** Authors:**
MAGDALENA NOCKOWSKA ROSIAK

** Abstract: **
This work is devoted to the study of the nonlinear second-order neutral difference equations with quasi-differences of the form
$$
\Delta \left( r_{n} \Delta \left( x_{n}+q_{n}x_{n-\tau}\right)\right)= a_{n}f(x_{n-\sigma})+b_n
$$
with respect to $(q_n)$. For $q_n\to1$, $q_n\in(0,1)$ the standard fixed point approach is insufficient 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 for the existence of the solution that belongs to $l^p$ space.

** Keywords: **
Nonlinear neutral difference equation, Krasnoselskii's fixed point theorem, approximation

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