Existence of periodic solutions for neutral nonlinear differential equations with variable delay

We use a variation of Krasnoselskii 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 with an example.