Existence of Periodic Solutions for Totally Nonlinear Neutral Differential Equations With Variable Delay

We use a modification of Krasnoselskii's fixed point theorem introduced by T. A. Burton (see [ 1] Theorem $3$) to show that the totally nonlinear neutral differential equation with variable delay{multline*}x(t)=-a(t)x³(t)+c(t)x(t-g(t))Q^{}( x( t-g(t)) )\\ +G(t,x³(t),x³(t-g(t))) ,{multline*}has a periodic solution. We invert this equation to construct a sum of a compact map and a large contraction which is suitable for applying the modification of Krasnoselskii's theorem. The results of [ 5] are generalized. Finally, an example is given to illustrate our result. 2000 Mathematics Subject Classification. 34K20, 45J05, 45D05