Existence and controllability results for neutral fractional Volterra-Fredholm integro-differential equations

This paper delves into the investigation of a Volterra-Fredholm integro-differential equation enhanced with Caputo fractional derivatives subject to specific order conditions.The study rigorously establishes the existence of solutions through the application of the Schauder fixed-point theorem.Furthermore, it encompasses neutral Volterra-Fredholm integro-differential equations, thereby extending the applicability of the findings.In addition, the paper explores the concept of controllability for the obtained solutions, offering valuable insights into how these solutions behave over extended time periods.