On the Aleksandrov-Rassias problem and the Hyers-Ulam-Rassias stability problem

Abstract

Let $X$ and $Y$ be normed linear spaces. A mapping $T:X \rightarrow Y$ is called preserving the distance $r$ if for all $x,y$ of $X$ with $\|x-y\|_X=r$ then $\|T(x)-T(y)\|=r$. In this paper, we provide an overall account of the development of the Aleksandrov problem, the Aleksandrov-Rassias problem for mappings which preserve distances and details for the Hyers-Ulam-Rassias stability problem.