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Abstract and Applied Analysis

Volume 2012 (2012), Article ID 310369, 19 pages

http://dx.doi.org/10.1155/2012/310369

## Vectorial Ekeland Variational Principles and Inclusion Problems in Cone Quasi-Uniform Spaces

^{1}School of Mathematical Science, Xuzhou Normal University, Xuzhou 221116, China^{2}Department of Mathematics and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea^{3}Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China

Received 2 December 2011; Revised 23 February 2012; Accepted 5 March 2012

Academic Editor: D. O'Regan

Copyright © 2012 Jiang Zhu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Some new vectorial Ekeland variational principles in cone quasi-uniform spaces are proved. Some new equivalent principles, vectorial quasivariational inclusion principle, vectorial quasi-optimization principle, vectorial quasiequilibrium principle are obtained. Also, several other important principles in nonlinear analysis are extended to cone quasi-uniform spaces. The results of this paper extend, generalize, and improve the corresponding results for Ekeland's variational principles of the directed vectorial perturbation type and other generalizations of Ekeland's variational principles in the setting of -type topological space and quasi-metric spaces in the literatures. Even in usual real metric spaces, some of our results are new.

#### 1. Introduction

Ekeland’s variational principle [1] is a forceful tool in nonlinear analysis, control theory, global analysis, and many others. In the last two decades, it has been further studied, extended, and applied to many fields in mathematics (see, e.g., [2–17, 19–21, 26, 28, 29] and the references therein). Especially, we want to emphasize that many generalizations of Ekeland’s variational principle to vector-valued functions have been recently obtained in [2–5, 8–10, 14–17] and the references therein. For example, in [2], the author proved some vectorial Ekeland’s variational principles for vector-valued functions defined on quasi-metric spaces. In [3, 5, 14], the authors proved some vectorial Ekeland’s variational principles for vector-valued functions defined on metric spaces. In [8], the authors proved some vector Ekeland’s variational principle in a -type topological space. However, all of these results are essentially Ekeland’s principle of the directed vectorial perturbation type. But, the study for the case of nondirected vectorial perturbation type has just been started in [15]. On the other hand, variational inclusion problem is a very important problem and it contains many important problems such as complementarity problems, minimax inequalities, equilibrium problems, saddle point problems, optimization theory, bilevel problems, mathematical programs with equilibrium constraints, variational inequalities, and fixed point problems (see, e.g., [18–21] and the references therein).

In this paper, we first prove some nondirected vectorial perturbation type Ekeland’s variational principles in cone quasi-uniform spaces, then by using our new principles, we introduce some variational inclusion principles and prove the equivalence among these results and other equivalents of our principle.

The paper is organized as follows: in Section 2, some properties of cone, some new equivalent characterizations of the pseudo-nuclear cone, and the definition of cone quasi-uniform spaces are given. In Section 3, we prove some new Ekeland’s variational principles for both non-directed vectorial perturbation type and directed vectorial perturbation type in the setting of general topological vector spaces for vector-valued functions defined on complete cone quasi-uniform spaces. Some of new equivalent principles, vectorial quasi-variational inclusion principle, vectorial quasi-optimization principle, and vectorial quasi-equilibrium principle are introduced and proved, which have a wide practical background in quasi-variational inclusion problems, quasi-optimization problems, and quasi-equilibrium problems (see, e.g., [21]). In addition to these new equivalent principles, generalized Caristi-Kirk type coincidence point theorem for multivalued maps defined on cone quasi-uniform spaces, nonconvex maximal element theorem for the family of multivalued maps defined on cone quasi-uniform spaces, generalized vectorial Takahashi nonconvex minimization theorem, and Oettli-Théra type theorem defined on cone quasi-uniform spaces are also presented. The results of this paper not only give some new Ekeland’s principles of the non-directed vectorial perturbation type and some new equivalent principles, but also extend and generalize or improve many corresponding results for Ekeland’s variational principles of the directed vectorial perturbation type and other generalizations of Ekeland’s variational principles in the setting of -type topological space and quasi-metric spaces in the literatures [2, 3, 5, 8, 14, 15]. Even in usual real metric spaces, some of our results are new.

#### 2. Preliminaries

Let be a topological vector space and (resp., ) denote the -neighborhood base with respect to the topology (resp., to the weak topology ). A subset is called a closed convex cone if is closed, and for all . A quasiorder (i.e., a reflexive and transitive relation) on can be defined by , that is, if and only if . We write whenever and . The quasi-order ≤ is called a partial order if it is antisymmetric. Let be the topological dual space of (i.e., is the set of all continuous linear functions on ) and let be the dual cone of , that is, .

We recall that a subset is said to be *-saturated* if , where is defined by
where . The cone is said to be *normal* if there is a -neighborhood base for consisting of -saturated sets.

The following Lemma 2.1 is an elementary result (refer to [22–24]) characterizing the concept of normality that will be used in our proof.

Lemma 2.1. *Let be a topological vector space and be a cone. Then the following propositions are equivalent.*(1)* is a normal cone.*(2)*There exists a -neighborhood base for consisting of sets for which implies .*(3)*For any two nets and in , if for all and converges to zero for , then converges to zero for .*(4)*For any -neighborhood of zero, there exists a -neighborhood of zero such that implies .*

The following notion of a pseudo-nuclear cone is a generalization of nuclear cone which has many applications in optimizations, fixed point theory, and the best approximation theory, and so forth.

*Definition 2.2 (see [25]). *Let be a topological vector space (not necessarily locally convex) and be a closed convex cone (not necessary pointed). If , then we denote . A set of the kind is called a *-*pseudoslice of , where , the dual space of . We say that is pseudo-nuclear if each -neighborhood in contains a -pseudoslice of .

To discuss the properties of pseudo-nuclear cone, we need the following existence theorem of a quasi-norms family which can determine the topology of .

Lemma 2.3 (see [26]). *Let be a topological vector space and be a balanced -neighborhood base for the topology . Then there exist a directed set and a quasi-norms family such that*(a)*for any , and , ;*(b)*for any , there exists such that and for any ;*(c)*for any , if , then for any ;*(d)* and determine the same topology ;*(e)*if is a topology ., for any , there exists a neighborhood of zero such that ), then*

The following lemma gives several equivalent conditions of pseudo-nuclearity.

Lemma 2.4. *Let be a topological vector space and a cone. Then the following assertions are equivalent.*(1)* is a pseudo-nuclear cone.*(2)*For all , there exists such that implies .*(3)*For all , there exists such that .*(4)*If is the family of quasi-norms which determines the topology of , then, for any , there exists such that*

*Proof. *It follows from Theorem 3.8 in [24] that (1) *⇒* (2).

(2) *⇒* (3) Since each neighborhood of zero with respect to the weak topology is also a neighborhood of zero with respect to the topology , for any , we have . From (2), we know that there exists such that implies . This and (4) of Lemma 2.1 imply that is a normal cone with respect to the weak topology . Since the weak topology is defined by semi-norms family , is a locally convex topological vector space. Therefore, with respect to the weak topology . From (2), it follows that, for any , there exists such that implies . Thus there exist and such that . Also, there exist such that for each . Let . Then and
Thus , that is, (3) holds.

(3) *⇒* (4) Suppose that is the family of *quasi-norms* which determines the topology of . Then, for any , . It follows from (3) that, for any , there exists such that . If and , then , that is, . If and , then, for any , , which implies that . Since is arbitrary, we know that and so for all , that is, (4) is true.

(4) *⇒* (1) By Lemma 2.3(d), and determine the same topology . Then, for any , there exist and positive numbers such that . By (4), there exists such that for all . Let and . Then and
This shows that (1) holds. This completes the proof.

To discuss Ekeland’s principles of the nondirected vectorial perturbation type, by enlightening of the work in [27], we define the following cone quasi-uniform space, which is an extension of the cone uniform space in [27]. About the discussion and applications of cone uniform space, one can refer to [15, 27–29].

*Definition 2.5. *Let be a nonempty set and be a directed set. Let be a topological vector space and be a cone. Let the family
satisfy the following conditions:(d1) for any , for all ;(d2)for any , if and only if ;(d3)for any , there exists with such that
Then is called a *family of cone *quasi-metrics on and is called a *cone *quasi-uniform* space*.

*Definition 2.6. *Let be a sequence in a cone quasi-uniform space .(1)The sequence is said to be convergent to a point if, for any and -neighborhood in , there exists a positive integer such that for any , that is, , which is denoted by .(2)The sequence is called a Cauchy sequence in if, for any and -neighborhood in , there exists a positive integer such that for any , that is, .(3)If every Cauchy sequence is convergent in , then is called a sequentially complete cone quasi-uniform space.(4)For any net in a cone quasi-uniform space , the convergence of the net and the Cauchy net can be defined similarly. If every Cauchy net is convergent in , then is called a complete cone quasi-uniform space.

*Remark 2.7. *If we take , then the cone quasi-uniform space is a generalization of -type topological spaces [30]. If , then reduces to a cone quasi-metric space . The notions on convergence and completeness in a cone quasi-uniform space are complicated because of the lack of symmetry of the cone quasi-metric. Similarly, we can define the other kinds of convergence and completeness in a cone quasi-uniform space. For more details, the reader can refer to same discussions in the case of quasi-metric spaces [31].

Recall that a cone has a *base* [8] or is *well based* [11] if there exists a convex subset with such that .

Lemma 2.8 (see [8]). *Let be a topological vector space and be a closed convex cone. Then the cone has a base if and only if
*

*Definition 2.9 (see [16]). *Let be a quasiordered set.(1) is said to be *totally ordered upperseparable* (resp., *totally ordered lower-separable*) if, for any totally ordered nonempty subset of , there exists an increasing sequence (resp., decreasing sequence) such that, for any , there exists satisfying (resp., ).(2) is said to be *totally ordered separable* if is both totally ordered upperseparable and totally ordered lower-separable.

Lemma 2.10 (see [16]). *Let be a partially ordered set such that every increasing sequence has an upper bound, let be a totally ordered upper-separable (resp., totally ordered lower-separable) partially ordered set, and let be an increasing (resp., decreasing) mapping. Then, for any , there exists such that for all . Moreover, if is strictly monotone, then is a maximal element.*

#### 3. Vectorial Ekeland’s Variational Principle

Let be a topological vector space, let be a closed convex cone in , let be a complete cone quasi-uniform space, and let be a nondecreasing function, that is, implies that . For a function , we first give the following conditions:(H1) for all ;(H2) for all ;(H3) for any fixed , the mapping is bound from below and

Is a closed subset in .

Lemma 3.1. *Let be a topological vector space with a topology and be a closed convex pseudo-nuclear cone or a closed point convex cone (i.e., ). Let be a complete cone quasi-uniform space and be a function satisfying (H1) and (H2). For any fixed , one defines a relation ≼ on as follows:
*

*Then is a partially ordered set.*

*Proof. *It is clear that . If and , then we have
This shows that and . It follows from the condition (H2) that
It follows from Lemma 2.3 that there exists a family of *quasi-norms* which defines the topology of . If is pseudo-nuclear, by Lemma 2.4, it follows that, for any , there exists with the property:
For any , we know that is an increasing function on . Then it follows from (3.4) that , and then . Thus, by (3.3), it follows that , for any and . It follows from (3.5) that for all , that is, . If is a closed convex cone, then the ordering relation ≤ is a partial ordering. It follows from (3.3) and (3.4) that and . Again by (3.3), it follows that for all , that is, . If and , then we have
which imply that and and so
By (d3) of Definition 2.5, it follows that, for any , there exists with such that and so
This shows that . Therefore, is a partial order on . This completes the proof.

First, we give a general Ekeland’s principle of the nondirected vectorial perturbation type as follows:

Theorem 3.2. *Let be a topological vector space with a topology and be a closed convex cone. Let be a complete cone quasi-uniform space and let be a function satisfying the conditions , , and . If the cone is pseudo-nuclear, then, for any , there exists such that*(1)* for all ;*(2)*for all with , there exists such that*

*Proof. *For any fixed , if is defined by (3.2), then Lemma 3.1 shows that is a partially ordered set.

First, we prove that any increasing net is a Cauchy net. It follows from Lemma 2.3 that there exists a family of quasi-norms which defines the topology of . Since is pseudo-nuclear, by Lemma 2.4, it follows that, for any , there exists with the property:
From the proof of Lemma 3.1, we can show that is a decreasing net and, for any ,
and so
It follows from that
Thus we have
and so, for any ,
It follows from the condition (H3) that is bounded from below. This and (3.15) imply that
It follows from (3.10) that, for any ,
This shows that is a Cauchy net in . The completeness implies that converges to some . It follows from for any that
By (H3), we get that
Thus , that is, is an upper bound of the net . Assume that is a totally ordered set. Then is also a directed set. If we represent by , where , then is an increasing net. By the results just proved above, we can know that has an upper bound. By the well-known Zorn’s lemma, the set has a maximal element , that is,
Therefore, it follows from that the conclusion (1) holds. For any with , if for all , then . Since is a maximal element, we have that , which is a contradiction. Therefore, conclusion (2) holds. This completes the proof.

If the cone has a base, we have the following result.

Theorem 3.3. *Let be a topological vector space with a topology and let be a closed convex cone. Let be a sequentially complete cone quasi-uniform space and let be a mapping satisfying the conditions , , and . If the cone has a base, then, for any , there exists such that*(1)* for all ;*(2)*for all with , there exists such that*

*Proof. *Since cone has a base, it follows from Lemma 2.3 that . Let . For any fixed , assume that is defined by (3.2). Similarly, as in the proofs of Lemma 3.1 and Theorem 3.2, we can prove that is a partially ordered set and any increasing sequence has an upper bound. The only difference is that is replaced by . Assume that and . Then we have
implies that there exists such that . Then we have
that is, . Let . Then is strictly increasing. Assume that is a totally ordered set. Then is also a totally set and, for any , if and only if . It follows from Theorem 2.3 in [16] that the set of real numbers is totally ordered separable and so there exist an increasing sequence and a decreasing sequence such that, for any , there exist positive integers and satisfying
Let and be two sequences in such that and . Since is strictly increasing, we know that is increasing and is decreasing. For any , by (3.24), we have
and so
Thus is totally ordered separable. By using Lemma 2.10, we can get that there exists a maximal element , that is,
Then, in the same way as in the proof of Theorem 3.2, we can prove that conclusions (1) and (2) holds. This completes the proof.

*Remark 3.4. *From the proof of Theorems 3.2 and 3.3 we can see that if cone is a pointed convex cone and replace the pseudo-nuclearity of with the regularity of (i.e., any monotone and bounded net in is convergent), then the conclusions of Theorems 3.2 and 3.3 still hold.

Clearly, our nondirected vectorial perturbation type Ekeland’s principle contains the directed vectorial perturbation type Ekeland’s principle as a special case. From Theorem 3.2, we have the following Theorem 3.5, which improves the corresponding results in [2, 3, 5, 8, 14].

Theorem 3.5. *Let be a complete -type topological space, let be a Hausdorff topological vector space, and let be a closed convex cone. Let be a function satisfying the conditions , , and . Let for all . If is pseudo-nuclear, or is a locally convex space and , then, for any , there exists such that*(1)* for all ;*(2)*for all with , there exists such that*

*Proof. *Let be a mapping defined by and
It is clearly that, for any net and ,
Since is a complete -type topological space, we have that is a complete cone quasi-uniform space. If is pseudo-nuclear, then the conclusion of Theorem 3.5 can be obtained by using Theorem 3.2 for . If is a locally convex space and , then, for each , we have . By using well-known separation theorem of convex sets, there exist a continuous linear function and a number , such that , and for any . This shows that and . If , then for any . So we have that . Then by letting , we get that for any . Thus, for each , there exists a continuous function , such that . For any fixed , if is defined by
By Lemma 3.1, is a partially ordered set. Similarly as in the proofs of Theorem 3.2, we can prove that any increasing net has an upper bound. The only difference is that is replaced by . Then, in the same way as in the proof of Theorem 3.2, we can prove that the conclusion of Theorem 3.5 holds. This completes the proof.

In the following Theorem 3.6, we introduce some new principles, which are equivalent with our Ekeland principle.

Theorem 3.6. *Let be a topological vector space with a topology , be a closed convex and pseudo-nuclear cone in and be a complete cone quasi-uniform space. Let be a function satisfying the conditions . Then, for any , the following conclusions hold and they are equivalent.*

(1)* (Vectorial Ekeland type variational principle in cone quasi-uniform spaces) There exists**such that, for all with , there exists satisfying the following:*(2)* (Vectorial quasivariational inclusion principle in cone quasi-uniform spaces Let be a vector space and be a nonempty convex subset. Let and be both multivalued mappings with nonempty values. Suppose that**holds, if and . Then there exists such that for any .*(3)* (Vectorial quasi-optimization principle in cone quasi-uniform spaces) Let be a vector space and be a nonempty convex subset. Let and be both multivalued mappings with nonempty values. Suppose that (3.34) holds if and . Then there exists such that for any .*(4)* (Vectorial quasi-optimization principle in cone quasi-uniform spaces Let be a vector space and be a nonempty convex subset. Let and be both multivalued maps with nonempty values. Suppose that (3.34) holds if and . Then there exists such that for any .*(5)*(Generalized Caristi-Kirk type coincidence point in cone quasi-uniform spaces Let be an index set. For each , let be a multivalued mapping, let be a nonempty subset of , and let be a surjective mapping. Suppose further that, if, for each with , there exist and such that**Then there exists a coincidence point of and , that is, such that .*(6)* (Nonconvex maximal element for a family of multivalued maps in cone quasi-uniform spaces Let be any index set and for each , be a multivalued mapping. Assume that, for each with , there exists with such that** Then there exists such that for each .*(7)* (Generalized Takahashi nonconvex minimization Theorem in cone quasi-uniform spaces Suppose that, for each with**there exists with such that (3.34) holds. Then there exists such that for all .*(8)* (Oettli-Théra type theorem in cone quasi-uniform spaces Let and suppose that, for any , there exists such that (3.34) holds. Then there exists .*

*Proof. *It follows from Theorem 3.2 that (1) holds.

(1) (2) By (1), there exists such that, for all with , there exists satisfying

If (2) does not hold for , then there exists with such that
which contradicts (3.38). Thus for any , that is, (2) holds.

(2) *⇒* (3) Let . Then implies that
This shows that and so, if and , then the conditions of (3) implies that (3.34) holds. It follows from (2) that there exists such that for any . This implies that for any . Thus for any , that is, (3) holds.

(3) (4) Let
If and , then . By the condition of (4), we have that (3.34) holds. This shows that the condition of (3) holds for . By the conclusion of (3), there exists such that for any . Thus , that is, (4) holds.

(4) (1) Let

If, for any and any scalar , we define
then is a vector space and is a cone in . Let be a multivalued mapping defined by
and be a multivalued mapping defined by for any . If
(note that ), then . This shows that (3.34) holds. It follows from (4) that there exists such that for any . Thus, for all with , there exists satisfying the following:
That is, (1) holds.

(1) (5) From (1), there exists such that, for all with , there exists satisfying the following:
Since is a surjective mapping, there exists such that . We claim that . If , by the hypotheses of (V), there exist and such that
which contradicts (3.47). Thus . implies that .

(5) (6) If the conclusion of (6) dose not hold, then, for any , there exists such that . By the hypotheses of (6), there exists with such that (3.34) holds. Let , : (the identity mapping of ) and
Then the conditions of (5) are satisfied for , and . Thus, from (5), it follows that there exists such that , which contradicts the definition of . Therefore, there exists such that for any .

(6) (7) Let
From this, we know that, if , then . By the hypotheses of (7), there exists with such that
By using (6), there exists such that for any , that is, for any .

(7) (8) Suppose that (7) and the hypothesis of (8) hold. Suppose that, for any , . By the hypothesis of (8), there exists such that
Equation (3.52) shows that the conditions of (7) naturally hold. Then there exists such that for any , which contradicts (3.52). Therefore, there exists .

(8) (1) Let
If , then , that is, there exists such that
This shows that the conditions of (8) hold and so there exists . shows that, for all with , there exists satisfying the following:
This completes the proof.

In the following, we provide some examples to illustrate our results.

*Example 3.7. *Let , , where . Then is a nuclear and regular cone if and a regular cone, but not a nuclear cone in if . However, for any , is not a solid cone (see, e.g., [32]). Let and define a mapping by . Then is a cone quasi-metric on . Since is complete, is also a complete quasi-metric space. Define a mapping by for any . Then, for each , is bound from below and
is a closed subset in . It follows from Theorem 3.2 and Remark 3.4 that, for any , there exists such that(1) for a.e. ;(2)for all with , there exists a measuable set with a positive measure such that

In fact, meets the demands.

*Example 3.8. *Let endowed with the topology defined by the semi-norms , where . The cone
is a nuclear cone, but not well based (see [33]). Let , define a mapping by
and a mapping by
Then is a complete cone quasi-metric space, for each , is bound from below and
is a closed subset in . It follows from Theorem 3.2 and Remark 3.4 that, for any , there exists such that
and, for all with ,

*Remark 3.9. *In above examples, the cone quasi-metric is not directed metric, the cone is not solid or not well based, thus, Example 3.7 and Example 3.8 show that our results are different from the results in [2–5, 8–10, 14–17].

#### Acknowledgments

First, the authors are very grateful to the referees for their careful reading of the paper, valuable comments and suggestions that enable us to revise and greatly improve this paper. Next, this paper was supported by the National Natural Science Foundation of China (11171286, 11126344) and Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No.: 2011-0021821).

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