Exponential instability of linear skew-product semiflows in terms of Banach function spaces. (English) Zbl 1063.34048

The authors prove necessary and sufficient conditions for “uniform exponential instability” of linear skew-product semiflows that extend several previous results on instability of evolution equations.


34G10 Linear differential equations in abstract spaces
34D05 Asymptotic properties of solutions to ordinary differential equations
47D06 One-parameter semigroups and linear evolution equations
Full Text: DOI


[1] Chicone, C, Latushkin, Y., Evolution Semigroups in Dynamical Systems and Differential Equations, Math. Surveys and Monographs 70, Amer. Math. Soc. 1999. · Zbl 0970.47027
[2] Chow, S. N., Leiva, H., Existence and roughness of the exponential dichotomy for linear skew-product semiflow in Banach space, J. Differential Equations 102 (1995), 429–477. · Zbl 0831.34067
[3] Chow, S. N., Leiva, H., Two definitions of exponential dichotomy for skew-product semiflow in Banach spaces, Proc. Amer. Math. Soc. 124 (1996), 1071–1081. · Zbl 0845.34064
[4] Datko, R., Uniform asymptotic stability of evolutionary processes in Banach spaces, SIAM J. Math. Anal. 3 (1972), 428–445. · Zbl 0241.34071
[5] Latushkin, Y., Montgomery-Smith, S., Randolph, T., Evolutionary semigroups and dichotomy of linear skew-product flows on locally compact spaces with Banach fibres, J. Differential Equations 125 (1996), 75–116. · Zbl 0881.47020
[6] Latushkin, Y., Schnaubelt, R., Evolution semigroups, translation algebras and exponential dichotomy of cocycles, J. Differential Equations 159 (1999), 321–369. · Zbl 0962.34035
[7] Meyer-Nieberg, P., Banach Lattices, Spinger Verlag, Berlin, Heidelberg, New York, 1991. · Zbl 0743.46015
[8] Megan, M., Sasu, A. L., Sasu, B., On uniform exponential stability of linear skew-product semiflows in Banach spaces, Bull. Belg. Math. Soc. Simon Stevin 9 (2002), 143–154. · Zbl 1032.34046
[9] Megan M., Sasu, B., Sasu A. L., On nonuniform exponential dichotomy of evolution operators in Banach spaces, Integral Equations Operator Theory 44 (2002), 71–78. · Zbl 1034.34056
[10] Megan, M., Sasu, A. L., Sasu, B., Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dynam. Systems 9 (2003), 383–397. · Zbl 1032.34048
[11] Megan, M., Sasu, A. L., Sasu, B., Theorems of Perron type for uniform exponential dichotomy of linear skew-product semiflows, Bull. Belg. Math. Soc. – Simon Stevin 10 (2003), 1–21. · Zbl 1045.34022
[12] Megan, M., Sasu, A. L., Sasu, B., Perron conditions for uniform exponential expansiveness of linear skew-product flows, Monatsh. Math. 138 (2003), 145–157. · Zbl 1023.34043
[13] Megan, M., Sasu, A. L., Sasu, B., Exponential expansiveness and complete admissibility for evolution families, to appear in Czech. Math. J. (2004). · Zbl 1080.34546
[14] Megan, M., Sasu, A. L., Sasu, B., Banach function spaces and exponential instability of evolution families, Arch. Math. (Brno) 39 (2003), 277–286. · Zbl 1116.34328
[15] Megan, M., Sasu, A. L., Sasu, B., Perron conditions for uniform exponential stability of linear skew-product semiflows on locally compact spaces, Acta Math. Univ. Comenian. 70 (2001), 229–240. · Zbl 1019.34059
[16] Van Minh, N., Räbiger, F., Schnaubelt, R., Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equations Operator Theory 32 (1998), 332–353. · Zbl 0977.34056
[17] Van Neerven, J. M. A. M., The Asymptotic Behaviour of Semigroups of Linear Operators, Operator Theory Adv. Appl. 88, Birkhäuser, Bassel, 1996. · Zbl 0905.47001
[18] Przyluski, K. M, Rolewicz, S., On stability of linear time-varying infinite-dimensional discrete-time systems, Systems Control Lett. 4 (1984), 307–315. · Zbl 0543.93057
[19] Rolewicz, S., On uniform N – equistability, J. Math. Anal. Appl. 115 (1986), 434–441. · Zbl 0597.34064
[20] Sacker, R. J., Sell, G. R., Dichotomies for linear evolutionary equations in Banach spaces, J. Differential Equations 113 (1994), 17–67. · Zbl 0815.34049
[21] Zabczyk, Z., Remarks on the control of discrete-time distributed parameter systems, SIAM J. Control Optim. 12 (1994), 721–735. · Zbl 0254.93027
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.