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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, Int. J. Math., 25, 1450024 (2014), 20 pages. doi: 10.1142/S0129167X14500244.  Google Scholar

[2]

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

[3]

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.  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, 2008. 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 Time-Variant Systems and Interpolation, Oper. Theory Adv. Appl., 56, Birkhäuser, 1992, pp. 90-119.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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.  Google Scholar

[14]

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

[15]

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

[16]

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.  Google Scholar

[17]

Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory, 58 (2007), 387-414.  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, Trudi Uralskogo Politekhnicheskogo Instituta, Mathematics, 51 (1954), 20-50.  Google Scholar

[20]

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.  Google Scholar

[21]

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

[22]

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

[23]

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.  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, Integral Equations Operator Theory, 32 (1998), 332-353. doi: 10.1007/BF01203774.  Google Scholar

[25]

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

[26]

K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer, Dordrecht, 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, J. Differential Equations, 252 (2012), 1723-1747. doi: 10.1016/j.jde.2011.07.026.  Google Scholar

[28]

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

[29]

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

[30]

S. Yu. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. 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, Vestnik St. Petersburg University: Mathematics, 44 (2011), 208-213. doi: 10.3103/S106345411103006X.  Google Scholar

[32]

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.  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, Integral Equations Operator Theory, 49 (2004), 405-418. doi: 10.1007/s00020-002-1268-7.  Google Scholar

[34]

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.  Google Scholar

[35]

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

[36]

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

[37]

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.  Google Scholar

[38]

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.  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, Int. J. Math., 25, 1450024 (2014), 20 pages. doi: 10.1142/S0129167X14500244.  Google Scholar

[2]

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

[3]

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.  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, 2008. 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 Time-Variant Systems and Interpolation, Oper. Theory Adv. Appl., 56, Birkhäuser, 1992, pp. 90-119.  Google Scholar

[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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.  Google Scholar

[14]

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

[15]

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

[16]

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.  Google Scholar

[17]

Yu. Latushkin, A. Pogan and R. Schnaubelt, Dichotomy and Fredholm properties of evolution equations, J. Operator Theory, 58 (2007), 387-414.  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, Trudi Uralskogo Politekhnicheskogo Instituta, Mathematics, 51 (1954), 20-50.  Google Scholar

[20]

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.  Google Scholar

[21]

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

[22]

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

[23]

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.  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, Integral Equations Operator Theory, 32 (1998), 332-353. doi: 10.1007/BF01203774.  Google Scholar

[25]

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

[26]

K. Palmer, Shadowing in Dynamical Systems. Theory and Applications, Kluwer, Dordrecht, 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, J. Differential Equations, 252 (2012), 1723-1747. doi: 10.1016/j.jde.2011.07.026.  Google Scholar

[28]

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

[29]

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

[30]

S. Yu. Pilyugin and S. Tikhomirov, Lipschitz shadowing implies structural stability, Nonlinearity, 23 (2010), 2509-2515. 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, Vestnik St. Petersburg University: Mathematics, 44 (2011), 208-213. doi: 10.3103/S106345411103006X.  Google Scholar

[32]

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.  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, Integral Equations Operator Theory, 49 (2004), 405-418. doi: 10.1007/s00020-002-1268-7.  Google Scholar

[34]

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.  Google Scholar

[35]

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

[36]

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

[37]

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.  Google Scholar

[38]

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.  Google Scholar

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