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 delaymultline*x^ (t) =-a (t) x^3 (t) +c (t) x^ (t-g (t) ) Q^ (x (t-g (t) ) ) \\ +G (t, x^3 (t), x^3 (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