Positive periodic solutions in neutral dynamic equations on a time scale

Abstract

Let $\mathbb{T}$ be a periodic time scale. We use a fixed point theorem due to Krasnosel'ski\u{\i} to show that the nonlinear neutral dynamic system \begin{displaymath} x^{\Delta}(t) = -a(t)x^{\sigma}(t)+ c(t)x^{\tilde{\Delta}}(\tau(t))+ q\left( t,x(\tau(t))\right),\; t \in \mathbb{T}, \end{displaymath} with delay $\tau(t)$ has a positive periodic solution. Here $x^\Delta$ is the $\Delta$-derivative on $\mathbb{T}$ and $x^{\tilde{\Delta}}$ is the $\Delta$-derivative on $\mathbb{\tau(T)}$.