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In this paper we analyze the existence of bounded solutions for a nonlinear second-order neutral difference equation, which is more general than other equations of this type studied recently. Moreover, our analysis is based under more general conditions than the required in other works, as we only assume the continuity of the involved functions. The main tool here are the so called measures of noncompactness, and more specifically, the celebrated Darbo fixed point theorem. Also we will state, in the specified sense, the stability of the solutions.