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

By Rabian Wangkeeree, Pakkapon Preechasilp. Journal of Inequalities and Applications, 2013

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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 . For a contraction f on C and t∈(0,1), let xt∈C be the unique fixed point of the contraction x↦tf(x)⊕(1−t)Tx; i.e., xt=tf(xt)⊕(1−t)Txt and xn+1=αnf(xn)⊕(1−αn)Txn,n≥0, where x0∈C is arbitrarily chosen and {αn}⊂(0,1) satisfies certain conditions. We prove that the iterative schemes {xt} and {xn} converge strongly to the same point x˜ such that x˜=PF(T)f(x˜), which is the unique solution of the variational inequality (VIP) 〈x˜fx˜→,xx˜→〉≥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. Appl. Math. 2012:421050, 2012. In fact, strong convergence theorems for two given iterative schemes are established in CAT(0) spaces without the property .

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