A Gronwall inequality via the generalized proportional fractional derivative with applications

Abstract

In this paper, we provide a new version for the Gronwall inequality in the frame of the generalized proportional fractional derivative. Prior to the main results, we introduce the generalized proportional fractional derivative and expose some of its features. As an application, we accommodate the newly defined derivative to prove the uniqueness and obtain a bound in terms of Mittag-Leffler function for the solutions of a nonlinear delay proportional fractional system. An example is presented to demonstrate the applicability of the theory.

Bounds of Generalized Proportional Fractional Integrals in General Form via Convex Functions and Their Applications

“…The Gronwall inequalities which involve the proportional fractional integral operator can be found in work of Alzabut et al [42]. Rahman et al [43] established the Minkowski inequality and other types of inequalities in the frame of the proportional fractional integrals.…”

Certain inequalities via generalized proportional Hadamard fractional integral operators

“…Recently Alzabut et al and Rahman et al [11,34] studied generalized proportional derivative and integral operators and established certain Gronwall and Minkowski inequalities involving the operators mentioned above.…”

Tempered Fractional Integral Inequalities for Convex Functions

“…Alzabut et al [45] recently studied the Gronwall inequalities by considering generalized proportional fractional derivative operator. Rahman et al [46] presented the Minkowski inequalities by employing proportional fractional integral.…”