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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 705126, 13 pages
http://dx.doi.org/10.1155/2013/705126
Research Article

New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

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

Received 29 May 2013; Revised 17 October 2013; Accepted 22 October 2013

Academic Editor: Chengming Huang

Copyright © 2013 Bin Zheng and Qinghua Feng. 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.

Abstract

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.