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

In this paper, we prove that every solution of the first order nonlinear neutral differential equation[x(t)−px(t−τ)]′+q(t)∏j=1m|x(t−σj)|βjsign[x(t−σ1)]=0,t⩾t0, oscillates if and only if∫t0∞q(s)exp[τ−1lnp(∑j=1mβj−1)s]ds=∞, when (∑j=1mβj−1)lnp<0, and∫t0∞q(s)ds=∞, when (∑j=1mβj−1)lnp>0, where p  , τ>0τ>0, βj>0βj>0, σj⩾0σj⩾0, j=1,2,…,mj=1,2,…,m, q∈C([t0,∞),[0,∞))q∈C([t0,∞),[0,∞)).