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

Abduljawad Anwar; Shayma Adil Murad; Abduljawad Anwar; Shayma Adil Murad

doi:10.3934/mmc.2024035

Mathematical Modelling and Control

2024, Volume 4, Issue 4: 439-458. doi: 10.3934/mmc.2024035

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Research article

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

Abduljawad Anwar ^,,
Shayma Adil Murad ^,

Department of Mathematics, College of Science, University of Duhok, Duhok 42001, Iraq

Received: 04 February 2024 Revised: 02 October 2024 Accepted: 10 November 2024 Published: 18 December 2024

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{ADM}} $) is used to construct the analytical approximate solutions for the problems. Finally, examples are given that illustrate the effectiveness of the theoretical results.
Citation: Abduljawad Anwar, Shayma Adil Murad. On the Ulam stability and existence of $ L^p $-solutions for fractional differential and integro-differential equations with Caputo-Hadamard derivative[J]. Mathematical Modelling and Control, 2024, 4(4): 439-458. doi: 10.3934/mmc.2024035

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Abstract

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{ADM}} $) is used to construct the analytical approximate solutions for the problems. Finally, examples are given that illustrate the effectiveness of the theoretical results.

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