Oscillation Criteria for Delay and Advanced Difference Equations with General Arguments

Dedicated to the memory of Professor Panayiotis D. Siafarikas, a very good friend and colleague. Consider the first-order delay difference equation ∆x(n) + p(n)x(τ(n)) = 0, n ≥ 0, and the first-order advanced difference equation ∇x(n) − p(n)x(µ(n)) = 0, n ≥ 1, [∆x(n) − p(n)x(ν(n)) = 0, n ≥ 0], where ∆ denotes the forward difference operator ∆x(n) = x(n+ 1) − x(n), ∇ denotes the backward difference operator ∇x(n) = x(n) − x(n − 1), {p(n)} is a sequence of nonnegative real numbers, {τ(n)} is a sequence of positive integers such that τ(n) ≤ n − 1, for all n ≥ 0, and {µ(n)} [{ν(n)}] is a sequence of positive integers such that µ(n) ≥ n+ 1 for all n ≥ 1, [ν(n) ≥ n+ 2 for all n ≥ 0]. A survey on the oscillation of all solutions to these equations is presented. Exam-ples illustrating the results are given