Solvability of a Higher-Order Nonlinear Neutral Delay Difference Equation

Abstract

The existence of bounded nonoscillatory solutions of a higher-order nonlinear neutral delay difference equation Δ a kn • • • Δ a 2n Δ a 1n Δ x n b n x n−d f n, x n−r1n , x n−r2n ,. .. , x n−rsn 0, n ≥ n 0 , where n 0 ≥ 0, d > 0, k > 0, and s > 0 are integers, {a in } n≥n0 i 1, 2,. .. , k and {b n } n≥n0 are real sequences, s j 1 {r jn } n≥n0 ⊆ Z, and f : {n : n ≥ n 0 } × R s → R 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 {b n } n≥n0. Moreover, these sufficient conditions guarantee that this equation has not only one bounded nonoscillatory solution but also uncountably many bounded nonoscillatory solutions.

Approximative solutions to difference equations of neutral type

“…By a solution of (E) we mean a sequence x: N(0) → R satisfying (E) for all large n. Asymptotic properties of solutions to neutral equations were investigated in many papers, see, for example, [1][2][3][4][5][6][7][8][9], [11][12][13].…”

“…The existence of solutions with prescribed asymptotic behavior to difference equations of neutral type is the theme of many papers. For example, the existence of non-oscillatory or positive or bounded solutions is investigated in the papers [3], [5][6][7][8][9] or in [21]. Our approach is based on an idea analogous to the idea used in the theory of ordinary difference equations, for example, of the form m x n = a n f (x n ) + b n .…”