Periodic Solutions in Shifts \(\delta _\pm \) for Impulsive Neutral Dynamic Equations with Infinite Delay on Time Scales

Let \(\Pi \) be a periodic time scale in shift \(\delta _\pm \) . We use the Krasnoselskii’s fixed point theorem to show that the impulsive neutral dynamic equations with infinite delay \(\displaystyle \left \{ \begin {array}{l} x^\Delta (t)= -A(t) x^\sigma (t)+f^\Delta \big (t,x(t-h(t))\big )\\ \qquad +\int _{-\infty }^{t}B(t,s)g\big (x(s)\big )\Delta s,\quad t_{k}\neq t\in \Pi ,\\ x(t^+_k)-x(t^-_k)= I_k\big (x(t_k)\big ),\quad k \in \mathbb {Z}^+. \end {array} \right . \) have a periodic solution in shift \(\delta _\pm \) . Under a slightly stringent conditions we show that the periodic solution in shifts \(\delta _\pm \) is unique using the contraction mapping principle.