Existence and stability results for a hybrid initial-value problem with ψ-Caputo fractional derivative

A hybrid initial-value problem incorporating the ψ-Caputo fractional derivative (ψ-CFD) was examined. The ψ-CFD provides a flexible framework for modeling memory effects on non-uniform time scales, while hybrid initial value problems capture complex systems with multiple interacting dynamics, making their combination particularly valuable for modeling biological networks, infectious disease control, and stochastic hybrid systems in engineering applications. We demonstrate the existence and uniqueness of solutions by employing the two popular fixed-point theorems (FPTs): the Leray–Schauder alternative and Banach’s FPT. The former ensures the presence of at least one solution to the problem, while the latter guarantees its uniqueness. Moreover, we establish the stability of the presented problem by employing Ulam–Hyers and Ulam–Hyers–Rassias stability concepts under appropriate conditions. An explicit example that meets all theoretical assumptions is presented to validate the existence, uniqueness, and stability conditions; numerical illustrations demonstrate how the contraction parameter for existence and uniqueness, as well as the stability parameter, remain within their theoretically-derived constrained ranges across the fractional order axis. This work advances existing literature by developing a unified analytical framework that simultaneously establishes existence, uniqueness, and dual stability concepts for fractional hybrid initial-value problems (FHIVPs) with ψ-CFD under a single set of assumptions.