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

Mathematics

卷:7

Weerawat Sudsutad¹  Jehad Alzabut²  Hamid Baghani³  Zeynep Kayar⁴ 

[1] Department of General Education, Faculty of Science and Health Technology, Navamindradhiraj University, Bangkok 10300, Thailand;

[2] Department of Mathematics and General Sciences, Prince Sultan, P.O. Box 66833, Riyadh 11586, Saudi Arabia;

[3] Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran;

[4] Department of Mathematics, Van Yuzuncu Yil University, 65080 Van, Turkey;

关键词: Gronwall–Bellman inequality;    proportional fractional derivative;    Riemann–Liouville and Caputo proportional fractional initial value problem;   

DOI  :  10.3390/math7080747

来源: DOAJ

【 摘 要 】

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.

【 授权许可】

Unknown