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Volume 14
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10.3390/axioms14110855
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

The Existence and Uniqueness of Mild Solutions for Fuzzy Hilfer Fractional Evolution Equations with Non-Local Conditions

by
Kholoud N. Alharbi
1,* and
Sanaa Alotaibi
2
1
Department of Mathematics, College of Science, Qassim University, Buraydah 52571, Saudi Arabia
2
Department of Mathematics, Faculty of Sciences and Arts, King Abdulaziz University, Rabigh 21911, Saudi Arabia
*
Author to whom correspondence should be addressed.
Axioms 2025, 14(11), 855; https://doi.org/10.3390/axioms14110855
Submission received: 17 October 2025 / Revised: 10 November 2025 / Accepted: 18 November 2025 / Published: 20 November 2025
(This article belongs to the Special Issue Fractional Differentiation and Applied Mathematics: Recent Advances and Applications)
Versions Notes

Abstract

In this paper, we investigate a fuzzy Hilfer fractional evolution equation of type 0<β<1 and order 1<α<2 subject to nonlocal conditions. Using the infinitesimal generator of a strongly continuous cosine family, we define a mild solution for the proposed system. The existence and uniqueness of such mild solutions are established through Schauder’s fixed-point theorem and the Banach contraction principle. An illustrative application to a fuzzy fractional wave equation is presented to demonstrate the effectiveness of the developed approach. The main contribution of this study lies in the unified treatment of fuzzy Hilfer fractional evolution equations under nonlocal conditions, which generalizes and extends several existing results and provides a solid analytical foundation for modeling systems with memory and uncertainty.
Keywords: fuzzy Hilfer fractional derivative; evolution equation; mild solution; Schauder’s fixed-point theorem fuzzy Hilfer fractional derivative; evolution equation; mild solution; Schauder’s fixed-point theorem

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MDPI and ACS Style

Alharbi, K.N.; Alotaibi, S. The Existence and Uniqueness of Mild Solutions for Fuzzy Hilfer Fractional Evolution Equations with Non-Local Conditions. Axioms 2025, 14, 855. https://doi.org/10.3390/axioms14110855

AMA Style

Alharbi KN, Alotaibi S. The Existence and Uniqueness of Mild Solutions for Fuzzy Hilfer Fractional Evolution Equations with Non-Local Conditions. Axioms. 2025; 14(11):855. https://doi.org/10.3390/axioms14110855

Chicago/Turabian Style

Alharbi, Kholoud N., and Sanaa Alotaibi. 2025. "The Existence and Uniqueness of Mild Solutions for Fuzzy Hilfer Fractional Evolution Equations with Non-Local Conditions" Axioms 14, no. 11: 855. https://doi.org/10.3390/axioms14110855

APA Style

Alharbi, K. N., & Alotaibi, S. (2025). The Existence and Uniqueness of Mild Solutions for Fuzzy Hilfer Fractional Evolution Equations with Non-Local Conditions. Axioms, 14(11), 855. https://doi.org/10.3390/axioms14110855

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