A generalized Gronwall inequality via $\psi$-Hilfer proportional fractional operators and its applications to nonlocal Cauchy-type system

This paper establishes a novel generalized Gronwall inequality concerning the $\psi$-Hilfer proportional fractional operators. Before proving the main results, the solution of the linear nonlocal coupled $\psi$-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.