Asymptotic stability in differential equations with unbounded delay

Abstract

In this paper we consider a functional differential equation of the form $$x'=F(t,x,\int_0^t C(at-s) x(s)\,ds)$$ where $a$ is a constant satisfying $0<a<\infty$. Thus, the integral represents the memory of past positions of the solution $x$. We make the assumption that $\int_0^\infty |C(t)|\, dt<\infty$ so that this is a fading memory problem and we are interested in studying the effects of that memory over all those values of $a$. Very different properties of solutions emerge as we vary $a$ and we are interested in developing an approach which handles them in a unified way.