On the Ulam stability and existence of $L^p$-solutions for fractional differential and integro-differential equations with Caputo-Hadamard derivative

In this paper, we investigate the existence and uniqueness of $L^p$-solutions for nonlinear fractional differential and integro-differential equations with boundary conditions using the Caputo-Hadamard derivative. By employing Hölder's inequality together with the Krasnoselskii fixed-point theorem and the Banach contraction principle, the study establishes sufficient conditions for solving nonlinear problems. The paper delves into preliminary results, the existence and uniqueness of $L^p$ solutions to the boundary value problem, and presents the Ulam-Hyers stability. Furthermore, it investigates the existence, uniqueness, and stability of solutions for fractional integro-differential equations. Through standard fixed-points and rigorous mathematical frameworks, this research contributes to the theoretical foundations of nonlinear fractional differential equations. Also, the Adomian decomposition method ( $\mathcal{A}\mathcal{D}\mathcal{M}$) is used to construct the analytical approximate solutions for the problems. Finally, examples are given that illustrate the effectiveness of the theoretical results.