- Research Article
- Open Access

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

- Yang Shujie
^{1}Email author and - Shi Bao
^{1}

**2010**:973731

https://doi.org/10.1155/2010/973731

© The Author(s) Yang Shujie and Shi Bao. 2010

**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.

## Keywords

- Banach Space
- Laplacian Operator
- Point Theory
- Trivial Solution
- Classical Condition

## 1. Introduction

where is a continuous function, , and .

where , , , , , , is continuous, .

where , , and , , , and are nonnegative real constants.

For any real function defined on the interval and any with , we denote by an element of defined by , .

where and , , , , is a continuous function, and for , , , and are nonnegative real constants, and for .

from ordinary difference equations. Here this condition connects the history with the single . This is suggested by the well-posedness of BVP (1.9), since the function depends on the term (i.e., past values of ).

As usual, a sequence is said to be a positive solution of BVP (1.9) if it satisfies BVP (1.9) and for with for .

## 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.

Then (1.9) has no positive solution.

Proof.

From , we know that is convex for .

- (1)
Consider that .

which is a contradiction to the convexity of .

which is a contradiction to the convexity of . If for , then is a trivial solution. So there exists a such that .

which is a contradiction to the convexity of .

- (2)
Consider that .

which is a contradiction to the convexity of .

- (1)
Consider that .

which is a contradiction to the convexity of .

If , similar to the above proof, we can also get a contradiction.

A contradiction to the convexity of follows.

- (2)
Consider that .

which is a contradiction to the convexity of .

We always assume that

( ) , and .

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

Theorem 2.2.

Proof.

where and are constants that will be determined.

where and are constants that will be determined.

where is defined in (2.14). That is, is the Green's function of BVP (2.13).

Remark 2.3.

Then .

Lemma 2.4.

Assume that ( ) 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.

Condition (H ) implies that (2.36) has a unique solution . Therefore for . 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 , .

with .

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

- (1)
if for , then ;

- (2)
if for , then .

Then is a Banach space endowed with norm and is a cone in .

Lemma 3.2.

For every , there is , such that

Proof.

Obviously, there is a , such that (3.11) holds.

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 .

Proof.

which implies that .

Lemma 3.4.

Suppose that ( ) holds. Then is completely continuous.

We assume that

(H )

(H ) .

We have the following main results.

Theorem 3.5.

Assume that ( )–( ) hold. Then BVP (3.1) has at least one positive solution if the following conditions are satisfied:

(H ) there exists a such that, for , if , then ;

(H ) there exists a such that, for , if , then

or

(H ) ;

(H ) there exists a such that, for , if , then ;

(H ) there exists an , such that, for , if , then

Proof.

So by (3.18) and (3.20), there exists one positive fixed point of operator with .

So by (3.22) and (3.24), there exists one positive fixed point of operator with .

Consequently, or is a positive solution of BVP (3.1).

Theorem 3.6.

Assume that ( )–( ) hold. Then BVP (3.1) has at least one positive solution if ( ) and ( ) or ( ) and ( ) hold.

Theorem 3.7.

Assume that ( )–( ) hold. Then BVP (3.1) has at least two positive solutions if (H ), (H ), and (H ) or (H ), (H ), and (H ) hold.

Theorem 3.8.

Assume that (H )–(H ) hold. Then BVP (3.1) has at least three positive solutions if (H )–(H ) hold.

Assume that , , and

( )

which satisfies

H .

Obviously, exists.

with and .

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.

where

By calculation, we can see that (H )–(H ) hold, then by Theorem 3.8, the BVP (3.33) has at least three positive solutions.

## 4. Eigenvalue Intervals

with .

Then solving the BVP (4.1) is equivalent to finding fixed points in . Obviously is completely continuous and keeps the invariant for .

respectively. We have the following results.

Theorem 4.1.

Assume that (H ), (H ), (H ),

(H ) ,

(H )

hold, where , then BVP (4.1) has at least one positive solution, where is a positive constant.

Proof.

Finally, we consider the assumption . By the definition of , there is

Theorem 4.2.

Assume that (H ),(H ), and (H ) hold. If or , then there is a such that for , BVP (4.1) has at least one positive solution.

Proof.

Then , where .

where .

which by combining with (4.21) completes the proof.

Example 4.3.

where , is some positive constant, .

By calculation, , , and ; let . Then by Theorem(4.1), for , 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

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