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

Abstract

Let $\mathbb{T}$ be a periodic time scale. Using a modification of Krasnoselskii's fixed point theorem due to Burton, we show that the totally nonlinear dynamic equation with variable delay
\[x^{\triangle}\left( t\right) =-a\left( t\right) x^{3}\left( \sigma\left( t\right) \right) +c(t)Q^{\widetilde{\triangle}}\left( x\left( t-r\left( t\right) \right) \right) +G\left( t,x^{3}\left( t\right) ,x^{3}\left( t-r\left( t\right) \right) \right) ,
\] where $t\in\mathbb{T}$, $f^{\triangle}$ is the $\triangle$-derivative on $\mathbb{T}$ and $f^{\widetilde{\triangle}}$ is the $\triangle$-derivative on
$\left( id-r\right) \left( \mathbb{T}\right) $, has a periodic solution. 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 \cite{d1,d2} and Ardjouni and Djoudi \cite{a2,a3}.