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

  • Changyou Wang1, 2, 3Email author,

    Affiliated with

    • Ruifang Wang2, 4,

      Affiliated with

      • Shu Wang3 and

        Affiliated with

        • Chunde Yang1

          Affiliated with

          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:
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ1_HTML.gif
          (1.1)
          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq1_HTML.gif is a real number, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq2_HTML.gif is continuous and http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ2_HTML.gif
          (1.2)

          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq4_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq5_HTML.gif denotes Caputo derivative, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq6_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq7_HTML.gif (i.e., http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq8_HTML.gif is singular at http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq10_HTML.gif of a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq11_HTML.gif is given by
          http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq13_HTML.gif of a continuous function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq14_HTML.gif is given by
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ4_HTML.gif
          (2.2)

          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq15_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq17_HTML.gif .

          Lemma 2.3 (see [28]).

          Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq20_HTML.gif , then
          http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ6_HTML.gif
          (2.4)

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

          Lemma 2.5 (see [9]).

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

          Lemma 2.6.

          Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq31_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq32_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq33_HTML.gif is a continuous function and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq34_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq35_HTML.gif is continuous function on http://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
          http://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 http://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 http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ10_HTML.gif
          (2.8)
          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq39_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq40_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq41_HTML.gif is continuous in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq42_HTML.gif , there exists a constant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq43_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq44_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq45_HTML.gif . Hence
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ11_HTML.gif
          (2.9)

          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq46_HTML.gif denotes the beta function. Thus, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq47_HTML.gif as http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq49_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq50_HTML.gif .

          By Lemma 2.4 we have
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ12_HTML.gif
          (2.10)
          From the boundary conditions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq51_HTML.gif , one has
          http://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
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq52_HTML.gif satisfies (2.7), then we have
          http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ16_HTML.gif
          (2.14)
          as well as
          http://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
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq53_HTML.gif , since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq54_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq55_HTML.gif we can obtain
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq56_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq57_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq58_HTML.gif .

          Let http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq60_HTML.gif is a nonempty closed subset of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq61_HTML.gif defined as http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq64_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq65_HTML.gif , where operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq66_HTML.gif is defined as
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq67_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq68_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq69_HTML.gif is a continuous function and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq70_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq71_HTML.gif is continuous function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq72_HTML.gif , then the operator http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq73_HTML.gif is completely continuous.

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

          Definition 2.9.

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

          then the function http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq92_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq93_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq94_HTML.gif is a continuous function with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq95_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq96_HTML.gif is a continuous function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq97_HTML.gif . Assume that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq99_HTML.gif , moreover,
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ22_HTML.gif
          (3.1)

          Proof.

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

          endowed with the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq100_HTML.gif , then we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq101_HTML.gif . Hence http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq105_HTML.gif .

          For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq106_HTML.gif , we have http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ24_HTML.gif
          (3.3)

          Hence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq108_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq109_HTML.gif , that is, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq111_HTML.gif has at least a fixed-point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq112_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq114_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq115_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq116_HTML.gif .

          Corollary 3.2.

          Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq117_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq118_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq119_HTML.gif is a continuous function with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq120_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq121_HTML.gif is a continuous function on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq123_HTML.gif , such that
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq124_HTML.gif , moreover
          http://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
          http://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
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq125_HTML.gif
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ29_HTML.gif
          (3.8)
          Namely, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq128_HTML.gif , moreover
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq129_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq130_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq131_HTML.gif , such that
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ31_HTML.gif
          (3.10)

          then when http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq136_HTML.gif . In fact, for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq137_HTML.gif , by assumption (3.10), we have
          http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq138_HTML.gif is a continuous function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq139_HTML.gif . Thus, when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq140_HTML.gif , the operator http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq144_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq146_HTML.gif of (1.2) existing on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq147_HTML.gif , the inequality http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq148_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq150_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq151_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq152_HTML.gif is a continuous function with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq153_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq154_HTML.gif is a continuous function on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq155_HTML.gif . Assume that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq157_HTML.gif such that
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq158_HTML.gif and minimal solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq159_HTML.gif of (1.2) on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq160_HTML.gif , moreover
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq161_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq163_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq165_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq166_HTML.gif from the following linear iteration process:
          http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq168_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq169_HTML.gif possess the following monotone property:
          http://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
          http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ38_HTML.gif
          (4.6)
          Letting http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq170_HTML.gif in (4.3) shows that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq171_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq172_HTML.gif satisfy the equations
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq173_HTML.gif   and  http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ40_HTML.gif
          (4.8)

          respectively, furthermore, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq175_HTML.gif then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq176_HTML.gif is the unique solution in http://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
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_Equ41_HTML.gif
          (4.9)

          where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq178_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq179_HTML.gif . It is obvious from http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq180_HTML.gif that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F297026/MediaObjects/13661_2010_Article_34_IEq181_HTML.gif , http://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

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