We consider the delay differential equation \(\dot x(t) = - \mu x(t) + f(x(t - \tau ))\), where µ, τ are positive parameters and f is a strictly monotone, nonlinear C1-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 τ* > 0 such that for every τ > τ* there is a unique periodic solution. The period function is the minimal period of the unique periodic solution as a function of τ > τ*. First we show that it is a monotone nondecreasing Lipschitz continuous function of τ with Lipschitz constant 2. As an application of our theorem we give a new proof of some recent results of Yi, Chen and Wu  about uniqueness and existence of periodic solutions of a system of delay differential equations.
Key words and phrases
delay differential equation discrete Lyapunov functional neural networks period function periodic orbit
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