One-local retracts and Banach operator pairs in metric spaces

In this paper, we first introduce the concept of NR-map and then use this concept to establish the existence of common fixed points for Banach operator pairs in the context of uniformly convex geodesic metric spaces. New proofs of main results (Theorems 2.1 and 3.5) of Chen and Li [J. Chen, Z. Li, Banach operator pair and common fixed points for nonexpansive maps, Nonlinear Anal. 74 (2011) 3086-3090] are presented. Further, we prove De Marr's theorem for the family of symmetric Banach operator pairs in metric spaces and R-trees for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity.