# American Institute of Mathematical Sciences

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).   Google Scholar [2] L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity,", Encyclopedia of Math. and Its Appl. \textbf{115}, 115 (2007).   Google Scholar [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.  Google Scholar [4] L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math. \textbf{1926}, 1926 (2008).   Google Scholar [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.  Google Scholar [6] C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs \textbf{70}, 70 (1999).   Google Scholar [7] Ju. Dalec'kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space,", Translations of Mathematical Monographs \textbf{43}, 43 (1974).   Google Scholar [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.  Google Scholar [9] B. Levitan and V. Zhikov, "Almost Periodic Functions and Differential Equations,", Cambridge University Press, (1982).   Google Scholar [10] J. Massera and J. Schäffer, Linear differential equations and functional analysis. I,, Ann. of Math. (2), 67 (1958), 517.  doi: 10.2307/1969871.  Google Scholar [11] J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces,", Pure and Applied Mathematics \textbf{21}, 21 (1966).   Google Scholar [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.  Google Scholar [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.  Google Scholar [14] V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197.   Google Scholar [15] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen,, Math. Z., 32 (1930), 703.  doi: 10.1007/BF01194662.  Google Scholar [16] Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Math. USSR-Izv., 10 (1976), 1261.  doi: 10.1070/IM1976v010n06ABEH001835.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] J. Schäffer, Function spaces with translations,, Math. Ann., 137 (1959), 209.  doi: 10.1007/BF01343353.  Google Scholar [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.  Google Scholar

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##### References:
 [1] L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", University Lecture Series \textbf{23}, 23 (2002).   Google Scholar [2] L. Barreira and Ya. Pesin, "Nonuniform Hyperbolicity,", Encyclopedia of Math. and Its Appl. \textbf{115}, 115 (2007).   Google Scholar [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.  Google Scholar [4] L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math. \textbf{1926}, 1926 (2008).   Google Scholar [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.  Google Scholar [6] C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs \textbf{70}, 70 (1999).   Google Scholar [7] Ju. Dalec'kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space,", Translations of Mathematical Monographs \textbf{43}, 43 (1974).   Google Scholar [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.  Google Scholar [9] B. Levitan and V. Zhikov, "Almost Periodic Functions and Differential Equations,", Cambridge University Press, (1982).   Google Scholar [10] J. Massera and J. Schäffer, Linear differential equations and functional analysis. I,, Ann. of Math. (2), 67 (1958), 517.  doi: 10.2307/1969871.  Google Scholar [11] J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces,", Pure and Applied Mathematics \textbf{21}, 21 (1966).   Google Scholar [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.  Google Scholar [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.  Google Scholar [14] V. Oseledets, A multiplicative ergodic theorem. Liapunov characteristic numbers for dynamical systems,, Trans. Moscow Math. Soc., 19 (1968), 197.   Google Scholar [15] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen,, Math. Z., 32 (1930), 703.  doi: 10.1007/BF01194662.  Google Scholar [16] Ya. Pesin, Families of invariant manifolds corresponding to nonzero characteristic exponents,, Math. USSR-Izv., 10 (1976), 1261.  doi: 10.1070/IM1976v010n06ABEH001835.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [20] J. Schäffer, Function spaces with translations,, Math. Ann., 137 (1959), 209.  doi: 10.1007/BF01343353.  Google Scholar [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.  Google Scholar
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