# Viscosity approximation methods for the implicit midpoint rule of nonexpansive mappings in CAT(0) Spaces

Volume 10, Issue 2, pp 386--394
Publication Date: February 20, 2017 Submission Date: August 25, 2016
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### Authors

Liang-cai Zhao - College of Mathematics, Yibin University, Yibin, Sichuan, 644007, China. 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, China. Gang Wang - College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China.

### Abstract

The purpose of this paper is to introduce the implicit midpoint rule of nonexpansive mappings in CAT(0) spaces. The strong convergence of this method is proved under certain assumptions imposed on the sequence of parameters. Moreover, it is shown that the limit of the sequence generated by the implicit midpoint rule solves an additional variational inequality. Applications to nonlinear Volterra integral equations and nonlinear variational inclusion problem are included. The results presented in the paper extend and improve some recent results announced in the current literature.

### Keywords

• Viscosity
• implicit midpoint rule
• nonexpansive mapping
• projection
• variational inequality
• CAT(0) space.

•  47H09
•  47J25

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