zbMATH — the first resource for mathematics

On the Hyers-Ulam-Rassias stability of generalized quadratic mappings in Banach modules. (English) Zbl 1046.39023
Let \(X,Y\) be Banach modules over a Banach \(\ast\)-algebra \(A\). A mapping \(Q:X\to Y\) is called \(A\)-quadratic if it satisfies \[ Q(x+y)+Q(x-y)=2Q(x)+2Q(y)\quad \text{and}\quad Q(ax)=aQ(x)a^{\ast} \] for all \(a\in A\), \(x,y\in X\). Two types of generalized \(A\)-quadratic mappings are considered and for both of them the stability is proved. The research was motivated by the stability results for the quadratic equation obtained by F. Skof [Rend. Semin. Mat. Fis. Milano 53, 113–129 (1983; Zbl 0599.39007)], P. W. Cholewa [Aequationes Math. 27, 76–86 (1984; Zbl 0549.39006)] and S. Czerwik [Abh. Math. Semin. Univ. Hamb. 62, 59–64 (1992; Zbl 0779.39003)] as well as by some ideas of Th. M. Rassias.

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46H25 Normed modules and Banach modules, topological modules (if not placed in 13-XX or 16-XX)
Full Text: DOI
[1] Cho, Y.J.; Lin, C.S.; Kim, S.S.; Misiak, A., Theory of 2-inner product spaces, (2001), Nova Science Huntington, New York · Zbl 1016.46002
[2] Cholewa, P.W., Remarks on the stability of functional equations, Aequationes math., 27, 76-86, (1984) · Zbl 0549.39006
[3] Czerwik, S., On the stability of the quadratic mapping in normed spaces, Abh. math. sem. univ. Hamburg, 62, 59-64, (1992) · Zbl 0779.39003
[4] Hyers, D.H., On the stability of the linear functional equation, Proc. natl. acad. sci. USA, 27, 222-224, (1941) · Zbl 0061.26403
[5] Lin, C.S., A-bilinear forms and generalized A-quadratic forms on unitary left A-modules, Bull. austral. math. soc., 39, 49-53, (1989) · Zbl 0684.46041
[6] Lin, C.S., Sesquilinear and quadratic forms on modules over \(∗\)-algebra, Publ. inst. math., 51, 81-86, (1992) · Zbl 0794.46043
[7] Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. amer. math. soc., 72, 297-300, (1978) · Zbl 0398.47040
[8] Rassias, Th.M., On the stability of the quadratic functional equation and its applications, Stud. univ. babes-bolyai, 18, 89-124, (1998) · Zbl 1009.39025
[9] Rassias, Th.M., The problem of S.M. Ulam for approximately multiplicative mappings, J. math. anal. appl., 246, 352-378, (2000) · Zbl 0958.46022
[10] Rassias, Th.M., On the stability of functional equations in Banach spaces, J. math. anal. appl., 251, 264-284, (2000) · Zbl 0964.39026
[11] Rassias, Th.M., On the stability of functional equations and a problem of Ulam, Acta appl. math., 62, 23-130, (2000) · Zbl 0981.39014
[12] Rassias, Th.M.; Šemrl, P., On the hyers – ulam stability of linear mappings, J. math. anal. appl., 173, 325-338, (1993) · Zbl 0789.46037
[13] Rassias, Th.M.; Shibata, K., Variational problem of some quadratic functionals in complex analysis, J. math. anal. appl., 228, 234-253, (1998) · Zbl 0945.30023
[14] Skof, F., Proprietà locali e approssimazione di operatori, Rend. sem. mat. fis. milano, 53, 113-129, (1983)
[15] Ulam, S.M., Problems in modern mathematics, (1960), Wiley New York · Zbl 0137.24201
[16] Vukman, J., Some results concerning the Cauchy functional equation in certain Banach algebra, Bull. austral. math. soc., 31, 137-144, (1985) · Zbl 0559.46024
[17] Vukman, J., Some functional equations in Banach algebras and an application, Proc. amer. math. soc., 100, 133-136, (1987) · Zbl 0623.46021
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.