Nonlinear integrodifferential equations anda priori bounds on periodic solutions

Abstract

This paper studies the existence of aperiodic solution of a nonlinear integrodifferential system of the form\(x'(t) = Dx(t) + f\left( {x(t)} \right) + \int\limits_{ - \infty }^t {k(t,s)g\left( {x(s)} \right)ds + p(t)} ,\), for each continuous periodic function p and under suitable assumptions on f, k and g. A topological transversality method is employed to obtain the existence of periodic solutions. This method relies ona priori bounds on periodic solutions. Several examples are provided where a variant of Liapunov's direct method is employed to obtaina priori bounds on periodic solutions.