- Research Article
- Open Access

# Vectorial Form of Ekeland-Type Variational Principle in Locally Convex Spaces and Its Applications

- S Eshghinezhad
^{1}and - M Fakhar
^{1}Email author

**2010**:276294

https://doi.org/10.1155/2010/276294

© S. Eshghinezhad and M. Fakhar. 2010

**Received: **23 June 2010

**Accepted: **2 November 2010

**Published: **21 November 2010

## Abstract

By using a Dane ' drop theorem in locally convex spaces we obtain a vectorial form of Ekeland-type variational principle in locally convex spaces. From this theorem, we derive some versions of vectorial Caristi-Kirk's fixed-point theorem, Takahashi's nonconvex minimization theorem, and Oettli-Théra's theorem. Furthermore, we show that these results are equivalent to each other. Also, the existence of solution of vector equilibrium problem is given.

## Keywords

## 1. Introduction and Preliminaries

A very important result in nonlinear analysis about the existence result for an approximate minimizer of a lower semicontinuous and bounded below function was first presented by Ekeland [1]. Known nowadays as Ekeland's variational principle (in short, EVP), it has significant applications in the geometry theory of Banach spaces, optimization theory, game theory, optimal control theory, dynamical systems, and so forth; see [1–11] and references therein. It is well known that EVP is equivalent to many famous results, namely, the Caristi-Kirk fixed-point theorem, the petal theorem, Phelp's lemma, Danês' drop theorem, Oettli-Théra's theorem and Takahashi's theorem, see, for example, [4, 6, 7, 10, 12–19]. Many authors have obtained EVP on complete metric spaces [1, 10, 19, 20] and in locally convex spaces [20–24]. Along with the development of vector optimization and motivated by the wide usefulness of EVP, many authors have been interested in obtaining this principle for vector-valued functions and set-valued mappings; see [3, 5, 8, 9, 11–15, 21, 22, 25]. Recently, this principle has been obtained for bifunctions and applied to solve equilibrium problem in nonconvex setting [10, 12, 14–16, 20, 26, 27]. Our goal in this paper is to obtain Ekeland's variational principle for vector-valued bifunctions in locally convex spaces. By using this result we derive the existence of solution of vector equilibrium problem in the setting of seminormed spaces. Also, we obtain vectorial Caristi-Kirk's fixed-point theorem, vectorial Takahashi's nonconvex minimization theorem and vectorial Oettli-Théra's theorem. Moreover, we show that these results and Dane ' drop theorem are equivalent to each other. Let us, introduce some known definitions and results which will be used in the sequel.

Let be a Hausdorff locally convex real vector space. A subset of is said to be a disc, if is bounded and absolutely convex. Let be the vector subspace spanned by , and be the Minkowski functional of , then is a normed space. If is a Banach space, then is called a Banach disc. A sequence in is said to be locally convergent to an element if there is a disc in such that the sequence is convergent to in and is said to be locally Cauchy if there is a disc in such that is a Cauchy sequence in . We say that is a locally complete space if every locally Cauchy sequence is locally convergent. This is equivalent to that each bounded subset of is contained in a certain Banach disc. A nonempty subset of is said to be locally complete if every locally Cauchy sequence in is locally convergent to a point in . The subset is said to be locally closed if for any locally convergent sequence in , its local limit point belongs to . It is well known that every sequentially complete locally convex space is locally complete and the converse is not true; see [28, 29].

Definition 1.1.

Let be a nonempty subset of a locally convex space , be a locally convex space ordered by the nontrivial closed convex cone . A vector-valued function is said to be

(1) -locally lower semicontinuous if for every the set is locally closed in ;

(2) -upper semicontinuous at if for any neighborhood of , there exists a neighborhood of such that , for all . If is -upper semicontinuous at each point of , then is said to be -upper semicontinuous on ;

(3) -bounded from below, if there exists such that for all .

It is well known that VEP includes fundamental mathematical problems like vector optimization, vector variational inequality, and vector complementarity problem. For further details on VEP, one can refer to [23, 24, 30–32].

The following lemma describes some properties of the Gerstewitz function and it will be used in the sequel. For its proof we refer the reader to [5, 6, 33].

Lemma 1.2.

For each and , the following statements are satisfied:

(v) is positively homogeneous and continuous on .

(vii) is monotone, that is, if , then .

In order to obtain a vectorial form of Ekeland-type variational principle we need the following result.

Theorem 1.3 (see [17]).

Let be a locally closed subset of a locally convex space and a locally closed, bounded convex subset of with . If either or is locally complete, then for each , there exists such that , where denotes the convex hull of .

## 2. Vectorial Ekeland-Type Variational Principle

Recently, Qiu [18] obtained some versions of Ekeland's variational principle in locally convex spaces, which only need to assume local completeness of some related sets. Motivated by this paper we obtain some versions of EVP for vector-valued bifunctions in locally convex spaces. These results extend Qiu's results to vector-valued bifunctions.

Throughout this section is a locally convex space, is locally closed subset of , is a family of seminorms generating the locally convex topology on , is a Hausdorff locally convex space ordered by a closed convex cone with and . We consider a vector-valued bifunction , a family of positive real numbers and the following assumptions:

(A3) is C-bounded from below for all .

(A4) is -locally lower semicontinuous for any .

(A5)There exists such that the set is locally complete.

(A6)The set is locally complete.

Notice that if assumptions (A1) and (A2) hold, then is called half distance. The following result is a vectorial form of Ekeland-type variational principle.

Theorem 2.1.

Suppose that assumptions (A1)–(A4) are satisfied. If either assumption (A5) or assumption (A6) holds, then for any , there exists such that

(ii)For any , there exists such that .

Proof.

Supposing that and , we consider the following two cases.

Case 1.

Case 2.

Remark 2.2.

In the above theorem, if assumption (A5) holds, then instead of assumption (A3), we can assume that is -bounded from below. Also, if assumption (A6) holds, assumption (A3) can be replaced by the following assumption: is -bounded from below for some .

As a consequence of the above theorem we can obtain the following result which is a vectorial version of Theorem 3.1 of [18].

Corollary 2.3.

Let be a function such that is -bounded from below and is -locally lower semicontinuous. Furthermore, let assumption (A6) holds or there exists such that the set is locally complete. Then there exists such that

(ii)for any , there exists such that .

Proof.

It is enough in Theorem 2.1 to consider for all .

In the following theorem we show that the previous results are equivalent to each other.

Theorem 2.4.

Corollary 2.3 implies Theorem 2.1.

Proof.

It is an easy task to derive the assumptions of Corollary 2.3 for the above function from the assumptions of Theorem 2.1. Therefore, there exists which satisfies the conditions (i) and (ii) of Corollary 2.3. Hence,

(ii)for any , there exists such that .

We extend by an additional element such that for all and for all .

By using Theorem 2.1 we obtain another version of vectorial form of Ekeland-type variational principle in which the perturbation function is the Minkowski functional of a bounded set.

Theorem 2.5.

Suppose that assumptions (A1)–(A4) are satisfied. Let
be a locally closed, bounded convex set containing 0 and *α* be a positive real number. Let
be locally complete or assumption (A5) holds. Then, for any
, there exists
such that:

Proof.

Since , then . Also, is -locally lower semicontinuous and is locally lower semicontinuous, then is closed in . Suppose that is restricted to . If is locally complete then is a Banach disk and is a Banach space. If the set is locally complete, then is a complete set in . Therefore, by Theorem 2.1 there exists such that:

Assuming that is a locally complete locally convex space, the condition on local completeness of some related subsets is automatically satisfied. However, we give the following examples of spaces which are not locally complete but the condition on local completeness of some related subsets is satisfied.

Example 2.6.

Let
be the space of all continuous functions defined on
. By Corollary 11-7-3, 11-7-4 of [29],
with *weak*-topology is quasi barreled but it is not barreled. Therefore, by Proposition 11-2-5 of [29],
with *weak**-topology is not locally complete.

But by Banach-Alaoglu theorem this set is *weak**-compact. Also
is separable, so *weak**-topology on unit ball
is metrizable. Hence, this set is locally complete.

Example 2.7.

Let be the space of all differentiable functions whose derivative is continuous. Then is not a complete space. Therefore, by Proposition [28] is not a locally complete space.

where is ordered by the cone . If we choose , then the set is locally complete.

## 3. Caristi-kirk's Fixed-Point Theorem, Takahashi's Nonconvex Minimization Theorem, and Oettli-Théra's Theorem and Equilibrium Problem

In this section, we obtain an existence result for solution of vector equilibrium problem in nonconvex setting. Also, some new versions of the vectorial Caristi-Kirk fixed-point theorem, vectorial Takahashi's nonconvex minimization theorem and the vectorial Oettli-Théra theorem are given.

Theorem 3.1.

Proof.

This is a contradiction.

In the above theorem, when is not necessarily weakly compact we have the following result. Since its proof is similar to Theorem 4 of [20], we omit it.

Theorem 3.2.

Let be a nonempty subset of a reflexive semi normed space . Suppose that is a function satisfying assumptions (A1)–(A5) together with some and is -upper semicontinuous for every . Let the following coercivity condition holds:

There exists a nonempty closed bounded subset of such that for all there exists with satisfying .

As a consequence of Theorems 2.1 and 2.5 we can obtain two versions of vectorial Caristi-Kirk's fixed-point theorem.

Theorem 3.3.

Then, there exists such that .

Proof.

Let be a point which satisfies in the parts (i) and (ii) with . We show that . If there exists and , then by Theorem 2.1 there is a such that and it is a contradiction.

The proof of the following results is similar to that of Theorem 3.3 and we omit it.

Theorem 3.4.

Then, there exists such that .

In the following we give two versions of vectorial Takahashi's nonconvex minimization.

Theorem 3.5.

Suppose that all of conditions of Theorem 2.1 are satisfied and for each with , there exists such that for any . Then, there exists such that that is .

Proof.

By Theorem 2.1 there exists such that for each there exists such that . If , then by assumption there exists such that , and this is a contradiction.

The proof of the following results is similar to that of Theorem 3.5 and we omit it.

Theorem 3.6.

Suppose that all of the conditions of Theorem 2.5 are satisfied and for each with , there exists such that . Then, there exists such that .

In the final step we obtain a vectorial form of Oettli-Théra type theorem.

Theorem 3.7.

Assume that all of the conditions of Theorem 2.1 are satisfied and . Let such that for every there exists such that and . Then .

Proof.

By Theorem 2.1 there exists such that satisfies the condition (ii) in Theorem 2.1. It is easy to see that .

## 4. Equivalences

In this section, we show that Dane ' drop theorem (Theorem 1.3), two versions of vectorial form of Ekeland-type variational principle, vectorial Caristi-Kirk's fixed-point theorem and vectorial Takahashi's nonconvex minimization theorem are equivalent. In order to show that Theorems 1.3 and 2.1 are equivalent to each other we need the following definition which was introduced by Cheng et al. [34].

Definition 4.1.

Theorem 4.2.

Theorems 2.1 and 1.3 are equivalent to each other.

Proof.

*α*which . It is easy to see that the assumptions (A1)–(A4) are satisfied. If or is locally complete, then is locally complete, so is locally complete. Therefore, by Theorem 2.5 there exists a point such that

Therefore, and so by (4.6), we conclude that .

Theorem 4.3.

Theorems 2.1, 2.5, 3.3, 3.5 and 3.7 are mutually equivalent.

- (1)
Theorem 2.5

*⇔*Theorem 2.1.

- (2)
Theorem 3.3

*⇔*Theorem 2.1.

- (3)
Theorem 3.5

*⇔*Theorem 2.1.

*⇒*Theorem 2.1. Without loss of generality we assume that . Let

Hence, , so by Theorem 3.5, there exists such that .

- (4)
Theorem 3.7

*⇔*Theorem 2.1.

Choose . If , then there exists such that . Therefore, assumption of Theorem 3.7 is satisfied. Hence, there exists . From the definition of the results (i) and (ii) of Theorem 2.1 are satisfied.

## Declarations

### Acknowledgment

M. FAKHAR was partially supported by the Center of Excellence for Mathematics, University of Isfahan.

## Authors’ Affiliations

## References

- Ekeland I:
**On the variational principle.***Journal of Mathematical Analysis and Applications*1974,**47:**324–353. 10.1016/0022-247X(74)90025-0MathSciNetView ArticleMATHGoogle Scholar - Aubin J-P, Ekeland I:
*Applied Nonlinear Analysis, Pure and Applied Mathematics (New York)*. John Wiley & Sons, New York, NY, USA; 1984:xi+518.Google Scholar - Chen GY, Huang XX:
**A unified approach to the existing three types of variational principles for vector valued functions.***Mathematical Methods of Operations Research*1998,**48**(3):349–357. 10.1007/s001860050032MathSciNetView ArticleMATHGoogle Scholar - Chen G-Y, Huang X, Yang X:
*Vector Optimization. Set-Valued and Variational Analysis, Lecture Notes in Economics and Mathematical Systems*.*Volume 541*. Springer, Berlin, Germany; 2005:x+306.MATHGoogle Scholar - Göpfert A, Tammer Chr, Zălinescu C:
**On the vectorial Ekeland's variational principle and minimal points in product spaces.***Nonlinear Analysis: Theory, Methods & Applications*2000,**39**(7):909–922. 10.1016/S0362-546X(98)00255-7MathSciNetView ArticleMATHGoogle Scholar - Göpfert A, Riahi H, Tammer Chr, Zălinescu C:
*Variational Methods in Partially Ordered Spaces, CMS Books in Mathematics*.*Volume 17*. Springer, New York, NY, USA; 2003:xiv+350.MATHGoogle Scholar - Hamel AH:
**Phelps' lemma, Daneš' drop theorem and Ekeland's principle in locally convex spaces.***Proceedings of the American Mathematical Society*2003,**131**(10):3025–3038. 10.1090/S0002-9939-03-07066-7MathSciNetView ArticleMATHGoogle Scholar - Isac G:
**Nuclear cones in product spaces, Pareto efficiency and Ekeland-type variational principles in locally convex spaces.***Optimization*2004,**53**(3):253–268. 10.1080/02331930410001720923MathSciNetView ArticleMATHGoogle Scholar - Isac G, Tammer Chr:
**Nuclear and full nuclear cones in product spaces: pareto efficiency and an Ekeland type variational principle.***Positivity*2005,**9**(3):511–539. 10.1007/s11117-004-2770-8MathSciNetView ArticleMATHGoogle Scholar - Lin L-J, Du W-S:
**Ekeland's variational principle, minimax theorems and existence of nonconvex equilibria in complete metric spaces.***Journal of Mathematical Analysis and Applications*2006,**323**(1):360–370. 10.1016/j.jmaa.2005.10.005MathSciNetView ArticleMATHGoogle Scholar - Gutiérrez C, Jiménez B, Novo V:
**A set-valued Ekeland's variational principle in vector optimization.***SIAM Journal on Control and Optimization*2008,**47**(2):883–903. 10.1137/060672868MathSciNetView ArticleMATHGoogle Scholar - Ansari QH:
**Vectorial form of Ekeland-type variational principle with applications to vector equilibrium problems and fixed point theory.***Journal of Mathematical Analysis and Applications*2007,**334**(1):561–575. 10.1016/j.jmaa.2006.12.076MathSciNetView ArticleMATHGoogle Scholar - Araya Y:
**Ekeland's variational principle and its equivalent theorems in vector optimization.***Journal of Mathematical Analysis and Applications*2008,**346**(1):9–16. 10.1016/j.jmaa.2008.04.055MathSciNetView ArticleMATHGoogle Scholar - Finet C, Quarta L:
**Vector-valued perturbed equilibrium problems.***Journal of Mathematical Analysis and Applications*2008,**343**(1):531–545. 10.1016/j.jmaa.2008.01.052MathSciNetView ArticleMATHGoogle Scholar - Finet C, Quarta L, Troestler C:
**Vector-valued variational principles.***Nonlinear Analysis: Theory, Methods & Applications*2003,**52**(1):197–218. 10.1016/S0362-546X(02)00103-7MathSciNetView ArticleMATHGoogle Scholar - Oettli W, Théra M:
**Equivalents of Ekeland's principle.***Bulletin of the Australian Mathematical Society*1993,**48**(3):385–392. 10.1017/S0004972700015847MathSciNetView ArticleMATHGoogle Scholar - Qiu J-H:
**Local completeness and drop theorem.***Journal of Mathematical Analysis and Applications*2002,**266**(2):288–297. 10.1006/jmaa.2001.7709MathSciNetView ArticleMATHGoogle Scholar - Qiu J-H:
**Local completeness, drop theorem and Ekeland's variational principle.***Journal of Mathematical Analysis and Applications*2005,**311**(1):23–39. 10.1016/j.jmaa.2004.12.045MathSciNetView ArticleMATHGoogle Scholar - Wu Z:
**Equivalent formulations of Ekeland's variational principle.***Nonlinear Analysis: Theory, Methods & Applications*2003,**55**(5):609–615. 10.1016/j.na.2003.07.009MathSciNetView ArticleMATHGoogle Scholar - Bianchi M, Kassay G, Pini R:
**Ekeland's principle for vector equilibrium problems.***Nonlinear Analysis: Theory, Methods & Applications*2007,**66**(7):1454–1464. 10.1016/j.na.2006.02.003MathSciNetView ArticleMATHGoogle Scholar - Chen G-Y, Yang XQ, Yu H:
**Vector Ekeland's variational principle in an -type topological space.***Mathematical Methods of Operations Research*2008,**67**(3):471–478. 10.1007/s00186-007-0205-6MathSciNetView ArticleMATHGoogle Scholar - Ha TXD:
**Some variants of the Ekeland variational principle for a set-valued map.***Journal of Optimization Theory and Applications*2005,**124**(1):187–206. 10.1007/s10957-004-6472-yMathSciNetView ArticleMATHGoogle Scholar - Bianchi M, Hadjisavvas N, Schaible S:
**Vector equilibrium problems with generalized monotone bifunctions.***Journal of Optimization Theory and Applications*1997,**92**(3):527–542. 10.1023/A:1022603406244MathSciNetView ArticleMATHGoogle Scholar - Fakhar M, Zafarani J:
**Generalized vector equilibrium problems for pseudomonotone multivalued bifunctions.***Journal of Optimization Theory and Applications*2005,**126**(1):109–124. 10.1007/s10957-005-2663-4MathSciNetView ArticleMATHGoogle Scholar - Zhu J, Zhong CK, Cho YJ:
**Generalized variational principle and vector optimization.***Journal of Optimization Theory and Applications*2000,**106**(1):201–217. 10.1023/A:1004619426652MathSciNetView ArticleMATHGoogle Scholar - Al-Homidan S, Ansari QH, Yao J-C:
**Some generalizations of Ekeland-type variational principle with applications to equilibrium problems and fixed point theory.***Nonlinear Analysis: Theory, Methods & Applications*2008,**69**(1):126–139. 10.1016/j.na.2007.05.004MathSciNetView ArticleMATHGoogle Scholar - Lin L-J, Du W-S:
**On maximal element theorems, variants of Ekeland's variational principle and their applications.***Nonlinear Analysis: Theory, Methods & Applications*2008,**68**(5):1246–1262. 10.1016/j.na.2006.12.018MathSciNetView ArticleMATHGoogle Scholar - Pérez Carreras P, Bonet J:
*Barrelled Locally Convex Spaces, North-Holland Mathematics Studies*.*Volume 131*. North-Holland, Amsterdam, The Netherlands; 1987:xvi+512.View ArticleGoogle Scholar - Jarchow H:
*Locally Convex Spaces*. B. G. Teubner, Stuttgart, Germany; 1981:548.View ArticleMATHGoogle Scholar - Fakhar M, Zafarani J:
**Equilibrium problems in the quasimonotone case.***Journal of Optimization Theory and Applications*2005,**126**(1):125–136. 10.1007/s10957-005-2664-3MathSciNetView ArticleMATHGoogle Scholar - Giannessi F (Ed):
*Vector Variational Inequalities and Vector Equilibria. Mathematical Theories, Nonconvex Optimization and Its Applications*.*Volume 38*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2000:xiv+523.MATHGoogle Scholar - Oettli W, Schläger D:
**Existence of equilibria for monotone multivalued mappings.***Mathematical Methods of Operations Research*1998,**48**(2):219–228. 10.1007/s001860050024MathSciNetView ArticleMATHGoogle Scholar - Gerstewitz C:
**Nichtkonvexe Dualität in der Vektoroptimierung.***Wissenschaftliche Zeitschrift der Technischen Hochschule für Chemie "Carl Schorlemmer", Leuna-Merseburg*1983,**25**(3):357–364.MathSciNetMATHGoogle Scholar - Cheng LX, Zhou Y, Zhang F:
**Danes' drop theorem in locally convex spaces.***Proceedings of the American Mathematical Society*1996,**124**(12):3699–3702. 10.1090/S0002-9939-96-03404-1MathSciNetView ArticleMATHGoogle Scholar

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