On the existence of bounded solutions for nonlinear second order neutral difference equations (1304.2501v2)

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}