Generalized Green’s functions for second-order discrete boundary-value problems with nonlocal boundary conditions

In this paper, generalized Green’s functions for second-order discrete boundaryvalue problems with nonlocal boundary conditions are investigated, where the necessary and sufficient existence condition of discrete Green’s function is not satisfied and nonlocal boundary conditions are described by linear functionals.


Introduction
In practice, problems often arise where we cannot measure data directly at the boundary. Then nonlocal boundary conditions (NBCs) instead of classical boundary conditions (BCs) are often formulated. During the last decade there has been a great interest in solving problems with NBCs by numerical methods. Let X n := {0, 1, 2, . . . , n} and F (X n ) := {u|u : X n → C} denote the space of complex linear functions with the basis {δ j : δ j i = δ j (i)}. We consider the space F * (X n ) of linear functionals in the space F (X n ). Let δ i , u = u i = u(i).
Let us consider the second-order discrete problem with NBCs L j , u := κ j , u − γ j κ j , u = 0, j = 1, 2, where L : F (X n ) → F (X n−2 ) = im L is a linear operator and L 1 , L 2 are linear functionals, let L = (L 1 , L 2 ). Many NBCs can be written in the form (2), where κ j , u , j = 1, 2, are classical parts and κ j , u , j = 1, 2, are nonlocal parts of BCs. If the unique solution of the problem (1)-(2) can be given by where (v j , w j ) Xn := n j=0 v j w j , v, w ∈ F (X n ), then the function G ∈ F (X n × X n−2 ) is called discrete Green's function of operator L with NBCs (2). According to S. Ro-man [2], the necessary and sufficient existence condition of discrete Green's function is where {u 1 , u 2 } is a fundamental system of homogeneous equation (1). In this paper we consider the problem (1)-(2) and generalize its discrete Green function when D(L)[u] = 0.

Moore-Penrose inverse
The problem (1)-(2) is equivalent to the linear system of equations Then the existence condition of discrete Green's function is equivalent to the condition det A = 0. Therefore, discrete Green's function can be constructed by If det A = 0, then discrete Green's function doesn't exist.

Definition 1.
A matrix X ∈ C n×m is called the Moore-Penrose inverse of A ∈ C m×n and denoted by A † , if it satisfies all Penrose equations where A * denotes the adjoint matrix of A, C m×n -m × n complex matrices.
The existence and uniqueness of the Moore-Penrose inverse was proved by Urquhart and Penrose, respectively [1]. It follows that the Moore-Penrose inverse of nonsingular matrix coincides with the ordinary inverse. According to this, we define generalized discrete Green's function of operator L with NBCs (2) by (4) for arbitrary c ∈ C n . Here P ker A is the orthogonal projector on ker A.
Thus, the general least squares solution is where c ∈ C n is an arbitrary vector, I is the identity matrix, and A (1,3) is any matrix, satisfying both Penrose equations AXA = A, (AX) * = AX.

Remark 1.
Because the Moore-Penrose inverse satisfies all four Penrose equations, it also satisfies the first and third Penrose equations. Thus, the vector (5) is always the general least squares solution for consistent or inconsistent linear system of equations, i.e., the vector (5) always minimizes the Euclidean norm of residual vector According to Remark 1, we define generalized discrete Green's functions for consistent and inconsistent linear systems of equations by the same formula (4).

Applications to problems with NBCs
Let us denote L j , u := n k=0 b k j u k , j = 1, 2.
Then the system (3) can be written in the extended matrix form Let dim ker A = dim ker A * = r ∈ {1, 2}. Then det A = 0. Let us suppose that the basic (n + 1 − r)-order minor of matrix A ∈ C (n+1)×(n+1) satisfies M n+1−r = det A = 0, where A = ( A ij ), i, j ∈ X n−r . Then the solution of ker A = {w ∈ C n+1 : Aw = 0} basis is equivalent to the solution of the problem where w = (w 0 , w 1 , . . . , w n−r ) T , g 1 i = 0, i ∈ X n−r−2 , g 1 n−r−1 = −a 2 n−r−1 w n+1−r , g 1 n−r = −δ r 2 (a 1 n−2 w n−1 + a 2 n−2 w n ) − δ r 1 b n 1 w n . It is easy to see that (6) is a restriction of discrete problem where matrix determinant equals to M n+1−r = 0. Therefore, D(l 1 , l 2 )[u] = 0. Then the solution of (6) is where G ij ∈ F (X n × X n−2 ) is discrete Green's function of operator L with NBCs l k , w = 0, k = 1, 2, and G i,n−1 : . Let e i = (δ i j ), i, j ∈ X n , be the standard basis of R n+1 . Then the kernel of A is composed of the vector Taking w n+1−i w n+1−j = δ i j , i, j = 1, r, we get the concrete basis of ker A. The solution of ker A * = {v ∈ C n+1 : A * v = 0} basis is equivalent to the solution of the problem where A * is the adjoint matrix of (6) matrix Similarly, we can show that the basis of ker A * is composed of vectors Then other vectors are orthogonalised by Gram-Schmidt orthogonalization process where (· , ·) denotes the standard inner product. Taking α i = x ′ i · y ′ i , i = 1, r, and applying Theorem 1, we get that the Moore-Penrose inverse of A is Then generalized discrete Green's function is and, according to (5), the general solution of (3) is given by Example 1. Let us consider differential equation with Bitsadze-Samarskij NBC x ∈ (0, 1), We introduce the mesh ω h = {x i = ih: i ∈ X n , nh = 1}. Suppose ξ coincides with a mesh point, i.e., ξ = sh. We consider such an approximation problem where The problem (9)-(10) has a unique solution and discrete Green's function if γ = 1 ξ . We consider the case, when discrete Green's function doesn't exist, i.e., γ = 1 ξ . Firstly, discrete Green's function of operator L with l 1 , u := u 0 = 0 and l 2 , u := u n = 0 is Then it follows from (7) and (8)  We can show that v 2 = 1 6 γ 2 h 2 (n − s) 2(n − s) ns 2 + 3 + ns + 1, s ∈ X n .

Conclusion
1. Generalized discrete Green's function always exists and is uniquely constructed.
3. If M n+1−r = 0, then generalized discrete Green's function can be described by the discrete Green function of the same discrete equation with simpler NBCs.