A Note on Scalar-Valued Gap Functions for Generalized Vector Variational Inequalities

and Applied Analysis 3 Since F and g are continuous,K(x) is a closed set. Moreover, since sol (GVVI) = ⋂ x∈K K (x) , (17) we get that sol (GVVI) is a closed set. The proof is complete. Taking g = 0 in Lemma 1, we can easily get the following result. Corollary 2 (see [9]). The following properties hold. (i) ⋃ ξ∈intR + sol (VI) ξ ⊂ sol (VVI) = ⋃ ξ∈R m + \{0} sol (VI) ξ . (18) (ii) If F i is a continuous function for every i, then sol (VVI) is a closed set. 3. Gap Functions for (GVVI) and (VVI) In this section, we propose some new gap functions for (GVVI). Now, we first introduce the definitions of gap functions for (GVVI) and (VVI). Definition 3. A real-valued function ψ : X → R is said to be a scalar-valued gap function of (GVVI) if it satisfies the following conditions: (i) ψ(x) ≥ 0, for any x ∈ K; (ii) ψ(x 0 ) = 0 if and only if x 0 ∈ K is a solution of (GVVI). Definition 4. A real-valued function φ : X → R is said to be a scalar-valued gap function of (VVI) if it satisfies the following conditions: (i) φ(x) ≥ 0, for any x ∈ K; (ii) φ(x 0 ) = 0 if and only if x 0 ∈ K is a solution of (VVI). Now, by using Lemma 1 and Corollary 2, we generalize the gap function introduced by Auslender [17] for scalar variational inequalities to the case of vector variational inequalities. The gap functions for (GVVI) and (VVI) are defined by ψ (x) = inf ξ∈S m {sup


Introduction
The concept of vector variational inequalities was firstly introduced by Giannessi [1] in a finite-dimensional space.Since then, extensive study of vector variational inequalities has been done by many authors in finite-or infinitedimensional spaces under generalized monotonicity and convexity assumptions.See [2][3][4][5][6][7][8][9][10] and the references therein.Among solution approaches for vector variational inequalities, scalarization is one of the most analyzed topics at least from the computational point of view; see [8][9][10].
Gap functions are very useful for solving vector variational inequalities.One advantage of the introduction of gap functions in vector variational inequalities is that vector variational inequalities can be transformed into optimization problems.Then, powerful optimization solution methods and algorithms can be applied for finding solutions of vector variational inequalities.Recently, some authors have investigated the gap functions for vector variational inequalities.Yang and Yao [11] introduced gap functions and established necessary and sufficient conditions for the existence of a solution of vector variational inequalities.Chen et al. [12] extended the theory of gap function for scalar variational inequalities to the case of vector variational inequalities.They also obtained the set-valued gap functions for vector variational inequalities.Li and Chen [13] introduced setvalued gap functions for a vector variational inequality and obtained some related properties.Li et al. [14] investigated differential and sensitivity properties of set-valued gap functions for vector variational inequalities and weak vector variational inequalities.Meng and Li [15] also investigated the differential and sensitivity properties of set-valued gap functions for Minty vector variational inequalities and Minty weak vector variational inequalities.
The purpose of this paper is to define a single variable gap function for generalized vector variational inequalities by using the scalarization approach.To this end, we first transform the generalized vector variational inequality into an equivalent scalar variational inequality by using the scalarization approach of [9].Then, we establish the relations between vector variational inequalities and variational inequalities.Finally, we apply the results to obtain gap functions for generalized vector variational inequalities.
In this paper, we consider the following generalized vector variational inequality (GVVI): The solution set of (GVVI) is denoted by sol (GVVI).
Lemma 1.The following properties hold.
(i) If   is an affine function for every , then, (ii) If   and   are continuous functions for every , then sol (GVVI) is a closed set.
Taking  = 0 in Lemma 1, we can easily get the following result.
(ii) If   is a continuous function for every , then sol (VVI) is a closed set.

Gap Functions for (GVVI) and (VVI)
In this section, we propose some new gap functions for (GVVI).Now, we first introduce the definitions of gap functions for (GVVI) and (VVI).
Definition 3. A real-valued function  :  →  is said to be a scalar-valued gap function of (GVVI) if it satisfies the following conditions: (i) () ≥ 0, for any  ∈ ; (ii) ( 0 ) = 0 if and only if  0 ∈  is a solution of (GVVI).
Definition 4. A real-valued function  :  →  is said to be a scalar-valued gap function of (VVI) if it satisfies the following conditions: (i) () ≥ 0, for any  ∈ ; (ii) ( 0 ) = 0 if and only if  0 ∈  is a solution of (VVI).Now, by using Lemma 1 and Corollary 2, we generalize the gap function introduced by Auslender [17] for scalar variational inequalities to the case of vector variational inequalities.The gap functions for (GVVI) and (VVI) are defined by respectively.The symbol   in the above expression denotes the unit simplex in   + ; that is, it is given as The use of   in the above expression is to stress the fact that the vector  ̸ = 0, and we just express the normalized version.Further, use of   has an advantage since if additionally  is compact and each   is convex for any  = 1, 2, . . ., , then, the functions  and  are finite.This means that  solves (GVI)  .Thus, using Lemma 1, we get that  is a solution of (GVVI).