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, 1-11

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 x t ∈ C be the unique fixed point of the contraction x ↦ t f ( x ) ⊕ ( 1 - t ) T x . We will show that if X is a CAT(0) space satisfying some property, then { x t } converge strongly to a fixed point of T which solves some variational inequality. Consider also the iteration process { x n } , where x 0 ∈ C is arbitrary and x n + 1 = α n f ( x n ) ⊕ ( 1 - α n ) T x n for n ≥ 1 , where { α n } ⊂ ( 0,1 ) . It is shown that under certain appropriate conditions on α n , { x n } converge strongly to a fixed point of T which solves some variational inequality.

Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:hin:jnljam:421050

DOI: 10.1155/2012/421050

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