Abstract
AbstractWe prove strong convergence of the viscosity approximation method for multivalued nonexpansive mappings in $\operatorname{CAT}(0)$ CAT ( 0 ) spaces. Our results generalize the results of Dhompongsa et al. (2012), Wangkeeree and Preechasilp (2013) and many others. Some related results in $\mathbb{R}$ R -trees are also given.
Highlights
1 Introduction One of the successful approximation methods for finding fixed points of nonexpansive mappings was given by Moudafi [ ]
The following scheme is known as the viscosity approximation method or Moudafi’s viscosity approximation method: x ∈ E arbitrarily chosen, xn+ = αnf + ( – αn)t(xn), n ∈ N, ( )
We note that the Halpern approximation method [ ], xn+ = αnu + ( – αn)t(xn), n ∈ N, where u is a fixed element in E, is a special case of the Moudafi one
Summary
One of the successful approximation methods for finding fixed points of nonexpansive mappings was given by Moudafi [ ]. Let E be a nonempty closed convex subset of a complete CAT( ) space X, t : E → E be a nonexpansive mapping with Fix(t) = ∅, and f : E → E be a contraction with constant k ∈ [ , ). → – →– ab, cd ρ (a, d) + ρ (b, c) – ρ (a, c) – ρ (b, d) for all a, b, c, d ∈ X
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