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International Journal of Mathematics and Mathematical Sciences
Volume 2010, Article ID 495138, 17 pages
http://dx.doi.org/10.1155/2010/495138
Research Article

Existence of Concave Positive Solutions for Boundary Value Problem of Nonlinear Fractional Differential Equation with -Laplacian Operator

Department of Mathematics, Xiangnan University, Chenzhou 423000, China

Received 27 July 2009; Accepted 9 March 2010

Academic Editor: Rodica Costin

Copyright © 2010 Jinhua Wang 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 the existence and multiplicity of concave positive solutions for boundary value problem of nonlinear fractional differential equation with -Laplacian operator , , , , , where , , , denotes the Caputo derivative, and is continuous function, , ,  . By using fixed point theorem, the results for existence and multiplicity of concave positive solutions to the above boundary value problem are obtained. Finally, an example is given to show the effectiveness of our works.

1. Introduction

As we know, boundary value problems of integer-order differential equations have been intensively studied; see [15] and therein. Recently, due to the wide development of its theory of fractional calculus itself as well as its applications, fractional differential equations have been constantly attracting attention of many scholars; see, for example, [615].

In [7], Jafari and Gejji used the adomian decomposition method for solving the existence of solutions of boundary value problem:

In [9], by using fixed point theorems on cones, Dehghani and Ghanbari considered triple positive solutions of nonlinear fractional boundary value problem: where is the standard Riemann-Liouvill derivative. But we think that Green’s function in [9] is wrong; if , then, Green's function cannot be decided by .

In [11], using fixed point theorems on cones, Zhang investigated the existence and multiplicity of positive solutions of the following problem: where is the Caputo fractional derivative.

In [12], by means of Schauder fixed-point theorem, Su and Liu studied the existence of nonlinear fractional boundary value problem involving Caputo's derivative:

To the best of our knowledge, the existence of concave positive solutions of fractional order equation is seldom considered and investigated. Motivated by the above arguments, the main objective of this paper is to investigate the existence and multiplicity of concave positive solutions of boundary value problem of fractional differential equation with -Laplacian operator as follows: where , , , denotes the Caputo derivative, and is continuous function, , , ,

By using fixed point theorem, some results for multiplicity of concave positive solutions to the above boundary value problems are obtained. Finally, an example is given to show the effectiveness of our works.

The rest of the paper is organized as follows. In Section 2, we will introduce some lemmas and definitions which will be used later. In Section 3, the multiplicity of concave positive solutions for the boundary value problem (1.5) will be discussed.

2. Basic Definitions and Preliminaries

Firstly we present here some necessary definitions and lemmas.

Definition 2.1 (see [6, 16, 17]). The fractional integral of order of a function is given by provided that the right side is pointwise defined on .

Definition 2.2 (see [6, 16, 17]). The Riemann-Liouville fractional derivative of order of a continuous function is given by where provided that the right side is pointwise defined on .

Definition 2.3 (see [17]). Caputo's derivative of order of a function is defined as provided that the right side is pointwise defined on .

Remark 2.4. The following properties are well known:(1), , (2), (3),
If is an integer, the derivative for order is understood in the sense of usual differentiation.

Definition 2.5. Let be a real Banach space over . A nonempty convex closed set is said to be a cone provided that(a), for all , (b), implies

Definition 2.6. Let be a real Banach space and let be a cone. A function is called a nonegative continuous concave functional if is continuous and for all and

Definition 2.7. Let be a real Banach space and let be a cone. A function is called a nonegative continuous convex functional if is continuous and for all and
Suppose that are two nonnegative continuous convex functionals satisfying where is a positive constant and
Let , be given constants, let be two nonnegative continuous convex functionals satisfying (2.6) and (2.7), and let be a nonnegative continuous concave functional on the cone . Define the bounded convex sets:

Lemma 2.8 (see [2, 5]). Let be a Banach space, let be a cone, and let , be given constants. Assume that , are two nonnegative continuous convex functionals on , such that (2.6) and (2.7) are satisfied; let be a nonnegative continuous concave functional on , such that for all and let be a completely continuous operator. Suppose the following:, for all , for all , for all with
Then has at least three fixed points , , and in . Further , and

Lemma 2.9 (see [12]). Assume that with a fractional derivative of order that belongs to . Then for some , where is the smallest integer greater than or equal to

Lemma 2.10. Let ; then the boundary value problem has a unique solution where

Proof. We may apply Lemma 2.9 to reduce (2.10) to an equivalent integral equation: From (2.14), we have and by (2.10) and (2.11), there are ,
Therefore, the unique solution of problem (2.10) and (2.11) is

Lemma 2.11. The function defined by (2.13) satisfies the following conditions:(1), for , ,(2) for ,

Proof. Since observing (2.13), we have .
Form (2.13), we obtain Clearly, , for , , we have that is increasing with respect to and therefore, , for , . () of Lemma 2.11 holds.
On the other hand, if then, If , then ; therefore, , for , . () of Lemma 2.11 holds.

3. Existence of Three Concave Positive Solutions

In this section, we study the existence of concave positive solution for problem (1.5).

Let . From Definitions 2.1 and 2.3, we obtain , , and , So, by () of Remark 2.4, we know that is continuous for all . Hence, for all , we can define

Lemma 3.1 (see [12]). is a Banach space.
Define the cone by .
Let the nonnegative continuous concave functional and the nonnegative continuous convex functionals , be defined on the cone by

Lemma 3.2 (see [1]). Let , ; then

Lemma 3.3. BVP (1.5) is equivalent to the integral equation

Proof. From BVP (1.5) and Lemma 2.9, we have By , there is , and then, Therefore, BVP(1.5) is equivalent to following problem: By Lemma 2.10, BVP (1.5) is equivalent to the integral equation (3.4).
Let be the operator defined by

Lemma 3.4. is completely continuous.

Proof. Let ; in view of nonnegativeness and continuity of and , we have , and is continuous:
Clearly, is continuous for .
By Remark 2.4 and noting (3.4) and (3.6), we have
So, is concave on and ; we obtain .
Let be bounded; that is, there exists a positive constant such that for all .
Let ; then, for all , we have So, for all , . Hence, is uniformly bounded.
Since is continuous on , it is uniformly continuous on . Thus for fixed and for any , there exists a constant , such that any and Therefore, That is to say, is equicontinuous. By the means of the Arzela-Ascoli Theorem, we have that is completely continuous. The proof is completed.

Let

Theorem 3.5. Suppose that there exist constants , , such that , and the following conditions hold: for ;, for for
Then, the BVP (1.5) has at least three concave positive solutions , , and such that

Proof. By Lemmas 3.3 and 3.4, we have that is completely continuous and problem (1.5) has a solution if and only if satisfies the operator equation .
Now, we show that all the conditions of Lemma 2.8 hold.
Step 1. We will show that
If , then ,
From , we have Then,
So,
Step 2. Let , . It is easy to see that , and . Consequently, .
If , then for all , , . By , we obtain , for
From Lemma 3.2, we have that is, , for all . This shows that condition of Lemma 2.8 holds.
Step 3. Let ; by , we have Similarly, we can prove that . of Lemma 2.8 holds.Step 4. Let , and ; we have
of Lemma 2.8 holds. Therefore, the BVP (1.5) has at least three positive solutions , , and satisfying The proof is completed.

Corollary 3.6. If there exist constants , and , , such that , for and the following conditions are satisfied: for , for ; then the problem (1.5) has at least concave positive solutions.

Proof. If , by Condition and Step 1 of the proof of Theorem 3.5, we can obtain that From the Schauder fixed-point theorem, the problem (1.5) has at least one fixed-point .
If , by Theorem 3.5, there exist at least three concave positive solutions , , and . By the induction method, we finish the proof.

Finally, we present an example to check our results.

Example 3.7. Consider the boundary value problem: where Let ; note that , , , , ; we have , , .
Choosing , , , , . It is easy to see that , and satisfying(1), (2)(3)
By Theorem 3.5, problem (3.21) has at least three concave positive solutions , , and satisfying

Acknowledgments

This work was jointly supported by the Natural Science Foundation of Hunan Provincial under Grants 09JJ3005 and 2009JT3042, the Construct Program of the Key Discipline in Hunan Province, and Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province.

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