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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ1_HTML.gif
(1.1)
with boundary value conditions
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ2_HTML.gif
(1.2)
where the operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq1_HTML.gif is the Jacobi operator
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ4_HTML.gif
(1.4)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq2_HTML.gif is a continuous function, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq3_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq5_HTML.gif -Laplacian operator
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ5_HTML.gif
(1.5)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq6_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq7_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq9_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq12_HTML.gif is continuous, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq14_HTML.gif -Laplacian operator:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ6_HTML.gif
(1.6)

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

For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq21_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq22_HTML.gif , let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ7_HTML.gif
(1.7)
Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq23_HTML.gif and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ8_HTML.gif
(1.8)

For any real function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq25_HTML.gif defined on the interval http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq26_HTML.gif and any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq27_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq28_HTML.gif , we denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq29_HTML.gif an element of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq30_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq31_HTML.gif , http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ9_HTML.gif
(1.9)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq33_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq34_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq35_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq36_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq37_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq38_HTML.gif is a continuous function, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq39_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq40_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq41_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq42_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq43_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq44_HTML.gif are nonnegative real constants, and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq45_HTML.gif for http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq47_HTML.gif with the single http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq49_HTML.gif depends on the term http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq50_HTML.gif (i.e., past values of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq51_HTML.gif ).

As usual, a sequence http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq53_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq54_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq55_HTML.gif for http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ11_HTML.gif
(2.1)
or
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq57_HTML.gif , we know that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq58_HTML.gif is convex for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq59_HTML.gif .

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

     
If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq62_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq63_HTML.gif . It follows that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq64_HTML.gif .

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq65_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq66_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq67_HTML.gif , then we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ14_HTML.gif
(2.4)
Hence
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq68_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq69_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq70_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq72_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq73_HTML.gif .

We assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq74_HTML.gif . Then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ16_HTML.gif
(2.6)
Hence
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq75_HTML.gif .

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

     
Now we have
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq78_HTML.gif .

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

     
If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq81_HTML.gif , then we obtain
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq82_HTML.gif .

If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq84_HTML.gif , and so http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq85_HTML.gif , then there exists a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq86_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq87_HTML.gif . Otherwise, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq88_HTML.gif is a trivial solution. Assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq89_HTML.gif , then
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ20_HTML.gif
(2.10)
which implies that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq90_HTML.gif follows.

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

     
Now we obtain
http://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 http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ23_HTML.gif
(2.13)

We always assume that

( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq94_HTML.gif ) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq95_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq96_HTML.gif and http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ24_HTML.gif
(2.14)
where
http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ27_HTML.gif
(2.17)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ28_HTML.gif
(2.18)
The three-point BVP
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq98_HTML.gif by http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ30_HTML.gif
(2.20)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq100_HTML.gif and http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ32_HTML.gif
(2.22)
By (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq102_HTML.gif ), we obtain http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq103_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq104_HTML.gif by solving the above equation:
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq105_HTML.gif by http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ35_HTML.gif
(2.25)

where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq107_HTML.gif and http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ37_HTML.gif
(2.27)
By (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq109_HTML.gif ), we can easily obtain
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ39_HTML.gif
(2.29)

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

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

Lemma 2.4.

Assume that ( http://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, http://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 http://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
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ42_HTML.gif
(2.32)
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq119_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ44_HTML.gif
(2.34)
which implies that
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ46_HTML.gif
(2.36)

Condition (H http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq121_HTML.gif . Therefore http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq122_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq124_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq125_HTML.gif .

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

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

Suppose that http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq128_HTML.gif is a Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq129_HTML.gif is a cone in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq130_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq131_HTML.gif . Furthermore, assume that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq132_HTML.gif is a completely continuous operator and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq133_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq135_HTML.gif  for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq136_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq137_HTML.gif ;

     
  2. (2)

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

     
Assume that http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ49_HTML.gif
(3.3)
Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq143_HTML.gif , it can be expressed as
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ50_HTML.gif
(3.4)
Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq145_HTML.gif . Then for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq146_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq147_HTML.gif and
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ51_HTML.gif
(3.5)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ52_HTML.gif
(3.6)

Then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq149_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq150_HTML.gif is a cone in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq151_HTML.gif .

For http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq152_HTML.gif , we have by (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq153_HTML.gif ) and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq154_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ53_HTML.gif
(3.7)
For every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq155_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq156_HTML.gif , by the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq157_HTML.gif and (3.5), if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq158_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ54_HTML.gif
(3.8)
If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq159_HTML.gif , we have, by (3.4),
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq160_HTML.gif , we obtain for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq161_HTML.gif
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq162_HTML.gif , there is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq163_HTML.gif , such that

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

Proof.

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

Obviously, there is a http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq169_HTML.gif by
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq170_HTML.gif .

Proof.

If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq171_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq173_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq174_HTML.gif , respectively. Thus, (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq175_HTML.gif ) yields
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq176_HTML.gif that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ61_HTML.gif
(3.15)

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

Lemma 3.4.

Suppose that ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq178_HTML.gif ) holds. Then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq180_HTML.gif ) http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq181_HTML.gif

(H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq182_HTML.gif ) http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq184_HTML.gif )–( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq186_HTML.gif ) there exists a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq187_HTML.gif such that, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq188_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq189_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq190_HTML.gif ;

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

or

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

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

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

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

Proof.

Assume that ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq208_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq209_HTML.gif ) hold. For every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq210_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq211_HTML.gif , thus
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ64_HTML.gif
(3.18)
For every http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq213_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq214_HTML.gif . Then by (3.13) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq215_HTML.gif ), we have
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq216_HTML.gif of operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq217_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq218_HTML.gif .

Assume that ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq219_HTML.gif )–( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq220_HTML.gif ) hold, for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq221_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq222_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq223_HTML.gif , by ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq224_HTML.gif ), we have
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ68_HTML.gif
(3.22)
For every http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq226_HTML.gif ,
http://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
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq227_HTML.gif of operator http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq228_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq229_HTML.gif .

Consequently, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq230_HTML.gif or http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq232_HTML.gif )–( http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq234_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq235_HTML.gif ) or ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq236_HTML.gif ) and ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq237_HTML.gif ) hold.

Theorem 3.7.

Assume that ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq238_HTML.gif )–( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq240_HTML.gif ), (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq241_HTML.gif ), and (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq242_HTML.gif )  or (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq243_HTML.gif ), (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq244_HTML.gif ), and (H http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq246_HTML.gif )–(H http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq248_HTML.gif )–(H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq249_HTML.gif ) hold.

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

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

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

which satisfies

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

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

Assume that http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ72_HTML.gif
(3.26)
where
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq259_HTML.gif ), (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq260_HTML.gif ), and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq261_HTML.gif , we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ74_HTML.gif
(3.28)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ75_HTML.gif
(3.29)
and, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq262_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ76_HTML.gif
(3.30)
Let
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq263_HTML.gif ), (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq264_HTML.gif ), and the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq265_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq266_HTML.gif for http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ78_HTML.gif
(3.32)

with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq268_HTML.gif and http://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:
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ79_HTML.gif
(3.33)
That is,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ80_HTML.gif
(3.34)
Then we obtain
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ81_HTML.gif
(3.35)
Let
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ82_HTML.gif
(3.36)

where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq271_HTML.gif )–(H http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq273_HTML.gif :
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ83_HTML.gif
(4.1)

with http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ84_HTML.gif
(4.2)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq275_HTML.gif be the solution of (3.3), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq276_HTML.gif . Then we have
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ85_HTML.gif
(4.3)
Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq277_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq278_HTML.gif be defined as the above. Define http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq279_HTML.gif by
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq280_HTML.gif . Obviously http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq281_HTML.gif is completely continuous and keeps the http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq282_HTML.gif invariant for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq283_HTML.gif .

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

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

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

hold, where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq293_HTML.gif ) holds. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq294_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq295_HTML.gif , there exists an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq296_HTML.gif sufficiently small, such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ88_HTML.gif
(4.6)
By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq297_HTML.gif , there is an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq298_HTML.gif , such that for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq299_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ89_HTML.gif
(4.7)
It follows that, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq300_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq301_HTML.gif ,
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ90_HTML.gif
(4.8)
For every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq302_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq303_HTML.gif , by (3.9), we have
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ92_HTML.gif
(4.10)
If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq304_HTML.gif then for a sufficiently small http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq305_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq306_HTML.gif Similar to the above, for every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq307_HTML.gif , we obtain by (3.10)
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ93_HTML.gif
(4.11)
If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq308_HTML.gif , choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq309_HTML.gif sufficiently large, such that
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ94_HTML.gif
(4.12)
By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq310_HTML.gif , there is an http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq311_HTML.gif such that, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq312_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq313_HTML.gif
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ95_HTML.gif
(4.13)
For every http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ96_HTML.gif
(4.14)
which implies that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq315_HTML.gif . By the definition of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq316_HTML.gif , there is

http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq317_HTML.gif , such that, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq318_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq319_HTML.gif ,
http://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    http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq320_HTML.gif , such that, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq321_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq322_HTML.gif In fact, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq323_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq324_HTML.gif and every http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq325_HTML.gif , http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ99_HTML.gif
(4.17)
which implies that
http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq327_HTML.gif ),(H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq328_HTML.gif ), and (H http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq329_HTML.gif ) hold. If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq330_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq331_HTML.gif , then there is a http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq332_HTML.gif such that for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq334_HTML.gif be given. Define
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ101_HTML.gif
(4.19)

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

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

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

For every http://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,
http://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
http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_Equ109_HTML.gif
(4.27)

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

By calculation, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq366_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq367_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq368_HTML.gif ; let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F973731/MediaObjects/13661_2010_Article_971_IEq369_HTML.gif . Then by Theorem(4.1), for http://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.