Abstract
By the early 1950s the theory of one-parameter semigroups of bounded linear operators on Banach spaces was established, and much further theory and diverse applications to many areas of mathematics and science developed rapidly. The theory was based on two main results, the wellposedness theorem and the generation theorem. By the early 1970s, analogues of these two theorems were developed for semigroups of nonlinear operators. Since then, extensions of the theory and deep and sometimes unexpected applications have continued to arise in a consistent and frequent pattern. This continues to blossom today, and a selection of these results is surveyed here.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arendt, W., Batty, C. J. K., Hieber, M., Neubrander, F.: Vector-valued Laplace transforms and Cauchy problems. Birkhäuser (2001).
Barbu, V.: Nonlinear semigroups and differential equations in Banach spaces. Noordhoff Intern. Publ. Leyden (1976).
Bátkai, A. and Piazzera, S.: Semigroups for delay equations. Birkhäuser (2005).
Belleni-Morante, A.: Applied nonlinear semigroups. An introduction. Chichester: John Wiley & Sons (1998).
Bobrowski, A.: Functional analysis for probability and stochastic processes. Cambridge University Press (2005).
Brezis, H.: Operateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North-Holland, Amsterdam (1973).
Butzer, P. L., Berens, H.: Semigroups of operators and approximation. Springer (1967).
Clement, Ph., Heijmans, H. J. A. M., Angenent, S., van Duijn, C. J., de Pagter, B.: One-parameter semigroups. North-Holland (1987).
Davies, E. B.: One-parameter semigroups. London Mathematical Society, Monographs (1980).
Eisner, T.: Stability of operators and operator semigroups. Birkhäuser (2010).
Engel, K.-J., Nagel, R.: One-parameter semigroups for linear evolution equations. Springer (2000).
Goldstein, J. A.: Semigroups of linear operators and applications. Oxford Mathematical Monographs. New York: Oxford University Press (1985).
Hille, E., Phillips R.: Functional analysis and semi-groups. American Mathematical Society, Providence, R. I. (1974).
Kantorovitz, S.: Topics in operator semigroups. Birkhäuser (2010).
Kato, T.: Perturbation theory for linear operators. Springer (1995).
Krein, S.: Linear equations in Banach spaces. Birkhäuser (1982).
Lax, P. D.: Functional analysis. Chichester: Wiley (2002).
McBride, A. C.: Semigroups of linear operatos: an introduction. Reseach Notes in Mathematics. Pitman (1987).
Morosanu, G.: Nonlinear evolution equations and applications. Springer (1988).
Nagel, R. (ed.): One-parameter semigroups of positive operators. Springer (1986).
Pavel, N. H.: Nonlinear evolution operators and semigroups. Springer (1987).
Pazy, A.: Semigroups of linear operators and applications to partial differential equations. Applied Mathematical Sciences, Springer (1983).
Reed, M., Simon, B.: Methods of modern Mathematical Physics: Functional analysis. Vol 1. Methods of Modern Mathematical Physics (1980).
Tanabe, H.: Equations of evolution. Monographs and Studies in Mathematics, Pitman (1979).
van Casteren, J. A.: Generators of strongly continuous semigroups. Research Notes in Mathematics, Pitman (1985).
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this paper
Cite this paper
Goldstein, J.A., Nagel, R. (2015). The Evolution of Operator Semigroups. In: Banasiak, J., Bobrowski, A., Lachowicz, M. (eds) Semigroups of Operators -Theory and Applications. Springer Proceedings in Mathematics & Statistics, vol 113. Springer, Cham. https://doi.org/10.1007/978-3-319-12145-1_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-12145-1_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-12144-4
Online ISBN: 978-3-319-12145-1
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)