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

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

MATHEMATICS, vol.7, no.8, 2019 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 7 Issue: 8
  • Publication Date: 2019
  • Doi Number: 10.3390/math7080747
  • Journal Name: MATHEMATICS
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus
  • Keywords: Gronwall-Bellman inequality, proportional fractional derivative, Riemann-Liouville and Caputo proportional fractional initial value problem
  • Van Yüzüncü Yıl University Affiliated: Yes

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.