Article Contents
Article Contents

# Admissibility versus nonuniform exponential behavior for noninvertible cocycles

• 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.
Mathematics Subject Classification: Primary: 34D09, 37D25.

 Citation:

•  [1] L. Barreira and Ya. Pesin, "Lyapunov Exponents and Smooth Ergodic Theory," University Lecture Series, 23, Amer. Math. Soc., 2002. [2] L. Barreira and C. Valls, "Stability of Nonautonomous Differential Equations," Lect. Notes in Math., 1926, Springer, 2008. [3] C. Chicone and Yu. Latushkin, "Evolution Semigroups in Dynamical Systems and Differential Equations," Mathematical Surveys and Monographs, 70, Amer. Math. Soc., 1999. [4] Ju. Dalec$'$kiĭ and M. Kreĭn, "Stability of Solutions of Differential Equations in Banach Space," Translations of Mathematical Monographs, 43, Amer. Math. Soc., 1974. [5] 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. [6] B. Levitan and V. Zhikov, "Almost Periodic Functions and Differential Equations," Cambridge University Press, 1982. [7] J. Massera and J. Schäffer, Linear differential equations and functional analysis. I, Ann. of Math., 67 (1958), 517-573.doi: 10.2307/1969871. [8] J. Massera and J. Schäffer, "Linear Differential Equations and Function Spaces," Pure and Applied Mathematics, 21, Academic Press, 1966. [9] M. Megan, B. Sasu and A. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces, Integral Equations Operator Theory, 44 (2002), 71-78.doi: 10.1007/BF01197861. [10] 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. [11] P. Ngoc and T. Naito, New characterizations of exponential dichotomy and exponential stability of linear difference equations, J. Difference Equ. Appl., 11 (2005), 909-918.doi: 10.1080/00423110500211947. [12] O. Perron, Die Stabilitätsfrage bei Differentialgleichungen, Math. Z., 32 (1930), 703-728.doi: 10.1007/BF01194662. [13] 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. [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), 405-418.doi: 10.1007/s00020-002-1268-7. [15] P. Preda, A. Pogan and C. Preda, Schäffer spaces and uniform exponential stability of linear skew-product semiflows, J. Differential Equations, 212 (2005), 191-207.doi: 10.1016/j.jde.2004.07.019. [16] 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. [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), 551-561 [18] A. Sasu and B. Sasu, Exponential dichotomy and $(l^p,l^q)$-admissibility on the half-line, J. Math. Anal. Appl., 316 (2006), 397-408.doi: 10.1016/j.jmaa.2005.04.047. [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), 332-353.doi: 10.1007/BF01203774.