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Article

Mixed Erdélyi–Kober and Caputo Fractional Differential Equations with Nonlocal Fractional Closed Boundary Conditions

by
Furkan Erkan
1,
Nuket Aykut Hamal
1,
Sotiris K. Ntouyas
2,* and
Bashir Ahmad
3
1
Department of Mathematics, Ege University, Bornova, Izmir 35100, Türkiye
2
Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
3
Nonlinear Analysis and Applied Mathematics (NAAM)—Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
*
Author to whom correspondence should be addressed.
Foundations 2026, 6(1), 12; https://doi.org/10.3390/foundations6010012
Submission received: 26 January 2026 / Revised: 15 February 2026 / Accepted: 4 March 2026 / Published: 6 March 2026
(This article belongs to the Special Issue Recent Advances in Fractional Differential Equations and Inclusions: 3rd Edition)
Versions Notes

Abstract

This work focuses on the analysis of a sequential fractional boundary value problem involving coupled Erdélyi–Kober and Caputo fractional differential operators, together with nonlocal boundary conditions of fractional type. The well-posedness of the problem is addressed by deriving conditions that ensure the existence and uniqueness of solutions. Uniqueness is obtained through an application of Banach’s contraction principle, whereas existence is established by employing Krasnosel’skiĭ’s fixed point approach and the nonlinear alternative of Leray–Schauder. Several numerical examples are presented to demonstrate and support the theoretical findings.
Keywords: Erdélyi–Kober fractional derivative; fractional integrals; existence and uniqueness; fixed point theorems Erdélyi–Kober fractional derivative; fractional integrals; existence and uniqueness; fixed point theorems

Share and Cite

MDPI and ACS Style

Erkan, F.; Hamal, N.A.; Ntouyas, S.K.; Ahmad, B. Mixed Erdélyi–Kober and Caputo Fractional Differential Equations with Nonlocal Fractional Closed Boundary Conditions. Foundations 2026, 6, 12. https://doi.org/10.3390/foundations6010012

AMA Style

Erkan F, Hamal NA, Ntouyas SK, Ahmad B. Mixed Erdélyi–Kober and Caputo Fractional Differential Equations with Nonlocal Fractional Closed Boundary Conditions. Foundations. 2026; 6(1):12. https://doi.org/10.3390/foundations6010012

Chicago/Turabian Style

Erkan, Furkan, Nuket Aykut Hamal, Sotiris K. Ntouyas, and Bashir Ahmad. 2026. "Mixed Erdélyi–Kober and Caputo Fractional Differential Equations with Nonlocal Fractional Closed Boundary Conditions" Foundations 6, no. 1: 12. https://doi.org/10.3390/foundations6010012

APA Style

Erkan, F., Hamal, N. A., Ntouyas, S. K., & Ahmad, B. (2026). Mixed Erdélyi–Kober and Caputo Fractional Differential Equations with Nonlocal Fractional Closed Boundary Conditions. Foundations, 6(1), 12. https://doi.org/10.3390/foundations6010012

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