Oscillatory and Asymptotic Behavior of a First-Order Neutral Equation of Discrete Type with Variable Several Delay under Δ Sign

Abstract EN

We obtain necessary and sufficient conditions so that every solution of neutral delay difference equation Δyn-∑j=1kpnjyn-mj+qnG(yσ(n))=fn oscillates or tends to zero as n→∞, where {qn} and {fn} are real sequences and G∈C(R,R), xG(x)>0, and m1,m2,…,mk are positive integers.

Here Δ is the forward difference operator given by Δxn=xn+1-xn, and {σn} is an increasing unbounded sequences with σn≤n.

This paper complements, improves, and generalizes some past and recent results.