Viscosity iteration method in CAT(0) spaces without the nice projection property

Abstract

© 2015, Kaewkhao et al. A complete CAT(0) space X is said to have the nice projection property (property 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∈X, m∈[x,y] implies (Formula presented.), where P<inf>L</inf> 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 N. Our result gives an affirmative answer to a problem raised by Piatek (Numer. Funct. Anal. Optim. 34:1245-1264, 2013).