Strong Convergence of Viscosity Approximation Methods for Nonexpansive Mappings in CAT(0) Spaces

Luo Yi Shi and Ru Dong Chen

Journal of Applied Mathematics, 2012, vol. 2012, issue 1

Abstract: Viscosity approximation methods for nonexpansive mappings in CAT(0) spaces are studied. Consider a nonexpansive self‐mapping T of a closed convex subset C of a CAT(0) space X. Suppose that the set Fix(T) of fixed points of T is nonempty. 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. We will show that if X is a CAT(0) space satisfying some property, then {xt} converge strongly to a fixed point of T which solves some variational inequality. Consider also the iteration process {xn}, where x0 ∈ C is arbitrary and xn+1 = αnf(xn)⊕(1 − αn)Txn for n ≥ 1, where {αn}⊂(0,1). It is shown that under certain appropriate conditions on αn, {xn} converge strongly to a fixed point of T which solves some variational inequality.

Date: 2012
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https://doi.org/10.1155/2012/421050

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2012:y:2012:i:1:n:421050

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