On weakly nonexpansive and *-nonexpansive multivalued mappings. (English) Zbl 0733.54010

The types of mapping to which the title refers were defined by T. Husain and A. Latif [Math. Jap. 33, No.3, 385-391 (1988; Zbl 0667.47028)] and by T. Husain and 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 Lami Dozo, Lim and Kirk. The paper also gives simple examples to show that there are \({}^*\)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 \({}^*non\)-expansive maps.


54C60 Set-valued maps in general topology
54H25 Fixed-point and coincidence theorems (topological aspects)
47H10 Fixed-point theorems