Abstract and Applied Analysis

Gronwall-Bellman Type Inequalities and Their Applications to Fractional Differential Equations

Jing Shao and Fanwei Meng

Full-text: Open access

Abstract

Some new weakly singular integral inequalities of Gronwall-Bellman type are established, which can be used in the qualitative analysis of the solutions to certain fractional differential equations.

Article information

Source
Abstr. Appl. Anal., Volume 2013 (2013), Article ID 217641, 7 pages.

Dates
First available in Project Euclid: 27 February 2014

Permanent link to this document
https://projecteuclid.org/euclid.aaa/1393512009

Digital Object Identifier
doi:10.1155/2013/217641

Mathematical Reviews number (MathSciNet)
MR3093753

Zentralblatt MATH identifier
1297.34010

Citation

Shao, Jing; Meng, Fanwei. Gronwall-Bellman Type Inequalities and Their Applications to Fractional Differential Equations. Abstr. Appl. Anal. 2013 (2013), Article ID 217641, 7 pages. doi:10.1155/2013/217641. https://projecteuclid.org/euclid.aaa/1393512009


Export citation

References

  • R. P. Agarwal, S. Deng, and W. Zhang, “Generalization of a retarded Gronwall-like inequality and its applications,” Applied Mathematics and Computation, vol. 165, no. 3, pp. 599–612, 2005.
  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier, Amsterdam, The Netherlands, 2006.
  • O. Lipovan, “A retarded Gronwall-like inequality and its applications,” Journal of Mathematical Analysis and Applications, vol. 252, no. 1, pp. 389–401, 2000.
  • Q.-H. Ma and J. Pečarić, “Some new explicit bounds for weakly singular integral inequalities with applications to fractional differential and integral equations,” Journal of Mathematical Analysis and Applications, vol. 341, no. 2, pp. 894–905, 2008.
  • H. Ye and J. Gao, “Henry-Gronwall type retarded integral inequalities and their applications to fractional differential equations with delay,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4152–4160, 2011.
  • B. G. Pachpatte, Integral and Finite Difference Inequalities and Applications, Elsevier, Oxford, UK, 2006.
  • H. Ye, J. Gao, and Y. Ding, “A generalized Gronwall inequality and its application to a fractional differential equation,” Journal of Mathematical Analysis and Applications, vol. 328, no. 2, pp. 1075–1081, 2007.
  • I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999.