Almost automorphic solutions of discrete delayed neutral system

TL;DR

This paper establishes new conditions for the existence of almost automorphic solutions in discrete delayed neutral systems, improving previous results by removing the boundedness assumption on the inverse matrix.

Contribution

It introduces novel existence criteria for almost automorphic solutions without requiring the inverse of the system matrix to be bounded, extending current theoretical understanding.

Findings

Proves existence of almost automorphic solutions under new conditions.

Removes the boundedness assumption on the inverse matrix $A(t)^{-1}$.

Provides examples demonstrating the applicability of the results.

Abstract

We study almost automorphic solutions of the discrete delayed neutral dynamic system% \[ x(t+1)=A(t)x(t)+\Delta Q(t,x(t-g(t)))+G(t,x(t),x(t-g(t))) \] by means of a fixed point theorem due to Krasnoselskii. Using discrete variant of exponential dichotomy and proving uniqueness of projector of discrete exponential dichotomy we invert the equation and obtain some limit results leading to sufficient conditions for the existence of almost automorphic solutions of the neutral system. Unlike the existing literature we prove our existence results without assuming boundedness of inverse matrix $A\left( t\right) ^{-1}$. Hence, we significantly improve the results in the existing literature. We provide two examples to illustrate effectiveness of our results. Finally, we also provide an existence result for almost periodic solutions of the system.