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Applications of order-clustered fixed point theorems to generalized saddle point problems and preordered variational inequalities
Fixed Point Theory and Applications volume 2014, Article number: 191 (2014)
Abstract
In this paper, we introduce the concepts of preordered Banach spaces, generalized saddle points and preordered variational inequalities. Then we apply the order-clustered fixed point theorems to prove the existence of solutions to these problems.
MSC:06A06, 47B48, 47B60, 49J40.
1 Introduction
In an ordinary game, the sets of strategies of the players or the decision makers may be equipped with a preorder, and the decision makers may consider having indifference (same) utilities at order equivalent elements. It is the motivation to investigate the properties of order equivalent elements and order equivalent classes in a preordered set, which are called order-clusters in [1].
Fixed point theory has played important roles in traditional equilibrium theory, variational inequalities and optimization theory (see [2–12]). Fixed point theorems have been applied in the proofs of the existence of solutions in equilibrium problems, variational inequalities, and other problems, in which the underlying spaces are topological spaces and the considered mappings must satisfy some continuity conditions. One is to extend these techniques from traditional variational analysis to analysis on ordered sets, in which there are order structures which may not be topological structures. Hence some new fixed point theorems on ordered sets must be developed.
For single-valued mappings, Tarski et al. provided a fixed point theorem on complete lattices and on chain-complete posets (see [1]). For set-valued mappings, Fujimoto in [13] generalized Tarski’s fixed point theorem from single-valued mappings to set-valued mappings on complete lattices. Very recently, in [6, 7], Li provided several versions of extensions of both the Abian-Brown fixed point theorem and the Fujimoto-Tarski fixed point theorem for set-valued mappings on chain-complete posets. These fixed point theorems have been applied to solve vector-complementarity problems and generalized Nash equilibrium problems.
To generalize the fixed point theorems from posets to preordered sets, Xie et al. in [1] introduced the concepts of order cluster and order-clustered fixed point for order-cluster preserving mappings. They also provided some order-clustered fixed point theorems on preordered sets, which have been applied to solve some generalized equilibrium problems for some strategic games with preordered preferences. We recall the fixed point theorem obtained in [1].
Let be a chain-complete preordered set and let be a set-valued mapping. Suppose that F satisfies the following two conditions:
A1. F is order-increasing upward.
A3. There is an element y in P with , for some .
Assume, in addition to conditions A1 and A3, F also satisfies one of the following conditions:
A2. is an inductive subset of P, for each .
A2′. is inductive with a finite number of maximal element clusters, for every .
A2″. has a maximum element, for every .
Then F has an order-clustered fixed point, that is, there are and with .
In this paper, we apply the above fixed point theorem to solve some generalized variational inequalities, in which the ranges of the considered mappings and the underlying spaces both are preordered sets. We also solve generalized saddle point problems and generalized equilibrium problems in preordered sets.
2 Order-clustered fixed point theorems on preordered sets
In this section, we recall and provide some concepts and properties of preordered sets and the concept of order-clustered fixed point of set-valued mapping on preordered sets, which is introduced in [1] by Xie et al. Here we closely follow the notations from [2, 7, 14–16] and [1].
Let P be a nonempty set. An ordering relation ≽ on P is said to be a preorder, whenever it satisfies the following two conditions:
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(reflexive) , for every .
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(transitive) and imply , for all .
Then P, together with the preorder ≽, is called a preordered set, which is denoted by . Furthermore, a preorder ≽ on a preordered set P is said to be a partial order if, in addition to the above two properties, it also satisfies
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3.
(antisymmetric) and imply , for every .
In this case, is simply called a poset.
Remark 2.1 It is worthy to mention for clarification that a preordered set equipped with the preorder ≽ on P is called a partially ordered system (p.o.s.) in [15].
Let be a preordered set and let A be a nonempty subset of P. Then an element u of P is called an upper bound of A if for each . The collection of all upper bounds of A is denoted by uppA. That is,
If an upper bound u of A is in A, then u is called a greatest element (or a maximum element) of A. The collection of all greatest elements (maximum elements) of A is denoted by maxA, that is,
A lower bound of A can be similarly defined and the collection of all lower bounds of A is written as
The collection of all smallest elements (minimum elements) of A is denoted by minA. That is,
If the set of all upper bounds of A has a smallest element, then we call it a supremum of A; and the collection of all supremum elements of A is denoted by supA or ∨A. That is,
An infimum of A is similarly defined as a greatest element of the set of all lower bounds of A, provided that it exists; and the collection of all infimum elements of A is denoted by infA or ∧A. We have
It is important to note that, from the above definitions, if A is a subset of a preordered set , then maxA and minA are subsets of A; and uppA, lowA, supA, and infA all are subsets of P. In particular, if is a poset, then maxA, minA, supA, and infA all are singletons, provided that they are nonempty. In this case, maxA, minA, supA, and infA are simply written by the contained elements, respectively.
An element is said to be a maximal element of a subset A of a preordered set if for any with implies . The collection of all maximal elements of A is denoted by MaxA, that is,
Similar to the definition of maximal element, a minimal element of A can be defined. The collection of all minimal elements of A is written as
Then every greatest element (or smallest element) of a subset A of a preordered set is a maximal element (or minimum element) of A; the converse does not hold. That is,
A subset C of a preordered set is said to be totally ordered, or a chain in P, whenever, for every pair , either or . Following [13] and [10], we have the following.
Definition 2.2 A nonempty subset A of a preordered set is said to be
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i.
inductive if every chain C in A possesses an upper bound in A, that is, ;
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ii.
chain-complete whenever, every chain C in A has a supremum element in A; that is, .
Let be a preordered set. For any , we denote the following ≽-intervals:
Let be a preordered set. For any , we say that x, y are ≽-order equivalent (≽-order indifference), which is denoted by , whenever both and hold. It is clear that ∼ is an equivalence relation on P. For any , let denote the order equivalent class (order indifference class) containing x, which is called a ≽-cluster (or simply an order cluster, or a cluster, if there is no confusion caused). Let or denote the collection of all order clusters in the preordered set . So , for every .
In the case that , we have , which is the order cluster containing z. It is worthy to note that both and can be considered as unions of order clusters.
Given two preordered sets and , we say that a single valued mapping is order-increasing (or order-preserving) if in X implies in U. An order-increasing single valued mapping is said to be strictly order-increasing whenever implies , where and are the strict parts of and , respectively.
We say that a set-valued mapping is order-increasing upward if in X and imply that there is such that . F is said to be order-increasing downward if in X and imply that there is such that . A set-valued mapping F is said to be order-increasing whenever F is both order-increasing upward and downward.
Order-increasing mappings from preordered set to preordered set have the following equivalent classes preserving properties (see [1]).
Lemma 2.3 Let and be two preordered sets and let be an order-increasing single valued mapping. Then, for every x in X, implies in U.
Definition 2.4 Let and be two preordered sets and let be a single valued mapping. f is said to be order-cluster-preserving, whenever
Lemma 2.5 Every order-preserving single valued mapping from a preordered set to another preordered set is cluster-preserving.
Definition 2.6 Let be a preordered set and let be a set-valued mapping. An element is called
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1.
a fixed point of F, whenever ;
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an -clustered fixed point (an order-clustered-fixed point, or simply a clustered fixed point) of F, whenever there is a such that .
Let be a chain-complete preordered set and let be a set-valued mapping. Following Fujimoto [13], we use the notation
Xie et al. in [1] proved the following results, which are extensions of the theorem proved by Li in [7] from chain-complete posets to chain-complete preordered sets.
Theorem 2.7 Let be a chain-complete preordered set and let be a set-valued mapping. Suppose that F satisfies the following two conditions:
A1. F is order-increasing upward.
A3. There is an element y in P with , for some .
Assume, in addition to conditions A1 and A3, F also satisfies one of the following conditions:
A2. is an inductive subset of P, for each .
A2′. is inductive with a finite number of maximal element clusters, for every .
A2″. has a maximum element, for every .
Then F has an order-clustered fixed point, that is, there are and with .
Remark 2.8 If is a chain-complete preordered set satisfying , then condition A3 in Theorem 2.7 will be automatically satisfied.
Proof In the case that , then for any , we must have , for every . □
3 Generalized saddle point problems for bifunctions on preordered sets
Let and be preordered sets. The product set of the preordered sets X and Y is ordered by the component-wise ordering, which is denoted by . That is, for any and
One can check that the coordinate ordering on induced by the preorders and defines a preordering relation on ; and hence is a preordered set. It has the following properties:
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If both and are chain-complete (inductive) preordered sets, then is also a chain-complete (inductive) preordered set.
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For any and , , here , , are understood to be the order clusters in , and , respectively.
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For any with , then we have
here, , , are the order intervals in , and , respectively.
Definition 3.1 Let , and be preordered sets. Let C, D be nonempty subsets of X and Y, respectively. Let be a mapping. A point is called a generalized saddle point of the mapping F if it satisfies
It is equivalent, for all and , to
for any , and .
Definition 3.2 Let , and be preordered sets. Let C, D be nonempty subsets of X and Y, respectively. Let be a mapping. F is said to be
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order-phase preserving for the first variable on C, whenever, for any in C,
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order-phase preserving for the second variable on D, whenever, for any in D,
From Definition 2.4, if a mapping F from to is order-cluster preserving, then it is order-cluster preserving for every single variable. That is,
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in C implies in U, for every given ;
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in D implies in U, for every given .
Example 3.3 Let be the set of real numbers equipped with the ordinary real order ≥. Let be a real function as
Then F is order-phase preserving for both variables on R.
Theorem 3.4 Let , and be preordered sets. Let C, D be nonempty chain-complete subsets of X and Y, respectively. Let be a mapping and satisfy the following conditions:
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is order-phase preserving for variables x and y on C and D, respectively.
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For every fixed , is a nonempty inductive subset of D with a finite number of maximal elements.
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For every fixed , is a nonempty inductive subset of C with a finite number of maximal elements.
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There is an element with , for some satisfying
and , for some satisfying
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F is order-cluster preserving on .
Then F has a generalized saddle point.
Proof For the given mapping , we define mappings and by
It is worthy to note that since and , the above definitions are equivalent to
Define by
From conditions 2 and 3, T is well defined.
At first, we show that, for any , implies . For any given , we need to prove . To this end, from that is equivalent to and , we have
and
From (1), we have , for all . Since , then from condition 1 in this theorem that is order-phase preserving for variable y on D, it implies , for all . That is, ; and therefore, . So we obtain
On the other hand, from (2), we have , for all . Since , then from condition 1 in this theorem that is order-phase preserving for the variable x on C, it implies , for all . That is, ; and therefore, . So we obtain
Combining (3) and (4), we get
That is, . It clearly implies that T is order-increasing upward.
We claim that an element is a maximal element of , if and only if p is a maximal element of and q is a maximal element of . In fact, if p is a maximal element of and q is a maximal element of , then, for any , we find that implies and implies . Then we find that implies . So is a maximal element of .
On the other hand, if p is not a maximal element of or q is not a maximal element of , say, p is not a maximal element of , then there is with , that is, . It implies that cannot be a maximal element of . Hence, from conditions 3 and 4 in this theorem, is an inductive preordered set with a finite number of maximal elements.
Let be the element in given in condition 4 of this theorem. Then there are points and satisfying , and ; that is,
Hence the mapping T from to satisfies all conditions in Theorem 2.7 with respect to condition A2′. As we apply the mapping T to the preordered set , which is a chain complete preordered set, from Theorem 2.7, T has an order-clustered fixed point, say . Let denote the order-cluster in containing the point . So there is an element such that . Then we have
It next will be shown that and both are order-cluster preserving on D and C, respectively. In fact, from condition 1, is order-phase preserving on C.
From , we get , that is,
From condition 5, F is order-cluster preserving. From , it implies . By (5), we have
From , we have , that is,
Since F is order-cluster preserving, from , it implies . From (7), we have
By combining (6) and (8), we get
Hence is a generalized saddle point of F on . □
If we apply the two mappings and for a mapping F, defined in the proof of Theorem 3.4, and apply their order-increasing properties, we can get the following theorem.
Theorem 3.5 Let , and be preordered sets. Let C, D be nonempty chain-complete subsets of X and Y, respectively. Let be a mapping and satisfy the following conditions:
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and both are order-increasing upward.
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For every fixed , is a nonempty inductive subset of D with a finite number of maximal elements.
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For every fixed , is a nonempty inductive subset of C with a finite number of maximal elements.
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There is an element with , for some and , for some .
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F is order-cluster preserving on .
Then F has a generalized saddle point.
Sketch of the proof Let be defined as in the proof of Theorem 3.4:
For any with , take an arbitrary element , we have and . Since and , from condition 1 in this theorem, there are and such that and . That is satisfying and . Hence T is an -increasing mapping from to . The rest of the proof is similar to the proof of Theorem 3.4. □
4 Generalized equilibrium problems of bifunctions on preordered sets
Definition 4.1 Let and be preordered sets. Let C be a nonempty subset of X. Let be a mapping. A point is called a generalized equilibrium of the mapping F if it satisfies
It is equivalent to the following, for all :
for any and .
For a given mapping , define a mapping by
We prove the following theorem for the existence of generalized equilibrium. The proof is similar to the proofs of Theorems 3.4 and 3.5.
Theorem 4.2 Let and be preordered sets. Let C be a nonempty chain-complete subset of X. Let be a mapping and satisfy the following conditions:
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is order-phase preserving on C, for every .
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For every fixed , is a nonempty inductive subset of C with a finite number of maximal elements.
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There is an element with , for some .
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F is order-cluster preserving on .
Then F has a generalized equilibrium in C.
Proof For the given mapping F, by condition 2 in this theorem, the mapping is well defined. It will next be shown that is order-increasing upward. For any with , for arbitrary , we have
That is, , for all . Since , then from condition 1 in this theorem we see that is order-phase preserving with respect to , it implies , for all . That is, . We obtain . It implies that is order-increasing upward. From condition 3 in this theorem, it is clearly seen that the elements , satisfy . Hence the mapping from C to satisfies all conditions in Theorem 2.7 with respect to condition A2′, as we consider the mapping is from the chain-complete preordered set to . We denote by the order cluster in containing . So has an order-clustered fixed point, say . So there is , such that . Then we have
which is equivalent to
Since F is order cluster preserving, from , it implies . By (9), we have
□
5 Order-clustered fixed point theorems and generalized preordered-variational inequalities
In this section, we consider preordered topological spaces and their properties. We also give some fixed point theorems on preordered topological spaces, which will be applied to solving generalized preordered-variational inequalities in preordered Banach spaces. The notations and terminologies used in this section are derived from Aliprantis and Burkinshaw [14], Carl and Heikkilä [2], Li [7], Ok [16], Xie et al. [1].
Let be a preordered set equipped with a topology τ (it is also a topological space). The topology τ is called a natural topology on with respect to the preorder ≽, whenever, for every , the ≽-intervals and are all τ-closed. A preordered set equipped with a natural topology τ is called a preordered topological space, and it is denoted by .
A preorder ≽ on a vector space P is said to be convex, whenever for any , the ≽-intervals and are all convex subsets in P.
We say that a Banach space endowed with a preorder ≽ is a preordered Banach space if its norm topology is a natural topology with respect to the preorder ≽ on B. A preordered Banach space is called a convex preordered Banach space if the preorder ≽ is convex on B.
Recall that every convex subset in a Banach space is closed in the norm topology if and only if it is closed in the weak topology. Then we have the following result.
Theorem 5.1 Let be a Banach space with norm . Let ≽ be a convex preorder on B. Then, with respect to the convex preorder ≽, the norm topology on B is a natural topology, if and only if the weak topology on B is a natural topology.
The connections between the chain-completeness and compactness of subsets in ordered topological spaces are very important topics in the theory of ordered topological spaces. In [6], Li proved that every nonempty compact subset of a partially ordered Hausdorff topological space is chain-complete. In this paper, we extend it to preordered Hausdorff topological spaces. The proof is similar to the proof of Theorem 2.3 in [6] and it is omitted here.
Theorem 5.2 Every nonempty compact subset of a preordered Hausdorff topological space is chain-complete.
Notice that nonempty bounded closed and convex subset of a reflexive Banach space is compact in the weak topology. From Theorems 5.1 and 5.2, we have the following.
Corollary 5.3 Let be a convex preordered reflexive Banach space. Then, for any nonempty bounded closed and convex subset P of B, is a chain-complete preordered set.
Applying Theorems 2.7, 5.1 and 5.2, we obtain the following result.
Theorem 5.4 Let be a convex preordered reflexive Banach space and let P be a nonempty bounded closed convex subset of B. Let be a set-valued mapping. Suppose that F satisfies the following two conditions:
A1. F is order-increasing upward.
A3. There is an element y in P with , for some .
Assume, in addition conditions A1 and A3, F also satisfies one of the conditions below:
A2. is an inductive subset of P, for each .
A2′. is inductive with a finite number of maximal element clusters, for every .
A2″. has a maximum element, for every .
Then F has an order-clustered fixed point, that is, there are and with .
Let and be Banach spaces. As usual, we denote by the collection of all linear mappings from B to U.
Definition 5.5 Let and be preordered Banach spaces and C a nonempty subset of B. Let be a mapping. A generalized preordered-variational inequality problem associated with C, F, and U, denoted by , is to find a point , such that
where 0 is understood as the origin of the Banach space U. Any such point is called a solution to the problem .
Notice that , then (10) is equivalent to
Definition 5.6 Let and be preordered Banach spaces, and C a nonempty subset of B. A mapping is said to be order-phase preserving on C, whenever for any in C, 0 implies , for any given .
Theorem 5.7 Let be a convex preordered reflexive Banach space and a preordered Banach space. Let C be a nonempty bounded closed convex subset of B. Let be a mapping and satisfy the following conditions:
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is preorder-phase preserving on C, for every .
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For every fixed , the set is a nonempty inductive subset of C with a finite number of maximal order clusters.
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There are elements , with , such that .
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is order-cluster preserving, for every .
Then the problem has a solution.
Proof Since is a convex preordered reflexive Banach space and C is a nonempty bounded closed convex subset of B, from Corollary 5.3, is chain-complete. Define a mapping by
From condition 2 in this theorem, is well defined. Similarly to the proof of Theorem 4.2, we can show that the mapping , from C to , satisfies all conditions in Theorem 5.4 with respect to condition A2′. So has an order-clustered fixed point, say . Then there is , such that . Then we have
which is equivalent to
Since F is order-cluster preserving, from , it implies . By (12), we have
Hence the point satisfies (11). It is equivalent to
□
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Acknowledgements
The authors sincerely thank the anonymous reviewers for their valuable suggestions, which improved the presentation of this paper. The first author was partially supported by the National Natural Science Foundation of China (11171137). The third author was partially supported by the National Natural Science Foundation of China (Grant 11101379).
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Xie, L., Li, J. & Yang, W. Applications of order-clustered fixed point theorems to generalized saddle point problems and preordered variational inequalities. Fixed Point Theory Appl 2014, 191 (2014). https://doi.org/10.1186/1687-1812-2014-191
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DOI: https://doi.org/10.1186/1687-1812-2014-191