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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,
Departamento de Matemática, Instituto Superior Técnico, UTL, 1049-001 Lisboa, Portugal
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, 2013, 33 (4) : 1297-1311. doi: 10.3934/dcds.2013.33.1297
L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," University Lecture Series, 23, Amer. Math. Soc., 2002. Google Scholar
L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations," Lect. Notes in Math., 1926, Springer, 2008. Google Scholar
C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical Surveys and Monographs, 70, Amer. Math. Soc., 1999. Google Scholar
Ju. Dalec$'$kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space," Translations of Mathematical Monographs, 43, Amer. Math. Soc., 1974. Google Scholar
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. Google Scholar
B. Levitan and V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge University Press, 1982. Google Scholar
J. Massera and J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math., 67 (1958), 517-573. doi: 10.2307/1969871. Google Scholar
J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces," Pure and Applied Mathematics, 21, Academic Press, 1966. Google Scholar
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. Google Scholar
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. Google Scholar
P. Ngoc and T. Naito, New characterizations of exponential dichotomy and exponential stability of linear difference equations, J. Difference Equ. Appl., 11 (2005), 909-918. doi: 10.1080/00423110500211947. Google Scholar
O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728. doi: 10.1007/BF01194662. Google Scholar
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. Google Scholar
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. Google Scholar
P. Preda, A. Pogan and C. Preda, Schäffer spaces and uniform exponential stability of linear skew-product semiflows, J. Differential Equations, 212 (2005), 191-207. doi: 10.1016/j.jde.2004.07.019. Google Scholar
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. Google Scholar
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), 551-561 Google Scholar
A. Sasu and B. Sasu, Exponential dichotomy and $(l^p,l^q)$-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408. doi: 10.1016/j.jmaa.2005.04.047. Google Scholar
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. Google Scholar
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Luis Barreira Claudia Valls