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

Abstract

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.

Keywords

• generalized gronwall inequality
• existence and uniqueness
• fixed point theorem
• $ \psi $-hilfer proportional fractional operators
• mittag-leffler-ulam-hyers stability