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 ▵ ( t ) = - a ( t ) x 3 ( σ ( t ) ) + c ( t ) x ▵ ∼ ( t - r ( t ) ) + G ( t , x 3 ( t ) , x 3 ( 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].