@article{Sudsutad2024, 
author = {Weerawat Sudsutad and Jutarat Kongson and Chatthai Thaiprayoon and Nantapat Jarasthitikulchai and Marisa Kaewsuwan},
title = {A generalized Gronwall inequality via  ψ-Hilfer proportional fractional operators and its applications to nonlocal Cauchy-type system},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {9},
pages = {24443-24479},
keywords = {generalized Gronwall inequality, existence and uniqueness, fixed point theorem, ψ-Hilfer proportional fractional operators, Mittag-Leffler-Ulam-Hyers stability},
url = {https://www.sciopen.com/article/10.3934/math.20241191},
doi = {10.3934/math.20241191},
abstract = {This paper establishes a novel generalized Gronwall inequality concerning the  ψ-Hilfer proportional fractional operators. Before proving the main results, the solution of the linear nonlocal coupled  ψ-Hilfer proportional Cauchy-type system with constant coefficients under the Mittag-Leffler kernel is created. The uniqueness result for the proposed coupled system is established using Banach's contraction mapping principle. Furthermore, a variety of the Mittag-Leffler-Ulam-Hyers stability of the solutions for the proposed coupled system is investigated. Finally, a numerical example is given to show the effectiveness and applicability of the obtained results, and graphical simulations in the case of linear systems are shown.}
}