Suppose 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.
Abkar and Eslamian (Nonlinear Anal. TMA, 74, 1835-1840 prove that if K is a nonempty bounded closed convex subset of a complete CAT(0) space X, t : K K is a single-valued quasi-nonexpansive mapping and T : K KC(K) is a multivalued mapping satisfying conditions (E) and (C l ) for some l (0, 1) such that t and T commute weakly, then there exists a point z K such that z = t(z) T(z). In this paper, we extend this result to the general setting of uniformly convex metric spaces. Nevertheless, condition (E) of T can be weakened to the strongly demiclosedness of I -T.
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 someλ∈(0,1). Our results properly contain the results of Abkar and Eslamian (2012), Akkasriworn et al. (2012), and many others.
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