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Invariant approximation for CAT(0) spaces. (English) Zbl 1188.54022
In the year 2008, {\it 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 $$(1/2) \|x-Tx\| \leq \|x-y\|,$$ 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 {\it N. Shahzad} and {\it 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
Full Text:
##### References:
 [1] Suzuki, T.: Fixed point theorems and convergence theorems for som generalized nonexpansive mappings, J. math. Anal. appl. 340, 1088-1095 (2008) · Zbl 1140.47041 · doi:10.1016/j.jmaa.2007.09.023 [2] Bridson, M.; Haefliger, A.: Metric spaces of non-positive curvature, (1999) · Zbl 0988.53001 [3] Dhompongsa, S.; Kirk, W. A.; Sims, B.: Fixed point of uniformly Lipschitzian mappings, Nonlinear anal. 65, 762-772 (2006) · Zbl 1105.47050 · doi:10.1016/j.na.2005.09.044 [4] Kirk, W. A.; Panyanak, B.: A concept of convergence in geodesic spaces, Nonlinear anal. 68, 3689-3696 (2008) · Zbl 1145.54041 · doi:10.1016/j.na.2007.04.011 [5] Goebel, K.; Kirk, W. A.: Iteration processes for nonexpansive mappings, Contemp. math. 21, 115-123 (1983) · Zbl 0525.47040 [6] Dhompongsa, S.; Kirk, W. A.; Panyanak, B.: Non-expansive set-valued mappings in metric and Banach spaces, J. nonlinear convex anal. 8, 35-45 (2007) · Zbl 1120.47043 [7] Choha, P.; Phon-On, A.: A note on fixed point sets in $CAT(0)$ spaces, J. math. Anal. appl. 320, 983-987 (2006) · Zbl 1101.54040 · doi:10.1016/j.jmaa.2005.08.006 [8] Dhompongsa, S.; Fupinwong, W.; Kaewkhao, A.: Common fixed point of nonexpansive semigroup and a convergence theorem for Mann iterations in geodesic metric spaces, Nonlinear anal. 65, 762-772 (2008) · Zbl 1191.47077 [9] Kirk, W. A.: Geodesic geometry and fixed point theory II, , 113-142 (2004) · Zbl 1083.53061 [10] A. Razani, H. Salahifard, Demiclosedness principle for total asymptotically nonexpansive mappings in CAT(0) spaces (submitted for publication) · Zbl 1188.54022 [11] Shahzad, N.: Invariant approximations in $CAT(0)$ spaces, Nonlinear anal. 70, 4338-4340 (2009) · Zbl 1167.47042 · doi:10.1016/j.na.2008.10.002 [12] Shahzad, N.; Markin, J.: Invariant approximations for commuting mappings in $CAT(0)$ and hyperconvex spaces, J. math. Anal. appl. 337, 1457-1464 (2008) · Zbl 1137.47043 · doi:10.1016/j.jmaa.2007.04.041