Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces

The purpose of this paper is to study the strong convergence theorems of Moudafi's viscosity approximation methods for a nonexpansive mapping T in CAT(0) spaces without the property P. 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)Tx; i.e., x t = tf (x t) ⊕ (1 – t)Tx t and x n+1 = α n f (x n) ⊕ (1 – α n)Tx n , n ≥ 0, where x 0 ∈ C is arbitrarily chosen and {α n } ⊂ (0, 1) satisfies certain conditions. We prove that the iterative schemes {x t } and {x n } converge strongly to the same point˜x such that˜x = P F(T) f (˜ x), which is the unique solution of the variational inequality (VIP) – – → ˜ xf˜x, – → x˜x ≥ 0, x ∈ F(T). By using the concept of quasilinearization, we remark that the proof is different from that of Shi and Chen in J. In fact, strong convergence theorems for two given iterative schemes are established in CAT(0) spaces without the property P.