Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation

Min Liu,Zhenyu Guo

doi:10.1155/2010/767620

Abstract

The existence of bounded nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation , , where , , , and are integers, and are real sequences, , and is a mapping, is studied. Some sufficient conditions for the existence of bounded nonoscillatory solutions of this equation are established by using Schauder fixed point theorem and Krasnoselskii fixed point theorem and expatiated through seven theorems according to the range of value of the sequence . Moreover, these sufficient conditions guarantee that this equation has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.

Highlights

Introduction and PreliminariesRecently, the interest in the study of the solvability of difference equations has been increasing see 1–17 and references cited therein
Some authors have paied their attention to various difference equations
Motivated and inspired by the papers mentioned above, in this paper, we investigate the following higher-order nonlinear neutral delay difference equation: Δ akn · · · Δ a2nΔ a1nΔ xn bnxn−d f n, xn−r1n, xn−r2n, . . . , xn−rsn 0, n ≥ n0, 1.11 where n0 ≥ sequences, 0, d s j1

Summary

Introduction and Preliminaries

The interest in the study of the solvability of difference equations has been increasing see 1–17 and references cited therein. Let Ω be a bounded closed convex subset of a Banach space X, and let T1, T2 : Ω → X satisfy T1x T2y ∈ Ω for each x, y ∈ Ω. The higher-order difference for a positive integer m is defined as Δmxn Δ Δm−1xn , Δ0xn xn. 1 ≤ j ≤ s, and lβ∞ denotes the set of real sequences defined on the set of positive integers lager than β where any individual sequence is bounded with respect to the usual supremum norm x supn≥β|xn| for x {xn}n≥β ∈ lβ∞. A bounded, uniformly Cauchy subset Ω of lβ∞ is relatively compact. By a solution of 1.11 , we mean a sequence {xn}n≥β with a positive integer N0 ≥ n0 d |α| such that 1.11 is satisfied for all n ≥ N0.

Existence of Nonoscillatory Solutions

Examples