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Zbl 1130.39027
Tan, Liyun; Xiang, Shuhuang
On the Aleksandrov-Rassias problem and the Hyers-Ulam-Rassias stability problem. (English)
[J] Banach J. Math. Anal. 1, No. 1, 11-22, electronic only (2007). ISSN 1735-8787/e

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.
[Jacek Chmielinski (Kraków)]

MSC 2000:: *39B82 Stability, separation, extension, and related topics
46B20 Geometry and structure of normed spaces
39-02 Research monographs (functional equations)
39B52 Functional equations for functions with more general domains

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

Cited in: Zbl 1254.46072

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