`Abstract and Applied AnalysisVolume 2011, Article ID 986575, 20 pageshttp://dx.doi.org/10.1155/2011/986575`
Research Article

Existence of Positive Solutions for m-Point Boundary Value Problem for Nonlinear Fractional Differential Equation

1Department of Mathematics, Faculty of Arts and Sciences, College of Education, P.O. Box 3771, Qassim-Unizah 51911, Saudi Arabia
2Department of Mathematics, Sciences Faculty for Girls, King Abdulaziz University, P.O. Box 30903, Jeddah 21487, Saudi Arabia

Received 23 January 2011; Revised 2 April 2011; Accepted 17 April 2011

Copyright © 2011 Moustafa El-Shahed and Wafa M. Shammakh. 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 investigate an m-point boundary value problem for nonlinear fractional differential equations. The associated Green function for the boundary value problem is given at first, and some useful properties of the Green function are obtained. By using the fixed point theorems of cone expansion and compression of norm type and Leggett-Williams fixed point theorem, the existence of multiple positive solutions is obtained.

1. Introduction

In recent years, the existence of positive solutions multipoint boundary value problems of fractional order differential equations has been studied by many authors using various methods (see [17]).

The study of multipoint boundary value problems for linear second-order ordinary differential equations was initiated by II'in and Moiseev [8, 9].

Since then, nonlinear multipoint boundary value problems have been studied by several authors (see [1014]). Recently, in [15], the authors have studied the existence of at least one positive solution for the following th-order three-point boundary value problem: where and may be singular at and .

Goodrich [16] considered the BVP for thehigher-dimensional fractional differential equation as follows: and a Harnack-like inequality associated with the Green function related to the above problem is obtained improving the results in [17].

Motivated by the aforementioned results and techniques in coping with those boundary value problems of fractional differential equations, we then turn to investigate the existence and multiplicity of positive solutions for the following BVP: where and are the Caputo fractional derivative.

In this paper, we study the existence of at least one positive solution, existence of two positive solutions associated with the BVP (1.3)-(1.4) by applying the fixed point theorems of cone expansion and compression of norm type, and the existence of at least three positive solutions for BVP (1.3)-(1.4) by using Leggett-Williams fixed point theorem.

The rest of the paper is organized as follows. In Section 2, we introduce some basic definitions and preliminaries used later. In Section 3, the existence of multipoint boundary value problem (1.3)-(1.4) will be discussed.

2. Preliminaries

In this section, we introduce definitions and preliminary facts which are used throughout this paper.

Definition 2.1 (see [18]). For a function , the Caputo derivative of fractional order is defined as

Definition 2.2 (see [18]). The standard Riemann-Liouville fractional derivative of order of a continuous function is given by where , provided that the integral on the right-hand side converges.

Definition 2.3 (see [18]). The Riemann-Liouville fractional integral of order of a function is given by provided that the integral on the right-hand side converges.

Definition 2.4 (see [19]). Let be a real Banach space. A nonempty closed convex set is called cone of if it satisfies the following conditions:(1); (2).

Definition 2.5. An operator is called completely continuous if it is continuous and maps bounded sets into precompact sets.

Theorem 2.6 (see [20]). Let be a Banach space and is a cone in . Assume that and are open subsets of with and . Let be completely continuous operator. In addition, suppose either (i)(ii)holds. Then, has a fixed point in .

Lemma 2.7. For , the general solution of the fractional differential equation is given by where .

Remark 2.8 (see [18]). In view of Lemma 2.7, it follows that for some .

Definition 2.9. The map is said to be a nonnegative continuous concave functional on a cone of a real Banach space provided that is continuous and for all .

Lemma 2.10 (see [21]). Let be a cone in a real Banach space is a nonnegative continuous concave functional on such that , for all , and . Suppose that is completely continuous and there exist positive constants such that (1), (2), (3), then has at least three fixed points , and with

Lemma 2.11. For a given and , the unique solution of the boundary value problem is given by where

Proof. Using Remark 2.8, for arbitrary constants , we have In view of the relations and for , we obtain Applying the boundary conditions (2.9), we find that then , and Substituting the values of the constants , in (2.12), we obtain

Lemma 2.12. , for .

Proof.

Lemma 2.13. The functions defined by (2.11) satisfy (i), (ii), for all ,(iii), where(iv), where

Proof. It is clear that (i) holds. So, we prove that (ii) is true.
(ii) For , in view of the expression for , it follows that for all , where .
If , we have If , then we have Thus Therefore, (iii) If , then If , then we have
(iv) Since , then is nonincreasing in , so

3. Main Results

Let us denote by the Banach space of all continuous real functions on endowed with the norm and the cone where , since , are constants do not depend on .

Let the nonnegative continuous concave functional on the cone be defined by .

Set by where are defined as in Lemma 2.11.

From (3.2) and Lemma 2.13, we have Hence, we have .

By standard argument, one can prove that is a completely continuous operator.

The Existence of One Positive Solution
We introduce the following definitions:
Theorem 3.1. Let be continuous on . If there exist two positive constants such that , , then the BVP (1.3)-(1.4) has at least a positive solution.Proof. We know that the operator defined by (3.2) is completely continuous.
(a) Let . For any , we have which implies that for every : which implies that
(b) Let . For any . We have which implies that In view of Theorem 2.6, has a fixed point which is a solution of the BVP (1.3)-(1.4).

The Existence of Two Positive Solutions
Theorem 3.2. Assume that all assumptions of Theorem 3.1, hold. Moreover, one assumes that also satisfies . Then, the BVP (1.3)-(1.4) has at least two positive solutions.Proof. At first, it follows from condition that Further, it follows from condition that Finally, since , there exists and such that Let and set , then implies, Therefore, we have Thus, from (3.6), (3.8), (3.13), and Theorem 2.6, has a fixed point , in and a fixed point , in . Both are positive solutions of BVP (1.3)-(1.4) and satisfy Theorem 3.3. Assume that be continuous on . If the following assumptions hold: , , there exists a constant such that ,then the BVP (1.3)-(1.4) has at least two positive solutions and such that Proof. At first, it follows from condition that we may choose such that where is defined as in Theorem 3.2. Set , and ; from (3.2) and Lemma 2.13, for , we have Therefore, we have Further, it follows from condition that there exists such that Let , set , then implies, Therefore, we have Finally, let and . By condition , we have which implies Thus, from (3.18), (3.21), (3.23), and Theorem 2.6, has a fixed point in and a fixed point , in . Both are positive solutions of BVP (1.3)-(1.4) and satisfy Theorem 3.4. Assume that be continuous on . If the following assumptions hold: (), (), ()there exists a constant such that ,then the BVP (1.3)-(1.4) has at least two positive solutions and such that The proof of Theorem 3.4 is very similar to that of Theorem 3.3 and therefore is omitted.Theorem 3.5. Assume that be continuous on . If the following assumptions hold: (), (), ()there exists a constant such that ,then the BVP (1.3)-(1.4) has at least two positive solutions and such that Proof. It follows from condition that we may choose such that Set , and ; from (3.2) and Lemma 2.13, for , we have Therefore, we have It follows from condition that there exists such that and we consider two cases.Case 1. Suppose that is unbounded, there exists such that for .Then, for and , we have Case 2. If is bounded, that is, for all , taking , for and , then we have Hence, in either case, we always may set such that Finally, set , then and and by condition and (3.2), we have Hence, we have Thus, from (3.29), (3.33), (3.36) and Theorem 2.6, has a fixed point in and a fixed point in .Both are positive solutions of BVP (1.3)-(1.4) and satisfy Theorem 3.6. Assume that be continuous on . If the following assumptions hold:, ,there exists a constant such that ,then the BVP (1.3)-(1.4) has at least two positive solutions and such that The proof of Theorem 3.6 is very similar to that of Theorem 3.5 and therefore omitted.

The Existence of Three Positive Solutions
Theorem 3.7. Let be continuous on . If there exist constants such that the following assumptions (i), (ii), (iii), hold, then BVP (1.3)-(1.4) has at least three positive solutions , and with Proof. We will show that all conditions of Lemma 2.10, are satisfied.
First, if , then . So, .
By condition (ii), we have which implies that . Hence .
Next, by using the analogous argument, it follows from condition (i) that if , then .
Choose , it is easy to see that .
Therefore, . On the other hand, if , then . By condition (iii), we have .
Hence, which implies that , for .
Finally, if and , then Thus, all the conditions of the Leggett-Williams fixed point theorem are satisfied by taking . Hence, the BVPs have at least three solutions in , that is, three positive solutions such that
Example 3.8. Consider the problem where , , , , , , , ,, , and .
Since is a monotone increasing function on , we take . We can get So, conditions and hold. By Theorem 3.1, the BVP (3.44) has at least one positive solution.

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