Order-theoretic aspects of metric fixed point theory (Q2761643)

The referred paper is a chapter of the ``Handbook of metric fixed point theory'' (W. A. Kirk and B. Sims (eds.)), Kluwer Academic Publishers (2001). It deals with the connections between metric methods and partial ordering techniques in fixed point theory. The author concentrates on the following problem: Given a space with a metric structure (uniform space, metric or Banach space) and a mapping satisfying some geometric conditions, a partial ordering is defined so that one of fundamental ordering principles -- the Knaster-Tarski theorem, Zermelo's theorem or the Tarski-Kantorovich theorem -- can be applied to deduce the existence of a fixed point. The approach is constructive because these principles are independent of the axiom of choice.NEWLINENEWLINENEWLINEIn Section 2 the author studies consequences of the Knaster-Tarski theorem. He presents a fixed point theorem for diametric contractions, for \(j\)-contractive mappings on a uniform space and for set-valued contractions. Section 3 is focused to Zermelo's theorem and its consequences. A common fixed point theorem for two mappings on a partially ordered set is established. Section 4 deals with the Tarski-Kantorovich theorem and its applications to obtain a new proofs of known results.NEWLINENEWLINENEWLINEThe paper contains very interesting relations between basic results in fixed point theory.NEWLINENEWLINEFor the entire collection see [Zbl 0970.54001].