Oscillation of deviating differential equations

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.

Keywords

• differential equation
• non-monotone argument
• oscillatory solution
• nonoscillatory solution
• grönwall inequality