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Zbl 0733.54010
Xu, Hongkun
On weakly nonexpansive and *-nonexpansive multivalued mappings. (English)
[J] Math. Jap. 36, No.3, 441-445 (1991). ISSN 0025-5513

The types of mapping to which the title refers were defined by {\it T. Husain} and {\it A. Latif} [Math. Jap. 33, No.3, 385-391 (1988; Zbl 0667.47028)] and by {\it T. Husain} and {\it E. Tarafdar} [Yokohama Math. J. 28, 1-6 (1980; Zbl 0456.47047)]. As the author points out, it is immediate from the definition of ``weakly non-expansive'' that it implies ``non-expansive'' in the usual sense with respect to the Hausdorff metric and hence that the work of the above mentioned authors dealing with this concept is subsumed under earlier work on non-expansive maps by {\it Lami Dozo}, {\it Lim} and {\it Kirk}. The paper also gives simple examples to show that there are ${}\sp*$non-expansive maps which are not non- expansive and vice versa (correcting a further statement of Husain and Latif). Finally, the paper contains a fixed-point theorem for ${}\sp*non$-expansive maps.
[A.C.Thompson (Halifax)]

MSC 2000:: *54C60 Set-valued maps
54H25 Fixed-point theorems in topological spaces
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces

Keywords: non-expansive maps

Citations: Zbl 0667.47028; Zbl 0456.47047

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