Existence of periodic solutions in shifts $\delta_{\pm}$ for neutral nonlinear dynamic systems

In this study, we focus on the existence of a periodic solution for the neutral nonlinear dynamic systems with delay% $$x^{\Delta}(t)=A(t)x(t)+Q^{\Delta}\left(t,x\left(\delta_{-}(s,t)\right) \right) +G\left(t,x(t),x\left(\delta_{-}(s,t)\right) \right) .$$ We utilize the new periodicity concept in terms of shifts operators, which allows us to extend the concept of periodicity to time scales where the additivity requirement $t\pm T\in\mathbb{T}$ for all $t\in\mathbb{T}$ and for a fixed $T>0,$ may not hold. More, importantly, the new concept will easily handle time scales that are not periodic in the conventional way such as; $\overline{q^{\mathbb{Z}}}$ and $\cup_{k=1}^{\infty}\left[ 3^{\pm k},2.3^{\pm k}\right] \cup\left\{0\right\} .$ Hence, we develop a tool that enables the investigation of periodic solutions of $q$-difference systems. Since we are dealing with systems, in order to convert our equation to an integral systems, we resort to the transition matrix of the homogeneous Floquet system $y^{\Delta}(t)=A(t)y(t)$ and then make use of Krasnoselskii's fixed point theorem to obtain a fixed point.