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

Abstract

Let $\mathbb{T}$ be a periodic time scale. The purpose of this paper is to use Krasnoselskii's fixed point theorem to prove the existence of positive periodic solutions on time scale of the nonlinear neutral dynamic equation with variable delay$$(x(t)-g(t, x(t-\tau(t))))^{\triangle}=r(t) x(t)-f(t, x(t-\tau(t))) .$$We invert this equation to construct a sum of a contraction and a compact map which is suitable for applying the Krasnoselskii's theorem. The results obtained here extend the works of Raffoul [17] and Ardjouni and Djoudi [3].