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

► In this work we use an interesting hybrid fixed point theorem due to Krasnoseskii–Burton. ► We prove the existence of periodic solutions for neutral nonlinear dynamic equation with variable delays. ► This study is made on a time scale which unifies theories of differential equations and difference equations. 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▵(t)=-a(t)x3(σ(t))+c(t)x▵∼(t-r(t))+G(t,x3(t),x3(t-r(t))),t∈T,where f▵ is the ▵-derivative on T and f▵∼ is the ▵-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 [8,9] and Ardjouni and Djoudi [2].