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Exponential dichotomies of evolution operators in Banach spaces

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Abstract

This paper considers three dichotomy concepts (exponential dichotomy, uniform exponential dichotomy and strong exponential dichotomy) in the general context of non-invertible evolution operators in Banach spaces. Connections between these concepts are illustrated. Using the notion of Green function, we give necessary conditions and sufficient ones for strong exponential dichotomy. Some illustrative examples are presented to prove that the converse of some implication type theorems are not valid.

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References

  1. Bainov, D.D., Kostadinov, S.I., Zabreiko, P.P.: Exponential dichotomy of linear impulsive equations in a Banach space. Internat. J. Theor. Phys. 28, 797–814 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  2. Barreira, L., Pesin, Ya.B.: Lyapunov exponents and smooth ergodic theory. Univ. Lecture Ser., vol. 23. American Mathematical Society, Providence (2002)

  3. Barreira, L., Valls, C.: Robustness of nonuniform exponential dichotomies in Banach spaces. J. Differ. Equ. 244, 2407–2447 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Barreira, L., Valls, C.: Quadratic Lyapunov functions and nonuniform exponential dichotomies. J. Differ. Equ. 246, 1235–1263 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barreira, L., Valls, C.: Robustness via Lyapunov functions. J. Differ. Eq. 246, 2891–2907 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Barreira, L., Valls, C.: Nonuniform exponential dichotomies and admissibility. Discrete Contin. Dyn. Syst. 30, 39–53 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  7. Caraballo, T., Real, J., Shaikhet, L.: Method of Lyapunov functionals construction in stability of delay evolution equations. J. Math. Anal. Appl. 334, 1130–1145 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chicone, C., Latushkin, Y.: Evolution semigroups in dynamical systems and differential equations. Mathematical Surveys and Monographs, vol. 70. American Mathematical Society, Providence (1999)

  9. Chow, S.N., Leiva, H.: Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces. Proc. Am. Math. Soc. 124, 1071–1081 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  10. Coppel, W.A.: Dichotomies in stability theory. Lecture Notes in Mathematics, vol. 629. Springer, Berlin (1978)

  11. Daleckiǐ, Ju.L., Kreǐn, M.G.: Stability of solutions of differential equations in Banach space. Translation of Mathematical Monographs, vol 43. American Mathematical Society, Providence (1974)

  12. Datko, R.: Uniform asymptotic stability of evolutionary processes in a Banach space. SIAM J. Math. Anal. 3, 428–445 (1972)

    Article  MATH  MathSciNet  Google Scholar 

  13. Hatvani, L.: On the asymptotic stability for functional differential equations by Lyapunov functionals. Nonlinear Anal. 47, 4333–4343 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  14. Huy, N.T.: Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line. J. Funct. Anal. 235, 330–354 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  15. Lupa, N., Megan, M.: Rolewicz type theorems for nonuniform exponential stability of evolution operators on the half-line. In: Gaşpar, D., Størmer, E., Timotin, D., Vasilescu, F.H. (eds.) Proceedings of the 23rd International Conference on Operator Theory, Timisoara, 29 June–4 July 2010. An Operator Theory Summer, pp. 63–68. Theta (2012)

  16. Lupa, N., Megan, M., Popa, I.L.: On weak exponential stability of evolution operators in Banach spaces. Nonlinear Anal. 73, 2445–2450 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Malisoff, M., Mazenc, F.: Constructions of strict Lyapunov functions. Communication Control Engineering Series. Springer, Berlin (2009)

  18. Massera, J.L., Schäffer, J.J.: Linear differential equations and function spaces. Pure and Applied Mathematics, vol 21. Academic Press, New York (1966)

  19. Megan, M., Buşe, C.: On uniform exponential dichotomy of observable evolution operators. Rend. Sem. Mat. Univ. Politec. Torino 50, 183–194 (2010)

    Google Scholar 

  20. Megan, M., Sasu, B., Sasu, A.L.: On nonuniform exponential dichotomy of evolution operators in Banach spaces. Integr. Equ. Oper. Theory 44, 71–78 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  21. Megan, M., Sasu, A.L., Sasu, B.: Perron conditions for uniform exponential expansiveness of linear skew-product flows. Monatsh. Math. 138, 145–157 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Van Minh, Nguyen, Räbiger, F., Schnaubelt, R.: Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line. Integr. Equ. Oper. Theory 32, 332–353 (1998)

    Article  MATH  Google Scholar 

  23. Mitropolsky, Yu.A., Samoilenko, A.M., Kulik, V.L.: Dichotomies and stability in nonautonomous linear systems. In: Stability and Control Theory Methods and Applications, vol. 14. Taylor & Francis, Boca Raton (2003)

  24. Nagel, R., Nickel, G.: Wellposedness for nonautonomous abstract Cauchy problems. Progr. Nonlinear Differ. Equ. Appl. 50, 279–293 (2002)

    MathSciNet  Google Scholar 

  25. Perron, O.: Die Stabilitätsfrage bei Differentialgleichungen. Math. Z. 32, 703–728 (1930)

    Article  MATH  MathSciNet  Google Scholar 

  26. Popescu, L.H.: Exponential dichotomy roughness on Banach spaces. J. Math. Anal. Appl. 314, 436–454 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Pötzsche, C.: Geometric theory of discrete nonautonomous dynamical systems. Lecture Notes in Mathematics, vol. 2002. Springer, Berlin (2010)

  28. Preda, P., Megan, M.: Nonuniform dichotomy of evolutionary processes in Banach spaces. Bull. Aust. Math. Soc. 27, 31–52 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  29. Preda, P., Megan, M.: Exponential dichotomy of evolutionary processes in Banach spaces. Czechoslovak Math. J. 35, 312–323 (1985)

    MathSciNet  Google Scholar 

  30. Preda, P., Pogan, A., Preda, C.: (\(L^p, L^q\))-admissibility and exponential dichotomy of evolutionary processes on the half-line. Integr. Equ. Oper. Theory 49, 405–418 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. Sacker, R.J., Sell, G.R.: Dichotomies for linear evolutionary equations in Banach spaces. J. Differ. Equ. 113, 17–67 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  32. Sasu, B.: Uniform dichotomy and exponential dichotomy of evolution families on the half-line. J. Math. Anal. Appl. 323, 1465–1478 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  33. Sasu, B.: Integral conditions for exponential dichotomy: a nonlinear approach. Bull. Sci. Math. 134, 235–246 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  34. Stoica, D.: Uniform exponential dichotomy of stochastic cocycles. Stoch. Process. Appl. 120, 1920–1928 (2010)

    Google Scholar 

  35. Zhang, J., Fan, M., Zhu, H.: Existence and roughness of exponential dichotomies of linear dynamic equations on time scales. Comput. Math. Appl. 59, 2658–2675 (2010)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

We would like to thank the referees for carefully reading our manuscript and for their helpful suggestions, which improve the quality of the paper.

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Author notes

  1. Mihail Megan

    Present address: Faculty of Mathematics and Computer Science, West University of Timişoara, Blvd. V. Pârvan 4, 300223 , Timisoara, Romania

Authors and Affiliations

  1. Faculty of Economics and Business Administration, West University of Timişoara, Blvd. Pestalozzi 16, 300115 , Timisoara, Romania

    Nicolae Lupa

  2. Academy of Romanian Scientists, Independenţei 54, 050094 , Bucharest, Romania

    Mihail Megan

Authors
  1. Nicolae Lupa
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  2. Mihail Megan
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Correspondence to Mihail Megan.

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Communicated by A. Constantin.

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Lupa, N., Megan, M. Exponential dichotomies of evolution operators in Banach spaces. Monatsh Math 174, 265–284 (2014). https://doi.org/10.1007/s00605-013-0517-y

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