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On the Robustness Property of Nonuniform Exponential Dichotomies

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Advances in Discrete Dynamical Systems, Difference Equations and Applications (ICDEA 2021)

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Abstract

Assume that a sequence \((A_n)_{n\in \mathbb Z}\) of bounded linear operators on a Banach space X admits a nonuniform exponential dichotomy with exponents \(\lambda >0\) and \(\epsilon \ge 0\). We formulate sufficient conditions under which a perturbed sequence \((A_n+B_n)_{n\in \mathbb Z}\) admits a nonuniform exponential dichotomy with the same exponents.

Supported in part by Croatian Science Foundation under the project IP-2019-04-1239 and by the University of Rijeka under the projects uniri-prirod-18-9 and uniri-pr-prirod-19-16.

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References

  1. Battelli, F., Fečkan, M.: On the exponents of exponential dichotomies. Mathematics 8, 651 (2020)

    Article  Google Scholar 

  2. Battelli, F., Franca, M., Palmer, K.J.: Exponential dichotomy for noninvertible linear difference equations. J. Differ. Equ. Appl. 27, 1657–1691 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  3. Barreira, L., Dragičević, D., Valls, C.: Exponential dichotomies with respect to a sequence of norms and admissibility. Internat. J. Math. 25, 1450024 (2014), 20 pp

    Google Scholar 

  4. Barreira, L., Dragičević, D., Valls, C.: Tempered exponential dichotomies: admissibility and stability under perturbations. Dyn. Syst. 31, 525–545 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barreira, L., Dragičević, D., Valls, C.: Nonuniform hyperbolicity and one-sided admissibility. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27, 235–247 (2016)

    Google Scholar 

  6. Barreira, L., Dragičević, D., Valls, C.: Admissibility and Hyperbolicity. SpringerBriefs in Mathematics, Springer, Cham (2018)

    Book  MATH  Google Scholar 

  7. Barreira, L., Silva, C., Valls, C.: Nonuniform behavior and robustness. J. Differ. Equ. 246, 3579–3608 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Barreira, L., Valls, C.: Stability of nonautonomous differential equations. In: Lecture Notes in Mathematics, vol. 1926. Springer, Berlin (2008)

    Google Scholar 

  9. Barreira, L., Valls, C.: Nonuniform cocycles: robustness of exponential dichotomies. Discret. Contin. Dyn. Syst. 32, 4111–4131 (2012)

    Article  MATH  Google Scholar 

  10. Bento, A.J., Silva, C.: Robustness of discrete nonuniform dichotomic behavior. arXiv:1308.6820

  11. Calamai, A., Franca, M.: Mel’nikov methods and homoclinic orbits in discontinuous systems. J. Dyn. Differ. Equ. 25, 733–764 (2013)

    Google Scholar 

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

    Google Scholar 

  13. Coppel, W.A.: Dichotomies in Stability Theory. In: Lecture Notes in Mathematics, vol. 629. Springer, Berlin (1978)

    Google Scholar 

  14. Daleckij, J.L., Krein, M.G.: Stability of Solutions of Differential Equations in Banach Space. American Mathematical Society, Providence, RI (1974)

    Google Scholar 

  15. Dragičević, D.: Admissibility and nonuniform polynomial dichotomies. Math. Nachr. 293, 226–243 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  16. Dragičević, D., Zhang, W.: Asymptotic stability of nonuniform behaviour. Proc. Am. Math. Soc. 147, 2437–2451 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  17. Massera, J.L., Schäffer, J.J.: Linear differential equations and functional analysis I. Ann. Math. 67, 517–573 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  18. Massera, J.L., Schäffer, J.J.: Linear Differential Equations and Function Spaces. Academic Press, New York (1966)

    MATH  Google Scholar 

  19. Naulin, R., Pinto, M.: Admissible perturbations of exponential dichotomy roughness. Nonlinear Anal. 31, 559–571 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  20. Palmer, K.: A perturbation theorem for exponential dichotomies. Proc. Roy. Soc. Edinburgh Sect. A 106, 25–37 (1987)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  22. Plis, V.A., Sell, G.R.: Robustness of exponential dichotomies in infinite-dimensional dynamical systems. J. Dyn. Differ. Equ. 11, 471–513 (1999)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Pötzsche, C.: Geometric Theory of Discrete Nonautonomous Dynamical Systems. Lecture Notes in Mathematics, vol. 2002. Springer, Berlin (2010)

    Google Scholar 

  25. Sasu, A.L.: Exponential dichotomy and dichotomy radius for difference equations. J. Math. Anal. Appl. 344, 906–920 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sasu, A.L., Sasu, B.: Strong exponential dichotomy of discrete nonautonomous systems: input-output criteria and strong dichotomy radius. J. Math. Anal. Appl. 504, 125373 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  27. Silva, C.M.: Admissibility and generalized nonuniform dichotomies for discrete dynamics. Commun. Pure Appl. Anal. 20, 3419–3443 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zhou, L., Lu, K., Zhang, W.: Roughness of tempered exponential dichotomies for infinite-dimensional random difference equations. J. Differ. Equ. 254, 4024–4046 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zhou, L., Zhang, W.: Admissibility and roughness of nonuniform exponential dichotomies for difference equations. J. Funct. Anal. 271, 1087–1129 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Faculty of Mathematics, University of Rijeka, Radmile Matejčić 2, 51000, Rijeka, Croatia

    Davor Dragičević

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  1. Davor Dragičević
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Correspondence to Davor Dragičević .

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Editors and Affiliations

  1. Department of Mathematics, Trinity University, San Antonio, TX, USA

    Saber Elaydi

  2. University of Rhode Island, Kingston, RI, USA

    Mustafa R. S. Kulenović

  3. University of Sarajevo, Sarajevo, Bosnia and Herzegovina

    Senada Kalabušić

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Dragičević, D. (2023). On the Robustness Property of Nonuniform Exponential Dichotomies. In: Elaydi, S., Kulenović, M.R.S., Kalabušić, S. (eds) Advances in Discrete Dynamical Systems, Difference Equations and Applications. ICDEA 2021. Springer Proceedings in Mathematics & Statistics, vol 416. Springer, Cham. https://doi.org/10.1007/978-3-031-25225-9_9

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