Equilibrium problems when the equilibrium condition is missing

Given a nonempty convex subset X of a topological vector space and a real bifunction f defined on X×X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \times X$$\end{document}, the associated equilibrium problem consists in finding a point x0∈X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x_0 \in X$$\end{document} such that f(x0,y)≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(x_0, y) \ge 0$$\end{document}, for all y∈X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y \in X$$\end{document}. A standard condition in equilibrium problems is that the values of f to be nonnegative on the diagonal of X×X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X \times X$$\end{document}. In this paper, we deal with equilibrium problems in which this condition is missing. For this purpose, we will need to consider, besides the function f, another one g:X×X→R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$g: X \times X \rightarrow \mathbb {R}$$\end{document}, the two bifunctions being linked by a certain compatibility condition. Applications to variational inequality problems, quasiequilibrium problems and vector equilibrium problems are given.

attempts led to the appearance of some concepts of generalized monotony (pseudomonotonicity, quasimonotonicity, properly quasimonotonicity) and to generalizations of the classical notion of upper semicontinuity (upper sign-continuity, local upper sign-continuity).
But, to the best of our knowledge, there is no existing result in the literature in which condition (F3) is missing. This is exactly the case we deal with in this paper. For this purpose, we will need to consider, besides the function f , another one g : X × X → R, the two bifunctions being linked by a certain compatibility condition. The method is not new. It can also be encountered in [7,14,17,25]. The compatibility condition used in this paper reads as follows: there exists a function h : X → X such that the following implication holds: x, y ∈ X, g(x, x) ≤ g(x, h(y)) ⇒ f (x, y) ≥ 0.
It is worth noting that the function h does not have to satisfy any continuity condition. The paper is organized as follows. Section 2 is devoted to the equilibrium problems when the equilibrium condition is replaced by the compatibility condition mentioned above. The existence of solutions is established first, for the case when the feasible set is compact and then, in the absence of compactness. In Sect. 3 we introduce a new concept of pseudomonotonicity that generalizes the classical one due to Karamardian. Under this condition of pseudomonotonicity, is established an existence result for the Minty variational inequality problem. We also obtain an existence theorem for quasi-equilibrium problems. Section 4 proposes a method, initiated by Oettli [25], to reduce a vector equilibrium problem to a scalar equilibrium problem.
At the end of this section we mention some notations, definitions and known results regarding set-valued mappings, which will be necessary during the paper. If X and Y are topological spaces and T : X ⇒ Y , we say that T is closed-valued if T (x) is a closed set for any x ∈ X . In the same way we define the terms compact-valued and convex-valued. If X and Y are topological spaces, a set-valued mapping T : X ⇒ Y is said to be: continuous, if it is both upper semicontinuous and lower semicontinuous; (iii) closed, if its graph (that is, the The following two lemmas collects well-known results concerning set-valued mappings. (i) [22] Let S, T : X ⇒ Y be two nonempty-valued mappings. If S is upper semicontinuous and compactvalued and T is closed, then S + T is a closed mapping. (ii) [9] If α : X → R is a continuous function and T : X ⇒ Y a compact closed set-valued mapping, then the set-valued mapping αT : In what follows, for a subset A of a topological vector space E, the standard notations conv A, cl A, int A designate respectively, the convex hull, the closure and the interior of A.

Equilibrium problems
The first theorem of this section is the main result of the paper. Theorem 2.1 Let X be a nonempty compact convex subset of a Hausdorff topological vector space and f, g be two real bifunctions defined on X × X . Assume that: (i) for each y ∈ X, the sets {x ∈ X : f (x, y) ≥ 0} and {(x, u) ∈ X × X : g(x, u) ≤ g(x, y)} are closed; (ii) for any x ∈ X , the function g(x, ·) is quasiconvex; (iii) there exists a function h : X → X such that the following implication holds: (1) Then, there exists x 0 ∈ X such that f (x 0 , y) ≥ 0, for all y ∈ X.
Proof For each y ∈ X , set Assume by way of contradiction that the conclusion of the theorem would be false. Then the family {D(y) : y ∈ X } covers the set X and it is composed by open subsets of X . Since X is compact, there exists a finite set {y 1 , . . . , y n } such that X = n i=1 D(y i ). Denote by z i = h(y i ) and by K the convex hull of the set {z 1 , . . . , z n }. Let {α 1 , . . . , α n } be a partition of unity on K subordinated to the open cover {D(y 1 ) ∩ K , . . . , D(y n ) ∩ K }.
Recall that this means that For each index i ∈ {1, . . . , n}, define the set-valued mapping G i : K ⇒ K , From (i), it follows easily that all the mappings G i are closed, hence upper semicontinuous and closed-valued. Moreover, the values of each mapping G i are nonempty (because, z i ∈ G i (x)) and convex (by (ii)). Consider the set-valued mapping T : K ⇒ K defined by Clearly, T has nonempty compact convex values. By Lemma 1.3, we infer that the set-valued mapping T is closed, hence upper semicontinuous. By the Kakutani fixed point theorem, there exists x ∈ K such that x ∈ T ( x). If then, x can be written as On the other hand, for each Remark 2.2 Note that condition (i) of Theorem 2.1 is satisfied when the bifunction f is upper semicontinuous in the first variable and g is jointly lower semicontinuous on X × X and upper semicontinuous in the first variable. But these conditions are only sufficient, not necessary.
Below we provide an example in which the assumptions of Theorem 2.1 are fulfilled even if g is neither lower semicontinuous on X × X , nor upper semicontinuous in the first variable.
Since the bifunction f is continuous in the first variable, the superlevel sets is a closed set.
We claim that assumption (iii) of Theorem 2.1 is satisfied.
In view of Theorem 2.1, the associated equilibrium problem has solutions. It can be easily verified that the unique solution is x 0 = 1. Note that on the diagonal of [−1, 1] × [−1, 1] the bifunction f takes both positive and negative values, consequently all existence results in which condition (F3) is needed, cannot be applied.
The study of equilibrium problems when the convex set X is not compact is usually based on a coercivity condition, as in the next theorem.

Theorem 2.4 Let X be a nonempty convex subset of a Hausdorff topological vector space and f, g : X × X → R two bifunctions such that:
(i) for each y ∈ X, the sets {x ∈ X : f (x, y) ≥ 0} and {(x, u) ∈ X × X : g(x, u) ≤ g(x, y)} are closed in X , respectively in X × X; (ii) for any x ∈ X , the function g(x, ·) is quasiconvex.
Moreover, suppose that there exist a compact convex subset C 0 of X , a function h : X → C 0 and a compact set K 0 ⊆ X such that condition (1) holds and Proof The proof goes along the same lines as the one of Theorem 4.3 in [5]. Denote by If C , C ∈ , then the set conv C ∪ C ) belongs also to , because the convex hull of the union of a finite family of compact convex sets is compact [1, Lemma 5.2.9]. Therefore, is a directed set relative to the order relation ⊆. Since the set K 0 is compact, we may assume that the net {x C } C∈ converges to some We now prove that f (x 0 , y) ≥ 0, for all y ∈ X . Take an arbitrary y ∈ X and denote by C y = conv C 0 ∪ {y} . Clearly, C y ∈ and for every C ∈ satisfying Remark 2.5 (a) Recall that a subset A of a topological space X is said to be compactly closed if for each compact subset K of X , the set A ∩ K is closed. It can be seen that assumption (i) in Theorem 2.4, can be replaced with a weaker condition, namely: for each y ∈ X , the sets {x ∈ X : f (x, y) ≥ 0} and {(x, u) ∈ X × X : g(x, u) ≤ g(x, y)} are compactly closed.
(b) A coercivity condition, introduced by Bianchi and Pini [10] and often used in equilibrium problems is the following: Observe that condition (iii) of Theorem 2.4 reduces to condition (CC) when K 0 = C 0 . Actually, the two conditions are equivalent if X is a convex subset of a Banach space. Indeed, in this case, if condition (iii) holds for the sets K 0 and C 0 then, by Mazur theorem, the closed convex hull of their union is compact and condition (CC) is satisfied for K = cl conv (K 0 ∪ C 0 ) . This reasoning does not work in any topological vector space. An example of compact set in an infinite dimensional space for which its closed convex hull is not compact can be found in [16].

Corollary 2.6
Let X be a nonempty convex subset of a Hausdorff topological vector space and f : X × X → R a bifunction that satisfies the following conditions: respectively in X × X; (ii) for any x ∈ X , the function f (x, ·) is quasiconvex; (iii) there exist a compact convex subset C 0 of X , a function h : X → C 0 and a compact set K 0 ⊆ X such that (iii 1 ) for each x ∈ X \K 0 there exists y ∈ C 0 such that f (x, y) < 0; (iii 2 ) the following implication holds: Proof It can be checked immediately that all the hypotheses of Theorem 2.4 are satisfied if the bifunction g is defined by Thus, Theorem 2.4 leads to the desired conclusion. To show that f also satisfies condition (iii) we will take In view of Corollary 2.6, the associated equilibrium problem has solutions. One can see that the solution set is [0, 1]. Let us note that neither for this example condition (F3) is fulfilled.

Variational inequalities and quasiequilibrium problems
Let E be a normed vector space and E * be its topological dual. Given a nonempty convex subset X of E and a set-valued mapping T : X ⇒ E * , the Minty variational inequality problem associated to X and T consists in finding a point x 0 ∈ X such that The Minty variational inequality problem was first considered by Debrunner and Flor in [13], where has been established a first existence result. With the help of Theorem 2.4, we shall obtain an existence theorem for this problem. To this aim we need to introduce first a new concept of pseudomonotonicity.
If T, Q : X ⇒ E * and h : X → X , we say that the set-valued mapping T is pseudomonotone with respect to the pair (Q, h) if for every (x, x * ) ∈ Gr Q, (y, y * ) ∈ Gr T , the following implication holds Clearly, when T = Q and h is the identity function on X , the above concept reduces to the classical notion of pseudomonotone operator.

Theorem 3.1 Let X and T be as above.
Assume that there exists a compact convex subset C 0 of X , a compact set K 0 ⊆ X and a weak* compact set-valued mapping Q : X ⇒ E * , norm-to-weak* lower semicontinuous and with nonempty weak* closed values such that (i) T is pseudomonotone with respect to (Q, h), for some function h : X → C 0 ; (ii) for each x ∈ X \K 0 there exists y ∈ C 0 such that inf y * ∈T (y) y * , y − x < 0.
Proof Since the set-valued mapping Q is weak* compact, its range Q(X ) is contained in a weak* compact subset L of E * . According to [6, Lemma 4.3] the duality pairing ·, · restricted to L × E is jointly continuous, when L is endowed with its weak* topology and E has its norm topology. Consider the bifunctions f, g : The desired conclusion results from Theorem 2.4 as soon as we show that all its assumptions are satisfied. Obviously, the sets {x ∈ X : f (x, y) ≥ 0} are closed in X , for all y ∈ X . Then, the bifunction g is quasiconvex (even convex) in the second variable as the upper envelope of the family of affine functions Let y ∈ X . Taking into account Remark 2.2, to prove that the set {(x, u) ∈ X × X : g(x, u) ≤ g(x, y)} is closed in X × X , it is sufficient to show that g is jointly lower semicontinuous on X × X and upper semicontinuous in the first variable. The upper semicontinuity of g(·, y) follows from [3,Theorem 2.5.2]. To prove that the bifunction g is lower semicontinuous on X × X , we show that for an arbitrary λ ∈ R, the set For any x * ∈ Q(x), since Q is norm-to-weak* lower semicontinuous, there exists a subsequence {x n i } of {x n } and a net {x * i } weak* convergent to x * , with Since the duality pairing ·, · is continuous in the product (norm × weak*) topology, we infer that x * , y − x} ≤ λ. Consequently, g(x, y) ≤ λ, hence the set M λ is closed in X × X .
Finally, let us note that, in view of (ii), the coerciveness condition (iii) of Theorem 2.2 is satisfied and thus the proof is complete.

Remark 3.2 A problem closely related to the Minty variational inequality problem is the so called Stampacchia variational inequality problem. This consists in finding a point
A solution x 0 of the Stampacchia variational inequality problem is called a strong solution, if the function x * that satisfies the above inequality does not depend on y. It is well-known (see, for instance, [18,Proposition 4.1]) that if the set-valued mapping T is upper sign-continuous (this means that, for all x ∈ X and v ∈ −x + X the following implication holds: , any solution of the Minty variational inequality problem is also solution for the Stampacchia variational inequality problem. Moreover, if the values of T are weak* compact and convex, it can be shown immediately that every solution of the Stampacchia variational inequality problem is actually a strong solution. Consequently, from Theorem 3.1 can be easily obtained a criterion for the existence of solutions (of strong solutions, respectively) for the Stampacchia variational inequality problem.
Given a bifunction f : X × X ⇒ R and a set-valued mapping T : X ⇒ X , we can formulate the following problem A problem like the one above is called a quasi-equilibrium problem. In general, to prove the existence of solution to a quasi-equilibrium problem, this is reduced to a fixed point problem (see, for instance [4,8,11,12,23,26]). The same method is used in what follows.

Theorem 3.3 Assume that X is a convex subset of a locally convex Hausdorff topological vector space,
T : X ⇒ X is a compact continuous set-valued mapping with nonempty closed and convex values and f, g : X × X → R are two bifunctions that satisfy the following conditions: (iv) for any x ∈ X , the function g(x, ·) is quasiconvex; (v) for all x, y ∈ X, g(x, x) ≤ g(x, y) ⇒ f (x, y) ≥ 0.
Proof Consider the set-valued mapping S : X ⇒ X , defined by From hypotheses, it follows that for any x ∈ X , the restrictions of f and g to T (x)×T (x) fulfill the assumptions of Theorem 2.1 (with h(y) = y for all y ∈ T (x)). Hence, there exists x ∈ T (x) such that f (x , y) ≥ 0, for all y ∈ T (x). Consequently, S has nonempty values. If x , x ∈ S(x) and λ ∈ [0, 1], since T (x) is a convex set, λx Further, we prove that S is a closed mapping. Let (x, x ) ∈ clS and {(x t , x t )} be a net in the graph of S converging to (x, x ). Then, for each index t, x t ∈ T (x t ) and, since T is a closed mapping (by Lemma 1.2 (i)), x ∈ T (x). If y ∈ T (x), since T is lower semicontinuous, there exists a subnet {x t i } of the net {x t } and a net {y i } converging to y, with Summing up, S is a closed mapping with nonempty convex values. Moreover, since T is compact, so will be S. From the Himmelberg fixed point theorem [19], S has a fixed point and thus the proof is complete.

Vector equilibrium problems
Following the method used in [14,17,25] we can study two types of vector equilibrium problems. Let us consider a locally convex Hausdorff topological vector space E with its topological dual E * and a closed convex cone C ⊂ E, with int C = ∅. We define on E the relations , ≺, , ⊀ as follows: e 0 ⇐⇒ e ∈ C; e ≺ 0 ⇐⇒ e ∈ − int C; e 0 ⇐⇒ e / ∈ C; The dual cone of C, denoted by C * , is defined as follows: where ·, · designates the duality pairing between E * and E. A subset B of the cone C * is called a base of C * , if 0 / ∈ clB and C * = λ>0 λB. It is known (see [21]) that, since C has nonempty interior, the dual cone C * has a weak* compact base B, and e 0 ⇔ ∀e * ∈ B, e * , e ≥ 0, respectively, Given a subset X of a topological vector space and a vector bifunction F : X × X → E we define two scalar bifunctions f m , f M : X × X → R as follows From (2) and (3), we infer that Based on these equivalents, from each result in Sect. 2 can be derived existence theorems for two types of vector equilibrium problems. Their conclusion will be, either there exists x 0 ∈ K 0 such that F(x 0 , y) 0, for all y ∈ X (for the first type) or, there exists x 0 ∈ K 0 such that F(x 0 , y) ⊀ 0, for all y ∈ X (for the second type). The problem of finding a point x 0 ∈ X that satisfies the first type (the second type, respectively) of conclusion is known in the literature as a vector equilibrium (weak vector equilibrium, respectively) problem. For illustration we give the next two results that corresponds to Theorem 2.4. Theorem 4.1 Let X be a nonempty convex subset of a Hausdorff topological vector space and F : X × X → E be a bifunction that satisfy the following condition: (i) for each y ∈ X, the set {x ∈ X : F(x, y) 0} is closed in X .
Assume that there exist a compact convex subset C 0 of X , a compact set K 0 ⊆ X , a bifunction g : X × X → R and a function h : X → C 0 such that (ii) for each y ∈ X, the set {(x, u) ∈ X × X : g(x, u) ≤ g(x, y)} is closed in X × X; (iii) for any x ∈ X , the function g(x, ·) is quasiconvex; (iv) for each x 0 ∈ X \K 0 there exists y ∈ C 0 such that F(x 0 , y) 0; (v) the following implication holds: Then, there exists x 0 ∈ K 0 such that F(x 0 , y) 0, for all y ∈ X. Theorem 4.2 Let X be a nonempty convex subset of a Hausdorff topological vector space and F : X × X → E be a bifunction that satisfy the following condition: (i) for each y ∈ X, the set {x ∈ X : F(x, y) ⊀ 0} is closed in X .
Assume that there exist a compact convex subset C 0 of X , a compact set K 0 ⊆ X , a bifunction g : X × X → R and a function h : X → C 0 such that (ii) for each y ∈ X, the set {(x, u) ∈ X × X : g(x, u) ≤ g(x, y)} is closed in X × X; (iii) for any x ∈ X , the function g(x, ·) is quasiconvex (iv) for each x 0 ∈ X \K 0 there exists y ∈ C 0 such that F(x 0 , y) ≺ 0; (v) the following implication holds: x, y ∈ X, g(x, x) ≤ g(x, h(y)) ⇒ F(x, y) ⊀ 0.
Then, there exists x 0 ∈ K 0 such that F(x 0 , y) ⊀ 0, for all y ∈ X. Remark 4.3 In vector optimization, we meet several concepts of cone continuity for vector-valued functions closely related by the classical concepts of lower and upper semicontinuity. Recall that (see [2]) a function ϕ : X → E is said to be C-upper semicontinuous at x 0 ∈ X , if for any neighborhood V ⊆ E of ϕ(x 0 ) there exists a neighborhood U of x 0 in X such that ϕ(x) ∈ V − C, for all x ∈ U . Furthermore, ϕ is said to be C-upper semicontinuous on X if it is C-upper semicontinuous at each x ∈ X . In [7] it is shown that conditions (i) in Theorems 4.1 and 4.2 are satisfied if for every y ∈ X , the function F(·, y) is C -upper semicontinuous on X .

Conclusions
In this paper, using fixed point techniques, are established existence results of solutions for equilibrium problems in both scalar and vector case. The particularity of our results consists in the fact that, unlike other results from the literature, the equilibrium condition is missing. Some example are given to highlight the importance of our achievements. In a future work our results could be exploited to study the existence of solutions for the variational-like inequality problems or for other problems that can be put in the format of an equilibrium problem in which the equilibrium condition does not appear.
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