Fixed Points of Single- and Set-Valued Mappings in Uniformly Convex Metric Spaces with No Metric Convexity

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 setvalued 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.

Geodesic Ptolemy spaces and fixed points

“…Likewise, the same holds for complete uniformly convex metric spaces with a monotone (or lower semi-continuous from the right) modulus of uniform convexity (see [5] for details).…”

“…A simple example of a reflexive metric space is a reflexive Banach space. Other examples include complete CAT(0) spaces, complete uniformly convex metric spaces with a monotone or a lower semi-continuous from the right modulus of uniform convexity (see [5,11]) and others.…”

“…However, as remarked in [5], it suffices that the class of admissible subsets is nested compact. By Theorem 3.4, it is immediate that in the context of a complete Ptolemy geodesic space with a uniformly continuous midpoint map, the class of admissible subsets is nested compact.…”

“…So far reflexivity of geodesic metric spaces has been proved only for uniformly convex spaces (see [5,11]). Therefore, a natural question at this point is to know if geodesic Ptolemy spaces with a uniformly continuous mipoint map are uniformly convex.…”

The fixed point property and unbounded sets in CAT(0) spaces

“…From the CN-inequality for CAT(0) spaces (see [1, pg. 163]) it is immediate that the modulus of convexity of a Hilbert space is a modulus of convexity for any CAT(0) space and so, in particular, any CAT(0) space is uniformly convex (see [6,7,8,11,15,19] for more on this fact). That is, if δ(a, r, ε) is the best modulus of convexity of a CAT(0) space then it must be the case that (5.7)…”