Rafael Espínola

Pythagorean Property and Best-Proximity Point Theorems

In this paper, a notion called proximally complete pair of subsets of a metric space is introduced, which weakens earlier notions in the theory of best-proximity points. By means of this notion, existence and convergence results of best-proximity points are proven for cyclic contraction mappings, which extent other recent results. By observing geometrical properties of Hilbert spaces, the so-called Pythagorean property is introduced. This property is employed to provide sufficient conditions for a cyclic map to be a cyclic contraction.

Fixed Point Theory in Hyperconvex Metric Spaces

In this chapter we propose a review of some of the most fundamental facts and properties on metric hyperconvexity in relation to Metric and Topological Fixed Point Theory. Hyperconvex metric spaces were introduced by Aronszajn and Panitchpakdi in 1956 in relation to the problem of extending uniformly continuous mappings defined between metric spaces. It was obvious from the very beginning that the structure given by the hyperconvexity of the metric to the space was a very rich one. As a consequence of that richness, a very profound and exhaustive Fixed Point Theory has been developed on hyperconvex metric spaces, especially from late eighties of the Twentieth Century by pioneering works due to Baillon, Sine and Soardi. This theory applies for single and multivalued mappings as well as for best-approximation results. Along 9 sections, we expose in a detailed and self-contained way the foundations of this theory. A final additional section, however, has been included to describe some of the newest trends on hyperconvexity and existence of fixed points.

The Knaster–Tarski theorem versus monotone nonexpansive mappings

Let

Existence and data dependence of fixed points for multivalued operators on gauge spaces

X

Nonexpansive Retractions in Hyperconvex Spaces

be a Hausdorff topological compact space with a partial order

FIXED POINTS AND APPROXIMATE FIXED POINTS IN PRODUCT SPACES

% \preceq

Nearest and farthest points in spaces of curvature bounded below

for which the order intervals are closed and let

Nonexpansive selections of metric projections in spaces of continuous functions

\mathcal{F}

Extension of compact mappings and N 0 -hyperconvexity

be a nonempty commutative family of monotone maps from

CROSSED CARTESIAN PRODUCT OF MULTIVALUED OPERATORS

X