On the Aleksandrov-Rassias problem and the Hyers-Ulam-Rassias stability problem. (English) Zbl 1130.39027

Tan, Liyun; Xiang, Shuhuang

Banach J. Math. Anal. 1, No. 1, 11-22 (2007).

This is a survey on two important problems for isometries. The first one concerns the conservative distances, i.e., description of mappings between metric (normed, euclidean) spaces preserving (in one or in both directions) a fixed distance. Mainly the results of Alexandrov, Beckmann and Quarles, Ciesielski, Mielnik, Šemrl and Rassias are presented.
The second part concerns the stability of functional equations and the stability of isometries in particular. It brings a review of results of Hyers and Ulam, Bourgin, Gruber, Gevirtz, Dolinar, Rassias and others. A particular emphasis is put on contributions to the above problems by Th. M. Rassias to whom the paper is dedicated.

Reviewer: Jacek Chmielinski (Kraków)

MSC:

39B82	Stability, separation, extension, and related topics for functional equations
46B20	Geometry and structure of normed linear spaces
39-02	Research exposition (monographs, survey articles) pertaining to difference and functional equations
39B52	Functional equations for functions with more general domains and/or ranges

Keywords:

isometry; conservative distance; stability of functional equations; survey paper