Order-Theoretic Aspects of Metric Fixed Point Theory

This chapter is intended to present connections between two branches of fixed point theory: The first, using metric methods which is the main subject of this Handbook, and the second, involving partial ordering techniques. We shall concentrate here on the following problem: Given a space with a metric structure (e.g., uniform space, metric space or Banach space) and a mapping satisfying some geometric conditions, define a partial ordering (depending on a structure of a space and/or a mapping) so that one of fundamental ordering principles — the Knaster-Tarski Theorem, Zermelo’s Theorem or the Tarski-Kantorovitch Theorem — can be applied to deduce the existence of a fixed point. We emphasize that all the above principles are independent of the Axiom of Choice (abbr., AC) so the above approach to metric fixed point theory is wholly constructive. It seems that such studies were initiated by H. Amann [5] and B. Fuchssteiner [33] in 1977. Subsequently, they were continued among others by S. Hayashi [37], R. Mańka ([59], [60]), R. Lemmert and P. Volkmann [58], A. Baranga [8], T. Büber and W. A. Kirk [20] and J. Jachymski ([41], [42], [43], [44], [46], [47]). On the other hand, some authors have also studied a reciprocal of the above problem: Given a partially ordered set and a mapping on it, define a metric depending on this order so that some theorems of metric fixed point theory could be applied. That was done recently by Y.-Z. Chen [22], who used Thompson’s [81] metric generated by an order. However, in this chapter, we shall not discuss these problems.