#### Abstract

We apply the extensions of the Abian-Brown fixed point theorem for set-valued mappings on chain-complete posets to examine the existence of generalized and extended saddle points of bifunctions defined on posets. We also study the generalized and extended equilibrium problems and the solvability of ordered variational inequalities on posets, which are equipped with a partial order relation and have neither an algebraic structure nor a topological structure.

#### 1. Introduction

Let be a topological vector space and let be a subset of . Let be a real valued function defined on . A point is called a saddle point of the function if it satisfies A point is called an equilibrium of the function if it satisfies

In optimization theory, the standard tools for dealing with the existence of saddle points and equilibria of a bifunction are fixed point theorems with respect to set-valued mappings, Fan-KKM theorem, variational inequalities, or others, where the considered bifunction must hold a certain type of continuity properties and the underlying subset must satisfy some geometrical conditions, such as convexity. The problems and have been extensively studied by many authors in many fields, such as variational inequality theory, complementarity problem theory, fixed point theory, convex analysis with applications to economic theory, and game theory (see [16]).

The concepts defined in and have been generalized by Giannessi [7], in 1980, to the cases that the income spaces of mappings are finite-dimensional vector spaces and the outcome spaces are ordered vector spaces that are posets equipped with both topological structure and algebraic structure. The extended problems are called vector optimization problems and vector variational inequalities. Since the underlying spaces in these problems are still equipped with both topological structure and algebraic structure, then they may also be solved by the standard techniques as in solving problems and .

In economic theory and social sciences, there are some examples that both of the income and outcome spaces of mappings are posets, which are equipped with neither topological structure nor algebraic structure. Then, in these circumstances, the optimization problems will be order-optimization problems, which are not normal optimization problems (with respect to real valued functions) and they cannot be solved by using the standard methods. The saddle points and equilibria of bifunction must be generalized from real valued functions to functions with values in ordered sets. In this case, more fixed point theorems on ordered sets must be acquired and some new analysis techniques dealing with mappings on ordered sets must be developed.

In [8], several extensions of the Abian-Brown fixed point theorem provided in [9] on posets are obtained, which are extensions from single valued mappings to set-valued mappings. Moreover, the author examined some nonmonetized noncooperative games where both the collections of the strategies and the ranges of the utilities for the players are posets and proved some existence theorems of extended and generalized Nash equilibria for nonmonetized, noncooperative games on chain-complete posets by applying the extensions.

On the other hand, in [10], Xie et al. generalized the extensions of the Abian-Brown fixed point theorem provided in [8] from chain-complete posets to chain-complete preordered sets for set-valued mappings. By using these generalizations and by applying the order-increasing upward property of set-valued mappings, they also prove several existence theorems of the extended and generalized Nash equilibria of nonmonetized noncooperative games on chain-complete preordered sets.

References [8, 10] mainly considered the existence of the fixed point of set-valued mappings in chain-complete posets and chain-complete preordered sets, respectively. The applications of these fixed point theorems were considered less, although [8, 10] gave some examples of nonmonetized noncooperative games on posets and preordered sets. To remedy the defect, we apply the extensions of the Abian-Brown fixed point theorem provided in [9] on posets to examine the existence of generalized and extended saddle points of bifunctions on posets. We also study the solvability of generalized and extended equilibrium problems of bifunctions and ordered variational inequalities on posets, which have neither an algebraic structure nor a topological structure.

#### 2. Several Extensions of the Abian-Brown Fixed Point Theorem on Posets

The order-increasing property of mappings is important for the considered mappings to have a fixed point. In this section, we recall the notations of the order-increasing property of mappings, which are used in [2, 9, 11, 12].

Let , be posets and let be a set-valued mapping. is said to be isotone or to be order-increasing upward, if in ; then, for any , there is a such that . is said to be order-increasing downward, if in ; then, for any , there is a such that . If is both order-increasing upward and order-increasing downward, then is said to be order-increasing. As a special case, a single valued mapping from a poset to a poset is said to be order-increasing whenever implies .

##### 2.1. The Abian-Brown Fixed Point Theorem

Let be a chain-complete poset and let : be an order-increasing single valued mapping. If there is an in with , then has a fixed point.

Similar to the well-known Kakutani contribution that extended the Brouwer fixed point theorem from single valued mappings to set-valued mappings in topological spaces, the main results of [8] extended the Abian-Brown fixed point theorem from single valued mappings to set-valued mappings in ordered sets, which are also the generalization of the Fujimoto-Tarski fixed point theorem from complete lattices to chain-complete posets. We recall the extensions obtained in [8] below.

Theorem 1. Let be a chain-complete poset and let be a set-valued mapping. If satisfies the following three conditions,(A1)F is order-increasing upward;(A2)the set , for some is an inductively ordered set, for each ;(A3)there is a y in P with , for some ,then has a fixed point; that is, there exists such that .

Theorem 2. Let be a chain-complete poset and let be a set-valued mapping, which satisfies conditions (A1) and (A3) given in Theorem 1. If any one of the following properties holds, is inductive with finite number of maximal elements, for every ; has a maximum element, for every ; is a chain-complete lattice, for each ,then has a fixed point.

#### 3. Generalized and Extended Saddle Points of Bifunctions on Posets

The generalized saddle points of bifunctions on Banach lattices were studied in [13]. In this section, we extend this concept to posets.

Definition 3. Let and be posets. Let be the coordinate ordering relation on the Cartesian product induced by the partial orders and . That is, for any and ,

The proof of the following lemma is straight forward and is omitted.

Lemma 4. Let and be posets. The coordinate ordering on induced by the partial orders and defines a partial ordering relation on ; and hence is a poset. Furthermore, if and are both (conditionally) chain-complete (inductive, Dedekind complete) posets, then is also a (conditionally) chain-complete (inductive, Dedekind complete) poset.

Definition 5. Let , , and be posets. Let and be nonempty subsets of and , respectively. Let be a mapping. A point is called a generalized saddle point of the mapping if it satisfies

Definition 6. Let , , and be posets. Let and be nonempty subsets of and , respectively. Let be a mapping. is said to be(1)order-negative with respect to , whenever for any in implies , for any ;(2)order-positive with respect to , whenever for any in implies , for any .For a given mapping , one defines mappings and by We prove the following theorem for the existence of generalized saddle point.

Theorem 7. Let , , and be posets. Let and be nonempty chain-complete subsets of and , respectively. Let be a mapping. If satisfies the following conditions,(1) is order-negative with respect to ;(2) is order-positive with respect to ;(3)for every fixed is a nonempty inductive subset of with finite number of maximal elements;(4)For every fixed is a nonempty inductive subset of with finite number of maximal elements;(5)There is an element with , for some , and , for some ,then has a generalized saddle point.

Proof. From Lemma 4, is a chain-complete poset. Define by From conditions and , is well defined.
Next we show that is order-increasing upward. To this end, we show that, for any and , implies . For any given , we have and ; that is, From (4), we have , for all . Since , then, from condition in this theorem that is order-positive with respect to , it implies , for all .
That is, . We obtain . So From (5), similar to the proof of (6), for , we can show that Combining (6) and (7), we get That is, . It implies that is order-increasing upward.
We claim that an element is a maximal element of , if and only if is a maximal element of and is a maximal element of . In fact, if is a maximal element of and is a maximal element of , then for any , we have or and or , which imply or . So is a maximal element of . On the other hand, if is not a maximal element of or is not a maximal element of , say, is not a maximal element of , then there is with ; that is, . It implies that cannot be a maximal element of . Hence, from Lemma 4 and conditions and in this theorem, is an inductive poset with finite number of maximal elements.
From condition of this theorem, it is clearly seen that the element with , for some and and for some , satisfies Hence the mapping from to satisfies all conditions in Theorem 2 with respect to condition .So has a fixed point; say ; that is, . Then we have and. So It is equivalent to

Definition 8. Let and be posets. Let and be nonempty subsets of and , respectively. Let be a mapping. A point is called an extended saddle point of the mapping if it satisfies

Definition 9. Let and be posets. Let and be nonempty subsets of and , respectively. Let be a mapping. is said to be(1)order-nonnegative with respect to , whenever for any in implies , for any ;(2)order-nonpositive with respect to , whenever for any in implies , for any .For a given mapping , define and by

Theorem 10. Let , and be posets. Let and be nonempty chain-complete subsets of and , respectively. Let be a mapping. If satisfies the following conditions, (1) is order-nonnegative with respect to ;(2) is order-nonpositive with respect to ;(3)for every fixed is a nonempty inductive subset of with finite number of maximal elements;(4)for every fixed is a nonempty inductive subset of with finite number of maximal elements;(5)there is an element with , for some and and for some ,then has an extended addle point.

Proof. Define by From conditions and in this theorem, is well defined. Next we show that is order-increasing upward. To this end, we show that, for any implies . For any given , we have and ; that is, Since , then, from condition in this theorem that is order-nonpositive with respect to . (14) implies , for all . We obtain . So Similar to (16), for , from condition in this theorem, we can show that Combining (16) and (17), we get . That is, . It implies that is order-increasing upward.
From the proof of Theorem 7 and conditions and in this theorem, we have that, for any is an inductive poset with finite number of maximal elements. From condition in this theorem, it is clearly seen that the element with , for some and and for some , satisfies and . Hence the mapping satisfies all conditions in Theorem 2 with respect to condition . So has a fixed point; say ; that is, . Then we have . So That is

#### 4. Generalized and Extended Equilibrium Problems of Bifunctions on Posets

Definition 11. Let and be posets. Let be a nonempty subset of . Let be a mapping. A point is called a generalized equilibrium of the mapping if it satisfies For a given mapping , define by
We prove the following theorem for the existence of generalized equilibrium. The proof is similar to the proofs of Theorems 7 and 10.

Theorem 12. Let and be posets. Let be a nonempty chain-complete subset of . Let be a mapping. If satisfies the following conditions, (1) is order-negative with respect to ;(2)for every fixed is a nonempty inductive subset of with finite number of maximal elements;(3)there is an element with , for some ,then has a generalized equilibrium.

Proof. Taking the mapping defined by (20), From condition in this theorem, is well defined.
Next we show that is order-increasing upward. To this end, we show that, for any implies . For any given , we have That is, , for all . Since , then, from condition in this theorem that is order-negative with respect to , it implies , for all . That is, . We obtain . Hence we have . It implies that is order-increasing upward. From condition in this theorem, it is clearly seen that the element with , for some , satisfies and . Hence the mapping from to satisfies all conditions in Theorem 2 with respect to condition . So has a fixed point; say ; that is, . Then we have , which is equivalent to

Definition 13. Let and be posets. Let be a nonempty subset of . Let be a mapping. A point is called an extended equilibrium of the mapping if it satisfies
It is clear to see that any generalized equilibrium of a mapping is an extended equilibrium of this mapping. For a given mapping ,define by

Theorem 14. Let and be posets. Let be a nonempty chain-complete subset of . Let be a mapping. If satisfies the following conditions, (1) is order-nonnegative with respect to ;(2)for every fixed is a nonempty inductive subset of with finite number of maximal elements;(3)there is an element with , for some ,then has an extended equilibrium.

Proof. Taking the mapping defined by (24), From condition in this theorem, the mapping is well defined.
Next we show that is order-increasing upward. To this end, we show that, for any implies . For any given , we have , for all . That is, Since , then, from condition in this theorem, is order-nonnegative with respect to , and (26) implies , for all . That is, , for all . We obtain . So, we have . Then the rest of the proof is similar to the proof of Theorem 12.

#### 5. Generalized and Extended Ordered Variational Inequalities on Posets

Let and be posets. We denote for the collection of all mappings from .

Definition 15. Let and be posets and a nonempty subset of . Let be a mapping. A generalized ordered variational inequality problem associated with , , and , denoted by , is to find a point , such that Such a point is called a generalized solution to the problem GOVI (C, F, U).

Definition 16. Let and be posets and a nonempty subset of. Let be a mapping. is said to be order-negative on , whenever, for any in implies , for any.

Theorem 17. Let and be posets. Let be a nonempty chain-complete subset of . Let be a mapping. If F satisfies the following conditions,
(1) is order-negative on ;(2)for every fixed , the poset is a nonempty inductive subsetof with finite number of maximal elements;(3)there is an element with , for some satisfying ,then the problem GOVI (, , ) has a generalized solution.

Proof . Define a mapping by From condition in this theorem, is well defined. Next we show that is order-increasing upward. To this end, we show that, for any implies . For any given , we have . That is, Since , then, from condition in this theorem that is order-negative on , (29) implies , for all . That is, . We obtain . Hence we have . It implies that is order-increasing upward. From condition in this theorem, it is clearly seen that the element , for some , satisfies and . Hence the mapping from to satisfies all conditions in Theorem 2 with respect to condition . So has a fixed point; say ; that is, . We have . It is equivalent to

Definition 18. Let and be posets and a nonempty subset of . Let be a mapping. An extended ordered variational inequality problem associated with , , and , denoted by , is to find a point , such that Such a point is called an extended solution to the problem .

Definition 19. Let and be posets and a nonempty subset of . Let be a mapping. is said to be order-nonnegative on , whenever, for any in implies , for any .

Theorem 20. Let and be posets. Let be a nonempty chain-complete subset of . Let be a mapping. If satisfies the following conditions, (1)is order-nonnegative on ;(2)for every fixed , for all , is a nonempty inductive subset of with finite number of maximal elements;(3)there is an element with , for some satisfying , for all ,then the problem has an extended solution.

Proof. Define a mapping by From condition in this theorem, is well defined. Next we show that is order-increasing upward. To this end, we show that, for any implies For any given , we have , for all . That is, Since , then, from condition in this theorem that is order-nonnegative on , (32) implies , for all . That is, , for all . We obtain . Hence we obtain . Then the rest of the proof is similar to the proof of Theorem 17.

#### 6. Concluding Remarks

In Sections 3 to 5 the existence of generalized and extended saddle points and equilibriums and the solvability of ordered variational inequalities are proved by applying fixed point Theorems 2 listed in Section 2 with respect to condition . As some special cases, the results in Sections 3 to 5 can be obtained by applying Theorem 2 with respect to condition or .

It is clearly seen that, from different fixed point theorems, one can obtain various results for solving some optimization problems. Note that all extensions of the Abian-Brown fixed point theorem listed in Section 2 are considered with chain-complete posets. There are many examples in applications that the underlying spaces are not chain-complete. This aspect can be more precisely demonstrated by the following useful example.

For any positive integer , let denote the -dimensional poset where is the -dimensional Euclidean space equipped with the coordinate partial order , which is defined as follows: for any whenever , for . Then is not chain-complete, and it is conditionally chain-complete.

So if we are able to get some fixed point theorem on conditionally chain-complete posets, then we can study some optimization problems under more general underlying spaces: conditionally chain-complete posets, which include all problems studied in Sections 3 to 5 in this paper as special cases.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The authors greatly appreciate the anonymous referees’ comments and suggestions, which improved the presentation of this paper.