A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative
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Department of Mathematics and General Sciences, Prince Sultan University, P.O. Box 66833, 11586 Riyadh, Saudi Arabia
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Department of General Education, Faculty of Science and Health Technology, Navamindradhiraj University, Bangkok 10300, Thailand
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Department of Mathematics, Van Yuzuncu Yil University, 65080 Van, Turkey
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Department of Mathematics, Faculty of Mathematics, University of Sistan and Baluchestan, Zahedan, Iran
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(8), 747; https://doi.org/10.3390/math7080747
Received: 19 July 2019
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Revised: 10 August 2019
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Accepted: 12 August 2019
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Published: 15 August 2019
(This article belongs to the Special Issue Inequalities)
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.
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Keywords:
Gronwall–Bellman inequality; proportional fractional derivative; Riemann–Liouville and Caputo proportional fractional initial value problem
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MDPI and ACS Style
Alzabut, J.; Sudsutad, W.; Kayar, Z.; Baghani, H. A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative. Mathematics 2019, 7, 747. https://doi.org/10.3390/math7080747
AMA Style
Alzabut J, Sudsutad W, Kayar Z, Baghani H. A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative. Mathematics. 2019; 7(8):747. https://doi.org/10.3390/math7080747
Chicago/Turabian StyleAlzabut, Jehad, Weerawat Sudsutad, Zeynep Kayar, and Hamid Baghani. 2019. "A New Gronwall–Bellman Inequality in Frame of Generalized Proportional Fractional Derivative" Mathematics 7, no. 8: 747. https://doi.org/10.3390/math7080747
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