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- Title
A Gronwall inequality and its applications to the Cauchy-type problem under ψ-Hilfer proportional fractional operators.
- Authors
Sudsutad, Weerawat; Thaiprayoon, Chatthai; Khaminsou, Bounmy; Alzabut, Jehad; Kongson, Jutarat
- Abstract
In this paper, we propose a generalized Gronwall inequality in the context of the ψ-Hilfer proportional fractional derivative. Using Picard's successive approximation and the definition of Mittag–Leffler functions, we construct the representation formula of the solution for the ψ-Hilfer proportional fractional differential equation with constant coefficient in the form of the Mittag–Leffler kernel. The uniqueness result is proved by using Banach's fixed-point theorem with some properties of the Mittag–Leffler kernel. Additionally, Ulam–Hyers–Mittag–Leffler stability results are analyzed. Finally, numerical examples are provided to demonstrate the theory's application.
- Subjects
GRONWALL inequalities; FRACTIONAL differential equations; KERNEL functions
- Publication
Journal of Inequalities & Applications, 2023, Vol 2023, Issue 1, p1
- ISSN
1025-5834
- Publication type
Article
- DOI
10.1186/s13660-023-02929-x