Abstract. In this paper we study the nonexpansivity of the so-called relatively nonexpansive mappings. A relatively nonexpansive mapping with respect to a pair of subsets (A, B) of a Banach space X is a mapping defined from A ∪ B into X such that T x − T y ≤ x − y for x ∈ A and y ∈ B. These mappings were recently considered in a paper by Eldred et al. (A, B) is nonexpansive. This fact will be used to improve one of the two main results from the aforementioned paper by Eldred et al. At that time we will also obtain several consequences regarding the strong continuity properties of relatively nonexpansive mappings and the relation between the two main results from the same work.
Abstract. In this paper we study the existence and uniqueness of best proximity points of cyclic contractions as well as the convergence of iterates to such proximity points. We take two different approaches, each one leading to different results that complete, if not improve, other similar results in the theory. Results in this paper stand for Banach spaces, geodesic metric spaces and metric spaces. We also include an appendix on CAT(0) spaces where we study the particular behavior of these spaces regarding the problems we are concerned with.
We study the existence of fixed points in the context of uniformly convex geodesic metric spaces, hyperconvex spaces and Banach spaces for single and multivalued mappings satisfying conditions that generalize the concept of nonexpansivity. Besides, we use the fixed point theorems proved here to give common fixed point results for commuting mappings.
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.
DEDICATED TO GEORGE LEITMANNThis paper is primarily concerned with the study of conditions on a hyperconvex subset D of a hyperconvex metric space M which assure that there exists a nonexpansive retraction R of M\D onto D which has the property that R M\D ⊂ ∂D A related question we take up is, when is such a retraction R proximinal, that is, when does R have the propertyMost of this research was conducted while the first author was visiting the University of Iowa. He thanks the faculty and staff of the Mathematics Department at Iowa for their kind hospitality. Also, the first and third authors acknowledge the support of DGICYT Research Project PB96-1338-C02-01. espínola, kirk, and lópez for each x ∈ M? Among other things, we show that if a subset D of a hyperconvex metricspace M has nonempty interior and is externally hyperconvex relative to M in a very weak sense, then there always exists a nonexpansive retraction of M onto D which maps M\D onto ∂D. We also show that any compact weakly externally hyperconvex subset of a hyperconvex M is a proximinal nonexpansive retract of M.
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