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Admissibility, a general type of Lipschitz shadowing and structural stability

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  • For a general one-sided nonautonomous dynamics defined by a sequence of linear operators, we consider the notion of a uniform exponential dichotomy and we characterize it completely in terms of the admissibility of a large class of function spaces. We apply those results to show that structural stability of a diffeomorphism is equivalent to a very general type of Lipschitz shadowing property. Our results extend those in [37] in various directions.
    Mathematics Subject Classification: Primary: 34D09, 34K12; Secondary: 37C50, 37D20.


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  • [1]

    L. Barreira, D. Dragičević and C. Valls, Exponential dichotomies with respect to a sequence of norms and admissibility, Int. J. Math., 25, 1450024 (2014), 20 pages.doi: 10.1142/S0129167X14500244.


    L. Barreira, D. Dragičević and C. Valls, Nonuniform hyperbolicity and admissibility, Adv. Nonlinear Stud., 14 (2014), 791-811.


    L. Barreira, D. Dragičević and C. Valls, Strong and weak $(L^p,L^q)$-admissibility, Bull. Sci. Math., 138 (2014), 721-741.doi: 10.1016/j.bulsci.2013.11.005.


    L. Barreira, D. Dragičević and C. Valls, Admissibility on the half line for evolution families, J. Anal. Math., to appear.


    L. Barreira and C. Valls, Stability of Nonautonomous Differential Equations, Lect. Notes. in Math. 1926, 2008.doi: 10.1007/978-3-540-74775-8.


    A. Ben-Artzi and I. Gohberg, Dichotomy of systems and invertibility of linear ordinary differential operators, in Time-Variant Systems and Interpolation, Oper. Theory Adv. Appl., 56, Birkhäuser, 1992, pp. 90-119.


    A. Ben-Artzi, I. Gohberg and M. Kaashoek, Invertibility and dichotomy of differential operators on a half-line, J. Dynam. Differential Equations, 5 (1993), 1-36.doi: 10.1007/BF01063733.


    C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations, Mathematical Surveys and Monographs 70, Amer. Math. Soc., 1999.doi: 10.1090/surv/070.


    C. Coffman and J. Schäffer, Dichotomies for linear difference equations, Math. Ann., 172 (1967), 139-166.


    W. Coppel, Dichotomies in Stability Theory, Lect. Notes. in Math. 629, Springer, 1978.


    Ju. Dalec'kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space, Translations of Mathematical Monographs 43, Amer. Math. Soc., 1974.


    D. Dragičević and S. Slijepčević, Characterization of hyperbolicity and generalized shadowing lemma, Dyn. Syst., 26 (2011), 483-502.doi: 10.1080/14689367.2011.606205.


    A. Fakhari, K. Lee, and K. Tajbakhsh, Diffeomorphisms with $L^p$-shadowing property, Acta Math. Sin., 27 (2011), 19-28.doi: 10.1007/s10114-011-0050-7.


    J. Hale, Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs 25, Amer. Math. Soc., 1988.


    D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lect. Notes in Math. 840, Springer, 1981.


    N. Huy, Exponential dichotomy of evolution equations and admissibility of function spaces on a half-line, J. Funct. Anal., 235 (2006), 330-354.doi: 10.1016/j.jfa.2005.11.002.


    Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory, 58 (2007), 387-414.


    B. Levitan and V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, 1982.


    A. D. Maizel, On stability of solutions of systems of differential equations, Trudi Uralskogo Politekhnicheskogo Instituta, Mathematics, 51 (1954), 20-50.


    R. Ma né, Characterizations of AS diffeomorphisms, in Geometry and Topology (eds. Jacob Palis and Manfredo do Carmo), Lecture Notes in Mathematics, Vol. 597, Springer Berlin, (1977), 389-394.


    J. Massera and J. Schäffer, Linear differential equations and functional analysis. I., Ann. of Math., 67 (1958), 517-573.


    J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics 21, Academic Press, 1966.


    N. Minh and N. Huy, Characterizations of dichotomies of evolution equations on the half-line, J. Math. Anal. Appl., 261 (2001), 28-44.doi: 10.1006/jmaa.2001.7450.


    N. Van Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line, Integral Equations Operator Theory, 32 (1998), 332-353.doi: 10.1007/BF01203774.


    K. Palmer, Exponential dichotomies and Fredholm operators, Proc. Amer. Math. Soc., 104 (1988), 149-156.doi: 10.2307/2047477.


    K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer, Dordrecht, 2000.doi: 10.1007/978-1-4757-3210-8.


    K. J. Palmer, S. Yu. Pilyugin and S. B. Tikhmirov, Lipschitz shadowing and structural stability of flows, J. Differential Equations, 252 (2012), 1723-1747.doi: 10.1016/j.jde.2011.07.026.


    O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.doi: 10.1007/BF01194662.


    S. Yu. Pilyugin, Shadowing in Dynamical Systems, Lecture Notes Math., vol. 1706, Springer, Berlin, 1999.


    S. Yu. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515.doi: 10.1088/0951-7715/23/10/009.


    S. Pilyugin, G. Volfson and D. Todorov, Dynamical systems with Lipschitz inverse shadowing properties, Vestnik St. Petersburg University: Mathematics, 44 (2011), 208-213.doi: 10.3103/S106345411103006X.


    V. A. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations, in Problems of Asymptotic Theory of Nonlinear Oscillations (Russian), Naukova Dumka, Kiev, (1977), 168-173.


    P. Preda, A. Pogan and C. Preda, $(L^p,L^q)$-admissibility and exponential dichotomy of evolutionary processes on the half-line, Integral Equations Operator Theory, 49 (2004), 405-418.doi: 10.1007/s00020-002-1268-7.


    A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations, J. Math. Anal. Appl., 344 (2008), 906-920.doi: 10.1016/j.jmaa.2008.03.019.


    G. Sell and Y. You, Dynamics of Evolutionary Equation Applied Mathematical Sciences 143, Springer, 2002.doi: 10.1007/978-1-4757-5037-9.


    S. Tikhomirov, Hölder shadowing on finite intervals, Ergodic Theory Dynam. Systems, (2014), http://dx.doi.org/10.1017/etds.2014.7


    D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in Shadowing Theory, Discrete Contin. Dyn. Syst., 33 (2013), 4187 - 4205.doi: 10.3934/dcds.2013.33.4187.


    W. Zhang, The Fredholm alternative and exponential dichotomies for parabolic equations, J. Math. Anal. Appl., 191 (1985), 180-201.doi: 10.1016/S0022-247X(85)71126-2.

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