# American Institute of Mathematical Sciences

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 - 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

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##### 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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