On Fixed Points of Maximalizing Mappings in Posets

Abstract

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 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 [4] 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.

2. Preliminaries

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 ]).

Lemma 2.1.

Given and , there exists a unique well-ordered chain in , called a w-o chain of -iterations, satisfying

(2.1)

If exists in , then , and .

The following result helps to analyze the w-o chain of -iterations.

Lemma 2.2.

Let and be nonempty subsets of . If and exist, then the equation

(2.2)

is valid whenever either of its sides is defined.

Proof.

The sets and have same upper bounds, which implies the assertion.

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.

Lemma 2.3.

Given and , let be the w-o chain of -iterations. Then the elements of have the following properties.

.

An element of has a successor in if and only if exists and , and then .

If is an initial segment of and exists, then .

If and is not a successor, then .

If exists, then .

Proof.

• (a) .

• (b) Assume first that , and that exists in . Applying (2.1), Lemma 2.2, and the definition of we obtain

(2.3)

Moreover, , by definition, whence .

Assume next that , that exists, and that . The previous proof implies the following

• (i) There is no element which satisfies .

Then , so that

(2.4)

Thus by (2.1). This result and (i) imply that .

• (c) 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).

• (d) 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 .

• (e) If exists, then by (c) when , whence .

In the case when we obtain the following result (cf. [7, Proposition ]).

Lemma 2.4.

Given and , there exists a unique well-ordered chain in , calleda w-o chain of -iterations of , satisfying

(2.5)

If , and if exists, then , and .

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

Let be a nonempty poset. As an application of Lemma 2.1 we will prove our first existence result.

Theorem 3.1.

A mapping has a fixed point if is maximalizing, that is, is a maximal element of for all , and if exists in for some where is the w-o chain of -iterations.

Proof.

If is the w-o chain of -iterations, and if exists in , then and by Lemma 2.1. Since is maximalizing, then , that is, is a fixed point of .

The next result is a consequence of Theorem 3.1. and Lemma 2.3(e).

Proposition 3.2.

Assume that is maximalizing. Given , let be the w-o chain of -iterations. If exists, it is a fixed point of if and only if exists.

Proof.

Assume that exists. It follows from Lemma 2.3(e) that . If is a fixed point of , that is, , then , and .

Assume conversely that exist. Applying (2.1) and Lemma 2.2 we obtain

(3.1)

Thus, by Theorem 3.1, is a fixed point of .

As a consequence of Proposition 3.2 we obtain the following result.

Corollary 3.3.

If nonempty chains of have supremums, if is maximalizing, and if exists for all , then for each the maximum of the w-o chain of -iterations exists and is a fixed point of .

Proof.

Let be the w-o chain of -iterations. The given hypotheses imply that both and exist. Thus the hypotheses of Proposition 3.2 are valid.

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.

Theorem 3.4.

Assume that is maximalizing, and that for some , and let be the w-o chain of -iterations of .

• (a) If exists, then , and is a fixed point of .

• (b) If exists, it is a fixed point of if and only if exists.

• (c) If nonempty chains of have supremums, and if exists for all , then exists, and is a fixed point of

The previous results have obvious duals, which imply the following results.

Theorem 3.5.

A mapping has a fixed point if is minimalizing, that is, is a minimal element of for all , and if exists in for some whenever is a nonempty chain in .

Theorem 3.6.

A minimalizing mapping has a fixed point if exists whenever is a nonempty chain in , and if for some .

Proposition 3.7.

A minimalizing mapping has a fixed point if every nonempty chain has the infimum in , and if exists for all .

Remark 3.8.

The hypothesis that is maximalizing can be weakened in Theorems 3.1 and 3.4 and in Proposition 3.2 to the form: is maximalizing, that is, is a maximal element of .

4. Examples and Remarks

We will first present an example of a maximalizing mapping whose fixed point is obtained as the maximum of the w-o chain of -iterations.

Example 4.1.

Let be a closed disc , ordered coordinate-wise. Let denote the greatest integer when . Define a function by

(4.1)

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

(4.2)

as long as is defined. Denoting , these successive approximations can be rewritten in the form

(4.3)

as long as and , and at least one of these inequalities is strict. Elementary calculations show that , for every . Thus (4.3) can be rewritten as

(4.4)

Since the function is increasing , then for every . Thus (4.4) can be reduced to the form

(4.5)

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 first elements of the w-o chain of -iterations can be estimated by the following Maple program (floor):

(1,1-floor(u) + floor(v)): (floor)/2: :

for to do (max, evalf(max); end do;

For instance, .

The verification of the following properties are left to the reader.

• (i) If , , and , then the elements of w-o chain of -iterations, after two first terms if , are of the form , , where is increasing and converges to . Thus is the maximum of and a fixed point of .

• (ii) If , , or , then .

• (iii) 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 .

• (iv) For any choice of the iterations and are not order related when . The sequence does not converge, and no subsequence of it converges to a fixed point of .

• (v) Denote . The function , defined by (4.1), satisfies and is maximalizing. The maximum of the w-o chain of -iterations with is , and is a fixed point of . If , then and are not comparable.

The following example shows that need not to have a fixed point if either of the hypothesis of Theorem 3.1 is not valid.

Example 4.2.

Denote and , where and are as in Example 4.1. Choose , and let be defined by (4.1). is maximalizing, but has no fixed points, since and . The last hypothesis of Theorem 3.1 is not satisfied.

Denoting , then the set is a complete join lattice, that is, every nonempty subset of has the supremum in . Let satisfy and . has no fixed points, but is not maximalizing, since .

Example 4.3.

The components , of the fixed point of in Example 4.1 form also a solution of the system

(4.6)

Moreover a Maple program introduced in Example 4.1 serves a method to estimate this solution. When , the estimate is , .

Remark 4.4.

The standard "solve" and "fsolve" commands of Maple 12 do not give a solution or its approximation for the system of Example 4.3.

In Example 4.1 the mapping is nonincreasing, nonextensive, nonascending, not semiincreasing upward, and noncontinuous.

Chain is compared in [10] 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 [10] (being identical with the set defined in [11]), 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 [10]). 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.