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

Abstract

Using the techniques connected with the measure of noncompactness we investigate the neutral difference equation of the following form \[\Delta \left( r_{n}\left( \Delta \left( x_{n}+p_{n}x_{n-k}\right) \right) ^{\gamma }\right) +q_{n}x_{n}^{\alpha }+a_{n}f(x_{n+1})=0,\] where $x\colon{\mathbb{N}}_{k}\rightarrow {\mathbb{R}}$, $a,p,q\colon {\mathbb{N}}_{0}\rightarrow {\mathbb{R}}$, $r\colon {\mathbb{N}}_{0}\rightarrow {\mathbb{R}} \setminus \{0\}$, $f\colon {\mathbb{R}}\rightarrow {\mathbb{R}}$ is continuous and $k$ is a given positive integer, $\alpha \geq 1$ is a ratio of positive integers with odd denominator, and $\gamma \leq 1$ is ratio of odd positive integers; ${\mathbb{N}}_{k}:=\left\{ k,k+1,\dots \right\}$. Sufficient conditions for the existence of a bounded solution are obtained. Also a special type of stability is studied.

Keywords

• difference equation
• emden--fowler equation
• measures of noncompactness
• darbo's fixed point theorem
• boundedness
• stability