continuous functions from into with the norm . Let be the Banach space of measurable functions which are Lebesgue integrable and normed by .

For a normed space , let

A multivalued map

(i) is convex (closed) valued if is convex (closed) for all ;(ii) is bounded on bounded sets if is bounded in for all (i.e., ;(iii) is called upper semicontinuous (u.s.c.) on if, for each , the set is a nonempty closed subset of and if, for each open set of containing , there exists an open neighborhood of such that ;(iv) is lower semicontinuous (l.s.c.) if the set is open for any open set in ;(v) is said to be completely continuous if is relatively compact for every ;(vi) is said to be measurable if, for every , the function

  is measurable;(vii)has a fixed point if there is such that . The fixed point set of the multivalued operator will be denoted by .

Definition 7. A multivalued map is said to be Carathéodory if

(i) is measurable for each ; (ii) is upper semicontinuous for almost all ;

Further a Carathéodory function is called -Carathéodory if

(iii) for each , there exists such that

 for all and for a. e. .

For each , define the set of selections of by

We define the graph of to be the set and recall two useful results regarding closed graphs and upper semicontinuity.

Lemma 8 (see [27, Proposition 1.2]). If is u.s.c., then is a closed subset of ; that is, for every sequence and , if, when , , , and , then . Conversely, if is completely continuous and has a closed graph, then it is upper semicontinuous.

Lemma 9 (see [30]). Let be a Banach space. Let be an -Carathéodory multivalued map, and let be a linear continuous mapping from to . Then the operator

is a closed graph operator in .

We recall the well-known nonlinear alternative of Leray-Schauder for multivalued maps.

Lemma 10 (nonlinear alternative for Kakutani maps [31]). Let be a Banach space, a closed convex subset of an open subset of , and . Suppose that is an upper semicontinuous compact map; here denotes the family of nonempty, compact convex subsets of . Then either

(i) has a fixed point in , or(ii)there is a and with .

Definition 11. Let be a subset of . is measurable if belongs to the -algebra generated by all sets of the form , where is Lebesgue measurable in and is Borel measurable in .

Definition 12. A subset of is decomposable if, for all , and measurable , the function , where stands for the characteristic function of .

Lemma 13 (see [32]). Let be a separable metric space, and let be a lower semicontinuous (l.s.c.) multivalued operator with nonempty closed and decomposable values. Then has a continuous selection; that is, there exists a continuous function (single-valued) such that for every .

Let be a metric space induced from the normed space . Consider given by

where and . Then is a metric space (see [33]).

Definition 14. A multivalued operator is called

(a)-Lipschitz if and only if there exists such that

(b) a contraction if and only if it is -Lipschitz with .

Lemma 15 (see [34]). Let be a complete metric space. If is a contraction, then .

3. Main Results

3.1. The Carathéodory Case

In this section, we are concerned with the existence of solutions for the problem (1) when the right-hand side has convex as well as nonconvex values. Initially, we assume that is a compact and convex valued multivalued map.

Theorem 16. Suppose that

the map is Carathéodory and has nonempty compact and convex values; there exist a continuous nondecreasing function and function such that

  for each ; there exists a number such that

with , where are defined in (17).

Then BVP (1) has at least one solution.

Proof. Let us introduce the operator as

for . We will show that the operator satisfies the assumptions of the nonlinear alternative of Leray-Schauder type. The proof consists of several steps. As a first step, we show that   is convex for each . For that, let . Then there exist such that, for each , we have

for . Let . Then, for each and putting , we have

Since is convex ( has convex values), therefore it follows that .

Next, we show that   maps bounded sets into bounded sets in . For a positive number , let be a bounded set in . Then, for each , there exists such that

Then

Now we show that maps bounded sets into equicontinuous sets of . Let with and , where , as above, is a bounded set of . For each , we obtain

Obviously the right-hand side of the above inequality tends to zero independently of as . As satisfies the above three assumptions, therefore it follows by Ascoli-Arzelá theorem that is completely continuous.

In our next step, we show that has a closed graph. Let , and . Then we need to show that . Associated with , there exists such that, for each ,

Thus we have to show that there exists such that, for each ,

Let us consider the continuous linear operator so that

Observe that

which tends to zero as .

Thus, it follows from Lemma 9 that is a closed graph operator. Further, we have . Since , it follows that

for some .

Finally, we discuss a priori bounds on solutions. Let be a solution of (1). Then there exists with such that, for , we have

Using the computations proving that maps bounded sets into bounded sets and the notations (17), we have

Consequently

In view of , there exists such that . Let us set

Note that the operator is upper semicontinuous and completely continuous. From the choice of , there is no such that for some . Consequently, by the nonlinear alternative of Leray-Schauder type [31], we deduce that has a fixed point which is a solution of the problem (1). This completes the proof.

3.2. The Lower Semicontinuous Case

Next, we study the case where is not necessarily convex valued. Our approach here is based on the nonlinear alternative of Leray-Schauder type combined with the selection theorem of Bressan and Colombo for lower semicontinuous maps with decomposable values.

Theorem 17. Assume that and the following conditions hold:

is a nonempty compact-valued multivalued map such that

(a) is measurable,(b) is lower semicontinuous for each ;

for each , there exists such that

Then the boundary value problem (1) has at least one solution on .

Proof. It follows from and that is of l.s.c. type [35]. Then from Lemma 13, there exists a continuous function such that for all .

Consider the problem

Observe that, if is a solution of (43), then is a solution to the problem (1). In order to transform the problem (43) into a fixed point problem, we define the operator as

It can easily be shown that is continuous and completely continuous. The remaining part of the proof is similar to that of Theorem 16. So we omit it. This completes the proof.

3.3. The Lipschitz Case

Now we prove the existence of solutions for the problem (1) with a nonconvex valued right-hand side by applying a fixed point theorem for multivalued map due to Covitz and Nadler [34].

Theorem 18. Assume that the following conditions hold:

is such that is measurable for each ; for almost all and with and for almost all .

Then the boundary value problem (1) has at least one solution on if , that is,

Proof. Observe that the set is nonempty for each by the assumption , so has a measurable selection (see Theorem [36]). Now we show that the operator satisfies the assumptions of Lemma 15. To show that for each , let be such that in . Then , and there exists such that, for each ,

As has compact values, we pass onto a subsequence to obtain that converges to in . Thus, and, for each ,

Hence, .

Next we show that there exists such that

Let and . Then there exists such that, for each ,

By , we have

So, there exists such that

Define by

Since the multivalued operator is measurable (Proposition [36]), there exists a function which is a measurable selection for . So , and for each , we have .

For each , let us define

Thus,

Hence,

Analogously, interchanging the roles of and , we obtain

Since is a contraction, it follows from Lemma 15 that has a fixed point which is a solution of (1). This completes the proof.

Remark 19. It is important to note that several new interesting special results of the present work can be obtained by fixing the parameters involved in the given problem. Some of these results are listed in the following.

(i) Our results correspond to the multivalued extension of the Dirichlet problem considered in [23] for .(ii) In case we take , we obtain the results for Langevin fractional differential inclusions with the three-point integral boundary conditions of the following form:

(iii) By taking , and , we get the results for Langevin fractional differential inclusions with four-point nonlocal integral boundary conditions of the following type:

  Note that we obtain the typical integral boundary condition in the limit .

Example 20. Consider the problem

where is a multivalued map given by

For , we have

Here ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  and . Clearly,

with . Using the given data, it is found that

Thus,

Clearly, all the conditions of Theorem 16 are satisfied. So there exists at least one solution of the problem (59) on .

Example 21. Consider the fractional inclusion boundary value problem (59) with given by

Then, we have

where . With and (from Example 20), it is found that

Since all the conditions of Theorem 18 are satisfied, therefore the problem (59) with given by (65) has at least one solution on .

Acknowledgments

The authors thank the referees for their useful comments. This work was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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