Nontrivial Periodic Solutions of Some Volterra Integral Equations

I. Introductory Remarks. My main purpose in this paper is to prove a bifurcation theorem for nontrivial periodic solutions of a general system of Volterra integral equations. The motivation for considering this problem can be found in models which arise in population dynamics, epidemiology and economics [1,3,6], an example of which appears in §5. The approach taken is that which is usually referred to as the method of Liapunov-Schmidt (or often called ~he method of alternative problems). This method, which is applicable in a very general setting, is outlined in §2 in a way suitable for the type of problems I have in mind. The fundamental ingredient for this approach in its application to many problems is a Fredholm alternative. A Fredholm alternative for systems of Volterra integral equations is proved in §3. The main bifurcation result (Theorem 4) appears in §4 and an application is given in §5 to a scalar model which has arisen in the mathematical theory of population growth and of epidemics.