A Gronwall inequality and its applications to the Cauchy-type problem under ψ-Hilfer proportional fractional operators

Autor:	Weerawat Sudsutad, Chatthai Thaiprayoon, Bounmy Khaminsou, Jehad Alzabut, Jutarat Kongson
Jazyk:	angličtina
Rok vydání:	2023
Předmět:	Generalized Gronwall inequality Fixed-point theorem ψ-Hilfer proportional fractional operators Ulam–Hyers–Mittag–Leffler stability Mathematics QA1-939
Zdroj:	Journal of Inequalities and Applications, Vol 2023, Iss 1, Pp 1-35 (2023)
Druh dokumentu:	article
ISSN:	1029-242X
DOI:	10.1186/s13660-023-02929-x
Popis:	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.
Databáze:	Directory of Open Access Journals
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