#### Abstract

We prove that Fan’s theorem is true for discontinuous increasing mappings in a real partially ordered reflexive, strictly convex, and smooth Banach space . The main tools of analysis are the variational characterizations of the generalized projection operator and order-theoretic fixed point theory. Moreover, we get some properties of the generalized projection operator in Banach spaces. As applications of our best approximation theorems, the fixed point theorems for non-self-maps are established and proved under some conditions. Our results are generalizations and improvements of the recent results obtained by many authors.

#### 1. Introduction

Let be a real Banach space with the dual space and a nonempty subset of . The set-valued mapping , is called the metric projection operator from onto . It is well known that the metric projection operator plays an important role in nonlinear functional analysis, optimization theory, fixed point theory, nonlinear programming, game theory, variational inequality, complementarity problems, and so forth.

In 1994, Alber [1] introduced the generalized projections and from Hilbert spaces to uniformly convex and uniformly smooth Banach spaces and studied their properties in detail. In [2], Li extended the generalized projection operator from uniformly convex and uniformly smooth Banach spaces to reflexive Banach spaces and studied some properties of the generalized projection operator with applications to solving the variational inequality in Banach spaces. Recently, Isac [3] and Nishimura and Ok [4] studied the order-theoretic approach towards establishing the solvability of variational inequality on a Hilbert lattice which is based on the fact that the metric projection operator is order-preserving if only if is a sublattice of . Very recently, Li and Ok [5] obtained the generalized projection operator is order-preserving in partially ordered Banach spaces.

Motivated and inspired by the above mentioned work, in this paper, we get the continuous property of generalized projection operator and increasing characterizations of in a partially ordered reflexive, strict convex, and smooth Banach space. Further, we consider the following Fan’s approximation theorem (Theorem 2 in [6]) through the variational characterization of . The normed space version of the theorem is as follows.

Theorem 1. *Let be a nonempty compact convex set in a normed linear space . If is a continuous map from into , then there exists a point in such that**The point is called a best approximation point of in .*

Fan’s theorem has been of great importance in nonlinear analysis, approximation theory, game theory, and minimax theorems. Various aspects of this theorem have been studied by many authors under different assumptions. For some related works, refer to [7–21] and the references therein.

In this paper, we obtain the existence of minimum best approximation point and maximum best approximation point in order interval. As an applications of our best approximation theorems, the fixed point theorems for non-self-maps are established under some conditions which do not need to require any continuous and compact conditions on .

The content of the present work can be summarized as follows. In Section 2, we review the definition of the generalized projection operator in Banach spaces and its basic properties. We also show some definitions in the partially ordered Banach space and some fundamental results for our theorems. In Section 3, we obtain the properties of the generalized projection operator in the partially ordered Banach space under some assumption. And we combine these results with an order-theoretic fixed point theorem to provide some of the best approximation theorems. Section 4 provides an application of these best approximation theorems to fixed point theory.

#### 2. Preliminaries

##### 2.1. The Partial Order

Suppose that is a real Banach space and is a nonempty closed convex cone of . By we denote the zero element of . We define a partial order with respect to by if and only if . We will write if and .

The cone is called normal if there is a number , such that for all implies . The cone is called regular if every increasing sequence which is bounded from above is convergent. That is, if is a sequence such that for some , then there is such that . Equivalently, the cone is regular if and only if every decreasing sequence which is bounded from below is convergent. It has been proved in Theorem 1.2.1 in [22] that every regular cone is normal.

A cone is called minihedral, if each two-element set has a least upper bound . Equivalently, the cone is minihedral if and only if each two-element set has a greatest lower bound . As is convenient, we denote as and as . And if exists for every nonempty and bounded from above , we say the cone is a strongly minihedral cone. If is a nonempty subset of which contains and for every , then is said to be subminihedral.

Let be a real partially ordered Banach space. Given such that , the set is called ordered interval. If the cone is minihedral, it is easy to see that is a subminihedral set of .

*Definition 2 (see [5]). *For any partially ordered spaces and , we say that a map is order-preserving if

*Definition 3 (see [23]). *Let be a partially ordered space and is convex; we say that a map is convex if

##### 2.2. Order-Dual

Let be a real partially ordered Banach space whose (topological) dual we denote by and a cone in . Recall that is called the dual cone of . The dual of is the partial order on defined as follows: If is a minihedral cone, it is well known that is a minihedral cone in . We now show that if and only if for every (see [24, Proposition 1.4.2]).

We denote by a Hilbert space whose norm satisfies where is defined by for each .

##### 2.3. The Generalized Projection Operator

Let be a real Banach space with the dual . We denote by the normalized duality mapping from to defined by for all , where denotes the generalized duality pairing between and . See [1] for basic characterizations of the normalized duality mapping.

Recall that a Banach space has the Kadec-Klee property, if for any sequence and with (weak convergence) and , then , as . It is well known that if is a uniformly convex Banach space, then has the Kadec-Klee property.

Let be a reflexive, strictly convex, and smooth Banach space and a nonempty closed convex subset of . Consider the Lyapunov functional defined by Following Alber [1], the generalized projection is a map that assigns to an arbitrary point the minimum point of the functional ; that is, , where is the solution to the minimization problem: existence and uniqueness of the operator follow from the properties of the functional and strict monotonicity of the mapping . It is obvious from the definition of functional that If is a Hilbert space, then and .

If is a reflexive, strictly convex, and smooth Banach space, then for , if and only if . It is sufficient to show that if then . From (10), we have . This implies that . From the definition of , one has . Therefore, we have . See [25, 26] for more details.

In [1], the generalized projection operators on arbitrary convex closed sets satisfy the following property.

The point is a generalized projection of on if and only if the following inequality is satisfied:

We denote , where and is Lyapunov functional in .

#### 3. Best Approximation Theorems

First we give the following properties of the generalized projection operators.

Lemma 4 (see [27]). *Let be a real partially ordered reflexive, strictly convex, and smooth Banach space with respect to a minihedral cone . Suppose is the dual cone of . The following statements are equivalent:**()**the normalized duality mapping is order-preserving;**()*

Lemma 5 (see [27]). *Let be a real partially ordered reflexive, strictly convex, and smooth Banach space with respect to a minihedral cone and satisfy condition (). Suppose that is closed convex subminihedral set of . Moreover, satisfies the condition:**()**Then, is increasing.*

*Remark 6. *The minihedral cones of many Banach spaces satisfy (). For example, if , every subminihedral set of (here partial order is defined coordinatewise) such that , then satisfies (); if , every subminihedral set of (here stands again for the coordinatewise ordering), such that , then satisfies (). See [5] for more details.

Lemma 7. *If is a uniformly convex and smooth Banach space and is a nonempty, closed, and convex subset of , then the generalized projection operator is continuous.*

*Proof. *Since is a uniformly convex and smooth Banach space, is single valued. Suppose , as , and suppose , and . From the inequalities and the hypothesis that , as , it yields is a bounded subset of . Since is reflexive, there exists a subsequence of ; without loss of the generality, we may assume it is itself, such that converges weakly to . From the properties of weakly convergence, we have . Moreover, and , which implies , as . Now we have Thus we have .

For any , one has . From the inequality , we have Therefore, Similar to the above argument, from inequality , we obtainAdding the above two inequalities side by side, we obtainSo If we use the inequalities and , similar to the above argument, we obtain In (18) and (19), taking , we have From the conditions that , as and is a smooth Banach space, we have , as . Using , as and combining (20), it yields , as . Since is a uniformly convex Banach space, then has the Kadec-Klee property. Therefore, we obtain , as . Thus this lemma is proved.

Lemma 8. *Let be a real partially ordered reflexive, strictly convex, and smooth Banach space with respect to and satisfy condition (). Suppose that is a minihedral cone and satisfies the condition:**()**Then, is increasing, and .*

*Proof. *Since () implies () and is subminihedral, from Lemma 5, is increasing. Next, we prove . To derive a contradiction, assume that there exists which does not satisfy ; that is, and . Then we have that is, Hence, As , we have and then,Since , from (23) and (25), we have And hence This contradicts (). Thus, . And hence, As is increasing, we have The assertion is proved.

Lemma 9. *Let be a real partially ordered reflexive, strictly convex, and smooth Banach space with respect to a minihedral cone and satisfy condition (). Suppose with and the following condition is satisfied:**()**Then, is increasing, and *

*Proof. *Following a similar argument as in the proof of Lemma 8, we obtain that is increasing and . And hence, As is increasing and , we have The proof is completed.

*Remark 10. *If is a partially ordered Hilbert space with respect to and a minihedral cone, () and () are satisfied.

From the above properties of the generalized projection operators and order-theoretic fixed point theorems, we can obtain the following best approximation theorems.

Theorem 11. *Let be a real partially ordered uniformly convex and smooth Banach space with respect to a minihedral cone and satisfy condition (). Suppose that is an increasing map. Moreover, satisfies the condition () and is relatively compact. Then, has a minimum best approximation point and a maximum best approximation point with respect to in , such that **where *

*Proof. *Define by . From Lemma 5, we get is increasing. It is easy to see and By Lemma 7, we know is continuous and is relatively compact. Thus satisfies all conditions of Theorem 2.1.4 in [22]. Then, has a minimum fixed point and a maximum fixed point and satisfies (32). Now we consider , ; that is, and . By the definition of , we get The assertion is proved.

Theorem 12. *Let be a real partially ordered reflexive, strictly convex, and smooth Banach space with respect to a normal and minihedral cone and satisfy condition (). Suppose that is an increasing map. Moreover, satisfies the condition (). Then, has a minimum best approximation point and a maximum best approximation point with respect to in . Moreover, if , (32) holds.*

*Proof. *Define by . From Lemma 5, we get is increasing. It is easy to see and Since is reflexive and is normal, is regular. Thus satisfies all conditions of Theorem 3.1.4 in [23]. Then, has a minimum fixed point and a maximum fixed point and satisfies (32). By the definition of , the assertion is proved.

*Remark 13. *In the above Theorem 11, is discontinuous map. And in Theorem 12, is discontinuous map and has no compact conditions.

*Example 14. *Let . Here stands for the coordinatewise ordering. It is easy to prove that all conditions in Theorem 12 hold. Given such that . Then, every increasing has a minimum best approximation point and a maximum best approximation point with respect to in .

Theorem 15. *Let be a real partially ordered reflexive, strictly convex, and smooth Banach space with respect to . If and the following conditions are satisfied,*(i)* is a normal, minihedral cone with satisfying () and ();*(ii)* is an increasing and convex map;*(iii)*there exists a such that ,**then, has a unique approximation point with respect to in . Moreover, if we take for , **where has nothing to do with .*

*Proof. *Define by . Since is convex and is increasing, for , we have Using Lemma 9 and , we obtain Thus is convex. And . Thus satisfies all conditions of Theorem 3.1.6 in [23]. Then, has a unique fixed point and satisfies (35). By the definition of , the assertion is proved.

#### 4. Fixed Point Theorems

In this section, we will prove some new fixed point theorems for non-self-maps by using results of Section 3.

Theorem 16. *Let be a real partially ordered uniformly convex and smooth Banach space with respect to a minihedral cone and satisfy condition (). Suppose that is an increasing map and is relative compact. Moreover, satisfies the condition () and **Then, has at least one fixed point in .*

*Proof. *By Theorem 11, has at least one best approximation point in ; that is, . From (11), we have We may use (38) to find a such that , and hence that is, Moreover, So we conclude that . It follows that . Moreover, as , and the inequality above must hold as an equality. We have . Therefore, . And thus The assertion is proved.

Following a similar argument as in the proof of Theorem 16, we can obtain the following fixed point theorems.

Theorem 17. *Let be a real partially ordered uniformly convex and smooth Banach space with respect to and satisfy condition (). Suppose that is a normal, minihedral cone and is an increasing map. Moreover, satisfies the condition () and (38). Then, has at least one fixed point in .*

*Example 18. *Let , the space of measurable functions which are the 2nd power summable on . Endow with the following norm and the cone : Given such that . It is easy to see that satisfies () and () holds in . Thus, by Theorem 17, every increasing satisfying (38) has at least one fixed point in .

Theorem 19. *Let be a real partially ordered reflexive, strictly convex, and smooth Banach space with respect to . If and the following conditions are satisfied,*(i)* is a normal, minihedral cone with satisfying (), () and (38);*(ii)* is an increasing and convex map;*(iii)*there exists such that ,**then, has a unique fixed point in . Moreover, if we take for ,**where has nothing to do with .*

#### Conflict of Interests

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

#### Acknowledgments

Dezhou Kong and Lishan Liu were supported financially by the National Natural Science Foundation of China (11371221), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001), and the Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province. Yonghong Wu was supported financially by the Australian Research Council through an ARC Discovery Project Grant.