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New Results on Periodic Solutions to Impulsive Nonautonomous Evolutionary Equations with Time Delays

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We obtain new results on periodic solutions for a class of nonautonomous impulsive evolutionary equations with time delays. Under suitable assumptions, such as the ultimate boundedness of the solutions of equations, we establish a theorem on periodic solutions to equations of this kind by using the Horn fixed-point theorem. At the end of the paper, we present an application to the case of nonautonomous impulsive partial differential equation with finite time delay.

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References

  1. P. Acquistapace, “Evolution operators and strong solution of abstract parabolic equations,” Different. Integr. Equat., 1, 433–457 (1988).

    MATH  MathSciNet  Google Scholar 

  2. P. Acquistapace and B. Terreni, “A unified approach to abstract linear nonautonomous parabolic equations,” Rend. Semin. Mat. Univ. Padova., 78, 47–107 (1987).

    MATH  MathSciNet  Google Scholar 

  3. N. U. Ahmed, “Measure solutions for impulsive evolution equations with measurable vector fields,” J. Math. Anal. Appl., 319, 74–93 (2006).

    Article  MATH  MathSciNet  Google Scholar 

  4. M. Benchohra, J. Henderson, and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publ. Corp., New York (2006).

    Book  MATH  Google Scholar 

  5. A. A. Boichuk and A. M. Samoilenko, Generalized Inverse Operators and Fredholm Boundary-Value Problems, VSP, Utrecht–Boston (2004).

    Book  MATH  Google Scholar 

  6. T. Burton, Stability and Periodic Solutions of Ordinary Differential Equations and Functional Differential Equations, Acad. Press, New York (1985).

    MATH  Google Scholar 

  7. T. A. Burton and B. Zhang, “Periodic solutions of abstract differential equations with infinite delay,” J. Different. Equat., 90, 357–396 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  8. O. Diekmann, S. van Gils, S. Lunel, and H. O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, 110, Appl. Math. Sci., Springer, Berlin (1995).

  9. K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, 194, GTM, Springer, Berlin (2000).

  10. K. Ezzinbi, J. Liu, and N. Minh, “Periodic solutions of impulsive evolution equations,” Int. J. Evolut. Equat., 4, 103–111 (2009).

    MATH  MathSciNet  Google Scholar 

  11. R. Grimmer, “Resolvent operators for integral equations in Banach space,” Trans. Amer. Math. Soc., 273, 333–349 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  12. D. Guo and X. Liu, “Extremal solutions of nonlinear impulsive integrodifferential equations in Banach spaces,” J. Math. Anal. Appl., 177, 538–552 (1993).

    Article  MATH  MathSciNet  Google Scholar 

  13. V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov, Theory of Impulsive Differential Equations, World Sci., Singapore; London (1989).

  14. Y. X. Li, “Existence and asymptotic stability of periodic solution for evolution equations with delays,” J. Funct. Anal., 261, 1309–1324 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  15. J. Liang, J. H. Liu, and T. J. Xiao, “Nonlocal impulsive problems for nonlinear differential equations in Banach spaces,” Math. Comput. Model., 49, 798–804 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  16. J. Liang, J. H. Liu, and T. J. Xiao, “Periodic solutions of delay impulsive differential equations,” Nonlin. Anal. T.M.A, 74, 6835–6842 (2011).

  17. J. H. Liu, “Bounded and periodic solutions of finite delay evolution equations,” Nonlin. Anal. T.M.A, 34, 101–111 (1998).

  18. J. H. Liu, “Periodic solutions of infinite delay evolution equations,” J. Math. Anal. Appl., 247, 627–644 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  19. J. H. Liu, “Bounded and periodic solutions of infinite delay evolution equations,” J. Math. Anal. Appl., 286, 705–712 (2003).

    Article  MATH  MathSciNet  Google Scholar 

  20. A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations [in Russian], Vyshcha Shkola, Kiev (1987).

    Google Scholar 

  21. X. Xiang and N. Ahmed, “Existence of periodic solutions of semilinear evolution equations with time lags,” Nonlin. Anal. T.M.A, 11, 1063–1070 (1992).

  22. J. M. Zhu, Y. C. Liu, and Z. X. Li, “The existence and attractivity of time periodic solutions for evolution equations with delays,” Nonlin. Anal. R.W.A, 9, 842–851 (2008).

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Authors and Affiliations

  1. Guangdong University, Foreign Studies, Guangzhou, 510420, China

    R. -N. Wang

  2. Nanchang University, Nanchang, Jiangxi, 330031, China

    P. -X. Zhu

Authors
  1. R. -N. Wang
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  2. P. -X. Zhu
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Corresponding authors

Correspondence to R. -N. Wang or P. -X. Zhu.

Additional information

Published in Neliniini Kolyvannya, Vol. 17, No. 4, pp. 476–488, October–December, 2014.

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Wang, R.N., Zhu, P.X. New Results on Periodic Solutions to Impulsive Nonautonomous Evolutionary Equations with Time Delays. J Math Sci 212, 412–425 (2016). https://doi.org/10.1007/s10958-015-2673-1

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  • DOI: https://doi.org/10.1007/s10958-015-2673-1

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