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Applications of orderclustered 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 orderclustered 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 orderclusters 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 singlevalued mappings, Tarski et al. provided a fixed point theorem on complete lattices and on chaincomplete posets (see [1]). For setvalued mappings, Fujimoto in [13] generalized Tarski’s fixed point theorem from singlevalued mappings to setvalued mappings on complete lattices. Very recently, in [6, 7], Li provided several versions of extensions of both the AbianBrown fixed point theorem and the FujimotoTarski fixed point theorem for setvalued mappings on chaincomplete posets. These fixed point theorems have been applied to solve vectorcomplementarity 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 orderclustered fixed point for ordercluster preserving mappings. They also provided some orderclustered 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 (P,\succcurlyeq ) be a chaincomplete preordered set and let F:P\to {2}^{P}\mathrm{\setminus}\{\mathrm{\varnothing}\} be a setvalued mapping. Suppose that F satisfies the following two conditions:
A1. F is orderincreasing upward.
A3. There is an element y in P with y\preccurlyeq u, for some u\in F(y).
Assume, in addition to conditions A1 and A3, F also satisfies one of the following conditions:
A2. \{z\in P:z\preccurlyeq u\text{for some}u\in F(x)\} is an inductive subset of P, for each x\in P.
A2′. F(x) is inductive with a finite number of maximal element clusters, for every x\in P.
A2″. F(x) has a maximum element, for every x\in P.
Then F has an orderclustered fixed point, that is, there are {x}^{\ast}\in P and w\in [{x}^{\ast}] with w\in F({x}^{\ast}).
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 Orderclustered fixed point theorems on preordered sets
In this section, we recall and provide some concepts and properties of preordered sets and the concept of orderclustered fixed point of setvalued 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:

1.
(reflexive) x\succcurlyeq x, for every x\in P.

2.
(transitive) x\succcurlyeq y and y\succcurlyeq z imply x\succcurlyeq z, for all x,y,z\in P.
Then P, together with the preorder ≽, is called a preordered set, which is denoted by (P,\succcurlyeq ). 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

3.
(antisymmetric) x\succcurlyeq y and y\succcurlyeq x imply x=y, for every x,y\in P.
In this case, (P,\succcurlyeq ) is simply called a poset.
Remark 2.1 It is worthy to mention for clarification that a preordered set (P,\succcurlyeq ) equipped with the preorder ≽ on P is called a partially ordered system (p.o.s.) in [15].
Let (P,\succcurlyeq ) 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 x\preccurlyeq u for each x\in A. 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 (P,\succcurlyeq ), then maxA and minA are subsets of A; and uppA, lowA, supA, and infA all are subsets of P. In particular, if (P,\succcurlyeq ) 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 y\in A is said to be a maximal element of a subset A of a preordered set (P,\succcurlyeq ) if for any z\in A with y\preccurlyeq z implies z\preccurlyeq y. 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 (P,\succcurlyeq ) is a maximal element (or minimum element) of A; the converse does not hold. That is,
A subset C of a preordered set (P,\succcurlyeq ) is said to be totally ordered, or a chain in P, whenever, for every pair x,y\in C, either x\preccurlyeq y or y\preccurlyeq x. Following [13] and [10], we have the following.
Definition 2.2 A nonempty subset A of a preordered set (P,\succcurlyeq ) is said to be

i.
inductive if every chain C in A possesses an upper bound in A, that is, \mathrm{\varnothing}\ne uppC\subseteq A;

ii.
chaincomplete whenever, every chain C in A has a supremum element in A; that is, \mathrm{\varnothing}\ne uppC\subseteq A.
Let (P,\succcurlyeq ) be a preordered set. For any z,w\in P, we denote the following ≽intervals:
Let (P,\succcurlyeq ) be a preordered set. For any x,y\in P, we say that x, y are ≽order equivalent (≽order indifference), which is denoted by x\sim y, whenever both x\succcurlyeq y and y\succcurlyeq x hold. It is clear that ∼ is an equivalence relation on P. For any x\in P, let [x] 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 P/\sim or \tilde{P} denote the collection of all order clusters in the preordered set (P,\succcurlyeq ). So x\in [x]\in \tilde{P}, for every x\in P.
In the case that z\sim w, we have [z,w]=[z], which is the order cluster containing z. It is worthy to note that both [z) and (w] can be considered as unions of order clusters.
Given two preordered sets (X,{\succcurlyeq}^{X}) and (U,{\succcurlyeq}^{U}), we say that a single valued mapping f:X\to U is orderincreasing (or orderpreserving) if x{\preccurlyeq}^{X}y in X implies f(x){\preccurlyeq}^{U}f(y) in U. An orderincreasing single valued mapping f:X\to U is said to be strictly orderincreasing whenever x{\prec}^{X}y implies f(x){\prec}^{U}f(x), where {\prec}^{X} and {\prec}^{U} are the strict parts of {\succcurlyeq}^{X} and {\succcurlyeq}^{U}, respectively.
We say that a setvalued mapping F:X\to {2}^{U}\mathrm{\setminus}\{\mathrm{\varnothing}\} is orderincreasing upward if x{\preccurlyeq}^{X}y in X and u\in F(x) imply that there is w\in F(y) such that u{\preccurlyeq}^{U}w. F is said to be orderincreasing downward if x{\preccurlyeq}^{X}y in X and w\in F(y) imply that there is u\in F(x) such that u{\preccurlyeq}^{U}w. A setvalued mapping F is said to be orderincreasing whenever F is both orderincreasing upward and downward.
Orderincreasing mappings from preordered set to preordered set have the following equivalent classes preserving properties (see [1]).
Lemma 2.3 Let (X,{\succcurlyeq}^{X}) and (U,{\succcurlyeq}^{U}) be two preordered sets and let f:X\to U be an orderincreasing single valued mapping. Then, for every x in X, y\in [x] implies f(y)\in [f(x)] in U.
Definition 2.4 Let (X,{\succcurlyeq}^{X}) and (U,{\succcurlyeq}^{U}) be two preordered sets and let f:X\to U be a single valued mapping. f is said to be orderclusterpreserving, whenever
Lemma 2.5 Every orderpreserving single valued mapping from a preordered set to another preordered set is clusterpreserving.
Definition 2.6 Let (X,{\succcurlyeq}^{X}) be a preordered set and let F:X\to {2}^{X}\mathrm{\setminus}\{\mathrm{\varnothing}\} be a setvalued mapping. An element x\in X is called

1.
a fixed point of F, whenever x\in F(x);

2.
an {\succcurlyeq}^{X}clustered fixed point (an orderclusteredfixed point, or simply a clustered fixed point) of F, whenever there is a w\in [x] such that w\in F(x).
Let (P,\succcurlyeq ) be a chaincomplete preordered set and let F:P\to {2}^{P}\mathrm{\setminus}\{\mathrm{\varnothing}\} be a setvalued 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 chaincomplete posets to chaincomplete preordered sets.
Theorem 2.7 Let (P,\succcurlyeq ) be a chaincomplete preordered set and let F:P\to {2}^{P}\mathrm{\setminus}\{\mathrm{\varnothing}\} be a setvalued mapping. Suppose that F satisfies the following two conditions:
A1. F is orderincreasing upward.
A3. There is an element y in P with y\preccurlyeq u, for some u\in F(y).
Assume, in addition to conditions A1 and A3, F also satisfies one of the following conditions:
A2. SF(x) is an inductive subset of P, for each x\in P.
A2′. F(x) is inductive with a finite number of maximal element clusters, for every x\in P.
A2″. F(x) has a maximum element, for every x\in P.
Then F has an orderclustered fixed point, that is, there are {x}^{\ast}\in P and w\in [{x}^{\ast}] with w\in F({x}^{\ast}).
Remark 2.8 If (P,\succcurlyeq ) is a chaincomplete preordered set satisfying \wedge P\ne \mathrm{\varnothing}, then condition A3 in Theorem 2.7 will be automatically satisfied.
Proof In the case that \wedge P\ne \mathrm{\varnothing}, then for any y\in \wedge P, we must have y\preccurlyeq u, for every u\in F(y). □
3 Generalized saddle point problems for bifunctions on preordered sets
Let (X,{\succcurlyeq}^{X}) and (Y,{\succcurlyeq}^{Y}) be preordered sets. The product set X\times Y of the preordered sets X and Y is ordered by the componentwise ordering, which is denoted by {\succcurlyeq}^{X\times Y}. That is, for any ({x}_{1},{y}_{1}) and ({x}_{2},{y}_{2})\in X\times Y
One can check that the coordinate ordering {\succcurlyeq}^{X\times Y} on X\times Y induced by the preorders {\succcurlyeq}^{X} and {\succcurlyeq}^{Y} defines a preordering relation on X\times Y; and hence (X\times Y,{\succcurlyeq}^{X\times Y}) is a preordered set. It has the following properties:

1.
If both (X,{\succcurlyeq}^{X}) and (Y,{\succcurlyeq}^{Y}) are chaincomplete (inductive) preordered sets, then (X\times Y,{\succcurlyeq}^{X\times Y}) is also a chaincomplete (inductive) preordered set.

2.
For any x\in X and y\in Y, [(x,y)]=[x]\times [y], here [(x,y)], [x], [y] are understood to be the order clusters in (X\times Y,{\succcurlyeq}^{X\times Y}), (X,{\succcurlyeq}^{X}) and (Y,{\succcurlyeq}^{Y}), respectively.

3.
For any (x,y),(s,t)\in X\times Y with (x,y){\preccurlyeq}^{Y}(s,t), then we have
((x,y)]=(x]\times (y],\phantom{\rule{1em}{0ex}}\phantom{\rule{1em}{0ex}}[(x,y))=[x)\times [y)\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}[(x,y),(s,t)]=[x,s]\times [y,t],
here, ((x,y)], (x], (y] are the order intervals in (X\times Y,{\succcurlyeq}^{X\times Y}), (X,{\succcurlyeq}^{X}) and (Y,{\succcurlyeq}^{Y}), respectively.
Definition 3.1 Let (X,{\succcurlyeq}^{X}), (Y,{\succcurlyeq}^{Y}) and (U;{\succcurlyeq}^{U}) be preordered sets. Let C, D be nonempty subsets of X and Y, respectively. Let F:C\times D\to U be a mapping. A point ({x}^{\ast},{y}^{\ast})\in C\times D is called a generalized saddle point of the mapping F if it satisfies
It is equivalent, for all x\in C and y\in D, to
for any u\in [F(x,{y}^{\ast})], {w}^{\ast}\in [F({x}^{\ast},{y}^{\ast})] and v\in [F({x}^{\ast},y)].
Definition 3.2 Let (X,{\succcurlyeq}^{X}), (Y,{\succcurlyeq}^{Y}) and (U;{\succcurlyeq}^{U}) be preordered sets. Let C, D be nonempty subsets of X and Y, respectively. Let F:C\times D\to U be a mapping. F is said to be

1.
orderphase preserving for the first variable on C, whenever, for any {x}_{2}{\succcurlyeq}^{X}{x}_{1} in C,
F({x}_{1},y){\preccurlyeq}^{U}F({x}_{1},t)\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}F({x}_{2},y){\preccurlyeq}^{U}F({x}_{2},t),\phantom{\rule{1em}{0ex}}\text{for any}y,t\in D; 
2.
orderphase preserving for the second variable on D, whenever, for any {y}_{2}{\succcurlyeq}^{Y}{y}_{1} in D,
F(x,{y}_{1}){\preccurlyeq}^{U}F(s,{y}_{1})\phantom{\rule{1em}{0ex}}\text{implies}\phantom{\rule{1em}{0ex}}F(x,{y}_{2}){\preccurlyeq}^{U}F(s,{y}_{2}),\phantom{\rule{1em}{0ex}}\text{for any}x,s\in C.
From Definition 2.4, if a mapping F from (C\times D,{\succcurlyeq}^{X\times Y}) to (U,{\succcurlyeq}^{U}) is ordercluster preserving, then it is ordercluster preserving for every single variable. That is,

1.
s\in [x] in C implies F(s,y)\in [F(x,y)] in U, for every given y\in D;

2.
t\in [y] in D implies F(x,t)\in [F(x,y)] in U, for every given x\in C.
Example 3.3 Let (R,\ge ) be the set of real numbers equipped with the ordinary real order ≥. Let F:R\times R\to R be a real function as
Then F is orderphase preserving for both variables on R.
Theorem 3.4 Let (X,{\succcurlyeq}^{X}), (Y,{\succcurlyeq}^{Y}) and (U;{\succcurlyeq}^{U}) be preordered sets. Let C, D be nonempty chaincomplete subsets of X and Y, respectively. Let F:C\times D\to U be a mapping and satisfy the following conditions:

1.
F(x,y) is orderphase preserving for variables x and y on C and D, respectively.

2.
For every fixed x\in C, \{t\in D:F(x,t)\in min\{F(x,y):y\in D\}\} is a nonempty inductive subset of D with a finite number of maximal elements.

3.
For every fixed y\in D, \{s\in C:F(s,y)\in max\{F(x,y):x\in C\}\} is a nonempty inductive subset of C with a finite number of maximal elements.

4.
There is an element ({x}^{\prime},{y}^{\prime})\in C\times D with {x}^{\prime}{\preccurlyeq}^{X}s, for some s\in C satisfying
F(s,{y}^{\prime})\in max\{F(x,{y}^{\prime}):x\in C\}
and {y}^{\prime}{\preccurlyeq}^{Y}t, for some t\in D satisfying

5.
F is ordercluster preserving on C\times D.
Then F has a generalized saddle point.
Proof For the given mapping F:C\times D\to U, we define mappings {\mathrm{\Upsilon}}_{F}:C\to {2}^{D} and {\mathrm{\Psi}}_{F}:D\to {2}^{C} by
It is worthy to note that since {\mathrm{\Upsilon}}_{F}(x)\subseteq D and {\mathrm{\Psi}}_{F}(y)\subseteq C, the above definitions are equivalent to
Define T:C\times D\to {2}^{C\times D}\mathrm{\setminus}\{\mathrm{\varnothing}\} by
From conditions 2 and 3, T is well defined.
At first, we show that, for any ({x}_{1},{y}_{1}),({x}_{2},{y}_{2})\in C\times D, ({x}_{2},{y}_{2}){\succcurlyeq}^{X\times Y}({x}_{1},{y}_{1}) implies T({x}_{1},{y}_{1})\subseteq T({x}_{2},{y}_{2}). For any given ({s}_{1},{t}_{1})\in T({x}_{1},{y}_{1}), we need to prove ({s}_{1},{t}_{1})\in T({x}_{2},{y}_{2}). To this end, from that ({s}_{1},{t}_{1})\in T({x}_{1},{y}_{1}) is equivalent to {s}_{1}\in {\mathrm{\Psi}}_{F}({y}_{1}) and {t}_{1}\in {\mathrm{\Upsilon}}_{F}({x}_{1}), we have
and
From (1), we have F(x,{y}_{1}){\preccurlyeq}^{U}F({s}_{1},{y}_{1}), for all x\in C. Since {y}_{2}{\succcurlyeq}^{Y}{y}_{1}, then from condition 1 in this theorem that F(x,y) is orderphase preserving for variable y on D, it implies F(x,{y}_{2}){\preccurlyeq}^{U}F({s}_{1},{y}_{2}), for all x\in C. That is, F({s}_{1},{y}_{2})\in max\{F(x,{y}_{2}):x\in C\}; and therefore, {s}_{1}\in {\mathrm{\Psi}}_{F}({y}_{2}). So we obtain
On the other hand, from (2), we have F({x}_{1},{t}_{1}){\preccurlyeq}^{U}F({x}_{1},y), for all y\in D. Since {x}_{2}{\succcurlyeq}^{X}{x}_{1}, then from condition 1 in this theorem that F(x,y) is orderphase preserving for the variable x on C, it implies F({x}_{2},{t}_{1}){\preccurlyeq}^{U}F({x}_{2},y), for all y\in D. That is, F({x}_{2},{t}_{1})\in min\{F({x}_{1},y):y\in D\}; and therefore, {t}_{1}\in {\mathrm{\Upsilon}}_{F}({x}_{2}). So we obtain
Combining (3) and (4), we get
That is, T({x}_{1},{y}_{1})\subseteq T({x}_{2},{y}_{2}). It clearly implies that T is orderincreasing upward.
We claim that an element (p,q) is a maximal element of {\mathrm{\Psi}}_{F}(y)\times {\mathrm{\Upsilon}}_{F}(x), if and only if p is a maximal element of {\mathrm{\Psi}}_{F}(y) and q is a maximal element of {\mathrm{\Upsilon}}_{F}(x). In fact, if p is a maximal element of {\mathrm{\Psi}}_{F}(y) and q is a maximal element of {\mathrm{\Upsilon}}_{F}(x), then, for any (s,t)\in {\mathrm{\Psi}}_{F}(y)\times {\mathrm{\Upsilon}}_{F}(x), we find that s{\succcurlyeq}^{X}p implies p{\succcurlyeq}^{X}s and t{\succcurlyeq}^{Y}q implies q{\succcurlyeq}^{Y}t. Then we find that (s,t){\succcurlyeq}^{X\times Y}(p,q) implies (p,q){\succcurlyeq}^{X\times Y}(s,t). So (p,q) is a maximal element of {\mathrm{\Psi}}_{F}(y)\times {\mathrm{\Upsilon}}_{F}(x).
On the other hand, if p is not a maximal element of {\mathrm{\Psi}}_{F}(y) or q is not a maximal element of {\mathrm{\Upsilon}}_{F}(x), say, p is not a maximal element of {\mathrm{\Psi}}_{F}(y), then there is s\in {\mathrm{\Psi}}_{F}(y) with s{\succ}^{X}p, that is, (s,q){\succ}^{X\times Y}(p,q). It implies that (p,q) cannot be a maximal element of {\mathrm{\Psi}}_{F}(y)\times {\mathrm{\Upsilon}}_{F}(x). Hence, from conditions 3 and 4 in this theorem, (T(x,y),{\succcurlyeq}^{X\times Y}) is an inductive preordered set with a finite number of maximal elements.
Let ({x}^{\prime},{y}^{\prime}) be the element in C\times D given in condition 4 of this theorem. Then there are points s\in {\mathrm{\Psi}}_{F}({y}^{\prime}) and t\in {\mathrm{\Upsilon}}_{F}({x}^{\prime}) satisfying {x}^{\prime}{\preccurlyeq}^{X}s, and {y}^{\prime}{\preccurlyeq}^{Y}t; that is,
Hence the mapping T from C\times D to {2}^{C\times D}\mathrm{\setminus}\{\mathrm{\varnothing}\} satisfies all conditions in Theorem 2.7 with respect to condition A2′. As we apply the mapping T to the preordered set (C\times D,{\succcurlyeq}^{X\times Y}), which is a chain complete preordered set, from Theorem 2.7, T has an orderclustered fixed point, say ({x}^{\ast},{y}^{\ast})\in C\times D. Let {[({x}^{\ast},{y}^{\ast})]}^{C\times D} denote the ordercluster in (C\times D,{\succcurlyeq}^{X\times Y}) containing the point ({x}^{\ast},{y}^{\ast}). So there is an element (u,v)\in {[({x}^{\ast},{y}^{\ast})]}^{C\times D}\subseteq C\times D such that (u,v)\in T({x}^{\ast},{y}^{\ast}). Then we have
It next will be shown that {\mathrm{\Psi}}_{F} and {\mathrm{\Upsilon}}_{F} both are ordercluster preserving on D and C, respectively. In fact, from condition 1, F(x,y) is orderphase preserving on C.
From u\in {\mathrm{\Psi}}_{F}({y}^{\ast}), we get F(u,{y}^{\ast})\in max\{F(x,{y}^{\ast}):x\in C\}, that is,
From condition 5, F is ordercluster preserving. From (u,v)\in [({x}^{\ast},{y}^{\ast})], it implies F(u,{y}^{\ast})\in [F({x}^{\ast},{y}^{\ast})]. By (5), we have
From v\in {\mathrm{\Upsilon}}_{F}({x}^{\ast}), we have F({x}^{\ast},v)\in min\{F({x}^{\ast},y):y\in D\}, that is,
Since F is ordercluster preserving, from (u,v)\in [({x}^{\ast},{y}^{\ast})], it implies F({x}^{\ast},v)\in [F({x}^{\ast},{y}^{\ast})]. From (7), we have
By combining (6) and (8), we get
Hence ({x}^{\ast},{y}^{\ast}) is a generalized saddle point of F on C\times D. □
If we apply the two mappings {\mathrm{\Upsilon}}_{F} and {\mathrm{\Psi}}_{F} for a mapping F, defined in the proof of Theorem 3.4, and apply their orderincreasing properties, we can get the following theorem.
Theorem 3.5 Let (X,{\succcurlyeq}^{X}), (Y,{\succcurlyeq}^{Y}) and (U;{\succcurlyeq}^{U}) be preordered sets. Let C, D be nonempty chaincomplete subsets of X and Y, respectively. Let F:C\times D\to U be a mapping and satisfy the following conditions:

1.
{\mathrm{\Upsilon}}_{F}:C\to {2}^{D} and {\mathrm{\Psi}}_{F}:D\to {2}^{C} both are orderincreasing upward.

2.
For every fixed x\in C, ({\mathrm{\Upsilon}}_{F}(x),{\succcurlyeq}^{Y}) is a nonempty inductive subset of D with a finite number of maximal elements.

3.
For every fixed y\in D, ({\mathrm{\Psi}}_{F}(y),{\succcurlyeq}^{X}) is a nonempty inductive subset of C with a finite number of maximal elements.

4.
There is an element ({x}^{\prime},{y}^{\prime})\in C\times D with {x}^{\prime}{\preccurlyeq}^{X}s, for some s\in {\mathrm{\Psi}}_{F}({y}^{\prime}) and {y}^{\prime}{\preccurlyeq}^{Y}t, for some t\in {\mathrm{\Upsilon}}_{F}({x}^{\prime}).

5.
F is ordercluster preserving on C\times D.
Then F has a generalized saddle point.
Sketch of the proof Let T:C\times D\to {2}^{C\times D}\mathrm{\setminus}\{\mathrm{\varnothing}\} be defined as in the proof of Theorem 3.4:
For any ({x}_{1},{y}_{1}),({x}_{2},{y}_{2})\in C\times D with ({x}_{2},{y}_{2}){\succcurlyeq}^{X\times Y}({x}_{1},{y}_{1}), take an arbitrary element ({s}_{1},{t}_{1})\in T({x}_{1},{y}_{1}), we have {s}_{1}\in {\mathrm{\Psi}}_{F}({y}_{1}) and {t}_{1}\in {\mathrm{\Upsilon}}_{F}({x}_{1}). Since {y}_{2}{\succcurlyeq}^{Y}{y}_{1} and {x}_{2}{\succcurlyeq}^{X}{x}_{1}, from condition 1 in this theorem, there are {s}_{2}\in {\mathrm{\Psi}}_{F}({y}_{2}) and {t}_{2}\in {\mathrm{\Upsilon}}_{F}({x}_{2}) such that {s}_{2}{\succcurlyeq}^{X}{s}_{1} and {t}_{2}{\succcurlyeq}^{Y}{t}_{1}. That is ({s}_{2},{t}_{2})\in C\times D satisfying ({s}_{2},{t}_{2}){\succcurlyeq}^{X\times Y}({s}_{1},{t}_{1}) and ({s}_{2},{t}_{2})\in T({x}_{2},{y}_{2}). Hence T is an {\succcurlyeq}^{X\times Y}increasing mapping from C\times D to {2}^{C\times D}\mathrm{\setminus}\{\mathrm{\varnothing}\}. 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 (X,{\succcurlyeq}^{X}) and (U;{\succcurlyeq}^{U}) be preordered sets. Let C be a nonempty subset of X. Let F:C\times C\to U be a mapping. A point {x}^{\ast}\in C is called a generalized equilibrium of the mapping F if it satisfies
It is equivalent to the following, for all x\in C:
for any u\in [F({x}^{\ast},x)] and {w}^{\ast}\in [F({x}^{\ast},{x}^{\ast})].
For a given mapping F:C\times C\to U, define a mapping \mathrm{\Gamma}:C\to {2}^{C} 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 (X,{\succcurlyeq}^{X}) and (U;{\succcurlyeq}^{U}) be preordered sets. Let C be a nonempty chaincomplete subset of X. Let F:C\times C\to U be a mapping and satisfy the following conditions:

1.
F(\cdot ,y) is orderphase preserving on C, for every y\in C.

2.
For every fixed x\in C, ({\mathrm{\Gamma}}_{F}(x),{\succcurlyeq}^{X}) is a nonempty inductive subset of C with a finite number of maximal elements.

3.
There is an element {x}^{\prime}\in C with {x}^{\prime}{\preccurlyeq}^{X}s, for some s\in {\mathrm{\Gamma}}_{F}({x}^{\prime}).

4.
F is ordercluster preserving on C\times C.
Then F has a generalized equilibrium in C.
Proof For the given mapping F, by condition 2 in this theorem, the mapping {\mathrm{\Gamma}}_{F}:C\to {2}^{C}\mathrm{\setminus}\{\mathrm{\varnothing}\} is well defined. It will next be shown that {\mathrm{\Gamma}}_{F} is orderincreasing upward. For any {x}_{1},{x}_{2}\in C with {x}_{2}{\succcurlyeq}^{X}{x}_{1}, for arbitrary {t}_{1}\in {\mathrm{\Gamma}}_{F}({x}_{1}), we have
That is, F({x}_{1},y){\preccurlyeq}^{U}F({x}_{1},{t}_{1}), for all y\in D. Since {x}_{2}{\succcurlyeq}^{X}{x}_{1}, then from condition 1 in this theorem we see that F(x,y) is orderphase preserving with respect to x\in C, it implies F({x}_{2},y){\preccurlyeq}^{U}F({x}_{2},{t}_{1}), for all y\in C. That is, F({x}_{2},{t}_{1})\in max\{F({x}_{2},y):y\in C\}. We obtain {t}_{1}\in {\mathrm{\Gamma}}_{F}({x}_{2}). It implies that {\mathrm{\Gamma}}_{F} is orderincreasing upward. From condition 3 in this theorem, it is clearly seen that the elements {x}^{\prime}\in C, s\in {\mathrm{\Gamma}}_{F}({x}^{\prime}) satisfy {x}^{\prime}\preccurlyeq s. Hence the mapping {\mathrm{\Gamma}}_{F} from C to {2}^{C}\mathrm{\setminus}\{\mathrm{\varnothing}\} satisfies all conditions in Theorem 2.7 with respect to condition A2′, as we consider the mapping is from the chaincomplete preordered set (C,{\succcurlyeq}^{X}) to {2}^{C}\mathrm{\setminus}\{\mathrm{\varnothing}\}. We denote by {[{x}^{\ast}]}_{C} the order cluster in (C,{\succcurlyeq}^{X}) containing {x}^{\ast}. So {\mathrm{\Gamma}}_{F} has an orderclustered fixed point, say {x}^{\ast}\in C. So there is u\in {[{x}^{\ast}]}_{C}\subseteq C, such that u\in {\mathrm{\Gamma}}_{F}({x}^{\ast}). Then we have
which is equivalent to
Since F is order cluster preserving, from u\in {[{x}^{\ast}]}_{C}, it implies F({x}^{\ast},u)\in [F({x}^{\ast},{x}^{\ast})]. By (9), we have
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5 Orderclustered fixed point theorems and generalized preorderedvariational 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 preorderedvariational 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 (P,\succcurlyeq ) be a preordered set equipped with a topology τ (it is also a topological space). The topology τ is called a natural topology on (P,\succcurlyeq ) with respect to the preorder ≽, whenever, for every z\in P, the ≽intervals [z) and (z] are all τclosed. A preordered set (P,\succcurlyeq ) equipped with a natural topology τ is called a preordered topological space, and it is denoted by (P,\tau ,\succcurlyeq ).
A preorder ≽ on a vector space P is said to be convex, whenever for any z\in P, the ≽intervals [z) and (z] are all convex subsets in P.
We say that a Banach space (B,\parallel \cdot \parallel ) 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 (B,\parallel \cdot \parallel ,\succcurlyeq ) 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 (B,\parallel \cdot \parallel ) be a Banach space with norm \parallel \cdot \parallel. 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 chaincompleteness 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 chaincomplete. 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 chaincomplete.
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 (B,\succcurlyeq ) be a convex preordered reflexive Banach space. Then, for any nonempty bounded closed and convex subset P of B, (P,\succcurlyeq ) is a chaincomplete preordered set.
Applying Theorems 2.7, 5.1 and 5.2, we obtain the following result.
Theorem 5.4 Let (B,\parallel \cdot \parallel ,\succcurlyeq ) be a convex preordered reflexive Banach space and let P be a nonempty bounded closed convex subset of B. Let F:P\to {2}^{P}\mathrm{\setminus}\{\mathrm{\varnothing}\} be a setvalued mapping. Suppose that F satisfies the following two conditions:
A1. F is orderincreasing upward.
A3. There is an element y in P with y\preccurlyeq u, for some u\in F(y).
Assume, in addition conditions A1 and A3, F also satisfies one of the conditions below:
A2. SF(x) is an inductive subset of P, for each x\in P.
A2′. F(x) is inductive with a finite number of maximal element clusters, for every x\in P.
A2″. F(x) has a maximum element, for every x\in P.
Then F has an orderclustered fixed point, that is, there are {x}^{\ast}\in P and w\in [{x}^{\ast}]\cap P with w\in F({x}^{\ast}).
Let (B,{\succcurlyeq}^{B}) and (U;{\succcurlyeq}^{U}) be Banach spaces. As usual, we denote by L(B,U) the collection of all linear mappings from B to U.
Definition 5.5 Let (B,{\succcurlyeq}^{B}) and (U;{\succcurlyeq}^{U}) be preordered Banach spaces and C a nonempty subset of B. Let F:C\to L(B,U) be a mapping. A generalized preorderedvariational inequality problem associated with C, F, and U, denoted by GPVI(F,C,U), is to find a point {x}^{\ast}\in C, such that
where 0 is understood as the origin of the Banach space U. Any such point {x}^{\ast}\in C is called a solution to the problem GPVI(F,C,U).
Notice that F({x}^{\ast})\in L(B,U), then (10) is equivalent to
Definition 5.6 Let (B,{\succcurlyeq}^{B}) and (U;{\succcurlyeq}^{U}) be preordered Banach spaces, and C a nonempty subset of B. A mapping F:C\to L(B,U) is said to be orderphase preserving on C, whenever for any {x}_{2}{\succcurlyeq}^{B}{x}_{1} in C, F({x}_{1})(y){\preccurlyeq}^{U} 0 implies F({x}_{2})(y){\preccurlyeq}^{U}0, for any given y\in B.
Theorem 5.7 Let (B,{\succcurlyeq}^{B}) be a convex preordered reflexive Banach space and (U;{\succcurlyeq}^{U}) a preordered Banach space. Let C be a nonempty bounded closed convex subset of B. Let F:C\to L(B,U) be a mapping and satisfy the following conditions:

1.
F(x):B\to U is preorderphase preserving on C, for every x\in C.

2.
For every fixed x\in C, the set (\{t\in C:F(x)(t)\in min\{F(x)(y):y\in C\}\},{\succcurlyeq}^{B}) is a nonempty inductive subset of C with a finite number of maximal order clusters.

3.
There are elements {x}^{\prime}, s\in C with {x}^{\prime}{\preccurlyeq}^{B}s, such that F({x}^{\prime})(s)\in min\{F({x}^{\prime})(y):y\in C\}.

4.
F(x):B\to U is ordercluster preserving, for every x\in C.
Then the problem GPVI(F,C,U) has a solution.
Proof Since (B,{\succcurlyeq}^{B}) is a convex preordered reflexive Banach space and C is a nonempty bounded closed convex subset of B, from Corollary 5.3, (C,{\succcurlyeq}^{B}) is chaincomplete. Define a mapping {\mathrm{\Lambda}}_{F} by
From condition 2 in this theorem, {\mathrm{\Lambda}}_{F}:C\to {2}^{C}\mathrm{\setminus}\{\mathrm{\varnothing}\} is well defined. Similarly to the proof of Theorem 4.2, we can show that the mapping {\mathrm{\Lambda}}_{F}, from C to {2}^{C}\mathrm{\setminus}\{\mathrm{\varnothing}\}, satisfies all conditions in Theorem 5.4 with respect to condition A2′. So {\mathrm{\Lambda}}_{F} has an orderclustered fixed point, say {x}^{\ast}\in C. Then there is u\in [{x}^{\ast}]\cap C, such that u\in {\mathrm{\Lambda}}_{F}({x}^{\ast}). Then we have
which is equivalent to
Since F is ordercluster preserving, from u\in [{x}^{\ast}]\cap C, it implies F({x}^{\ast})(u)\in [F({x}^{\ast})({x}^{\ast})]. By (12), we have
Hence the point {x}^{\ast} 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 orderclustered fixed point theorems to generalized saddle point problems and preordered variational inequalities. Fixed Point Theory Appl 2014, 191 (2014). https://doi.org/10.1186/168718122014191
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DOI: https://doi.org/10.1186/168718122014191