Existence Theorems for Vector Equilibrium Problems via Quasi-Relative Interior

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.


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 saddlepoint 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][4][5][6][7][8][9][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 quasirelative 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.

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 * := { * ∈ * | * ( ) ≥ 0, ∀ ∈ } . (1) We recall the definitions of quasi-interior and quasirelative interior of a convex set and some useful properties of the quasi-relative interior notion.
Definition 2 (see [17]). Let be a nonempty convex subset of . The quasi-relative interior of is the set qri := { ∈ | cl cone( − y) is a linear subspace of } .
In a separable Banach space the quasi-relative interior of any nonempty closed convex set is nonempty (cf. [17]).

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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.
The normal cone of a convex subset of at 0 ∈ 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]. For other generalizations of the classical interior we refer the reader to [19][20][21][22]. Whenever the interior of the set is nonempty, then int = qri (see [17,Corollary 2.14]). If is finite dimensional, then qri = 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].
Lemma 7 (see [17]). Let be a convex cone in a separable locally convex space . Then ∈ qi if and only if ∈ qri and cl ( − ) = .
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
For other separation theorems which involve the quasirelative interior we refer the reader to [25].
Definition 10. A function : → is said to be generalized -subconvexlike on if cone ( )+ qri is convex.

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.
Proof. Suppose by contradiction that (VEP) has no solutions; that is, for any ∈ there exists 0 ∈ such that This leads us to which implies that 0 ∈ cone ( , 0 ) + qri .
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 (27) where > 0 is a real constant, [0, ] is an interval, ( , ) is a -finite measure space, and ∈ [1, ∞). The next result deals with stronger assumptions than the ones presented in Theorem 13.

Existence Results for Vector Optimization Problems
Let = , and let the function : → . In this section we study the vector optimization problem, (VOP) According to [26], the point ( ) is called quasi-relative minimal point of the set ( ), that is, while is a quasi-relative minimizer of (VOP), that is, By Theorem 13 and Corollary 14 we have the following results.
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 (VEP) admits a solution, which implies that problem (VOP) has a solution, and the proof is completed.

Then, problem (VOP) 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.
Since ( ) − ( ) + = ( − , 0) + 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 int ̸ = 0, then it is equal to qri . The set for all ∈ (0, 1], while for = 1 Thus, and assumption (iii) is checked.

Existence Results for Vector Variational Inequalities
Let be a non-empty convex subset of a vector space, let = , and let the operator : → L( , ), where L( , ) denotes the set of all linear and continuous functions defined on with values on . Throughout this section we study the Minty vector variational inequality: (MVI) Definition 19. We say that a point 0 ∈ is a quasi-relative solution of (MVI) if By Theorem 13 and Corollary 14 we have the following results.
Proof. Let the function : × → be defined by The Scientific World Journal 5 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.

Then, problem (MVI) admits a quasi-relative solution.
In what follows we turn our attention to existence results for the Stampacchia vector variational inequality (SVI) Definition 22. We say that a point 0 ∈ is a quasi-relative solution of (SVI) if Next we recall a definition concerning vector variational inequalities (see [27]). (ii) is * -upper hemicontinuous.