The factorization technique for difference equations

TL;DR

This paper introduces a factorization method for solving multidimensional difference equations with continuous variables, utilizing boundary value problem techniques from elliptic pseudo differential equations to establish solvability in Sobolev--Slobodetskii spaces.

Contribution

It proposes a novel approach to analyze difference equations using boundary value problem methods from elliptic pseudo differential equations, ensuring unique solutions in Sobolev--Slobodetskii spaces.

Findings

Established unique solvability for boundary value problems of difference equations.

Applied elliptic pseudo differential equation techniques to difference equations.

Extended the theory to multidimensional difference equations with continuous variables.

Abstract

We study multidimensional difference equations with a continual variable in the Sobolev--Slobodetskii spaces. Using ideas and methods of the theory of boundary value problems for elliptic pseudo differential equations we suggest to consider certain boundary value problems for such difference equations. Special boundary conditions permit to prove unique solvability for these boundary value problems in appropriate Sobolev--Slobodetskii spaces.