Fixed points and stability for quadratic mappings in \(\beta\)-normed left Banach modules on Banach algebras
Suppose that \(B\) is a unital Banach algebra with norm \(|.|\), \(B_{1}=\{a\in B: |a|=1\}\), \(\mathbb{X}\) a \(\beta\)-normed left \(B\)-module, \(\mathbb{Y}\) a \(\beta\)-normed left Banach \(B\)-module, \(f:\mathbb{X}\rightarrow \mathbb{Y}\) a mapping and \(n\geq 2\) a fixed integer. The authors consider the following functional equation \[ D_{b}f(x_{1},\dots,x_{n})=0, \] where \[ \begin{split} D_{b}f(x_{1},\dots,x_{n}) =&f\Big(\sum^{n}_{i=1}a_ibx_i\Big)+\sum^{n-1}_{i=1}\sum^{n}_{j=i+1}\Big[f(a_ibx_i+a_jbx_j)+2f(a_ibx_i-a_jbx_j)\Big]\\&- (3n-2)\sum^{n}_{i=1}a^2_{i}b^2f(x_{i}) \end{split} \] for all \(x_{1},\dots,x_n\in \mathbb{X}\) and \(b\in B_{1}\), where \(a_{1},\dots,a_{n} \in \mathbb{Z}\setminus\{0\}\) and some \(\ell\in \{1 , 2 ,\dots, n-1\}\), \(a_{\ell}\neq \pm 1\) and \(a_{n}=1\). The main result reads as follows. Suppose that \(f : \mathbb{X} \to \mathbb{Y}\) satisfies \(f(0)=0\) and \[ \|D_{b}f(x_{1},\dots,x_{n})\|_\beta\leq\varphi(x_{1},\dots,x_{n}) \] for all \(x_{1},\dots,x_n\in {\mathbb X}\) and all \(b\in B_{1}\), where \(\varphi:{\mathbb X}^{n}\rightarrow[0,\infty)\) is a control mapping satisfying \[ \varphi(a_\ell x_1,\dots,a_\ell x_n)\leq a_\ell^{2\beta} L~\varphi(x_1,\dots,x_n) \] for all \(x_{1},\dots,x_n\in {\mathbb X}\) and \(0<L<1\). Then there exists a unique quadratic mapping \(Q: \mathbb{X} \to \mathbb{Y}\) such that \(Q\) approximates \(f\). Moreover, if \(f(tx)\) is continuous in \(t\in \mathbb{R}\) for each fixed \(x\in \mathbb{X}\), then \(Q\) is \(B\)-quadratic, i.e., \(Q(bx)=b^2Q(x)\) for all \(x\in \mathbb{X}\) and all \(b\in B\).
scientific article; zbMATH DE number 6124396