Electron. J. Diff. Eqns., Vol. 2008(2008), No. 113, pp. 1-15.

Oscillation and asymptotic behaviour of a higher order neutral differential equation with positive and negative coefficients

Basak Karpuz, Laxmi Narayan Padhy, Radhanath Rath

Abstract:
In this paper, we obtain necessary and sufficient conditions so that every solution of
$$ \big(y(t)- p(t) y(r(t))\big)^{(n)}+ q(t)G( y(g(t)))-u(t)H(y(h(t))) = f(t) $$
oscillates or tends to zero as $t \to \infty$, where $n$ is an integer $n \geq 2$, $q>0$, $u\geq 0$. Both bounded and unbounded solutions are considered in this paper. The results hold also when $u\equiv 0$, $f(t)\equiv 0$, and $G(u)\equiv u$. This paper extends and generalizes some recent results.

Submitted April 4, 2008. Published August 20, 2008.
Math Subject Classifications: 34C10, 34C15, 34K40.
Key Words: Oscillatory solution; neutral differential equation; asymptotic behaviour.

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Basak Karpuz
Department of Mathematics, Facaulty of Science and arts,
A.N.S. Campus, Afyon Kocatepe University, 03200 Afyonkarahisar, Turkey
email: bkarpuz@gmail.com
Laxmi Narayan Padhy
Department of Computer Science and Engineering, K.I.S.T,
Bhubaneswar Orissa, India
email: ln_padhy_2006@yahoo.co.in
Radhanath Rath
Department of Mathematics, Khallikote Autonomous College
Berhampur, 760001 Orissa, India
email: radhanathmath@yahoo.co.in

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