2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada.
Electron. J. Diff. Eqns., Conference 12, 2005, pp. 171-180.

Oscillation criteria for functional differential equations

Ioannis P. Stavroulakis

Abstract:
Consider the first-order linear delay differential equation
$$ x'(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0}, $$
and the second-order linear delay equation
$$ x''(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0}, $$
where $p$ and $\tau $ are continuous functions on $[t_{0},\infty )$, $p(t)$ greater than 0, $\tau (t)$ is non-decreasing, $\tau (t)\leq t$ for $t\geq t_{0}$ and $\lim_{t\to \infty }\tau (t)=\infty $. Several oscillation criteria are presented for the first-order equation when
$$ 0<\liminf_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds\leq \frac{1}{e} \quad \hbox{and}\quad \limsup_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds<1, $$
and for the second-order equation when
$$ \liminf_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds \leq \frac{1}{e}\quad \hbox{and}\quad \limsup_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds<1\,. $$

Published April 20, 2005.
Math Subject Classifications: 34K11, 34C10.
Key Words: Oscillation; delay differential equations.

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Ioannis P. Stavroulakis
Department of Mathematics
University of Ioannina
451 10 Ioannina, Greece
email: ipstav@cc.uoi.gr

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