Periodic Solution for Pair of Matrix Sylvester Volterra Integro-Dynamic System on Time Scales

In this paper, we study the periodic solution for a pair of matrix Sylvester Volterra integro-dynamic (VID) systems on time scales. We use the vectorization operator to transform the matrix Sylvester VID into the Kronecker product VID system on time scales. This approach allows us to demonstrate the conditions under which periodic solutions exist for these complex systems. By leveraging the Banach fixed point theorem, we can effectively address the intricacies involved in the periodicity of solutions within the context of time scales. We can establish new results regarding the existence of periodic solutions for the VID system on time scales through Krasnoselskii's fixed point theorem.