Approximate controllability of Caputo fractional Volterra–Fredholm integral equations under nonlocal conditions and impulsive effects

Abstract

This paper investigates the approximate controllability of a system of fractional control differential equations. The system involves Caputo fractional derivatives, Volterra–Fredholm integral equations, and impulsive effects. The analysis is carried out in the framework of Banach spaces under nonlocal conditions of order r ∈ (1, 2). The main objective is to establish sufficient conditions for the approximate controllability of the proposed control problem. Assuming that the associated linear system is approximately controllable, the analysis relies on tools from fractional calculus, Krasnoselskii’s fixed point theorem, and the theory of resolvent operators. The obtained results extend and improve several existing results in the related literature. Finally, two illustrative examples are presented to demonstrate the applicability of the theoretical findings.