Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 01, pp. 1-7.

Title: On oscillation and asymptotic behaviour of a  neutral differential
      equation of first order with positive and negative coefficients

Authors: Radhanath  Rath  (Khallikote Autonomous College, Orissa, India)
         Prayag Prasad Mishra (Silicin Inst. of Technology, Orissa, India)
	 Laxmi Narayan Padhy (K.I.S.T, Orissa, India)
	 
Abstract:
 In this paper sufficient conditions are obtained so that every
 solution of
 $$
 (y(t)- p(t)y(t-\tau))'+ Q(t)G(y(t-\sigma))-U(t)G(y(t-\alpha)) = f(t)
 $$
 tends to zero or to $\pm \infty$ as $t$ tends to $\infty$, where
 $\tau ,\sigma ,\alpha$ are positive real numbers,
 $p,f\in C([0,\infty),R),Q,U\in C([0,\infty),[0,\infty))$, and
 $G\in C(R,R)$, $G$ is non decreasing with $xG(x)>0$ for
 $ x\neq 0$.
 The  two primary assumptions in this paper are
 $\int_{t_0}^{\infty}Q(t)=\infty$ and
 $\int_{t_0}^{\infty}U(t)<\infty$. The results  hold when $G$
 is linear, super linear,or sublinear and also hold when
 $f(t) \equiv 0$.  This paper  generalizes and improves some
 of the recent results in [5,7,8,10]. 
 
 
Submitted November 2, 2006. Published January 2, 2007
Math Subject Classifications: 34C10, 34C15, 34K40.
Key Words: Oscillatory solution; nonoscillatory solution;
           asymptotic behaviour.