April  2011, 30(1): 39-53. doi: 10.3934/dcds.2011.30.39

Nonuniform exponential dichotomies and admissibility

1. 

Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa

2. 

Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa

Received  January 2010 Revised  May 2010 Published  February 2011

In this paper we consider the relation between the notions of exponential stability and admissibility, in the general context of nonuniform exponential behavior. In particular, we show that with respect to certain adapted norms related to the nonuniform behavior, if any $L^p$ space, with $p\in(1,\infty]$, is admissible for a given evolution process, then this process is a nonuniform exponential dichotomy. In addition, for each nonuniform exponential dichotomy we provide a collection of admissible Banach spaces, also defined in terms of the adapted norms.
Citation: Luis Barreira, Claudia Valls. Nonuniform exponential dichotomies and admissibility. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 39-53. doi: 10.3934/dcds.2011.30.39
References:
[1]

L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", University Lecture Series \textbf{23}, 23 (2002).

[2]

L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity,", Encyclopedia of Math. and Its Appl. \textbf{115}, 115 (2007).

[3]

L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension,, Israel J. Math., 116 (2000), 29. doi: 10.1007/BF02773211.

[4]

L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math. \textbf{1926}, 1926 (2008).

[5]

L. Barreira and C. Valls, Admissibility for nonuniform exponential contractions,, J. Differential Equations, 249 (2010), 2889. doi: 10.1016/j.jde.2010.06.010.

[6]

C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs \textbf{70}, 70 (1999).

[7]

Ju. Dalec'kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space,", Translations of Mathematical Monographs \textbf{43}, 43 (1974).

[8]

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

[9]

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

[10]

J. Massera and J. Schäffer, Linear differential equations and functional analysis. I,, Ann. of Math. (2), 67 (1958), 517. doi: 10.2307/1969871.

[11]

J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces,", Pure and Applied Mathematics \textbf{21}, 21 (1966).

[12]

M. Megan, B. Sasu and A. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces,, Integral Equations Operator Theory, 44 (2002), 71. doi: 10.1007/BF01197861.

[13]

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

[14]

V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197.

[15]

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

[16]

Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Math. USSR-Izv., 10 (1976), 1261. doi: 10.1070/IM1976v010n06ABEH001835.

[17]

P. Preda and M. Megan, Nonuniform dichotomy of evolutionary processes in Banach spaces,, Bull. Austral. Math. Soc., 27 (1983), 31. doi: 10.1017/S0004972700011473.

[18]

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. doi: 10.1007/s00020-002-1268-7.

[19]

P. Preda, A. Pogan and C. Preda, Schäffer spaces and exponential dichotomy for evolutionary processes,, J. Differential Equations, 230 (2006), 378. doi: 10.1016/j.jde.2006.02.004.

[20]

J. Schäffer, Function spaces with translations,, Math. Ann., 137 (1959), 209. doi: 10.1007/BF01343353.

[21]

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. doi: 10.1007/BF01203774.

show all references

References:
[1]

L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", University Lecture Series \textbf{23}, 23 (2002).

[2]

L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity,", Encyclopedia of Math. and Its Appl. \textbf{115}, 115 (2007).

[3]

L. Barreira and J. Schmeling, Sets of "non-typical" points have full topological entropy and full Hausdorff dimension,, Israel J. Math., 116 (2000), 29. doi: 10.1007/BF02773211.

[4]

L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math. \textbf{1926}, 1926 (2008).

[5]

L. Barreira and C. Valls, Admissibility for nonuniform exponential contractions,, J. Differential Equations, 249 (2010), 2889. doi: 10.1016/j.jde.2010.06.010.

[6]

C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs \textbf{70}, 70 (1999).

[7]

Ju. Dalec'kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space,", Translations of Mathematical Monographs \textbf{43}, 43 (1974).

[8]

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

[9]

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

[10]

J. Massera and J. Schäffer, Linear differential equations and functional analysis. I,, Ann. of Math. (2), 67 (1958), 517. doi: 10.2307/1969871.

[11]

J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces,", Pure and Applied Mathematics \textbf{21}, 21 (1966).

[12]

M. Megan, B. Sasu and A. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces,, Integral Equations Operator Theory, 44 (2002), 71. doi: 10.1007/BF01197861.

[13]

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

[14]

V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197.

[15]

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

[16]

Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Math. USSR-Izv., 10 (1976), 1261. doi: 10.1070/IM1976v010n06ABEH001835.

[17]

P. Preda and M. Megan, Nonuniform dichotomy of evolutionary processes in Banach spaces,, Bull. Austral. Math. Soc., 27 (1983), 31. doi: 10.1017/S0004972700011473.

[18]

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. doi: 10.1007/s00020-002-1268-7.

[19]

P. Preda, A. Pogan and C. Preda, Schäffer spaces and exponential dichotomy for evolutionary processes,, J. Differential Equations, 230 (2006), 378. doi: 10.1016/j.jde.2006.02.004.

[20]

J. Schäffer, Function spaces with translations,, Math. Ann., 137 (1959), 209. doi: 10.1007/BF01343353.

[21]

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. doi: 10.1007/BF01203774.

[1]

Luis Barreira, Claudia Valls. Admissibility versus nonuniform exponential behavior for noninvertible cocycles. Discrete & Continuous Dynamical Systems - A, 2013, 33 (4) : 1297-1311. doi: 10.3934/dcds.2013.33.1297

[2]

Mihail Megan, Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential dichotomy for evolution families. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 383-397. doi: 10.3934/dcds.2003.9.383

[3]

António J.G. Bento, Nicolae Lupa, Mihail Megan, César M. Silva. Integral conditions for nonuniform $μ$-dichotomy on the half-line. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3063-3077. doi: 10.3934/dcdsb.2017163

[4]

Luis Barreira, Claudia Valls. Delay equations and nonuniform exponential stability. Discrete & Continuous Dynamical Systems - S, 2008, 1 (2) : 219-223. doi: 10.3934/dcdss.2008.1.219

[5]

Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential trichotomy of dynamical systems. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2929-2962. doi: 10.3934/dcds.2014.34.2929

[6]

Luis Barreira, Claudia Valls. Characterization of stable manifolds for nonuniform exponential dichotomies. Discrete & Continuous Dynamical Systems - A, 2008, 21 (4) : 1025-1046. doi: 10.3934/dcds.2008.21.1025

[7]

Adina Luminiţa Sasu, Bogdan Sasu. Exponential trichotomy and $(r, p)$-admissibility for discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3199-3220. doi: 10.3934/dcdsb.2017170

[8]

Éder Rítis Aragão Costa. An extension of the concept of exponential dichotomy in Fréchet spaces which is stable under perturbation. Communications on Pure & Applied Analysis, 2019, 18 (2) : 845-868. doi: 10.3934/cpaa.2019041

[9]

Christian Pötzsche. Dichotomy spectra of triangular equations. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 423-450. doi: 10.3934/dcds.2016.36.423

[10]

Luis Barreira, Claudia Valls. Growth rates and nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2008, 22 (3) : 509-528. doi: 10.3934/dcds.2008.22.509

[11]

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

[12]

Yakov Pesin. On the work of Dolgopyat on partial and nonuniform hyperbolicity. Journal of Modern Dynamics, 2010, 4 (2) : 227-241. doi: 10.3934/jmd.2010.4.227

[13]

Jana Rodriguez Hertz. Genericity of nonuniform hyperbolicity in dimension 3. Journal of Modern Dynamics, 2012, 6 (1) : 121-138. doi: 10.3934/jmd.2012.6.121

[14]

Luis Barreira, Claudia Valls. Regularity of center manifolds under nonuniform hyperbolicity. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 55-76. doi: 10.3934/dcds.2011.30.55

[15]

Kristin Dettmers, Robert Giza, Rafael Morales, John A. Rock, Christina Knox. A survey of complex dimensions, measurability, and the lattice/nonlattice dichotomy. Discrete & Continuous Dynamical Systems - S, 2017, 10 (2) : 213-240. doi: 10.3934/dcdss.2017011

[16]

Luis Barreira, Claudia Valls. Center manifolds for nonuniform trichotomies and arbitrary growth rates. Communications on Pure & Applied Analysis, 2010, 9 (3) : 643-654. doi: 10.3934/cpaa.2010.9.643

[17]

Thorsten Hüls. Numerical computation of dichotomy rates and projectors in discrete time. Discrete & Continuous Dynamical Systems - B, 2009, 12 (1) : 109-131. doi: 10.3934/dcdsb.2009.12.109

[18]

Hassan Emamirad, Arnaud Rougirel. A functional calculus approach for the rational approximation with nonuniform partitions. Discrete & Continuous Dynamical Systems - A, 2008, 22 (4) : 955-972. doi: 10.3934/dcds.2008.22.955

[19]

Boris Kalinin, Anatole Katok and Federico Rodriguez Hertz. New progress in nonuniform measure and cocycle rigidity. Electronic Research Announcements, 2008, 15: 79-92. doi: 10.3934/era.2008.15.79

[20]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (15)

Other articles
by authors

[Back to Top]