Exponential expansiveness and complete admissibility for evolution families. (English) Zbl 1080.34546

Summary: Connections between uniform exponential expansiveness and complete admissibility of the pair \((c_0({\mathbb N}, X),c_0({\mathbb N}, X))\) are studied. A discrete version for a theorem due to N. Van Minh, F. Räbiger and R. Schnaubelt is presented. Equivalent characterizations of Perron type for uniform exponential expansiveness of evolution families in terms of complete admissibility are given.


34E05 Asymptotic expansions of solutions to ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
Full Text: DOI EuDML Link


[1] A. Ben-Artzi, I. Gohberg and M. A. Kaashoek: Invertibility and dichotomy of Differential operators on the half-line. J. Dynam. Differential Equations 5 (1993), 1-36. · Zbl 0771.34011
[2] C. Chicone and Y. Latushkin: Evolution Semigroups in Dynamical Systems and Differ-ential Equations. Math. Surveys and Monographs, Vol. 70. Amer. Math. Soc., 1999. · Zbl 0970.47027
[3] S. N. Chow and H. Leiva: Existence and roughness of the exponential dichotomy for linear skew-product semi ows in Banach spaces. J. Differential Equations 120 (1995), 429-477. · Zbl 0831.34067
[4] J. Daleckii and M. G. Krein: Stability of Solutions of Differential Equations in Banach Spaces. Trans. Math. Monographs 43. AMS, Providence, 1974.
[5] D. Henry: Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, New York, 1981. · Zbl 0456.35001
[6] Y. Latushkin and T. Randolph: Dichotomy of Differential equations on Banach spaces and algebra of weighted translation operators. Integral Equations Operator Theory 23 (1995), 472-500. · Zbl 0839.47026
[7] Y. Latushkin and R. Schnaubelt: Evolution semigroups, translation algebras and expo-nential dichotomy of cocycles. J. Differential Equations 159 (1999), 321-369. · Zbl 0962.34035
[8] M. Megan, A. L. Sasu and B. Sasu: On uniform exponential stability of periodic evolu-tion operators in Banach spaces. Acta Math. Univ. Comenian. 69 (2000), 97-106. · Zbl 0955.34037
[9] M. Megan, A. L. Sasu and B. Sasu: On uniform exponential stability of linear skew-product semi ows in Banach spaces. Bull. Belg. Math. Soc. Simon Stevin 9 (2002), 143-154. · Zbl 1032.34046
[10] M. Megan, A. L. Sasu and B. Sasu: Discrete admissibility and exponential dichotomy for evolution families. Discrete Contin. Dynam. Systems 9 (2003), 383-397. · Zbl 1032.34048
[11] M. Megan, B. Sasu and A. L. Sasu: On nonuniform exponential dichotomy of evolution operators in Banach spaces. Integral Equations Operator Theory 44 (2002), 71-78. · Zbl 1034.34056
[12] N. Van Minh, F. R?biger and R. Schnaubelt: Exponential stability, exponential expan-siveness and exponential dichotomy of evolution equations on the half line. Integral Equations Operator Theory 32 (1998), 332-353. · Zbl 0977.34056
[13] A. Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equa-tions. Springer-Verlag, Berlin-Heidelberg-New York, 1983. · Zbl 0516.47023
[14] V. A. Pliss and G. R. Sell: Robustness of exponential dichotomies in infinite-dimensional dynamical systems. J. Dynam. Differential Equations 3 (1999), 471-513. · Zbl 0941.37052
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.