Global Attractivity in Nonlinear Higher Order Difference Equations in Banach Algebras

TL;DR

This paper extends the analysis of nonlinear higher order difference equations to Banach algebras, providing new conditions for solutions to converge to zero and establishing global attractivity in a more general algebraic setting.

Contribution

It generalizes existing real-valued results to Banach algebras and introduces a new reduction of order technique to derive broader convergence conditions.

Findings

Established new sufficient conditions for convergence to zero in Banach algebras.

Extended the parameter ranges for global attractivity of the origin.

Applied reduction of order to nonlinear difference equations in algebraic contexts.

Abstract

Nonlinear higher order difference equations with linear arguments (containing linear forms within nonlinear maps of the space) are well-defined on Banach algebras. The scalar forms of these equations (i.e., with real variables and parameters) have appeared frequently in the literature. By generalizing existing results from real numbers to algebras and using a new result on reduction of order, new sufficient conditions are obtained for the convergence to zero of all solutions of nonlinear difference equations with linear arguments. Where reduction of order is possible, these conditions extend the ranges of parameters for which the origin is a global attractor even when all variables and parameters are real numbers.