Open Access

Linear Impulsive Periodic System with Time-Varying Generating Operators on Banach Space

Advances in Difference Equations20072007:026196

DOI: 10.1155/2007/26196

Received: 3 May 2007

Accepted: 28 August 2007

Published: 30 October 2007

Abstract

A class of the linear impulsive periodic system with time-varying generating operators on Banach space is considered. By constructing the impulsive evolution operator, the existence of -periodic -mild solution for homogeneous linear impulsive periodic system with time-varying generating operators is reduced to the existence of fixed point for a suitable operator. Further the alternative results on -periodic -mild solution for nonhomogeneous linear impulsive periodic system with time-varying generating operators are established and the relationship between the boundness of solution and the existence of -periodic -mild solution is shown. The impulsive periodic motion controllers that are robust to parameter drift are designed for a given periodic motion. An example given for demonstration.

[1234567891011121314151617181920212223]

Authors’ Affiliations

(1)
Department of Computer, College of Computer Science and Technology, Guizhou University
(2)
Department of Mathematics, College of Science, Guizhou University

References

  1. Amann H: Periodic solutions of semilinear parabolic equations. In Nonlinear Analysis: A Collection of Papers in Honor of Erich H. Rothe. Academic Press, New York, NY, USA; 1978:1–29.Google Scholar
  2. Guo D: Periodic boundary value problems for second order impulsive integro-differential equations in Banach spaces. Nonlinear Analysis: Theory, Methods & Applications 1997,28(6):983–997. 10.1016/S0362-546X(97)82855-6MathSciNetView ArticleGoogle Scholar
  3. Yong L, Fuzhong C, Zhenghua L, Wenbin L: Periodic solutions for evolution equations. Nonlinear Analysis: Theory, Methods & Applications 1999,36(3):275–293. 10.1016/S0362-546X(97)00626-3MATHMathSciNetView ArticleGoogle Scholar
  4. Erbe LH, Liu X: Existence of periodic solutions of impulsive differential systems. Journal of Applied Mathematics and Stochastic Analysis 1991,4(2):137–146. 10.1155/S1048953391000102MATHMathSciNetView ArticleGoogle Scholar
  5. Liu JH: Bounded and periodic solutions of finite delay evolution equations. Nonlinear Analysis: Theory, Methods & Applications 1998,34(1):101–111. 10.1016/S0362-546X(97)00606-8MATHMathSciNetView ArticleGoogle Scholar
  6. Li Y, Xing W: Existence of positive periodic solution of a periodic cooperative model with delays and impulses. International Journal of Mathematics and Mathematical Sciences 2006, 2006: 16 pages.MathSciNetGoogle Scholar
  7. Benkhalti R, Ezzinbi K: Periodic solutions for some partial functional differential equations. Journal of Applied Mathematics and Stochastic Analysis 2004,2004(1):9–18. 10.1155/S1048953304212011MATHMathSciNetView ArticleGoogle Scholar
  8. Sattayatham P, Tangmanee S, Wei W: On periodic solutions of nonlinear evolution equations in Banach spaces. Journal of Mathematical Analysis and Applications 2002,276(1):98–108. 10.1016/S0022-247X(02)00378-5MATHMathSciNetView ArticleGoogle Scholar
  9. Wang JR: Linear impulsive periodic system on Banach space. Proceedings of the 4th International Conference on Impulsive and Hybrid Dynamical Systems (ICIDSA '07), July 2007, Nanning, China 5: 20–25.Google Scholar
  10. Wang G, Yan J: Existence of periodic solution for first order nonlinear neutral delay equations. Journal of Applied Mathematics and Stochastic Analysis 2001,14(2):189–194. 10.1155/S1048953301000144MATHMathSciNetView ArticleGoogle Scholar
  11. Xiang X, Ahmed NU: Existence of periodic solutions of semilinear evolution equations with time lags. Nonlinear Analysis: Theory, Methods & Applications 1992,18(11):1063–1070. 10.1016/0362-546X(92)90195-KMATHMathSciNetView ArticleGoogle Scholar
  12. Baĭnov D, Simeonov P: Impulsive Differential Equations: Periodic Solutions and Applications, Pitman Monographs and Surveys in Pure and Applied Mathematics. Volume 66. Longman Scientific & Technical, Harlow, UK; 1993:x+228.Google Scholar
  13. Chu H, Ding S: On the existence of a periodic solution of a nonlinear ordinary differential equation. International Journal of Mathematics and Mathematical Sciences 1998,21(4):767–774. 10.1155/S0161171298001070MATHMathSciNetView ArticleGoogle Scholar
  14. Liu X: Impulsive stabilization and applications to population growth models. The Rocky Mountain Journal of Mathematics 1995,25(1):381–395. 10.1216/rmjm/1181072290MATHMathSciNetView ArticleGoogle Scholar
  15. Yang T: Impulsive Control Theory, Lecture Notes in Control and Information Sciences. Volume 272. Springer, Berlin, Germany; 2001:xx+348.Google Scholar
  16. Naito T, van Minh N, Miyazaki R, Shin JS: A decomposition theorem for bounded solutions and the existence of periodic solutions of periodic differential equations. Journal of Differential Equations 2000,160(1):263–282. 10.1006/jdeq.1999.3673MATHMathSciNetView ArticleGoogle Scholar
  17. Wei W, Xiang X: Anti-periodic solutions for first and second order nonlinear evolution equations in Banach spaces. Journal of Systems Science and Complexity 2004,17(1):96–108.MATHMathSciNetGoogle Scholar
  18. Lakshmikantham V, Baĭnov DD, Simeonov PS: Theory of Impulsive Differential Equations, Series in Modern Applied Mathematics. Volume 6. World Scientific, Teaneck, NJ, USA; 1989:xii+273.View ArticleGoogle Scholar
  19. Ahmed NU: Some remarks on the dynamics of impulsive systems in Banach spaces. Dynamics of Continuous Discrete and Impulsive Systems. Series A 2001,8(2):261–274.MATHMathSciNetGoogle Scholar
  20. Liu JH: Nonlinear impulsive evolution equations. Dynamics of Continuous, Discrete and Impulsive Systems 1999,6(1):77–85.MATHMathSciNetGoogle Scholar
  21. Xiang X, Wei W, Jiang Y: Strongly nonlinear impulsive system and necessary conditions of optimality. Dynamics of Continuous, Discrete and Impulsive Systems. Series A 2005,12(6):811–824.MATHMathSciNetGoogle Scholar
  22. Xiang X, Wei W: Mild solution for a class of nonlinear impulsive evolution inclusions on Banach space. Southeast Asian Bulletin of Mathematics 2006,30(2):367–376.MATHMathSciNetGoogle Scholar
  23. Ahmed NU: Semigroup Theory with Applications to Systems and Control, Pitman Research Notes in Mathematics Series. Volume 246. Longman Scientific & Technical, Harlow, UK; 1991:x+282.Google Scholar

Copyright

© JinRong Wang et al. 2007

This article is published under license to BioMed Central Ltd. 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.