Advertisement

Semigroup Forum

, Volume 96, Issue 2, pp 241–252 | Cite as

Nonuniform exponential dichotomy for linear skew-product semiflows over semiflows

  • Ciprian Preda
  • Oana Romina Onofrei
Research Article
  • 78 Downloads

Abstract

The aim of this paper is to obtain necessary and sufficient conditions for the existence of a nonuniform exponential dichotomy over a general class of linear skew-product semiflows (over semiflows) on a Banach space. We extend Datko’s classical result to the case of the exponential nonuniform dichotomy of linear skew-product semiflows over semiflows on a Banach space, by using Lyapunov norms.

Keywords

Semiflows Linear skew-product semiflows Cocycle Nonuniform exponentially growth Nonuniform exponential dichotomy Datko–Pazy theorem 

References

  1. 1.
    Barreira, L., Valls, C.: Stability of Nonautonomous Differential Equations. Lecture Notes in Mathematics, vol. 1926. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  2. 2.
    Barreira, L., Valls, C.: Admissibility for nonuniform exponential contractions. J. Differ. Equ. 249, 2889–2904 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barreira, L., Valls, C.: Nonuniform exponential dichotomies and admissibility. Discrete Contin. Dyn. Syst. 30(1), 39–53 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Datko, R.: Extending a theorem of Liapunov to Hilbert Spaces. J. Math. Anal. Appl. 32, 610–616 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Datko, R.: Uniform asymptotic stability of evolutionary processes in Banach space. SIAM J. Math. Anal. 3, 428–445 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Massera, J.L., Schäffer, J.J.: Linear differential equations and functional analysis, I. Ann. Math. 67(3), 517–573 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Massera, J.L., Schäffer, J.J.: Linear differential equations and function spaces. Academic Press, New York (1966)zbMATHGoogle Scholar
  8. 8.
    Pazy, A.: On the applicability of Lyapunov’s theorem in Hilbert space. SIAM J. Math. Anal. 3, 291–294 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Preda, C., Preda, P., Bătăran, F.: An extension of a theorem of R. Datko to the case of (non)uniform exponential stability of linear skewproduct semiflows. J. Math. Anal. Appl. 425(2), 1148–1154 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Preda, C., Preda, P., Crăciunescu, A.: Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of solutions of variational equations. J. Funct. Anal 258, 729–757 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Preda, C., Preda, P., Crăciunescu, A.: A version of a theorem of R. Datko for nonuniform exponential contractions. J. Math. Anal. Appl. 385, 572–581 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Preda, C., Preda, P., Petre, A.: On the asymptotic behavior of an exponentially bounded, strongly continuous cocycle over a semiflow. Commun. Pure Appl. Anal. 8, 1637–1645 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Preda, C., Preda, P., Praţa, C.: An extension of some theorems of L. Barreira and C. Valls for the nonuniform exponential dichotomous evolution operators. J. Math. Anal. Appl. 388, 1090–1106 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Preda, P., Megan, M.: Exponential dichotomy of strongly discontinuous semigroups. Bull. Aust. Math. Soc. 30, 435–448 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Preda, P., Megan, M.: Exponential dichotomy of evolutionary processes in Banach Spaces. Czechoslov. Math. J. 35(110), 312–323 (1985)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Rolewicz, S.: On uniform N-equistability. J. Math. Anal. Appl. 115, 434–441 (1986)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.West University of TimisoaraTimisoaraRomania

Personalised recommendations