A periodicity theorem for autonomous functional differential equations

Abstract

Many periodicity problems in the theory of control, biological behavior, econometrics and other active areas of scientific research can be formulated in terms of differential equations in which the derivatives depend upon previous states of the systems. Such history dependent differential equatioas are called functional differential equations. In this paper, we shall consider autonomous functional differential equations for which the derivatives at time t depend upon values of the solution on the finite time interval [t T, t], where Y is a fixed, positive number. The main result is a periodicity theorem for such autonomous functional differential equations. The theorem will be applied to examples of first and second order differential difference equations. The problem of determining conditions under which a functional differential equation will have a periodic solution has been undertaken by G. S. Jones in a series of papers, [I], [2], and [3]= He formulated certain asymptotic fixed point theorems which were applicable to operators arising in the study of functional differential equations of the type considered here. For a given equation, he defined an operator A with domain and range in an infinite dimensional Banach space B, such that a fixed point of A corresponded to a periodic solution of the equation. The problem thus is reduced to showing that the operator has a nontrivial fixed point in B. Woweverr, the equations considered by Jones (and here) have the origin of B as a critical point, and the operator A has the origin as a fixed point. This means that the origin of B must be removed from the set in which fixed points of A may lie. The approach used by Jones was to apply generalizations of standard fixed point theorems to the operator A. Wowever, in order to