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.,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 pointx such thatx = P F(T) f (x), which is the unique solution of the variational inequality (VIP)By using the concept of quasilinearization, we remark that the proof is different from that of Shi and Chen in J. Appl. Math. 2012Math. :421050, 2012. In fact, strong convergence theorems for two given iterative schemes are established in CAT(0) spaces without the property P.
Let C be a nonempty, closed, and convex subset of a complete CAT(0) space X and let T be an asymptotically nonexpansive mapping of C into itself such that the set of common fixed points of T is nonempty. We introduce the iterative schemes for finding the common fixed point of an asymptotically nonexpansive mapping which is the unique solution of some variational inequalities. The strong convergence theorem of the proposed iterative schemes is established. Our result improves and generalizes several other results in the literature.
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