Stability of functional equations and properties of groups
Abstract
Investigating Hyers–Ulam stability of the additive Cauchy equation with domain in a group G, in order to obtain an additive function approximating the given almost additive one we need some properties of G, starting from commutativity to others more sophisticated. The aim of this survey is to present these properties and compare, as far as possible, the classes of groups
involved.
Keywords
additive functional equation; quadratic functional equation; stability; amenable group; weak commutativity
References
2. R. Badora, B. Przebieracz, P. Volkmann, On Tabor groupoids and stability of some functional equations, Aequationes Math. 87 (2014), no. 1–2, 165–171.
3. A. Bahyrycz, Forti's example of an unstable homomorphism equation, Aequationes Math. 74 (2007), no. 3, 310–313.
4. J. Dixmier, Les moyennes invariantes dans les semi-groups et leurs applications, Acta. Sci. Math. Szeged 12 (1950), 213–227.
5. G.-L. Forti, The stability of homomorphisms and amenability, with applications to functional equations, Abh. Math. Sem. Univ. Hamburg 57 (1987), 215–226.
6. G.-L. Forti, Hyers–Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), no. 1–2, 143–190.
7. G.-L. Forti, J. Sikorska, Variations on the Drygas equation and its stability, Nonlinear Anal. 74 (2011), no. 2, 343–350.
8. Z. Gajda, Z. Kominek, On separation theorems for subadditive and superadditive functionals, Studia Math. 100 (1991), no. 1, 25–38.
9. F.P. Greenleaf, Invariant Means on Topological Groups, Van Nostrand Mathematical Studies 16, Van Nostrand Reinhold Co., New York–Toronto–London–Melbourne, 1969.
10. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.
11. S.V. Ivanov, The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput. 4 (1994), no. 1–2, 1–308.
12. W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory, Dover Publications, Inc., New York, 1976.
13. P.S. Novikov, S.I. Adian, Infinite periodic groups. I–III. (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 212–244, 251–254, 709–731.
14. A.Yu. Ol'shanskii, On the question of the existence of an invariant mean on a group. (Russian), Uspekhi Mat. Nauk 35 (1980), no. 4(214), 199–200.
15. A.Yu. Ol'shanskii, The Novikov–Adian theorem. (Russian), Mat. Sb. (N.S.) 118(160) (1982), no. 2, 203–235.
16. D.V. Osin, Uniform non-amenability of free Burnside groups, Arch. Math. (Basel) 88 (2007), no. 5, 403–412.
17. J. Rätz, On approximately additive mappings, in: E.F. Beckenbach (ed.), General Inequalities. 2, Proc. Second Internat. Conf., Oberwolfach 1978, Birkhäuser, Basel, 1980, pp. 233–251.
18. E. Shulman, Group representations and stability of functional equations, J. London Math. Soc. (2) 54 (1996), no. 1, 111–120.
19. E. Shulman, Addition theorems and related geometric problems of group representation theory, in: J. Brzdęk et al. (eds.), Recent Developments in Functional Equations and Inequalities, Banach Center Publ., vol. 99, Polish Acad. Sci. Inst. Math.,Warsaw, 2013, pp. 155–172.
20. F. Skof, Proprietà locali e approssimazione di operatori. Geometry of Banach spaces and related topics (Milan, 1983), Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129 (1986).
21. L. Székelyhidi, Note on a stability theorem, Canad. Math. Bull. 25 (1982), no. 4, 500–501.
22. L. Székelyhidi, Remark 17, 22sd International Symposium on Functional Equations, Oberwolfach 1984, Aequationes Math. 29 (1985), no. 1, 95.
23. J. Tabor, Remark 18, 22sd International Symposium on Functional Equations, Oberwolfach 1984, Aequationes Math. 29 (1985), no. 1, 96.
24. I. Toborg, Tabor groups with finiteness conditions, Aequationes Math. 90 (2016), no. 4, 699–704.
25. D. Yang, Remarks on the stability of Drygas' equation and the Pexider–quadratic equation, Aequationes Math. 68 (2004), no. 1–2, 108–116.
26. D. Yang, The stability of the quadratic functional equation on amenable groups, J. Math. Anal. Appl. 291 (2004), no. 2, 666–672.
Università degli Studi di Milano, Dipartimento di Matematica, Italy Italy
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