ABSTRACT

Recall that, for any convex subset C of a Banach space X with diam(C) > 0 and for any multi-index α = (α1, . . . , αn), there is an α-nonexpansive mapping T : C → C that is not nonexpansive. We also know that a mapping T is αnonexpansive if and only if it is nonexpansive with respect to the metric d defined by

d(x, y) = n∑ j=1

 n∑ i=j

αi

∥∥T j−1x− T j−1y∥∥ . Moreover, every time we prove the fixed point theorem for nonexpansive mappings, we get a fixed point theorem for some class of α-nonexpansive mappings.