Generalized Green's functions for \(m\)th-order discrete nonlocal problems (Q2627901)

This paper is conerned with linear discrete problems of the form \[ \begin{aligned} a_{i}^{m}u_{i+m}+\cdots+a_{i}^{1}u_{i+1}+a_{i}^{0}u_{i} &= f_{i},\text{ \;\;~}a_{i}^{0},a_{i}^{m}\neq 0,\,i\in 0,1,\dots,n-m \\ \sum\limits_{j=0}^{n}L_{k}^{j}u_{j} &=g_{k},\text{ \;\;}k=1,\dots,m,\end{aligned} \] where the data and unknown are complex numbers. The discrete problem can be written as a matrix system \(Au=b\) (with \(b=\left(f_{0},\dots,f_{n-m},g_{1},\dots,g_{m}\right) ^{T}\)), and its Moore-Penrose inverse \(A^{\dag }\) exists and is unique and the corresponding unique minimum norm least squares solution is \(u^{o}=A^{\dag }b.\) The vector \(u^{o}\) can be written in the form \[ u^{o}=G^{g}\begin{pmatrix} f_{0} \\ \vdots \\ f_{n-m}\end{pmatrix} +\sum\limits_{k=1}^{m}v^{g,k}g_{k}. \] The matrix \(G^{g}\) is then called the generalized discrete Green's function and the \(m\) vectors \(v^{g,1},v^{g,2},\dots,v^{g,m}\) are said to form the generalized biorthogonal fundamental system. It is shown that the generalized discrete Green's function coincides with the true Green's function of the original problem in case the latter exists. Furthermore, many other properties of the gneralized discrete Green's function are given. In partiuclar, applications to difference equations with nonlocal conditions and Bitsadze-Samarskii conditions are derived.