Existence and Uniqueness of Solutions for a Semilinear Functional Dynamic Equation with Infinite Delay and Impulses

This work is devoted to the prove of the existence of solutions for a semilinear retarded differential equation with infinite delay and impulses on timescales, which is done by using a version of the Arzela-Ascoli theorem on time-scales, and applying the Leray-Schauder alternative. After that, the uniqueness of solutions is proved by applying a version of Gronwall's inequality for impulsive differential equations, and finally, the continuation of solutions is proved.