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

Viscosity approximation methods for nonexpansive mappings in CAT 0 spaces are studied. Consider a nonexpansive self-mapping T of a closed convex subsetC of a CAT 0 spaceX. 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.