Mathematica Bohemica, Vol. 145, No. 4, pp. 435-448, 2020
Oscillation of deviating differential equations
George E. Chatzarakis
Received January 8, 2019. Published online January 21, 2020.
Abstract: Consider the first-order linear delay (advanced) differential equation $x'(t)+p(t)x( \tau(t)) =0\quad(x'(t)-q(t)x(\sigma(t)) =0),\quad t\geq t_0$, where $p$ $(q)$ is a continuous function of nonnegative real numbers and the argument $\tau(t)$ $(\sigma(t))$ is not necessarily monotone. Based on an iterative technique, a new oscillation criterion is established when the well-known conditions $\limsup\limits_{t\rightarrow\infty}\int_{\tau(t)}^tp(s) {\rm d}s>1\quad\biggl(\limsup\limits_{t\rightarrow\infty}\int_t^{\sigma(t)}q(s) {\rm d}s>1\bigg)$ and $\liminf_{t\rightarrow\infty}\int_{\tau(t)}^tp(s) {\rm d}s>\frac1{\ee}\quad\biggl(\liminf_{t\rightarrow\infty}\int_t^{\sigma(t)}q(s) {\rm d}s>\frac1{\ee}\bigg)$ are not satisfied. An example, numerically solved in MATLAB, is also given to illustrate the applicability and strength of the obtained condition over known ones.
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Affiliations: George E. Chatzarakis, Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), 14121 N. Heraklio, Athens, Greece, e-mail: geaxatz@otenet.gr, gea.xatz@aspete.gr
