Open Access

Green's Function for Discrete Second-Order Problems with Nonlocal Boundary Conditions

Boundary Value Problems20102011:767024

DOI: 10.1155/2011/767024

Received: 1 June 2010

Accepted: 9 November 2010

Published: 24 November 2010

Abstract

We investigate a second-order discrete problem with two additional conditions which are described by a pair of linearly independent linear functionals. We have found the solution to this problem and presented a formula and the existence condition of Green's function if the general solution of a homogeneous equation is known. We have obtained the relation between two Green's functions of two nonhomogeneous problems. It allows us to find Green's function for the same equation but with different additional conditions. The obtained results are applied to problems with nonlocal boundary conditions.

1. Introduction

The study of boundary-value problems for linear differential equations was initiated by many authors. The formulae of Green's functions for many problems with classical boundary conditions are presented in [1]. In this book, Green's functions are constructed for regular and singular boundary-value problems for ODEs, the Helmholtz equation, and linear nonstationary equations. The investigation of semilinear problems with Nonlocal Boundary Conditions (NBCs) and the existence of their positive solutions are well founded on the investigation of Green's function for linear problems with NBCs [27]. In [8], Green's function for a differential second-order problem with additional conditions, for example, NBCs, has been investigated.

In this paper, we consider a discrete difference equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq1_HTML.gif . This equation is analogous to the linear differential equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ2_HTML.gif
(1.2)

In order to estimate a solution of a boundary value problem for a difference equation, it is possible to use the representation of this solution by Green's function [9].

In [10], Bahvalov et al. established the analogy between the finite difference equations of one discrete variable and the ordinary differential equations. Also, they constructed a Green's function for a grid boundary-value problem in the simplest case (Dirichlet BVP).

The direct method for solving difference equations and an iterative method for solving the grid equations of a general form and their application to difference equations are considered in [11, 12]. Various variants of Thomas' algorithm (monotone, nonmonotone, cyclic, etc.) for one-dimensional three-pointwise equations are described. Also, modern economic direct methods for solving Poisson difference equations in a rectangle with boundary conditions of various types are stated.

Chung and Yau [13] study discrete Green's functions and their relationship with discrete Laplace equations. They discuss several methods for deriving Green's functions. Liu et al. [14] give an application of the estimate to discrete Green's function with a high accuracy analysis of the three-dimensional block finite element approximation.

In this paper, expressions of Green's functions for (1.1) have been obtained using the method of variation of parameters [12]. The advantage of this method is that it is possible to construct the Green's function for a nonhomogeneous equation (1.1) with the variable coefficients https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq2_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq3_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq4_HTML.gif and various additional conditions (e.g., NBCs). The main result of this paper is formulated in Theorem 4.1, Lemma 5.3, and Theorem 5.4. Theorem 4.1 can be used to get the solution of an equation with a difference operator with any two linearly independent additional conditions if the general solution of a homogeneous equation is known. Theorem 5.4 gives an expression for Green's function and allows us to find Green's function for an equation with two additional conditions if we know Green's function for the same equation but with different additional conditions. Lemma 5.3 is a partial case of this theorem if we know the special Green's function for the problem with discrete (initial) conditions. We apply these results to BVPs with NBCs: first, we construct the Green's function for classical BCs, then we can construct Green's function for a problem with NBCs directly (Lemma 5.3) or via Green's function for a classical problem (Theorem 5.4). Conditions for the existence of Green's function were found. The results of this paper can be used for the investigation of quasilinear problems, conditions for positiveness of Green's functions, and solutions with various BCs, for example, NBCs.

The structure of the paper is as follows. In Section 2, we review the properties of functional determinants and linear functionals. We construct a special basis of the solutions in Section 3 and introduce some functions that are independent of this basis. The expression of the solution to the second-order linear difference equation with two additional conditions is obtained in Section 4. In Section 5, discrete Green's function definitions of this problem are considered. Then a Green's function is constructed for the second-order linear difference equation. Applications to problems with NBCs are presented in Section 6.

2. Notation

We begin this section with simple properties of determinants. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq5_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq6_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq7_HTML.gif .

For all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq9_HTML.gif , the equality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ3_HTML.gif
(2.1)

is valid. The proof follows from the Laplace expansion theorem [8].

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq10_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq11_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq12_HTML.gif be a linear space of real (complex) functions. Note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq13_HTML.gif and functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq15_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq16_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq17_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq18_HTML.gif is a Kronecker symbol: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq19_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq20_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq21_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq22_HTML.gif ), form a basis of this linear space. So, for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq23_HTML.gif , there exists a unique choice of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq24_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq25_HTML.gif . If we have the vector-function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq26_HTML.gif , then we consider the matrix function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq27_HTML.gif and its functional determinant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq28_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ4_HTML.gif
(2.2)
The Wronskian determinant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq29_HTML.gif in the theory of difference equations is denoted as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ5_HTML.gif
(2.3)
Let (if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq30_HTML.gif )
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ6_HTML.gif
(2.4)

We define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq31_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq32_HTML.gif . Note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq33_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq34_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq35_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq36_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq37_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ7_HTML.gif
(2.5)

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq38_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq39_HTML.gif , then we get https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq40_HTML.gif . So, the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq41_HTML.gif is invariant with respect to the basis https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq42_HTML.gif and we write https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq43_HTML.gif .

Lemma 2.1.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq44_HTML.gif , then the equality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ8_HTML.gif
(2.6)

is valid.

Proof.

If we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq45_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq46_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq47_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq48_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq49_HTML.gif , in (2.1), then we get equality (2.6).

Corollary 2.2.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq50_HTML.gif , then the equality
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ9_HTML.gif
(2.7)

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq51_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq52_HTML.gif is valid.

We consider the space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq53_HTML.gif of linear functionals in the space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq54_HTML.gif , and we use the notation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq55_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq56_HTML.gif for the functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq57_HTML.gif value of the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq58_HTML.gif . Functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq59_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq60_HTML.gif form a dual basis for basis https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq61_HTML.gif . Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq62_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq63_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq64_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq66_HTML.gif , then we can define the linear functional (direct product) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq67_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ10_HTML.gif
(2.8)
We define the matrix
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ11_HTML.gif
(2.9)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq68_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq69_HTML.gif , and the determinant
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ12_HTML.gif
(2.10)
For example,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ13_HTML.gif
(2.11)

Let the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq70_HTML.gif be linearly independent.

Lemma 2.3.

Functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq71_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq72_HTML.gif are linearly independent on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq73_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq74_HTML.gif .

Proof.

We can investigate the case where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq75_HTML.gif . The functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq76_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq77_HTML.gif are linearly independent if the equality https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq78_HTML.gif is valid only for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq79_HTML.gif . We can rewrite this equality as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq80_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq81_HTML.gif . A system of functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq82_HTML.gif is the basis of the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq83_HTML.gif , and the above-mentioned equality is equivalent to the condition below
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ14_HTML.gif
(2.12)
Thus, the functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq84_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq85_HTML.gif are linearly independent if and only if the vectors
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ15_HTML.gif
(2.13)
are linearly independent. But these vectors are linearly independent if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ16_HTML.gif
(2.14)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq86_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq87_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq88_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ17_HTML.gif
(2.15)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ18_HTML.gif
(2.16)

3. Special Basis in a Two-Dimensional Space of Solutions

Let us consider a homogeneous linear difference equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ19_HTML.gif
(3.1)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq89_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq90_HTML.gif a be two-dimensional linear space of solutions, and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq91_HTML.gif be a fixed basis of this linear space. We investigate additional equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ20_HTML.gif
(3.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq92_HTML.gif are linearly independent linear functionals, and we use the notation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq93_HTML.gif . We introduce new functions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ21_HTML.gif
(3.3)
For these functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq94_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq95_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq96_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq97_HTML.gif . So, the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq98_HTML.gif satisfies equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq99_HTML.gif , and the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq100_HTML.gif satisfies equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq101_HTML.gif . Components of the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq102_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq103_HTML.gif in the basis https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq104_HTML.gif are
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ22_HTML.gif
(3.4)
respectively. It follows that the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq105_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq106_HTML.gif are linearly independent if and only if
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ23_HTML.gif
(3.5)

But this determinant is zero if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq107_HTML.gif . We combine Lemma 2.3 and these results in the following lemma.

Lemma 3.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq108_HTML.gif be the basis of the linear space https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq109_HTML.gif . Then the following propositions are equivalent:

(1)the functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq110_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq111_HTML.gif are linearly independent;

(2)the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq112_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq113_HTML.gif are linearly independent;

(3) https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq114_HTML.gif .

If we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq115_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq116_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq117_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq118_HTML.gif , in formula (2.1), then we get
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ24_HTML.gif
(3.6)
The left-hand side of this equality is equal to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ25_HTML.gif
(3.7)
Finally, we have (see (3.3))
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ26_HTML.gif
(3.8)
Similarly we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ27_HTML.gif
(3.9)

Lemma 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq119_HTML.gif be a fundamental system of homogeneous equation (3.1). Then equality (3.9) is valid, and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ28_HTML.gif
(3.10)

Propositions in Lemma 3.1 are equivalent to the condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq120_HTML.gif .

Corollary 3.3.

If functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq121_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq122_HTML.gif are linearly independent, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq123_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ29_HTML.gif
(3.11)
that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq124_HTML.gif , then the two bases https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq125_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq126_HTML.gif are biorthogonal:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ30_HTML.gif
(3.12)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ31_HTML.gif
(3.13)

Remark 3.4.

Propositions in Lemma 3.1 are valid if we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq127_HTML.gif instead of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq128_HTML.gif .

Remark 3.5.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq129_HTML.gif is another fundamental system and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq130_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq131_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ32_HTML.gif
(3.14)

(see (2.15)). So, the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq132_HTML.gif is invariant with respect to the basis https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq133_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq134_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq135_HTML.gif .

4. Discrete Difference Equation with Two Additional Conditions

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq136_HTML.gif be the solutions of a homogeneous equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ33_HTML.gif
(4.1)
Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq137_HTML.gif is the solution of (4.1), that is,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ34_HTML.gif
(4.2)

For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq138_HTML.gif , this equality shows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq139_HTML.gif , and we arrive at the conclusion that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq140_HTML.gif (the case where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq141_HTML.gif are linearly dependent solutions) or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq142_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq143_HTML.gif (the case of the fundamental system).

In this section, we consider a nonhomogeneous difference equation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ35_HTML.gif
(4.3)
with two additional conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ36_HTML.gif
(4.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq144_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq145_HTML.gif are linearly independent functionals.

4.1. The Solution to a Nonhomogeneous Problem with Additional Homogeneous Conditions

A general solution of (4.1) is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq146_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq147_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq148_HTML.gif are arbitrary constants and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq149_HTML.gif is the fundamental system of this homogeneous equation. We replace the constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq150_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq151_HTML.gif by the functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq152_HTML.gif (Method of Variation of Parameters [12]), respectively. Then, by substituting
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ37_HTML.gif
(4.5)
into (4.3) and denoting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq153_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq154_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq155_HTML.gif [12], we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ38_HTML.gif
(4.6)
The functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq156_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq157_HTML.gif are solutions of the homogeneous equation (4.1). Consequently,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ39_HTML.gif
(4.7)
Denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq158_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq159_HTML.gif . We derive ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq160_HTML.gif )
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ40_HTML.gif
(4.8)
Then we rewrite equality (4.7) as ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq161_HTML.gif by definition)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ41_HTML.gif
(4.9)
We can take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq162_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq163_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq164_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq165_HTML.gif , and we obtain the following systems:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ42_HTML.gif
(4.10)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq166_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq167_HTML.gif are linearly independent, the determinant https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq168_HTML.gif is not equal to zero and system (4.10) has a unique solution
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ43_HTML.gif
(4.11)
Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ44_HTML.gif
(4.12)
and the formula for solution of nonhomogeneous equation (with the conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq169_HTML.gif ) is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ45_HTML.gif
(4.13)
for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq170_HTML.gif . We introduce a function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq171_HTML.gif :
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ46_HTML.gif
(4.14)
Then we rewrite (4.13) and the conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq173_HTML.gif as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ47_HTML.gif
(4.15)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq174_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq175_HTML.gif . So, we derive a formula for the general solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq176_HTML.gif . We use this formula for the special basis https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq177_HTML.gif (see (3.11)). In this case, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ48_HTML.gif
(4.16)
Let there be homogeneous conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ49_HTML.gif
(4.17)
So, by substituting general solution (4.16) into homogeneous additional conditions, we find (see (3.12))
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ50_HTML.gif
(4.18)
Next we obtain a formula for solution in the case of difference equation with two additional homogeneous conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ51_HTML.gif
(4.19)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq178_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq179_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq180_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq181_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq182_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq183_HTML.gif .

4.2. A Homogeneous Equation with Additional Conditions

Let us consider the homogeneous equation (4.1) with the additional conditions (4.4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ52_HTML.gif
(4.20)
We can find the solution
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ53_HTML.gif
(4.21)

to this problem if the general solution is inserted into the additional conditions.

The solution of nonhomogeneous problems is of the form https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq184_HTML.gif (see (4.19) and (4.21)). Thus, we get a simple formula for solving problem (4.3)-(4.4).

Theorem 4.1.

The solution of problem (4.3)-(4.4) can be expressed by the formula
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ54_HTML.gif
(4.22)

Formula (4.22) can be effectively employed to get the solutions to the linear difference equation, with various https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq185_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq186_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq187_HTML.gif , any right-hand side function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq188_HTML.gif , and any functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq189_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq190_HTML.gif and any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq191_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq192_HTML.gif , provided that the general solution of the homogeneous equation is known. In this paper, we also use (4.22) to get formulae for Green's function.

4.3. Relation between Two Solutions

Next, let us consider two problems with the same nonhomogeneous difference equation with a difference operator as in the previous subsection
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ55_HTML.gif
(4.23)
and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq193_HTML.gif . The difference https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq194_HTML.gif satisfies the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ56_HTML.gif
(4.24)
Thus, it follows from formula (4.21) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ57_HTML.gif
(4.25)
or
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ58_HTML.gif
(4.26)

and we can express the solution of the second problem (4.23) via the solution of the first problem.

Corollary 4.2.

The relation
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ59_HTML.gif
(4.27)

between the two solutions of problems (4.23) is valid.

Proof.

If we expand the determinant in (4.27) according to the last row, then we get formula (4.26).

Remark 4.3.

The determinant in formula (4.27) is equal to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ60_HTML.gif
(4.28)
In this way, we can rewrite (4.27) as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ61_HTML.gif
(4.29)

Note that in this formula the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq195_HTML.gif is in the first term only and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq196_HTML.gif is invariant with regard to the basis https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq197_HTML.gif .

5. Green's Functions

5.1. Definitions of Discrete Green's Functions

We propose a definition of Green's function (see [9, 12]). In this section, we suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq198_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq199_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq200_HTML.gif be a linear operator, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq201_HTML.gif . Consider an operator equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq202_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq203_HTML.gif is unknown and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq204_HTML.gif is given. This operator equation, in a discrete case, is equivalent to the system of linear equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ62_HTML.gif
(5.1)
that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq205_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq206_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq207_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq208_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq209_HTML.gif . We have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq210_HTML.gif . In the case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq211_HTML.gif , we must add additional conditions if we want to get a unique solution. Let us add https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq212_HTML.gif homogeneous linear equations
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ63_HTML.gif
(5.2)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq213_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq214_HTML.gif , and denote
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ64_HTML.gif
(5.3)
We have a system of linear equations https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq215_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq216_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq217_HTML.gif . The necessary condition for a unique solution is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq218_HTML.gif . Additional equations (5.2) define the linear operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq219_HTML.gif and the additional operator equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq220_HTML.gif , and we have the following problem:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ65_HTML.gif
(5.4)
If solution of (5.4) allows the following representation:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ66_HTML.gif
(5.5)

then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq221_HTML.gif is called Green's function of operator https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq222_HTML.gif with the additional condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq223_HTML.gif . Green's function exists if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq224_HTML.gif . This condition is equivalent to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq225_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq226_HTML.gif . In this case, we can easily get an expression for Green's function in representation (5.5) from the Kramer formula or from the formula for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq227_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq228_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq229_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq230_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq231_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq232_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq233_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq234_HTML.gif (or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq235_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq236_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq237_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq238_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq239_HTML.gif . So, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq240_HTML.gif is a unique solution of problem (5.4) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq241_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq242_HTML.gif .

Example 5.1.

In the case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq243_HTML.gif , formula (5.5) can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ67_HTML.gif
(5.6)

The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq244_HTML.gif is an example of Green's function for (4.3) with discrete (initial) conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq245_HTML.gif . In the case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq246_HTML.gif , formula (5.6) is the same as (4.15), https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq247_HTML.gif .

Remark 5.2.

Let us consider the case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq248_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq249_HTML.gif , where the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq250_HTML.gif is defined on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq251_HTML.gif , then we use the shifted Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq252_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ68_HTML.gif
(5.7)
For finite-difference schemes, discrete functions are defined in points https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq253_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq254_HTML.gif . In this paper, we introduce meshes
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ69_HTML.gif
(5.8)
with the step sizes https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq255_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq256_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq257_HTML.gif , and a semi-integer mesh
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ70_HTML.gif
(5.9)
with the step sizes https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq258_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq259_HTML.gif . We define the inner product
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ71_HTML.gif
(5.10)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq260_HTML.gif , and the following mesh operators:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ72_HTML.gif
(5.11)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq261_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq262_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq263_HTML.gif , then we define the Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq264_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ73_HTML.gif
(5.12)
For many applications another discrete Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq265_HTML.gif is used [9, 11]
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ74_HTML.gif
(5.13)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq266_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq267_HTML.gif . The relations between these functions are
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ75_HTML.gif
(5.14)

So, if we know the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq268_HTML.gif , then we can calculate https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq269_HTML.gif , and vice versa. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq270_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq271_HTML.gif ), then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq272_HTML.gif coincides with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq273_HTML.gif .

Note that the Wronskian determinant can be defined by the following formula (see [10]):
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ76_HTML.gif
(5.15)

5.2. Green's Functions for a Linear Difference Equation with Additional Conditions

Let us consider the nonhomogeneous equation (4.3) with the operator: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq274_HTML.gif , where additional homogeneous conditions define the subspace https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq275_HTML.gif .

Lemma 5.3.

Green's function for problem (4.3) with the homogeneous additional conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq276_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq277_HTML.gif , where functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq278_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq279_HTML.gif are linearly independent, is equal to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ77_HTML.gif
(5.16)

Proof.

In the previous section, we derived a formula of the solution (see Theorem 4.1 for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq280_HTML.gif )
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ78_HTML.gif
(5.17)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq281_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq282_HTML.gif . So, Green's function is equal to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ79_HTML.gif
(5.18)
We have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ80_HTML.gif
(5.19)

too. If we expand this determinant according to the last row and divide by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq283_HTML.gif , then we get the right-hand side of (5.18). The lemma is proved.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq284_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq285_HTML.gif , then we get that Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq286_HTML.gif , that is, it is invariant with respect to the basis https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq287_HTML.gif .

For the theoretical investigation of problems with NBCs, the next result about the relations between Green's functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq288_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq289_HTML.gif of two nonhomogeneous problems
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ81_HTML.gif
(5.20)

with the same https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq290_HTML.gif , is useful.

Theorem 5.4.

If Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq291_HTML.gif exists and the functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq292_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq293_HTML.gif are linearly independent, then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ82_HTML.gif
(5.21)

Proof.

We have equality (4.26) (the case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq294_HTML.gif )
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ83_HTML.gif
(5.22)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq295_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ84_HTML.gif
(5.23)
So, Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq296_HTML.gif is equal to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ85_HTML.gif
(5.24)

A further proof of this theorem repeats the proof of Lemma 5.3 (we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq297_HTML.gif instead of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq298_HTML.gif ).

Remark 5.5.

Instead of formula (5.18), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ86_HTML.gif
(5.25)
We can write the determinant in formula (5.21) in the explicit way
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ87_HTML.gif
(5.26)
Formulaes (5.25) and (5.26) easily allow us to find Green's function for an equation with two additional conditions if we know Green's function for the same equation, but with other additional conditions. The formula
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ88_HTML.gif
(5.27)

can be used to get the solutions of the equations with a difference operator with any two linear additional (initial or boundary or nonlocal boundary) conditions if the general solution of a homogeneous equation is known.

6. Applications to Problems with NBC

Let us investigate Green's function for the problem with nonlocal boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ89_HTML.gif
(6.1)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ90_HTML.gif
(6.2)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ91_HTML.gif
(6.3)

We can write many problems with nonlocal boundary conditions (NBC) in this form, where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq299_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq300_HTML.gif , is a classical part and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq301_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq302_HTML.gif , is a nonlocal part of boundary conditions.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq303_HTML.gif , then problem (6.1)–(6.3) becomes classical. Suppose that there exists Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq304_HTML.gif for the classical case. Then Green's function exists for problem (6.1)–(6.3) if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq305_HTML.gif . For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq306_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq307_HTML.gif , we derive
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ92_HTML.gif
(6.4)
Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq308_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq309_HTML.gif , we can rewrite formula (5.26) as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ93_HTML.gif
(6.5)

Example 6.1.

Let us consider the differential equation with two nonlocal boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ94_HTML.gif
(6.6)
We introduce a mesh https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq310_HTML.gif (see (5.8)). Denote https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq311_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq312_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq313_HTML.gif . Then problem (6.6) can be approximated by a finite-difference problem (scheme)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ95_HTML.gif
(6.7)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ96_HTML.gif
(6.8)

We suppose that the points https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq314_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq315_HTML.gif are coincident with the grid points, that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq316_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq317_HTML.gif .

We rewrite (6.7) in the following form:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ97_HTML.gif
(6.9)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ98_HTML.gif
(6.10)
We can take the following fundamental system: https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq318_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq319_HTML.gif . Then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ99_HTML.gif
(6.11)
As a result, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ100_HTML.gif
(6.12)
For a problem with the boundary conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq320_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq321_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ101_HTML.gif
(6.13)
and we express Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq322_HTML.gif of the Dirichlet problem via Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq323_HTML.gif of the initial problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ102_HTML.gif
(6.14)
We derive expressions for "classical" Green's function
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ103_HTML.gif
(6.15)
or (see (5.7) and (5.13))
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ104_HTML.gif
(6.16)

Remark 6.2.

Note that the index of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq324_HTML.gif on the right-hand side of (6.9) is shifted (cf. (6.1)).

Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq325_HTML.gif is the same as in [10], and it is equal to Green's function
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ105_HTML.gif
(6.17)

for differential problem (6.6) at grid points in the case https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq326_HTML.gif .

For a "nonlocal" problem with the boundary conditions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq327_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq328_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ106_HTML.gif
(6.18)
It follows from (6.5) that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ107_HTML.gif
(6.19)
if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq329_HTML.gif . Green's function does not exist for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq330_HTML.gif . By substituting Green's function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq331_HTML.gif for the problem with the classical boundary conditions into the above equation, we obtain Green's function for the problem with nonlocal boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ108_HTML.gif
(6.20)
This formula corresponds to the formula of Green's function for differential problem (6.6) (see [4])
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ109_HTML.gif
(6.21)

Example 6.3.

Let us consider the problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ110_HTML.gif
(6.22)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq332_HTML.gif .

Problem (6.22) can be approximated by the difference problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ111_HTML.gif
(6.23)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq333_HTML.gif are approximations of the weight functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq334_HTML.gif in integral boundary conditions, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq335_HTML.gif is a quadrature formula for the integral https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq336_HTML.gif approximation (e.g., trapezoidal formula https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq337_HTML.gif ).

The expression of Green's function for the problem with the classical boundary conditions ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq338_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq339_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq340_HTML.gif ) is described in Example 6.1. The existence condition of Green's function for problem (6.23) is https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq341_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ112_HTML.gif
(6.24)
(such a condition was obtained for problem (6.23) in [15, 16]) and Green's function is equal to (see Theorem 5.4)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ113_HTML.gif
(6.25)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq342_HTML.gif is defined by (6.15).

Green's function for differential problem (6.22) was derived in [8]. For this problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ114_HTML.gif
(6.26)

if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq343_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq344_HTML.gif is defined by formula (6.17).

Remark 6.4.

We could substitute (6.15) into (6.25) and obtain an explicit expression of Green's function. However, it would be quite complicated, and we will not write it out. Note that, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq345_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq346_HTML.gif , then discrete problem (6.23) is the same as (6.7)-(6.8). For example, it happens if a trapezoidal formula is used for the approximation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq347_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq348_HTML.gif and we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq349_HTML.gif . It is easy to see that we could obtain the same expression for Green's function (6.19) in this case.

Example 6.5.

Let us consider a difference problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ115_HTML.gif
(6.27)
A condition for the existence of the Green's function (fundamental system https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq350_HTML.gif ) is
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ116_HTML.gif
(6.28)
We consider three types ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq351_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq352_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq353_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq354_HTML.gif ; https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq355_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq356_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq357_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq358_HTML.gif ) of discrete boundary conditions
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ117_HTML.gif
(6.29)

All the cases yield https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq359_HTML.gif . Consequently, Green's function for the three problems does not exist.

7. Conclusions

Green's function for problems with additional conditions is related with Green's function of a similar problem, and this relation is expressed by formulae (5.26). Green's function exists if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq360_HTML.gif . If we know Green's function for the problem with additional conditions and the fundamental basis of a homogeneous difference equation, then we can obtain Green's function for a problem with the same equation but with other additional conditions. It is shown by a few examples for problems with NBCs that but formulae (5.26) can be applied to a very wide class of problems with various boundary conditions as well as additional conditions.

All the results of this paper can be easily generalized to the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq361_HTML.gif -order difference equation with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq362_HTML.gif additional functional conditions. The obtained results are similar to a differential case [8, 17].

Authors’ Affiliations

(1)
Institute of Mathematics and Informatics, Vilnius University

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Copyright

© S. Roman and A. Štikonas. 2011

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.