Open Access

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

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: https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq3_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq4_HTML.gif and https://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 https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ2_HTML.gif
(1.2)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ3_HTML.gif
(1.3)

where https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq8_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq10_HTML.gif and https://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:
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ6_HTML.gif
(1.6)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ7_HTML.gif
(1.7)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq12_HTML.gif and https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq14_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq15_HTML.gif are nonnegative constants with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq16_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq19_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq20_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq21_HTML.gif

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

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

https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq27_HTML.gif be a Banach space with norm https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq30_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq31_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq32_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq33_HTML.gif is the zero element of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq34_HTML.gif

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

(3) a cone https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq40_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq41_HTML.gif implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq42_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq43_HTML.gif

(4) a cone https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq45_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq46_HTML.gif is nonempty.

Definition 2.2.

Let https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq48_HTML.gif an operator, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq49_HTML.gif Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq50_HTML.gif is called a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq51_HTML.gif -concave operator if
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq53_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq54_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq55_HTML.gif -concave increasing operator. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq56_HTML.gif has a unique fixed point in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq58_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq59_HTML.gif are defined as in (A1) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq60_HTML.gif . Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq61_HTML.gif the BVP consisting of the equation
https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ11_HTML.gif
(2.3)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ12_HTML.gif
(2.4)

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ13_HTML.gif
(2.5)
Then
https://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
https://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
https://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
https://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
https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq62_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq63_HTML.gif

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

(3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq66_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq68_HTML.gif and https://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) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq70_HTML.gif if at least one of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq71_HTML.gif approaches https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq72_HTML.gif ;

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

(3) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq76_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq77_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq79_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq80_HTML.gif is a Banach space, where https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq82_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq83_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq84_HTML.gif . Furthermore, such a solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq85_HTML.gif satisfies the following properties:

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

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

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq92_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq93_HTML.gif is a normal solid cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq94_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq95_HTML.gif For any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq96_HTML.gif , if we define an operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq97_HTML.gif as follows:
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq99_HTML.gif is a fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq100_HTML.gif .

Now, we will prove that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq102_HTML.gif

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

In fact, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq105_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ22_HTML.gif
(3.4)

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

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

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq110_HTML.gif and https://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
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ23_HTML.gif
(3.5)

which indicates that https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq113_HTML.gif has a unique fixed point https://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 https://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,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ24_HTML.gif
(3.6)

which together with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq116_HTML.gif for https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq118_HTML.gif Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ25_HTML.gif
(3.7)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq119_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq120_HTML.gif We assert that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq121_HTML.gif Suppose on the contrary that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq122_HTML.gif Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq123_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq124_HTML.gif -concave increasing operator and for given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq125_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq126_HTML.gif is strictly increasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq127_HTML.gif , we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq128_HTML.gif Thus, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq129_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq130_HTML.gif And so,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ27_HTML.gif
(3.9)

which indicates that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq131_HTML.gif is strictly increasing in https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq133_HTML.gif we first suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq134_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq135_HTML.gif From (P2), we know that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ28_HTML.gif
(3.10)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ29_HTML.gif
(3.11)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq136_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq137_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq138_HTML.gif If we define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ30_HTML.gif
(3.12)
then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq139_HTML.gif and
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq140_HTML.gif implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ32_HTML.gif
(3.14)
So,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_Equ33_HTML.gif
(3.15)
Therefore,
https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq141_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F106962/MediaObjects/13661_2010_Article_895_IEq142_HTML.gif shows that
https://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
https://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.