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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.

MSC:
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)
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