Approximating Fixed Points of Nonexpansive Nonself Mappings in CAT(0) Spaces

Suppose that K is a nonempty closed convex subset of a complete CAT(0) space X with the nearest point projection P from X onto K. Let T:K→X be a nonexpansive nonself mapping with F(T):={x∈K:Tx=x}≠∅. Suppose that {xn} is generated iteratively by x1∈K, xn+1=P((1−αn)xn⊕αnTP[(1−βn)xn⊕βnTxn]), n≥1, where {αn} and {βn} are real sequences in [ε,1−ε] for some ε∈(0,1). Then {xn}Δ-converges to some point x∗ in F(T). 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

A Note on Common Fixed Point Results in Uniformly Convex Hyperbolic Spaces

It is shown that the notion of mappings satisfying condition (K) introduced by Akkasriworn et al. (2012) is weaker than the notion of asymptotically quasi-nonexpansive mappings in the sense of Qihou (2001) and is weaker than the notion of pointwise asymptotically nonexpansive mappings in the sense of Kirk and Xu (2008). We also obtain a common fixed point for a commuting pair of a mapping satisfying condition (K) and a multivalued mapping satisfying condition (Cλ) for some λ∈(0,1). Our results properly contain the results of Abkar and Eslamian (2012), Akkasriworn et al. (2012), and many others

Strong and <inline-formula> <graphic file="1029-242X-2009-730132-i1.gif"/></inline-formula> Convergence Theorems for Multivalued Mappings in <inline-formula> <graphic file="1029-242X-2009-730132-i2.gif"/></inline-formula> Spaces

<p/> <p>We show strong and <inline-formula> <graphic file="1029-242X-2009-730132-i3.gif"/></inline-formula> convergence for Mann iteration of a multivalued nonexpansive mapping whose domain is a nonempty closed convex subset of a CAT(0) space. The results we obtain are analogs of Banach space results by Song and Wang [2009, 2008]. Strong convergence of Ishikawa iteration are also included.</p

Approximating Fixed Points of Nonexpansive Nonself Mappings in CAT(0) Spaces

Suppose that is a nonempty closed convex subset of a complete CAT(0) space with the nearest point projection from onto . Let be a nonexpansive nonself mapping with . Suppose that is generated iteratively by , , , where and are real sequences in for some . Then -converges to some point in . 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.</p