Journal of Applied Mathematics

New Integral Inequalities with Weakly Singular Kernel for Discontinuous Functions and Their Applications to Impulsive Fractional Differential Systems

Jing Shao

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

Some new integral inequalities with weakly singular kernel for discontinuous functions are established using the method of successive iteration and properties of Mittag-Leffler function, which can be used in the qualitative analysis of the solutions to certain impulsive fractional differential systems.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 252946, 5 pages.

Dates
First available in Project Euclid: 2 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305753

Digital Object Identifier
doi:10.1155/2014/252946

Mathematical Reviews number (MathSciNet)
MR3212493

Citation

Shao, Jing. New Integral Inequalities with Weakly Singular Kernel for Discontinuous Functions and Their Applications to Impulsive Fractional Differential Systems. J. Appl. Math. 2014 (2014), Article ID 252946, 5 pages. doi:10.1155/2014/252946. https://projecteuclid.org/euclid.jam/1425305753


Export citation

References

  • A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204, Elsevier Science, Amsterdam, The Netherlands, 2006.
  • V. Lakshmikantham, S. Leela, and D. J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, 2009.
  • I. Podlubny, Fractional Differential Equations, vol. 198, Academic Press, San Diego, Calif, USA, 1999.
  • K. Diethelm, The Analysis of Fractional Differential Equations, vol. 2004, Springer, Berlin, Germany, 2010.
  • Z. Denton and A. S. Vatsala, “Fractional integral inequalities and applications,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1087–1094, 2010.
  • 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.
  • 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.
  • J. Shao and F. Meng, “Gronwall-Bellman type inequalities and their applications to fractional differential equations,” Abstract and Applied Analysis, vol. 2013, Article ID 217641, 7 pages, 2013.
  • A. Halanay and D. Wexler, Qualitative Theory of Impulsive Systems, Academia Română, Bucharest, Romania, 1968.
  • A. M. Samoilenko and N. Perestyuk, Differential Equations with Impulse Effect, Visha Shkola, Kyiv, Ukraine, 1987.
  • V. Lakshmikantham, D. D. Baĭnov, and P. S. Simeonov, Theory of Impulsive Differential Equations, vol. 6 of Series in Modern Applied Mathematics, World Scientific Publishing, Teaneck, NJ, USA, 1989.
  • D. Xu, Y. Hueng, and L. Ling, “Existence of positive solutions of an impulsive delay fishing model,” Bulletin of Mathematical Analysis and Applications, vol. 3, no. 2, pp. 89–94, 2011.
  • S. D. Borysenko, M. Ciarletta, and G. Iovane, “Integro-sum inequalities and motion stability of systems with impulse perturbations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 62, no. 3, pp. 417–428, 2005.
  • A. Gallo and A. M. Piccirillo, “About new analogies of Gronwall-Bellman-Bihari type inequalities for discontinuous functions and estimated solutions for impulsive differential systems,” Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 5, pp. 1550–1559, 2007.
  • A. Gallo and A. M. Piccirillo, “On some generalizations Bellman-Bihari result for integro-functional inequalities for discontinuous functions and their applications,” Boundary Value Problems, vol. 2009, Article ID 808124, 14 pages, 2009.
  • K. Balachandran, S. Kiruthika, and J. J. Trujillo, “On fractional impulsive equations of Sobolev type with nonlocal condition in Banach spaces,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1157–1165, 2011.
  • H. Jiang, “Existence results for fractional order functional differential equations with impulse,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3477–3483, 2012.
  • X. Li, F. Chen, and X. Li, “Generalized anti-periodic boundary value problems of impulsive fractional differential equations,” Communications in Nonlinear Science and Numerical Simulation, vol. 18, no. 1, pp. 28–41, 2013.
  • L. Mahto, S. Abbas, and A. Favini, “Analysis of Caputo impulsive fractional order differential equations with applications,” International Journal of Differential Equations, vol. 2013, Article ID 704547, 11 pages, 2013. \endinput