IJPAM: Volume 78, No. 7 (2012)

OSCILLATORY AND ASYMPTOTIC BEHAVIOR OF
SOLUTIONS OF SECOND ORDER NEUTRAL DELAY
DIFFERENTIAL EQUATIONS WITH ``MAXIMA''

E. Thandapani$^1$, V. Ganesan$^2$
$61$Ramanujan Institute for Advanced Study in Mathematics
University of Madras
Chennai, 600005, INDIA
$^2$Department of Mathematics
Periyar University
Salem, 636 011, INDIA


Abstract. The authors establish some new criteria for the oscillation and asymptotic behavior of all solutions of the equation.
\begin{gather*}
(a(t)(x(t)+p(t)x(\tau(t)))^\prime)^\prime + q(t) \max_{[\sigma(t),t]} x^{\alpha}(s) = 0,\quad t \geq t_0 \geq 0,
\end{gather*}
where $a(t)> 0$, $q(t) \geq 0$, $\tau(t) \leq t $, $\sigma(t) \leq t$, $\alpha$ is the ratio of odd positive integers, and $\int_0^\infty \frac{dt}{a(t)} < \infty$. Examples are included to illustrate the results.

Received: April 17, 2012

AMS Subject Classification: 34K11, 34K99

Key Words and Phrases: second order differential equations, maxima, oscillation

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 78
Issue: 7