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Additive selections and the stability of the Cauchy functional equation

Published online by Cambridge University Press:  17 February 2009

Roman Badora ,
Roman Ger  and
Zsolt Páles
Roman Badora
Affiliation:
Institute of Mathematics, Silesian University, ul. Bankowa 14, PL-40-007 Katowice, Poland; e-mail: [email protected].
Roman Ger
Affiliation:
Institute of Mathematics, Silesian University, ul. Bankowa 14, PL-40-007 Katowice, Poland; e-mail: [email protected].
Zsolt Páles
Affiliation:
Institute of Mathematics and Informatics, University of Debrecen, H-4010 Debrecen, Pf. 12, Hungary; e-mail: [email protected].
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Abstract

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The main result of this paper offers a necessary and sufficient condition for the existence of an additive selection of a weakly compact convex set-valued map defined on an amenable semigroup. As an application, we obtain characterisations of the solutions of several functional inequalities, including that of quasi-additive functions.

Type
Research Article
Information
The ANZIAM Journal , Volume 44 , Issue 3 , January 2003 , pp. 323 - 337
Copyright
Copyright © Australian Mathematical Society 2003

References

[1]Badora, R., “On some generalized invariant means and their applications to the stability of Hyers-Ulam type”, Ann. Polon. Math. 58 (1993) 147–159.CrossRefGoogle Scholar
[2]Baran, M., “The graph of a quasi-additive function”, Aequationes Math. 39 (1990) 129–133.CrossRefGoogle Scholar
[3]Day, M. M., “Amenable semigroups”, Illinois J. Math. 1 (1957) 509–544.CrossRefGoogle Scholar
[4]Forti, G.-L., “Hyers-Ulam stability of functional equations in several variables”, Aequationes Math. 50 (1995) 143–190.CrossRefGoogle Scholar
[5]Gajda, Z., “Invariant means and representation of semigroups in the theory of functional equations”, Prace Naukowe Uniwersytetu Ślaskiego w Katowicach 1273, 1992.Google Scholar
[6]Ger, R., “The singular case in the stability behaviour of linear mappings”, Grazer Math. Ber. 316 (1992) 59–70.Google Scholar
[7]Ger, R., “A survey of recent results on stability of functional equations”, in Proc. of the 4th International Conference on Functional Equations and Inequalities, (Pedagogical University in Cracow, 1994) 5–36.Google Scholar
[8]Hewitt, E. and Ross, K. A., Abstract harmonic analysis, Die Grundlehren der Mathematischen Wissenschaften 115 (Springer, Berlin, 1963).Google Scholar
[9]Holmes, R. B., Geometric functional analysis and its applications, Graduate Texts in Math. 24 (Springer, New York, 1975).CrossRefGoogle Scholar
[10]Hyers, D. H., “On the stability of linear functional equations”, Proc. Nat. Acad. Sci. USA 27 (1941) 222–224.CrossRefGoogle Scholar
[11]Hyers, D. H., Isac, G. and Rassias, Th. M., Stability of functional equations in several variables, (Birkhäuser, Boston, 1998).CrossRefGoogle Scholar
[12]Hyers, D. H. and Rassias, Th. M., “Approximate homomorphisms”, Aequationes Math. 44 (1992) 125–153.CrossRefGoogle Scholar
[13]Kuczma, M., An introduction to the theory of functional equations and inequalities (Państwowe Wydawnictwo Naukowe and Silesian University, Warszawa–Kraków–Katowice, 1985).Google Scholar
[14]Nikodem, K. and Páles, Zs., “A characterization of midpoint-quasiaffine functions”, Publ. Math. Debrecen 52 (1998) 575–595.CrossRefGoogle Scholar
[15]Nikodem, K., Páles, Zs. and Wąsowicz, Sz., “Abstract separation theorems of Rodé type and their applications”, Ann. Polon. Math. 72 (1999) 207–218.CrossRefGoogle Scholar
[16]Páles, Zs., “Generalized stability of the Cauchy functional equation”, Aequationes Math. 56 (1998) 222–232.CrossRefGoogle Scholar
[17]Ramachandran, B. and Lau, K.-S., Functional equations in probability theory, Probability and Mathematical Statistics (Academic Press Inc., Boston, MA, 1991).Google Scholar
[18]Rao, R. C. and Shanbhag, D. N., Choquet-Deny type functional equations with applications to stochastic models, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics (John Wiley & Sons, Chichester, 1994).Google Scholar
[19]Székelyhidi, L.. “A note on Hyers's theorem”, C. R. Math. Rep. Acad. Sci. Canada 8 (1986) 127–129.Google Scholar
[20]Tabor, J., “On functions behaving like additive functions”, Aequationes Math. 35 (1988) 164–185.CrossRefGoogle Scholar
[21]Tabor, J., “Quasi-additive functions”, Aequationes Math. 39 (1990) 179–197.CrossRefGoogle Scholar
[22]Yosida, K., Functional analysis, Grundlehren der Mathematischen Wissenschaften 123 (Springer, New York, 1980).Google Scholar