Modified viscosity type iteration for total asymptotically nonexpansive mappings in CAT(0) spaces and its application to optimization problems

Volume 11, Issue 2, pp 288--302 http://dx.doi.org/10.22436/jnsa.011.02.10

Publication Date: February 07, 2018 Submission Date: October 22, 2017 Revision Date: November 26, 2017 Accteptance Date: December 01, 2017

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Authors

Wiyada Kumam - Program in Applied Statistics, Department of Mathematics and Computer Science, Faculty of Science and Technology, Rajamangala University of Technology Thanyaburi, Thanyaburi, Pathumthani 12110, Thailand. Nuttapol Pakkaranang - KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. Poom Kumam - KMUTTFixed Point Research Laboratory, Department of Mathematics, Room SCL 802 Fixed Point Laboratory, Science Laboratory Building, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140, Thailand. - KMUTT-Fixed Point Theory and Applications Research Group, Theoretical and Computational Science Center (TaCS), Science Laboratory Building, Facuty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), 126 Pracha-Uthit Road, Bang Mod, Thung Khru, Bangkok 10140,, Thailand. - Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 40402, Taiwan.

Abstract

In this paper, we introduce a modified two-step viscosity iteration process for total asymptotically nonexpansive mappings in CAT(0) spaces. We prove strong convergence of the proposed iteration process to a fixed point of total asymptotically nonexpansive mappings in CAT(0) spaces, which also shows that the limit of the sequence generated by proposed iteration process solves the solution of the variational inequality. We also provide illustrating a numerical example for supporting our main results. Moreover, we show the existence of solutions of our consequently results for some applications.

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Keywords

• Viscosity approximation methods
• total asymptotically nonexpansive mapping
• variational inequality
• CAT(0) spaces

MSC

•  47H09
•  47J20
•  47J25

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