Abstract
AbstractSuppose that "Equation missing" is a nonempty closed convex subset of a complete CAT(0) space "Equation missing" with the nearest point projection "Equation missing" from "Equation missing" onto "Equation missing". Let "Equation missing" be a nonexpansive nonself mapping with "Equation missing". Suppose that "Equation missing" is generated iteratively by "Equation missing", "Equation missing", "Equation missing", where "Equation missing" and "Equation missing" are real sequences in "Equation missing" for some "Equation missing". Then "Equation missing""Equation missing"-converges to some point "Equation missing" in "Equation missing". This is an analog of a result in Banach spaces of Shahzad (2005) and extends a result of Dhompongsa and Panyanak (2008) to the case of nonself mappings.
Highlights
A metric space X is a CAT 0 space if it is geodesically connected and if every geodesic triangle in X is at least as “thin” as its comparison triangle in the Euclidean plane
He showed that every nonexpansive single-valued mapping defined on a bounded closed convex subset of a complete CAT 0 space always has a fixed point
If T : K → X is a nonexpansive mapping with nonempty fixed point set, and if {xn} is generated iteratively by x1 ∈ K, xn 1 P 1 − αn xn ⊕ αnT P 1 − βn xn ⊕ βnT xn, 1.1 where {αn} and {βn} are real sequences in ε, 1 − ε for some ε ∈ 0, 1, we show that the sequence {xn} defined by 1.1 Δ-converges to a fixed point of T
Summary
A metric space X is a CAT 0 space if it is geodesically connected and if every geodesic triangle in X is at least as “thin” as its comparison triangle in the Euclidean plane. Fixed point theory in a CAT 0 space was first studied by Kirk see 6, 7 .
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