Periodica Mathematica Hungarica

, Volume 63, Issue 2, pp 173–190 | Cite as

The period function of a delay differential equation and an application

  • Ábel GarabEmail author
  • Tibor Krisztin


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 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 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 [14] 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 

Mathematics subject classification numbers

34K13 92B20 


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Copyright information

© Akadémiai Kiadó, Budapest, Hungary 2011

Authors and Affiliations

  1. 1.Bolyai InstituteUniversity of SzegedSzegedHungary

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