Bounded solutions and asymptotic stability to nonlinear second-order neutral difference equations with quasi-differences

TL;DR

This paper investigates bounded solutions and asymptotic stability of nonlinear second-order neutral difference equations with quasi-differences, employing fixed point and approximation methods to establish existence results.

Contribution

It introduces a combined fixed point and approximation approach for analyzing solutions when standard methods are insufficient, especially for $q_n o 1$.

Findings

Established conditions for bounded solutions with $q_n o 1$

Derived existence criteria in $l^p$ spaces using Krasnoselskii's theorem

Extended analysis to cases with $|q_n|<2^{1-p}$

Abstract

This work is devoted to the study of the nonlinear second-order neutral difference equations with quasi-differences of the form $$ \Delta \left( r_{n} \Delta \left( x_{n}+q_{n}x_{n-\tau}\right)\right)= a_{n}f(x_{n-\sigma})+b_n%, \ n\geq n_0 $$ with respect to $(q_n)$. For $q_n\to1$, $q_n\in(0,1)$ the standard fixed point approach is not sufficed to get the existence of the bounded solution, so we combine this method with an approximation technique to achieve our goal. Moreover, for $p\ge 1$ and $\sup|q_n|<2^{1-p}$ using Krasnoselskii's fixed point theorem we obtain sufficient conditions of the existence of the solution which belongs to $l^p$ space.