Ulam stability of functional equations in 2-Banach spaces via the fixed point method
Let \(X\) be a linear space and \(Y\) be a \(2\)-Banach space over the field \(\mathbb{F}=\mathbb{R}\) or \(\mathbb{C}\). Let \(a_{11}, a_{12},\dots, a_{n1}, a_{n2}\in \mathbb{F}, a_{1,j_1,\dots,j_n}, \dots, a_{n,j_1,\dots,j_n}\in\mathbb{F}\) for \(j_1,\dots, j_n\in\{-1, 1\}\) and \(A_{i_1\dots,i_n}\in \mathbb{F}\) for \(i_1,\dots, i_n\in \{1, 2\}\) be given scalars. Set \(A := \sum_ {i_1,\dots, i_n\in\{1,2\}} A_{i_1,\dots,i_n}\) and assume that \(|A|>1\). The authors prove the Ulam stability of the following functional equations \[f(a_{11}x_{11} + a_{12}x_{12}, \dots, a_{n1}x_{n1} + a_{n2}x_{n2}) =\sum_ {i_1,\dots, i_n\in\{1,2\}} A_{i_1,\dots,i_n}f(x_{1i_1},\dots, x_{ni_n}) \] and \[ \begin{multlined} \sum_ {j_1,\dots, j_n\in\{-1,1\}}f( a_{1, j_1},\dots,j_n(x_{11}+j_1x_{12}),\dots, a_{n, j_1},\dots, j_n(x_{n1}+j_nx_{n2}))\\ =\sum_ {i_1,\dots, i_n\in\{1,2\}} A_{i_1,\dots,i_n}f(x_{1i_1},\dots, x_{ni_n}) \end{multlined} \] by using the fixed point method.
scientific article; zbMATH DE number 7381141
scientific article