Open Access

Existence of Solutions for Fourth-Order Four-Point Boundary Value Problem on Time Scales

Boundary Value Problems20092009:491952

DOI: 10.1155/2009/491952

Received: 11 April 2009

Accepted: 28 July 2009

Published: 19 August 2009

Abstract

We present an existence result for fourth-order four-point boundary value problem on time scales. Our analysis is based on a fixed point theorem due to Krasnoselskii and Zabreiko.

1. Introduction

Very recently, Karaca [1] investigated the following fourth-order four-point boundary value problem on time scales:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ1_HTML.gif
(1.1)

for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq1_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq3_HTML.gif And the author made the following assumptions:

(A1) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq5_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq6_HTML.gif

(A2) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq8_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq9_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq10_HTML.gif

The following key lemma is provided in [1].

Lemma 1.1 (see [1, Lemma  2.5]).

Assume that conditions ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq11_HTML.gif ) and ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq12_HTML.gif ) are satisfied. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq13_HTML.gif then the boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ2_HTML.gif
(1.2)
has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ3_HTML.gif
(1.3)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ4_HTML.gif
(1.4)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ5_HTML.gif
(1.5)
Here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq14_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq15_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq16_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq17_HTML.gif are given as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ6_HTML.gif
(1.6)
Unfortunately, this lemma is wrong. Without considering the whole interval https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq18_HTML.gif the author only considers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq19_HTML.gif in the Green's function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq20_HTML.gif Thus, the expression of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq21_HTML.gif (1.3) which is a solution to BVP (1.2) is incorrect. In fact, if one takes https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq22_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq23_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq24_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq25_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq26_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq27_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq28_HTML.gif then (1.1) reduces to the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ7_HTML.gif
(1.7)
The counterexample is given by [2], from which one can see clearly that [1, Lemma  2.5] is wrong. If one takes https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq29_HTML.gif , here https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq30_HTML.gif is a constant, then (1.1) reduces to the following fourth-order four-point boundary value problem on time scales:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ8_HTML.gif
(1.8)

The purpose of this paper is to establish some existence criteria of solution for BVP (1.8) which is a special case of (1.1). The paper is organized as follows. In Section 2, some basic time-scale definitions are presented and several preliminary results are given. In Section 3, by employing a fixed point theorem due to Krasnoselskii and Zabreiko, we establish existence of solutions criteria for BVP (1.8). Section 4 is devoted to an example illustrating our main result.

2. Preliminaries

The study of dynamic equations on time scales goes back to its founder Hilger [3] and it is a new area of still fairly theoretical exploration in mathematics. In the recent years boundary value problem on time scales has received considerable attention [46]. And an increasing interest in studying the existence of solutions to dynamic equations on time scales is observed, for example, see [716].

For convenience, we first recall some definitions and calculus on time scales, so that the paper is self-contained. For the further details concerning the time scales, please see [1719] which are excellent works for the calculus of time scales.

A time scale https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq31_HTML.gif is an arbitrary nonempty closed subset of real numbers https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq32_HTML.gif . The operators https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq33_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq34_HTML.gif from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq35_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq36_HTML.gif

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ9_HTML.gif
(2.1)

are called the forward jump operator and the backward jump operator, respectively.

For all https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq37_HTML.gif we assume throughout that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq38_HTML.gif has the topology that it inherits from the standard topology on https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq39_HTML.gif The notations https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq40_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq41_HTML.gif and so on, will denote time-scale intervals

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ10_HTML.gif
(2.2)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq42_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq43_HTML.gif

Definition 2.1.

Fix https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq44_HTML.gif Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq45_HTML.gif Then we define https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq46_HTML.gif to be the number (if it exists) with the property that given https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq47_HTML.gif there is a neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq48_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq49_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ11_HTML.gif
(2.3)

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq50_HTML.gif is called derivative of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq51_HTML.gif

Definition 2.2.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq52_HTML.gif then we define the integral by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ12_HTML.gif
(2.4)
We say that a function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq53_HTML.gif is regressive provided
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ13_HTML.gif
(2.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq54_HTML.gif which is called graininess function. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq55_HTML.gif is a regressive function, then the generalized exponential function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq56_HTML.gif is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ14_HTML.gif
(2.6)
for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq57_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq58_HTML.gif is the cylinder transformation, which is defined by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ15_HTML.gif
(2.7)
Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq59_HTML.gif be two regressive functions, then define
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ16_HTML.gif
(2.8)

The generalized function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq60_HTML.gif has then the following properties.

Lemma 2.3 (see [18]).

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq61_HTML.gif are two regressive functions, then

(i) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq62_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq63_HTML.gif

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq64_HTML.gif

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq65_HTML.gif

(iv) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq66_HTML.gif

(v) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq67_HTML.gif

(vi) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq68_HTML.gif

(vii) https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq69_HTML.gif

The following well-known fixed point theorem will play a very important role in proving our main result.

Theorem 2.4 (see [20]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq70_HTML.gif be a Banach space, and let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq71_HTML.gif be completely continuous. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq72_HTML.gif is a bounded linear operator such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq73_HTML.gif is not an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq74_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ17_HTML.gif
(2.9)

Then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq75_HTML.gif has a fixed point in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq76_HTML.gif

Throughout this paper, let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq77_HTML.gif be endowed with the norm by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ18_HTML.gif
(2.10)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq78_HTML.gif And we make the following assumptions:

() https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq80_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq81_HTML.gif

() https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq83_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq84_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq85_HTML.gif

() https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq87_HTML.gif

Set

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ19_HTML.gif
(2.11)

For convenience, we denote

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ20_HTML.gif
(2.12)

First, we present two lemmas about the calculus on Green functions which are crucial in our main results.

Lemma 2.5.

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq88_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq89_HTML.gif are satisfied. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq90_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq91_HTML.gif is a solution of the following boundary value problem (BVP):
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ21_HTML.gif
(2.13)
if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ22_HTML.gif
(2.14)
where the Green's function of (2.13) is as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ23_HTML.gif
(2.15)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq92_HTML.gif are given as (2.12), respectively.

Proof.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq93_HTML.gif is a solution of (2.13), setting
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ24_HTML.gif
(2.16)
then it follows from the first equation of (2.13) that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ25_HTML.gif
(2.17)
Multiplying (2.17) by https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq94_HTML.gif and integrating from https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq95_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq96_HTML.gif we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ26_HTML.gif
(2.18)
Similarly, by (2.18), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ27_HTML.gif
(2.19)
Then substituting (2.18) into (2.19), we get for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq97_HTML.gif that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ28_HTML.gif
(2.20)
Substituting this expression for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq98_HTML.gif into the boundary conditions of (2.13). By some calculations, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ29_HTML.gif
(2.21)
Then substituting (2.21) into (2.20), we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ30_HTML.gif
(2.22)
By interchanging the order of integration and some rearrangement of (2.22), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ31_HTML.gif
(2.23)

Thus, we obtain (2.14) consequently.

On the other hand, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq99_HTML.gif satisfies (2.14), then direct differentiation of (2.14) yields
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ32_HTML.gif
(2.24)

And it is easy to know that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq100_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq101_HTML.gif satisfies (2.13).

Corollary 2.6.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq102_HTML.gif then BVP (2.13) reduces to the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ33_HTML.gif
(2.25)
From Lemma 2.5, BVP (2.25) has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ34_HTML.gif
(2.26)
where the Green's function of (2.25) is as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ35_HTML.gif
(2.27)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ36_HTML.gif
(2.28)
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ37_HTML.gif
(2.29)

Proof.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq103_HTML.gif is a solution of (2.25), take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq104_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq105_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq106_HTML.gif Hence, from (2.20) we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ38_HTML.gif
(2.30)
Substituting this expression for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq107_HTML.gif into the boundary conditions of (2.25). By some calculations, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ39_HTML.gif
(2.31)
where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq108_HTML.gif is given as (2.28). Then substituting (2.31) into (2.30), we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ40_HTML.gif
(2.32)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq109_HTML.gif are as in (2.29), respectively. By some rearrangement of (2.32), we obtain (2.26) consequently.

From the proof of Corollary 2.6, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq110_HTML.gif take https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq111_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq112_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq113_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq114_HTML.gif we get the following result.

Corollary 2.7.

The following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ41_HTML.gif
(2.33)
has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ42_HTML.gif
(2.34)
where the Green's function of (2.33) is as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ43_HTML.gif
(2.35)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ44_HTML.gif
(2.36)
After some rearrangement of (2.35), one obtains
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ45_HTML.gif
(2.37)

Remark 2.8.

Green function (2.37) associated with BVP (2.33) which is a special case of (2.13) is coincident with part of [21, Lemma  1].

Lemma 2.9.

Assume that conditions ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq115_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq116_HTML.gif ) are satisfied. If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq117_HTML.gif then boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ46_HTML.gif
(2.38)
has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ47_HTML.gif
(2.39)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ48_HTML.gif
(2.40)

and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq118_HTML.gif is given in Lemma 2.5.

Proof.

Consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ49_HTML.gif
(2.41)

The Green's function associated with the BVP (2.41) is https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq119_HTML.gif . This completes the proof.

Remark 2.10.

In [1, Lemma  2.5], the solution of (1.2) is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ50_HTML.gif
(2.42)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq120_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq121_HTML.gif are given as (1.4) and (1.5), respectively. In fact, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq122_HTML.gif is incorrect. Thus, we give the right form of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq123_HTML.gif as the special case https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq124_HTML.gif in our Lemma 2.9.

3. Main Results

Theorem 3.1.

Assume ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq125_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq126_HTML.gif ) are satisfied. Moreover, suppose that the following condition is satisfied:

https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq127_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq128_HTML.gif are continuous, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq129_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ51_HTML.gif
(3.1)
and there exists a continuous nonnegative function https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq130_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq131_HTML.gif If
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ52_HTML.gif
(3.2)
where
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ53_HTML.gif
(3.3)

then BVP (1.8) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq132_HTML.gif .

Proof.

Define an operator https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq133_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ54_HTML.gif
(3.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq134_HTML.gif is given by (2.40). Then by Lemmas 2.5 and 2.9, it is clear that the fixed points of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq135_HTML.gif are the solutions to the boundary value problem (1.8). First of all, we claim that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq136_HTML.gif is a completely continuous operator, which is divided into 3 steps.

Step 1 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq137_HTML.gif is continuous).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq138_HTML.gif be a sequence such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq139_HTML.gif then we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ55_HTML.gif
(3.5)

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq140_HTML.gif are continuous, we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq141_HTML.gif which yields https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq142_HTML.gif That is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq143_HTML.gif is continuous.

Step 2 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq144_HTML.gif maps bounded sets into bounded sets in https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq145_HTML.gif ).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq146_HTML.gif be a bounded set. Then, for https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq147_HTML.gif and any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq148_HTML.gif we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ56_HTML.gif
(3.6)

By virtue of the continuity of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq149_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq150_HTML.gif , we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq151_HTML.gif is bounded uniformly, and so https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq152_HTML.gif is a bounded set.

Step 3 ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq153_HTML.gif maps bounded sets into equicontinuous sets of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq154_HTML.gif ).

Let https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq155_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq156_HTML.gif then
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ57_HTML.gif
(3.7)

The right hand side tends to uniformly zero as https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq157_HTML.gif Consequently, Steps 1–3 together with the Arzela-Ascoli theorem show that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq158_HTML.gif is completely continuous.

Now we consider the following boundary value problem:
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ58_HTML.gif
(3.8)
Define
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ59_HTML.gif
(3.9)

Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq159_HTML.gif is a completely continuous bounded linear operator. Moreover, the fixed point of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq160_HTML.gif is a solution of the BVP (3.8) and conversely.

We are now in the position to claim that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq161_HTML.gif is not an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq162_HTML.gif

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq163_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq164_HTML.gif then (3.8) has no nontrivial solution.

If https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq165_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq166_HTML.gif suppose that the BVP (3.8) has a nontrivial solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq167_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq168_HTML.gif then we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ60_HTML.gif
(3.10)
which yields
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ61_HTML.gif
(3.11)
On the other hand, we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ62_HTML.gif
(3.12)

From the above discussion (3.11) and (3.12), we have https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq169_HTML.gif . This contradiction implies that boundary value problem (3.8) has no trivial solution. Hence, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq170_HTML.gif is not an eigenvalue of https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq171_HTML.gif

At last, we show that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ63_HTML.gif
(3.13)
By https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq172_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq173_HTML.gif then for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq174_HTML.gif there exist a https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq175_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ64_HTML.gif
(3.14)

Set https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq176_HTML.gif and select https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq177_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq178_HTML.gif

Denote
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ65_HTML.gif
(3.15)
Thus for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq179_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq180_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq181_HTML.gif it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ66_HTML.gif
(3.16)
In a similar way, we also conclude that for any https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq182_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ67_HTML.gif
(3.17)
Therefore,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ68_HTML.gif
(3.18)
On the other hand, we get
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ69_HTML.gif
(3.19)
Combining (3.18) with (3.19), we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ70_HTML.gif
(3.20)

Theorem 2.4 guarantees that boundary value problem (1.8) has a solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq183_HTML.gif It is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq184_HTML.gif when https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq185_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq186_HTML.gif In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq187_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq188_HTML.gif will lead to a contradiction, which completes the proof.

4. Application

We give an example to illustrate our result.

Example 4.1.

Consider the fourth-order four-pint boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ71_HTML.gif
(4.1)
Notice that https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq189_HTML.gif To show that (4.1) has at least one nontrivial solution we apply Theorem 3.1 with https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq190_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq191_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq192_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq193_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq194_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq195_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq196_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq197_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq198_HTML.gif Obviously, ( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq199_HTML.gif )–( https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq200_HTML.gif ) are satisfied. And
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ72_HTML.gif
(4.2)

Since https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq201_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq202_HTML.gif we have the following.

By simple calculation we have
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ73_HTML.gif
(4.3)
On the other hand, we notice that
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ74_HTML.gif
(4.4)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_Equ75_HTML.gif
(4.5)

That is, https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq203_HTML.gif is satisfied. Thus, Theorem 3.1 guarantees that (4.1) has at least one nontrivial solution https://static-content.springer.com/image/art%3A10.1155%2F2009%2F491952/MediaObjects/13661_2009_Article_849_IEq204_HTML.gif

Declarations

Acknowledgments

The authors would like to thank the referees for their valuable suggestions and comments. This work is supported by the National Natural Science Foundation of China (10771212) and the Natural Science Foundation of Jiangsu Education Office (06KJB110010).

Authors’ Affiliations

(1)
School of Mathematical Science, Yangzhou University
(2)
Department of Mathematics, Huaiyin Normal University

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© Dandan Yang et al. 2009

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