Electron. J. Diff. Eqns., Vol. 2007(2007), No. 163, pp. 1-14.

Non-oscillatory behaviour of higher order functional differential equations of neutral type

Radhanath Rath, Niyati Misra, Prayag Prasad Mishra, Laxmi Narayan Padhy

Abstract:
In this paper, we obtain sufficient conditions so that the neutral functional differential equation
$$\displaylines{ \big[r(t) [y(t)-p(t)y(\tau (t))]'\big]^{(n-1)} + q(t) G(y(h(t))) = f(t) }$$
has a bounded and positive solution. Here $n\geq 2$; $q,\tau, h$ are continuous functions with $q(t) \geq 0$; $h(t)$ and $\tau(t)$ are increasing functions which are less than $t$, and approach infinity as $t \to \infty$. In our work, $r(t) \equiv 1$ is admissible, and neither we assume that $G$ is non-decreasing, that $xG(x) > 0$ for $x \neq 0$, nor that $G$ is Lipschitzian. Hence the results of this paper generalize many results in [1] and [4]-[8].

Submitted September 24, 2007. Published November 30, 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
Niyati Misra
Department of Mathematics
Berhampur University
Berhampur, 760007 Orissa, India
email: niyatimath@yahoo.co.in
Prayag Prasad Mishra
Department of Mathematics
Silicin Institute of Technology
Bhubaneswar, Orissa, India
email: prayag@silicon.ac.in
Laxmi Narayan Padhy
Department of Computer Science and Engineering, K.I.S.T,
Bhubaneswar Orissa, India
email: ln_padhy_2006@yahoo.co.in

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