Abstract and Applied Analysis
Volume 2015 (2015), Article ID 205452, 10 pages
http://dx.doi.org/10.1155/2015/205452

Research Article

New Existence Results for Fractional Integrodifferential Equations with Nonlocal Integral Boundary Conditions

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

¹Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department 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 15 June 2014; Accepted 14 October 2014

Academic Editor: Erdal Karapinar

Copyright © 2015 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 consider a boundary value problem of fractional integrodifferential equations with new nonlocal integral boundary conditions of the form: , and . According to these conditions, the value of the unknown function at the left end point is proportional to its value at a nonlocal point while the value at an arbitrary (local) point is proportional to the contribution due to a substrip of arbitrary length . These conditions appear in the mathematical modelling of physical problems when different parts (nonlocal points and substrips of arbitrary length) of the domain are involved in the input data for the process under consideration. We discuss the existence of solutions for the given problem by means of the Sadovski fixed point theorem for condensing maps and a fixed point theorem due to O’Regan. Some illustrative examples are also presented.

1. Introduction

We consider a boundary value problem of fractional differential equations with nonlocal integral boundary conditions given by

where denotes the Caputo fractional derivative of order , is a given continuous function, , , and , , , are real constants.

Here we remark that the boundary conditions introduced in the problem (1) are of nonlocal strip type and describe the situation when the receptors at the end points of the boundary are influenced by the nonlocal contributions due to interior points and strips of the domain for the problem. For practical examples, see [1, 2]. The problem (1) can also be termed as a five-point nonlocal fractional boundary value problem.

In recent years, several aspects of fractional boundary value problems, ranging from theoretical analysis to numerical simulation, have been investigated. The nonlocal nature of fractional order differential operators has significantly contributed to the popularity and development of the subject. As a matter of fact, this characteristic of such operators help to understand the memory and hereditary properties of many useful materials and processes. For details and applications of fractional differential equations in physical and technical sciences such as biology, physics, biophysics, chemistry, statistics, economics, blood flow phenomena, control theory, and signal and image processing, see [3–6]. For some recent works on nonlocal fractional boundary value problems, we refer the reader to the papers [7–11], while the results based on monotone method for such problems can be found in [12, 13]. In [14], the limit properties of positive solutions of fractional boundary value problems have been discussed. Fractional differential inclusions supplemented with different kinds of boundary conditions have also been studied by several researchers, for instance, see [15–20].

The paper is organized as follows. In Section 2, we recall some basic definitions from fractional calculus and establish an auxiliary lemma which plays a pivotal role in the sequel. Section 3.1 contains an existence result for the problem (1) which is established by applying Sadovskii’s fixed point theorem for condensing maps. In Section 3.2, we show the existence of solutions for the problem (1) by means of a fixed point theorem due to O’Regan.

2. Preliminaries

In this section, some basic definitions on fractional calculus and an auxiliary lemma are presented [3, 4].

Definition 1. The Riemann-Liouville fractional integral of order for a continuous function is defined as

provided the integral exists.

Definition 2. For at least -times continuously differentiable function , the Caputo derivative of fractional order is defined as

where denotes the integer part of the real number .

Lemma 3. For any the unique solution of the linear fractional boundary value problem

is

where

Proof. It is well known that the general solution of the fractional differential equation in (4) can be written as

where are arbitrary constants.
Applying the given boundary conditions, we obtain the following system

from which we get

Substituting the values of in (7), we get (5). This completes the proof.

3. Existence Results

We denote by the Banach space of all continuous functions from endowed with the norm defined by . Also by we denote the Banach space of measurable functions which are Lebesgue integrable and normed by .

In the following we will give two existence results for the problem (1), one with the help of Sadovskii’s fixed point theorem and the other based on a fixed point theorem due to O’Regan in [21].

3.1. Existence Results via Sadovskii’s Fixed Point Theorem

Definition 4. Let be a bounded set in metric space ; then Kuratowskii measure of noncompactness, , is defined as covered by finitely many sets such that the diameter of each set .

Definition 5 (see [22]). Let be a bounded and continuous operator on Banach space . Then is called a condensing map if for all bounded sets , where denotes the Kuratowski measure of noncompactness.

Lemma 6 (see [23, Example 11.7]). The map is a -set contraction with and is thus condensing, if

(i) are operators on the Banach space ;(ii) is -contractive; that is,

for all and fixed ;(iii) is compact.

Theorem 7 (see [24]). Let be a convex, bounded, and closed subset of a Banach space and let be a condensing map. Then has a fixed point.

In view of Lemma 3, we define an operator by

where

Theorem 8. Let be continuous functions satisfying the following conditions.

(H ₁) satisfies the Lipschitz condition:

(H ₂)there exist a function and a nondecreasing function such that

Then the boundary value problem (1) has at least one solution provided that

Proof. Let be a closed bounded and convex subset of , where will be fixed later. We define a map as

where and are defined by (12) and (13), respectively. Notice that the problem (1) is equivalent to a fixed point problem .
Step 1   . For that, we set and select , where

Using , for , we get

where we have used the following relations:

Consequently

which implies that .
Step 2 ( is continuous and -contractive). To show the continuity of for , let us consider a sequence converging to . Then, by the assumption , we have

Next, we show that is -contractive. For , we get

By the given assumption

it follows that is -contractive.
Step 3 ( is compact). In Step 1, it has been shown that is uniformly bounded. Now we show that maps bounded sets into equicontinuous sets of . Let with and . Then we obtain

Obviously the right hand side of the above inequality tends to zero independently of as . Therefore it follows by the Arzelá-Ascoli theorem that is completely continuous. Thus is compact on .
Step 4 ( is condensing). Since is continuous, -contractive and is compact, by Lemma 6, with is a condensing map on .
Consequently, by Theorem 7, the map has a fixed point which, in turn, implies that the problem (1) has a solution.

Example 9. Consider a nonlocal integral boundary value problem of fractional integrodifferential equations given by

where , and .

Clearly as , and with and . Furthermore , and the condition (16) yields . Thus all the conditions of Theorem 8 are satisfied and consequently the problem (26) has a solution.

3.2. Existence Results via O’Regan’s Fixed Point Theorem

Our next existence result relies on a fixed point theorem due to O’Regan in [21].

Lemma 10. Denote by an open set in a closed, convex set of a Banach space . Assume . Also assume that is bounded and that is given by , in which is continuous and completely continuous and is a nonlinear contraction (i.e., there exists a nonnegative nondecreasing function satisfying for , such that for all ). Then, either

(C1) has a fixed point , or(C2)there exist a point and with , where and , respectively, represent the closure and boundary of  .

For convenience we set

Let

and denote the maximum number by

Theorem 11. Let be continuous functions. Assume that

(A ₁)there exist a nonnegative function and a nondecreasing function such that

(A ₂)there exist a positive constant and a continuous function such that and for all and ;(A ₃), where .

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

Proof. By the assumption (A₃), there exists a number such that

We will show that the operators and defined by (12) and (13), respectively, satisfy all the conditions of Lemma 10. The proof consists of a series of steps.
Step 1 (the operator is continuous and completely continuous). We first show that is bounded. For any , we have

Thus the operator is uniformly bounded. For any , we have

which is independent of and tends to zero as . Thus, is equicontinuous. Hence, by the Arzelá-Ascoli theorem, is a relatively compact set. Now, let with . Then the limit is uniformly valid on . From the uniform continuity of on the compact set , it follows that is uniformly valid on . Hence as . This shows the continuity of .
Step 2 (the operator is contractive). Consider

This, together with (A₂), implies that

so is a nonlinear contraction.
Step 3 (the set is bounded). Using the inequality

we have

for any . This, with the boundedness of the set , implies that the set is bounded.
Step 4 (finally, it will be shown that the case (C2) in Lemma 10 does not hold). On the contrary, we suppose that (C2) holds. Then, we have that there exist and such that . So, we have and

Using the assumptions (A₁) and (A₂), we get

which leads to a contradiction:

Thus the operators and satisfy all the conditions of Lemma 10. Hence, the operator has at least one fixed point , which is the solution of the problem (1). This completes the proof.

Example 12. Consider a nonlocal integral boundary value problem of fractional integrodifferential equations given by

where , and .

Observe that , and . Further, we set , and . With the given data, it is found that , and

Clearly, all the conditions of Theorem 11 are satisfied and hence there exists a solution for the problem (41).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

This article was funded by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia. The authors, therefore, acknowledge technical and financial support of KAU.

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