On the asymptotic behavior of the pantograph equations

Abstract

Our aim is studing the asymptotic behaviour of the solutions of the equation $\dot x(t) = -a(t)x(t)+a(t)x(pt)$ where $0<p<1$ is a constant. This equation is a special case of the so called pantograph equations of the form $\dot x(t) = -a(t)x(t)+b(t)x(p(t))$. First we prove an asymptotic estimate of the solutions of the later equation, then using this result we show the asymptotic behavior of the solutions of the former equation. In particular, we prove that all solutions are asymptotically logarithmically periodic.