Existence and Approximate Controllability for Higher-Order Hilfer Fractional Evolution Equations via Fixed-Point and Sequence Approaches

This study addresses the existence and approximate controllability of a type of higher-order Hilfer fractional evolution differential (HOHFED) system with time delays in Banach spaces. Using the properties of the Mittag–Leffler function, cosine families, and Hilfer-type fractional differential operators, we first demonstrate the existence and uniqueness of mild solutions using a fixed-point method. Furthermore, a sequential technique is proposed to establish adequate conditions for approximate controllability. A detailed example is provided to illustrate the applicability and effectiveness of the theoretical results.