## Oscillation criteria for delay equations

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- by M. Kon, Y. G. Sficas and I. P. Stavroulakis
- Proc. Amer. Math. Soc.
**128**(2000), 2989-2997 - DOI: https://doi.org/10.1090/S0002-9939-00-05530-1
- Published electronically: April 28, 2000
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## Abstract:

This paper is concerned with the oscillatory behavior of first-order delay differential equations of the form \begin{eqnarray} x’ (t)+p(t)x({\tau }(t))=0, \quad t\geq t_{0}, \end{eqnarray} 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 \[ k=\liminf _{t{\rightarrow }{\infty }} \int _{\tau (t)}^{t}p(s)ds \quad \mbox {and} \quad L=\limsup _{t{\rightarrow }{\infty }} \int _{\tau (t)}^{t}p(s)ds. \] 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 \[ L>2k+\frac {2}{{\lambda }_{1}}-1 \] holds, where ${\lambda _1}$ is the smaller root of the equation $\lambda =e^{k \lambda }$.## References

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## Bibliographic 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
- Received by editor(s): December 4, 1998
- Published electronically: April 28, 2000
- Communicated by: Hal L. Smith
- © Copyright 2000 American Mathematical Society
- Journal: Proc. Amer. Math. Soc.
**128**(2000), 2989-2997 - MSC (1991): Primary 34K15; Secondary 34C10
- DOI: https://doi.org/10.1090/S0002-9939-00-05530-1
- MathSciNet review: 1694869

Dedicated: Dedicated to Professor V. A. Staikos on the occasion of his 60th birthday