Hostname: page-component-8448b6f56d-t5pn6 Total loading time: 0 Render date: 2024-04-19T23:13:24.098Z Has data issue: false hasContentIssue false
  • English
  • Français

Amenability and Fixed Point Properties of Semitopological Semigroups in Modular Vector Spaces

Part of: Abstract harmonic analysis Nonlinear operators and their properties

Published online by Cambridge University Press:  16 December 2019

Khadime Salame
Khadime Salame*
Affiliation:
Diourbel, Senegal Email: khadime.salame1313@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we initiate the study of fixed point properties of amenable or reversible semitopological semigroups in modular spaces. Takahashi’s fixed point theorem for amenable semigroups of nonexpansive mappings, and T. Mitchell’s fixed point theorem for reversible semigroups of nonexpansive mappings in Banach spaces are extended to the setting of modular spaces. Among other things, we also generalize another classical result due to Mitchell characterizing the left amenability property of the space of left uniformly continuous functions on semitopological semigroups by introducing the notion of a semi-modular space as a generalization of the concept of a locally convex space.

Keywords

almost periodic functionamenabilitymodular functionmodular spacesemigroup

MSC classification

Primary: 47H10: Fixed-point theorems
Secondary: 43A07: Means on groups, semigroups, etc.; amenable groups 43A60: Almost periodic functions on groups and semigroups and their generalizations (recurrent functions, distal functions, etc.); almost automorphic functions
Type
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 3 , September 2020 , pp. 692 - 704
Check for updates
Copyright
Š Canadian Mathematical Society 2019

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abdou, A. A. N. and Khamsi, M. A., Fixed point theorems in modular spaces. J. Nonlinear Sci. Appl. 10(2017), 4046–4057. https://doi.org/10.22436/jnsa.010.08.01CrossRefGoogle Scholar
Alspach, D., A fixed point free nonexpansive map. Proc. Amer. Math. Soc. 82(1981), 423–424. https://doi.org/10.2307/2043954CrossRefGoogle Scholar
Berglund, J. F., Junghenn, H. D., and Milnes, P., Analysis on semigroups. Function spaces, compactifications, representations. Canadian Mathematical Society Series of Monographs and Advanced Texts, A Wiley-Interscience Publication, John Wiley & Sons, New York, 1989.Google Scholar
Browder, F. E., Nonexpansive nonlinear operators in a Banach space. Proc. Nat Acad. Sci. USA 54(1966), 1041–1044. https://doi.org/10.1073/pnas.54.4.1041CrossRefGoogle Scholar
DeMarr, R., Common fixed points for commuting contraction mappings. Pacific J. Math. 13(1963), 1139–1141.10.2140/pjm.1963.13.1139Google Scholar
Granirer, E. and Lau, A. T., Invariant means on locally compact groups. Illinois J. Math. 15(1971), 249–257.10.1215/ijm/1256052712CrossRefGoogle Scholar
Hewitt, E., On two problems of Urysohn. Ann. of Math. 47(1946), 503–509. https://doi.org/10.2307/1969089CrossRefGoogle Scholar
Khamsi, M. A. and Kozlowski, W. M., Fixed point theory in modular function spaces. Birkhäuser/Springer, Cham, 2015. https://doi.org/10.1007/978-3-319-14051-3CrossRefGoogle Scholar
Kirk, W. A., A fixed point theorem for mappings which do not increase distances. Amer. Math. Monthly 72(1965), 1004–1006. https://doi.org/10.2307/2313345CrossRefGoogle Scholar
Kozlowski, W. M., Modular function spaces. Monographs and Textbooks in Pure and Applied Mathematics, 122, Marcel Dekker, Inc, New York, 1988.Google Scholar
Kumam, P., Fixed point theorems for nonexpansive mappings in modular spaces. Arch. Math. 40(2004), 345–353.Google Scholar
Lau, A. T.-M., Invariant means on almost periodic functions and fixed point properties. Rocky Mountain J. Math. 3(1973), 69–76. https://doi.org/10.1216/RMJ-1973-3-1-69CrossRefGoogle Scholar
Lau, A. T.-M. and Zhang, Y., Fixed point properties of semigroups of non-expansive mappings. J. Funct. Anal. 254(2008), 2534–2554. https://doi.org/10.1016/j.jfa.2008.02.006CrossRefGoogle Scholar
Lau, A. T.-M. and Zhang, Y., Fixed point properties for semigroups of nonlinear mappings and amenability. J. Funct. Anal. 263(2012), 2949–2977. https://doi.org/10.1016/j.jfa.2012.07.013CrossRefGoogle Scholar
Mitchell, T., Topological semigroups and fixed points. Illinois J. Math. 14(1970), 630–641.10.1215/ijm/1256052955CrossRefGoogle Scholar
Mitchell, T., Fixed point of reversible semigroups of non-expansive mappings. Kodai Math. Sem. Rep. 22(1970), 322–323.10.2996/kmj/1138846168CrossRefGoogle Scholar
Musielak, J., Orlicz spaces and modular spaces. Lecture notes in Mathematics, 1034, Springer-Verlag, Berlin, 1983. https://doi.org/10.1007/BFb0072210CrossRefGoogle Scholar
Nakano, H., Modular semi-ordered spaces. Maruzen, Tokyo, 1950.Google Scholar
Orlicz, W., Über konjugierte exponentenfolgen. Studia Math. 3(1931), 200–211.10.4064/sm-3-1-200-211CrossRefGoogle Scholar
Salame, K., Amenable semigroups of nonlinear operators in uniformly convex Banach spaces. Bull. Aust. Math. Soc. 99(2019), 284–292. https://doi.org/10.1017/S0004972718001077CrossRefGoogle Scholar
Salame, K., A characterization of 𝜎-extremely amenable semitopological semigroups. J. Nonlinear Convex Anal. 19(2018), 1443–1458.Google Scholar
Takahashi, W., Fixed point theorem for amenable semigroups of nonexpansive mappings. Kodai Math. Rep. 21(1969), 383–386.10.2996/kmj/1138845984Google Scholar