Gap functions and Hausdorff continuity of solution mappings to parametric strong vector quasiequilibrium problems

In this paper, we consider parametric strong vector quasiequilibrium problems in Hausdorff topological vector spaces. Firstly, we introduce parametric gap functions for these problems, and study the continuity property of these functions. Next, we present two key hypotheses related to the gap functions for the considered problems and also study characterizations of these hypotheses. Then, afterwards, we prove that these hypotheses are not only sufficient but also necessary for the Hausdorff lower semicontinuity and Hausdorff continuity of solution mappings to these problems. Finally, as applications, we derive several results on Hausdorff (lower) continuity properties of the solution mappings in the special cases of variational inequalities of the Minty type and the Stampacchia type.

1. Introduction. One of the classes of problems in optimization, which has attracted attention of mathematicians all over the world, is the class of equilibrium problem. The equilibrium problem was named by Blum and Oettli [19] as a generalization of the variational inequality and optimization problems. This model has been proved to contain also other important problems related to optimization, namely, minimax problems, complementarity problems, Nash equilibrium, fixedpoint and coincidence-point problems, traffic network problems, etc. During the last two decades, there have been many papers devoted to equilibrium and related problems. The first and the most important topic is the existence conditions for this class of problems (see, e.g., [24,27,29,30,36,40,49], and the references therein). Another important topic is the stability and sensitivity analysis, including semicontinuity, continuity, Hölder/Lipschitz continuity and differentiability properties 66 LAM QUOC ANH AND NGUYEN VAN HUNG of the solution mappings to equilibrium and related problems. In fact, differentiability of the solution mappings is a rather high level of regularity and is somehow close to the Lipschitz continuous property (due to the Rademacher theorem). For generalized differentiation properties and applications to mathematical programs with equilibrium constraints, we would like to refer the reader to the two volume book by Mordukhovich [45,46]. For the efficient use of the solution mappings, of course the higher level of regularity such as Hölder/Lipschitz continuity (see, e.g., [1,2,4,5,7,8,10,11,15,16,17,38,39]) they possess, the better they are. However, to have a certain property of the solution mapping, usually the problem data needs to possess the same level of the corresponding property, and this assumption about the data is often not satisfied in practice. In addition, in a number of practical situations such as mathematical models for competitive economies, the semicontinuity of the solution mapping is enough for the efficient use of the models. Hence, the study of the semicontinuity and continuity properties of solution mappings in the sense of Berge and Hausdorff is among the most interesting and important topic in the stability and sensitivity analysis of equilibrium problem, see, e.g., [3,6,9,12,28,33,34,42,43,47,48,52].
It is well known that gap functions is an efficient approach to study existence conditions for problems related to optimization. Recently, gap functions have been used in stability conditions for such problems. In 1997, Zhao [50] introduced the key hypothesis (H 1 ) related to a gap function of optimization problem and showed that (H 1 ) was a sufficient condition for the Hausdorff lower semicontinuity of the solution mapping to the parametric nonlinear optimization problem. Then, by giving some characterizations of the hypothesis (H 1 ), Kien [32] sharpened the main results of Zhao. Motivated by [32,50], Li and Chen [41] and Chen and Li [20] presented hypothesis (H g ) and employed it to obtain sufficient conditions for the solution mapping to the parametric weak vector variational inequality in Banach spaces to be Hausdorff lower semicontinuous. By applying an assumption similar to (H g ), Chen et al. [21] improved and extended the results in [20] to the generalized vector quasivariational inequality. Answering an open question put forward in [21], Zhong and Huang [51] proved that the hypothesis (H g ) was a sufficient and necessary condition for Hausdorff lower semicontinuity of solution mapping to the set-valued weak vector variational inequality in Banach spaces. Recently, the gap function method and the hypothesis (H g ) were employed in [37] to derive the Hausdorff continuity property of the solution mapping of the scalar quasivariational inequality of the Minty type in finite dimensional spaces. Because of technical difficulties, as far as we know, there has been a single paper Zhong and Huang [52] devoted to the Hausdorff lower semicontinuity property of the solution map to equilibrium problems (in fact, the authors introduced hypothesis (H g ) , similar to the one in [51], then the authors proved that the hypothesis (H g ) was not only sufficient but also necessary for the Hausdorff lower semicontinuity of the solution mapping to the parametric weak vector quasiequilibrium problem in Banach spaces), and there have not been any works on Hausdorff lower semicontinuity and Hausdorff continuity of the solution mappings to strong equilibrium and related problems using above mentioned approaches.
Motivated and inspired by the above observations, in this paper, we use the gap function method to study the continuity property of solution mappings to parametric strong vector quasiequilibrium problems in Hausdorff topological vector spaces.
We introduce gap functions of such problems and investigate their properties. Similar to the above mentioned papers, we also present two key hypotheses related to the gap functions of the problems and study some characterizations of these hypotheses. Then, we prove that these hypotheses are not only sufficient but also necessary for the Hausdorff lower semicontinuity and Hausdorff continuity of solution mappings to the considered problems. As applications, we derive several results on the special cases of variational inequalities of the Minty type and the Stampacchia type. Our results are new and improve some key results in the literatures (e.g., [20,32,37,41,50]).
The paper is organized as follows. Sect. 2 is devoted to the setting of parametric quasiequilibrium problems and some preliminary results which are needed in the sequel. In Sect. 3, we introduce the gap functions for these problems and give some properties of these functions. In Sect. 4, we propose two key hypotheses (H p (γ 0 )) and (H h (γ 0 )) and obtain some characterizations of these hypotheses. Then we show that these conditions are sufficient and necessary for the solution mappings to the considered problems to be Hausdorff lower semicontinuous and Hausdorff continuous. As applications, we derive some results for the special cases of quasivariational inequalities of the Minty type and the Stampacchia type in the last section.

2.
Preliminaries. Let X, Y, Z, P be Hausdorff topological vector spaces, A ⊆ X, B ⊆ Y and Γ ⊆ P be nonempty subsets, and let C be a closed convex cone in Z with intC = ∅. Let K : A × Γ ⇒ A, T : A × Γ ⇒ B be multifunctions and f : A × B × A × Γ → Z be an equilibrium function, i.e., f (x, t, x, γ) = 0 for all x ∈ A, t ∈ B, γ ∈ Γ. Motivated and inspired by variational inequalities in the sense of Minty and Stampacchia, we consider the following two parametric strong vector quasiequilibrium problems. (QEP 1 ) Find x ∈ K(x, γ) such that f (x, t, y, γ) ∈ C, ∀y ∈ K(x, γ), ∀t ∈ T (y, γ).
Since the existence of solutions has been well studied, throughout the article, we assume that S 1 (γ) = ∅ and S 2 (γ) = ∅, for each γ in a neighborhood of the reference point. To provide our motivations for these settings, we discuss some special cases of the problems.
and (QEP 2 ) reduce to the variational inequalities of the Minty type and Stampacchia type (in short, (MVIP) and (SVIP)), respectively, studied in [25,31,35]. Firstly, we recall some basic definitions and some of their properties, see [9,13,18]. Let X, Y be topological vector spaces, A ⊂ X be a nonempty compact subset, and G : X ⇒ Y be a multifunction. G is said to be lower semicontinuous We say that G satisfies a certain property on a subset A ⊂ X if G satisfies it at every point of A.
Lemma 2.1. ( [13,18]). Let X and Y be topological vector spaces and G : X ⇒ Y be a multifunction.
and only if for each net {x α } ⊂ X which converges to x 0 and for each y 0 ∈ G(x 0 ), there is y α ∈ G(x α ) such that y α → y 0 . (iv) If G has compact values, then G is usc at x 0 if and only if, for each net {x α } ⊂ X which converges to x 0 and for each net 26,44]) For any fixed e ∈ intC, y ∈ Y and the nonlinear scalarization function ξ e : Y → R defined by ξ e (y) := min{r ∈ R : y ∈ re − C}, we have (i) ξ e is a continuous and convex function on Y ; 3. Gap functions for (QEP 1 ) and (QEP 2 ). In this section, we introduce the parametric gap functions for (QEP 1 ) and (QEP 2 ). Then we study some properties of these functions which will be needed in Sect. 4. In the rest of this section, we assume that f is continuous on Definition 3.1. A function g : A × Γ → R is said to be a parametric gap function for problem (QEP 1 ) ((QEP 2 ), respectively), if: Now we suppose that K and T have compact valued in a neighborhood of the reference point. We define two functions p : and Since K(x, γ) and T (x, γ) are compact sets for any (x, γ) ∈ A × Γ, ξ e and f are continuous, p and h are well-defined.
(a) It is easy to see that ϕ(x, t, γ) ≥ 0. Indeed, suppose to the contrary that there which is a contradiction. Hence,
The following result gives sufficient conditions for the parametric gap function p to be continuous. Proof. Since the proof techniques are similar, we discuss only the continuity of h. Firstly, we prove that h is lower semicontinuous on A × Γ. Let a ∈ R, and where ψ is defined as in Theorem 3.2. Since ξ e is continuous and K is continuous with compact values on A × Γ, Proposition 23, in Section 1 of Chapter 3 [13], deduces that ψ is continuous. By the compactness of T , there exists t α ∈ T (x α , γ α ) such that Let y 0 ∈ K(x 0 , γ 0 ) be arbitrary. As K is lower semicontinuous on A × Γ, there exists y α ∈ K(x α , γ α ) such that y α → y 0 . Since y α ∈ K(x α , γ α ), we have Since T is upper semicontinuous with compact values on A × Γ, we can assume that, there exists t 0 ∈ T (x 0 , γ 0 ) such that t α → t 0 . From the continuity of f and ξ e , taking the limit in (3), we have By the arbitrariness of y 0 , it follows from (4) that for some t 0 ∈ T (x 0 , γ 0 ). Hence, This proves that, for a ∈ R, the lower level set {(x, γ) : h(x, γ) ≤ a} is closed. Therefore, h is lower semicontinuous on A × Γ. Next, we will show that h is upper semicontinuous on A × Γ. Let a ∈ R, ξ e (−f (x α , t, y, γ α )) ≥ a, ∀t ∈ T (x α , γ α ).
Let t 0 ∈ T (x 0 , γ 0 ) be arbitrary. As T is lower semicontinuous on A × Γ, there exists Combining the compactness of K and the continuity of f and ξ e , there exists y α ∈ K(x α , γ α ) such that Since K is upper semicontinuous with compact values, we can assume that y α → y 0 , for some y 0 ∈ K(x 0 , γ 0 ) (take a subnet of {y α } if necessary). Taking the limit in (6), we obtain and hence, Since t 0 ∈ T (x 0 , γ 0 ) is arbitrary, it follows from (7) that i.e., for each a ∈ R, the upper level set {(x, γ) : h(x, γ) ≥ a} is closed. Hence, h is upper semicontinuous on A × Γ.
Since S 2 is closed on Γ, S 2 (γ) is a closed subset of A for all γ ∈ Γ. Hence, S 2 is compact-valued as A is compact.
As examples, we provide the following examples to illustrate the essentialness of the assumptions of Theorem 4.1(i).
which is not usc at γ = 0. otherwise.
Since K is usc with compact values and A is compact, we conclude that K is closed. So, for each γ ∈ Γ, E(λ) is a closed subset of A, and hence it is compact. Therefore, for each α, E(γ α ) \ (S 1 (γ α ) + U ) is compact. Since p is continuous, there exists . Clearly, (11) implies the fact that lim γα→γ0 p(x α , γ α ) = 0.
The following examples show that (H p (γ 0 )) in Theorem 4.2 is essential.

5.
Applications. Since the equilibrium problems contain optimization problems, variational inequalities, fixed-point and coincidence-point problems, the complementarity problems, the network traffic problems, etc, the results of the previous sections can be employed to derive the corresponding results for such special cases. In this section, as an application, we consider the special case of quasivariational inequalities of the types of Minty and Stampacchia, and apply some typical obtained results to this special setting. Let X, Y, Z, A, B, C, K, T be as in Sect. 2, L(X; Y ) be the space of all linear continuous operators from X into Y and g : A × Λ → A be a vector function. t, x denotes the value of a linear operator t ∈ L(X; Y ) at x ∈ X.
For each γ ∈ Γ, we denote the solution sets of the problems (MQVI) and (SQVI) by Φ(γ) and Ψ(γ), respectively. We assume that these solution sets are nonempty in a neighborhood of a reference point.
The following results are derived from the main results of Sect. 4.

Corollary 1.
Assume that A is compact, K and T are continuous with compact values in A × Γ, and g is continuous in A × Γ. Then, (i) Φ is Hausdorff lower semicontinuous on Γ if and only if (H p (γ 0 )) holds for all γ 0 ∈ Γ, (ii) Ψ is Hausdorff lower semicontinuous on Γ if and only if (H h (γ 0 )) holds for all γ 0 ∈ Γ.
Proof. Combining the continuity of g and (12), we conclude that f is continuous.
It is easy to see that all the assumptions of Theorem 4.2 are satisfied, and hence by applying this theorem we obtain the conclusions of Corollary 1.

Remark 3. (a)
In special case, X = R n , Y = R m , C = R + , g(x, γ) = x, the problem (MQVI) reduces to the parametric scalar quasivariational inequality of the Minty type which was studied in [37]. As our parametric gap function p(x, γ) is new and different from the existing ones in the literature, Corollary 1(i) is new and extends Theorem 4.4 in [37]. Moreover, our hypothesis (H p ) is not only sufficient but also necessary for the Hausdorff lower semicontinuity of the solution map to (MQVI), while the assumption (H g ) in [37] is only sufficient for the corresponding property of the solution map to the considered problem. (b) As far as we know, there have not been any works on Hausdorff lower semicontinuity of the solution map to parametric strong quasivariational inequalities of the Stampacchia type, and hence, even for the special case, Corollary 1(ii) is new.
For winding up this section, we present the sufficient and necessary conditions for Hausdorff continuity of the solution mappings to the problems (MQVI) and (SQVI).