Abstract
This paper is for the purpose of introducing and studying a class of new two-step viscosity iteration approximation methods for finding fixed points of set-valued nonexpansive mappings in CAT(0) spaces. By means of some properties and characteristic to CAT(0) space and using Cauchy-Schwarz inequality and Xu’s inequality, strong convergence theorems of the new two-step viscosity iterative process for set-valued nonexpansive and contraction operators in complete CAT(0) spaces are provided. The results of this paper improve and extend the corresponding main theorems in the literature.
Highlights
In [1], the fixed point theory in CAT(0) spaces was first introduced and studied by Kirk
On the other hand, fixed point theory for set-valued mappings has been applied to applied sciences, game theory, and optimization theory
This promotes the rapid development of fixed point theory for single-valued operators in CAT(0) spaces, and it is natural and meaningful to extensively study fixed point theory of set-valued operators
Summary
In [1], the fixed point theory in CAT(0) spaces was first introduced and studied by Kirk. Kirk [1] presented that each nonexpansive (single-valued) mapping on a bounded closed convex subset of a complete CAT(0) space always has a fixed point. On the other hand, fixed point theory for set-valued mappings has been applied to applied sciences, game theory, and optimization theory. This promotes the rapid development of fixed point theory for single-valued (set-valued) operators in CAT(0) spaces, and it is natural and meaningful to extensively study fixed point theory of set-valued operators. We refer to [5,6,7,8,9,10,11,12,13,14] and the references therein
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