A new Gronwall-Bellman inequality in frame of generalized proportional fractional derivative

Abstract

New versions of a Gronwall-Bellman inequality in the frame of the generalized (Riemann-Liouville and Caputo) proportional fractional derivative are provided. Before proceeding to the main results, we define the generalized Riemann-Liouville and Caputo proportional fractional derivatives and integrals and expose some of their features. We prove our main result in light of some efficient comparison analyses. The Gronwall-Bellman inequality in the case of weighted function is also obtained. By the help of the new proposed inequalities, examples of Riemann-Liouville and Caputo proportional fractional initial value problems are presented to emphasize the solution dependence on the initial data and on the right-hand side.