Alzabut, JehadSudsutad, WeerawatKayar, ZeynepBaghani, Hamid2025-05-102025-05-1020192227-739010.3390/math70807472-s2.0-85071179055https://doi.org/10.3390/math7080747https://hdl.handle.net/20.500.14720/11443Sudsutad, Weerawat/0000-0002-6762-813X; Baghani, Mostafa/0000-0001-6695-3128; Alzabut, Prof. Dr. Jehad/0000-0002-5262-1138New 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.eninfo:eu-repo/semantics/openAccessGronwall-Bellman InequalityProportional Fractional DerivativeRiemann-Liouville And Caputo Proportional Fractional Initial Value ProblemArticle78Q1Q2WOS:000482856500069