Approximating fixed points of nonexpansive mappings in CAT(0) spaces

Abstract

Let C be a nonempty closed convex subset of a complete CAT(0) space and T : C → C be a nonexpansive mapping with F(T) := {x ∈ C : Tx = x} ≠ ø. Suppose {xn} is generated iteratively by x1 ∈ C, xn+1 = tnT[snTxn (1 - sn)xn] (1 - tn)xn for all n ≥ 1, where {tn} and {sn} are real sequences in [0, 1] such that one of the following two conditions is satisfied : (i) tn ∈ [a, b] and sn ∈ [0, b] for some a, b with 0 < a < b < 1 , (ii) tn ∈ [a, 1] and sn ∈ [a, b] for some a, b with 0 < a ≤ b < 1. Then the sequence {xn} δ-converges to a fixed point of T. This is an analog of a result on weak convergence theorem in Banach spaces of Takahashi and Kim [W. Takahashi and G. E. Kim, Approximating fixed points of nonexpansive mappings in Banach spaces, Math. Japonica. 48 no. 1 (1998), 1-9]. Strong convergence of the iterative sequence {xn} is also discussed.