Some New Results on Itô–Doob Hadamard Fractional Stochastic Pantograph Equations in \(L^p\) Spaces

In this paper, we investigate the well-posedness, regularity, Ulam-Hyers stability, and averaging principle for solutions of Itô–Doob stochastic Hadamard fractional pantograph equations of order \(\alpha {\in } ({1 /2},1)\) $\alpha \in (1/2,1)$ in \(L^p\) $L^p$ spaces with \(p{\ge }2\) $p\ge 2$ . We first establish the existence and uniqueness of results by applying the Banach fixed point theorem, and demonstrate the continuous dependence of the solutions on the initial values. We then prove the time regularity of the solutions to the proposed equations and show that the solutions are ( \(\alpha {-} \tfrac{1}{2}\) $\alpha -\frac{1}{2}$ )-Hölder continuous. Next, we analyze the Ulam–Hyers stability of the considered equations utilizing the Gronwall inequality. Finally, we study the averaging principle for the studied equations under a general averaging condition. Our results are illustrated with two examples.