On approximately additive functions
Abstract
In the present paper we find a linear operator on a function space, essentially larger than the space of all bounded functions on an amenable semigroup, which behaves like an invariant mean. This leads to an extension of the Hyers-Ulam stability theorem for Cauchy's functional equation in the case of vector-valued mappings defined on amenable semigroups.
References
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2. R. Badora, On some generalized invariant means and almost approximately additive functions, Publ. Math. Debrecen 44 (1994), 123-135.
3. G.L. Forti, On an alternative functional equation related to the Cauchy equation, Aequationes Math. 24 (1982), 195-206.
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2. R. Badora, On some generalized invariant means and almost approximately additive functions, Publ. Math. Debrecen 44 (1994), 123-135.
3. G.L. Forti, On an alternative functional equation related to the Cauchy equation, Aequationes Math. 24 (1982), 195-206.
4. Z. Gajda, On the stability of additive mappings, Internat. J. Math. and Math. Sci. 14 (1991), 431-434.
5. Z. Gajda, Generalized invariant means and their application to the stability of homomorphisms, (preprint).
6. R. Ger, On functional inequalities stemming from stability questions, General Inequalities 6, ISNM 103, Birkhäuser Verlag, Basel - Boston - Stuttgart, 1992, 227-240.
7. R. Ger, The singular case in the stability behaviour of linear mappings, Grazer Math. Ber. 316 (1992), 59-70.
8. F.P. Greenleaf, Invariant means on topological groups and their applications, Van Nostrand Mathematical Studies 16, New York - Toronto - London - Melbourne, 1969.
9. D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.
10. D.H. Hyers, Th.M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125-153.
11. S.M. Ulam, Problems in modern mathematics, Science Edition, Wiley, New York, 1960.
BadoraR. (1994). On approximately additive functions. Annales Mathematicae Silesianae, 8, 111-126. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14223
Roman Badora
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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