Fixed points and stability of a nonconvolution equation

Abstract

In this note we consider an equation of the form x′(t) = - ∫ t-rta(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).