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The stability of homomorphisms and amenability, with applications to functional equations. (English) Zbl 0619.39012

Following a well known result of D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.264)], the concept of stability for the homomorphisms from a group into a Banach space is defined. Some consequences of the stability are proved and then the connections between the stability of the homomorphisms and the amenability of the group are investigated. The results obtained are used for solving some alternative Cauchy functional equations.

MSC:

39B52 Functional equations for functions with more general domains and/or ranges
39B99 Functional equations and inequalities
43A07 Means on groups, semigroups, etc.; amenable groups

Citations:

Zbl 0061.264
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References:

[1] K. Baron, Functions with differences in subspaces, Manuscript.
[2] I. Fenyö, Osservazioni su alcuni teoremi di D. H. Hyers, Ist. Lombardo Accad. Sci. Lett. Rend. A 114 (1980), 235–242.
[3] G. L. Forti, On an alternative functional equation related to the Cauchy equation, Aequationes Math. 24 (1982), 195–206. · Zbl 0517.39007 · doi:10.1007/BF02193044
[4] G. L. Forti -L. Paganoni, A method for solving a conditional Cauchy equation on abelian groups, Ann. Mat. Pura Appl. (IV) 127 (1981), 79–99. · Zbl 0494.39005 · doi:10.1007/BF01811720
[5] Z. Gajda, On the stability of the Cauchy equation on semigroups, Manuscript. · Zbl 0658.39006
[6] R. Ger, On a method of solving of conditional Cauchy equations, Univ. Beograd. Publ. Elektrotechn. Fak. Ser. Mat. Fiz. No. 544–576 (1976), 159–165.
[7] F. P. Greenleaf, Invariant means on topological groups, Van Nostrand Mathematical Studies 16, New York, Toronto, London, Melbourne, 1969. · Zbl 0174.19001
[8] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. · Zbl 0061.26403 · doi:10.1073/pnas.27.4.222
[9] D. H. Hyers, The stability of homomorphisms and related topics, Global Analysis- Analysis on Manifold (Ed. T. M. Rassias), Teubner-Texte zur Mathematik, Band 57, Teubner Verlagsgesellschaft, Leipzig 1983, 140–153. · Zbl 0517.22001
[10] M. Kuczma, Functional equations on restricted domains, Aequationes Math. 18 (1978), 1–34. · Zbl 0386.39002 · doi:10.1007/BF01844065
[11] Z. Moszner, Sur la stabilité de l’équation d’homomorphisme, Aequationes Math. 29 (1985), 290–306. · Zbl 0583.39012 · doi:10.1007/BF02189833
[12] L. Paganoni, Soluzione di una equazione funzionale su dominio ristretto, Boll. Un. Mat. Ital. (5) 17-B (1980), 979–993.
[13] L. Paganoni, On an alternative Cauchy equation, Aequationes Math. 29 (1985), 214–221. · Zbl 0583.39007 · doi:10.1007/BF02189830
[14] J. Ratz, On approximate additive mappings, General Inequalities 2 (Ed. E. F. Becken- bach), ISNM vol. 47, Birkhäuser, Basel, 1980, 233–251.
[15] L. Székelyhidi, Note on Hyers’ Theorem, C. R. Math. Rep. Acad. Sci. Canada, 8 (1986), 127–129.
[16] L. Székelyhidi, The Fréchet equation and Hyers theorem on noncommutative semigroups, Manuscript.
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