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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Oscillation criteria for delay equations


Authors: M. Kon, Y. G. Sficas and I. P. Stavroulakis
Journal: Proc. Amer. Math. Soc. 128 (2000), 2989-2997
MSC (1991): Primary 34K15; Secondary 34C10
Published electronically: April 28, 2000
MathSciNet review: 1694869
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Abstract:

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form

$\displaystyle x^{\prime}(t)+p(t)x({\tau}(t))=0, \quad t\geq t_{0},$     (1)

where $p, {\tau} \in C([t_{0}, \infty), \mathbb{R}^+), \mathbb{R}^+=[0, \infty), \tau (t)$ is non-decreasing, $\tau (t) <t$ for $t \geq t_{0}$ and $\lim_{t{\rightarrow}{\infty}} \tau (t) = \infty$. Let the numbers $k$ and $L$ be defined by

\begin{displaymath}k=\liminf_{t{\rightarrow}{\infty}} \int_{\tau (t)}^{t}p(s)ds ... ... L=\limsup_{t{\rightarrow}{\infty}} \int_{\tau (t)}^{t}p(s)ds. \end{displaymath}

It is proved here that when $L<1$ and $0<k \leq \frac{1}{e}$ all solutions of Eq. (1) oscillate in several cases in which the condition

\begin{displaymath}L>2k+\frac{2}{{\lambda}_{1}}-1 \end{displaymath}

holds, where ${\lambda _1}$ is the smaller root of the equation $\lambda =e^{k \lambda}$.


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Additional Information

M. Kon
Affiliation: Department of Mathematics, Boston University, Boston, Massachusetts 02215
Email: mkon@math.bu.edu

Y. G. Sficas
Affiliation: Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece

I. P. Stavroulakis
Affiliation: Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
Email: ipstav@cc.uoi.gr

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05530-1
PII: S 0002-9939(00)05530-1
Keywords: Oscillation, delay differential equations
Received by editor(s): December 4, 1998
Published electronically: April 28, 2000
Dedicated: Dedicated to Professor V. A. Staikos on the occasion of his 60th birthday
Communicated by: Hal L. Smith
Article copyright: © Copyright 2000 American Mathematical Society