Necessary and sufficient conditions for oscillation of first-order nonlinear neutral difference equations

In this paper, we prove that every solution of the first-order nonlinear neutral difference equation △(xn−pxn−τ)+qn∏j=1m|xn−σj|βjsign(xn−σ1)=0,n≥n0 oscillates if and only if ∑s=n0∞qsexp[τ−1lnp(∑j=1mβj−1)s]=∞, when (∑j=1mβj−1)lnp<0, and ∑s=n0∞qs=∞, when (∑j=1mβj−1)lnp>0, where p,βj>0, τ>0τ>0 and σj≥0σj≥0 are integers, j=1,2,…,mj=1,2,…,m, qn≥0,n≥0qn≥0,n≥0.