Electron. J. Diff. Eqns., Vol. 2007(2007), No. 01, pp. 1-7.

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

Radhanath Rath, Prayag Prasad Mishra, Laxmi Narayan Padhy

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 $xG(x)$ greater than 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.

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Radhanath Rath
Department of Mathematics
Khallikote Autonomous College
Berhampur, 760001 Orissa, India
email: radhanathmath@yahoo.co.in
Prayag Prasad Mishra
Department of Mathematics
Silicin Institute of Technology
Bhubaneswar, Orissa, India
email: prayag@silicon.ac.in
Laxmi Narayan Padhi
Department of Mathematics, K.I.S.T
Bhubaneswar Orissa, India
email: ln_padhy_2006@yahoo.co.in

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