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`Abstract and Applied AnalysisVolume 2014, Article ID 902054, 9 pageshttp://dx.doi.org/10.1155/2014/902054`
Research Article

Existence and Uniqueness Results for Hadamard-Type Fractional Differential Equations with Nonlocal Fractional Integral Boundary Conditions

1Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok 10800, Thailand

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

Received 1 April 2014; Accepted 10 June 2014; Published 26 June 2014

Copyright © 2014 Phollakrit Thiramanus 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 study the existence and uniqueness of solutions for a fractional boundary value problem involving Hadamard-type fractional differential equations and nonlocal fractional integral boundary conditions. Our results are based on some classical fixed point theorems. Some illustrative examples are also included.

1. Introduction

In this paper, we investigate the following Hadamard boundary value problem: where denotes the Hadamard fractional derivative of order is a continuous function, ,  , for all ,  ,  ,  , and is the Hadamard fractional integral of order .

We mention that integral boundary conditions are encountered in various applications such as population dynamics, blood flow models, chemical engineering, cellular systems, heat transmission, plasma physics, and thermoelasticity.

Condition (2) is a general form of the integral boundary conditions considered in [1] and covers many special cases. For example, if , for all and  , then condition (2) reduces to

Fractional differential equations provide appropriate models for describing real world problems, which cannot be described using classical integer order differential equations. The theory of fractional differential equations has received much attention over the past years and has become an important field of investigation due to its extensive applications in numerous branches of physics, economics, and engineering sciences [25]. Some recent contributions to the subject can be seen in [1, 620] and references cited therein.

It has been noticed that most of the work on this topic is based on Riemann-Liouville and Caputo type fractional differential equations. Another kind of fractional derivatives that appears side by side to Riemann-Liouville and Caputo derivatives in the literature is the fractional derivative due to Hadamard introduced in 1892 [21], which differs from the preceding ones in the sense that the kernel of the integral (in the definition of Hadamard derivative) contains logarithmic function of arbitrary exponent. Details and properties of Hadamard fractional derivative and integral can be found in [2, 2226]. For some recent results on Hadamard boundary value problem we refer to [27, 28].

We establish a variety of results for the problem (1)-(2) by using classical fixed point theorems. The first result, Theorem 4, relies on Banach contraction mapping principle and concerns an existence and uniqueness result for the solutions of the problem (1)-(2). A second existence and uniqueness result is proved in Theorem 7, via nonlinear contractions and a fixed point theorem due to Boyd and Wong. Existence results are proved in the third result, Theorem 9, by using Krasnoselskii fixed point theorem, and in the fourth result, Theorem 12, by using nonlinear alternative of Leray-Schauder type.

The paper is organized as follows. In Section 2, we recall some preliminary concepts that we need in the sequel and prove a preliminary lemma. Section 3 contains the main results for the problem (1)-(2). In Section 4, some illustrative examples are discussed.

2. Preliminaries

In this section, we introduce some notations and definitions of fractional calculus [2] and present preliminary results needed in our proofs later.

Definition 1. The Hadamard derivative of fractional order for a function is defined as where denotes the integer part of the real number , , and is the Gamma function.

Definition 2. The Hadamard fractional integral of order for a function is defined by provided the integral exists.

For convenience, we set

Lemma 3. Let , , , and for , , and . The unique solution of the following fractional differential equation, subject to the boundary condition, is given by the integral equation

Proof. Applying the Hadamard fractional integral of order to both sides of (7), we have where .

The condition of implies . Therefore, For any , by Definition 2, it follows that The second condition of (8) with (12) leads to Substituting the value of a constant into (11), we obtain (9) as required. The proof is completed.

3. Main Results

Let denote the Banach space of all continuous functions from to endowed with the norm defined by . As in Lemma 3, we define an operator by with . It should be noticed that problem (1)-(2) has solutions if and only if the operator has fixed points.

For the sake of convenience, we put

The first existence and uniqueness result is based on the Banach contraction mapping principle.

Theorem 4. Let be a continuous function satisfying the assumption that (H1)there exists a constant such that , for each and . If where is given by (15), then the boundary value problem (1)-(2) has a unique solution on .

Proof. We transform the problem (1)-(2) into a fixed point problem,  , where the operator is defined by (14). By using Banach’s contraction mapping principle, we will show that has a fixed point which is a unique solution of problem (1)-(2).

We set and choose

Now, we show that , where . For any , we have It follows that .

For and for each , we have The above result implies that . As , therefore is a contraction. Hence, by the Banach contraction mapping principle, we deduce that has a fixed point which is the unique solution of the problem (1)-(2).

Next, we give the second existence and uniqueness result by using nonlinear contractions.

Definition 5. Let be a Banach space and let be a mapping. is said to be a nonlinear contraction if there exists a continuous nondecreasing function such that and for all with the property

Lemma 6 (see [29]). Let be a Banach space and let be a nonlinear contraction. Then has a unique fixed point in .

Theorem 7. Let be a continuous function satisfying the assumption (H2),  ,  ,  where   is continuous and a constant is defined by Then the boundary value problem (1)-(2) has a unique solution.

Proof. We define the operator as (14) and a continuous nondecreasing function by Note that the function satisfies and for all .

For any and for each , we have This implies that . Therefore is a nonlinear contraction. Hence, by Lemma 6 the operator has a fixed point which is the unique solution of the problem (1)-(2).

Next, we give an existence result by using Krasnoselskii’s fixed point theorem.

Lemma 8 (Krasnoselskii’s fixed point theorem [30]). Let M be a closed, bounded, convex, and nonempty subset of a Banach space X. Let A, B be the operators such that (a) , whenever ; (b) A is compact and continuous; (c) B is a contraction mapping. Then there exists such that z = Az + Bz.

Theorem 9. Assume that is a continuous function satisfying assumption . In addition we suppose that (H3), and . If then the boundary value problem (1)-(2) has at least one solution on .

Proof. We define and choose a suitable constant as where is defined by (15). Furthermore, we define the operators and on as For , we have This shows that . It follows from assumption together with (24) that is a contraction mapping. Since the function is continuous, we have that the operator is continuous. It is easy to verify that Therefore, is uniformly bounded on .

Next, we prove the compactness of the operator . Let us set ; consequently we get which is independent of and tends to zero as . Thus, is equicontinuous. So is relatively compact on . Hence, by the Arzelá-Ascoli theorem, is compact on . Thus, all the assumptions of Lemma 8 are satisfied. So the boundary value problem (1)-(2) has at least one solution on . The proof is completed.

Remark 10. In the above theorem we can interchange the roles of the operators and to obtain a second result replacing (24) by the following condition:

Now, our last existence result is based on Leray-Schauder’s nonlinear alternative.

Theorem 11 (nonlinear alternative for single-valued maps [31]). Let E be a Banach space, C a closed, convex subset of  E, U an open subset of  C, and  . Suppose that is a continuous, compact (i.e., is a relatively compact subset of C) map. Then either (i)F has a fixed point in or(ii)there is a (the boundary of U in C) and , with .

Theorem 12. Assume that is a continuous function. In addition we suppose that(H4)there exists a continuous nondecreasing function and a function such that (H5)there exists a constant such that where is defined by (15). Then the boundary value problem (1)-(2) has at least one solution on .

Proof. Firstly, we will show that the operator , defined by (14), maps bounded sets (balls) into bounded sets in . For a positive number , let be a bounded ball in . Then for , we have Therefore, we conclude that .

Secondly, we show that maps bounded sets into equicontinuous sets of . Let , with and . Then we have Obviously, the right hand side of the above inequality tends to zero independently of as . Therefore it follows from the Arzelá-Ascoli theorem that is completely continuous.

Let be a solution. Then, for , following the similar computations as in the first step, we have Consequently, we have In view of (H5), there exists such that . Let us set Note that the operator is continuous and completely continuous. From the choice of , there is no such that for some . Consequently, by nonlinear alternative of Leray-Schauder type (Theorem 11) we deduce that has a fixed point in , which is a solution of the boundary value problem (1)-(2). This completes the proof.

4. Examples

Example 1. Consider the following boundary value problem for Hadamard fractional differential equation:

Here  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  and  . Since then (H1) is satisfied with . We can show that Hence, by Theorem 4, the boundary value problem (38) has a unique solution on .

Example 2. Consider the following boundary value problem for Hadamard fractional differential equation:

Here ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  and  . We choose and that Clearly, Hence, by Theorem 7, the boundary value problem (41) has a unique solution on .

Example 3. Consider the following boundary value problem for Hadamard fractional differential equation:

Here ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  ,  and  . Clearly, Choosing and , we can show that which implies that . Hence, by Theorem 12, the boundary value problem (44) has at least one solution on .

Conflict of Interests

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

Acknowledgments

The research of Phollakrit Thiramanus is supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand. The research of Jessada Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand. Sotiris K. Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

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