On the Hyers-Ulam-Rassias stability of generalized quadratic mappings in Banach modules.
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 \textit{F. Skof} [Rend. Semin. Mat. Fis. Milano 53, 113--129 (1983; Zbl 0599.39007)], \textit{P. W. Cholewa} [Aequationes Math. 27, 76--86 (1984; Zbl 0549.39006)] and \textit{S. Czerwik} [Abh. Math. Semin. Univ. Hamb. 62, 59--64 (1992; Zbl 0779.39003)] as well as by some ideas of \textit{Th. M. Rassias}.
scientific article; zbMATH DE number 2056506
scientific article