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Thieme, Horst R.

Balanced exponential growth of operator semigroups. (English) Zbl 0943.47032

J. Math. Anal. Appl. 223, No. 1, 30-49 (1998).
A \(C_0\)-semigroup \(S(t)\), \(t\geq 0\), on a Banach space \(X\) (weakly, strongly, uniformly) approaches balanced (or asynchronous) exponential growth if there exists some \(s\in\mathbb{R}\) such that \[ P= \lim_{t\to\infty} e^{-st}S(t) \] exists (in the weak, strong, uniform operator topology) and \(P\) is not the \(0\) operator. The author characterizes the strong and uniform approach to balanced exponential growth and derives applicable sufficient conditions.

Cited in 17 Documents

MSC:

47D06 One-parameter semigroups and linear evolution equations

Keywords:

dual semigroups; essentially norm-continuous semigroups; Cauchy problems; positive perturbations; \(C_0\)-semigroup; exponential growth
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\textit{H. R. Thieme}, J. Math. Anal. Appl. 223, No. 1, 30--49 (1998; Zbl 0943.47032)
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References:

[1] Arendt, W.; Batty, C. J.K., Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306, 837-852 (1988) · Zbl 0652.47022
[2] Arendt, W.; Prüss, J., Vector-valued Tauberian theorems and asymptotic behavior of linear Volterra equations, SIAM J. Math. Anal., 23, 412-448 (1992) · Zbl 0765.45009
[3] Batty, C. J.K.; Phóng, Vũ Quôc, Stability of individual elements under one-parameter semigroups, Trans. Amer. Math. Soc., 322, 805-818 (1990) · Zbl 0711.47023
[4] Bell, G. I., Cell growth and division III. Conditions for balanced exponential growth in a mathematical model, Biophys. J., 8, 431-444 (1968)
[5] Clément, Ph., Generation of linear semigroups, (Clément, Ph.; Heijmans, H. J.A. M.; Angenent, S.; van Duijn, C. J.; de Pagter, B., One-Parameter Semigroups (1987), North-Holland: North-Holland Amsterdam), 47-92
[6] Clément, Ph.; Heijmans, H. J.A. M.; Angenent, S.; van Duijn, C. J.; de Pagter, B., One-Parameter Semigroups (1987), North-Holland: North-Holland Amsterdam · Zbl 0636.47051
[7] de Pagter, B., A characterization of sun-reflexivity, Math. Ann., 283, 511-518 (1989) · Zbl 0696.47039
[9] Feller, W., An Introduction to Probability Theory and Its Applications (1966), Wiley: Wiley New York · Zbl 0138.10207
[10] Greiner, G., Spectral theory of positive semigroups on \(C_0X\), (Nagel, R., One-Parameter Semigroups of Positive Operators. One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184 (1986), Springer-Verlag: Springer-Verlag New York), 163-203
[11] Greiner, G., Compact and quasi-compact semigroups, (Nagel, R., One-Parameter Semigroups of Positive Operators. One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184 (1986), Springer-Verlag: Springer-Verlag New York), 209-218
[12] Greiner, G., Spectral theory of positive semigroups on Banach lattices, (Nagel, R., One-Parameter Semigroups of Positive Operators. One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184 (1986), Springer-Verlag: Springer-Verlag New York), 292-332
[13] Greiner, G.; Nagel, R., Spectral theory of semigroups on Banach Spaces, (Nagel, R., One-Parameter Semigroups of Positive Operators. One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184 (1986), Springer-Verlag: Springer-Verlag New York), 60-97
[14] Greiner, G.; Nagel, R., Convergence of positive semigroups, (Nagel, R., One-Parameter Semigroups of Positive Operators. One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184 (1986), Springer-Verlag: Springer-Verlag New York), 342-355
[15] Greiner, G.; Neubrander, F., Stability of positive semigroups on Banach lattices, (Nagel, R., One-Parameter Semigroups of Positive Operators. One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184 (1986), Springer-Verlag: Springer-Verlag New York), 334-342
[16] Gyllenberg, M., The size and scar distribution of the yeast, Saccharomyces cerevisiae, J. Math. Biol., 24, 81-101 (1986) · Zbl 0593.92016
[17] Heijmans, H. J.A. M., Perron-Frobenius theory for positive semigroups, (Clément, Ph.; Heijmans, H. J.A. M.; Angenent, S.; van Duijn, C. J.; de Pagter, B., One-Parameter Semigroups (1987), North-Holland: North-Holland Amsterdam), 193-211 · Zbl 0627.93031
[18] Heijmans, H. J.A. M., Two examples from structured population dynamics, (Clément, Ph.; Heijmans, H. J.A. M.; Angenent, S.; van Duijn, C. J.; de Pagter, B., One-Parameter Semigroups (1987), North-Holland: North-Holland Amsterdam), 235-244 · Zbl 0605.47043
[19] Heijmans, H. J.A. M., Some results from spectral theory, (Clément, Ph.; Heijmans, H. J.A. M.; Angenent, S.; van Duijn, C. J.; de Pagter, B., One-Parameter Semigroups (1987), North-Holland: North-Holland Amsterdam), 281-291 · Zbl 0627.93031
[20] Heijmans, H. J.A. M.; de Pagter, B., Asymptotic behavior, (Clément, Ph.; Heijmans, H. J.A. M.; Angenent, S.; van Duijn, C. J.; de Pagter, B., One-Parameter Semigroups (1987), North-Holland: North-Holland Amsterdam), 213-233
[21] Hille, E.; Phillips, R. S., Functional Analysis and Semi-Groups (1957), Amer. Math. Soc: Amer. Math. Soc Providence
[22] Iannelli, M., Mathematical Theory of Age-Structured Population Dynamics. Mathematical Theory of Age-Structured Population Dynamics, Applied Mathematics Monographs (C.N.R.), 7 (1995), Giardini Editori e Stampatori: Giardini Editori e Stampatori Pisa
[23] Jagers, P., Branching Processes with Biological Applications (1975), Wiley: Wiley New York · Zbl 0356.60039
[24] Jagers, P., Balanced exponential growth: What does it mean and when is it there, (Valleron, A.; MacDonald, P., Biomathematics and Cell Kinetics, Development in Cell Biology (1978), Elsevier: Elsevier New York), 21-29
[26] Jamison, B., Asymptotic behavior of successive iterates of continuous functions under a Markov operator, J. Math. Anal. Appl., 9, 203-214 (1964) · Zbl 0133.10701
[27] Kato, T., Perturbation Theory for Linear Operators (1976), Springer-Verlag: Springer-Verlag New York
[28] Krengel, U., Ergodic Theorems (1985), De Gruyter: De Gruyter Berlin · Zbl 0471.28011
[29] Ljubich, Y. I.; Phóng, Vũ Quôc, Asymptotic stability of linear differential equations in Banach spaces, Studia Math., 88, 37-42 (1988) · Zbl 0639.34050
[31] Martinez, J.; Mazon, J. M., \(C_0\), Semigroup Forum, 52, 213-224 (1996) · Zbl 0927.47029
[32] Metz, J. A.J.; Diekmann, O., The Dynamics of Physiologically Structured Populations. The Dynamics of Physiologically Structured Populations, Lecture Notes in Biomathematics, 68 (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0614.92014
[33] Neubrander, F., Asymptotics of semigroups on Banach spaces, (Nagel, R., One-Parameter Semigroups of Positive Operators. One-Parameter Semigroups of Positive Operators, Lecture Notes in Mathematics, 1184 (1986), Springer-Verlag: Springer-Verlag New York), 98-116
[34] Painter, P. R.; Marr, A. G., Mathematics of microbial populations, Ann. Rev. Microbiol., 22, 519-548 (1968)
[35] Pazy, A., Semigroups of Linear Operators and Applications to Partial Differential Equations (1983), Springer-Verlag: Springer-Verlag New York · Zbl 0516.47023
[36] Schaefer, H. H., Topological Vector Spaces (1966), Springer-Verlag: Springer-Verlag New York · Zbl 0141.30503
[37] Schaefer, H. H., Banach Lattices (1974), Springer-Verlag: Springer-Verlag New York · Zbl 0291.46008
[38] Thieme, H. R., Positive perturbations of dual and integrated semigroups, Adv. Math. Sci. Appl., 6, 445-507 (1996) · Zbl 0860.47027
[41] van Neerven, J. M.A. M., The Adjoint of a Semigroup of Linear Operators. The Adjoint of a Semigroup of Linear Operators, Lecture Notes in Mathematics, 1529 (1992), Springer-Verlag: Springer-Verlag New York · Zbl 0895.47031
[42] van Neerven, J. M.A. M., The Asymptotic Behaviour of Semigroups of Linear Operators (1996), Birkhäuser: Birkhäuser Basel · Zbl 0892.47040
[43] Webb, G. F., Theory of Nonlinear Age-Dependent Population Dynamics (1985), Dekker: Dekker New York · Zbl 0555.92014
[44] Webb, G. F., An operator-theoretic formulation of asynchronous exponential growth, Trans. Amer. Math. Soc., 303, 751-763 (1987) · Zbl 0654.47021
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