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Invariant approximation for CAT(0) spaces. (English) Zbl 1188.54022

In the year 2008, T. Suzuki [J. Math. Anal. Appl. 340, No. 2, 1088–1095 (2008; Zbl 1140.47041)] showed the following:

Let $T$ be a mapping on a subset $C$ of a Banach space $E$. Then $T$ is said to satify condition (C) if

$\left(1/2\right)\parallel x-Tx\parallel \le \parallel x-y\parallel ,$

for all $x,y\in C$. In this paper, the authors apply this to the solution of the problem for commuting pairs consisting of a single valued mapping and a multivalued mapping in CAT(0) spaces. For this a multivalued version of condition (C) in CAT(0) spaces has been defined.

In this way the authors improve and extend the results of N. Shahzad and J. Markin, in [J. Math. Anal. Appl. 337, No. 2, 1457–1464 (2008; Zbl 1137.47043)].

##### MSC:
 54H25 Fixed-point and coincidence theorems in topological spaces 41A50 Best approximation, Chebyshev systems 47H10 Fixed point theorems for nonlinear operators on topological linear spaces
##### Keywords:
fixed points; condition (C); CAT(0) spaces; commuting mappings