The Two Types Of Sub-linear Neutral Difference Equations Of Vibration

In this paper, our objective is to study the oscillation of the sublinear neutral difference equations as follows:where Δ is the forward difference operator: Δx(n) = x(n + 1) — x(n).First we give some necessary and sufficient conditions for the oscillation of(1) by Lebesguge's dominated convergence theorem and Schauder — Tychonoff fixed point theorem. Then we use Lebesguge's dominated convergence theorem to obtain some sufficient conditions for (2) and study the existence of a bounded positive solution. Give some conditions for the bounded solutions of (2).