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Article

Keywords:
evolution family; uniform exponential expansiveness; complete admissibility
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 Van Minh, Räbiger and Schnaubelt is presented. Equivalent characterizations of Perron type for uniform exponential expansiveness of evolution families in terms of complete admissibility are given.
References:
[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. DOI 10.1007/BF01063733 | MR 1205452
[2] C.  Chicone and Y. Latushkin: Evolution Semigroups in Dynamical Systems and Differential Equations. Math. Surveys and Monographs, Vol. 70. Amer. Math. Soc., , 1999. DOI 10.1090/surv/070 | MR 1707332
[3] S. N. Chow and H.  Leiva: Existence and roughness of the exponential dichotomy for linear skew-product semiflows in Banach spaces. J. Differential Equations 120 (1995), 429–477. DOI 10.1006/jdeq.1995.1117 | MR 1347351
[4] J. Daleckii and M. G. Krein: Stability of Solutions of Differential Equations in Banach Spaces. Trans. Math. Monographs 43. AMS, Providence, 1974. MR 0352639
[5] D. Henry: Geometric Theory of Semilinear Parabolic Equations. Springer-Verlag, New York, 1981. MR 0610244 | 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. DOI 10.1007/BF01203919 | MR 1361056
[7] Y. Latushkin and R. Schnaubelt: Evolution semigroups, translation algebras and exponential dichotomy of cocycles. J.  Differential Equations 159 (1999), 321–369. DOI 10.1006/jdeq.1999.3668 | MR 1730724
[8] M. Megan, A. L. Sasu and B.  Sasu: On uniform exponential stability of periodic evolution operators in Banach spaces. Acta Math. Univ. Comenian. 69 (2000), 97–106. MR 1796790
[9] M.  Megan, A. L.  Sasu and B. Sasu: On uniform exponential stability of linear skew-product semiflows in Banach spaces. Bull. Belg. Math. Soc. Simon Stevin 9 (2002), 143–154. DOI 10.36045/bbms/1102715145 | MR 1905653
[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. MR 1952381
[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. DOI 10.1007/BF01197861 | MR 1913424
[12] 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 | MR 1652689
[13] A.  Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer-Verlag, Berlin-Heidelberg-New York, 1983. MR 0710486 | 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. MR 1693858
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