Strong Convergence of Viscosity Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

Abstract

Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self-mapping T of a closed convex subset C of a CAT(0) space X. Suppose that the set Fix(T) of fixed points of T is nonempty. For a contraction f on C and t∈(0,1), let xt∈C be the unique fixed point of the contraction x↦tf(x)⊕(1-t)Tx. We will show that if X is a CAT(0) space satisfying some property, then {xt} converge strongly to a fixed point of T which solves some variational inequality. Consider also the iteration process {xn}, where x0∈C is arbitrary and xn+1=αnf(xn)⊕(1-αn)Txn for n≥1, where {αn}⊂(0,1). It is shown that under certain appropriate conditions on αn,{xn} converge strongly to a fixed point of T which solves some variational inequality.