Periodic matrix Sylvester Volterra integro-dynamic system on time scales

This study examines the periodic solution of the matrix Sylvester Volterra integro-dynamic system on periodic time scales. We establish the system's continuity to guarantee uniform convergence. If the limiting system is uniformly confined, the original system and its limiting counterpart yield similar outcomes. We establish the necessary and sufficient requirements for the integrability of matrix Sylvester Volterra integro-dynamic systems on time scales, employing Fubini's theorem, and concentrate on the existence and uniqueness of periodic solutions by the application of the Lipschitz condition. The study also investigates how these factors affect the system's behaviour across various time scales. An illustrative case is provided to the theoretical results, supplemented by graphical representations that elucidate the behaviour of the solutions and reinforce the findings.