Exact and Approximate Controllability for Backward Fractional Integro‐Differential Equations With History‐Dependent Operators

This paper is devoted to the analysis of controllability for a class of backward fractional integro‐differential equations involving history‐dependent operators, which arise naturally in systems with memory effects. The study begins with the formulation of an appropriate functional framework, within which the concept of approximate controllability is established by means of Schauders fixed‐point theorem, without imposing restrictive compactness or Lipschitz‐type conditions. Building on this foundation, the investigation is extended to exact controllability through a more refined analytical approach, where Mönch′s fixed‐point theorem is employed in conjunction with suitable operator‐theoretic techniques to derive verifiable sufficient conditions. The results obtained significantly generalize existing contributions in the literature by accommodating the combined effects of fractional dynamics and history dependence. Finally, a representative example is presented to illustrate the applicability, validity, and effectiveness of the proposed theoretical framework in practical settings.