Admissibility, a general type of Lipschitz shadowing and structural stability

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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

1.	Department of Mathematics, University of Rijeka, 51000 Rijeka

Received May 2014 Revised October 2014 Published March 2015

Abstract
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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.

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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