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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.
MSC:
54C60Set-valued maps (general topology)
54H25Fixed-point and coincidence theorems in topological spaces
47H10Fixed point theorems for nonlinear operators on topological linear spaces