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Journal of Applied Mathematics
Volume 2013, Article ID 341706, 8 pages
Research Article

Some New Gronwall-Bellman-Type Inequalities Based on the Modified Riemann-Liouville Fractional Derivative

School of Science, Shandong University of Technology, Zibo, Shandong 255049, China

Received 22 May 2013; Accepted 28 August 2013

Academic Editor: Francisco J. Marcellán

Copyright © 2013 Bin Zheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


By use of the properties of the modified Riemann-Liouville fractional derivative, some new Gronwall-Bellman-type inequalities are researched. First, we derive some new explicit bounds for the unknown functions lying in these inequalities, which are of different forms from some existing bounds in the literature. Then, we apply the results established to research the boundedness, uniqueness, and continuous dependence on the initial value for the solution to a certain fractional differential equation.