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

Document Type : Regular research papers

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1 Department of Mathematics, Faculty of Science, Alexandria University, Egypt.

2 Department of Mathematics, Faculty of Education, Alexandria University, Alexandria, Egypt

3 Department 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.

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