On the Generalized Hyers–Ulam Stability in Multi-Banach Spaces Associated to a Jensen-type Additive Mapping

Abstract

Using the fixed point method, we prove the generalized Hyers–Ulam–Rassias stability of the following functional equation in multi-Banach spaces: $$\begin{aligned} f\left(\frac{\sum_{i=1}^{n}r_ix_i}{k}\right)+ \sum_{1 \le i < j \le n}f\left(\frac{r_ix_i +r_jx_j}{k}\right) = \frac{n}{k}\sum_{i=1}^{n}r_if(x_i).\end{aligned}$$ (1) The concept of generalized Hyers–Ulam–Rassias stability originated from Th. M. Rassias’ stability theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72, 297–300, 1978.