Solution Map Methods for Stability Analysis of Linear and Nonlinear Volterra Difference Equations

摘要

This paper is concerned with the qualitative behaviour of solutions to Volterra difference equations. These equations have unbounded order and they cannot normally be treated using techniques adapted from the theory of difference equations of fixed finite order. We focus on questions of stability of solutions and we present a unified theory based on the use of a solution map operator that applies to nonlinear equations as well as linear equations and that extends our previous work on difference equations of finite order. We conclude the paper by showing how the approach can be applied to both linear and nonlinear problems with fading memory and also to analyse certain numerical methods for Volterra integral and integro-differential equations.