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

Admissibility versus nonuniform exponential behavior for noninvertible cocycles

Luis Barreira 1, and  Claudia Valls 2,

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.
Keywords: nonuniform exponential dichotomies., Admissibility.
Mathematics Subject Classification: Primary: 34D09, 37D2.
Citation: 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
References:
[1]

L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", University Lecture Series, 23 (2002).   Google Scholar

[2]

L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math., 1926 (2008).   Google Scholar

[3]

C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs, 70 (1999).   Google Scholar

[4]

Ju. Dalec$'$kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space,", Translations of Mathematical Monographs, 43 (1974).   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces,", Pure and Applied Mathematics, 21 (1966).   Google Scholar

[9]

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

[10]

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

[11]

P. Ngoc and T. Naito, New characterizations of exponential dichotomy and exponential stability of linear difference equations,, J. Difference Equ. Appl., 11 (2005), 909.  doi: 10.1080/00423110500211947.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

P. Preda, A. Pogan and C. Preda, Schäffer spaces and uniform exponential stability of linear skew-product semiflows,, J. Differential Equations, 212 (2005), 191.  doi: 10.1016/j.jde.2004.07.019.  Google Scholar

[16]

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

[17]

A. Sasu and B. Sasu, Discrete admissibility, $l^p$-spaces and exponential dichotomy on the real line,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13 (2006), 551.   Google Scholar

[18]

A. Sasu and B. Sasu, Exponential dichotomy and $(l^p,l^q)$-admissibility on the half-line,, J. Math. Anal. Appl., 316 (2006), 397.  doi: 10.1016/j.jmaa.2005.04.047.  Google Scholar

[19]

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

show all references

References:
[1]

L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory,", University Lecture Series, 23 (2002).   Google Scholar

[2]

L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations,", Lect. Notes in Math., 1926 (2008).   Google Scholar

[3]

C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations,", Mathematical Surveys and Monographs, 70 (1999).   Google Scholar

[4]

Ju. Dalec$'$kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space,", Translations of Mathematical Monographs, 43 (1974).   Google Scholar

[5]

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

[6]

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

[7]

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

[8]

J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces,", Pure and Applied Mathematics, 21 (1966).   Google Scholar

[9]

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

[10]

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

[11]

P. Ngoc and T. Naito, New characterizations of exponential dichotomy and exponential stability of linear difference equations,, J. Difference Equ. Appl., 11 (2005), 909.  doi: 10.1080/00423110500211947.  Google Scholar

[12]

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

[13]

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

[14]

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

[15]

P. Preda, A. Pogan and C. Preda, Schäffer spaces and uniform exponential stability of linear skew-product semiflows,, J. Differential Equations, 212 (2005), 191.  doi: 10.1016/j.jde.2004.07.019.  Google Scholar

[16]

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

[17]

A. Sasu and B. Sasu, Discrete admissibility, $l^p$-spaces and exponential dichotomy on the real line,, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 13 (2006), 551.   Google Scholar

[18]

A. Sasu and B. Sasu, Exponential dichotomy and $(l^p,l^q)$-admissibility on the half-line,, J. Math. Anal. Appl., 316 (2006), 397.  doi: 10.1016/j.jmaa.2005.04.047.  Google Scholar

[19]

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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