Viscosity iteration method in CAT(0) spaces without the nice projection property
Attapol Kaewkhao, Bancha Panyanak, Suthep Suantai · Journal of Inequalities and Applications · 2015
A complete CAT(0) space X is said to have the nice projection property (property $\mathcal{N}$ for short) if its metric projection onto a geodesic segment preserves points on each geodesic segment, that is, for any geodesic segment L in X and $x,y\in X$ , $m\in[x,y]$ implies $P_{L}(m)\in[P_{L}(x), P_{L}(y)]$ , where $P_{L}$ denotes the metric projection from X onto L. In this paper, we prove a strong convergence theorem of a two-step viscosity iteration method for nonexpansive mappings in CAT(0) spaces without the condition on the property $\mathcal{N}$ . Our result gives an affirmative answer to a problem raised by Piatek (Numer. Funct. Anal. Optim. 34:1245-1264, 2013).