Open Access

Solutions and Green's Functions for Boundary Value Problems of Second-Order Four-Point Functional Difference Equations

Boundary Value Problems20102010:973731

DOI: 10.1155/2010/973731

Received: 23 April 2010

Accepted: 11 July 2010

Published: 28 July 2010

Abstract

We consider the Green's functions and the existence of positive solutions for a second-order functional difference equation with four-point boundary conditions.

1. Introduction

In recent years, boundary value problems (BVPs) of differential and difference equations have been studied widely and there are many excellent results (see Gai et al. [1], Guo and Tian [2], Henderson and Peterson [3], and Yang et al. [4]). By using the critical point theory, Deng and Shi [5] studied the existence and multiplicity of the boundary value problems to a class of second-order functional difference equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ1_HTML.gif
(1.1)
with boundary value conditions
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ2_HTML.gif
(1.2)
where the operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq1_HTML.gif is the Jacobi operator
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ3_HTML.gif
(1.3)
Ntouyas et al. [6] and Wong [7] investigated the existence of solutions of a BVP for functional differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq2_HTML.gif is a continuous function, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq3_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq4_HTML.gif .

Weng and Guo [8] considered the following two-point BVP for a nonlinear functional difference equation with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq5_HTML.gif -Laplacian operator
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ5_HTML.gif
(1.5)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq12_HTML.gif is continuous, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq13_HTML.gif .

Yang et al. [9] considered two-point BVP of the following functional difference equation with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq14_HTML.gif -Laplacian operator:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ6_HTML.gif
(1.6)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq15_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq16_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq17_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq18_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq19_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq20_HTML.gif are nonnegative real constants.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq21_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq22_HTML.gif , let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ7_HTML.gif
(1.7)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq23_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq24_HTML.gif are both Banach spaces endowed with the max-norm
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ8_HTML.gif
(1.8)

For any real function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq25_HTML.gif defined on the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq26_HTML.gif and any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq27_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq28_HTML.gif , we denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq29_HTML.gif an element of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq30_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq31_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq32_HTML.gif .

In this paper, we consider the following second-order four-point BVP of a nonlinear functional difference equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ9_HTML.gif
(1.9)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq34_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq35_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq36_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq37_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq38_HTML.gif is a continuous function, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq39_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq40_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq41_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq42_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq43_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq44_HTML.gif are nonnegative real constants, and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq45_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq46_HTML.gif .

At this point, it is necessary to make some remarks on the first boundary condition in (1.9). This condition is a generalization of the classical condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ10_HTML.gif
(1.10)

from ordinary difference equations. Here this condition connects the history https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq47_HTML.gif with the single https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq48_HTML.gif . This is suggested by the well-posedness of BVP (1.9), since the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq49_HTML.gif depends on the term https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq50_HTML.gif (i.e., past values of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq51_HTML.gif ).

As usual, a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq52_HTML.gif is said to be a positive solution of BVP (1.9) if it satisfies BVP (1.9) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq53_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq54_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq55_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq56_HTML.gif .

2. The Green's Function of (1.9)

First we consider the nonexistence of positive solutions of (1.9). We have the following result.

Lemma 2.1.

Assume that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ11_HTML.gif
(2.1)
or
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ12_HTML.gif
(2.2)

Then (1.9) has no positive solution.

Proof.

From https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq57_HTML.gif , we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq58_HTML.gif is convex for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq59_HTML.gif .

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq60_HTML.gif is a positive solution of (1.9) and (2.1) holds.
  1. (1)

    Consider that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq61_HTML.gif .

     
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq62_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq63_HTML.gif . It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ13_HTML.gif
(2.3)

which is a contradiction to the convexity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq64_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq65_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq66_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq67_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ14_HTML.gif
(2.4)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ15_HTML.gif
(2.5)

which is a contradiction to the convexity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq68_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq69_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq70_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq71_HTML.gif is a trivial solution. So there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq72_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq73_HTML.gif .

We assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq74_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ16_HTML.gif
(2.6)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ17_HTML.gif
(2.7)

which is a contradiction to the convexity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq75_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq76_HTML.gif , similar to the above proof, we can also get a contradiction.
  1. (2)

    Consider that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq77_HTML.gif .

     
Now we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ18_HTML.gif
(2.8)

which is a contradiction to the convexity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq78_HTML.gif .

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq79_HTML.gif is a positive solution of (1.9) and (2.2) holds.
  1. (1)

    Consider that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq80_HTML.gif .

     
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq81_HTML.gif , then we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ19_HTML.gif
(2.9)

which is a contradiction to the convexity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq82_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq83_HTML.gif , similar to the above proof, we can also get a contradiction.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq84_HTML.gif , and so https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq85_HTML.gif , then there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq86_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq87_HTML.gif . Otherwise, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq88_HTML.gif is a trivial solution. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq89_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ20_HTML.gif
(2.10)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ21_HTML.gif
(2.11)

A contradiction to the convexity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq90_HTML.gif follows.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq91_HTML.gif , we can also get a contradiction.
  1. (2)

    Consider that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq92_HTML.gif .

     
Now we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ22_HTML.gif
(2.12)

which is a contradiction to the convexity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq93_HTML.gif .

Next, we consider the existence of the Green's function of equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ23_HTML.gif
(2.13)

We always assume that

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq94_HTML.gif ) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq95_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq96_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq97_HTML.gif .

Motivated by Zhao [10], we have the following conclusions.

Theorem 2.2.

The Green's function for second-order four-point linear BVP (2.13) is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ24_HTML.gif
(2.14)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ25_HTML.gif
(2.15)

Proof.

Consider the second-order two-point BVP
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ26_HTML.gif
(2.16)
It is easy to find that the solution of BVP (2.16) is given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ27_HTML.gif
(2.17)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ28_HTML.gif
(2.18)
The three-point BVP
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ29_HTML.gif
(2.19)
can be obtained from replacing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq98_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq99_HTML.gif in (2.16). Thus we suppose that the solution of (2.19) can be expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ30_HTML.gif
(2.20)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq101_HTML.gif are constants that will be determined.

From (2.18) and (2.20), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ31_HTML.gif
(2.21)
Putting the above equations into (2.19) yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ32_HTML.gif
(2.22)
By (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq102_HTML.gif ), we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq103_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq104_HTML.gif by solving the above equation:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ33_HTML.gif
(2.23)
By (2.19) and (2.20), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ34_HTML.gif
(2.24)
The four-point BVP (2.13) can be obtained from replacing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq105_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq106_HTML.gif in (2.19). Thus we suppose that the solution of (2.13) can be expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ35_HTML.gif
(2.25)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq108_HTML.gif are constants that will be determined.

From (2.24) and (2.25), we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ36_HTML.gif
(2.26)
Putting the above equations into (2.13) yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ37_HTML.gif
(2.27)
By (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq109_HTML.gif ), we can easily obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ38_HTML.gif
(2.28)
Then by (2.17), (2.20), (2.23), (2.25), and (2.28), the solution of BVP (2.13) can be expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ39_HTML.gif
(2.29)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq110_HTML.gif is defined in (2.14). That is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq111_HTML.gif is the Green's function of BVP (2.13).

Remark 2.3.

By (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq112_HTML.gif ), we can see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq113_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq114_HTML.gif . Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ40_HTML.gif
(2.30)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq115_HTML.gif .

Lemma 2.4.

Assume that ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq116_HTML.gif ) holds. Then the second-order four-point BVP (2.13) has a unique solution which is given in (2.29).

Proof.

We need only to show the uniqueness.

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq117_HTML.gif in (2.29) is a solution of BVP (2.13). Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq118_HTML.gif is another solution of BVP (2.13). Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ41_HTML.gif
(2.31)
Then by (2.13), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ42_HTML.gif
(2.32)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ43_HTML.gif
(2.33)
From (2.32) we have, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq119_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ44_HTML.gif
(2.34)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ45_HTML.gif
(2.35)
Combining (2.33) with (2.35), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ46_HTML.gif
(2.36)

Condition (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq120_HTML.gif ) implies that (2.36) has a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq121_HTML.gif . Therefore https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq122_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq123_HTML.gif . This completes the proof of the uniqueness of the solution.

3. Existence of Positive Solutions

In this section, we discuss the BVP (1.9).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq124_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq125_HTML.gif .

We rewrite BVP (1.9) as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ47_HTML.gif
(3.1)

with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq126_HTML.gif .

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq127_HTML.gif is a solution of the BVP (3.1). Then it can be expressed as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ48_HTML.gif
(3.2)

Lemma 3.1 (see Guo et al. [11]).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq128_HTML.gif is a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq129_HTML.gif is a cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq130_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq131_HTML.gif . Furthermore, assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq132_HTML.gif is a completely continuous operator and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq133_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq134_HTML.gif . Thus, one has the following conclusions:
  1. (1)

    if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq135_HTML.gif  for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq136_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq137_HTML.gif ;

     
  2. (2)

    if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq138_HTML.gif  for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq139_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq140_HTML.gif .

     
Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq141_HTML.gif . Then (3.1) may be rewritten as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ49_HTML.gif
(3.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq142_HTML.gif be a solution of (3.3). Then by (3.2) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq143_HTML.gif , it can be expressed as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ50_HTML.gif
(3.4)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq144_HTML.gif be a solution of BVP (3.1) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq145_HTML.gif . Then for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq146_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq147_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ51_HTML.gif
(3.5)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ52_HTML.gif
(3.6)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq148_HTML.gif is a Banach space endowed with norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq149_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq150_HTML.gif is a cone in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq151_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq152_HTML.gif , we have by (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq153_HTML.gif ) and the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq154_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ53_HTML.gif
(3.7)
For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq155_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq156_HTML.gif , by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq157_HTML.gif and (3.5), if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq158_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ54_HTML.gif
(3.8)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq159_HTML.gif , we have, by (3.4),
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ55_HTML.gif
(3.9)
hence by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq160_HTML.gif , we obtain for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq161_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ56_HTML.gif
(3.10)

Lemma 3.2.

For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq162_HTML.gif , there is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq163_HTML.gif , such that

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ57_HTML.gif
(3.11)

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq164_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq165_HTML.gif , by the definitions of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq166_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq167_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ58_HTML.gif
(3.12)

Obviously, there is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq168_HTML.gif , such that (3.11) holds.

Define an operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq169_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ59_HTML.gif
(3.13)

Then we may transform our existence problem of positive solutions of BVP (3.1) into a fixed point problem of operator (3.13).

Lemma 3.3.

Consider that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq170_HTML.gif .

Proof.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq171_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq173_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq174_HTML.gif , respectively. Thus, (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq175_HTML.gif ) yields
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ60_HTML.gif
(3.14)
It follows from the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq176_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ61_HTML.gif
(3.15)

which implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq177_HTML.gif .

Lemma 3.4.

Suppose that ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq178_HTML.gif ) holds. Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq179_HTML.gif is completely continuous.

We assume that

(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq180_HTML.gif ) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq181_HTML.gif

(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq182_HTML.gif ) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq183_HTML.gif .

We have the following main results.

Theorem 3.5.

Assume that ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq184_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq185_HTML.gif ) hold. Then BVP (3.1) has at least one positive solution if the following conditions are satisfied:

(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq186_HTML.gif ) there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq187_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq188_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq189_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq190_HTML.gif ;

(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq191_HTML.gif ) there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq192_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq193_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq194_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq195_HTML.gif

or

(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq196_HTML.gif ) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq197_HTML.gif ;

(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq198_HTML.gif ) there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq199_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq200_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq201_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq202_HTML.gif ;

(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq203_HTML.gif ) there exists an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq204_HTML.gif , such that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq205_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq206_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq207_HTML.gif

where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ62_HTML.gif
(3.16)

Proof.

Assume that ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq208_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq209_HTML.gif ) hold. For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq210_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq211_HTML.gif , thus
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ63_HTML.gif
(3.17)
which implies by Lemma 3.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ64_HTML.gif
(3.18)
For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq212_HTML.gif , by (3.8)–(3.10) and Lemma 3.2, we have, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq213_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq214_HTML.gif . Then by (3.13) and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq215_HTML.gif ), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ65_HTML.gif
(3.19)
which implies by Lemma 3.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ66_HTML.gif
(3.20)

So by (3.18) and (3.20), there exists one positive fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq216_HTML.gif of operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq217_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq218_HTML.gif .

Assume that ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq219_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq220_HTML.gif ) hold, for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq221_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq222_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq223_HTML.gif , by ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq224_HTML.gif ), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ67_HTML.gif
(3.21)
Thus we have from Lemma 3.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ68_HTML.gif
(3.22)
For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq225_HTML.gif , by (3.8)–(3.10), we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq226_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ69_HTML.gif
(3.23)
Thus we have from Lemma 3.1 that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ70_HTML.gif
(3.24)

So by (3.22) and (3.24), there exists one positive fixed point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq227_HTML.gif of operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq228_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq229_HTML.gif .

Consequently, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq230_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq231_HTML.gif is a positive solution of BVP (3.1).

Theorem 3.6.

Assume that ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq232_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq233_HTML.gif ) hold. Then BVP (3.1) has at least one positive solution if ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq234_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq235_HTML.gif ) or ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq236_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq237_HTML.gif ) hold.

Theorem 3.7.

Assume that ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq238_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq239_HTML.gif ) hold. Then BVP (3.1) has at least two positive solutions if (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq240_HTML.gif ), (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq241_HTML.gif ), and (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq242_HTML.gif )  or (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq243_HTML.gif ), (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq244_HTML.gif ), and (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq245_HTML.gif ) hold.

Theorem 3.8.

Assume that (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq246_HTML.gif )–(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq247_HTML.gif ) hold. Then BVP (3.1) has at least three positive solutions if (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq248_HTML.gif )–(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq249_HTML.gif ) hold.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq250_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq251_HTML.gif , and

( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq252_HTML.gif ) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq253_HTML.gif

Define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq254_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ71_HTML.gif
(3.25)

which satisfies

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq255_HTML.gif H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq256_HTML.gif .

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq257_HTML.gif exists.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq258_HTML.gif is a solution of (1.9). Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ72_HTML.gif
(3.26)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ73_HTML.gif
(3.27)
By (1.9), (3.26), (3.27), (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq259_HTML.gif ), (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq260_HTML.gif ), and the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq261_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ74_HTML.gif
(3.28)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ75_HTML.gif
(3.29)
and, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq262_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ76_HTML.gif
(3.30)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ77_HTML.gif
(3.31)
Then by (3.27), (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq263_HTML.gif ), (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq264_HTML.gif ), and the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq265_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq266_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq267_HTML.gif Thus, the BVP (1.9) can be changed into the following BVP:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ78_HTML.gif
(3.32)

with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq268_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq269_HTML.gif .

Similar to the above proof, we can show that (1.9) has at least one positive solution. Consequently, (1.9) has at least one positive solution.

Example 3.9.

Consider the following BVP:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ79_HTML.gif
(3.33)
That is,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ80_HTML.gif
(3.34)
Then we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ81_HTML.gif
(3.35)
Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ82_HTML.gif
(3.36)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq270_HTML.gif

By calculation, we can see that (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq271_HTML.gif )–(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq272_HTML.gif ) hold, then by Theorem 3.8, the BVP (3.33) has at least three positive solutions.

4. Eigenvalue Intervals

In this section, we consider the following BVP with parameter https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq273_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ83_HTML.gif
(4.1)

with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq274_HTML.gif .

The BVP (4.1) is equivalent to the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ84_HTML.gif
(4.2)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq275_HTML.gif be the solution of (3.3), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq276_HTML.gif . Then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ85_HTML.gif
(4.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq277_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq278_HTML.gif be defined as the above. Define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq279_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ86_HTML.gif
(4.4)

Then solving the BVP (4.1) is equivalent to finding fixed points in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq280_HTML.gif . Obviously https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq281_HTML.gif is completely continuous and keeps the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq282_HTML.gif invariant for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq283_HTML.gif .

Define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ87_HTML.gif
(4.5)

respectively. We have the following results.

Theorem 4.1.

Assume that (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq284_HTML.gif ), (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq285_HTML.gif ), (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq286_HTML.gif ),

(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq287_HTML.gif )   https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq288_HTML.gif ,

(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq289_HTML.gif )    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq290_HTML.gif

hold, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq291_HTML.gif , then BVP (4.1) has at least one positive solution, where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq292_HTML.gif is a positive constant.

Proof.

Assume that condition (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq293_HTML.gif ) holds. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq294_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq295_HTML.gif , there exists an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq296_HTML.gif sufficiently small, such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ88_HTML.gif
(4.6)
By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq297_HTML.gif , there is an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq298_HTML.gif , such that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq299_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ89_HTML.gif
(4.7)
It follows that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq300_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq301_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ90_HTML.gif
(4.8)
For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq302_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq303_HTML.gif , by (3.9), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ91_HTML.gif
(4.9)
Therefore by (3.13) and Lemma 3.2, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ92_HTML.gif
(4.10)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq304_HTML.gif then for a sufficiently small https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq305_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq306_HTML.gif Similar to the above, for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq307_HTML.gif , we obtain by (3.10)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ93_HTML.gif
(4.11)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq308_HTML.gif , choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq309_HTML.gif sufficiently large, such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ94_HTML.gif
(4.12)
By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq310_HTML.gif , there is an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq311_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq312_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq313_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ95_HTML.gif
(4.13)
For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq314_HTML.gif , by (3.8)–(3.10) and (3.13), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ96_HTML.gif
(4.14)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ97_HTML.gif
(4.15)

Finally, we consider the assumption https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq315_HTML.gif . By the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq316_HTML.gif , there is

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq317_HTML.gif , such that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq318_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq319_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ98_HTML.gif
(4.16)
We now show that there is    https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq320_HTML.gif , such that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq321_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq322_HTML.gif In fact, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq323_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq324_HTML.gif and every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq325_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq326_HTML.gif hence in a similar way, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ99_HTML.gif
(4.17)
which implies that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ100_HTML.gif
(4.18)

Theorem 4.2.

Assume that (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq327_HTML.gif ),(H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq328_HTML.gif ), and (H https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq329_HTML.gif ) hold. If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq330_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq331_HTML.gif , then there is a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq332_HTML.gif such that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq333_HTML.gif , BVP (4.1) has at least one positive solution.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq334_HTML.gif be given. Define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ101_HTML.gif
(4.19)

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq335_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq336_HTML.gif .

For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq337_HTML.gif , we know that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq338_HTML.gif . By the definition of operator https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq339_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ102_HTML.gif
(4.20)
It follows that we can take https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq340_HTML.gif such that, for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq341_HTML.gif and all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq342_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ103_HTML.gif
(4.21)
Fix https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq343_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq344_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq345_HTML.gif , we obtain a sufficiently large https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq346_HTML.gif such that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq347_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ104_HTML.gif
(4.22)
It follows that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq348_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq349_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ105_HTML.gif
(4.23)
For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq350_HTML.gif , by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq351_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq352_HTML.gif and the definition of Lemma 3.2, there exists a https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq353_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq354_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq355_HTML.gif , thus https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq356_HTML.gif . Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ106_HTML.gif
(4.24)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq357_HTML.gif there is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq358_HTML.gif , such that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq359_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq360_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ107_HTML.gif
(4.25)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq361_HTML.gif .

For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq362_HTML.gif , by(3.8)–(3.10) and Lemma 3.2,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ108_HTML.gif
(4.26)

which by combining with (4.21) completes the proof.

Example 4.3.

Consider the BVP(3.33) in Example 3.9 with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ109_HTML.gif
(4.27)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq363_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq364_HTML.gif is some positive constant, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq365_HTML.gif .

By calculation, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq366_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq367_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq368_HTML.gif ; let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq369_HTML.gif . Then by Theorem(4.1), for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq370_HTML.gif , the above equation has at least one positive solution.

Declarations

Acknowledgments

The authors would like to thank the editor and the reviewers for their valuable comments and suggestions which helped to significantly improve the paper. This work is supported by Distinguished Expert Science Foundation of Naval Aeronautical and Astronautical University.

Authors’ Affiliations

(1)
Institute of Systems Science and Mathematics, Naval Aeronautical and Astronautical University

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Copyright

© The Author(s) Yang Shujie and Shi Bao. 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.