Carathéodory's Theorem for a Nonlocal Boundary Value Problems of a Functional Integro-Fractional Differential Equations
El-Sayed, A., El-Sayed, W., Hagag, S., Issa, S. (2025). Carathéodory's Theorem for a Nonlocal Boundary Value Problems of a Functional Integro-Fractional Differential Equations. EKB Journal Management System, 16(1), 1-13. doi: 10.21608/jfca.2025.358118.1156
Ahmed M. A. El-Sayed; Wagdy G. El-Sayed; Shimaa Hagag; shenouda Issa. "Carathéodory's Theorem for a Nonlocal Boundary Value Problems of a Functional Integro-Fractional Differential Equations". EKB Journal Management System, 16, 1, 2025, 1-13. doi: 10.21608/jfca.2025.358118.1156
El-Sayed, A., El-Sayed, W., Hagag, S., Issa, S. (2025). 'Carathéodory's Theorem for a Nonlocal Boundary Value Problems of a Functional Integro-Fractional Differential Equations', EKB Journal Management System, 16(1), pp. 1-13. doi: 10.21608/jfca.2025.358118.1156
El-Sayed, A., El-Sayed, W., Hagag, S., Issa, S. Carathéodory's Theorem for a Nonlocal Boundary Value Problems of a Functional Integro-Fractional Differential Equations. EKB Journal Management System, 2025; 16(1): 1-13. doi: 10.21608/jfca.2025.358118.1156

Carathéodory's Theorem for a Nonlocal Boundary Value Problems of a Functional Integro-Fractional Differential Equations

Journal of Fractional Calculus and Applications
Volume 16, Issue 1, 2025, Page 1-13  XML PDF (211.67 K)
Document Type: Regular research papers
DOI: 10.21608/jfca.2025.358118.1156
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Authors
Ahmed M. A. El-Sayedorcid 1; Wagdy G. El-Sayed1; Shimaa Hagag2; shenouda Issa email orcid 3
1Department of Mathematics, Faculty of Science, Alexandria University, Egypt.
2Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt
3Department of Mathematics and computer Science, Faculty of Science, Alexandria University
Abstract
This paper analyzes the problem linked to the existence of solutions of a nonlocal two-point boundary value problem concerning an ordinary integro-fractional differential equation. Under certain conditions, utilizing Carathéodory’s criteria in conjunction with an iterative approach using the Arzelà–Ascoli theorem as well as the Lebesgue Dominated Convergence theorem, we show that at least one solution exists. Additionally, we show the Hyers-Ulam stability of the problem, which means that the solution is stable with respect to small perturbations. Imposing a Lipschitz restriction on the nonlinear component, we consider the continuous dependence of the unique solution on the given causes. A range of special cases, namely different boundary value problems that encompass a wide variety of models in physics, biology, engineering, and economics, are analyzed. Moreover, examples are presented for both nonlinear and discontinuous forcing functions, which illustrate the practical utility and broad scope of application of our theory and allow the researchers to study the solvability of a new class of functional equations.
Keywords
Functional integro-fractional differential equations; Nonlocal boundary value problems; Existence and uniqueness; Hyers-Ulam stability; Continuous dependence
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