An affirmative answer to the open questions on the viscosity approximation methods for nonexpansive mappings in CAT(0) spaces

Volume 9, Issue 6, pp 4563--4570 http://dx.doi.org/10.22436/jnsa.009.06.94

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Authors

Shih-Sen Chang - Center for General Education, China Medical University, Taichung 40402, Taiwan. Lin Wang - College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, P. R. China. Gang Wang - College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, P. R. China. Lijuan Qin - Department of Mathematics, Kunming University, Kunming, Yunnan 650214, P. R. China.

Abstract

We prove a strong convergence theorem of a two-step viscosity iteration method for nonexpansive mappings in CAT(0) spaces without the nice projection property \(\mathbb{N}\) and the restriction of the contraction constant \(k \in [0; \frac{1}{2} )\). Our result gives an affrrmative answer to the open questions raised by Piatek [B. Piatek, Numer. Funct. Anal. Optim., 34 (2013), 1245-1264], and Kaewkhao et al. [A. Kaewkhao, B. Panyanak, S. Suantai, J. Inequal. Appl., 2015 (2015), 9 pages].

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Keywords

• Viscosity approximation method
• fixed point
• strong convergence
• multivalued nonexpansive mapping
• the nice projection property \(\mathbb{N}\)
• CAT(0) space.

MSC

•  47J25
•  47H09
•  47H10

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