arXiv subjects
Publications and source records attributed to John M. Rassias.
Using the fixed point theorem we establish the Hyers-Ulam-Rassias stability of the generalized Pexider functional equation $$\frac{1}{\mid K\mid}\sum_{k\in K}f(x+k\cdot y)=g(x)+h(y),\;\;x,y\in E$$ from a normed space $E$ into a complete $β$-normed space $F$, where $K$ is a finite abelian subgroup of the automorphism group of the group $(E,+)$.
In this paper we present a new criterion on characterization of real inner product spaces. We conclude that a real normed space $(X, \|...\|)$ is an inner product space if $$\sum_{ε_i \in \{-1,1\}} \|x_1 + \sum_{i=2}^kε_ix_i\|^2=\sum_{ε_i \in \{-1,1\}} (\|x_1\| + \sum_{i=2}^kε_i\|x_i\|)^2,$$ for some positive integer $k\geq 2$ and all $x_1, ..., x_k \in X$. Conversely, if $(X, \|...\|)$ is an inner product space, then the equality above holds for all $k\geq 2$ and all $x_1, ..., x_k \in X$.