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Abstract and Applied Analysis

Volume 2014 (2014), Article ID 207547, 12 pages

http://dx.doi.org/10.1155/2014/207547

## Existence Theory for -Antiperiodic Boundary Value Problems of Sequential -Fractional Integrodifferential Equations

^{1}Department of Mathematics, Texas A&M University, Kingsville, TX 78363-8202, USA^{2}Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 21 February 2014; Accepted 3 April 2014; Published 30 April 2014

Academic Editor: Erdal Karapinar

Copyright © 2014 Ravi P. Agarwal 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 and uniqueness of solutions for a new class of sequential -fractional integrodifferential equations with -antiperiodic boundary conditions. Our results rely on the standard tools of fixed-point theory such as Krasnoselskii's fixed-point theorem, Leray-Schauder nonlinear alternative, and Banach's contraction principle. An illustrative example is also presented.

#### 1. Introduction

We consider a-antiperiodic boundary value problem of sequential-fractional integrodifferential equations given by where and denote the fractional-derivative of the Caputo type, , denotes Riemann-Liouville integral withbeing given continuous functions, andbeing real constants.

The aim of the present study is to establish some existence and uniqueness results for the problem (1) by means of Krasnoselskii’s fixed-point theorem, Leray-Schauder nonlinear alternative, and Banach’s contraction principle. Though the tools employed in this work are standard, yet their exposition in the framework of the given problem is new.

Fractional calculus has developed into a popular mathematical modelling tool for many real world phenomena occurring in physical and technical sciences, see, for example, [1–4]. A fractional-order differential operator distinguishes itself from an integer-order differential operator in the sense that it is nonlocal in nature and can describe the memory and hereditary properties of some important and useful materials and processes. This feature has fascinated many researchers and several results ranging from theoretical analysis to asymptotic behavior and numerical methods for fractional differential equations have been established. For some recent work on the topic, see [5–12] and references therein.

The mathematical modeling of linear control systems, concerning the controllability of systems consisting of a set of well-defined interconnected objects, is based on the linear systems of divided difference functional equations. The controllability in mathematical control theory studies the concepts such as controllability of the state, controllability of the output, controllability at the origin, and complete controllability. The-difference equations play a key role in the control theory as these equations are always completely controllable and appear in the-optimal control problem [13]. The variational-calculus is known as a generalization of the continuous variational calculus due to the presence of an extra-parameterwhose nature may be physical or economical. The study of the-uniform lattice rely on the-Euler equations. In other words, it suffices to solve the-Euler-Lagrange equation for finding the extremum of the functional involved instead of solving the Euler-Lagrange equation [14]. One can find more details in a series of papers [15–21].

The subject of fractional-difference (-fractional) equations is regarded as fractional analogue of-difference equations and has recently gained a considerable attention. For examples and details, we refer the reader to the works [22–33] and references therein while some earlier work on the subject can be found in [34–36]. The present work is motivated by recent interest in the study of fractional-order differential equations.

#### 2. Preliminaries on Fractional -Calculus

Let us describe the notations and terminology for-fractional calculus [35].

For a real parameter, a-real number denoted byis defined by

The-analogue of the Pochhammer symbol (-shifted factorial) is defined as

The-analogue of the exponent is

The-gamma functionis defined as where . Observe that .

*Definition 1 (see [35]). *Letbe a function defined onThe fractional -integral of the Riemann-Liouville type of orderisand

Observe that the above-integral reduces to the following one for. Further details of-integrals and fractional-integrals can be found respectively in Section 1.3 and Section 4.2 of the text [35].

*Remark 2. *The semigroup property holds for-fractional integration (Proposition 4.3 [35]):
Further, it has been shown in Lemma 6 of [37] that

Before giving the definition of fractional-derivative, we recall the concept of-derivative.

Letbe a real valued function defined on a-geometric set(). Then the-derivative of a function is defined as For, the-derivative at zero is defined forby Provided that the limit exists and does not depend on.

Furthermore,

*Definition 3 (see [35]). *The Caputo fractional-derivative of orderis defined by
where is the smallest integer greater than or equal to

Next we enlist some properties involving Riemann-Liouville-fractional integral and Caputo fractional-derivative (Theorem 5.2 [35]):

Now we establish a lemma that plays a key role in the sequel.

Lemma 4. *For a given, the boundary value problem
**
is equivalent to the-integral equation
*

*Proof. *It is well known that the solution of-fractional equation in (15) can be written as
Differentiating (17), we obtain

Using the boundary conditions (15) in (17) and (18) and solving the resulting expressions forand , we get

Substituting the values ofandin (17) yields the solution (16). The converse follows in a straightforward manner. This completes the proof.

Let denote the Banach space of all continuous functions frominto endowed with the usual norm defined by.

In view of Lemma 4, we define an operator as

Observe that the problem (1) has solutions only if the operator equationhas fixed points.

#### 3. Main Results

For the forthcoming analysis, the following conditions are assumed. are continuous functions such thatand, for all .There existwith, for all , where. For computational convenience, we set

Our first existence result is based on Krasnoselskii’s fixed point theorem.

Lemma 5 (see, Krasnoselskii [38]). *Letbe a closed, convex, bounded, and nonempty subset of a Banach spaceLetbe the operators such that (i)whenever; (ii)is compact and continuous; and (iii)is a contraction mapping. Then there existssuch that*

Theorem 6. *Let be continuous functions satisfying .Furthermore, whereis given by (22) andThen the problem (1) has at least one solution on.*

*Proof. *Consider the set, whereis given by

Define operatorsandonas
For, we find that

Thus,Continuity ofandimply that the operator is continuous. Also,is uniformly bounded onas

Now, we prove the compactness of the operatorIn view of , we define
Consequently, for, we have
which is independent ofand tends to zero as. Thus,is relatively compact on. Hence, by the Arzelá-Ascoli Theorem,is compact onNow, we shall show thatis a contraction.

From and for, we have
where we have used (22). In view of the assumption, the operatoris a contraction. Thus, all the conditions of Lemma 5 are satisfied. Hence, by the conclusion of Lemma 5, the problem (1) has at least one solution on.

Our next result is based on Leray-Schauder nonlinear alternative.

Lemma 7 (nonlinear alternative for single valued maps, see [39]). *Letbe a Banach space,a closed, convex subset of an open subset of, andSuppose that is a continuous, compact is a relatively compact subset of) map. Then either *(i)* has a fixed point in , or*(ii)*there is a(the boundary ofin) andwith*

Theorem 8. *Let be continuous functions and the following assumptions hold: **there exist functions, and nondecreasing functionssuch that for all **there exists a constantsuch that
**
Then the boundary value problem (1) has at least one solution on*

*Proof. *Consider the operator defined by (20). The proof consists of several steps. (i)It is easy to show that is continuous.(ii) maps bounded sets into bounded sets in .

For a positive number, let be a bounded set in and. Then, we have

This shows that .(iii) maps bounded sets into equicontinuous sets of .

Let with and , where is a bounded set of . 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.(iv)Letbe a solution of the given problem such thatforThen, for, it follows by the procedure used to establish (ii) that
Consequently, we have
In view of, there existssuch that. Let us set
Note that the operator is continuous and completely continuous. From the choice of, there is nosuch thatfor some. In consequence, by the nonlinear alternative of Leray-Schauder type (Lemma 7), we deduce that has a fixed pointwhich is a solution of the problem (1). This completes the proof.

Finally we show the existence of a unique solution of the given problem by applying Banach’s contraction mapping principle (Banach fixed-point theorem).

Theorem 9. *Suppose that the assumption holds and
**
where, andare given by (21) and. Then the boundary value problem (1) has a unique solution.*

*Proof. *Fix , where are finite numbers given by. Selecting , we show that , where . For , we have

This shows that For , we obtain
Sinceby the given assumption, therefore is a contraction. Hence, it follows by Banach’s contraction principle that the problem (1) has a unique solution.

*Example 10. *Consider a-fractional integrodifferential equation with-antiperiodic boundary conditions given by
where , +, With the given data,and

Clearly,, and the condition implies that . Thus all the assumptions of Theorem 8 are satisfied. Hence, the conclusion of Theorem 8 applies to the problem (39).

*Example 11. *Consider the following-fractional-antiperiodic boundary value problem:
where , With the given data, it is found thatas , Clearly . Moreover, and . Using the given values, it is found that. Thus all the assumptions of Theorem 9 are satisfied. Hence, by the conclusion of Theorem 9, there exists a unique 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 work was partially supported by Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.

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