A New Gronwall-Bellman Inequality in Frame of Generalized Proportional Fractional Derivative

Alzabut, Jehad; Sudsutad, Weerawat; Kayar, Zeynep; Baghani, Hamid

doi:10.3390/math7080747

A New Gronwall-Bellman Inequality in Frame of Generalized Proportional Fractional Derivative

Atıf İçin Kopyala

Alzabut J., Sudsutad W., Kayar Z. , Baghani H.

MATHEMATICS, cilt.7, 2019 (SCI İndekslerine Giren Dergi)

Cilt numarası: 7 Konu: 8
Basım Tarihi: 2019
Doi Numarası: 10.3390/math7080747
Dergi Adı: MATHEMATICS

Özet

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.