The Scientific World Journal

Volume 2013, Article ID 150130, 6 pages

http://dx.doi.org/10.1155/2013/150130

## Existence Theorems for Vector Equilibrium Problems via Quasi-Relative Interior

Technical University of Cluj-Napoca, Department of Mathematics, Cluj-Napoca, G. Bariţiu 25-28, 400027 Jud. Cluj, Romania

Received 7 April 2013; Accepted 25 April 2013

Academic Editors: L. B. Ciric, M. Hajarian, and Y. Xia

Copyright © 2013 Capătă Adela Elisabeta. 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

The aim of this paper is to present new existence theorems for solutions of vector equilibrium problems, by using weak interior type conditions and weak convexity assumptions.

#### 1. Introduction

Starting with the paper of Blum and Oettli [1], the field of equilibrium problems is intensively studied by researchers, and many papers dealing with vector equilibrium problems were written. The interest of the researchers on this topic is due to the fact that equilibrium problems represent a natural and unified framework for other problems, such as optimization problems, variational inequality problems, and saddle-point problems, problems which until now were separately studied and have applications in physics, economics, and so forth, (see, for instance, [2]). Since then, a large variety of vector equilibrium problems were considered and the authors studied the existence of solutions (see, for instance, [3–10]), well posedness (see, for instance, [11, 12]), and sensitivity analysis (see, for instance, [13, 14]).

Most of the existence results are based on the hypothesis of nonemptyness of the ordering cone. In this paper, based on weak convexity assumptions defined by the means of quasi-relative interior of a convex set introduced in [15] and by using a quite recent separation theorem, whose statement was given in [16], we establish existence theorems for solutions of vector equilibrium problems. Then, the results are applied to vector optimization problems and to vector variational inequalities.

The paper is organized as follows. In Section 2 we recall some notions and auxiliary results that we need throughout this paper. Then, in Section 3 we present the main results of this paper, whose statements are given in the terms of quasi-relative interior and weak convexity assumptions. These results are applied to a vector optimization problem and to vector variational inequalities in Sections 4 and 5, respectively.

#### 2. Preliminaries

Let be a separated locally convex space, and let be a nontrivial convex cone. Having , the dual space of , the dual cone of is

We recall the definitions of quasi-interior and quasi-relative interior of a convex set and some useful properties of the quasi-relative interior notion.

*Definition 1. * Let be a nonempty convex subset of . The quasi interior of is the set

*Definition 2 (see [17]). *Let be a nonempty convex subset of . The quasi-relative interior of is the set

In a separable Banach space the quasi-relative interior of any nonempty closed convex set is nonempty (cf. [17]). For the proof of the following properties and other useful properties of the quasi-relative interior of a convex set, we refer the reader to [17, 18], respectively.

Proposition 3. *Let and be nonempty convex subsets of , , and . Then the following statements are true:*(i)qri qri qri;(ii) qri qri;(iii) qri qri qri;(iv) qri qri;* **If * qri, *then *(v) qri;(vi)qri.

The normal cone of a convex subset of at is defined as By means of the normal cone, the next characterizations of the abovementioned interior notions hold, and they can be found in [16, 17].

Theorem 4. *Let be a nonempty convex subset of and . Then if and only if . *

Theorem 5. *Let be a nonempty convex subset of and . Then if and only if is a linear subspace of . *

For other generalizations of the classical interior we refer the reader to [19–22]. Whenever the interior of the set is nonempty, then int (see [17, Corollary 2.14]). If is finite dimensional, then ri (see [17]), where by ri we understand the relative interior of , that is, the interior with respect to the affine hull.

The following characterization for the quasi-interior of a convex cone holds, and it can be found in [23].

Proposition 6. *Let be a convex cone in a locally convex space , and let . Then if and only if
*

Lemma 7 (see [17]). *Let be a convex cone in a separable locally convex space . Then if and only if and . *

Lemma 8 (see [17]). *Let be a convex cone in a locally convex space . Then
**
for all . *

The statement of the next theorem is due to [16], where it was proved for normed spaces, and, later on, it was proved for separated locally convex spaces by [24].

Theorem 9. *Let be a nonempty convex subset of and . Then, there exists such that for all . *

For other separation theorems which involve the quasi-relative interior we refer the reader to [25].

*Definition 10. *A function is said to be generalized -subconvexlike on if cone is convex.

Proposition 11. *If is convex, then cone is also convex. *

*Proof. *Let and let . So, there exist , , and such that
*Case *1. If and , then , and
*Case *2. If or , denote
So, we have
In what follows we will prove that . For this, we evaluate
Dividing this equality by the positive number it yields
By the above equalities and the convexity of , we have
Thus,
whence multiplying this relation by we obtain the conclusion.

#### 3. Existence Results

Throughout this paper we study the following strong vector equilibrium problem: where and are non-empty sets, , and has a non-empty quasi-relative interior.

*Definition 12. *We say that a point is a quasi-relative solution of if

Theorem 13. *Suppose that the following conditions are satisfied: *(i)*for every , the function is generalized -subconvexlike on ;*(ii)*;*(iii)*for every , is not a linear subspace of . ** Then, problem admits a quasi-relative solution. *

*Proof. *Suppose by contradiction that has no solutions; that is, for any there exists such that
This leads us to
which implies that
Assumption (i) assures the convexity of the set , and, together with the relation (18) and assumption (iii), it gives
The separation Theorem 9 of convex sets assures the existence of a nonzero functional such that
So
Taking , by and (21) we have
which gives
Further, for any , dividing by relation (21) it implies
Now letting in the above relation we get
Since , by assumption (ii), Lemma 7, and Proposition 6, we get
which is a contradiction to (25), and this completes the proof.

This theorem allows us to obtain existence results for important practical spaces, whose ordering cones have empty interiors, but nonempty quasi-relative interiors. This is the case of the Banach space with the positive cone where is a real constant, is an interval, is a -finite measure space, and .

The next result deals with stronger assumptions than the ones presented in Theorem 13.

Corollary 14. * Suppose that the following conditions are satisfied: *(i)*for every , the function is generalized -subconvexlike on ;*(ii)*;*(iii)*for every , .** Then, problem admits a quasi-relative solution. *

* Proof. *Since and
for every , by the definition of the quasi-relative interior, we get
is not a linear subspace of . So all the assumption of Theorem 13 are satisfied, and the conclusion follows now by this theorem.

*Remark 15. *In Theorem 13 and Corollary 14, according to Proposition 11, the generalized -subconvexlike assumption of the function on can be replaced by the convexity of the set .

#### 4. Existence Results for Vector Optimization Problems

Let , and let the function . In this section we study the vector optimization problem, According to [26], the point is called quasi-relative minimal point of the set , that is, while is a quasi-relative minimizer of , that is,

By Theorem 13 and Corollary 14 we have the following results.

Theorem 16. * Suppose that the following conditions are satisfied: *(i)*for every , the function is generalized -subconvexlike on ;*(ii)*;*(iii)*for every , is not a linear subspace of . ** Then, problem admits a quasi-relative solution. *

*Proof. * Define the function by
It is easy to see that all the assumptions of the Theorem 13 are satisfied by this function . So, problem admits a solution, which implies that problem has a solution, and the proof is completed.

Corollary 17. * Suppose that the following conditions are satisfied: *(i)*for every , the function is generalized subconvexlike on ;*(ii)*;*(iii)*for every , .** Then, problem admits a quasi-relative solution. *

To show that the set of functions which satisfies the assumptions of Corollary 17 is nonempty, we give the following example.

*Example 18. * Let , , , and . We have that
Since and is a convex set, we deduce that
is also convex and that, by Proposition 11, the first assumption of Corollary 17 is verified. The second assumption is obviously satisfied, and we still have to verify its third assumption. Because , then it is equal to . The set
for all , while for
Thus,
and assumption (iii) is checked.

#### 5. Existence Results for Vector Variational Inequalities

Let be a non-empty convex subset of a vector space, let , and let the operator , where denotes the set of all linear and continuous functions defined on with values on . Throughout this section we study the Minty vector variational inequality:

*Definition 19. *We say that a point is a quasi-relative solution of if

By Theorem 13 and Corollary 14 we have the following results.

Theorem 20. * Suppose that the following conditions are satisfied: *(i)*;*(ii)*for every , is not a linear subspace of . ** Then, problem admits a quasi-relative solution. *

*Proof. * Let the function be defined by
Obviously fulfills assumptions (ii) and (iii) of Theorem 13. It remains to show that the first assumption of this theorem is also verified by this function .

Since is a convex set, by the definition of the function we deduce that is a convex set for every . By this we get
convex, which together with Proposition 11 gives the conclusion, and proof is completed.

Corollary 21. *Suppose that the following conditions are satisfied:*(i)*;*(ii)*for every , .** Then, problem admits a quasi-relative solution. *

In what follows we turn our attention to existence results for the Stampacchia vector variational inequality

*Definition 22. *We say that a point is a quasi-relative solution of if

Next we recall a definition concerning vector variational inequalities (see [27]).

*Definition 23. *Let . The operator is said to be -upper hemicontinuous if is a convex set, and, for all the function
is upper semicontinuous at .

Theorem 24. * Let , let , and suppose that the following conditions are satisfied: *(i)* for all ;*(ii)* is -upper hemicontinuous. ** Then, problem admits a quasi-relative solution.*

*Proof . *By assumption (i)
whence, by using Lemma 8, we deduce that
that is, is a quasi-relative solution of . In what follows we will prove that is also a quasi-relative solution of .

Because is convex, then
for all and all , and by (43), we get
For each , there exists, according to (ii), a number such that
This, together with (46), gives
Since the inequality holds for each , we deduce that
whence, by using Lemma 8, we get
which completes the proof.

#### References

- E. Blum and W. Oettli, “From optimization and variational inequalities to equilibrium problems,”
*The Mathematics Student*, vol. 63, no. 1–4, pp. 123–145, 1994. View at Google Scholar - J. P. Aubin,
*Optima and Equilibria, An Introduction to Nonlinear Analysis*, Springer, 1993. - Q. H. Ansari, I. V. Konnov, and J. C. Yao, “On generalized vector equilibrium problems,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 47, no. 1, pp. 543–554, 2001. View at Publisher · View at Google Scholar · View at Scopus - Q. H. Ansari, W. Oettli, and D. Schläger, “A generalization of vectorial equilibria,”
*Mathematical Methods of Operations Research*, vol. 46, no. 2, pp. 147–152, 1997. View at Publisher · View at Google Scholar - Q. H. Ansari, X. Q. Yang, and J. C. Yao, “Existence and duality of implicit vector variational problems,”
*Numerical Functional Analysis and Optimization*, vol. 22, no. 7-8, pp. 815–829, 2001. View at Publisher · View at Google Scholar · View at Scopus - M. Bianchi, N. Hadjisavvas, and S. Schaible, “Vector equilibrium problems with generalized monotone bifunctions,”
*Journal of Optimization Theory and Applications*, vol. 92, no. 3, pp. 527–542, 1997. View at Google Scholar · View at Scopus - M. Bianchi, G. Kassay, and R. Pini, “Ekeland's principle for vector equilibrium problems,”
*Nonlinear Analysis, Theory, Methods and Applications*, vol. 66, no. 7, pp. 1454–1464, 2007. View at Publisher · View at Google Scholar · View at Scopus - G. Bigi, A. Capătă, and G. Kassay, “Existence results for strong vector equilibrium problems and their applications,”
*Optimization*, vol. 61, no. 5, pp. 567–583, 2012. View at Publisher · View at Google Scholar - A. Capătă, “Existence results for proper efficient solutions of vector equilibrium problems and applications,”
*Journal of Global Optimization*, vol. 51, no. 4, pp. 657–675, 2011. View at Publisher · View at Google Scholar - A. Capătă and G. Kassay, “On vector equilibrium problems and applications,”
*Taiwanese Journal of Mathematics*, vol. 15, no. 1, pp. 365–380, 2011. View at Google Scholar - M. Bianchi, G. Kassay, and R. Pini, “Well-posedness for vector equilibrium problems,”
*Mathematical Methods of Operations Research*, vol. 70, no. 1, pp. 171–182, 2009. View at Publisher · View at Google Scholar - J. W. Chen, Y. J. Cho, and Z. Wan, “The existence of solutions and wellposedness for bilevel mixed equilibrium problems in Banach spaces,”
*Taiwanese Journal of Mathematics*, vol. 17, pp. 724–748, 2013. View at Google Scholar - K. Kimura and J. C. Yao, “Sensitivity analysis of vector equilibrium problems,”
*Taiwanese Journal of Mathematics*, vol. 12, no. 3, pp. 649–669, 2008. View at Google Scholar · View at Scopus - K. Kimura and J. C. Yao, “Sensitivity analysis of solution mappings of parametric generalized quasi vector equilibrium problems,”
*Taiwanese Journal of Mathematics*, vol. 12, no. 9, pp. 2233–2268, 2008. View at Google Scholar · View at Scopus - Z. A. Zhou and X. M. Yang, “Optimality conditions of generalized subconvexlike set-valued optimization problems based on the quasi-relative interior,”
*Journal of Optimization Theory and Applications*, vol. 150, no. 2, pp. 327–340, 2011. View at Publisher · View at Google Scholar · View at Scopus - P. Daniele, S. Giuffrè, G. Idone, and A. Maugeri, “Infinite dimensional duality and applications,”
*Mathematische Annalen*, vol. 339, no. 1, pp. 221–239, 2007. View at Publisher · View at Google Scholar · View at Scopus - J. M. Borwein and A. S. Lewis, “Partially finite convex programming. Part I: quasi relative interiors and duality theory,”
*Mathematical Programming*, vol. 57, no. 1–3, pp. 15–48, 1992. View at Publisher · View at Google Scholar · View at Scopus - J. Borwein and R. Goebel, “Notions of relative interior in Banach spaces,”
*Journal of Mathematical Sciences*, vol. 115, no. 4, pp. 2542–2553, 2003. View at Publisher · View at Google Scholar - R. B. Holmes,
*Geometric Functional Analysis and Its Applications*, Springer, Berlin, Germany, 1975. - M. A. Limber and R. K. Goodrich, “Quasi interiors, lagrange multipliers, and
*L*^{p}spectral estimation with lattice bounds,”*Journal of Optimization Theory and Applications*, vol. 78, no. 1, pp. 143–161, 1993. View at Publisher · View at Google Scholar · View at Scopus - R. T. Rockafellar,
*Conjugate Duality and Optimization*, vol. 16 of*CBMS Regional Conference Series*, Society for Industrial and Applied Mathematics, Philadelphia, Pa, USA, 1974. - C. Zălinescu,
*Convex Analysis in General Vector Spaces*, World Scientific Publishing, Singapore, 2002. View at Publisher · View at Google Scholar - X. H. Gong, “Optimality conditions for Henig and globally proper efficient solutions with ordering cone has empty interior,”
*Journal of Mathematical Analysis and Applications*, vol. 307, no. 1, pp. 12–31, 2005. View at Publisher · View at Google Scholar · View at Scopus - R. I. Boţ, E. R. Csetnek, and G. Wanka, “Regularity conditions via quasi-relative interior in convex programming,”
*SIAM Journal on Optimization*, vol. 19, no. 1, pp. 217–233, 2008. View at Publisher · View at Google Scholar · View at Scopus - F. Cammaroto and B. D. Bella, “Separation theorem based on the quasirelative interior and application to duality theory,”
*Journal of Optimization Theory and Applications*, vol. 125, no. 1, pp. 223–229, 2005. View at Publisher · View at Google Scholar · View at Scopus - T. Q. Bao and B. S. Mordukhovich, “Relative Pareto minimizers for multiobjective problems: existence and optimality conditions,”
*Mathematical Programming A*, vol. 122, no. 2, pp. 301–347, 2010. View at Publisher · View at Google Scholar - X. H. Gong, “Efficiency and Henig efficiency for vector equilibrium problems,”
*Journal of Optimization Theory and Applications*, vol. 108, no. 1, pp. 139–154, 2001. View at Publisher · View at Google Scholar · View at Scopus