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

Abstract

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.

Keywords

• Generalized Gronwall inequality
• Fixed-point theorem
• ψ-Hilfer proportional fractional operators
• Ulam–Hyers–Mittag–Leffler stability