A Gronwall inequality via the generalized proportional fractional derivative with applications. (English) Zbl 1499.26092
Summary: 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.
MSC:
| 26D15 | Inequalities for sums, series and integrals |
| 26A33 | Fractional derivatives and integrals |
| 33E12 | Mittag-Leffler functions and generalizations |
Keywords:
proportional Gronwall inequality; uniqueness of solution; delay proportional fractional system; bound for the solutionReferences:
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