Periodic and Nonnegative Periodic Solutions of Nonlinear Neutral Dynamic Equations on a Time Scale

Let T be a periodic time scale. We use Krasnoselskii--Burton's fixed point theorem to show new results on the existence of periodic and nonnegative periodic solutions of nonlinear neutral dynamic equation with variable delay of the form$x^{\Delta }(t)=-a(t)h(x^{\sigma }(t))+Q(t,x(t-\tau (t)))^{\Delta}+G(t,x(t),x(t-\tau (t))),\text{ }t\in \mathbb{T}.$We invert the given equation to obtain an equivalent integral equation from which we define a fixed point mapping written as a sum of a large contraction and a completely continuous map. The Caratheodory condition is used for the functions $Q$ and $G$. The results obtained here extend the work of Mesmouli, Ardjouni and Djoudi [16].