On Fixed Points of Maximalizing Mappings in Posets
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Research Article On Fixed Points of Maximalizing Mappings in Posets S. Heikkila¨ Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, 90014 Oulu, Finland Correspondence should be addressed to S. Heikkil¨a, [email protected] Received 7 October 2009; Accepted 16 November 2009 Academic Editor: Mohamed A. Khamsi Copyright q 2010 S. Heikkil¨a. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We use chain methods to prove fixed point results for maximalizing mappings in posets. Concrete examples are also presented.
1. Introduction According to Bourbaki’s fixed point theorem cf. 1, 2 a mapping G from a partially ordered set X X, ≤ into itself has a fixed point if G is extensive, that is, x ≤ Gx for all x ∈ X, and if every nonempty chain of X has the supremum in X. In 3, Theorem 3 the existence of a fixed point is proved for a mapping G : X → X which is ascending, that is, Gx ≤ y implies Gx ≤ Gy. It is easy to verify that every extensive mapping is ascending. In 4 the existence of a fixed point of G is proved if a ≤ Ga for some a ∈ X, and if G is semi-increasing upward, that is, Gx ≤ Gy whenever x ≤ y and Gx ≤ y. This property holds, for instance, if G is ascending or increasing, that is, Gx ≤ Gy whenever x ≤ y. In this paper we prove further generalizations to Bourbaki’s fixed point theorem by assuming that a mapping G : X → X is maximalizing, that is, Gx is a maximal element of {x, Gx} for all x ∈ X. Concrete examples of maximalizing mappings G which have or do not have fixed points are presented. Chain methods introduced in 5, 6 are used in the proofs. These methods are also compared with three other chain methods.
2. Preliminaries A nonempty set X, equipped with a reflexive, antisymmetric, and transitive relation ≤ in X × X, is called a partially ordered set poset. An element b of a poset X is called an upper
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Fixed Point Theory and Applications
bound of a subset A of X if x ≤ b for each x ∈ A. If b ∈ A, we say that b is the greatest element of A, and denote b max A. A lower bound of A and the least element, min A, of A are defined similarly, replacing x ≤ b above by b ≤ x. If the set of all upper bounds of A has the least element, we call it the supremum of A and denote it by sup A. We say that y is a maximal element of A if y ∈ A, and if z ∈ A and y ≤ z imply that y z. The infimum of A, inf A, and a minimal element of A are defined similarly. A subset W of X is called a chain if x ≤ y or y ≤ x for all x, y ∈ W. We say that W is well ordered if nonempty subsets of W have least elements. Every well-ordered set is a chain. Let X be a nonempty poset. A basis to our considerations is the following chain method cf. 6, Lemma 2. Lemma 2.1. Given G : X → X and a ∈ X, there exists a unique well-ordered chain C in X, called a w-o chain of aG-iterations, satisfying x∈C
iff x sup
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