On Fixed Points of Maximalizing Mappings in Posets
© S. Heikkilä. 2010
Received: 7 October 2009
Accepted: 16 November 2009
Published: 18 November 2009
We use chain methods to prove fixed point results for maximalizing mappings in posets. Concrete examples are also presented.
According to Bourbaki's fixed point theorem (cf. [1, 2]) a mapping from a partially ordered set into itself has a fixed point if is extensive, that is, for all , and if every nonempty chain of has the supremum in . In [3, Theorem ] the existence of a fixed point is proved for a mapping which isascending, that is, implies . It is easy to verify that every extensive mapping is ascending. In  the existence of a fixed point of is proved if for some , and if issemi-increasing upward, that is, whenever and . This property holds, for instance, if is ascending orincreasing, that is, whenever .
In this paper we prove further generalizations to Bourbaki's fixed point theorem by assuming that a mapping ismaximalizing, that is, is a maximal element of for all . Concrete examples of maximalizing mappings 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.
A nonempty set , equipped with a reflexive, antisymmetric, and transitive relation in , is called a partially ordered set (poset). An element of a poset is called an upper bound of a subset of if for each . If , we say that is thegreatest element of , and denote . A lower bound of and the least element, , of are defined similarly, replacing above by . If the set of all upper bounds of has the least element, we call it thesupremum of and denote it by . We say that is amaximal element of if , and if and imply that . The infimum of , , and a minimal element of are defined similarly. A subset of is called a chain if or for all . We say that is well ordered if nonempty subsets of have least elements. Every well-ordered set is a chain.
Let be a nonempty poset. A basis to our considerations is the following chain method (cf. [6, Lemma ]).
is valid whenever either of its sides is defined.
A subset of a chain is called aninitial segment of if , and imply . If is well ordered, then every element of which is not the possible maximum of has a successor: , in . The next result gives a characterization of elements of the w-o chain of -iterations.
Assume that is an initial segment of , and that exists. If there is such that , then , so that . Assume next that every element of has the successor in . Since by (b), then . This holds for all . Since , then is an upper bound of . If is an upper bound of , then for every . Thus is an upper bound of , whence . But then , so that by (2.1).
Assume that , and that is not a successor of any element of . Obviously, is an upper bound of . Let be an upper bound of . If , then also since is not a successor. Because by (b), then . This holds for every . Since also , then is an upper bound of . Thus . This holds for every upper bound of , whence .
In the case when we obtain the following result (cf. [7, Proposition ]).
Lemma 2.4 is in fact a special case of Lemma 2.1, since the assumption implies that equals to the w-o chain of -iterations. As for the use of in fixed point theory and in the theory of discontinuous differential and integral equations, see, for example, [8, 9] and the references therein.
3. Main Results
The next result is a consequence of Theorem 3.1. and Lemma 2.3(e).
As a consequence of Proposition 3.2 we obtain the following result.
The results of Lemma 2.3 are valid also when is replaced by the w-o chain of -iterations of . As a consequence of these results and Lemma 2.4 we obtain the following generalizations to Bourbaki's fixed point theorem.
The previous results have obvious duals, which imply the following results.
4. Examples and Remarks
It is easy to verify that , and that is maximalizing. To find a fixed point of , choose . It follows from Lemma 2.3(b) that the first elements of the w-o chain of -iterations are successive approximations
The sequence is strictly increasing, whence also is strictly increasing by (4.5). Thus the set is an initial segment of . Moreover, , and if , then . Since is bounded above by , then exists, and . Thus , and it belongs to , whence by Lemma 2.3(c). To determine , notice that by (4.5). Thus , or equivalently, , so that . Since , then by Lemma 2.3(c). Because is a maximal element of , then . Moreover, , so that is a fixed point of .
The verification of the following properties are left to the reader.
If , then and , , where the sequences and are bounded and increasing. The limit of is the smaller real root of ; , and the limit of is . Moreover and , whence no subsequence of the iteration converges to a fixed point of .
The standard "solve" and "fsolve" commands of Maple 12 do not give a solution or its approximation for the system of Example 4.3.
Chain is compared in  with three other chains which generalize the sequence of ordinary iterations , and which are used to prove fixed point results for . These chains are the generalized orbit defined in  (being identical with the set defined in ), the smallest admissible set containing (cf. [12–14]), and the smallest complete -chain containing (cf. [10, 15]). If is extensive, and if nonempty chains of have supremums, then , and is their cofinal subchain (cf. [10, Corollary ]). The common maximum of these four chains is a fixed point of . This result implies Bourbaki's Fixed Point Theorem.
On the other hand, if the hypotheses of Theorem 3.4 hold and , then and are not necessarily comparable. The successor of such an in is by [14, Proposition ]. In such a case the chains , and attain neither nor any fixed point of . For instance when in Example 4.1, then , where is the w-o chain of -iterations. Since , then does not exist, (see ). Thus only attains a fixed point of as its maximum. As shown in Example 4.1, the consecutive elements of the iteration sequence are unordered, and their limits are not fixed points of . Hence, in these examples also finite combinations of chains used in [16, Theorem ] to prove a fixed point result are insufficient to attain a fixed point of .
Neither the above-mentioned four chains nor their duals are available to find fixed points of if and are unordered. For instance, they cannot be applied to prove Theorems 3.1 and 3.5 or Propositions 3.2 and 3.7.
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