Abstract Nonlinear differential equations involving fractional operators of complex order are considered in this paper. In particular, we focus on equations defined by the complex-order ψ-Hilfer fractional derivative. Allowing the order of differentiation to be complex makes it possible to describe memory-dependent effects together with oscillatory behaviour, which cannot be achieved using standard real-order fractional derivatives alone. The primary focus of the analysis is the solvability of the proposed problem. By applying the Banach and Schauder fixed-point theorems, sufficient conditions for the existence and uniqueness of solutions are obtained. In addition, the existence of positive solutions is discussed, and several specific cases are examined. Furthermore, the Hyers–Ulam (H–U) stability of the solutions is investigated, showing that small perturbations in the given data do not lead to significant changes in the corresponding solutions. To illustrate the theoretical results, a concrete example is presented. The results obtained here may be viewed as a natural extension of several known results for real-order fractional differential equations (FDEs) to the case of the complex-order ψ-Hilfer derivative.
Verma et al. (Wed,) studied this question.