Fixed Points for Multivalued Mappings in Uniformly Convex Metric Spaces

In 1974, Lim 1 developed a result concerning the existence of fixed points for multivalued nonexpansive self-mappings in uniformly convex Banach spaces. This result was extended to nonself-mappings satisfying the inwardness condition independently by Downing and Kirk 2 and Reich 3 . This result was extended to weak inward mappings independently by Lim 4 and Xu 5 . Recently, Dhompongsa et al. 6 presented an analog of Lim-Xu’s result in CAT 0 spaces. In this note, we extend the result to uniformly convex metric spaces which improve results of both Lim-Xu and Dhompongsa et al. In addition, we also give a new proof of a result of Lim 7 by using Caristi’s theorem 8 . Finally, we give some basic properties of fixed point sets for quasi-nonexpansive mappings for these spaces.