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

Jehad Alzabut, Weerawat Sudsutad, Zeynep Kayar and Hamid Baghani
Additional contact information
Jehad Alzabut: Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia
Weerawat Sudsutad: Department of General Education, Faculty of Science and Health Technology, Navamindradhiraj University, Bangkok 10300, Thailand
Zeynep Kayar: Department of Mathematics, Van Yuzuncu Yil University, 65080 Van, Turkey
Hamid Baghani: Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran

Mathematics, 2019, vol. 7, issue 8, 1-15

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.

Keywords: Gronwall–Bellman inequality; proportional fractional derivative; Riemann–Liouville and Caputo proportional fractional initial value problem (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (2)

Downloads: (external link)
https://www.mdpi.com/2227-7390/7/8/747/pdf (application/pdf)
https://www.mdpi.com/2227-7390/7/8/747/ (text/html)

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:gam:jmathe:v:7:y:2019:i:8:p:747-:d:258017

Access Statistics for this article

Mathematics is currently edited by Ms. Emma He

More articles in Mathematics from MDPI
Bibliographic data for series maintained by MDPI Indexing Manager ().