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

In this paper, we obtain sufficient conditions so that the neutral functional differential equation
   \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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