Lithuanian Mathematical Journal

, Volume 58, Issue 2, pp 167–184 | Cite as

Continuous-time dichotomies without unstable invariant manifolds for autonomous system

  • Ciprian Preda
  • Florin Bǎtǎran


It is known that a differential equation (t) = Ax(t) on the Banach space X (we assume the well-posedness, i.e., A generates a C0-semigroup {T(t)}t ≥ 0) is hyperbolic if X can be decomposed as X = X1X2 so that solutions (∙) starting from X1 (respectively, from X2) decay exponentially in forward time (respectively, in backward time). Hyperbolicity forces the solutions that start from X2 to exist for negative time (or, equivalently, the semigroup generated by A to extend to a C0-group on X2). We generalize this notion by replacing the exponential decay in negative time for the solutions starting in X2 with an exponential blow-up in positive time (we call this an exponential dichotomy). It is obvious that hyperbolicity implies the existence of an exponential dichotomy, but the converse is not valid (we point out an example in this context). We obtain a unified treatment of admissibility-type conditions guaranteeing the existence of an exponential dichotomy and complete characterizations of the hyperbolicity of autonomous differential equations.


C0-semigroups Schäffer spaces admissibility hyperbolicity 


34D05 47D06 93D20 


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  1. 1.
    W.A. Coppel, Dichotomies in Stability Theory, Lect. Notes Math., Vol. 629, Springer, New York, 1978.Google Scholar
  2. 2.
    J.L. Daleckij and M.G. Krein, Stability of Solutions of Differential Equations in Banach spaces, Transl. Math. Monogr., Vol. 43, AMS, Providence, RI, 1974.Google Scholar
  3. 3.
    M.S. Elbialy, Locally Lipschitz perturbations of bi-semigroups, Commun. Pure Appl. Anal., 9(2):327–349, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    M.S. Elbialy, Stable and unstable manifolds for hyperbolic bi-semigroups, J. Funct. Anal., 262(5):2516–2560, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    M.A. Kaashoek and S.M. Verduyn Lunel, An integrability condition on the resolvent for hyperbolicity of the semigroup, J. Differ. Equations, 112(2):374–406, 1994.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Y. Latushkin and B. Layton, The optimal gap condition for invariant manifolds, Discrete Contin. Dyn. Syst., 5(2):233–268, 1999.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Y. Latushkin and A. Pogan, The dichotomy theorem for evolution bi-families, J. Differ. Equations, 245(8):2267–2306, 2008.MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Y. Latushkin, A. Pogan, and R. Schnaubelt, Dichotomies and Fredholm properties of evolution equations, J. Oper. Theory, 58(2):387–414, 2007.zbMATHGoogle Scholar
  9. 9.
    T. Li, Die Stabilitätsfrage bei Differenzengleichungen, Acta Math., 63:99–141, 1934.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    J.L. Massera and J.J. Schäffer, Linear differential equations and functional analysis. I, Ann.Math. (2), 67(3):517–573, 1958.Google Scholar
  11. 11.
    J.L. Massera and J.J. Schäffer, Linear Differential Equations and Function Spaces, Academic Press, New York, 1966.zbMATHGoogle Scholar
  12. 12.
    R.O. Moşincat, C. Preda, and P. Preda, Dichotomies with no invariant unstable manifolds for autonomous equations, J. Funct. Spaces Appl., 2012:527647, 2012, available from:
  13. 13.
    N. van Minh, F. Räbiger, and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line, Integral Equations Oper. Theory, 32(3):332–353, 1998.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    J.M.A.M. van Neerven, The Asymptotic Behaviour of Semigroups of Linear Operators, Oper. Theory: Adv. Appl., Vol. 88, Birkhäuser, Basel, 1996.Google Scholar
  15. 15.
    J.M.A.M van Neerven, Characterizations of exponential stability of a semigroup of operators in terms of its action by convolution on vector-valued function spaces over ℝ+, J. Differ. Equations, 124(2):324–342, 1996.Google Scholar
  16. 16.
    A. Pazy, Semigroups of operators and Applications to Partial Differential Equations, Springer, Berlin, 1983.CrossRefzbMATHGoogle Scholar
  17. 17.
    O. Perron, Die Stabilitätsfrage bei Differentialgeighungen, Math. Z., 32:703–728, 1930.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    V.Q. Phóng, On the exponential stability and dichotomy of C 0-semigroups, Stud. Math., 132(2):141–149, 1999.CrossRefzbMATHGoogle Scholar
  19. 19.
    C. Preda and P. Preda, On the asymptotic behavior of the solutions of autonomous equations without unstable invariant manifolds, Comput. Math. Appl., 64(1):35–47, 2012.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    C. Preda, P. Preda, and A. Craciunescu, Criterions for detecting the existence of the exponential dichotomies in the asymptotic behavior of the solutions of variational equations, J. Funct. Anal., 258(3):729–757, 2010.MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    P. Preda, A. Pogan, and C. Preda, On the Perron problem for the exponential dichotomy of C 0-semigroups, Acta Math. Univ. Comen., New Ser., 72(2):207–212, 2003.Google Scholar
  22. 22.
    P. Preda, A. Pogan, and C. Preda, (L p , L q)-admissibility and exponential dichotomy of evolutionary processes on the half-line, Integral Equations Oper. Theory, 49(3):405–418, 2004.Google Scholar
  23. 23.
    P. Preda, A. Pogan, and C. Preda, Schäffer spaces and uniform exponential stability of linear skew-product semiflows, J. Differ. Equations, 212(1):191–207, 2005.CrossRefzbMATHGoogle Scholar
  24. 24.
    P. Preda, A. Pogan, and C. Preda, Schäffer spaces and exponential dichotomy for evolutionary processes, J. Differ. Equations, 230(1):378–391, 2006.CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.West University of TimişoaraTimişoaraRomania

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