April  2013, 33(4): 1297-1311. doi: 10.3934/dcds.2013.33.1297

Admissibility versus nonuniform exponential behavior for noninvertible cocycles

1. 

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

2. 

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

Received  October 2011 Revised  January 2012 Published  October 2012

We study the relation between the notions of exponential dichotomy and admissibility for a nonautonomous dynamics with discrete time. More precisely, we consider $\mathbb{Z}$-cocycles defined by a sequence of linear operators in a Banach space, and we give criteria for the existence of an exponential dichotomy in terms of the admissibility of the pairs $(\ell^p,\ell^q)$ of spaces of sequences, with $p\le q$ and $(p,q)\ne(1,\infty)$. We extend the existing results in several directions. Namely, we consider the general case of nonuniform exponential dichotomies; we consider $\mathbb{Z}$-cocycles and not only $\mathbb{N}$-cocycles; and we consider exponential dichotomies that need not be invertible in the stable direction. We also exhibit a collection of admissible pairs of spaces of sequences for any nonuniform exponential dichotomy.
Citation: Luis Barreira, Claudia Valls. Admissibility versus nonuniform exponential behavior for noninvertible cocycles. Discrete & Continuous Dynamical Systems, 2013, 33 (4) : 1297-1311. doi: 10.3934/dcds.2013.33.1297
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show all references

References:
[1]

University Lecture Series, 23, Amer. Math. Soc., 2002.  Google Scholar

[2]

Lect. Notes in Math., 1926, Springer, 2008.  Google Scholar

[3]

Mathematical Surveys and Monographs, 70, Amer. Math. Soc., 1999.  Google Scholar

[4]

Translations of Mathematical Monographs, 43, Amer. Math. Soc., 1974.  Google Scholar

[5]

J. Funct. Anal., 235 (2006), 330-354. doi: 10.1016/j.jfa.2005.11.002.  Google Scholar

[6]

Cambridge University Press, 1982.  Google Scholar

[7]

Ann. of Math., 67 (1958), 517-573. doi: 10.2307/1969871.  Google Scholar

[8]

Pure and Applied Mathematics, 21, Academic Press, 1966.  Google Scholar

[9]

Integral Equations Operator Theory, 44 (2002), 71-78. doi: 10.1007/BF01197861.  Google Scholar

[10]

J. Math. Anal. Appl., 261 (2001), 28-44. doi: 10.1006/jmaa.2001.7450.  Google Scholar

[11]

J. Difference Equ. Appl., 11 (2005), 909-918. doi: 10.1080/00423110500211947.  Google Scholar

[12]

Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662.  Google Scholar

[13]

Bull. Austral. Math. Soc., 27 (1983), 31-52. doi: 10.1017/S0004972700011473.  Google Scholar

[14]

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

[15]

J. Differential Equations, 212 (2005), 191-207. doi: 10.1016/j.jde.2004.07.019.  Google Scholar

[16]

J. Differential Equations, 230 (2006), 378-391. doi: 10.1016/j.jde.2006.02.004.  Google Scholar

[17]

Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13 (2006), 551-561  Google Scholar

[18]

J. Math. Anal. Appl., 316 (2006), 397-408. doi: 10.1016/j.jmaa.2005.04.047.  Google Scholar

[19]

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

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