Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 163, pp. 1-14.
Title: Non-oscillatory behaviour of higher order functional
differential equations of neutral type
Authors: Radhanath Rath (Khallikote Autonomous College, Orissa, India)
Niyati Misra (Berhampur Univ., Orissa, India)
Prayag Prasad Mishra (Silicin Inst. of Tech., Orissa, India}
Laxmi Narayan Padhy (Berhampur Univ., Orissa, India)
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.