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

  • Svetlana Roman1Email author and

    Affiliated with

    • Artūras Štikonas1

      Affiliated with

      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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ1_HTML.gif
      (1.1)
      where http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq2_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq3_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq5_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq6_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq7_HTML.gif .

      For all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq9_HTML.gif , the equality
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq10_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq11_HTML.gif . http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq13_HTML.gif and functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq14_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq15_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq16_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq17_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq18_HTML.gif is a Kronecker symbol: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq19_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq20_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq21_HTML.gif if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq23_HTML.gif , there exists a unique choice of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq24_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq25_HTML.gif . If we have the vector-function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq26_HTML.gif , then we consider the matrix function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq27_HTML.gif and its functional determinant http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq28_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ4_HTML.gif
      (2.2)
      The Wronskian determinant http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ5_HTML.gif
      (2.3)
      Let (if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq30_HTML.gif )
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ6_HTML.gif
      (2.4)

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

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

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq38_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq39_HTML.gif , then we get http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq40_HTML.gif . So, the function http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq42_HTML.gif and we write http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq43_HTML.gif .

      Lemma 2.1.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq44_HTML.gif , then the equality
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq45_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq46_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq47_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq48_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq50_HTML.gif , then the equality
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ9_HTML.gif
      (2.7)

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

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

      Let the functions http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq71_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq72_HTML.gif are linearly independent on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq73_HTML.gif if and only if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq75_HTML.gif . The functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq76_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq77_HTML.gif are linearly independent if the equality http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq78_HTML.gif is valid only for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq79_HTML.gif . We can rewrite this equality as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq80_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq81_HTML.gif . A system of functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq82_HTML.gif is the basis of the http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ14_HTML.gif
      (2.12)
      Thus, the functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq84_HTML.gif , http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ16_HTML.gif
      (2.14)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq86_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq87_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq88_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ17_HTML.gif
      (2.15)
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ19_HTML.gif
      (3.1)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq89_HTML.gif . Let http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ20_HTML.gif
      (3.2)
      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq93_HTML.gif . We introduce new functions
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ21_HTML.gif
      (3.3)
      For these functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq94_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq95_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq96_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq97_HTML.gif . So, the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq98_HTML.gif satisfies equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq99_HTML.gif , and the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq100_HTML.gif satisfies equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq101_HTML.gif . Components of the functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq102_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq103_HTML.gif in the basis http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq104_HTML.gif are
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq105_HTML.gif , http://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
      http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq110_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq111_HTML.gif are linearly independent;

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

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

      If we take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq115_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq116_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq117_HTML.gif , http://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
      http://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
      http://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))
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ26_HTML.gif
      (3.8)
      Similarly we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ27_HTML.gif
      (3.9)

      Lemma 3.2.

      Let http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq120_HTML.gif .

      Corollary 3.3.

      If functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq121_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq122_HTML.gif are linearly independent, that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq123_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ29_HTML.gif
      (3.11)
      that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq124_HTML.gif , then the two bases http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq125_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq126_HTML.gif are biorthogonal:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ30_HTML.gif
      (3.12)
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq127_HTML.gif instead of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq128_HTML.gif .

      Remark 3.5.

      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq129_HTML.gif is another fundamental system and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq130_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq131_HTML.gif , then
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq133_HTML.gif : http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq134_HTML.gif , http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ33_HTML.gif
      (4.1)
      Then http://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,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ34_HTML.gif
      (4.2)

      For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq138_HTML.gif , this equality shows that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq140_HTML.gif (the case where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq141_HTML.gif are linearly dependent solutions) or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq142_HTML.gif for all http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ35_HTML.gif
      (4.3)
      with two additional conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ36_HTML.gif
      (4.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq144_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq146_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq147_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq148_HTML.gif are arbitrary constants and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq150_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq151_HTML.gif by the functions http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq153_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq154_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq155_HTML.gif [12], we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ38_HTML.gif
      (4.6)
      The functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq156_HTML.gif and http://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,
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ39_HTML.gif
      (4.7)
      Denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq158_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq159_HTML.gif . We derive ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq160_HTML.gif )
      http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq161_HTML.gif by definition)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ41_HTML.gif
      (4.9)
      We can take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq162_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq163_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq164_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq165_HTML.gif , and we obtain the following systems:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ42_HTML.gif
      (4.10)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq166_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq167_HTML.gif are linearly independent, the determinant http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ43_HTML.gif
      (4.11)
      Then
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq169_HTML.gif ) is
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ45_HTML.gif
      (4.13)
      for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq170_HTML.gif . We introduce a function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq171_HTML.gif :
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq172_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq173_HTML.gif as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ47_HTML.gif
      (4.15)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq174_HTML.gif , http://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 http://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 http://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
      http://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
      http://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))
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ51_HTML.gif
      (4.19)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq178_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq179_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq180_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq181_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq182_HTML.gif , http://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)
      http://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
      http://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 http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq185_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq186_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq187_HTML.gif , any right-hand side function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq188_HTML.gif , and any functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq189_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq190_HTML.gif and any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq191_HTML.gif , http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ55_HTML.gif
      (4.23)
      and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq193_HTML.gif . The difference http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq194_HTML.gif satisfies the problem
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ57_HTML.gif
      (4.25)
      or
      http://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
      http://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
      http://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
      http://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 http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq198_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq199_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq200_HTML.gif be a linear operator, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq201_HTML.gif . Consider an operator equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq202_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq203_HTML.gif is unknown and http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ62_HTML.gif
      (5.1)
      that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq205_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq206_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq207_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq208_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq209_HTML.gif . We have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq210_HTML.gif . In the case http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq212_HTML.gif homogeneous linear equations
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ63_HTML.gif
      (5.2)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq213_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq214_HTML.gif , and denote
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq215_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq216_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq219_HTML.gif and the additional operator equation http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq220_HTML.gif , and we have the following problem:
      http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ66_HTML.gif
      (5.5)

      then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq222_HTML.gif with the additional condition http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq223_HTML.gif . Green's function exists if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq224_HTML.gif . This condition is equivalent to http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq225_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq227_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq228_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq229_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq230_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq231_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq232_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq233_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq234_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq235_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq236_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq237_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq238_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq239_HTML.gif . So, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq241_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq242_HTML.gif .

      Example 5.1.

      In the case http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ67_HTML.gif
      (5.6)

      The function http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq245_HTML.gif . In the case http://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), http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq248_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq249_HTML.gif , where the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq250_HTML.gif is defined on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq252_HTML.gif
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq253_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq254_HTML.gif . In this paper, we introduce meshes
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ69_HTML.gif
      (5.8)
      with the step sizes http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq255_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq256_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq257_HTML.gif , and a semi-integer mesh
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ70_HTML.gif
      (5.9)
      with the step sizes http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq258_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq259_HTML.gif . We define the inner product
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ71_HTML.gif
      (5.10)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq260_HTML.gif , and the following mesh operators:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ72_HTML.gif
      (5.11)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq261_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq262_HTML.gif , where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq264_HTML.gif
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq265_HTML.gif is used [9, 11]
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ74_HTML.gif
      (5.13)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq266_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq267_HTML.gif . The relations between these functions are
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq268_HTML.gif , then we can calculate http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq269_HTML.gif , and vice versa. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq270_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq271_HTML.gif ), then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq272_HTML.gif coincides with http://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]):
      http://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: http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq276_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq277_HTML.gif , where functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq278_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq279_HTML.gif are linearly independent, is equal to
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq280_HTML.gif )
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ78_HTML.gif
      (5.17)
      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq281_HTML.gif , http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ79_HTML.gif
      (5.18)
      We have
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq284_HTML.gif , where http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq288_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq289_HTML.gif of two nonhomogeneous problems
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ81_HTML.gif
      (5.20)

      with the same http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq291_HTML.gif exists and the functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq292_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq293_HTML.gif are linearly independent, then
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq294_HTML.gif )
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ83_HTML.gif
      (5.22)
      If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq295_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ84_HTML.gif
      (5.23)
      So, Green's function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq296_HTML.gif is equal to
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq297_HTML.gif instead of http://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
      http://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
      http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ89_HTML.gif
      (6.1)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ90_HTML.gif
      (6.2)
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq299_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq300_HTML.gif , is a classical part and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq301_HTML.gif , http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq305_HTML.gif . For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq306_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq307_HTML.gif , we derive
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ92_HTML.gif
      (6.4)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq308_HTML.gif , http://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
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ94_HTML.gif
      (6.6)
      We introduce a mesh http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq310_HTML.gif (see (5.8)). Denote http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq311_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq312_HTML.gif for http://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)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ95_HTML.gif
      (6.7)
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq314_HTML.gif , http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq316_HTML.gif , http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ97_HTML.gif
      (6.9)
      where
      http://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: http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq318_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq319_HTML.gif . Then
      http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq320_HTML.gif we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq321_HTML.gif ,
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq323_HTML.gif of the initial problem
      http://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
      http://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))
      http://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 http://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 http://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
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq327_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq328_HTML.gif ,
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ107_HTML.gif
      (6.19)
      if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq330_HTML.gif . By substituting Green's function http://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
      http://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])
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ110_HTML.gif
      (6.22)

      where http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ111_HTML.gif
      (6.23)

      where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq333_HTML.gif are approximations of the weight functions http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq334_HTML.gif in integral boundary conditions, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq336_HTML.gif approximation (e.g., trapezoidal formula http://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 ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq338_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq339_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq341_HTML.gif , where
      http://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)
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ113_HTML.gif
      (6.25)

      where http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ114_HTML.gif
      (6.26)

      if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq343_HTML.gif , where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq345_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq347_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq348_HTML.gif and we take http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq350_HTML.gif ) is
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ116_HTML.gif
      (6.28)
      We consider three types ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq351_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq352_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq353_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq354_HTML.gif ; http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq355_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq356_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq357_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq358_HTML.gif ) of discrete boundary conditions
      http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_Equ117_HTML.gif
      (6.29)

      All the cases yield http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F767024/MediaObjects/13661_2010_Article_58_IEq361_HTML.gif -order difference equation with http://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.