Journal of Function Spaces

Volume 2015, Article ID 201946, 7 pages

http://dx.doi.org/10.1155/2015/201946

## The Generalized Green’s Function for Boundary Value Problem of Second Order Difference Equation

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 30 June 2014; Accepted 27 July 2014

Academic Editor: Kishin Sadarangani

Copyright © 2015 Xiaoling Han and Juanjuan Huang. 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.

#### Abstract

Let and . In this paper, by building the generalized Green’s function for the problems, we study the solvability of the S-L problem , , , and the periodic S-L problem , , where the parameter is an eigenvalue of the linear problem , , or the problem , , and , is defined and real valued on , and in the periodic S-L problem we have .

#### 1. Introduction

There are many results about the solvability of S-L problem and the periodic S-L problem where , , and is defined and real valued on , and . Consider , , in the periodic S-L problem we have .

If the parameter is not an eigenvalue of the linear problem or the periodic S-L problem then the linear operator is reversible; we can get an equivalent summation form for the problem (1) or problem (2) by using Green’s function; there are many results in this case; we refer readers to [1–4]. However, when is an eigenvalue, we cannot get the summation form in usual way. So, in this paper we introduce the concept of generalized Green’s function and construct the generalized Green function for the two problems. On this base we still can convert the originally irreversible difference operator into summation form. Moreover, we also obtain the necessary and sufficient condition for the solvability of problems (1) and (2). We refer readers to [5–10] for more details about discrete equation.

#### 2. Preliminaries

We first wish to collect some basic lemmas that will be important to us in what follows.

*Definition 1 (see [5]). *Let and be solutions of ; then

Lemma 2 (see [5]). *If and are defined on , then
**
for .*

By summing both sides of (6) from to , we get the following lemma.

Lemma 3 (see [5]). *Assume that and are defined on . Then
*

*Lemma 4 (see [5]). If and are solutions of , then , , where C is a constant.*

*Remark 5. *If and are linearly dependent, then ; if they are linearly independent, then .

*3. The Main Results for Problem (1)*

*Theorem 6. The semihomogeneous boundary value problem (1) has a solution if and only if
where is an eigenfunction corresponding to the eigenvalue .*

*Proof. *Let be the solution of problem (1) and the nontrivial solution for homogeneous problem (3); using
and , , we have
Since and satisfy the boundary condition of (1), then

On the other hand, choose a solution for , satisfying , ; then
By conditions , , , and (9), observe that
then
that is, . Thus, is a solution of problem (9), and this completes the proof.

*In what follows, we begin to build the generalized Green function.*

*Theorem 7. If the S-L problem (3) has a solution , and , then there exists a unique function which satisfies the following properties:(1);
(2);
(3);
(4). The function having the above properties is called the generalized Green function of the problem (1).*

*Proof. *Choose another solution of which is linearly independent with , such that

Let be any one particular solution of . Thus, by the structure of general solution for nonhomogeneous difference equations, we know that
where can be contained by the properties of .

Firstly, let satisfy the boundary conditions
By the fact that and , we get that
Since and are linearly independent, thus, and , so we get that and are
then
However, from (16) and we get
By (21), we get
In the following we will prove that (20) is consistent with (22).

By
we get
From
and we get
Now, we get
that is, (20) and (22) are consistent.

In order to determine and , by , we obtain ; putting into , we get
then

All these show that we can get , , , and uniquely one by one; then is unique too; that is, the function constructed by the above steps has all the properties of generalized Green’s function. This completes the proof.

*Example 8. *Consider the generalized Green function of the S-L problem

*Proof. *From the conditions and , we can get
It is easy to verify that
is a particular solution of the following equation:
Note that
So
then we can get the generalized Green function

Now, we prove that by the extended Green function it still can convert the originally irreversible difference operator into reversible form, and its summation form is unique.

*Theorem 9. Let be the generalized Green function of the S-L problem (2), and satisfying is a solution of the S-L problem, and . Then the solution of the semihomogeneous boundary value problem
can be expressed by
*

*Proof. *By the fact that
then

To prove the uniqueness, it needs to be proved that we cannot find and which satisfy except and ; if there exists , then by Theorem 6, we know that is also an eigenfunction of the S-L problem; this shows that and are linearly dependent, which completes the proof.

*4. The Main Results for Problem (2)*

*4. The Main Results for Problem (2)*

*In what follows, we build the generalized Green function of the periodic S-L problem.*

*Theorem 10. If the periodic S-L problem (2) has a solution , and choosing a linearly independent solution with for , then there exists a function which satisfies the following properties:(1);
(2);
(3);
(4) and .The function having the above properties is called the generalized Green function to the problem (2).*

*Proof. *Choose so that

Let be any one particular solution to ; then
where can be contained by the properties of .

Firstly, let satisfy the boundary conditions
By the fact that and in the periodic problem we have ; then .

Thus, can be contained; that is, by the conditions (3) and (4), we get
In order to determine , by , we have ; by , we have ; putting into and respectively, it shows that
Combine the above equations; and can be determined; then and can be determined, so also can be determined; that is, the function set by the above steps has all the properties of generalized Green’s function. This completes the proof.

*Remark 11. *Because the choice of in the above structure of generalized Green’s function is not unique, so the generalized Green function of periodic S-L problem (2) is changed with , to have different form.

*Example 12. *Consider the generalized Green function of the periodic S-L problem

*Proof. *Let , and from
we can let , (where ). It is easy to verify that
is a particular solution of the following equation:
By Theorem 10, we have
then
So we can get the generalized Green function

However, if we put , then the generalized Green function is

*Be similar to Theorem 9, we can get the following result; however, because the generalized Green function of the periodic S-L problem (2) is not unique, summation form of difference operator is also not unique.*

*Theorem 13. Let be the generalized Green function of the periodic S-L problem (2); then the solution of the semihomogeneous boundary value problem
can be expressed by
*

*Conflict of Interests*

*Conflict of Interests**The authors declare that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgments*

*Acknowledgments**The authors are grateful to the anonymous referees for their valuable suggestions. This research was supported by the National Natural Science Foundation of China (11101335 and 11201378).*

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