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.