Solvability of a Second Order Nonlinear Neutral Delay Difference Equation

and Applied Analysis 3 Let l∞ β denote the Banach space of all bounded sequences in Zβ with norm ‖x‖ sup n∈Zβ |xn| for x {xn}n∈Zβ ∈ l∞ β , B d,D { x {xn}n∈Zβ ∈ l∞ β : ‖x − d‖ ≤ D } for d {d}n∈Zβ ∈ l∞ β , D > 0 2.2 represent the closed ball centered at d and with radius D in l∞ β . By a solution of 1.1 , we mean a sequence {xn}n∈Zβ with a positive integer T ≥ n0 τ |β| such that 1.1 is satisfied for all n ≥ T . As is customary, a solution of 1.1 is said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is said to be nonoscillatory. Lemma 2.1 2 . A bounded, uniformly Cauchy subset Y of l∞ β is relatively compact. Lemma 2.2 Krasnoselskii fixed point theorem 10 . Let Y be a nonempty bounded closed convex subset of a Banach space X and S,G : Y → X mappings such that Sx Gy ∈ Y for every pair x, y ∈ Y . If S is a contraction and G is completely continuous, then