# zbMATH — the first resource for mathematics

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.

##### MSC:
 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:
##### References:
 [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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.