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

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.


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