An application of measures of noncompactness in the investigation of boundedness of solutions of second-order neutral difference equations

摘要

The purpose of this paper is to investigate a nonlinear second-order neutral difference equation of the form Δ ( r n Δ ( x n + p n x n ? k ) ) + a n f ( x n ) = 0 , where x : N 0 → R , a : N 0 → R , p , r : N 0 → R ? { 0 } , f : R → R is a continuous function, and k is a given positive integer. Sufficient conditions for the existence of a bounded solution of this equation are obtained. Also, stability and asymptotic stability of this equation are studied. Additionally, the Emden-Fowler difference equation is considered as a special case of the above equation. The obtained results are illustrated by examples. MSC:39A10, 39A22, 39A30.