Asymptotic behavior of higher-order neutral difference equations with general arguments

Abstract

We study the asymptotic behavior of solutions of the higher-order neutral difference equation $$ {\varDelta^m}\left[ {x(n)+cx\left( {\tau (n)} \right)} \right]+p(n)x\left( {\sigma (n)} \right)=0,\quad \mathbb{N}\mathrel\backepsilon m\geq 2,\quad n\geq 0, $$ where τ (n) is a general retarded argument, σ(n) is a general deviated argument, c ∈ $ \mathbb{R} $ ; (p(n)) n ≥ 0 is a sequence of real numbers, ∆ denotes the forward difference operator ∆x(n) = x(n+1) - x(n); and ∆ j denotes the jth forward difference operator ∆ j (x(n) = ∆ (∆ j-1(x(n))) for j = 2, 3,…,m. Examples illustrating the results are also given.