Existence of periodic solutions for nonlinear neutral dynamic equations with variable delay on a time scale

Let T be a periodic time scale. The purpose of this paper is to use a modification of Krasnoselskii's fixed point theorem due to Burton to prove the existence of periodic solutions on time scale of the nonlinear dynamic equation with variable delay. x(Delta)(t) = -a(t)x(3)(sigma(t)) + c(t)x((Delta) over tilde)(t - r(t)) + G(t,x(3)(t), x(3)(t - r(t))), t is an element of T. where f(Delta) is the Delta-derivative on T and f((Delta) over tilde) is the Delta-derivative on (id - r)(T). We invert this equation to construct a sum of a compact map and a large contraction which is suitable for applying the Burton-Krasnoselskii's theorem. The results obtained here extend the works of Deham and Djoudi 18,91 and Ardjouni and Djoudi [2]. (C) 2011 Elsevier B.V. All rights reserved.