LIAPUNOV FUNCTIONALS AND PERIODICITY IN INTEGRAL EQUATIONS

Abstract

Liapunov's direct method has been used very effectively for a hundred years on various types of differential equations. It has not, however, been used with much success on non-differentiated equations. In this paper we construct a Liapunov function for a nonlinear integral equation with an infinite delay which is nonconvolution type. From that Liapunov function we deduce conditions for boundedness, stability, and the existence of periodic solutions. The kernel of the integral equation is a perturbation of a positive kernel and there are estimates showing how large the perturbation can be. The advantage of the Liapunov approach over classical methods for integral equations is the simplicity of analysis, once a Liapunov function is constructed.