J. Appl. Math. & Computing Vol. 21(2006), No. 1 - 2, pp. 99 - 118 Website: http://jamc.net OSCILLATION OF NONLINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS ELMETWALLY M. ELABBASY, TAHER S. HASSAN * AND SAMIR H. SAKER Abstract. In this paper, we study the oscillatory behavior of first order nonlinear neutral delay differential equations. Several new sufficient condi- tions which ensure that all solutions are oscillatory are given. The obtained results extend and improve several known results in the literature. Some examples are considered to illustrate the main results. AMS Mathematics Subject Classification : 34K15, 34C10. Key words and phrases : Oscillation, nonoscillation, neutral delay of differ- ential equations. 1. Introduction In this paper, we shall consider the first order nonlinear neutral delay differ- ential equation (x(t) − q(t)x(t − σ)) ′ + f (t, x(τ (t))) = 0, t ≥ t 0 , (1) where q,τ ∈ C ([t 0 , ∞), R + ), σ ∈ (0, ∞) τ (t) < t, lim t→∞ τ (t)= ∞, (2) and n  i=1 i  j=1 1 q(t j ) −→ ∞ as n −→ ∞, (3) f ∈ C ([t 0 , ∞) × R, R), uf (t, u) ≥ 0, q(t) = 1. (4) In connection with the nonlinear function f (t, u) in (1) we suppose that the following assumption (H) holds : Received October 10, 2005. Revised January 16, 2006. ∗ Corresponding author. c  2006 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 99