Rafa Espínola

A new approach to relatively nonexpansive mappings

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 ∥Tx-Ty∥ < ∥x-y∥ for x ∈ A and y ∈ B. These mappings were recently considered in a paper by Eldred et al. (Proximinal normal structure and relatively nonexpansive mappings, Studia Math. 171 (3) (2005), 283-293) to obtain a generalization of Kirks Fixed Point Theorem. In this work we show that, for certain proximinal pairs (A, B), there exists a natural semimetric for which any relatively nonexpansive mapping with respect to (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.

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

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.

Common Fixed Points for Multimaps in Metric Spaces

We discuss the existence of common fixed points in uniformly convex metric spaces for single-valued pointwise asymptotically nonexpansive or nonexpansive mappings and multivalued nonexpansive, -nonexpansive, or -semicontinuous maps under different conditions of commutativity.

On the Structure of Minimal Sets of Relatively Nonexpansive Mappings

In this article, we study the structure of minimal sets of relatively non-expansive mappings. We consider the cyclic and the noncyclic cases and show that results alike to the celebrated Goebel-Karlovitz lemma for non-expansive self-mappings can be obtained for relatively non-expansive mappings.

Applications of fixed point theorems in the theory of invariant subspaces

We survey several applications of fixed point theorems in the theory of invariant subspaces. The general idea is that a fixed point theorem applied to a suitable map yields the existence of invariant subspaces for an operator on a Banach space.MSC:47A15, 47H10.