Abstract
We study the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces. We also study the existence of fixed points for set-valued nonexpansive mappings in the same class of spaces. Our results do not assume convexity of the metric which makes a big difference when studying the existence of fixed points for set-valued mappings.
Highlights
This paper is motivated by the recent paper 1
We study the existence of fixed points and convergence of iterates for asymptotic pointwise contractions in uniformly convex metric spaces
We study the existence of fixed points for setvalued nonexpansive mappings in the same class of spaces
Summary
This paper is motivated by the recent paper 1. In 1 the authors study different questions related to fixed points of asymptotic pointwise contractive/nonexpansive mappings in CAT 0 spaces. The notion of asymptotic pointwise contractions was introduced in 6 It was studied in 7 where, by means of ultrapower techniques, different results about the. Finding fixed point for set-valued nonexpansive mappings in uniformly convex metric spaces was first studied by Shimizu and Takahashi 9 , where the existence of fixed points was guaranteed under stronger conditions on the modulus of convexity and the additional condition of metric convexity of the space. The fact that we do not have that the metric are convex will make the problem more complicated and this will take us to impose new conditions on the modulus of convexity which we will relate with the geometry of the space
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