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May  2015, 14(3): 861-880. doi: 10.3934/cpaa.2015.14.861

Admissibility, a general type of Lipschitz shadowing and structural stability

Davor Dragičević 1,

1. 

Department of Mathematics, University of Rijeka, 51000 Rijeka

Received  May 2014 Revised  October 2014 Published  March 2015

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.
Keywords: Admissibility, exponential dichotomies, shadowing, structural stability..
Mathematics Subject Classification: Primary: 34D09, 34K12; Secondary: 37C50, 37D2.
Citation: Davor Dragičević. Admissibility, a general type of Lipschitz shadowing and structural stability. Communications on Pure & Applied Analysis, 2015, 14 (3) : 861-880. doi: 10.3934/cpaa.2015.14.861
References:
[1]

L. Barreira, D. Dragičević and C. Valls, Exponential dichotomies with respect to a sequence of norms and admissibility,, \emph{Int. J. Math.}, 25 (2014).  doi: 10.1142/S0129167X14500244.  Google Scholar

[2]

L. Barreira, D. Dragičević and C. Valls, Nonuniform hyperbolicity and admissibility,, \emph{Adv. Nonlinear Stud.}, 14 (2014), 791.   Google Scholar

[3]

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

[4]

L. Barreira, D. Dragičević and C. Valls, Admissibility on the half line for evolution families,, \emph{J. Anal. Math.}, ().   Google Scholar

[5]

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

[6]

A. Ben-Artzi and I. Gohberg, Dichotomy of systems and invertibility of linear ordinary differential operators,, in \emph{Time-Variant Systems and Interpolation, (1992), 90.   Google Scholar

[7]

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

[8]

C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations,, Mathematical Surveys and Monographs 70, (1999).  doi: 10.1090/surv/070.  Google Scholar

[9]

C. Coffman and J. Schäffer, Dichotomies for linear difference equations,, \emph{Math. Ann.}, 172 (1967), 139.   Google Scholar

[10]

W. Coppel, Dichotomies in Stability Theory,, Lect. Notes. in Math. 629, (1978).   Google Scholar

[11]

Ju. Dalec'kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space,, Translations of Mathematical Monographs 43, (1974).   Google Scholar

[12]

D. Dragičević and S. Slijepčević, Characterization of hyperbolicity and generalized shadowing lemma,, \emph{Dyn. Syst.}, 26 (2011), 483.  doi: 10.1080/14689367.2011.606205.  Google Scholar

[13]

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

[14]

J. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs 25, (1988).   Google Scholar

[15]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lect. Notes in Math. 840, (1981).   Google Scholar

[16]

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

[17]

Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations,, \emph{J. Operator Theory}, 58 (2007), 387.   Google Scholar

[18]

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

[19]

A. D. Maizel, On stability of solutions of systems of differential equations,, \emph{Trudi Uralskogo Politekhnicheskogo Instituta, 51 (1954), 20.   Google Scholar

[20]

R. Ma né, Characterizations of AS diffeomorphisms,, in \emph{Geometry and Topology} (eds. Jacob Palis and Manfredo do Carmo), (1977), 389.   Google Scholar

[21]

J. Massera and J. Schäffer, Linear differential equations and functional analysis. I.,, \emph{Ann. of Math.}, 67 (1958), 517.   Google Scholar

[22]

J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces,, Pure and Applied Mathematics 21, (1966).   Google Scholar

[23]

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

[24]

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

[25]

K. Palmer, Exponential dichotomies and Fredholm operators,, \emph{Proc. Amer. Math. Soc.}, 104 (1988), 149.  doi: 10.2307/2047477.  Google Scholar

[26]

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

[27]

K. J. Palmer, S. Yu. Pilyugin and S. B. Tikhmirov, Lipschitz shadowing and structural stability of flows,, \emph{J. Differential Equations}, 252 (2012), 1723.  doi: 10.1016/j.jde.2011.07.026.  Google Scholar

[28]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen,, \emph{Math. Z.}, 32 (1930), 703.  doi: 10.1007/BF01194662.  Google Scholar

[29]

S. Yu. Pilyugin, Shadowing in Dynamical Systems,, Lecture Notes Math., (1706).   Google Scholar

[30]

S. Yu. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability,, \emph{Nonlinearity}, 23 (2010), 2509.  doi: 10.1088/0951-7715/23/10/009.  Google Scholar

[31]

S. Pilyugin, G. Volfson and D. Todorov, Dynamical systems with Lipschitz inverse shadowing properties,, \emph{Vestnik St. Petersburg University: Mathematics}, 44 (2011), 208.  doi: 10.3103/S106345411103006X.  Google Scholar

[32]

V. A. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations,, in \emph{Problems of Asymptotic Theory of Nonlinear Oscillations} (Russian), (1977), 168.   Google Scholar

[33]

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

[34]

A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations,, \emph{J. Math. Anal. Appl.}, 344 (2008), 906.  doi: 10.1016/j.jmaa.2008.03.019.  Google Scholar

[35]

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

[36]

S. Tikhomirov, Hölder shadowing on finite intervals,, \emph{Ergodic Theory Dynam. Systems}, (2014).   Google Scholar

[37]

D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in Shadowing Theory,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013).  doi: 10.3934/dcds.2013.33.4187.  Google Scholar

[38]

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

show all references

References:
[1]

L. Barreira, D. Dragičević and C. Valls, Exponential dichotomies with respect to a sequence of norms and admissibility,, \emph{Int. J. Math.}, 25 (2014).  doi: 10.1142/S0129167X14500244.  Google Scholar

[2]

L. Barreira, D. Dragičević and C. Valls, Nonuniform hyperbolicity and admissibility,, \emph{Adv. Nonlinear Stud.}, 14 (2014), 791.   Google Scholar

[3]

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

[4]

L. Barreira, D. Dragičević and C. Valls, Admissibility on the half line for evolution families,, \emph{J. Anal. Math.}, ().   Google Scholar

[5]

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

[6]

A. Ben-Artzi and I. Gohberg, Dichotomy of systems and invertibility of linear ordinary differential operators,, in \emph{Time-Variant Systems and Interpolation, (1992), 90.   Google Scholar

[7]

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

[8]

C. Chicone and Yu. Latushkin, Evolution Semigroups in Dynamical Systems and Differential Equations,, Mathematical Surveys and Monographs 70, (1999).  doi: 10.1090/surv/070.  Google Scholar

[9]

C. Coffman and J. Schäffer, Dichotomies for linear difference equations,, \emph{Math. Ann.}, 172 (1967), 139.   Google Scholar

[10]

W. Coppel, Dichotomies in Stability Theory,, Lect. Notes. in Math. 629, (1978).   Google Scholar

[11]

Ju. Dalec'kiĭ and M. Kreĭn, Stability of Solutions of Differential Equations in Banach Space,, Translations of Mathematical Monographs 43, (1974).   Google Scholar

[12]

D. Dragičević and S. Slijepčević, Characterization of hyperbolicity and generalized shadowing lemma,, \emph{Dyn. Syst.}, 26 (2011), 483.  doi: 10.1080/14689367.2011.606205.  Google Scholar

[13]

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

[14]

J. Hale, Asymptotic Behavior of Dissipative Systems,, Mathematical Surveys and Monographs 25, (1988).   Google Scholar

[15]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lect. Notes in Math. 840, (1981).   Google Scholar

[16]

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

[17]

Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations,, \emph{J. Operator Theory}, 58 (2007), 387.   Google Scholar

[18]

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

[19]

A. D. Maizel, On stability of solutions of systems of differential equations,, \emph{Trudi Uralskogo Politekhnicheskogo Instituta, 51 (1954), 20.   Google Scholar

[20]

R. Ma né, Characterizations of AS diffeomorphisms,, in \emph{Geometry and Topology} (eds. Jacob Palis and Manfredo do Carmo), (1977), 389.   Google Scholar

[21]

J. Massera and J. Schäffer, Linear differential equations and functional analysis. I.,, \emph{Ann. of Math.}, 67 (1958), 517.   Google Scholar

[22]

J. Massera and J. Schäffer, Linear Differential Equations and Function Spaces,, Pure and Applied Mathematics 21, (1966).   Google Scholar

[23]

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

[24]

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

[25]

K. Palmer, Exponential dichotomies and Fredholm operators,, \emph{Proc. Amer. Math. Soc.}, 104 (1988), 149.  doi: 10.2307/2047477.  Google Scholar

[26]

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

[27]

K. J. Palmer, S. Yu. Pilyugin and S. B. Tikhmirov, Lipschitz shadowing and structural stability of flows,, \emph{J. Differential Equations}, 252 (2012), 1723.  doi: 10.1016/j.jde.2011.07.026.  Google Scholar

[28]

O. Perron, Die Stabilitätsfrage bei Differentialgleichungen,, \emph{Math. Z.}, 32 (1930), 703.  doi: 10.1007/BF01194662.  Google Scholar

[29]

S. Yu. Pilyugin, Shadowing in Dynamical Systems,, Lecture Notes Math., (1706).   Google Scholar

[30]

S. Yu. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability,, \emph{Nonlinearity}, 23 (2010), 2509.  doi: 10.1088/0951-7715/23/10/009.  Google Scholar

[31]

S. Pilyugin, G. Volfson and D. Todorov, Dynamical systems with Lipschitz inverse shadowing properties,, \emph{Vestnik St. Petersburg University: Mathematics}, 44 (2011), 208.  doi: 10.3103/S106345411103006X.  Google Scholar

[32]

V. A. Pliss, Bounded solutions of inhomogeneous linear systems of differential equations,, in \emph{Problems of Asymptotic Theory of Nonlinear Oscillations} (Russian), (1977), 168.   Google Scholar

[33]

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

[34]

A. L. Sasu, Exponential dichotomy and dichotomy radius for difference equations,, \emph{J. Math. Anal. Appl.}, 344 (2008), 906.  doi: 10.1016/j.jmaa.2008.03.019.  Google Scholar

[35]

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

[36]

S. Tikhomirov, Hölder shadowing on finite intervals,, \emph{Ergodic Theory Dynam. Systems}, (2014).   Google Scholar

[37]

D. Todorov, Generalizations of analogs of theorems of Maizel and Pliss and their application in Shadowing Theory,, \emph{Discrete Contin. Dyn. Syst.}, 33 (2013).  doi: 10.3934/dcds.2013.33.4187.  Google Scholar

[38]

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

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