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 → tf (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 nf (x n) ⊕ (1-α n)tx 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. copyright © 2012 luo yi shi and ru dong chen.