Mild solutions and controllability of $(k,\phi )$-Hilfer fractional delay differential equations with history-dependent operators

The objective of this work was to investigate a class of Hilfer-type fractional differential equations governed by the interaction of semigroup operators, history-dependent mechanisms, and probability density functions, with particular attention to their analytical and control properties. First, the existence of mild solutions was established through semigroup theory and fixed-point arguments tailored to fractional dynamics. We then addressed the controllability problem for the corresponding $(k,\phi )$-Hilfer fractional delay differential equation, taking into account the influence of memory and delay effects induced by history-dependent operators. By combining Mönch's fixed-point theorem with the measure of noncompactness, a set of sufficient conditions for controllability was obtained. This approach not only captures the complexity of the system but also deepens the understanding of how fractional-order behavior and past-state dependence affect controllability.