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) m j=1 x(t −σj ) βj sign x(t −σ1) = 0, t t0, oscillates if and only if ∞ t0 q(s) exp τ−1 lnp m j=1 βj −1 s ds=∞, when ( m j=1 βj −1) lnp <0, and ∞ t0 q(s) ds=∞, when ( m j=1 βj −1) lnp >0, where p, τ >0, βj > 0, σj 0, j = 1, 2, . . . , m, q ∈ C([t0,∞), [0,∞)). © 2005 Published by Elsevier Inc.