Approximating Fixed Points of Nonexpansive Mappings in CAT(0) Spaces
Thanomsak Laokul, Bancha Panyanak · 2009
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} = ∅. 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.