A periodicity threshold theorem for a volterra integral equation

Abstract

A nonlinear Volterra integral equation, which arises in biological applications, with kernel of the form a(s)p(t-s) is studied with a(s) a positive periodic function. Conditions are given under which solutions converge to zero and under which solutions are asymptotic to a positive periodic function. With certain nonlinearities these conditions can be stated as a threshold theorem with the mean value of a(s) being the threshold parameter. Using the methods presented here one can compute permissible bounds on the amplitude of a(s).