Abstract and Applied Analysis

Volume 2012 (2012), Article ID 723507, 17 pages

doi:10.1155/2012/723507

Research Article

Existence of Positive Solution for Semipositone Fractional Differential Equations Involving Riemann-Stieltjes Integral Conditions

Wei Wang¹ and Li Huang²

¹Water Transportation Planning & Logistics Engineering Institute, College of Harbor, Coastal and Offshore Engineering, Hohai University, Jiangsu, Nanjing 210098, China

²National Research Center for Resettlement, Hohai University, Jiangsu, Nanjing 210098, China

Received 13 May 2012; Accepted 11 July 2012

Academic Editor: Yong Hong Wu

Copyright © 2012 Wei Wang and Li Huang. 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.

The existence of at least one positive solution is established for a class of semipositone fractional differential equations with Riemann-Stieltjes integral boundary condition. The technical approach is mainly based on the fixed-point theory in a cone.

1. Introduction

In this paper, we discuss the existence of positive solutions for the following singular semipositone fractional differential equation with nonlocal condition:

where , , and , is the standard Riemann-Liouville derivative. denotes the Riemann-Stieltjes integral, and is a function of bounded variation. is continuous, is Lebesgue integrable.

Differential equations of fractional order have been recently proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering. Indeed, we can find numerous applications in physics, engineering (like traffic, transportation, logistics, etc.), mechanics, chemistry, and so forth, (see [1–5]). There has been a significant development in the study of fractional differential equations in recent years, see the monographs of Kilbas et al. [6], Lakshmikantham et al. [7], Podlubny [4], Samko et al. [8], the survey by Agarwal et al. [9] and some recent results [10–14].

On the other hand, the nonlocal condition given by a Riemann-Stieltjes integral is due to Webb and Infante in [15–17] and gives a unified approach to many BVPs in [15, 16]. Motivated by [15–17], Hao et al. [18] studied the existence of positive solutions for th-order singular nonlocal boundary value problem:

where can be singular at , also can be singular at , but there is no singularity at . The existence of positive solutions of the BVP (1.2) is obtained by means of the fixed point index theory in cones.

More recently, Zhang [19] considered the following problem whose nonlinear term and boundary condition contain integer order derivatives of unknown functions

where is the standard Riemann-Liouville fractional derivative of order , may be singular at and may be singular at , . By using fixed point theorem of the mixed monotone operator, the unique existence result of positive solution to problem (1.3) was established. And then, Goodrich [20] was concerned with a partial extension of the problem (1.3) by extending boundary conditions

and the author derived the Green’s function for the problem (1.4) and showed that it satisfies certain properties, then by using cone theoretic techniques, a general existence theorem for (1.4) was obtained when satisfies some growth conditions.

Recently, Rehman and Khan [21] investigated the multipoint boundary value problems for fractional differential equations of the form:

where , , , with . By using the Schauder fixed point theorem and the contraction mapping principle, the authors established the existence and uniqueness of nontrivial solutions for the BVP (1.5) provided that the nonlinear function is continuous and satisfies certain growth conditions. Since covers the multipoint BVP and integral BVP as special case, the fractional differential equations with the Riemann-Stieltjes integral condition also were extensively studied by many authors, see [22, 23]. In [23], Zhang and Han considered the existence of positive solution of the following singular fractional differential equation:

where and can be a signed measure. Some growth conditions were adopted to guarantee that (1.6) has an unique positive solution, moreover, the authors also gave the iterative sequence of the solution, an error estimation, and the convergence rate of the positive solution.

Fractional differential equations like (1.1), with nonlinearities which are allowed to change sign and boundary conditions which contain nonlocal condition given by a Riemann-Stieltjes integral with a signed measure, are rarely studied. This type problems are referred to as semipositone problems in the literature, which arise naturally in chemical reactor theory [24]. In the recent work [25], by constructing a modified function, Zhang and Liu studied the existence of positive solution of a class of semipositone singular second-order Dirichlet boundary value problem, and when is superlinear, a sufficient condition for the existence of positive solution is obtained under the simple assumptions.

Motivated by the above work, in this paper, we establish the existence of positive solutions for the semipositone fractional differential equations (1.1) when is superlinear and involves fractional derivatives of unknown functions.

2. Preliminaries and Lemmas

Definition 2.1 (see [4, 5]). The Riemann-Liouville fractional integral of order of a function is given by

provided that the right-hand side is pointwise defined on .

Definition 2.2 (see [4, 5]). The Riemann-Liouville fractional derivative of order of a function is given by

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

Remark 2.3. If with order , then

Proposition 2.4 (see [4, 5]). (1) If , then

(2) If , then

Proposition 2.5 (see [4, 5]). Let , and is integrable, then

where , is the smallest integer greater than or equal to .

Let , by standard discuss, one easily reduces the BVP (1.1) to the following modified problems,

and the BVP (2.7) is equivalent to the BVP (1.1).

Lemma 2.6 (see [26]). Given , then the problem,

has the unique solution

where is the Green function of the BVP (2.8) and is given by

Lemma 2.7 (see [26]). For any , satisfies:

By Lemma 2.6, the unique solution of the problem,

is . Let

and define

as in [25], one can get that the Green function for the nonlocal BVP (2.7) is given by

Throughout paper one always assumes the following holds.

is a increasing function of bounded variation such that for and , where is defined by (2.13).

Define

One has the following Lemma.

Lemma 2.8. Let and hold, then the unique solution of the linear problem,

satisfies

where

Proof. By (2.11) and that is a increasing function of bounded variation, we have

Consequently,

Remark 2.9. (1) satisfies

(2)

In fact, since , the left side of (1) clearly holds. For right side of (1), from , one gets

thus we have

(2) is obvious from (2.11).

Now define a function for any by

and consider the following approximate problem of the BVP (2.7):

Lemma 2.10. Suppose is a positive solution of the problem (2.27) and satisfies , then is a positive solution of the problem (2.7), consequently, also is a positive solution of the BVP (1.1).

Proof. In fact, if is a positive solution of the BVP (2.27) such that for any , then, from (2.27) and the definition of , we have

Let , then we have

and , which implies that

Since , then (2.27) is transformed to (2.7), that is, is a positive solution of the BVP (2.27). By (2.7), is a positive solution of the BVP (1.1).

It is well known that the BVP (2.27) is equivalent to the fixed points of the mapping given by

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

for any . Let

then is a cone of .

For the convenience in presentation, we now present some assumptions to be used in the rest of the paper.

For any fixed and for any , there exist constants , such that, for all ,

, for any , and

where

and are defined by (2.13), (2.19), and (2.23), respectively.

Remark 2.11. If , the reversed inequality of (2.34) holds, that is, for all , one has

Lemma 2.12. Suppose - holds, for any fixed , is nondecreasing in on and nonincreasing in on ; and for any ,

Proof. Let . If , obviously holds. If , let , then . It follows from that

Thus is nondecreasing in on .

On the other hand, for any , if , obviously holds. If , let , then . By , we have

Thus is nonincreasing in on .

Now choose and , then by Remark 2.11, we have

Thus for any , and any , we have

Therefore

Lemma 2.13. Assume that – holds. Then is well defined. Furthermore, is a completely continuous operator.

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

By (2.44) and -, we have

which implies that the operator is well defined.

Next let , for any , by (2.11), we have

On the other hand, by (2.11), (2.22), and (2.46), we also have

So we have

which yields that .

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

Lemma 2.14 (see [27]). Let be a real Banach space, be a cone. Assume 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 Results

Theorem 3.1. Suppose – hold. Then the BVP (1.1) has at least one positive solutions , and satisfies

Proof. Let

and . Then, for any , notice that , we have

By and , one has

Therefore,

On the other hand, choose a real number such that

From (2.38), there exists such that, for any ,

Take

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

So for any , by (3.7)-(3.9), we have

Thus

By Lemma 2.14, has at least one fixed points such that .

In the end,

Equation (3.12) implies that , and

So by Lemma 2.10, the BVP (1.1) has at least one positive solution , and satisfies (3.13).

Example 3.2. Consider the following semipositone boundary value problem with fractional order :

Let

then the BVP (3.14) becomes the 4-Point BVP with coefficients

Then the BVP (3.14) has at least one positive solution , and satisfies

Proof. Obviously, , and

On the other hand, we have

Thus , and is increasing. So holds.

Take

then

For any , we also have

Then holds.

In the end, we notice

then

Moreover,

which implies that holds.

According to Theorem 3.1, the BVP (3.14) has at least one positive solution , and satisfies

Acknowledgments

The paper is Funded by the Humanities and Social Sciences Foundation of Ministry of Education (no. 09YJC630056), the National Natural Science Foundation of China (no. 51009060, no. 50909042), the Fundamental Research Funds for the Central Universities (no. 2009B13414), and the Priority Academic Program Development of Jiangsu Higher Education Institutions(Coastal Development Conservancy).

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