Existence of Solutions for Generalized Vector Quasi-Equilibrium Problems by Scalarization Method with Applications

Abstract

The aim of this paper is to study generalized vector quasi-equilibrium problems (GVQEPs) by scalarization method in locally convex topological vector spaces. A general nonlinear scalarization function for set-valued mappings is introduced, its main properties are established, and some results on the existence of solutions of the GVQEPs are shown by utilizing the introduced scalarization function. Finally, a vector variational inclusion problem is discussed as an application of the results of GVQEPs.

1. Introduction

Recently, various vector equilibrium problems were investigated by adopting many different methods, such as the scalarization method (e.g., [1, 2]), the recession method (e.g., [3]), and duality method (e.g., [4]).

The scalarization method is an important and efficacious tool of translating the vector problems into the scalar problems. In 1992, Chen [5] translated a vector variational inequality into a classical variational inequality by providing a kind of solution conceptions with variable domination structures. Gerth and Weidner [6] solved a vector optimization problem by introducing a scalarization function with a variable and Gong [1] dealt with vector equilibrium problems by using the scalarization function defined in [6]. By constructing new nonlinear scalarization functions with two variables, Chen and Yang [7] discussed a vector variational inequality and Chen et al. [2] investigated a generalized vector quasi-equilibrium problem (GVQEP), respectively. In addition, the authors in [8, 9] studied the systems of vector equilibrium problems by the scalarization method since the gap functions, indeed established by the nonlinear scalarization function defined in [2], were adopted.

In this paper, we will discuss the GVQEPs by utilizing scalarization method. The essential preliminaries are listed in Section 2. On the basis of the works in [2, 6, 7], a general nonlinear scalarization function of a set-valued mapping is produced under a variable ordering structure and its main properties are discussed in Section 3. The results of properties for the general nonlinear scalarization function generalized the corresponding ones in [2]. In Section 4, some results on the existence of solutions of the GVQEPs are proved by employing the scalarization function introduced in Section 3. The GVQEPs are different from the one in [10] and include the one in [2] as a special case. It is worth mentioning that the existence results of solutions for the GVQEPs extend the corresponding one in [2]. Finally, a vector variational inclusion problem (VVIP) is given as an application of the GVQEPs in Section 5.

Suppose that 𝑋 and 𝑌 are topological vector spaces (TVSs). The subset 𝐷⊂𝑌 is called a cone, if 𝜆𝑥∈𝐷 for all 𝑥∈𝐷 and 𝜆>0. A cone 𝐷⊂𝑌 is said to be proper, if 𝐷≠𝑌. 𝐴 is called 𝐷-closed [11] if 𝐴+𝐾 is closed and 𝐷-bonded [11] if for each neighborhood 𝑈 of zero in 𝑌, there exists 𝜆>0 such that 𝐴⊂𝑈+𝐷. A set-valued mapping 𝐺∶𝑋→2^𝑌 is called strict, if 𝐺(𝑥)≠∅ for any 𝑥∈𝑋. Throughout this paper, ℝ denotes by the set of the real numbers. Several notations are also listed as follows: 𝐺(𝐴)=_𝑢∈𝐴𝐺(𝑢),,,𝑥−𝐵=𝑥−𝑦∶𝑦∈𝐵𝐴−𝐵=𝑥−𝑦∶𝑥∈𝐴,𝑦∈𝐵 (1) where 𝐴,𝐵⊂𝑋 are nonempty, 𝑥∈𝑋 and 𝐺∶𝑋→2^𝑌. Incidentally, every TVS is Hausdorff (see [12]).

2. Preliminaries

Let 𝑋 and 𝑌 be topological spaces and 𝐸⊂𝑋 a nonempty subset. A function 𝑔∶𝑋→ℝ is said to be upper semicontinuous (usc for brevity) on 𝑋, if {𝑢∈𝑋∶𝑔(𝑢)<𝜆} is open for each 𝜆∈ℝ; to be lower semicontinuous (lsc for brevity), {𝑢∈𝑋∶𝑔(𝑢)>𝜆} is open for each 𝜆∈ℝ. In addition, some known notions of continuity and closeness for a set-valued mapping are given (see [13]). A set-valued mapping 𝐺∶𝐸→2^𝑌 is said to be usc at 𝑢₀∈𝐸, if for any neighborhood 𝑁(𝐺(𝑢₀)) of 𝐺(𝑢₀), there exists a neighborhood 𝐵(𝑢₀) of 𝑢₀ such that 𝐺(𝑢)⊂𝑁(𝐺(𝑢₀)) for all 𝑢∈𝐵(𝑢₀); to be lsc at 𝑢₀∈𝐸, if for any 𝑦₀∈𝐺(𝑢₀) and any neighborhood 𝑁(𝑦₀) of 𝑦₀, there exists a neighborhood 𝐵(𝑢₀) of 𝑢₀ such that 𝐺(𝑢)∩𝑁(𝑦₀)≠∅ for all 𝑢∈𝐵(𝑢₀); to be usc (resp., lsc) on 𝐸, if 𝐺 is usc (resp., lsc) at each 𝑢∈𝐸; to be continuous at 𝑢₀∈𝐸 (resp., on 𝐸), if 𝐺 is usc and lsc at 𝑢₀ (resp., on 𝐸); to be closed, if its graph Graph(𝐺)={(𝑢,𝑦)∈𝐸×𝑌∶𝑦∈𝐺(𝑢)} is closed in 𝐸×𝑌.

Let 𝑌 and 𝑍 be real TVSs, 𝐴⊂𝑌 a nonempty subset, and 𝐷 a convex cone in 𝑌. 𝑦∈𝐴 is called, vector minimal point (resp., weakly vector minimal point) of 𝐴, if 𝑦−𝑦∉−𝐷⧵{0} (resp., 𝑦−𝑦∉−int𝐷) for each 𝑦∈𝐴. The set of vector minimal points (resp., weakly vector minimal points) of 𝐴 is denoted by Min_𝐷𝐴   (resp., 𝑤Min_𝐷𝐴).

Definition 1. Let 𝐵⊂𝑍 be nonempty convex subset. 𝐺∶𝐵→2^𝑌 is called generalized 𝐷-quasiconvex, if for any 𝑦∈𝑌, the set {𝑢∈𝐵∶𝐺(𝑢)⊂𝑦−𝐷} is convex (here, ∅ is regarded as a convex set).

The generalized 𝐷-quasiconvexity and the 𝐷-quasiconvexity introduced in [14] for the set-valued mappings are both generalizations of 𝐷-quasiconvexity for the single-valued mapping introduced in [11], but they are indeed distinct. See the following example.

Example 2. Let 𝑌=ℝ²,𝑍=𝐵=ℝ, and 𝐷=ℝ²₊ and define 𝐺₁,𝐺₂∶𝐵→2^𝑌 as 𝐺₁(𝑢)=𝑥₁,𝑥₂∶𝑥₁=𝑢,−𝑢²≤𝑥₂,𝐺≤0₂(𝑢)=𝑥₁,𝑥₂∶𝑥₁=𝑢,0≤𝑥₂≤|sin𝑢|. (2) Then 𝐺₁ is generalized 𝐷-quasiconvex but not 𝐷-quasiconvex, while 𝐺₂ is just the reverse. Indeed, for any fixed 𝑦=(𝑥₁,𝑥₂)∈𝑌, both the sets 𝑢∈𝐵∶𝐺₁(𝑢)⊂𝑦−𝐷=∅,if𝑥₂<0,𝑥₁∈ℝ,−∞,𝑥₁,otherwise,𝑢∈𝐵∶𝐺₂(𝑢)∩𝑦−𝐷≠∅=∅,if𝑥₂<0,𝑥₁∈ℝ,−∞,𝑥₁,otherwise (3) are convex. But the set 𝑢∈B∶𝐺₁(𝑢)∩((2,−1)−𝐷)≠∅=(−∞,−1]∪[1,2] (4) is not convex and neither is the set 𝑢∈𝐵∶𝐺₂(𝑢)⊂(0,0)−𝐷={−𝑛𝜋∶𝑛=0,1,2,…}. (5)

Lemma 3 (see [15]). Let 𝑋 and 𝑌 be topological spaces and 𝑆∶𝑋→2^𝑌 a set-valued mapping. (1) If 𝑆 is usc with closed values, then 𝑆 is closed.(2) If 𝑋 is compact and 𝑆 is usc with compact values, then 𝑆(𝑋) is compact.

Lemma 4 (see [13]). Let 𝑋 and 𝑌 be Hausdorff topological spaces and 𝐸⊂𝑋 a nonempty compact set and let ℎ∶𝐸×𝑌→ℝ be a function and 𝐺∶𝐸→2^𝑌 a set-valued mapping. If ℎ is continuous on 𝐸×𝑌 and 𝐺 is continuous with compact values, then the marginal function 𝑉(𝑢)=sup_{𝑦∈𝐺(𝑢)}ℎ(𝑢,𝑦) is continuous and the marginal set-valued mapping 𝛿(𝑢)={𝑦∈𝐺(𝑢)∶𝑉(𝑢)=ℎ(𝑢,𝑦)} is usc.

Lemma 5 (Kakutani, see [13, Theorem 13 in Section 4 Chapter 6]). Let 𝐸 be a nonempty compact and convex subset of a locally convex TVS 𝑋. If Φ∶𝐸→2^𝐸 is usc and Φ(𝑥) is a nonempty, convex, and closed subset for any 𝑥∈𝐸, then there exists 𝑥∈𝐸 such that 𝑥∈Φ(𝑥).

3. A General Nonlinear Scalarization Function

From now on, unless otherwise specified, let 𝑋,𝑌, and 𝑍 be real TVSs and 𝐸⊂𝑋 and 𝐹⊂𝑍 nonempty subsets. Let 𝐶∶𝐸→2^𝑌 be a set-valued mapping such that for any 𝑥∈𝐸, 𝐶(𝑥) is a proper, closed, and convex cone with int𝐶(𝑥)≠∅.

In this section, suppose that 𝐺∶𝐹→2^𝑌 is a strict mapping with compact values and 𝑒∶𝐸→𝑌 is a vector-valued mapping with 𝑒(𝑥)∈int𝐶(𝑥),forall𝑥∈𝐸. Obviously, 𝐺(𝑢) is 𝐶(𝑥)-closed [11, Definition 3.1 and Proposition 3.3] for each 𝑥∈𝐸 and 𝑢∈𝐹. Then, in view of [16, Lemma 3.1], we can define a general nonlinear scalarization function of G as follows.

Definition 6. The general nonlinear scalarization function 𝜉_𝐺∶𝐸×𝐹→ℝ, of 𝐺 is defined as 𝜉_𝐺(){(𝑢)(𝑥)(𝑥,∀()𝑥,𝑢=min𝜆∈ℝ∶𝐺⊂𝜆𝑒−𝐶)}𝑥,𝑢∈𝐸×𝐹. (6)

Remark 7. If 𝑋=𝑌=𝑍=𝐸=𝐹 and 𝐺(𝑢)={𝑢},forall𝑢∈𝐹, then the general nonlinear scalarization function 𝜉_𝐺 of 𝐺 becomes the nonlinear scalarization function defined in [2].

Example 8. Let 𝑋=𝑍=ℝ,𝑌=ℝ², and 𝐸=𝐹=[0,+∞)⊂𝑋 and let 𝐺,𝐶∶𝐸→2^𝑌,𝑒∶𝐸→𝑌 define as 𝐺(𝑢)=)}𝐶(𝑥)=)3{(𝑈,𝑉∶𝑈−𝑢≤𝑉≤𝑢−𝑈,0≤𝑈≤𝑢,∀𝑢∈𝐹,(𝑈,𝑉∶0≤𝑉≤2𝑒(𝑥)=()+𝑥𝑈,𝑈≥0,∀𝑥∈𝐸,1,1,∀𝑥∈𝐸, (7) respectively. We can see that 𝜉_𝐺(𝑥,𝑢)=(1+(2/(1+2𝑥)))𝑢.

Definition 9. Let 𝐷⊂𝑌 be a cone, 𝐻∶𝐹→2^𝑌, and 𝜍∶𝐹→ℝ. 𝜍 is called monotone (resp., strictly monotone) with respect to (wrt for brevity) 𝐻, if for any 𝑢₁,𝑢₂∈𝐹, 𝐻(𝑢₁)−𝐻(𝑢₂)⊂int𝐷 implies that 𝜍(𝑢₂)≤𝜍(𝑢₁)(resp.,𝜍(𝑢₂)<𝜍(𝑢₁)).

Obviously, the strict monotonicity wrt 𝐻 implies the monotonicity wrt 𝐻. Moreover, if 𝐻(𝑢)={𝑢},forall𝑢∈𝐹, then the (strict) monotonicity wrt 𝐻 of 𝜍 is equivalent to the (strict) monotonicity under the general order structures. The following examples illuminate the relationship between the monotonicity wrt 𝐻 and the monotonicity in the normal sense when 𝐻 is not the identity mapping.

Example 10. Let 𝑌=𝑍=ℝ,𝐹=[0,10], and 𝐷=ℝ₊ and let 𝜍(𝑢)=[],(],𝐻(𝑢)=[][),[][](].𝑢,𝑢∈0,50,𝑢∈5,10𝑢,𝑢+1,𝑢∈0,50,6,𝑢=5,0,10−𝑢,𝑢∈5,10 (8) Then 𝐻(𝑢₁)−𝐻(𝑢₂)⊂int𝐷 implies that 𝜍(𝑢₁)>𝜍(𝑢₂). Thus 𝜍 is strictly monotone wrt 𝐻, while 𝜍 is not monotone in the normal sense.

Example 11. Let 𝑌=𝑍=ℝ,𝐹=[0,5], and 𝐷=ℝ₊ and let 𝜍(𝑢)=𝑢,forall𝑢∈𝐹, and 𝐻(𝑢)=[−𝑢−1,−𝑢],forall𝑢∈𝐹. Then 𝜍 is strictly monotone in the normal sense, but 𝜍 is not monotone wrt 𝐻. As the case stands, 𝐻(𝑢₁)−𝐻(𝑢₂)⊂int𝐷 is equivalent to 𝑢₁,𝑢₂∈𝐹 and 𝑢₂−𝑢₁>1, which just results in 𝜍(𝑢₁)<𝜍(𝑢₂).
Now some main properties of the general nonlinear scalarization function are established. First, according to [16, Proposition 3.1], we have the following.

Theorem 12. For each 𝜆∈ℝ,𝑥∈𝐸 and 𝑢∈𝐹, the following assertions hold.(1)𝜉_𝐺(𝑥,𝑢)<𝜆⇔𝐺(𝑢)⊂𝜆𝑒(𝑥)−int𝐶(𝑥). (2)𝜉_𝐺(𝑥,𝑢)≤𝜆⇔𝐺(𝑢)⊂𝜆𝑒(𝑥)−𝐶(𝑥).

Theorem 13. 𝜉_𝐺 is strictly monotone wrt 𝐺 in the second variable.

Proof. Letting 𝑢₁,𝑢₂∈𝐹 such that 𝐺(𝑢₁)−𝐺(𝑢₂)⊂int𝐶(𝑥) and 𝜆=𝜉_𝐺(𝑥,𝑢₁), we have 𝐺𝑢₂⊂𝐺𝑢₁(𝑥)(𝑥)(𝑥)(𝑥)(𝑥)(𝑥).−int𝐶⊂𝜆𝑒−𝐶−int𝐶byDefinition6⊂𝜆𝑒−int𝐶 (9) It is from this assertion that 𝜉_𝐺(𝑥,𝑢₂)<𝜆=𝜉_𝐺(𝑥,𝑢₁) by Theorem 12 (1).

Theorem 14. Suppose that 𝑒 is continuous.(1) If 𝑊 and 𝐺 are usc on 𝐸 and 𝐹, respectively, where 𝑊(𝑥)=𝑌⧵int𝐶(𝑥),∀𝑥∈𝐸, (10) then 𝜉_𝐺 is usc on 𝐸×𝐹.(2) If 𝐶 is usc on 𝐸 and 𝐺 is lsc on 𝐹, then 𝜉_𝐺 is lsc on 𝐸×𝐹.

Proof. (1) It is sufficient to attest the fact that for each 𝜆∈ℝ, the set 𝐴=(𝑥,𝑢)∈𝐸×𝐹∶𝜉_𝐺(𝑥,𝑢)≥𝜆 (11) is closed. As a matter of fact, for any sequence {(𝑥_𝑘,𝑢_𝑘)} in 𝐴 such that (𝑥_𝑘,𝑢_𝑘)→(𝑥₀,𝑢₀) as 𝑘→∞, 𝜉_𝐺(𝑥_𝑘,𝑢_𝑘)≥𝜆, that is, 𝐺(𝑢_𝑘)⊄𝜆𝑒(𝑥_𝑘)−int𝐶(𝑥_𝑘), by Theorem 12 (1). Obviously, 𝐹₀={𝑢,𝑢₁,𝑢₂,…}⊂𝐹 is compact and so is 𝐺(𝐹₀) by Lemma 3 (2). Then there exists 𝑦_𝑘∈𝐺(𝑢_𝑘)⊂𝐺(𝐹₀) such that 𝑦_𝑘∉𝜆𝑒(𝑥_𝑘)−int𝐶(𝑥_𝑘) and a subsequence {𝑦_𝑘𝑖} of {𝑦_𝑘}, such that 𝑦_𝑘𝑖→𝑦₀∈𝐺(𝐹₀) as 𝑖→∞ and 𝜆𝑒𝑥_𝑘𝑖−𝑦_𝑘𝑖∉int𝐶𝑥_𝑘𝑖,thatis,𝜆𝑒𝑥_𝑘𝑖−𝑦_𝑘𝑖∈𝑊𝑥_𝑘𝑖. (12) Obviously, 𝐺 is usc with closed values, which implies that 𝐺 is closed by Lemma 3 (1). Hence, 𝑦₀∈𝐺(𝑢₀). Similarly, 𝑊 is closed. Letting 𝑖→∞ in (12) and applying the continuity of 𝑒 and the closeness of 𝑊, we obtain that 𝜆𝑒(𝑥₀)−𝑦₀∈𝑊(𝑥₀). Namely, 𝑦₀∉𝜆𝑒(𝑥₀)−int𝐶(𝑥₀). Thus 𝐺(𝑢₀)⊄𝜆𝑒(𝑥₀)−int𝐶(𝑥₀), which is equivalent to 𝜉_𝐺(𝑥₀,𝑢₀)≥𝜆 by Theorem 12 (1). Therefore, 𝜆∈𝐴 and 𝐴 is closed.
(2) It's enough to argue that for each 𝜆∈ℝ, the set 𝐵=(𝑥,𝑢)∈𝐸×𝐹∶𝜉_𝐺(𝑥,𝑢)≤𝜆 (13) is closed. In fact, for any sequence {(𝑥_𝑘,𝑢_𝑘)} in 𝐵 such that (𝑥_𝑘,𝑢_𝑘)→(𝑥₀,𝑢₀) as 𝑘→∞, by Theorem 12 (2), 𝜉_𝐺(𝑥_𝑘,𝑢_𝑘)≤𝜆, equivalently, 𝐺(𝑢_𝑘)⊂𝜆𝑒(𝑥_𝑘)−𝐶(𝑥_𝑘). For any 𝑦₀∈𝐺(𝑢₀), there exists 𝑦_𝑘∈𝐺(𝑢_𝑘) such that 𝑦_𝑘→𝑦₀ as 𝑘→∞ according to the equivalent definition of lower semicontinuity of 𝐺 (see [13], page 108). Also, 𝜆𝑒𝑥_𝑘−𝑦_𝑘∈𝐶𝑥_𝑘. (14) Linking the continuity of 𝑒 with the closeness of 𝐶 inferred from Lemma 3 (1) and letting 𝑘→∞ in (14), we get 𝜆𝑒(𝑥₀)−𝑦₀∈𝐶(𝑥₀) and 𝐺(𝑢₀)⊂𝜆𝑒(𝑥₀)−𝐶(𝑥₀). Consequently, Theorem 12 (2) leads to 𝜉_𝐺(𝑥₀,𝑢₀)≤𝜆 which amounts to (𝑥₀,𝑢₀)∈𝐵. 𝐵 is closed.
If 𝑋=𝑌=𝑍=𝐸=𝐹 (𝐹 is not required to be compact) and 𝐺(𝑢)={𝑢},forall𝑢∈𝐹, then Theorem 14 becomes [2, Theorem 2.1]. Actually, [2, Theorem 2.1] can be regarded as the case where 𝐺 is continuous in Theorem 14. In addition, Example 2.1 (resp., Example 2.2) in [2] shows that if 𝐶 (resp., 𝑊) is not usc, maybe the general nonlinear scalarization function fails to be lsc (resp., usc) under all the other assumptions. Now the following example demonstrates that the assumption of the upper semicontinuity (resp., lower semicontinuity) of 𝐺 is necessary in Theorem 14 (1) (resp., (2)) even if 𝐶 is continuous.

Example 15. Let 𝑋=𝑍=ℝ, Y=ℝ², 𝐸=ℝ₊, and 𝐹=[−10,10]⊂𝑍, and let 𝑒(𝑥)=()𝐶(𝑥)=)31,1,∀𝑥∈𝐸,(𝑈,𝑉∶0≤𝑉≤2+𝑥𝑈,𝑈≥0,∀𝑥∈𝐸. (15)(16)
(1) Define 𝐺(𝑢)=[](],[].𝑢,𝑢+1,𝑢∈0,100,𝑢∈−10,0 (17) Evidently, 𝐺 is not usc on 𝐹. After simply calculating, 𝜉_𝐺(𝑥,𝑢)=𝑢+1,𝑥∈𝐸,0<𝑢≤10,0,𝑥∈𝐸,−10≤𝑢≤0. (18)𝜉_𝐺 is not usc on 𝐸×𝐹 due to the fact that {(𝑥,𝑢)∈𝐸×𝐹∶𝜉(𝑥,𝑢)≥1}=ℝ₊×(0,10] is not closed.
(2) Consider the following mapping: 𝐺(𝑢)=[][],[).𝑢,𝑢+1,𝑢∈0,100,𝑢∈−10,0 (19) Obviously, 𝐺 is not lsc on 𝐹. Also, 𝜉_𝐺 fails to be lsc on 𝐸×𝐹, where 𝜉_𝐺(𝑥,𝑢)=𝑢+1,𝑥∈𝐸,0≤𝑢≤10,0,𝑥∈𝐸,−10≤𝑢<0. (20)

4. Existence Results on Solutions of the GVQEPs

In this section, further suppose that 𝑋,𝑌, and 𝑍 are locally convex. Let 𝑔∶𝐹→𝐸 be a vector-valued mapping. Now two GVQEPs are characterized as follows: GVQEP1: Seek 𝑥∈𝑃(𝑥) and 𝑧∈𝑄(𝑥) such that 𝑓(𝑣,𝑧)−𝑓(𝑥,𝑧)⊄−int𝐶𝑔(𝑧),∀𝑣∈𝑃(𝑥), (21) where 𝑃∶𝐸→2^𝐸,𝑄∶𝐸→2^𝐹, and 𝑓∶𝐸×𝐹→2^𝑌 are set-valued mappings. GVQEP2: Find 𝑥∈𝑃(𝑥) and 𝑧∈𝑄(𝑥) such that 𝑓(𝑣,𝑥,𝑧)−𝑓(𝑥,𝑥,𝑧)⊄−int𝐶𝑔(𝑧),∀𝑣∈𝑃(𝑥), (22) where 𝑃∶𝐸→2^𝐸,𝑄∶𝐸→2^𝐹, and 𝑓∶𝐸×𝐸×𝐹→2^𝑌 are set-valued mappings.

It's worth noting that the GVQEP considered in [2] is just the special case of the GVQEP2 (when 𝑓 is single valued).

Lemma 16. Let 𝑒∶𝐸→𝑌 be a vector-valued mapping such that 𝑒(𝑥)∈int𝐶(𝑥) for all 𝑥∈𝐸 and 𝑓∶𝐸×𝐹→2^𝑌 a set-valued mapping and define 𝑓_𝑧(𝑥)=𝑓(𝑥,𝑧) for each 𝑧∈𝐹. If for each 𝑧∈𝐹, 𝑓_𝑧 is generalized 𝐶(𝑔(𝑧))-quasiconvex with compact values, then 𝑥↦ℎ(𝑥,𝑧) is ℝ₊-quasiconvex, where ℎ(𝑥,𝑧)=𝜉_𝑓𝑧(𝑔(𝑧),𝑥) and 𝜉_𝑓𝑧 is the general nonlinear scalarization function of 𝑓_𝑧.

Proof. It's sufficient to testify that 𝐿_𝑧(𝜆)⊂𝑋 is convex, where 𝐿_𝑧(𝜆)=𝑥∈𝐸∶𝜉_𝑓𝑧𝑔(𝑧),𝑥≤𝜆 (23) for each 𝑧∈𝐹 and 𝜆∈ℝ. Indeed, for any 𝑥₁,𝑥₂∈𝐿_𝑧(𝜆), 𝜉_𝑓𝑧𝑔(𝑧),𝑥_𝑖≤𝜆,𝑖=1,2,thatis,𝑓_𝑧𝑥_𝑖⊂𝜆𝑒𝑔(𝑧)−𝐶𝑔(𝑧),𝑖=1,2. (24) Clearly, for each 𝑧∈𝐸,𝑥₁,𝑥₂∈𝑀, where 𝑀=𝑥∈𝐸∶𝑓_𝑧(𝑥)⊂𝜆𝑒𝑔(𝑧)−𝐶𝑔(𝑧). (25)𝑀 is convex by reason of the generalized 𝐶(𝑔(𝑧))-quasiconvexity of 𝑓_𝑧 and so 𝑥^∼=𝑡𝑥₁+(1−𝑡)𝑥₂∈𝑀 for all 0≤𝑡≤1. Hence, 𝑓𝑥_𝑧^∼⊂𝜆𝑒𝑔(𝑧)−𝐶𝑔(𝑧), (26) which is equal to 𝜉_𝑓𝑧(𝑔(𝑧),𝑥^∼)≤𝜆 by Theorem 12 (2). Thus, 𝑥^∼∈𝐿_𝑧(𝜆) and 𝐿_𝑧(𝜆) is convex.

Now a result on existence of solutions of the GVQEP1 is verified by making use of the general nonlinear scalarization function defined in Section 3.

Theorem 17. Let 𝐸 and 𝐹 be compact and convex subsets and 𝑃∶𝐸→2^𝐸,𝑄∶𝐸→2^𝐹, and 𝑓∶𝐸×𝐹→2^𝑌 set-valued mappings. For each 𝑧∈𝐹, define 𝑓_𝑧(𝑥)=𝑓(𝑥,𝑧). Suppose that the following conditions are fulfilled:(a)𝑥↦int𝐶(𝑥) has a continuous select 𝑒∶𝐸→𝑌;(b)both 𝐶 and 𝑊 are usc on 𝐸, where 𝑊(𝑥)=𝑌⧵int𝐶(𝑥),∀𝑥∈𝐸; (27)(c)𝑔 is continuous, 𝑓 and 𝑃 are strict and continuous, and 𝑄 is strict and usc;(d)for each 𝑧∈𝐹,𝑓_𝑧 is generalized 𝐶(𝑔(𝑧))-quasiconvex;(e)for each (𝑥,𝑧)∈𝐸×𝐹, 𝑓(𝑥,𝑧) is compact and for each 𝑥∈𝐸, both 𝑃(𝑥) and 𝑄(𝑥) are closed and convex.Then the GVQEP1 has a solution (𝑥,𝑧)∈𝐸×𝐹; that is, there exist 𝑥∈𝑃(𝑥) and 𝑧∈𝑄(𝑥) such that 𝑓(𝑣,𝑧)−𝑓(𝑥,𝑧)⊄−int𝐶𝑔(𝑧),∀𝑣∈𝑃(𝑥). (28) Further assume that 𝑓(𝑥,𝑧)⊂𝐶(𝑔(𝑧)) for each 𝑥∈𝐸 and 𝑧∈𝑄(𝑥). Then 𝑓(𝑣,𝑧)⊄−int𝐶𝑔(𝑧),∀𝑣∈𝑃(𝑥). (29)

Proof. Denote 𝐾=𝐸×𝐹 and 𝑤=(𝑥,𝑧) where 𝑥∈𝐸 and 𝑧∈𝐹, and define 𝑔^∼∶𝐾→𝐸 as 𝑔^∼(𝑤)=𝑔^∼(𝑥,𝑧)=𝑔(𝑧). (30) Obviously, 𝑔^∼ is continuous on 𝐾. The general nonlinear scalarization function of 𝑓𝜉_𝑓(𝑢,𝑤)=min𝜆∈ℝ∶𝑓(𝑤)⊂𝜆𝑒(𝑢)−𝐶(𝑢) (31) is continuous on 𝐸×𝐾 by Theorem 14. Set ℎ(𝑥,𝑧)=𝜉_𝑓𝑧(𝑔(𝑧),𝑥), where 𝜉_𝑓𝑧 is the general nonlinear scalarization function of 𝑓_𝑧. Since ℎ(𝑤)=ℎ(𝑥,𝑧)=min𝜆∈ℝ∶𝑓_𝑧(𝑥)𝑔(𝑧)𝑔(𝑧)()⊂𝜆𝑒−𝐶=min𝜆∈ℝ∶𝑓𝑥,𝑧⊂𝜆𝑒𝑔^∼(𝑥,𝑧)−𝐶𝑔^∼()(𝑤)𝑥,𝑧=min𝜆∈ℝ∶𝑓⊂𝜆𝑒𝑔^∼(𝑤)−𝐶𝑔^∼(𝑤)=𝜉𝑔_𝑓^∼(𝑤),𝑤, (32)ℎ is continuous on 𝐾 by virtue of the continuity of 𝜉_𝑓 and 𝑔^∼. Define 𝑃^∼∶𝐸×𝐹→2^𝐸×𝐹 as 𝑃^∼(𝑥,𝑧)=𝑃(𝑥)×{𝑧},∀(𝑥,𝑧)∈𝐸×𝐹. (33) Obviously, 𝑃^∼ has compact values and is continuous. Define a set-valued mapping 𝛿∶𝐸×𝐹→2^𝐸×𝐹 as 𝛿(𝑥,𝑧)=(𝑢,𝑧)∈𝑃(𝑥)×{𝑧}∶ℎ(𝑢,𝑧)=min_{𝑣∈𝑃(𝑥)}ℎ(=)(𝑥)×{𝑧}()𝑣,𝑧)(𝑢,𝑧∈𝑃∶−ℎ𝑢,𝑧=max_{𝑣∈𝑃(𝑥)}(=)∈−ℎ𝑣,𝑧)(𝑢,𝑎𝑃^∼(𝑥,𝑧)∶−ℎ(𝑢,𝑎)=max_{()∈𝑣,𝑏∼𝑃()𝑥,𝑧}−ℎ(𝑣,𝑏). (34) By Lemma 4, 𝛿(𝑥,𝑧) is usc. Let 𝜂(𝑥,𝑧)=𝑢∈𝑃(𝑥)∶ℎ(𝑢,𝑧)=min_{𝑣∈𝑃(𝑥)}ℎ(𝑣,𝑧). (35) Then 𝛿(𝑥,𝑧)=𝜂(𝑥,𝑧)×{𝑧}. The upper semicontinuity of 𝜂 is obvious. Now we show that for each (𝑥,𝑧)∈𝐸×𝐹,𝜂(𝑥,𝑧) is convex and closed. Detailedly,(i) Define 𝜆=min_{𝑣∈𝑃(𝑥)}ℎ(𝑣,𝑧). (36)Then for any 𝑢₁,𝑢₂∈𝜂(𝑥,𝑧), ℎ(𝑢_𝑖,𝑧)=𝜆,𝑖=1,2. Clearly, 𝑢^∼=𝑡𝑢₁+(1−𝑡)𝑢₂∈𝑃(𝑥) for all 𝑡∈[0,1] since 𝑃(𝑥) is convex. In view of condition (d) and Lemma 16, 𝑥↦ℎ(𝑥,𝑧) is ℝ₊-quasiconvex, which deduces that for each 𝑧∈𝐹, the set 𝐿_𝑧(𝜆)={𝑥∈𝐸∶ℎ(𝑥,𝑧)≤𝜆} (37) is convex. Thus, ℎ(𝑢^∼,𝑧)≤𝜆. It follows from the definition of 𝜆 that ℎ(𝑢^∼,𝑧)=𝜆, which results in 𝑢^∼∈𝜂(𝑥,𝑧) and the convexity of 𝜂(𝑥,𝑧).(ii) For each (𝑥,𝑧)∈𝐸×𝐹 and for any sequence {𝑢_𝑘}⊂𝜂(𝑥,𝑧) such that 𝑢_𝑘→𝑢₀ as 𝑘→∞, ℎ𝑢_𝑘,𝑧=min_{𝑣∈𝑃(𝑥)}ℎ(𝑣,𝑧),∀𝑘. (38)Since ℎ is continuous, ℎ𝑢₀,𝑧=lim_𝑘→∞ℎ𝑢_𝑘,𝑧=min_{𝑣∈𝑃(𝑥)}ℎ(𝑣,𝑧). (39) Thus 𝑢₀∈𝜂(𝑥,𝑧) and 𝜂(𝑥,𝑧) is closed.
Now consider a set-valued mapping Φ∶𝐸×𝐹→2^𝐸×𝐹 prescribed as Φ(𝑥,𝑧)=𝜂(𝑥,𝑧)×𝑄(𝑥),∀(𝑥,𝑧)∈𝐸×𝐹. (40) It's easy to check that Φ is usc on 𝐸×𝐹 and for each (𝑥,𝑧)∈𝐸×𝐹,Φ(𝑥,𝑧) is convex and closed. By Lemma 5, Φ exists a fixed point (𝑥,𝑧), that is to say, 𝑥∈𝑃(𝑥),𝑧∈𝑄(𝑥),ℎ(𝑥,𝑧)=min_{𝑣∈𝑃(𝑥)}ℎ(𝑣,𝑧). (41) Hence, for each 𝑣∈𝑃(𝑥), ℎ(𝑣,𝑧)≥ℎ(𝑥,𝑧). In other words, 𝜉_𝑓𝑧𝑔(𝑧),𝑣≥𝜉_𝑓𝑧𝑔(𝑧),𝑥, (42) which deduces that 𝑥∈𝑃(𝑥),𝑧∈𝑄(𝑥), and 𝑓(𝑣,𝑧)−𝑓(𝑥,𝑧)=𝑓_𝑧(𝑣)−𝑓_𝑧(𝑥)⊄−int𝐶𝑔(𝑧),∀𝑣∈𝑃(𝑥), (43) by Theorem 13. The first conclusion is proved. Further assume that 𝑓(𝑥,𝑧)⊂𝐶(𝑔(𝑧)) for each 𝑥∈𝐸 and 𝑧∈𝑄(𝑥). If the second conclusion is not true; namely, there exists 𝑣∈𝑃(𝑥) such that 𝑓(𝑣,𝑧)⊂−int𝐶(𝑔(𝑧)), then 𝑓(𝑣,𝑧)−𝑓(𝑥,𝑧)⊂−int𝐶𝑔(𝑧)−𝐶𝑔(𝑧)=−int𝐶𝑔(𝑧). (44) This is absurd.

If mapping 𝐶 in Theorem 17 satisfies that for each 𝑦∈𝑌, (int𝐶)⁻¹(𝑦)={𝑥∈𝐸∶𝑦∈int𝐶(𝑥)} is open, then 𝑥↦int𝐶(𝑥) must exist a continuous selection by Browder Selection Theorem [17]. Especially, when 𝑓 is single valued in Theorem 17, a result is stated as follows.

Corollary 18. Let 𝐸 and 𝐹 be compact and convex subsets, 𝑃∶𝐸→2^𝐸 and 𝑄∶𝐸→2^𝐹 set-valued mappings and 𝑓∶𝐸×𝐹→𝑌 a vector-valued mapping. For each 𝑧∈𝐹, define 𝑓_𝑧(𝑥)=𝑓(𝑥,𝑧). Suppose that the following conditions are in force:(a)𝑥↦int𝐶(𝑥) has a continuous select 𝑒∶𝐸→𝑌;(b) both C and 𝑊 are usc on 𝐸, where 𝑊(𝑥)=𝑌⧵int𝐶(𝑥),∀𝑥∈𝐸; (45)(c)𝑓 and 𝑔 are continuous, 𝑃 is strict and continuous, and 𝑄 is strict and usc;(d) for each 𝑧∈𝐹,𝑓_𝑧 is 𝐶(𝑔(𝑧))-quasiconvex;(e) for each 𝑥∈𝐸, both 𝑃(𝑥) and 𝑄(𝑥) are closed and convex.Then there exist 𝑥∈𝑃(𝑥) and 𝑧∈𝑄(𝑥) such that 𝑓(𝑥,𝑧)∈𝑤Min_{𝐶(𝑔(𝑧))}𝑓(𝑃(𝑥),𝑧). (46) Further suppose that 𝑓(𝑥,𝑧)∈𝐶(𝑔(𝑧)) for each 𝑥∈𝐸 and 𝑧∈𝑄(𝑥). Then 𝑓(𝑃(𝑥),𝑧)∩−int𝐶𝑔(𝑧)=∅. (47)

Theorem 19. Let 𝐸 and 𝐹 be compact and convex subsets and 𝑃∶𝐸→2^𝐸,𝑄∶𝐸→2^𝐹, and 𝑓∶𝐸×𝐸×𝐹→2^𝑌 set-valued mappings. For each (𝑥,𝑧)∈𝐸×𝐹, define 𝑓_𝑥𝑧(𝑢)=𝑓(𝑢,𝑥,𝑧). Suppose that the following conditions hold:(a)𝑥↦int𝐶(𝑥) has a continuous select 𝑒∶𝐸→𝑌;(b) both 𝐶 and 𝑊 are usc on 𝐸, where 𝑊(𝑥)=𝑌⧵int𝐶(𝑥),∀𝑥∈𝐸; (48)(c)𝑔 is continuous, 𝑓 and 𝑃 are strict and continuous, and 𝑄 is strict and usc;(d) for each (𝑥,𝑧)∈𝐸×𝐹, 𝑓_𝑥𝑧 is generalized 𝐶(𝑔(𝑧))-quasiconvex;(e) for each (𝑢,𝑥,𝑧)∈𝐸×𝐸×𝐹, 𝑓(𝑢,𝑥,𝑧) is compact and for each 𝑥∈𝐸, both 𝑃(𝑥) and 𝑄(𝑥) are closed and convex.Then the GVQEP2 exists a solution (𝑥,𝑧)∈𝐸×𝐹; namely, there exist 𝑥∈𝑃(𝑥) and 𝑧∈𝑄(𝑥) such that 𝑓(𝑣,𝑥,𝑧)−𝑓(𝑥,𝑥,𝑧)⊄−int𝐶𝑔(𝑧),∀𝑣∈𝑃(𝑥). (49) Furthermore, if 𝑓(𝑥,𝑥,𝑧)⊂𝐶(𝑔(𝑧)) for each 𝑥∈𝐸 and 𝑧∈𝑄(𝑥), then 𝑓(𝑣,𝑥,𝑧)⊄−int𝐶𝑔(𝑧),∀𝑣∈𝑃(𝑥). (50)

Proof. Denoting 𝐾=𝐸×𝐹, we see that 𝐾 is compact. Define 𝑔^∼∶𝐾→𝐸 and 𝑆∶𝐸→2^𝐾 as 𝑔^∼(𝑤)=𝑔^∼(𝑥,𝑧)=𝑔(𝑧),forall𝑤=(𝑥,𝑧)∈𝐾 and 𝑆(𝑥)={𝑥}×𝑄(𝑥),forall𝑥∈𝐸, respectively. Then 𝑔^∼ is continuous on 𝐾 and for each 𝑤∈𝐾, 𝑓_𝑤 is generalized 𝐶(𝑔^∼(𝑤))-quasiconvex according to condition (d). Since 𝑄 is strict and usc with compact values, so is 𝑆. Replacing 𝐹,𝑔, and 𝑄 in Theorem 17 by 𝐾,𝑔^∼, and 𝑆, respectively, we see that these conclusions are true.

The result below follows from Theorem 19 immediately by further assuming that 𝑓 is single-valued.

Corollary 20 (see [2]). Let 𝐸 and 𝐹 be compact and convex subsets, 𝑃∶𝐸→2^𝐸 and 𝑄∶𝐸→2^𝐹 set-valued mappings, and 𝑓∶𝐸×𝐸×𝐹→𝑌 a vector-valued mapping. For each (𝑥,𝑧)∈𝐸×𝐹, define 𝑓_𝑥𝑧(𝑢)=𝑓(𝑢,𝑥,𝑧). The following assumptions are in operation:(a)𝑥↦int𝐶(𝑥) has a continuous select 𝑒∶𝐸→𝑌;(b)both 𝐶 and 𝑊 are usc on 𝐸, where 𝑊(𝑥)=𝑌⧵int𝐶(𝑥),∀𝑥∈𝐸; (51)(c)𝑓 and 𝑔 are continuous, 𝑃 is strict and continuous, and 𝑄 is strict and usc;(d)for each (𝑥,𝑧)∈𝐸×𝐹, 𝑓_𝑥𝑧 is 𝐶(𝑔(𝑧))-quasiconvex;(e)for each 𝑥∈𝐸, both 𝑃(𝑥) and 𝑄(𝑥) are closed and convex.Then there exist 𝑥∈𝑃(𝑥) and 𝑧∈𝑄(𝑥) such that 𝑓(𝑥,𝑥,𝑧)∈𝑤Min_{𝐶(𝑔(𝑧))}𝑓(𝑃(𝑥),𝑥,𝑧). (52) Additionally, on the assumption that 𝑓(𝑥,𝑥,𝑧)∈𝐶(𝑔(𝑧)) for each 𝑥∈𝐸 and 𝑧∈𝑄(𝑥), 𝑓(𝑃(𝑥),𝑥,𝑧)∩−int𝐶𝑔(𝑧)=∅. (53)

5. An Application of GVQEP1: A VVIP

Let 𝑋 be a real 𝐹-space (𝑋 is called an 𝐹-space [14], if it is a TVS such that its topology is induced by a complete invariant metric), 𝑌 a real locally convex TVS, 𝐸⊂𝑋 a nonempty subset and 𝐷⊂𝑌 a proper, closed, and convex cone with int𝐷≠∅ and let 𝑙∶𝐸→2^{𝐿(𝑋,𝑌)} be strict and continuous with compact values. A VVIP is described as follows: to find 𝑥∈𝐸 such that ⟨𝑙(𝑥),𝑥−𝑥⟩⊄−int𝐷,∀𝑥∈𝐸, (54) where ⟨𝑙(𝑧),𝑥⟩=⟨𝑝,𝑥⟩∶𝑝∈𝑙(𝑧). (55) The VVIP was investigated in [18, 19]. If 𝑙 is single valued, then the VVIP becomes a vector variational inequality problem, which was discussed in [20–22]. Note that 𝑋 is indeed locally convex.

Assume that 𝐸 is compact and convex and ⟨𝑙(𝑥),𝑥⟩ is a singleton for each 𝑥∈𝐸.

Let 𝐶(𝑥)𝑓()(𝑧)()𝑃(𝑥)𝑄(𝑥)={𝑥}𝑋=𝑍,𝐸=𝐹,=𝐷,∀𝑥∈𝐸,𝑔∶𝐸→𝑌beanycontinuousmapping,𝑥,𝑧=⟨𝑙,𝑥⟩,∀𝑥,𝑧∈𝐸×𝐸,=𝐸,∀𝑥∈𝐸,,∀𝑥∈𝐸 (56) in Theorem 17. Then the GVQEP1 reduces to the VVIP. All the assumptions of Theorem 17 are verified as follows.

(a) and (b) Since 𝐷 is a constant cone, 𝑥↦int𝐷 has a continuous selection and both 𝐶 and 𝑊 are usc.

(c) Obviously, 𝑓 is strict, 𝑔 is continuous, 𝑃 is strict and continuous, and 𝑄 is strict and usc. In addition, 𝑓 is continuous with compact values.

In fact, for each fixed (𝑥₀,𝑧₀)∈𝐸×𝐸, letting 𝑚(𝑝)=⟨𝑝,𝑥₀⟩, we see that 𝑚 is continuous with compact values. So 𝑓(𝑥₀,𝑧₀)=𝑚(𝑙(𝑧₀)) is compact by the compactness of 𝑙(𝑧₀) and Lemma 3 (2). Moreover, take 𝐿(𝑝,𝑥,𝑧)=⟨𝑝,𝑥⟩ and 𝐻(𝑥,𝑧)=𝑙(𝑧). Obviously, 𝐻 is continuous with compact values. 𝐿 is continuous in view of the bilinearity of itself and [11, Theorem 2.17]. Thus 𝑓(𝑥,𝑧)=_{𝑝∈𝐻(𝑥,𝑧)}𝐿𝑝,𝑥,𝑧 (57) is continuous by [13, Theorem 1].

(d) For each 𝑧∈𝐸, 𝑥↦𝑓(𝑥,𝑧) is generalized 𝐷-quasiconvex by its linearity.

(e) The assertion that 𝑓 has compact values was verified in (c). Clearly, for each 𝑥∈𝐸, both 𝑃(𝑥) and 𝑄(𝑥) are closed and convex.

Thus there exist (𝑥,𝑧)∈𝐸×𝐸, 𝑥∈𝑃(𝑥)=𝐸, and 𝑧∈𝑄(𝑥)={𝑥} such that 𝑓(𝑥,𝑥)−𝑓(𝑥,𝑥)(⊄−int𝐷,∀𝑥∈𝐸,thatis,⟨𝑙𝑥),𝑥⟩−⟨𝑙(𝑥),𝑥⟩⊄−int𝐷,∀𝑥∈𝐸. (58) Since ⟨𝑙(𝑥),𝑥⟩ is a singleton for each 𝑥∈𝐸, ⟨𝑙(𝑥),𝑥−𝑥⟩⊄−int𝐷,∀𝑥∈𝐸, (59) which implies that 𝑥 is a solution of VVIP.

Incidentally, the set-valued mapping 𝑙 satisfying the conditions above exists. For instance, let 𝑋=ℝ²,𝑌=ℝ, and 𝐸=𝑥=𝑥₁,𝑥₂∶𝑥²₁+𝑥²₂≤1⊂𝑋. (60) Define 𝑙 by 𝑙(𝑥)=𝜆𝑥₂,−𝜆𝑥₁∶|𝜆|≤1,∀𝑥=𝑥₁,𝑥₂∈𝐸. (61) Clearly, 𝑙 is strict and continuous with compact values. Moreover, ⟨𝑙(𝑥),𝑥⟩={0} for each 𝑥∈𝐸.

Acknowledgment

The work was supported by both the Doctoral Fund of innovation of Beijing University of Technology (2012) and the 11th graduate students Technology Fund of Beijing University of Technology (No. ykj-2012-8236).