# Strong convergence of modified viscosity implicit approximation methods for asymptotically nonexpansive mappings in complete CAT(0) spaces

Volume 17, Issue 3, pp 345-354 Publication Date: July 13, 2017

### Authors

Nuttapol Pakkaranang - Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
Poom Kumam - Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
Yeol Je Cho - Department of Mathematics Education and the RINS, Gyeongsang National University, Jinju 660-701, Korea.
Plern Saipara - Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
Anantachai Padcharoen - Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.
Chatuphol Khaofong - Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand.

### Abstract

In this paper, we introduce a modified viscosity implicit iteration for asymptotically nonexpansive mappings in complete CAT(0) spaces. Under suitable conditions, we prove some strong convergence to a fixed point of an asymptotically nonexpansive mapping in a such space which is also the solution of variational inequality. Moreover, we illustrate some numerical example of our main results. Our results extend and improve some recent result of Yao et al. [Y.-H. Yao, N. Shahzad, Y.-C. Liou, Fixed Point Theory Appl., 2015 (2015), 15 pages] and Xu et al. [H.-K. Xu, M. A. Alghamdi, N. Shahzad, Fixed Point Theory Appl., 2015 (2015), 12 pages].

### Keywords

• Asymptotically nonexpansive mapping
• projection
• viscosity
• implicit iterative rule
• variational inequality
• CAT(0) spaces.

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