Jump to ContentJump to Main Navigation
Show Summary Details
More options …

Nonautonomous Dynamical Systems

formerly Nonautonomous and Stochastic Dynamical Systems

Editor-in-Chief: Diagana, Toka

Managing Editor: Cánovas, Jose

1 Issue per year


Mathematical Citation Quotient (MCQ) 2016: 0.56


Emerging Science

Open Access
Online
ISSN
2353-0626
See all formats and pricing
More options …

Periodic Solutions for Nonlinear Evolution Equations with Non-instantaneous Impulses

Michal Fečkan
  • Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynská dolina, 842 48 Bratislava, Slovakia, and Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, 814 73 Bratislava, Slovakia
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ JinRong Wang
  • Department of Mathematics, Guizhou University, Guiyang, Guizhou 550025, P.R. China School of Mathematics and Computer Science, Guizhou Normal College, Guiyang, Guizhou 550018, P.R. China
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Yong Zhou
Published Online: 2014-08-15 | DOI: https://doi.org/10.2478/msds-2014-0004

Abstract

In this paper, we consider periodic solutions for a class of nonlinear evolution equations with non-instantaneous impulses on Banach spaces. By constructing a Poincaré operator, which is a composition of the maps and using the techniques of a priori estimate, we avoid assuming that periodic solution is bounded like in [1-4] and try to present new sufficient conditions on the existence of periodic mild solutions for such problems by utilizing semigroup theory and Leray-Schauder's fixed point theorem. Furthermore, existence of a global compact connected attractor for the Poincaré operator is derived.

Keywords: Nonlinear evolution equations; Non-instantaneous impulses; Periodic solutions; Attractor; Existence

References

  • [1] J. H. Liu, Bounded and periodic solutions of differential equations in Banach space, Appl. Math. Comput., 65(1994), 141-150.Google Scholar

  • [2] J. H. Liu, Bounded and periodic solutions of semilinear evolution equations, Dynam. Syst. Appl., 4(1995), 341-350.Google Scholar

  • [3] J. H. Liu, Bounded and periodic solutions of finite delay evolution equations, Nonlinear Anal.:TMA, vol. 34(1998), 101-111.Google Scholar

  • [4] J. H. Liu, T. Naito, N. V. Minh, Bounded and periodic solutions of infinite delay evolution equations, J. Math. Anal. Appl., 286(2003), 705-712.Google Scholar

  • [5] E. Hernández, D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc, 141(2013), 1641-1649.Google Scholar

  • [6] M. Pierri, D. O'Regan, V. Rolnik, Existence of solutions for semi-linear abstract differential equations with not instantaneous impulses, Appl. Math. Comput., 219(2013), 6743-6749.Google Scholar

  • [7] D. D. Bainov, P. S. Simeonov, Impulsive differential equations: periodic solutions and applications, New York, 1993.Google Scholar

  • [8] A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, vol.14 of World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, World Scientific, Singapore, 1995.Google Scholar

  • [9] M. Benchohra, J. Henderson, S. Ntouyas, Impulsive differential equations and inclusions, vol. 2 of Contemporary Mathematics and Its Applications, Hindawi, New York, NY, USA, 2006.Google Scholar

  • [10] X. Xiang, N. U. Ahmed, Existence of periodic solutions of semilinear evolution equations with time lags, Nonlinear Anal.:TMA, 18(1992), 1063-1070.Google Scholar

  • [11] P. Sattayatham, S. Tangmanee, W. Wei, On periodic solutions of nonlinear evolution equations in Banach spaces, Journal of Mathematical Analysis and Applications, J. Math. Anal. Appl., 276(2002), 98-108.Google Scholar

  • [12] J. Wang, X. Xiang, Y. Peng, Periodic solutions of semilinear impulsive periodic system on Banach space, Nonlinear Anal.:TMA, 71(2009), e1344-e1353.Google Scholar

  • [13] Z. Liu, Anti-periodic solutions to nonlinear evolution equations, J. Funct. Anal., 258(2011), 2026-2033.Google Scholar

  • [14] Y. Li, Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Funct. Anal., 261(2011), 1309-1324.Web of ScienceGoogle Scholar

  • [15] P. Kokocki, Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Math. Anal. Appl., 392(2012), 55-74.Google Scholar

  • [16] N. U. Ahmed, Semigroup theory with applications to systems and control, vol.246, Pitman Research Notes in Mathematics Series, Longman Scientific and Technical, Harlow, UK, 1991.Google Scholar

  • [17] J. K. Hale, Stability and gradient dynamical systems, Rev. Mat. Complut. 17(2003), 7-57.Google Scholar

  • [18] J. K. Hale, Asymptotic behaviour of dissipative systems, AMS, Providence, Rhode Islans, 1988.Google Scholar

About the article

MSC: 34C25; 34K30; 35R12


Received: 2014-01-28

Accepted: 2014-04-30

Published Online: 2014-08-15


Citation Information: Nonautonomous Dynamical Systems, ISSN (Online) 2299-3193, DOI: https://doi.org/10.2478/msds-2014-0004.

Export Citation

© 2014 Michal Fečkan et. al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

Comments (0)

Please log in or register to comment.
Log in