Uniqueness and Parameter Dependence of Positive Solution of Fourth-Order Nonhomogeneous BVPs

  • Jian-Ping Sun1Email author and

    Affiliated with

    • Xiao-Yun Wang1

      Affiliated with

      Boundary Value Problems20102010:106962

      DOI: 10.1155/2010/106962

      Received: 23 February 2010

      Accepted: 11 July 2010

      Published: 28 July 2010

      Abstract

      We investigate the following fourth-order four-point nonhomogeneous Sturm-Liouville boundary value problem: http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq3_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq4_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq5_HTML.gif are nonnegative parameters. Some sufficient conditions are given for the existence and uniqueness of a positive solution. The dependence of the solution on the parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq6_HTML.gif is also studied.

      1. Introduction

      Boundary value problems (BVPs for short) consisting of fourth-order differential equation and four-point homogeneous boundary conditions have received much attention due to their striking applications. For example, Chen et al. [1] studied the fourth-order nonlinear differential equation
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ1_HTML.gif
      (1.1)
      with the four-point homogeneous boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ2_HTML.gif
      (1.2)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ3_HTML.gif
      (1.3)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq7_HTML.gif . By means of the upper and lower solution method and Schauder fixed point theorem, some criteria on the existence of positive solutions to the BVP (1.1)–(1.3) were established. Bai et al. [2] obtained the existence of solutions for the BVP (1.1)–(1.3) by using a nonlinear alternative of Leray-Schauder type. For other related results, one can refer to [35] and the references therein.

      Recently, nonhomogeneous BVPs have attracted many authors' attention. For instance, Ma [6, 7] and L. Kong and Q. Kong [810] studied some second-order multipoint nonhomogeneous BVPs. In particular, L. Kong and Q. Kong [10] considered the following second-order BVP with multipoint nonhomogeneous boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ4_HTML.gif
      (1.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq8_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq9_HTML.gif are nonnegative parameters. They derived some conditions for the above BVP to have a unique solution and then studied the dependence of this solution on the parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq10_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq11_HTML.gif . Sun [11] discussed the existence and nonexistence of positive solutions to a class of third-order three-point nonhomogeneous BVP. The authors in [12] studied the multiplicity of positive solutions for some fourth-order two-point nonhomogeneous BVP by using a fixed point theorem of cone expansion/compression type. For more recent results on higher-order BVPs with nonhomogeneous boundary conditions, one can see [1316].

      Inspired greatly by the above-mentioned excellent works, in this paper we are concerned with the following Sturm-Liouville BVP consisting of the fourth-order differential equation:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ5_HTML.gif
      (1.5)
      and the four-point nonhomogeneous boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ6_HTML.gif
      (1.6)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ7_HTML.gif
      (1.7)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq12_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq13_HTML.gif are nonnegative parameters. Under the following assumptions:

      (A1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq14_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq15_HTML.gif are nonnegative constants with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq16_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq17_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq18_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq19_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq20_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq21_HTML.gif

      (A2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq22_HTML.gif is continuous and monotone increasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq23_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq24_HTML.gif ;

      (A3) there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq25_HTML.gif such that

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ8_HTML.gif
      (1.8)

      we prove the uniqueness of positive solution for the BVP (1.5)–(1.7) and study the dependence of this solution on the parameters http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq26_HTML.gif .

      2. Preliminary Lemmas

      First, we recall some fundamental definitions.

      Definition 2.1.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq27_HTML.gif be a Banach space with norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq28_HTML.gif . Then

      (1) a nonempty closed convex set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq29_HTML.gif is said to be a cone if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq30_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq31_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq32_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq33_HTML.gif is the zero element of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq34_HTML.gif

      (2) every cone http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq35_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq36_HTML.gif defines a partial ordering in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq37_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq38_HTML.gif

      (3) a cone http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq39_HTML.gif is said to be normal if there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq40_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq41_HTML.gif implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq42_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq43_HTML.gif

      (4) a cone http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq44_HTML.gif is said to be solid if the interior http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq45_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq46_HTML.gif is nonempty.

      Definition 2.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq47_HTML.gif be a solid cone in a real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq48_HTML.gif an operator, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq49_HTML.gif Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq50_HTML.gif is called a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq51_HTML.gif -concave operator if
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ9_HTML.gif
      (2.1)

      Next, we state a fixed point theorem, which is our main tool.

      Lemma 2.3 (see [17]).

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq52_HTML.gif is a normal solid cone in a real Banach space http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq53_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq54_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq55_HTML.gif -concave increasing operator. Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq56_HTML.gif has a unique fixed point in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq57_HTML.gif

      The following two lemmas are crucial to our main results.

      Lemma 2.4.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq58_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq59_HTML.gif are defined as in (A1) and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq60_HTML.gif . Then for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq61_HTML.gif the BVP consisting of the equation
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ10_HTML.gif
      (2.2)
      and the boundary conditions (1.6) and (1.7) has a unique solution
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ11_HTML.gif
      (2.3)
      where
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ12_HTML.gif
      (2.4)

      Proof.

      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ13_HTML.gif
      (2.5)
      Then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ14_HTML.gif
      (2.6)
      By (2.5) and (1.6), we know that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ15_HTML.gif
      (2.7)
      On the other hand, in view of (2.5) and (1.7), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ16_HTML.gif
      (2.8)
      So, it follows from (2.6) and (2.8) that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ17_HTML.gif
      (2.9)
      which together with (2.7) implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ18_HTML.gif
      (2.10)

      Lemma 2.5.

      Assume that (A1) holds. Then

      (1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq62_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq63_HTML.gif

      (2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq64_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq65_HTML.gif

      (3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq66_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq67_HTML.gif

      3. Main Result

      For convenience, we denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq69_HTML.gif . In the remainder of this paper, the following notations will be used:

      (1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq70_HTML.gif if at least one of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq71_HTML.gif approaches http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq72_HTML.gif ;

      (2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq73_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq74_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq75_HTML.gif ;

      (3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq76_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq77_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq78_HTML.gif and at least one of them is strict.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq79_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq80_HTML.gif is a Banach space, where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq81_HTML.gif is defined as usual by the sup norm.

      Our main result is the following theorem.

      Theorem 3.1.

      Assume that (A1)–(A3) hold. Then the BVP (1.5)–(1.7) has a unique positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq82_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq83_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq84_HTML.gif . Furthermore, such a solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq85_HTML.gif satisfies the following properties:

      (P1) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq86_HTML.gif

      (P2) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq87_HTML.gif is strictly increasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq88_HTML.gif , that is,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ19_HTML.gif
      (3.1)
      (P3) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq89_HTML.gif is continuous in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq90_HTML.gif , that is, for any given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq91_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ20_HTML.gif
      (3.2)

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq92_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq93_HTML.gif is a normal solid cone in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq94_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq95_HTML.gif For any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq96_HTML.gif , if we define an operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq97_HTML.gif as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ21_HTML.gif
      (3.3)

      then it is not difficult to verify that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq98_HTML.gif is a positive solution of the BVP (1.5)–(1.7) if and only if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq99_HTML.gif is a fixed point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq100_HTML.gif .

      Now, we will prove that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq101_HTML.gif has a unique fixed point by using Lemma 2.3.

      First, in view of Lemma 2.5, we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq102_HTML.gif

      Next, we claim that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq103_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq104_HTML.gif -concave operator.

      In fact, for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq105_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq106_HTML.gif it follows from (3.3) and (A3) that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ22_HTML.gif
      (3.4)

      which shows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq107_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq108_HTML.gif -concave.

      Finally, we assert that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq109_HTML.gif is an increasing operator.

      Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq110_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq111_HTML.gif By (3.3) and (A2), we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ23_HTML.gif
      (3.5)

      which indicates that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq112_HTML.gif is increasing.

      Therefore, it follows from Lemma 2.3 that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq113_HTML.gif has a unique fixed point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq114_HTML.gif which is the unique positive solution of the BVP (1.5)–(1.7). The first part of the theorem is proved.

      In the rest of the proof, we will prove that such a positive solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq115_HTML.gif satisfies properties (P1), (P2), and (P3).

      First,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ24_HTML.gif
      (3.6)

      which together with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq116_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq117_HTML.gif implies (P1).

      Next, we show (P2). Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq118_HTML.gif Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ25_HTML.gif
      (3.7)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq119_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq120_HTML.gif We assert that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq121_HTML.gif Suppose on the contrary that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq122_HTML.gif Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq123_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq124_HTML.gif -concave increasing operator and for given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq125_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq126_HTML.gif is strictly increasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq127_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ26_HTML.gif
      (3.8)
      which contradicts the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq128_HTML.gif Thus, we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq129_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq130_HTML.gif And so,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ27_HTML.gif
      (3.9)

      which indicates that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq131_HTML.gif is strictly increasing in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq132_HTML.gif .

      Finally, we prove (P3). For any given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq133_HTML.gif we first suppose that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq134_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq135_HTML.gif From (P2), we know that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ28_HTML.gif
      (3.10)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ29_HTML.gif
      (3.11)
      Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq136_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq137_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq138_HTML.gif If we define
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ30_HTML.gif
      (3.12)
      then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq139_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ31_HTML.gif
      (3.13)
      which together with the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq140_HTML.gif implies that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ32_HTML.gif
      (3.14)
      So,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ33_HTML.gif
      (3.15)
      Therefore,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ34_HTML.gif
      (3.16)
      In view of (3.10) and (3.16), we obtain that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ35_HTML.gif
      (3.17)
      which together with the fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq141_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq142_HTML.gif shows that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ36_HTML.gif
      (3.18)
      Similarly, we can also prove that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ37_HTML.gif
      (3.19)

      Hence, (P3) holds.

      Declarations

      Acknowledgment

      Supported by the National Natural Science Foundation of China (10801068).

      Authors’ Affiliations

      (1)
      Department of Applied Mathematics, Lanzhou University of Technology

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      Copyright

      © Jian-Ping Sun and Xiao-Yun Wang. 2010

      This article is published under license to BioMed Central Ltd. 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.