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 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 x t ∈ C be the unique fixed point of the contraction x ↦ t f ( x ) ⊕ ( 1 - t ) T x . We will show that if X is a CAT(0) space satisfying some property, then { x t } converge strongly to a fixed point of T which solves some variational inequality. Consider also the iteration process { x n } , where x 0 ∈ C is arbitrary and x n + 1 = α n f ( x n ) ⊕ ( 1 - α n ) T x n for n ≥ 1 , where { α n } ⊂ ( 0,1 ) . It is shown that under certain appropriate conditions on α n , { x n } converge strongly to a fixed point of T which solves some variational inequality.