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Spectra Based on Bohl Exponents and Bohl Dichotomy for Nonautonomous Difference Equations

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Abstract

For nonautonomous linear difference equations with bounded coefficients on \(\mathbb {N}\) which have a bounded inverse, we introduce two different notions of spectra and discuss their relation to the well-known exponential dichotomy spectrum. The first new spectral notion is called Bohl spectrum and is based on an extended notion of the concept of Bohl exponents. The second new spectral notion is called Bohl dichotomy spectrum and is based on a relaxed version of exponential dichotomy called Bohl dichotomy. We prove spectral theorems and show that the Bohl dichotomy spectrum is the closure of the Bohl spectrum and also a subset of the exponential dichotomy spectrum. We discuss the spectra of upper triangular systems and how they relate to the spectra of their diagonal entries. An example illustrates the subtle differences between the different notions of spectra.

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References

  1. Aulbach, B., Siegmund, S.: The dichotomy spectrum for noninvertible systems of linear difference equations. J. Differ. Equ. Appl. 7(6), 895–913 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Aulbach, B., Siegmund, S.: A spectral theory for nonautonomous difference equations. In: Proceedings of the 5th International Conference of Difference Equations and Application (Temuco, Chile, 2000), pp. 45–55 (2002)

  3. Babiarz, A., Czornik, A., Niezabitowski, M.: Relations between Bohl exponents and general exponent of discrete linear time-varying systems. J. Differ. DEqu. Appl. 25(4), 560–572 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barabanov, E.A., Konyukh, A.V.: Bohl exponents of linear differential systems. Mem. Differ. Equ. Math. Phys. 24, 151–158 (2001)

    MathSciNet  MATH  Google Scholar 

  5. Barreira, L.: Lyapunov Exponents. Springer, Cham (2017)

    MATH  Google Scholar 

  6. Barreira, L., Valls, C.: On two notions of exponential dichotomy. Dyn. Syst. 33(4), 708–721 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ben-Artzi, A., Gohberg, I.: Dichotomy, discrete Bohl exponents, and spectrum of block weighted shifts. Integr. Equ. Oper. Theory 14(5), 613–677 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bohl, P.: Über Differentialgleichungen. J. Reine Angew. Math. 144, 284–318 (1913)

    MATH  Google Scholar 

  9. Daleckii, Y.L., Krein, M.G.: Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs, vol. 43. American Mathematical Society, Providence (1974)

    Google Scholar 

  10. Doan, T.S., Palmer, K.J., Rasmussen, M.: The Bohl spectrum for linear nonautonomous differential equations. J. Dyn. Differ. Equ. 29(4), 1459–1485 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hinrichsen, D., Pritchard, A.J.: Mathematical Systems Theory I. Springer, Berlin (2005)

    Book  MATH  Google Scholar 

  12. Izobov, N.A.: Lyapunov Exponents and Stability. Cambridge Scientific Publishers, Cottenham (2012)

    MATH  Google Scholar 

  13. Megginson, R.E.: An Introduction to Banach Space Theory. Graduate Texts in Mathematics, vol. 183. Springer, New York (1998)

    MATH  Google Scholar 

  14. Pötzsche, C.: Geometric Theory of Discrete Nonautonomous Dynamical Systems. Springer, Berlin (2010)

    Book  MATH  Google Scholar 

  15. Pötzsche, C.: Fine structure of the dichotomy spectrum. Integr. Equ. Oper. Theory 73(1), 107–151 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  16. Russ, E.: Dichotomy spectrum for difference equations in Banach spaces. J. Differ. Equ. Appl. 23, 574–617 (2017)

    Article  MathSciNet  MATH  Google Scholar 

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Funding

The research of A. Czornik was supported by the Polish National Agency for Academic Exchange according to the decision PPN/BEK/2020/1/00188/UO/00001.

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Authors and Affiliations

  1. Faculty of Automatic Control, Electronics and Computer Science, Silesian University of Technology, Gliwice, Poland

    Adam Czornik

  2. Institute of Analysis, Faculty of Mathematics, TU Dresden, Dresden, Germany

    Konrad Kitzing & Stefan Siegmund

Authors
  1. Adam Czornik
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  2. Konrad Kitzing
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  3. Stefan Siegmund
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All authors have contributed equally to the article.

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Correspondence to Konrad Kitzing.

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Czornik, A., Kitzing, K. & Siegmund, S. Spectra Based on Bohl Exponents and Bohl Dichotomy for Nonautonomous Difference Equations. J Dyn Diff Equat (2023). https://doi.org/10.1007/s10884-023-10311-0

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  • DOI: https://doi.org/10.1007/s10884-023-10311-0

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