Abstract
The purpose of this paper is to prove some new fixed point theorem and common fixed point theorems of a commuting family of order-preserving mappings defined on an ordered set, which unify and generalize some relevant fixed point theorems.
1. Introduction and Preliminaries
The fixed points theory has experienced a great development in recent years, among enormous results which have enriched this theory, Tarski’s theorem on ordered sets is “any application which preserves order on a complete lattice admits a fixed point and the set of its fixed points is a complete lattice.”
In our previous work, we have introduced the concept of complete T-lattice to show that a complete lattice is a special case of a complete T-lattice. So, the reader may wonder if we can get a generalization of Tarski’s theorem.
In one of our articles [1, 2], we proved that any application that preserves the order on a complete T-lattice has a fixed point. In this work, we give a structure of the set of these fixed points and some other results.
So, we begin by describing the relevant notation and terminology. Let be a partially ordered set and a nonempty subset. Recall that an upper (resp. lower) bound for is an element with (resp. ) for each ; the least upper (resp. greatest-lower) bound of will be denoted (resp. ).
Next, let be a partially ordered set with the least element and greatest element and be a given operator on reversing the order such that for all . We consider the following subsets and of , and , so and are not empty (since and ). In what follows, let us say that the set a complete T-lattice if every subset of admits the greatest lower (resp. least upper) bound as soon as (resp. ), where designates the image of under the map for , and we denote by if for every , , we have for and be two subsets of .
2. Complete Sub-T-Lattice and Fixed Point Theorems
2.1. Complete Sub-T-Lattice
Inspired by the success of the complete lattice subset concept introduced by Birkhoff [3] in 1940, we propose a similar concept in the partially ordered sets. The aim of the concept is to structure the set of fixed points of an increasing application on an ordered set.
Let be a partially ordered set and be a given operator on reversing the order such that for all .
Let us denote by the restriction of the ordered relation of on with a subset of .
Definition 1. We say that is a complete sub-T-lattice of if admits a least and a greatest element and every party of exists as soon as (resp. ): (1)Every finite subset of the set ordered by the usual order relation is a sub-T-lattice with for all with is the least element of (2)We consider Figure 1: We takeand the following order relation ≤ defined by:for all,, andfor. Moreover, we have and where for ; for ; and for . for , Let , so () is a complete sub-T-lattice
Recall. Let us note that (resp. ) is equal to the greatest-lower (resp. least-upper) bound of in for every party of .
Proposition 2. Let be a complete sub-T-lattice with the least element and the greatest element. Then, for every party (resp. ) (resp. ) exists in (resp. in ).
Proof. Let be a nonempty party of . Let us define the set. We have , since because in this case , and for every , we have ; thus, for all which shows that . Consequently, exists since is a complete sub-T-lattice. As we have for all , therefore which implies . If was a party of , the proof of existence of in would be symmetric.
By this proposition, we can easily prove that each chain of a complete sub-T-lattice has the least upper bound and the greatest lower bound; indeed, we assume that is an increasing chain if ; then, the exists in by Proposition 2, but if , we have for all which shows that and since is a chain of a complete sub-T-lattice then exists. If is a decreasing sequence, the proof of existence of is symmetric.
Recall that a chain of is maximal if there is an element of comparable to every element of then .
In the following, we define and the for any called intervals.
If is a complete T-lattice; then, is a complete sub-T-lattice, and we can show that each interval of is a complete sub-T-lattice, indeed. We have for all , which implies that is the least element and is the greatest element of .
Let be a nonempty set such that . Since is a complete lattice, so exists in . We will only prove that exists in . Then, we have for all , which gives . Consequently, . The proof for the existence of the greatest-lower if follows identically.
In the same fashion, we can prove that is a complete sub-T-lattice.
Proposition 3. In a complete sub-T-lattice, any decreasing family of nonempty intervals has a nonempty intersection and it is an interval.
Proof. Let be a complete sub-T-lattice and a decreasing family of nonempty intervals in , so we have and . Consequently, is an increasing sequence, and is a decreasing sequence; furthermore, we have for all ; indeed, if and if . Therefore, since is a complete T-lattice which shows that and exist in .
For all , we have . Consequently,.
2.2. Fixed Point Theorems
The theory of fixed points is concerned with the conditions which guarantee that a map of a set into itself admits one or more fixed points, that there are pointsfor which.
Now, let be an ordered set and be a given operator on reversing the order such that or for all .
We take a part of and a monotone maps ; we note the set of fixed points of by .
Theorem 4. If is a complete sub-T-lattice with the least element and the greatest element , then, is a nonempty complete sub-T-lattice.
Indeed, we have ; this shows that is a nonempty set. Furthermore, we also have for any and for any chain of , we put ; for all , we have apply the mapping , that gives so . Therefore, satisfies the assumptions of Zorn’s lemma. Hence, for every , there exists a maximal element such that . We claim that if is a maximal element of , then which gives us which implies that . According to the maximality of , we have which proves that .
To show that is a complete sub-T-lattice, let It is clear that since . For any chain of , exists in , and we have for all which implies . We applyand we obtain; hence,that proves. Consequently, we have the existence of a maximal element of according to Zorn’s lemma. We claim that if is a maximal element of , then for all . In the same fashion, we applythat givesfor allthat implies. It is easy from here to see that in fact we have by the maximality of and for all that shows that is the least element of . The proof of existence of the greatest element of is symmetrical.
Now, let be a nonempty set such that . We want to find , the greatest-lower bound of in . Let us put . For all , , as . Thus . Therefore, the decreasing sequence tends to a fixed point of which will be the greatest lower bounds (or the greatest-lower bound) of in . The construction of is the least upper bound of in if is symmetric. Thereafter, is a complete sub-T-lattice.
3. Common Fixed Points for Commuting Family of Order-Preserving Mappings
3.1. Main Results
In this section, we give a generalization of Tarski’s theorem [4] that implies that any finite commuting family of order-preserving mappings has a common fixed point.
The existence of a common fixed point for a finite family of order-preserving applications is related to the intersection of a finite decreasing sequence of complete sub-T-lattices.
Theorem 5. Let be a complete T-lattice. Then, any finite commuting family of order-preserving mappings (monotone mappings) , , has a common fixed point. Moreover, if we denote by the set of the common fixed points, then is a complete sub-T-lattice of .
Proof. We take , where . Let be a complete T-lattice so the set of fixed points of is a complete sub-T-lattice by Theorem 4. As , we have that implies for all . Therefore we can restrict the maps to , and since is a complete sub-T-lattice, the set of fixed points of in is a complete sub-T-lattice. It is easy to see that the family such that for all with is a decreasing sequence.
Now, we consider the map
As the map is monotone and the set is a complete T-lattice, then there exists such that that means that that gives and . Consequently, .
It remains therefore to show that is a complete sub-T-lattice. For this, it is easy to see that which shows that is a complete T-lattice.
3.2. Common Fixed Points
In the following, we investigate the existence of a common fixed point of a commuting family of order-preserving mappings defined on a complete sub-T-lattice. The proof of our result follows the ideas of Baillon [5] developed in hyperconvex metric spaces and Abu-Sbeih and Khamsi [6] on the partially ordered sets. It is astonishing that we will develop their ideas in the case of the complete sub-T-trellis despite the difficulty of the demonstration which comes from the fact that in complete sub-T-trellis, we do not always have the existence of sup and inf of its parts.
We have the following result in partially ordered sets.
Let be a partially ordered set and be a given operator on reversing the order such that for all . Then,
Theorem 6. For any decreasing family of nonempty complete sub-T-lattice subsets of , where is a directed index set, we have is not empty and it is a complete sub-T-lattice.
Proof. Consider the family The set is not empty since . If we order by the inclusion relation, every chain of has a lower bound since in a complete sub-T-lattice, any decreasing family of nonempty intervals has a nonempty intersection and it is an interval by Proposition 3. Therefore, satisfies the assumptions of Zorn’s lemma. Hence, for every , there exists a minimal element such that . We claim that if is minimal, then each is a singleton. Indeed, let us fix . We know that . We consider the new family Our assumptions on and imply that . Moreover, we have for any . Since is minimal, we get for any . In particular, we have If , then we must have and . Therefore, we proved the existence of such that
It is easy from here to show that in fact we have by the minimality of , which proves our claim. Clearly, we have for any which implies is not empty.
We will prove that is a complete sub-T-lattice. First, we start by proving the existence of the least and the greatest element of for that we consider the set:
For the same reason above, we have: (1) is not empty since (2) satisfies the assumptions of Zorn’s lemma. Hence, for every , there exists a minimal element such that (3)If is a minimal element of , then for all according to the minimality of which gives us for all . Hence the result we are searching
Secondly, let be nonempty such that . We will only prove that the sup A exists in . The proof for the existence of the infimum follows identically if . For any , we have . Since is a complete sub-T-lattice, then exists in and the family is an increasing chain.
Now, we consider the set for . Then, is a nonempty complete sub-T-lattice of . It is easy to see that the family is a decreasing chain of complete sub-T-lattice. Hence, is a not empty interval and . Obviously, we have which completes the proof of Theorem 6.
As a consequence of this theorem, we obtain the following common fixed point result.
Theorem 7. If is a complete T-lattice, then any commuting family of order-preserving mappings , , has a common fixed point. Moreover, if we noted by the set of the common point fixed points, then is a complete sub-T-lattice of .
Proof. First, note that the fixed point theorem (Theorem2) implies that any finite commuting family of order-preserving mappings, has a common fixed point. Moreover, if we denote bythe set of the common fixed points, i.e.,it is a complete sub-T-lattice. Let . Clearly, is downward directed (where the order on is the set inclusion). For any , the set of common fixed point set of the mappings , , is a nonempty complete sub-T-lattice. Clearly, the family is decreasing. The theorem above implies that is nonempty and it is a complete sub-T-lattice.
The commutativity assumption may be relaxed using a new concept discovered in [6] (see also [7, 8]). Of course, this new concept was initially defined in the metric setting; therefore, we will extend it to the case of partially ordered sets in the next work where we will arrive at a result similar to De Marr’s result [9] without compactness assumption of the domain.
4. Conclusion
In this paper, we have extended some Tarski’s theorems of the fixed point into ordered sets by new fixed point theorems. The original proof of fixed point for complete T-lattice [2] is beautiful and elegant but nonconstructive and somewhat uninformative. In [3], we have given a constructive proof that generalizes the Tarski’s version results. In this paper, we have given a structure to the set of fixed points of an increasing application on an ordered set and we have investigated the existence of a common fixed point of a commuting family of order-preserving mappings defined on a complete sub-T-lattice. Our next work concerns the development of some fixed point theorems in hyperconvex metric spaces.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors were grateful to all the members of the Laboratory of Algebra and Analysis and Applications L3A.