Jensen's functional equation in multi-normed spaces (Q993614)

Let \(E\) be a linear space and let \((F^n,\|\cdot\|_{n})_{n\in\mathbb{N}}\) be a multi-Banach space. In such a setting, the Hyers-Ulam stability of Jensen's functional equation is proved. Suppose that a mapping \(f: E\to F\) is an approximate solution of Jensen's equation, i.e., \(f(0)=0\) and, with some \(\alpha\geq 0\), \[ \sup_{k\in\mathbb{N}} \left\| \left( f\left(\frac{x_1+y_1}{2}\right)-\frac{f(x_1)+f(y_1)}{2},\dots, f\left(\frac{x_k+y_k}{2}\right)-\frac{f(x_k)+f(y_k)}{2} \right) \right\|_{k} \leq\alpha \] for all \(x_i,y_i\in E\). Then, there exists a unique additive mapping \(T: E\to F\) such that \[ \sup_{k\in\mathbb{N}}\left\|(f(x_1)-T(x_1),\dots,f(x_k)-T(x_k))\right\|_{k}\leq 2\alpha \] for all \(x_i\in E\). Additionally, the stability on a restricted domain is proved as well as an asymptotic behavior.