Fixed points and stability of a nonconvolution equation

Abstract

In this note we consider an equation of the form \[ x’(t)=-\int ^{t}_{t-r} a(t,s)g(x(s))ds\] and give conditions on $a$ and $g$ to ensure that the zero solution is asymptotically stable. When applied to the classical case of $a(t,s)=a(t-s)$, these conditions do not require that $a(r)=0$, nor do they involve the sign of $a(t)$ or the sign of any derivative of $a(t)$.