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Periodic Solutions for Nonlinear Evolution Equations with Non-instantaneous Impulses

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
1
Numer
1
Opis fizyczny
Daty
otrzymano
2014-01-28
zaakceptowano
2014-04-30
online
2014-08-15
Twórcy
  • 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, Michal.Feckan@fmph.uniba.sk
autor
  • 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, sci.jrwang@gzu.edu.cn
autor
  • Department of Mathematics, Xiangtan University, Xiangtan, Hunan 411105, P.R. China, yzhou@xtu.edu.cn
Bibliografia
  • [1] J. H. Liu, Bounded and periodic solutions of differential equations in Banach space, Appl. Math. Comput., 65(1994), 141-150.
  • [2] J. H. Liu, Bounded and periodic solutions of semilinear evolution equations, Dynam. Syst. Appl., 4(1995), 341-350.
  • [3] J. H. Liu, Bounded and periodic solutions of finite delay evolution equations, Nonlinear Anal.:TMA, vol. 34(1998), 101-111.
  • [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.
  • [5] E. Hernández, D. O'Regan, On a new class of abstract impulsive differential equations, Proc. Amer. Math. Soc, 141(2013), 1641-1649.
  • [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.
  • [7] D. D. Bainov, P. S. Simeonov, Impulsive differential equations: periodic solutions and applications, New York, 1993.
  • [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.
  • [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.
  • [10] X. Xiang, N. U. Ahmed, Existence of periodic solutions of semilinear evolution equations with time lags, Nonlinear Anal.:TMA, 18(1992), 1063-1070.
  • [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.
  • [12] J. Wang, X. Xiang, Y. Peng, Periodic solutions of semilinear impulsive periodic system on Banach space, Nonlinear Anal.:TMA, 71(2009), e1344-e1353.
  • [13] Z. Liu, Anti-periodic solutions to nonlinear evolution equations, J. Funct. Anal., 258(2011), 2026-2033.[WoS]
  • [14] Y. Li, Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Funct. Anal., 261(2011), 1309-1324.[WoS]
  • [15] P. Kokocki, Existence and asymptotic stability of periodic solution for evolution equations with delays, J. Math. Anal. Appl., 392(2012), 55-74.
  • [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.
  • [17] J. K. Hale, Stability and gradient dynamical systems, Rev. Mat. Complut. 17(2003), 7-57.
  • [18] J. K. Hale, Asymptotic behaviour of dissipative systems, AMS, Providence, Rhode Islans, 1988.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_msds-2014-0004
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