Analyzing existence, uniqueness, and stability of neutral fractional Volterra-Fredholm integro-differential equations

This paper explores the investigation of a Volterra-Fredholm integro-differential equation that incorporates Caputo fractional derivatives and adheres to specific order conditions. The study rigorously establishes both the existence and uniqueness of analytical solutions by applying the Banach principle. Additionally, it presents a unique outcome regarding the existence of at least one solution, supported by exacting conditions derived from the Krasnoselskii fixed point theorem. Furthermore, the paper encompasses neutral Volterra-Fredholm integro-differential equations, thus extending the applicability of the findings. Additionally, the paper explores the concept of Ulam stability for the obtained solutions, providing valuable insights into their long-term behavior. To emphasis the practical significance and reliability of the results, an illustrative example is included, effectively demonstrating the applicability of the theoretical discoveries.