Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2012 (2012), Article ID 214609, 14 pages
http://dx.doi.org/10.1155/2012/214609
Research Article

Some New Difference Inequalities and an Application to Discrete-Time Control Systems

1Department of Mathematics, Hechi University, Guangxi, Yizhou 546300, China
2Department of Electrical and Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117583

Received 1 July 2012; Accepted 17 September 2012

Academic Editor: Jong Hae Kim

Copyright © 2012 Hong Zhou et al. 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.

Linked References

  1. T. H. Gronwall, “Note on the derivatives with respect to a parameter of the solutions of a system of differential equations,” Annals of Mathematics, vol. 20, no. 4, pp. 292–296, 1919. View at Publisher · View at Google Scholar
  2. R. Bellman, “The stability of solutions of linear differential equations,” Duke Mathematical Journal, vol. 10, pp. 643–647, 1943. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. D. S. Mitrinović, J. E. Pečarić, and A. M. Fink, Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic, Dordrecht, The Netherlands, 1991.
  4. D. Baĭnov and P. Simeonov, Integral Inequalities and Applications, vol. 57, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1992.
  5. B. G. Pachpatte, Inequalities for Differential and Integral Equations, vol. 197, Academic Press, New York, USA, 1998.
  6. W. Zhang and S. Deng, “Projected Gronwall-Bellman's inequality for integrable functions,” Mathematical and Computer Modelling, vol. 34, no. 3-4, pp. 393–402, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  8. B.-I. Kim, “On some Gronwall type inequalities for a system integral equation,” Bulletin of the Korean Mathematical Society, vol. 42, no. 4, pp. 789–805, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. O. Lipovan, “Integral inequalities for retarded Volterra equations,” Journal of Mathematical Analysis and Applications, vol. 322, no. 1, pp. 349–358, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  10. W.-S. Cheung, “Some new nonlinear inequalities and applications to boundary value problems,” Nonlinear Analysis A, vol. 64, no. 9, pp. 2112–2128, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. R. P. Agarwal, C. S. Ryoo, and Y.-H. Kim, “New integral inequalities for iterated integrals with applications,” Journal of Inequalities and Applications, vol. 2007, Article ID 24385, 18 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. W.-S. Wang, “A generalized retarded Gronwall-like inequality in two variables and applications to BVP,” Applied Mathematics and Computation, vol. 191, no. 1, pp. 144–154, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. R. P. Agarwal, Y.-H. Kim, and S. K. Sen, “New retarded integral inequalities with applications,” Journal of Inequalities and Applications, vol. 2008, Article ID 908784, 15 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. W.-S. Wang and C.-X. Shen, “On a generalized retarded integral inequality with two variables,” Journal of Inequalities and Applications, vol. 2008, Article ID 518646, 9 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. C.-J. Chen, W.-S. Cheung, and D. Zhao, “Gronwall-bellman-type integral inequalities and applications to BVPs,” Journal of Inequalities and Applications, vol. 2009, Article ID 258569, 15 pages, 2009. View at Publisher · View at Google Scholar · View at Scopus
  16. A. Abdeldaim and M. Yakout, “On some new integral inequalities of Gronwall-Bellman-Pachpatte type,” Applied Mathematics and Computation, vol. 217, no. 20, pp. 7887–7899, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  17. B. G. Pachpatte, “Finite difference inequalities and discrete time control systems,” Indian Journal of Pure and Applied Mathematics, vol. 9, no. 12, pp. 1282–1290, 1978. View at Google Scholar · View at Zentralblatt MATH
  18. W.-S. Cheung and J. Ren, “Discrete non-linear inequalities and applications to boundary value problems,” Journal of Mathematical Analysis and Applications, vol. 319, no. 2, pp. 708–724, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  19. B. G. Pachpatte, Integral and Finite Difference Inequalities and Applications, vol. 205 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006.
  20. Q.-H. Ma and W.-S. Cheung, “Some new nonlinear difference inequalities and their applications,” Journal of Computational and Applied Mathematics, vol. 202, no. 2, pp. 339–351, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  21. W.-S. Wang, “A generalized sum-difference inequality and applications to partial difference equations,” Advances in Difference Equations, vol. 2008, Article ID 695495, 12 pages, 2008. View at Google Scholar · View at Zentralblatt MATH
  22. W.-S. Wang, “Estimation on certain nonlinear discrete inequality and applications to boundary value problem,” Advances in Difference Equations, vol. 2009, Article ID 708587, 8 pages, 2009. View at Google Scholar · View at Zentralblatt MATH
  23. X.-M. Zhang and Q.-L. Han, “Delay-dependent robust H filtering for uncertain discrete-time systems with time-varying delay based on a finite sum inequality,” IEEE Transactions on Circuits and Systems II, vol. 53, no. 12, pp. 1466–1470, 2006. View at Publisher · View at Google Scholar · View at Scopus
  24. K. L. Zheng, S. M. Zhong, and M. Ye, “Discrete nonlinear inequalities in time control systems,” in Proceedings of the International Conference on Apperceiving Computing and Intelligence Analysis (ICACIA '09), pp. 403–406, 2009.