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 and Continuous Dynamical Systems, 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 23, Amer. Math. Soc., 2002.

[2]

L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity," Encyclopedia of Math. and Its Appl. 115, Cambridge Univ. Press, 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), 29-70. doi: 10.1007/BF02773211.

[4]

L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations," Lect. Notes in Math. 1926, Springer, 2008.

[5]

L. Barreira and C. Valls, Admissibility for nonuniform exponential contractions, J. Differential Equations, 249 (2010), 2889-2904. 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 70, Amer. Math. Soc., 1999.

[7]

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

[8]

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.

[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), 517-573. doi: 10.2307/1969871.

[11]

J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces," Pure and Applied Mathematics 21, Academic Press, 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), 71-78. 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), 28-44. 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), 197-221.

[15]

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

[16]

Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 10 (1976), 1261-1305. doi: 10.1070/IM1976v010n06ABEH001835.

[17]

P. Preda and M. Megan, Nonuniform dichotomy of evolutionary processes in Banach spaces, Bull. Austral. Math. Soc., 27 (1983), 31-52. 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), 405-418. 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), 378-391. doi: 10.1016/j.jde.2006.02.004.

[20]

J. Schäffer, Function spaces with translations, Math. Ann., 137 (1959), 209-262. 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), 332-353. doi: 10.1007/BF01203774.

show all references

References:
[1]

L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," University Lecture Series 23, Amer. Math. Soc., 2002.

[2]

L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity," Encyclopedia of Math. and Its Appl. 115, Cambridge Univ. Press, 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), 29-70. doi: 10.1007/BF02773211.

[4]

L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations," Lect. Notes in Math. 1926, Springer, 2008.

[5]

L. Barreira and C. Valls, Admissibility for nonuniform exponential contractions, J. Differential Equations, 249 (2010), 2889-2904. 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 70, Amer. Math. Soc., 1999.

[7]

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

[8]

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.

[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), 517-573. doi: 10.2307/1969871.

[11]

J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces," Pure and Applied Mathematics 21, Academic Press, 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), 71-78. 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), 28-44. 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), 197-221.

[15]

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

[16]

Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents, Math. USSR-Izv., 10 (1976), 1261-1305. doi: 10.1070/IM1976v010n06ABEH001835.

[17]

P. Preda and M. Megan, Nonuniform dichotomy of evolutionary processes in Banach spaces, Bull. Austral. Math. Soc., 27 (1983), 31-52. 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), 405-418. 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), 378-391. doi: 10.1016/j.jde.2006.02.004.

[20]

J. Schäffer, Function spaces with translations, Math. Ann., 137 (1959), 209-262. 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), 332-353. doi: 10.1007/BF01203774.

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