Convergence and Divergence of the Solutions of a Neutral Difference Equation

Abstract

We investigate the asymptotic behavior of the solutions of a neutral type difference equation of the form Δ[𝑥(𝑛)+𝑐𝑥(𝜏(𝑛))]+𝑝(𝑛)𝑥(𝜎(𝑛))=0, where 𝜏(𝑛) is a general retarded argument, 𝜎(𝑛) is a general deviated argument (retarded or advanced), 𝑐∈ℝ, (𝑝(𝑛))𝑛≥0 is a sequence of positive real numbers such that 𝑝(𝑛)≥𝑝, 𝑝∈ℝ+, and Δ denotes the forward difference operator Δ𝑥(𝑛)=𝑥(𝑛+1)−𝑥(𝑛). Also, we examine the asymptotic behavior of the solutions in case they are continuous and differentiable with respect to 𝑐.