Abstract and Applied Analysis

Volume 2013 (2013), Article ID 847184, 7 pages

http://dx.doi.org/10.1155/2013/847184

Research Article

Lower and Upper Solutions Method for Positive Solutions of Fractional Boundary Value Problems

R. Darzi,¹ B. Mohammadzadeh,² A. Neamaty,³ and D. Bǎleanu4,5,6

¹Department of Mathematics, Neka Branch, Islamic Azad University, P.O. Box 48411-86114, Neka, Iran

²Department of Mathematics, Sari Branch, Islamic Azad University, P.O. Box 48161-19318, Sari, Iran

³Department of Mathematics, University of Mazandaran, P.O. Box 47416-95447, Babolsar, Iran

⁴Department of Mathematics and Computer Science, Faculty of Art and Sciences, Cankaya University, Yenimahalle, 06810 Ankara, Turkey

⁵Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia

⁶Institute of Space Sciences, P.O. Box, MG-23, R 76900 Magurele, Bucharest, Romania

Received 10 May 2013; Revised 11 July 2013; Accepted 1 August 2013

Academic Editor: Juan J. Trujillo

Copyright © 2013 R. Darzi 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 apply the lower and upper solutions method and fixed-point theorems to prove the existence of positive solution to fractional boundary value problem , , , , , , where denotes Riemann-Liouville fractional derivative, β is positive real number, , and is continuous on . As an application, one example is given to illustrate the main result.

1. Introduction

In the recent years, fractional calculus has been one of the most interesting issues that have attracted many scientists, especially in the fields of mathematics and engineering sciences. Many natural phenomena can be presented by boundary value problems of fractional differential equations. Many authors in different fields such as chemical physics, fluid flows, electrical networks, and viscoelasticity try to present a model of these phenomena by boundary value problems of fractional differential equations [1–4]. In order to achieve extra information in fractional calculus, interested readers can refer to more valuable books that are written by other authors [5–20].

The existence and multiplicity of solutions or positive solutions of nonlinear fractional differential equation (FDE) by the use of fixed point theorems, Leray-Shauder theory, and so forth are mentioned in some papers [6, 8, 12, 20, 21]. Few papers have considered the boundary value problems of fractional differential equations [12, 14]. By the use of some fixed point theorems on cones, Zhang [15] obtained the existence of positive solution for the equation

with the boundary conditions

In [22], Liang and Zhang applied lower and upper solutions method and fixed point theorems to obtain some results on the existence of positive solutions for the following BVPs:

where denotes Riemann-Liouville fractional derivative.

In this paper, we investigate the existence of positive solution for a nonlocal BVP of FDE,

using lower and upper solutions method and fixed point theorem, where denotes standard Riemann-Liouville fractional derivative, , and .

The main result of this paper can be seen in Theorem 10. In Theorem 10, we use the following conditions:

is nondecreasing with respect to , for , there exist a positive constant such that , for all , and the Schauder fixed-point theorem to show that problem (4)-(5) has a positive solution.

2. Basic Definitions and Preliminaries

In this section, we present the necessary definitions and lemmas that will be used to prove our new results.

Definition 1 (see [5, 6]). The Riemann-Liouville fractional integral of order of a function is defined by

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

Definition 2 (see [5, 6]). The Riemann-Liouville fractional derivative of order of a function is defined by

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

Definition 3 (see [5, 6]). A function is called a lower solution of problem (4)-(5) if satisfies

Definition 4 (see [7, 8]). A function is called an upper solution of problem (4)-(5) if satisfies

Lemma 5 (see [7, 8]). Let . Then the fractional differential equation

has

as a unique solution.

Lemma 6 (see [7, 8]). Let with a fractional derivative of order , , that belongs to . Then

Lemma 7. If , then for , the problem,

has a unique positive solution

Proof. We can apply Lemma 6 to reduce (13) to an equivalent integral equation

for some . From and in (14), we have . On the other hand, yields

Then, the unique solution of problem is given by . Obviously, if on . The proof is complete.

3. Main Result

In this section, we present and prove our main result.

Lemma 8. Suppose that . Given that , the Green function for the problem (13)-(14) is given by

Proof. By Lemma 7, for , we have

For , we have

The proof is complete.

Lemma 9. Suppose that and is a positive solution of (4)-(5). Then

where

where , , and and are two constants.

Proof. Since , there exists so that for . We define

Therefore, we have

On the other hand, by direct computation, we get

This completes the proof of the lemma.

Theorem 10. The fractional boundary value problem (4)-(5) has a positive solution if the conditions ()–() are satisfied.

Proof. Suppose that

and . We show that and are lower and upper solutions of (4)-(5), respectively. From Lemma 7, is a positive solution of the following problem:

We know that . Now, using the assumption of the theorem, we get

Therefore, from and since , the following relations satisfy

Consequently

Since and satisfy the boundary conditions, and are lower and upper solutions of (4)-(5), respectively. Now, we suppose that

and prove that FBVP,

has a solution. Consider operator , with , where is defined as in Lemma 8. It is easy to see that is continuous in . Since is nondecreasing in (from), for , we have

So, there exists a positive constant , such that . We will show that the operator is equicontinuous.

Case  1. If ,

Case  2. If ,

Therefore, the operator is equicontinuous, and by Arzela-Ascoli theorem, is a compact operator. Now, the Schauder fixed-point theorem [23] shows that the operator has a fixed-point theorem and so FBVP (32)-(33) has a solution. Finally, we will prove that FBVP (4)-(5) has a positive solution. Suppose that is a solution of FBVP (32)-(33). Since the function is nondecreasing in , we have

Assuming ,

By Lemma 7, ; that is, for . Similarly, for . Therefore is a positive solution of FBVP (4)-(5). The proof is complete.

Example 11. Consider the following fractional boundary value problem:

where

For , we have . Therefore

Now, by Theorem 10, we obtain that the FBVP (39) has a positive solution.

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