Existence and Stability for Fractional Differential Equations with a ψ –Hilfer Fractional Derivative in the Caputo Sense

Wenchang He, Yuhang Jin, Luyao Wang, Ning Cai and Jia Mu ()
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Wenchang He: School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China
Yuhang Jin: School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China
Luyao Wang: School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China
Ning Cai: School of Intelligent Engineering and Automation, Beijing University of Posts and Telecommunications, Beijing 100876, China
Jia Mu: School of Mathematics and Computer Science, Northwest Minzu University, Lanzhou 730030, China

Mathematics, 2024, vol. 12, issue 20, 1-12

Abstract: This article aims to explore the existence and stability of solutions to differential equations involving a ψ -Hilfer fractional derivative in the Caputo sense, which, compared to classical ψ -Hilfer fractional derivatives (in the Riemann–Liouville sense), provide a clear physical interpretation when dealing with initial conditions. We discovered that the ψ -Hilfer fractional derivative in the Caputo sense can be represented as the inverse operation of the ψ -Riemann–Liouville fractional integral, and used this property to prove the existence of solutions for linear differential equations with a ψ -Hilfer fractional derivative in the Caputo sense. Additionally, we applied Mönch’s fixed-point theorem and knowledge of non-compactness measures to demonstrate the existence of solutions for nonlinear differential equations with a ψ -Hilfer fractional derivative in the Caputo sense, and further discussed the Ulam–Hyers–Rassias stability and semi-Ulam–Hyers–Rassias stability of these solutions. Finally, we illustrated our results through case studies.

Keywords: regularized ? -Hilfer derivative; existence; Ulam–Hyers–Rassias stability; semi-Ulam–Hyers–Rassias stability (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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