On the existence of bounded solutions for nonlinear second order neutral difference equations

TL;DR

This paper investigates the existence of bounded solutions for a class of nonlinear second order neutral difference equations using measure of noncompactness techniques, extending previous results and exploring stability properties.

Contribution

It introduces new sufficient conditions for bounded solutions of nonlinear neutral difference equations and develops methods to analyze stability and asymptotic behavior.

Findings

Established conditions for bounded solutions

Studied stability and asymptotic stability

Generalized earlier results on difference equations

Abstract

\noindent Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form \begin{equation*} \Delta \left(r_{n}\left(\Delta \left(x_{n}+p_{n}x_{n-k}\right) \right) ^{\gamma}\right) +q_{n}x_{n}^{\alpha}+a_{n}f(x_{n})=0. \end{equation*}% where $x:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$, $a,p,q:{\mathbb{N}}%_{0}\rightarrow {\mathbb{R}}$, $r:{\mathbb{N}}_{0}\rightarrow {\mathbb{R}}% \setminus \{0\}$, $f\colon {\mathbb{R}}\rightarrow {\mathbb{R}}$ is a continuous function, and $k$ is a given positive integer, $\gamma \leq 1$ is ratio of odd positive integers, $\alpha $ is a nonnegative constant. %$\sum a_{n}\left(t\right)$ converges uniformly on ${\mathbb{R}}$. %Here $\bN_0\colon =\left\{0,1,2, \dots \right\}$ and $\bN_k \colon = \left\{k, k+1, -k+2, \dots \right\}$ where $k$ is a given positive integer. Sufficient conditions for the existence of a bounded solution are obtained. Also a special type of stability and asymptotic stability are studied. Some earlier results are generalized. We note that the solution which we obtain does not directly correspond to a fixed point of a certain continuous operator since it is partially iterated. The method which we develop allows for considering through techniques connected with the measure of noncompactness also difference equations with memory. {\small \textbf{Keywords} Difference equation, measures of noncompactness, Darbo's fixed point theorem, boundedness, stability} {\small \textbf{AMS Subject classification} 39A10, 39A22, 39A30}