Qualitative Analysis of Stochastic Caputo–Katugampola Fractional Differential Equations

Stochastic pantograph fractional differential equations (SPFDEs) combine three intricate components: stochastic processes, fractional calculus, and pantograph terms. These equations are important because they allow us to model and analyze systems with complex behaviors that traditional differential equations cannot capture. In this study, we achieve significant results for these equations within the context of Caputo–Katugampola derivatives. First, we establish the existence and uniqueness of solutions by employing the contraction mapping principle with a suitably weighted norm and demonstrate that the solutions continuously depend on both the initial values and the fractional exponent. The second part examines the regularity concerning time. Third, we illustrate the results of the averaging principle using techniques involving inequalities and interval translations. We generalize these results in two ways: first, by establishing them in the sense of the Caputo–Katugampola derivative. Applying condition β=1, we derive the results within the framework of the Caputo derivative, while condition β→0+ yields them in the context of the Caputo–Hadamard derivative. Second, we establish them in Lp space, thereby generalizing the case for p=2.