THE PERIOD FUNCTION OF A DELAY DIFFERENTIAL EQUATION AND AN APPLICATION

We consider the delay differential equation (x) Overdot = - mu x(t)+ f(x(t-tau)),, where A mu, tau are positive parameters and f is a strictly monotone, nonlinear C (1)-function satisfying f(0) = 0 and some convexity properties. It is well known that for prescribed oscillation frequencies (characterized by the values of a discrete Lyapunov functional) there exists tau* > 0 such that for every tau > tau* there is a unique periodic solution. The period function is the minimal period of the unique periodic solution as a function of tau > tau*. First we show that it is a monotone nondecreasing Lipschitz continuous function of tau with Lipschitz constant 2. As an application of our theorem we give a new proof of some recent results of Yi, Chen and Wu [14] about uniqueness and existence of periodic solutions of a system of delay differential equations.