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On the Mild Solutions of Second-Order Θ-Caputo Fractional Boundary Value Problems

Author

Listed:
  • Mouataz Billah Mesmouli

    (Department of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi Arabia)

  • Abdelouaheb Ardjouni

    (Department of Mathematics, Faculty of Sciences and Technology, University of Souk Ahras, P.O. Box 1553, Souk Ahras 41000, Algeria)

  • Loredana Florentina Iambor

    (Department of Mathematics and Computer Science, University of Oradea, Universitatii nr. 1, 410087 Oradea, Romania)

  • Taher S. Hassan

    (Department of Mathematics, College of Science, University of Ha’il, Ha’il 2440, Saudi Arabia
    Jadara University Research Center, Jadara University, Irbid 21110, Jordan
    Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt)

Abstract

In this paper, we study a class of second-order fractional boundary value problems involving Θ-Caputo derivatives of different orders. By reformulating the problem to an integral equation, we introduce an appropriate notion of a mild solution in the Θ-fractional framework. Existence results are obtained via Krasnoselskii’s fixed point theorem, while uniqueness is established using the Banach contraction principle under suitable Lipschitz-type conditions. The obtained results extend several earlier works on Caputo, Hadamard–Caputo, and Riemann–Liouville fractional derivatives. Two examples are presented to illustrate the applicability of the theoretical results.

Suggested Citation

  • Mouataz Billah Mesmouli & Abdelouaheb Ardjouni & Loredana Florentina Iambor & Taher S. Hassan, 2026. "On the Mild Solutions of Second-Order Θ-Caputo Fractional Boundary Value Problems," Mathematics, MDPI, vol. 14(9), pages 1-13, April.
    • Handle: RePEc:gam:jmathe:v:14:y:2026:i:9:p:1434-:d:1927778
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