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

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)

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

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

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).