Theoretical Results on the pth Moment of ϕ-Hilfer Stochastic Fractional Differential Equations with a Pantograph Term

Here, we establish significant results on the well-posedness of solutions to stochastic pantograph fractional differential equations (SPFrDEs) with the ϕ-Hilfer fractional derivative. Additionally, we prove the smoothness theorem for the solution and present the averaging principle result. Firstly, the contraction mapping principle is applied to determine the existence and uniqueness of the solution. Secondly, continuous dependence findings are presented under the condition that the coefficients satisfy the global Lipschitz criteria, along with regularity results. Thirdly, we establish results for the averaging principle by applying inequalities and interval translation techniques. Finally, we provide numerical examples and graphical results to support our findings. We make two generalizations of these findings. First, in terms of the fractional derivative, our established theorems and lemmas are consistent with the Caputo operator for ϕ(t) = t, a=1. Our findings match the Riemann–Liouville fractional operator for ϕ(t)=t, a=0. They agree with the Hadamard and Caputo–Hadamard fractional operators when ϕ(t)=ln(t), a=0 and ϕ(t)=ln(t), a=1, respectively. Second, regarding the space, we are make generalizations for the case p=2.