Existence of periodic solutions for neutral nonlinear differential equations with variable delay. (English) Zbl 1203.34110

Hafsia, Deham; Ahcéne, Djoudi

Electron. J. Differ. Equ. 2010, Paper No. 127, 8 p. (2010).

Summary: We use a variation of Krasnoselskii’s fixed point theorem introduced by Burton to show that the nonlinear neutral differential equation \[ x'(t)=-a(t)x^3(t)+c(t)x'(t-g(t))+G(t,x^3(t-g(t)) \] has a periodic solution. Since this equation is nonlinear, the variation of parameters can not be applied directly; we add and subtract a linear term to transform the differential into an equivalent integral equation suitable for applying a fixed point theorem. Our result is illustrated by an example.

MSC:

34K13	Periodic solutions to functional-differential equations
34K40	Neutral functional-differential equations
47N20	Applications of operator theory to differential and integral equations

Keywords:

periodic solution; nonlinear neutral differential equation; large contraction; integral equation