Existence, stability and controllability of non-linear adjoint Volterra–Fredholm–Hammerstein integro-dynamic system with delays and impulses on hybrid time domains

Abstract In this article, we studied the existence, uniqueness, stability and controllability of solution of non-linear Volterra–Fredholm–Hammerstein adjoint integro-delay dynamic system with impulses on time scales. The Banach fixed point theorem is employed to demonstrate both the existence and uniqueness of solutions. The Hyers–Ulam stability was verified by applying Grönwall’s inequality on time scale. To address controllability within the presented framework, control function and Picard operator are utilized. To deal with the difficulties in getting the desired results, some assumptions are made. Finally, our theoretical results were supported through two numerical examples and simulations on different time scales.