Open Access

Positive Solution of Singular Boundary Value Problem for a Nonlinear Fractional Differential Equation

Boundary Value Problems20112011:297026

DOI: 10.1155/2011/297026

Received: 16 August 2010

Accepted: 9 January 2011

Published: 17 January 2011

Abstract

The method of upper and lower solutions and the Schauder fixed point theorem are used to investigate the existence and uniqueness of a positive solution to a singular boundary value problem for a class of nonlinear fractional differential equations with non-monotone term. Moreover, the existence of maximal and minimal solutions for the problem is also given.

1. Introduction

Fractional differential equation can be extensively applied to various disciplines such as physics, mechanics, chemistry, and engineering, see [13]. Hence, in recent years, fractional differential equations have been of great interest, and there have been many results on existence and uniqueness of the solution of boundary value problems for fractional differential equations, see [47]. Especially, in [8] the authors have studied the following type of fractional differential equations:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq1_HTML.gif is a real number, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq2_HTML.gif is continuous and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq3_HTML.gif is the fractional derivative in the sense of Riemann-Liouville. Recently, Qiu and Bai [9] have proved the existence of a positive solution to boundary value problems of the nonlinear fractional differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ2_HTML.gif
(1.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq5_HTML.gif denotes Caputo derivative, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq6_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq7_HTML.gif (i.e., https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq8_HTML.gif is singular at https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq9_HTML.gif ). Their analysis relies on Krasnoselskii's fixed-point theorem and nonlinear alternative of Leray-Schauder type in a cone. More recently, Caballero Mena et al. [10] have proved the existence and uniqueness of a positive and non-decreasing solution to this problem by a fixed-point theorem in partially ordered sets. Other related results on the boundary value problem of the fractional differential equations can be found in the papers [1123]. A study of a coupled differential system of fractional order is also very significant because this kind of system can often occur in applications [2426].

However, in the previous works [9, 10], the nonlinear term has to satisfy the monotone or other control conditions. In fact, the nonlinear fractional differential equation with non-monotone term can respond better to impersonal law, so it is very important to weaken control conditions of the nonlinear term. In this paper, we mainly investigate the fractional differential (1.2) without any monotone requirement on nonlinear term by constructing upper and lower control function and exploiting the method of upper and lower solutions and Schauder fixed-point theorem. The existence and uniqueness of positive solution for (1.2) is obtained. Some properties concerning the maximal and minimal solutions are also given. This work is motivated by the above references and my previous work [27]. This paper is organized as follows. In Section 2, we recall briefly some notions of the fractional calculus and the theory of the operators for integration and differentiation of fractional order. Section 3 is devoted to the study of the existence and uniqueness of positive solution for (1.2) utilizing the method of upper and lower solutions and Schauder fixed-point theorem. The existence of maximal and minimal solutions for (1.2) is given in Section 4.

2. Preliminaries and Notations

For the convenience of the reader, we present here the necessary definitions and properties from fractional calculus theory, which are used throughout this paper.

Definition 2.1.

The Riemann-Liouville fractional integral of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq10_HTML.gif of a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq11_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ3_HTML.gif
(2.1)

provided that the right-hand side is pointwise defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq12_HTML.gif .

Definition 2.2.

The Caputo fractional derivative of order https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq13_HTML.gif of a continuous function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq14_HTML.gif is given by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ4_HTML.gif
(2.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq15_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq16_HTML.gif , provided that the right-hand side is pointwise defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq17_HTML.gif .

Lemma 2.3 (see [28]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq19_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq20_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ5_HTML.gif
(2.3)

Lemma 2.4 (see [28]).

The relation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ6_HTML.gif
(2.4)

is valid when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq21_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq22_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq23_HTML.gif .

Lemma 2.5 (see [9]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq24_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq25_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq26_HTML.gif is a continuous function and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq27_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq28_HTML.gif is continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq29_HTML.gif , then the function
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ7_HTML.gif
(2.5)
is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq30_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ8_HTML.gif
(2.6)

Lemma 2.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq31_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq32_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq33_HTML.gif is a continuous function and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq34_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq35_HTML.gif is continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq36_HTML.gif , then the boundary value problems (1.2) are equivalent to the Volterra integral equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ9_HTML.gif
(2.7)

Proof.

From Lemma 2.5, the Volterra integral equation (2.7) is well defined. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq37_HTML.gif satisfies the boundary value problems (1.2), then applying https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq38_HTML.gif to both sides of (1.2) and using Lemma 2.3, one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ10_HTML.gif
(2.8)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq39_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq40_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq41_HTML.gif is continuous in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq42_HTML.gif , there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq43_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq44_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq45_HTML.gif . Hence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ11_HTML.gif
(2.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq46_HTML.gif denotes the beta function. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq47_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq48_HTML.gif . In the similar way, we can prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq49_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq50_HTML.gif .

By Lemma 2.4 we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ12_HTML.gif
(2.10)
From the boundary conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq51_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ13_HTML.gif
(2.11)
Therefore, it follows from (2.8) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ14_HTML.gif
(2.12)

Namely, (2.7) follows.

Conversely, suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq52_HTML.gif satisfies (2.7), then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ15_HTML.gif
(2.13)
From Lemmas 2.3 and 2.4 and Definition 2.2, one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ16_HTML.gif
(2.14)
as well as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ17_HTML.gif
(2.15)
Thus, from (2.12), (2.14), and (2.15), it is follows that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ18_HTML.gif
(2.16)

Namely, (1.2) holds. The proof is therefore completed.

Remark 2.7.

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq53_HTML.gif , since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq54_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq55_HTML.gif we can obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ19_HTML.gif
(2.17)

Hence, it is follow from (2.6) that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq56_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq57_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq58_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq59_HTML.gif is the Banach space endowed with the infinity norm, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq60_HTML.gif is a nonempty closed subset of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq61_HTML.gif defined as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq62_HTML.gif . The positive solution which we consider in this paper is a function such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq63_HTML.gif .

According to Lemma 2.6, (1.2) is equivalent to the fractional integral equation (2.7). The integral equation (2.7) is also equivalent to fixed-point equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq64_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq65_HTML.gif , where operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq66_HTML.gif is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ20_HTML.gif
(2.18)

then we have the following lemma.

Lemma 2.8 (see [9]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq67_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq68_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq69_HTML.gif is a continuous function and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq70_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq71_HTML.gif is continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq72_HTML.gif , then the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq73_HTML.gif is completely continuous.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq74_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq75_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq76_HTML.gif is a continuous function, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq77_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq78_HTML.gif is continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq79_HTML.gif . Take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq80_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq81_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq82_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq83_HTML.gif , we define the upper-control function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq84_HTML.gif , and lower-control function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq85_HTML.gif , it is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq86_HTML.gif are monotonous non-decreasing on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq87_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq88_HTML.gif .

Definition 2.9.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq89_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq90_HTML.gif , and satisfy, respectively
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ21_HTML.gif
(2.19)

then the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq91_HTML.gif are called a pair of order upper and lower solutions for (1.2).

3. Existence and Uniqueness of Positive Solution

Now, we give and prove the main results of this paper.

Theorem 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq92_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq93_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq94_HTML.gif is a continuous function with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq95_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq96_HTML.gif is a continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq97_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq98_HTML.gif are a pair of order upper and lower solutions of (1.2), then the boundary value problem (1.2) has at least one solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq99_HTML.gif , moreover,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ22_HTML.gif
(3.1)

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ23_HTML.gif
(3.2)

endowed with the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq100_HTML.gif , then we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq101_HTML.gif . Hence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq102_HTML.gif is a convex, bounded, and closed subset of the Banach space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq103_HTML.gif . According to Lemma 2.8, the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq104_HTML.gif is completely continuous. Then we need only to prove https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq105_HTML.gif .

For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq106_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq107_HTML.gif . In view of Remark 2.7, Definition 2.9, and the definition of control function, one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ24_HTML.gif
(3.3)

Hence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq108_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq109_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq110_HTML.gif . According to Schauder fixed-point theorem, the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq111_HTML.gif has at least a fixed-point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq112_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq113_HTML.gif . Therefore the boundary value problem (1.2) has at least one solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq114_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq115_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq116_HTML.gif .

Corollary 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq117_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq118_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq119_HTML.gif is a continuous function with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq120_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq121_HTML.gif is a continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq122_HTML.gif . Assume that there exist two distinct positive constant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq123_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ25_HTML.gif
(3.4)
then the boundary value problem (1.2) has at least a positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq124_HTML.gif , moreover
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ26_HTML.gif
(3.5)

Proof.

By assumption (3.4) and the definition of control function, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ27_HTML.gif
(3.6)
Now, we consider the equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ28_HTML.gif
(3.7)
From Lemmas 2.5 and 2.6, (3.7) has a positive continuous solution on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq125_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ29_HTML.gif
(3.8)
Namely, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq126_HTML.gif is a upper solution of (1.2). In the similar way, we obtain https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq127_HTML.gif is the lower solution of (1.2). An application of Theorem 3.1 now yields that the boundary value problem (1.2) has at least a positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq128_HTML.gif , moreover
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ30_HTML.gif
(3.9)

Theorem 3.3.

If the conditions in Theorem 3.1 hold. Moreover for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq129_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq130_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq131_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ31_HTML.gif
(3.10)

then when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq132_HTML.gif , the boundary value problem (1.2) has a unique positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq133_HTML.gif .

Proof.

According to Theorem 3.1, if the conditions in Theorem 3.1 hold, then the boundary value problems (1.2) have at least a positive solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq134_HTML.gif . Hence we need only to prove that the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq135_HTML.gif defined in (2.18) is the contraction mapping in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq136_HTML.gif . In fact, for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq137_HTML.gif , by assumption (3.10), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ32_HTML.gif
(3.11)

Note that, from Lemma 2.5, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq138_HTML.gif is a continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq139_HTML.gif . Thus, when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq140_HTML.gif , the operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq141_HTML.gif is the contraction mapping. Then by Banach contraction fixed-point theorem, the boundary value problem (1.2) has a unique positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq142_HTML.gif .

4. Maximal and Minimal Solutions Theorem

In this section, we consider the existence of maximal and minimal solutions for (1.2).

Definition 4.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq143_HTML.gif be a solution of (1.2) in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq144_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq145_HTML.gif is said to be a maximal solution of (1.2), if for every solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq146_HTML.gif of (1.2) existing on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq147_HTML.gif , the inequality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq148_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq149_HTML.gif , holds. A minimal solution may be defined similarly by reversing the last inequality.

Theorem 4.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq150_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq151_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq152_HTML.gif is a continuous function with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq153_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq154_HTML.gif is a continuous function on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq155_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq156_HTML.gif is monotone non-decreasing with respect to the second variable, and there exist two positive constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq157_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ33_HTML.gif
(4.1)
Then there exist maximal solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq158_HTML.gif and minimal solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq159_HTML.gif of (1.2) on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq160_HTML.gif , moreover
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ34_HTML.gif
(4.2)

Proof.

It is easy to know from Corollary 3.2  that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq161_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq162_HTML.gif are the upper and lower solutions of (1.2), respectively. Then by using https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq163_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq164_HTML.gif as a pair of coupled initial iterations we construct two sequences https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq165_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq166_HTML.gif from the following linear iteration process:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ35_HTML.gif
(4.3)
It is easy to show from the monotone property of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq167_HTML.gif and the condition (4.1)  that the sequences https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq168_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq169_HTML.gif possess the following monotone property:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ36_HTML.gif
(4.4)
The above property implies that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ37_HTML.gif
(4.5)
exist and satisfy the relation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ38_HTML.gif
(4.6)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq170_HTML.gif in (4.3) shows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq171_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq172_HTML.gif satisfy the equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ39_HTML.gif
(4.7)
It is easy to verify that the limits https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq173_HTML.gif   and  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq174_HTML.gif are maximal and minimal solutions of (1.2) in
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ40_HTML.gif
(4.8)

respectively, furthermore, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq175_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq176_HTML.gif is the unique solution in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq177_HTML.gif , and hence the proof is completed.

Finally, we give an example to illuminate our results.

Example 4.3.

We consider the fractional order differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ41_HTML.gif
(4.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq178_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq179_HTML.gif . It is obvious from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq180_HTML.gif that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq181_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq182_HTML.gif . By Corollary 3.2, then (4.9) has a positive solution. Nevertheless it is easy to prove that the conclusions of [9, 10] cannot be applied to the above example.

Declarations

Acknowledgments

The authors are grateful to the referee for the comments. This work is supported by Natural Science Foundation Project of CQ CSTC (Grants nos. 2008BB7415, 2010BB9401) of China, Ministry of Education Project (Grant no. 708047) of China, Science and Technology Project of Chongqing municipal education committee (Grant no. KJ100513) of China, the NSFC (Grant no. 51005264) of China.

Authors’ Affiliations

(1)
College of Mathematics and Physics, Chongqing University of Posts and Telecommunications
(2)
Key Laboratory of Network Control & Intelligent Instrument, Chongqing University of Posts and Telecommunications, Ministry of Education
(3)
College of Applied Sciences, Beijing University of Technology
(4)
Automation Institute, Chongqing University of Posts and Telecommunications

References

  1. Mainardi F: The fundamental solutions for the fractional diffusion-wave equation. Applied Mathematics Letters 1996, 9(6):23-28. 10.1016/0893-9659(96)00089-4View ArticleMathSciNetGoogle Scholar
  2. Buckwar E, Luchko Y: Invariance of a partial differential equation of fractional order under the Lie group of scaling transformations. Journal of Mathematical Analysis and Applications 1998, 227(1):81-97. 10.1006/jmaa.1998.6078View ArticleMathSciNetGoogle Scholar
  3. Zhu ZY, Li GG, Cheng CJ: Quasi-static and dynamical analysis for viscoelastic Timoshenko beam with fractional derivative constitutive relation. Applied Mathematics and Mechanics 2002, 23(1):1-12. 10.1007/BF02437724View ArticleGoogle Scholar
  4. Nahušev AM: The Sturm-Liouville problem for a second order ordinary differential equation with fractional derivatives in the lower terms. Doklady Akademii Nauk SSSR 1977, 234(2):308-311.MathSciNetGoogle Scholar
  5. Aleroev TS: The Sturm-Liouville problem for a second-order differential equation with fractional derivatives in the lower terms. Differentsial'nye Uravneniya 1982, 18(2):341-342.MathSciNetGoogle Scholar
  6. Zhang S: Existence of solution for a boundary value problem of fractional order. Acta Mathematica Scientia B 2006, 26(2):220-228. 10.1016/S0252-9602(06)60044-1View ArticleGoogle Scholar
  7. Zhang S: Positive solutions for boundary-value problems of nonlinear fractional differential equations. Electronic Journal of Differential Equations 2006, 36: 1-12.Google Scholar
  8. Bai ZB, Lü HS: Positive solutions for boundary value problem of nonlinear fractional differential equation. Journal of Mathematical Analysis and Applications 2005, 311(2):495-505. 10.1016/j.jmaa.2005.02.052View ArticleMathSciNetGoogle Scholar
  9. Qiu T, Bai Z: Existence of positive solutions for singular fractional differential equations. Electronic Journal of Differential Equations 2008, 149: 1-9.MathSciNetGoogle Scholar
  10. Caballero Mena J, Harjani J, Sadarangani K: Existence and unqiueness of positive and nondecreasing solutions for a class of singular fractional boundary value problems. Boundary Value Problems 2009, 2009:-10.Google Scholar
  11. Ahmad B, Nieto JJ: Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions. Boundary Value Problems 2009, 2009:-11.Google Scholar
  12. Chang Y-K, Nieto JJ: Some new existence results for fractional differential inclusions with boundary conditions. Mathematical and Computer Modelling 2009, 49(3-4):605-609. 10.1016/j.mcm.2008.03.014View ArticleMathSciNetGoogle Scholar
  13. Ahmad B, Nieto JJ: Existence results for a coupled system of nonlinear fractional differential equations with three-point boundary conditions. Computers & Mathematics with Applications 2009, 58(9):1838-1843. 10.1016/j.camwa.2009.07.091View ArticleMathSciNetGoogle Scholar
  14. Su X: Boundary value problem for a coupled system of nonlinear fractional differential equations. Applied Mathematics Letters 2009, 22(1):64-69. 10.1016/j.aml.2008.03.001View ArticleMathSciNetGoogle Scholar
  15. Zhang SQ: Positive solutions to singular boundary value problem for nonlinear fractional differential equation. Computers & Mathematics with Applications 2010, 59(3):1300-1309.View ArticleMathSciNetGoogle Scholar
  16. Ahmad B: Existence of solutions for irregular boundary value problems of nonlinear fractional differential equations. Applied Mathematics Letters 2010, 23(4):390-394. 10.1016/j.aml.2009.11.004View ArticleMathSciNetGoogle Scholar
  17. Ahmad B, Sivasundaram S: Existence of solutions for impulsive integral boundary value problems of fractional order. Nonlinear Analysis. Hybrid Systems 2010, 4(1):134-141. 10.1016/j.nahs.2009.09.002View ArticleMathSciNetGoogle Scholar
  18. Daftardar-Gejji V, Jafari H: Analysis of a system of nonautonomous fractional differential equations involving Caputo derivatives. Journal of Mathematical Analysis and Applications 2007, 328(2):1026-1033. 10.1016/j.jmaa.2006.06.007View ArticleMathSciNetGoogle Scholar
  19. Momani S, Qaralleh R: An efficient method for solving systems of fractional integro-differential equations. Computers & Mathematics with Applications 2006, 52(3-4):459-470. 10.1016/j.camwa.2006.02.011View ArticleMathSciNetGoogle Scholar
  20. Hosseinnia SH, Ranjbar A, Momani S: Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part. Computers & Mathematics with Applications 2008, 56(12):3138-3149. 10.1016/j.camwa.2008.07.002View ArticleMathSciNetGoogle Scholar
  21. Abdulaziz O, Hashim I, Momani S: Solving systems of fractional differential equations by homotopy-perturbation method. Physics Letters. A 2008, 372(4):451-459. 10.1016/j.physleta.2007.07.059View ArticleMathSciNetGoogle Scholar
  22. Abdulaziz O, Hashim I, Momani S: Application of homotopy-perturbation method to fractional IVPs. Journal of Computational and Applied Mathematics 2008, 216(2):574-584. 10.1016/j.cam.2007.06.010View ArticleMathSciNetGoogle Scholar
  23. Hashim I, Abdulaziz O, Momani S: Homotopy analysis method for fractional IVPs. Communications in Nonlinear Science and Numerical Simulation 2009, 14(3):674-684. 10.1016/j.cnsns.2007.09.014View ArticleMathSciNetGoogle Scholar
  24. Chen Y, An H-L: Numerical solutions of coupled Burgers equations with time- and space-fractional derivatives. Applied Mathematics and Computation 2008, 200(1):87-95. 10.1016/j.amc.2007.10.050View ArticleMathSciNetGoogle Scholar
  25. Gafiychuk V, Datsko B, Meleshko V: Mathematical modeling of time fractional reaction-diffusion systems. Journal of Computational and Applied Mathematics 2008, 220(1-2):215-225. 10.1016/j.cam.2007.08.011View ArticleMathSciNetGoogle Scholar
  26. Deng WH, Li CP: Chaos synchronization of the fractional Lü system. Physica A 2005, 353(1–4):61-72.View ArticleGoogle Scholar
  27. Wang C: Existence and stability of periodic solutions for parabolic systems with time delays. Journal of Mathematical Analysis and Applications 2008, 339(2):1354-1361. 10.1016/j.jmaa.2007.07.082View ArticleMathSciNetGoogle Scholar
  28. Samko SG, Kilbas AA, Marichev OI: Fractional Integrals and Derivatives. Gordon and Breach Science, Yverdon, Switzerland; 1993:xxxvi+976.Google Scholar

Copyright

© Changyou Wang et al. 2011

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