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On the generalized Hyers–Ulam–Rassias stability in Banach modules over a \(C^{*}\)-algebra. (English) Zbl 1053.46028
In 1940, S. M. Ulam [“Problems in modern mathematics. First published under the title ‘A collection of mathematical problems’ ” (Science Editions. New York: John Wiley and Sons, Inc.) (1964; Zbl 0137.24201)] posed the following problem concerning the stability of group homomorphisms: Under what condition does there exist an additive mapping near an approximately additive mapping? In 1941, D. H. Hyers [Proc. Natl. Acad. Sci. USA 27, 222-224 (1941; Zbl 0061.26403)] gave a partial solution of Ulam’s problem in the context of Banach spaces. In 1978, Th. M. Rassias [Proc. Am. Math. Soc. 72, 297-300 (1978; Zbl 0398.47040)] generalized Hyers’ result. Over the last decades, several stability problems for functional equations have been investigated in the spirit of Hyers–Ulam–Rassias; cf. S. Czerwik (ed.) [“Stability of Functional Equations of Ulam–Hyers–Rassias Type” (Hadronic Press, ISBN 1-57485-057-1) (2003)]. A generalized Hyers–Ulam–Rassias stability of the quadratic functional equation \[ f(\sum_ {i=1}^ nx_ i)+\sum_ {1\leq i<j\leq n} f(x_ i-x_ j)=n\sum_ {i=1}^ n f(x_ i) \] was investigated by J.-H. Bae and K.-W. Jun [J. Math. Anal. Appl. 258, 183-193 (2001; Zbl 0983.39013)]. In the present paper, the authors extend the generalized Hyers–Ulam–Rassias stability of the above \(n\)-dimensional quadratic functional equation to Banach modules over a \(C^{*}\)-algebra.
C. G. Park [J. Math. Anal. Appl. 291, 214-223 (2004; Zbl 1046.39023)] recently treated the stability of generalized quadratic mappings in Banach modules.

46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
39B82 Stability, separation, extension, and related topics for functional equations
39B72 Systems of functional equations and inequalities
39B52 Functional equations for functions with more general domains and/or ranges
46L05 General theory of \(C^*\)-algebras
Full Text: DOI
[1] Bonsall, F.; Duncan, J., Complete normed algebras, (1973), Springer-Verlag New York · Zbl 0271.46039
[2] Bae, J.-H.; Jun, K.-W., On the generalized hyers – ulam – rassias stability of an n-dimensional quadratic functional equation, J. math. anal. appl., 258, 183-193, (2001) · Zbl 0983.39013
[3] Bae, J.-H.; Jun, K.-W.; Lee, Y.-H., On the hyers – ulam – rassias stability of a pexiderized quadratic equation, Math. inequal. appl., 7, 63-77, (2004) · Zbl 1050.39033
[4] Borelli, C.; Forti, G.L., On a general hyers – ulam-stability result, Internat. J. math. math. sci., 18, 229-236, (1995) · Zbl 0826.39009
[5] Cholewa, P.W., Remarks on the stability of functional equations, Aequationes math., 27, 76-86, (1984) · Zbl 0549.39006
[6] Czerwik, S., On the stability of the quadratic mapping in normed spaces, Abh. math. sem. univ. Hamburg, 62, 59-64, (1992) · Zbl 0779.39003
[7] Găvruta, P., A generalization of the hyers – ulam – rassias stability of approximately additive mappings, J. math. anal. appl., 184, 431-436, (1994) · Zbl 0818.46043
[8] Hyers, D.H., On the stability of the linear functional equation, Proc. nat. acad. sci. USA, 27, 222-224, (1941) · Zbl 0061.26403
[9] Hyers, D.H.; Isac, G.; Rassias, Th.M., Stability of functional equations in several variables, (1998), Birkhäuser Basel · Zbl 0894.39012
[10] Jun, K.-W.; Lee, Y.-H., On the hyers – ulam – rassias stability of a pexiderized quadratic inequality, Math. inequal. appl., 4, 93-118, (2001) · Zbl 0976.39031
[11] Kadison, R.; Pedersen, G., Means and convex combinations of unitary operators, Math. scand., 57, 249-266, (1985) · Zbl 0573.46034
[12] Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. amer. math. soc., 72, 297-300, (1978) · Zbl 0398.47040
[13] Rassias, Th.M., On the stability of functional equations and a problem of Ulam, Acta appl. math., 62, 23-130, (2000) · Zbl 0981.39014
[14] Th.M. Rassias, On the stability of the quadratic functional equation, Mathematica, in press · Zbl 1281.39036
[15] Rassias, Th.M.; Semrl, P., On the hyers – ulam stability of linear mappings, J. math. anal. appl., 173, 325-338, (1993) · Zbl 0789.46037
[16] Rassias, Th.M.; Tabor, J., What is left of hyers – ulam stability?, J. natur. geom., 1, 65-69, (1992) · Zbl 0757.47032
[17] Skof, F., Proprietà locali e approssimazione di operatori, Rend. sem. mat. fis. milano, 53, 113-129, (1983)
[18] Ulam, S.M., Problems in modern mathematics, (1960), Wiley New York, Chapter VI · Zbl 0137.24201
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