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

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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