Abstract and Applied Analysis

Volume 2014 (2014), Article ID 925010, 9 pages

http://dx.doi.org/10.1155/2014/925010

Research Article

Positive Solutions for the Eigenvalue Problem of Semipositone Fractional Order Differential Equation with Multipoint Boundary Conditions

Ge Dong

Department of Basic Teaching, Shanghai Jianqiao College, Shanghai 201319, China

Received 17 January 2014; Accepted 14 February 2014; Published 15 April 2014

Academic Editor: Xinguang Zhang

Copyright © 2014 Ge Dong. 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 of positive solution for the eigenvalue problem of semipositone fractional order differential equation with multipoint boundary conditions by using known Krasnosel'skii's fixed point theorem. Some sufficient conditions that guarantee the existence of at least one positive solution for eigenvalues   sufficiently small and sufficiently large are established.

1. Introduction

In this paper, we study the existence of positive solutions to the following eigenvalue problem of semipositone fractional order differential equation with multipoint boundary conditions:

where , , , with , is a positive parameter, and are the standard Rieman-Liouville derivative. Throughout the paper, we assume that is semipositone; that is, is continuous and there exists , such that , for any .

The multipoint boundary value problems (BVPs for short) for ordinary differential equations arise in a variety of different applied mathematics and physics. Recently, Feng and Bai [1] investigated the existence of positive solutions for a semipositone second-order multipoint boundary value problem:

By using Krasnosel'skii's fixed point theorem, some sufficient conditions that guarantee the existence of at least one positive solution are obtained. In [2], a -type conjugate boundary value problem for the nonlinear fractional differential equation,

is considered. Based on the nonlinear alternative of Leray-Schauder type and Krasnosel'skii's fixed-point theorems, the existence of positive solution of the semipositone boundary value problems (3) for a sufficiently small was given. In recent paper [3], Zhang et al. established the existence of multiple positive solutions for a general higher order fractional differential equation with derivatives and a negatively Carathèodory perturbed term:

Some local and nonlocal growth conditions were adopted to guarantee the existence of at least two positive solutions for the higher order fractional differential equation (4). For the recent work in application, the reader is referred to [4–20].

Inspired by the above work, in this paper we study the existence of positive solutions to the semipositone BVP (1). Here we also emphasize that the main results of this paper contain not only the cases for sufficiently small, but also for sufficiently large, which is different from [2, 3].

2. Preliminaries and Lemmas

Definition 1 (see [21–24]). The fractional integral of order of a function is given by

provided that the right-hand side is pointwisely on .

Definition 2 (see [21–24]). The Riemann-Liouville fractional derivative of order of a function : is given by

where , denotes the integer part of the number , and , provided that the right-hand side is defined on .

Lemma 3 (see [21–24]). Assuming that with a fractional derivative of order , then

where .

Lemma 4 (see [3]). Suppose that . Then the following boundary value problem

has a unique solution

where

is the Green function of the boundary value problem (8) and

Lemma 5 (see [2]). The function in Lemma 4 has the following properties:

(R1), for ;(R2), for ;(R3), for , where

Lemma 6. The following boundary value problem

has a unique solution , which satisfies

Proof. By Lemma 4, the unique solution of (13) is

So

and by , we have , so

The basic space used in this paper is , where is the set of real numbers. Obviously, the space is a Banach space if it is endowed with the norm as follows:

for any . Let

and then is a cone of .

Now let ; then the boundary value problem (1) is equivalent to the following boundary value problem:

Define a modified function for any by

and consider

Lemma 7. The BVP (1) and the BVP (22) are equivalent. Moreover, if is a positive solution of the problem (22) and satisfies , , then is a positive solution of the boundary value problem (1).

Proof. Since is a positive solution of the BVP (22) such that for any , we have

Let , and then we have

Substitute (24) into (23), that is (20), which implies that is a positive solution of the BVP (1).

It follows from Lemma 4 that the BVP (22) is equivalent to the integral equation

Thus it is sufficient to find fixed points for the mapping defined by

Lemma 8. is a completely continuous operator.

Proof. For any fixed , there exists a constant such that , and

Take

then

This implies that the operator is bounded.

Next for any , by Lemma 5, we have

On the other hand, it follows from Lemma 5, , and that

So, by (30) and (31), we have

which yields that .

At the end, using standard arguments, according to the Ascoli-Arzela Theorem, one can show that is completely continuous. Thus is a completely continuous operator.

Lemma 9 (see [25]). Let be a real Banach space, and let be a cone. Assume that are two bounded open subsets of with , and let be a completely continuous operator such that either

(1) and , or(2) and .

Then has a fixed point in .

3. Main Result

Define

Theorem 10. Suppose that

Then there exists a constant such that, for any , the BVP (1) has at least one positive solution.

Proof. Choosing with , then

Let

For any , , and sufficiently small such that , we have

Therefore,

On the other hand, take

and choose a large enough such that

By (33), we know that is an unbounded continuous function. Therefore, for any , there exists a constant such that

Choosing

then . Let . Then for any and for any , we have

Consequently, for , it follows from (43) that

and then by (41) and (44), for , we get

So for any and , by (45), we have

Thus, we have

By Lemma 9, has a fixed point such that .

From

we have

Thus

By Lemma 7 and (50), the boundary value problem (1) has at least one positive solution. The proof of Theorem 10 is completed.

Theorem 11. Suppose that

and there exist constants and such that

Then there exists a constant such that, for any , the BVP (1) has at least one positive solution.

Proof. Choosing

and let . Then for any ,, and , we have

so for any and , by (52)–(55), we have

Thus, we have

According to (51), it is clear that

Let us choose such that

Then there exists a large enough such that

Thus, by (60), if

then

Now denote that

and choose

Then .

Next let . Then for any and for any , we have

which implies that

By Lemma 9, has at least a fixed points such that .

It follows from that

By Lemma 7 and (67), the boundary value problem (1) has at least one positive solution. The proof of Theorem 11 is completed.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors were supported financially by “Chen Guang” Project supported by Shanghai Municipal Education Development Foundation (10CGB25) and Shanghai Universities for Outstanding Young Teachers Scientific Research Selection and Training Special Fund (sjq08011).

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