Abstract and Applied Analysis

Volume 2013 (2013), Article ID 813903, 5 pages

http://dx.doi.org/10.1155/2013/813903

Research Article

Nonlocal Integrodifferential Boundary Value Problem for Nonlinear Fractional Differential Equations on an Unbounded Domain

Lihong Zhang,¹ Bashir Ahmad,² Guotao Wang,¹ Ravi P. Agarwal,2,3 Maryem Al-Yami,² and Wafa Shammakh²

¹School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China

²Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

³Department of Mathematics, Texas A and M University, Kingsville, TX 78363-8202, USA

Received 8 May 2013; Accepted 10 June 2013

Academic Editor: Dumitru Baleanu

Copyright © 2013 Lihong Zhang 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.

This paper investigates the existence of nonnegative solutions for nonlinear fractional differential equations with nonlocal fractional integrodifferential boundary conditions on an unbounded domain by means of Leray-Schauder nonlinear alternative theorem. An example is discussed for the illustration of the main work.

1. Introduction

Recent studies on fractional differential equations, appeared in several special issues and books, reveal an extensive development of various aspects of the subject. One of the reasons for the popularity of fractional calculus is the nonlocal behavior of fractional-order operators in contrast to the classical integer-order operators. This characteristic has motivated many experts on modelling to introduce the concept of fractional modelling by taking into account the ideas of fractional calculus. Examples include various disciplines of science and engineering such as physics, chemistry, biomathematics, dynamical processes in porous media, dynamics of earthquakes, material viscoelastic theory, and control theory of dynamical systems. Furthermore, the outcome of certain experimentations indicate that integral and derivative operators of fractional order possess some characteristics related to complex systems having long memory in time. For details and examples, we refer the reader to the works in [1–7].

Boundary value problems of fractional-order differential equations have been extensively investigated during the last few years, and a variety of results on the topic have been established. A great deal of the work on fractional boundary value problems involves local/nonlocal boundary conditions on bounded and unbounded domains; for example, see [8–26].

In this paper, we study a new class of problems on fractional differential equations with nonlocal boundary conditions on unbounded domains. Precisely, we consider the following problem:

where denotes Riemann-Liouville fractional derivative of order , , and .

2. Preliminaries

In this section, we present some useful definitions and related theorems.

Definition 1 (see [4]). The Riemann-Liouville fractional derivative of order for a continuous function is defined by

Provided that the right hand side is pointwise defined on and is the integer part of .

Definition 2 (see [4]). The Riemann-Liouville fractional integral of order for a function is defined as

provided that such integral exists.

Theorem 3 (see [27) (Leray-Schauder  nonlinear alternative). Let be a convex subset of a Banach space, and let be an open subset of with . Then every completely continuous map has at least one of the following two properties:

(1)has a fixed point in ;(2)there is an and with .

Theorem 4 (see [28]). Let be a bounded set. Then is relatively compact in if the following conditions hold:

(i)for any , is equicontinuous on any compact interval of ;(ii)for any , there exists a constant such that

for any and .

Now we list the assumptions needed in the sequel.

: . : there exist nonnegative functions , defined on and a constant such that

3. Some Lemmas

This section contains some preliminary works that we need to establish the main result for problem (1).

Lemma 5. Let with . For , the associated linear fractional boundary value problem,

has a unique solution given by

where

Proof. It is well known that the fractional equation in (6) is equivalent to the integral equation:

where are arbitrary constants. From (10), we have

Using the given boundary conditions in (10), we find that and

where

Substituting the values of , into (10) gives

where is defined by (8).

Remark 6. In view of the assumption , Green's function satisfies the properties:

(1), (2)

For the forthcoming analysis, we introduce a space

equipped with the norm

Notice that is a Banach space.

Define an operator as follows:

Observe that problem (1) has a solution only if the operator has a fixed point.

Lemma 7. If , hold, then the operator is completely continuous.

Proof. We divide the proof into several steps.

(i) The operator is uniformly bounded. Let be any bounded subset of ; then there exists a constant such that . By , we have

This shows that is uniformly bounded.

(ii) is continuous. Take such that , , and as . Then, by , we have

where is defined by (15).

By the Lebesgue dominated convergence theorem and continuity of , we obtain

Taking the limit , we get

Therefore, is continuous.

(iii) is equicontinuous. We consider two cases.

(a) Let be any compact interval, and let be such that . Let be any bounded subset of ; then for any , we have

Since is continuous on , we have that is a uniformly continuous function on the compact set . Moreover, for , we have that this function only depends on ; in consequence it is uniformly continuous on . So we have that for all and , the following property holds.

For all there is such that if , then . By this, together with (23), and the fact that

we can get that is equicontinuous on .

(b) In fact, when , we have

From this, it is not difficult to verify that for any given , there exists a constant such that

for any and . Hence, is equiconvergent at .

Thus the conclusion of Theorem 4 applies that is relatively compact on . So, is completely continuous. This completes the proof.

4. Main Results

Theorem 8. Assume that and with hold. If there exists such that

with given by (15), then problem (1) has a solution satisfying

Proof. Let . For , if there exist such that , then we have

This implies that

which contradicts (27). By Lemma 7 and Theorem 3, we conclude that problem (1) has a solution satisfying

This completes the proof.

In the next, we formulate existence results for the cases and . We do not provide the proof of these results as it is similar to that of Theorem 8. For that, we denote with and , respectively, by and .

Theorem 9. Let the assumptions and hold. Then problem (1) has a solution satisfying

where with given by (15).

Theorem 10. Suppose that and hold and that there exists with given by (15). Then problem (1) has a solution such that

5. Example

Example 1. With , , and , we consider the following boundary value problem:

Clearly the condition holds as , . Letting , , we find that

This shows that holds true. Finally, fixing , it can easily be verified that the condition (27) is satisfied. Thus all the conditions of Theorem 8 are satisfied. Therefore, by Theorem 8, problem (27) has a solution such that

Acknowledgments

The work was supported by the Natural Science Foundation for Young Scientists of Shanxi Province, China (no. 2012021002-3). This paper was funded by King Abdulaziz University, under Grant no. (130-1-1433/HiCi). The authors, therefore, acknowledge technical and financial support of KAU. The authors also thank the reviewers for their useful comments.

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