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.