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Oscillation of differential equations with non-monotone retarded arguments

Part of: Functional-differential and differential-difference equations

Published online by Cambridge University Press:  01 March 2016

George E. Chatzarakis  and
Özkan Öcalan
George E. Chatzarakis
Affiliation:
Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), N. Heraklion, 14121 Athens, Greece email geaxatz@otenet.gr
Özkan Öcalan
Affiliation:
Akdeniz University, Faculty of Science, Department of Mathematics, 07058 Antalya, Turkey email ozkanocalan@akdeniz.edu.tr

Abstract

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Consider the first-order retarded differential equation

$$\begin{eqnarray}x^{\prime }(t)+p(t)x({\it\tau}(t))=0,\quad t\geqslant t_{0},\end{eqnarray}$$
where $p(t)\geqslant 0$ and ${\it\tau}(t)$ is a function of positive real numbers such that ${\it\tau}(t)\leqslant t$ for $t\geqslant t_{0}$, and $\lim _{t\rightarrow \infty }{\it\tau}(t)=\infty$. Under the assumption that the retarded argument is non-monotone, a new oscillation criterion, involving $\liminf$, is established when the well-known oscillation condition
$$\begin{eqnarray}\liminf _{t\rightarrow \infty }\int _{{\it\tau}(t)}^{t}p(s)\,ds>\frac{1}{e}\end{eqnarray}$$
 is not satisfied. An example illustrating the result is also given.

MSC classification

Primary: 34K11: Oscillation theory
Secondary: 34K06: Linear functional-differential equations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Issue 1 , 2016 , pp. 98 - 104
Copyright
© The Author(s) 2016 

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