On classification of singular matrix difference equations of mixed order

TL;DR

This paper classifies singular matrix difference equations of mixed order using a Weyl-type method, providing criteria for limit point cases and illustrating the influence of off-diagonal coefficients with examples.

Contribution

It introduces a classification framework for singular matrix difference equations of mixed order and establishes new limit point criteria based on coefficients.

Findings

Classification of equations via Weyl's method

Limit point criteria in terms of coefficients

Influence of off-diagonal coefficients demonstrated

Abstract

This paper is concerned with singular matrix difference equations of mixed order. The existence and uniqueness of initial value problems for these equations are derived, and then the classification of them is obtained with a similar classical Weyl's method by selecting a suitable quasi-difference. An equivalent characterization of this classification is given in terms of the number of linearly independent square summable solutions of the equation. The influence of off-diagonal coefficients on the classification is illustrated by two examples. In particular, two limit point criteria are established in terms of coefficients of the equation.