Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 916089, 8 pages http://dx.doi.org/10.1155/2013/916089 Research Article Existence of Solutions for Generalized Vector Quasi-Equilibrium Problems by Scalarization Method with Applications De-ning Qu1,2 and Cao-zong Cheng1 1 2 College of Applied Science, Beijing University of Technology, Beijing 100124, China College of Mathematics, Jilin Normal University, Siping, Jilin 136000, China Correspondence should be addressed to De-ning Qu; qudening@126.com Received 26 November 2012; Accepted 6 February 2013 Academic Editor: Ngai-Ching Wong Copyright © 2013 D.-n. Qu and C.-z. Cheng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 π and π are topological vector spaces (TVSs). The subset π· ⊂ π is called a cone, if ππ₯ ∈ π· for all π₯ ∈ π· and π > 0. A cone π· ⊂ π is said to be proper, if π· =ΜΈ π. π΄ is called π·-closed [11] if π΄ + πΎ is closed and π·-bonded [11] if for each neighborhood π of zero in π, there exists π > 0 such that π΄ ⊂ π + π·. A set-valued mapping πΊ : π → 2π is called strict, if πΊ(π₯) =ΜΈ 0 for any π₯ ∈ π. Throughout this paper, R denotes by the set of the real numbers. Several notations are also listed as follows: πΊ (π΄) = β πΊ (π’) , π’∈π΄ 2 Abstract and Applied Analysis π₯ − π΅ = {π₯ − π¦ : π¦ ∈ π΅} , π΄ − π΅ = {π₯ − π¦ : π₯ ∈ π΄, π¦ ∈ π΅} , (1) where π΄, π΅ ⊂ π are nonempty, π₯ ∈ π and πΊ : π → 2π . Incidentally, every TVS is Hausdorff (see [12]). 2. Preliminaries Let π and π be topological spaces and πΈ ⊂ π a nonempty subset. A function π : π → R is said to be upper semicontinuous (usc for brevity) on π, if {π’ ∈ π : π(π’) < π} is open for each π ∈ R; to be lower semicontinuous (lsc for brevity), {π’ ∈ π : π(π’) > π} is open for each π ∈ R. In addition, some known notions of continuity and closeness for a set-valued mapping are given (see [13]). A set-valued mapping πΊ : πΈ → 2π is said to be usc at π’0 ∈ πΈ, if for any neighborhood π(πΊ(π’0 )) of πΊ(π’0 ), there exists a neighborhood π΅(π’0 ) of π’0 such that πΊ(π’) ⊂ π(πΊ(π’0 )) for all π’ ∈ π΅(π’0 ); to be lsc at π’0 ∈ πΈ, if for any π¦0 ∈ πΊ(π’0 ) and any neighborhood π(π¦0 ) of π¦0 , there exists a neighborhood π΅(π’0 ) of π’0 such that πΊ(π’) ∩ π(π¦0 ) =ΜΈ 0 for all π’ ∈ π΅(π’0 ); to be usc (resp., lsc) on πΈ, if πΊ is usc (resp., lsc) at each π’ ∈ πΈ; to be continuous at π’0 ∈ πΈ (resp., on πΈ), if πΊ is usc and lsc at π’0 (resp., on πΈ); to be closed, if its graph Graph(πΊ) = {(π’, π¦) ∈ πΈ × π : π¦ ∈ πΊ(π’)} is closed in πΈ × π. Let π and π be real TVSs, π΄ ⊂ π a nonempty subset, and Μ ∈ π΄ is called, vector minimal point π· a convex cone in π. π¦ Μ ∉ −π· \ {0} (resp., weakly vector minimal point) of π΄, if π¦ − π¦ Μ ∉ − int π·) for each π¦ ∈ π΄. The set of vector (resp., π¦ − π¦ minimal points (resp., weakly vector minimal points) of π΄ is denoted by Minπ·π΄ (resp., π€Minπ·π΄). Definition 1. Let π΅ ⊂ π be nonempty convex subset. πΊ : π΅ → 2π is called generalized π·-quasiconvex, if for any π¦ ∈ π, the set {π’ ∈ π΅ : πΊ(π’) ⊂ π¦ − π·} is convex (here, 0 is regarded as a convex set). The generalized π·-quasiconvexity and the π·-quasiconvexity introduced in [14] for the set-valued mappings are both generalizations of π·-quasiconvexity for the single-valued mapping introduced in [11], but they are indeed distinct. See the following example. Example 2. Let π = R2 , π = π΅ = R, and π· = R2+ and define πΊ1 , πΊ2 : π΅ → 2π as πΊ1 (π’) = {(π₯1 , π₯2 ) : π₯1 = π’, −π’2 ≤ π₯2 ≤ 0} , πΊ2 (π’) = {(π₯1 , π₯2 ) : π₯1 = π’, 0 ≤ π₯2 ≤ |sin π’|} . are convex. But the set {π’ ∈ B : πΊ1 (π’) ∩ ((2, −1) − π·) =ΜΈ 0} = (−∞, −1] ∪ [1, 2] (4) is not convex and neither is the set {π’ ∈ π΅ : πΊ2 (π’) ⊂ (0, 0) − π·} = {−ππ : π = 0, 1, 2, . . .} . (5) Lemma 3 (see [15]). Let π and π be topological spaces and π : π → 2π a set-valued mapping. (1) If π is usc with closed values, then π is closed. (2) If π is compact and π is usc with compact values, then π(π) is compact. Lemma 4 (see [13]). Let π and π be Hausdorff topological spaces and πΈ ⊂ π a nonempty compact set and let β : πΈ × π → R be a function and πΊ : πΈ → 2π a set-valued mapping. If β is continuous on πΈ × π and πΊ is continuous with compact values, then the marginal function π(π’) = supπ¦∈πΊ(π’) β(π’, π¦) is continuous and the marginal set-valued mapping πΏ(π’) = {π¦ ∈ πΊ(π’) : π(π’) = β(π’, π¦)} is usc. Lemma 5 (Kakutani, see [13, Theorem 13 in Section 4 Chapter 6]). Let πΈ be a nonempty compact and convex subset of a locally convex TVS π. If Φ : πΈ → 2πΈ is usc and Φ(π₯) is a nonempty, convex, and closed subset for any π₯ ∈ πΈ, then there exists π₯ ∈ πΈ such that π₯ ∈ Φ(π₯). 3. A General Nonlinear Scalarization Function From now on, unless otherwise specified, let π, π, and π be real TVSs and πΈ ⊂ π and πΉ ⊂ π nonempty subsets. Let πΆ : πΈ → 2π be a set-valued mapping such that for any π₯ ∈ πΈ, πΆ(π₯) is a proper, closed, and convex cone with int πΆ(π₯) =ΜΈ 0. In this section, suppose that πΊ : πΉ → 2π is a strict mapping with compact values and π : πΈ → π is a vector-valued mapping with π(π₯) ∈ int πΆ(π₯), for all π₯ ∈ πΈ. Obviously, πΊ(π’) is πΆ(π₯)-closed [11, Definition 3.1 and Proposition 3.3] for each π₯ ∈ πΈ and π’ ∈ πΉ. Then, in view of [16, Lemma 3.1], we can define a general nonlinear scalarization function of G as follows. (2) Then πΊ1 is generalized π·-quasiconvex but not π·-quasiconvex, while πΊ2 is just the reverse. Indeed, for any fixed π¦ = (π₯1 , π₯2 ) ∈ π, both the sets 0, { { { if π₯2 < 0, π₯1 ∈ R, {π’ ∈ π΅ : πΊ1 (π’) ⊂ π¦ − π·} = { { { (−∞, π₯1 ] , { otherwise, 0, { { { if π₯2 < 0, π₯1 ∈ R, {π’ ∈ π΅ : πΊ2 (π’) ∩ (π¦ − π·) =ΜΈ 0} = { { { (−∞, π₯1 ] , { otherwise (3) Definition 6. The general nonlinear scalarization function ππΊ : πΈ × πΉ → R, of πΊ is defined as ππΊ (π₯, π’) = min {π ∈ R : πΊ (π’) ⊂ ππ (π₯) − πΆ (π₯)} , ∀ (π₯, π’) ∈ πΈ × πΉ. (6) Remark 7. If π = π = π = πΈ = πΉ and πΊ(π’) = {π’}, for all π’ ∈ πΉ, then the general nonlinear scalarization function ππΊ of πΊ becomes the nonlinear scalarization function defined in [2]. Abstract and Applied Analysis 3 Example 8. Let π = π = R, π = R2 , and πΈ = πΉ = [0, +∞) ⊂ π and let πΊ, πΆ : πΈ → 2π , π : πΈ → π define as πΊ (π’) = {(π, π) : π − π’ ≤ π ≤ π’ − π, 0 ≤ π ≤ π’} , 3 πΆ (π₯) = {(π, π) : 0 ≤ π ≤ ( + π₯) π, π ≥ 0} , 2 π (π₯) = (1, 1) , ∀π’ ∈ πΉ, ∀π₯ ∈ πΈ, ∀π₯ ∈ πΈ, (7) respectively. We can see that ππΊ(π₯, π’) = (1 + (2/(1 + 2π₯)))π’. Proof. Letting π’1 , π’2 ∈ πΉ such that πΊ(π’1 ) − πΊ(π’2 ) ⊂ int πΆ(π₯) and π = ππΊ(π₯, π’1 ), we have πΊ (π’2 ) ⊂ πΊ (π’1 ) − int πΆ (π₯) ⊂ ππ (π₯) − πΆ (π₯) − int πΆ (π₯) (by Definition 6) (9) ⊂ ππ (π₯) − int πΆ (π₯) . It is from this assertion that ππΊ(π₯, π’2 ) < π = ππΊ(π₯, π’1 ) by Theorem 12 (1). Theorem 14. Suppose that π is continuous. Definition 9. Let π· ⊂ π be a cone, π» : πΉ → 2π , and π : πΉ → R. π is called monotone (resp., strictly monotone) with respect to (wrt for brevity) π», if for any π’1 , π’2 ∈ πΉ, π»(π’1 ) − π»(π’2 ) ⊂ int π· implies that π(π’2 ) ≤ π(π’1 ) (resp., π(π’2 ) < π(π’1 )). (1) If π and πΊ are usc on πΈ and πΉ, respectively, where Obviously, the strict monotonicity wrt π» implies the monotonicity wrt π». Moreover, if π»(π’) = {π’}, for all π’ ∈ πΉ, then the (strict) monotonicity wrt π» of π is equivalent to the (strict) monotonicity under the general order structures. The following examples illuminate the relationship between the monotonicity wrt π» and the monotonicity in the normal sense when π» is not the identity mapping. (2) If πΆ is usc on πΈ and πΊ is lsc on πΉ, then ππΊ is lsc on πΈ × πΉ. Example 10. Let π = π = R, πΉ = [0, 10], and π· = R+ and let π’, π’ ∈ [0, 5] , π (π’) = { 0, π’ ∈ (5, 10] , [π’, π’ + 1] , π’ ∈ [0, 5) , { { π» (π’) = {[0, 6] , π’ = 5, { 10 − π’] , π’ ∈ (5, 10] . [0, { (8) Then π»(π’1 ) − π»(π’2 ) ⊂ int π· implies that π(π’1 ) > π(π’2 ). Thus π is strictly monotone wrt π», while π is not monotone in the normal sense. Example 11. Let π = π = R, πΉ = [0, 5], and π· = R+ and let π(π’) = π’, for all π’ ∈ πΉ, and π»(π’) = [−π’−1, −π’], for all π’ ∈ πΉ. Then π is strictly monotone in the normal sense, but π is not monotone wrt π». As the case stands, π»(π’1 ) − π»(π’2 ) ⊂ int π· is equivalent to π’1 , π’2 ∈ πΉ and π’2 − π’1 > 1, which just results in π(π’1 ) < π(π’2 ). 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 π ∈ R, π₯ ∈ πΈ and π’ ∈ πΉ, the following assertions hold. (1) ππΊ(π₯, π’) < π ⇔ πΊ(π’) ⊂ ππ(π₯) − int πΆ(π₯). (2) ππΊ(π₯, π’) ≤ π ⇔ πΊ(π’) ⊂ ππ(π₯) − πΆ(π₯). Theorem 13. ππΊ is strictly monotone wrt πΊ in the second variable. π (π₯) = π \ int πΆ (π₯) , ∀π₯ ∈ πΈ, (10) then ππΊ is usc on πΈ × πΉ. Proof. (1) It is sufficient to attest the fact that for each π ∈ R, the set π΄ = {(π₯, π’) ∈ πΈ × πΉ : ππΊ (π₯, π’) ≥ π} (11) is closed. As a matter of fact, for any sequence {(π₯π , π’π )} in π΄ such that (π₯π , π’π ) → (π₯0 , π’0 ) as π → ∞, ππΊ(π₯π , π’π ) ≥ π, that is, πΊ(π’π ) ⊂ΜΈ ππ(π₯π ) − int πΆ(π₯π ), by Theorem 12 (1). Obviously, πΉ0 = {π’, π’1 , π’2 , . . .} ⊂ πΉ is compact and so is πΊ(πΉ0 ) by Lemma 3 (2). Then there exists π¦π ∈ πΊ(π’π ) ⊂ πΊ(πΉ0 ) such that π¦π ∉ ππ(π₯π ) − int πΆ(π₯π ) and a subsequence {π¦ππ } of {π¦π }, such that π¦ππ → π¦0 ∈ πΊ(πΉ0 ) as π → ∞ and ππ (π₯ππ ) − π¦ππ ∉ int πΆ (π₯ππ ) , that is, ππ (π₯ππ ) − π¦ππ ∈ π (π₯ππ ) . (12) Obviously, πΊ is usc with closed values, which implies that πΊ is closed by Lemma 3 (1). Hence, π¦0 ∈ πΊ(π’0 ). Similarly, π is closed. Letting π → ∞ in (12) and applying the continuity of π and the closeness of π, we obtain that ππ(π₯0 ) − π¦0 ∈ π(π₯0 ). Namely, π¦0 ∉ ππ(π₯0 ) − int πΆ(π₯0 ). Thus πΊ(π’0 ) ⊂ΜΈ ππ(π₯0 ) − int πΆ(π₯0 ), which is equivalent to ππΊ(π₯0 , π’0 ) ≥ π by Theorem 12 (1). Therefore, π ∈ π΄ and π΄ is closed. (2) It’s enough to argue that for each π ∈ R, the set π΅ = {(π₯, π’) ∈ πΈ × πΉ : ππΊ (π₯, π’) ≤ π} (13) is closed. In fact, for any sequence {(π₯π , π’π )} in π΅ such that (π₯π , π’π ) → (π₯0 , π’0 ) as π → ∞, by Theorem 12 (2), ππΊ(π₯π , π’π ) ≤ π, equivalently, πΊ(π’π ) ⊂ ππ(π₯π ) − πΆ(π₯π ). For any π¦0 ∈ πΊ(π’0 ), there exists π¦π ∈ πΊ(π’π ) such that π¦π → π¦0 as π → ∞ according to the equivalent definition of lower semicontinuity of πΊ (see [13], page 108). Also, ππ (π₯π ) − π¦π ∈ πΆ (π₯π ) . (14) Linking the continuity of π with the closeness of πΆ inferred from Lemma 3 (1) and letting π → ∞ in (14), we get ππ(π₯0 )− π¦0 ∈ πΆ(π₯0 ) and πΊ(π’0 ) ⊂ ππ(π₯0 ) − πΆ(π₯0 ). Consequently, 4 Abstract and Applied Analysis Theorem 12 (2) leads to ππΊ(π₯0 , π’0 ) ≤ π which amounts to (π₯0 , π’0 ) ∈ π΅. π΅ is closed. If π = π = π = πΈ = πΉ (πΉ is not required to be compact) and πΊ(π’) = {π’}, for all π’ ∈ πΉ, then Theorem 14 becomes [2, Theorem 2.1]. Actually, [2, Theorem 2.1] can be regarded as the case where πΊ is continuous in Theorem 14. In addition, Example 2.1 (resp., Example 2.2) in [2] shows that if πΆ (resp., π) 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 πΊ is necessary in Theorem 14 (1) (resp., (2)) even if πΆ is continuous. Example 15. Let π = π = R, Y = R2 , πΈ = R+ , and πΉ = [−10, 10] ⊂ π, and let π (π₯) = (1, 1) , ∀π₯ ∈ πΈ, 3 πΆ (π₯) = {(π, π) : 0 ≤ π ≤ ( + π₯) π, π ≥ 0} , 2 (15) ∀π₯ ∈ πΈ. (16) GVQEP2: Find π₯ ∈ π(π₯) and π§ ∈ π(π₯) such that π (V, π₯, π§) − π (π₯, π₯, π§) ⊂ΜΈ − int πΆ (π (π§)) , ∀V ∈ π (π₯) , where π : πΈ → 2πΈ , π : πΈ → 2πΉ , and π : πΈ×πΈ×πΉ → 2π are set-valued mappings. It’s worth noting that the GVQEP considered in [2] is just the special case of the GVQEP2 (when π is single valued). Lemma 16. Let π : πΈ → π be a vector-valued mapping such that π(π₯) ∈ int πΆ(π₯) for all π₯ ∈ πΈ and π : πΈ × πΉ → 2π a set-valued mapping and define ππ§ (π₯) = π(π₯, π§) for each π§ ∈ πΉ. If for each π§ ∈ πΉ, ππ§ is generalized πΆ(π(π§))-quasiconvex with compact values, then π₯ σ³¨→ β(π₯, π§) is R+ -quasiconvex, where β(π₯, π§) = πππ§ (π(π§), π₯) and πππ§ is the general nonlinear scalarization function of ππ§ . Proof. It’s sufficient to testify that πΏ π§ (π) ⊂ π is convex, where πΏ π§ (π) = {π₯ ∈ πΈ : πππ§ (π (π§) , π₯) ≤ π} (1) Define πΊ (π’) = { [π’, π’ + 1] , π’ ∈ (0, 10] , 0, π’ ∈ [−10, 0] . (17) π’ + 1, ππΊ (π₯, π’) = { 0, π₯ ∈ πΈ, 0 < π’ ≤ 10, π₯ ∈ πΈ, −10 ≤ π’ ≤ 0. ππΊ is not usc on πΈ × πΉ due to the fact that {(π₯, π’) ∈ πΈ × πΉ : π(π₯, π’) ≥ 1} = R+ × (0, 10] is not closed. (2) Consider the following mapping: [π’, π’ + 1] , π’ ∈ [0, 10] , πΊ (π’) = { 0, π’ ∈ [−10, 0) . (19) Obviously, πΊ is not lsc on πΉ. Also, ππΊ fails to be lsc on πΈ × πΉ, where π’ + 1, ππΊ (π₯, π’) = { 0, π₯ ∈ πΈ, 0 ≤ π’ ≤ 10, π₯ ∈ πΈ, −10 ≤ π’ < 0. (20) 4. Existence Results on Solutions of the GVQEPs In this section, further suppose that π, π, and π are locally convex. Let π : πΉ → πΈ be a vector-valued mapping. Now two GVQEPs are characterized as follows: GVQEP1: Seek π₯ ∈ π(π₯) and π§ ∈ π(π₯) such that π (V, π§) − π (π₯, π§) ⊂ΜΈ − int πΆ (π (π§)) , ∀V ∈ π (π₯) , π = 1, 2, that is, ππ§ (π₯π ) ⊂ ππ (π (π§)) − πΆ (π (π§)) , (18) (23) for each π§ ∈ πΉ and π ∈ R. Indeed, for any π₯1 , π₯2 ∈ πΏ π§ (π), πππ§ (π (π§) , π₯π ) ≤ π, Evidently, πΊ is not usc on πΉ. After simply calculating, (22) π = 1, 2. (24) Clearly, for each π§ ∈ πΈ, π₯1 , π₯2 ∈ π, where π = {π₯ ∈ πΈ : ππ§ (π₯) ⊂ ππ (π (π§)) − πΆ (π (π§))} . (25) π is convex by reason of the generalized πΆ(π(π§))Μ = π‘π₯1 + (1 − π‘)π₯2 ∈ π quasiconvexity of ππ§ and so π₯ for all 0 ≤ π‘ ≤ 1. Hence, ππ§ (Μ π₯) ⊂ ππ (π (π§)) − πΆ (π (π§)) , (26) Μ ) ≤ π by Theorem 12 (2). Thus, which is equal to πππ§ (π(π§), π₯ Μ ∈ πΏ π§ (π) and πΏ π§ (π) is convex. π₯ 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 πΈ and πΉ be compact and convex subsets and π : πΈ → 2πΈ , π : πΈ → 2πΉ , and π : πΈ × πΉ → 2π set-valued mappings. For each π§ ∈ πΉ, define ππ§ (π₯) = π(π₯, π§). Suppose that the following conditions are fulfilled: (a) π₯ σ³¨→ int πΆ(π₯) has a continuous select π : πΈ → π; (b) both πΆ and π are usc on πΈ, where (21) where π : πΈ → 2πΈ , π : πΈ → 2πΉ , and π : πΈ × πΉ → 2π are set-valued mappings. π (π₯) = π \ int πΆ (π₯) , ∀π₯ ∈ πΈ; (27) (c) π is continuous, π and π are strict and continuous, and π is strict and usc; (d) for each π§ ∈ πΉ, ππ§ is generalized πΆ(π(π§))-quasiconvex; Abstract and Applied Analysis 5 (e) for each (π₯, π§) ∈ πΈ × πΉ, π(π₯, π§) is compact and for each π₯ ∈ πΈ, both π(π₯) and π(π₯) are closed and convex. Then the GVQEP1 has a solution (π₯, π§) ∈ πΈ × πΉ; that is, there exist π₯ ∈ π(π₯) and π§ ∈ π(π₯) such that π (V, π§) − π (π₯, π§) ⊂ΜΈ − int πΆ (π (π§)) , ∀V ∈ π (π₯) . (28) Further assume that π(π₯, π§) ⊂ πΆ(π(π§)) for each π₯ ∈ πΈ and π§ ∈ π(π₯). Then π (V, π§) ⊂ΜΈ − int πΆ (π (π§)) , ∀V ∈ π (π₯) . (29) Proof. Denote πΎ = πΈ × πΉ and π€ = (π₯, π§) where π₯ ∈ πΈ and Μ : πΎ → πΈ as π§ ∈ πΉ, and define π Μ (π€) = π Μ (π₯, π§) = π (π§) . π (30) Μ is continuous on πΎ. The general nonlinear Obviously, π scalarization function of π ππ (π’, π€) = min {π ∈ R : π (π€) ⊂ ππ (π’) − πΆ (π’)} (31) is continuous on πΈ × πΎ by Theorem 14. Set β(π₯, π§) = πππ§ (π(π§), π₯), where πππ§ is the general nonlinear scalarization function of ππ§ . Since β (π€) = β (π₯, π§) = min {π ∈ R : ππ§ (π₯) ⊂ ππ (π (π§)) − πΆ (π (π§))} = min {π ∈ R : π (π₯, π§) ⊂ ππ (Μ π (π₯, π§)) − πΆ (Μ π (π₯, π§))} π (π€) , π€) , = ππ (Μ (32) Μ. β is continuous on πΎ by virtue of the continuity of ππ and π Μ : πΈ × πΉ → 2πΈ×πΉ as Define π ∀ (π₯, π§) ∈ πΈ × πΉ. π = min β (V, π§) . Then for any π’1 , π’2 ∈ π(π₯, π§), β(π’π , π§) = π, π = 1, 2. Clearly, Μ = π‘π’1 + (1 − π‘)π’2 ∈ π(π₯) for all π‘ ∈ [0, 1] since π(π₯) is π’ convex. In view of condition (d) and Lemma 16, π₯ σ³¨→ β(π₯, π§) is R+ -quasiconvex, which deduces that for each π§ ∈ πΉ, the set πΏ π§ (π) = {π₯ ∈ πΈ : β (π₯, π§) ≤ π} (ii) For each (π₯, π§) ∈ πΈ × πΉ and for any sequence {π’π } ⊂ π(π₯, π§) such that π’π → π’0 as π → ∞, β (π’π , π§) = min β (V, π§) , (38) Since β is continuous, β (π’0 , π§) = lim β (π’π , π§) = min β (V, π§) . π→∞ V∈π(π₯) (39) Thus π’0 ∈ π(π₯, π§) and π(π₯, π§) is closed. Now consider a set-valued mapping Φ : πΈ × πΉ → 2πΈ×πΉ prescribed as Φ (π₯, π§) = π (π₯, π§) × π (π₯) , π₯ ∈ π (π₯) , ∀ (π₯, π§) ∈ πΈ × πΉ. (40) π§ ∈ π (π₯) , β (π₯, π§) = min β (V, π§) . V∈π(π₯) πππ§ (π (π§) , V) ≥ πππ§ (π (π§) , π₯) , V∈π(π₯) − β (V, π)}. π (V, π§) − π (π₯, π§) = ππ§ (V) − ππ§ (π₯) ∀V ∈ π (π₯) , (43) by Theorem 13. The first conclusion is proved. Further assume that π(π₯, π§) ⊂ πΆ(π(π§)) for each π₯ ∈ πΈ and π§ ∈ π(π₯). If the second conclusion is not true; namely, there exists V ∈ π(π₯) such that π(V, π§) ⊂ − int πΆ(π(π§)), then π (V, π§) − π (π₯, π§) ⊂ − int πΆ (π (π§)) − πΆ (π (π§)) (34) By Lemma 4, πΏ(π₯, π§) is usc. Let (42) which deduces that π₯ ∈ π(π₯), π§ ∈ π(π₯), and ⊂ΜΈ − int πΆ (π (π§)) , = {(π’, π§) ∈ π (π₯) × {π§} : −β (π’, π§) = max − β (V, π§)} (41) Hence, for each V ∈ π(π₯), β(V, π§) ≥ β(π₯, π§). In other words, V∈π(π₯) max ∀π. V∈π(π₯) πΏ (π₯, π§) = {(π’, π§) ∈ π (π₯) × {π§} : β (π’, π§) = min β (V, π§)} Μ (V,π)∈π(π₯,π§) (37) is convex. Thus, β(Μ π’, π§) ≤ π. It follows from the definition of π Μ ∈ π(π₯, π§) and the convexity that β(Μ π’, π§) = π, which results in π’ of π(π₯, π§). (33) Μ has compact values and is continuous. Define a Obviously, π set-valued mapping πΏ : πΈ × πΉ → 2πΈ×πΉ as Μ (π₯, π§) : −β (π’, π) = = {(π’, π) ∈ π (36) V∈π(π₯) It’s easy to check that Φ is usc on πΈ × πΉ and for each (π₯, π§) ∈ πΈ × πΉ, Φ(π₯, π§) is convex and closed. By Lemma 5, Φ exists a fixed point (π₯, π§), that is to say, = min {π ∈ R : π (π€) ⊂ ππ (Μ π (π€)) − πΆ (Μ π (π€))} Μ (π₯, π§) = π (π₯) × {π§} , π (i) Define = − int πΆ (π (π§)) . (44) This is absurd. π (π₯, π§) = {π’ ∈ π (π₯) : β (π’, π§) = min β (V, π§)} . V∈π(π₯) (35) Then πΏ(π₯, π§) = π(π₯, π§) × {π§}. The upper semicontinuity of π is obvious. Now we show that for each (π₯, π§) ∈ πΈ × πΉ, π(π₯, π§) is convex and closed. Detailedly, If mapping πΆ in Theorem 17 satisfies that for each π¦ ∈ π, (int πΆ)−1 (π¦) = {π₯ ∈ πΈ : π¦ ∈ int πΆ(π₯)} is open, then π₯ σ³¨→ int πΆ(π₯) must exist a continuous selection by Browder Selection Theorem [17]. Especially, when π is single valued in Theorem 17, a result is stated as follows. 6 Abstract and Applied Analysis Corollary 18. Let πΈ and πΉ be compact and convex subsets, π : πΈ → 2πΈ and π : πΈ → 2πΉ set-valued mappings and π : πΈ × πΉ → π a vector-valued mapping. For each π§ ∈ πΉ, define ππ§ (π₯) = π(π₯, π§). Suppose that the following conditions are in force: (a) π₯ σ³¨→ int πΆ(π₯) has a continuous select π : πΈ → π; (b) both C and π are usc on πΈ, where π (π₯) = π \ int πΆ (π₯) , ∀π₯ ∈ πΈ; (45) (c) π and π are continuous, π is strict and continuous, and π is strict and usc; (d) for each π§ ∈ πΉ, ππ§ is πΆ(π(π§))-quasiconvex; (e) for each π₯ ∈ πΈ, both π(π₯) and π(π₯) are closed and convex. Then there exist π₯ ∈ π(π₯) and π§ ∈ π(π₯) such that π (π₯, π§) ∈ π€MinπΆ(π(π§)) π (π (π₯) , π§) . (46) Further suppose that π(π₯, π§) ∈ πΆ(π(π§)) for each π₯ ∈ πΈ and π§ ∈ π(π₯). Then π (π (π₯) , π§) ∩ (− int πΆ (π (π§))) = 0. (47) Theorem 19. Let πΈ and πΉ be compact and convex subsets and π : πΈ → 2πΈ , π : πΈ → 2πΉ , and π : πΈ×πΈ×πΉ → 2π set-valued mappings. For each (π₯, π§) ∈ πΈ × πΉ, define ππ₯π§ (π’) = π(π’, π₯, π§). Suppose that the following conditions hold: (a) π₯ σ³¨→ int πΆ(π₯) has a continuous select π : πΈ → π; (b) both πΆ and π are usc on πΈ, where π (π₯) = π \ int πΆ (π₯) , ∀π₯ ∈ πΈ; (48) (c) π is continuous, π and π are strict and continuous, and π is strict and usc; (d) for each (π₯, π§) ∈ πΈ × πΉ, ππ₯π§ is generalized πΆ(π(π§))quasiconvex; (e) for each (π’, π₯, π§) ∈ πΈ × πΈ × πΉ, π(π’, π₯, π§) is compact and for each π₯ ∈ πΈ, both π(π₯) and π(π₯) are closed and convex. Then the GVQEP2 exists a solution (π₯, π§) ∈ πΈ × πΉ; namely, there exist π₯ ∈ π(π₯) and π§ ∈ π(π₯) such that π (V, π₯, π§) − π (π₯, π₯, π§) ⊂ΜΈ − int πΆ (π (π§)) , ∀V ∈ π (π₯) . (49) The result below follows from Theorem 19 immediately by further assuming that π is single-valued. Corollary 20 (see [2]). Let πΈ and πΉ be compact and convex subsets, π : πΈ → 2πΈ and π : πΈ → 2πΉ set-valued mappings, and π : πΈ × πΈ × πΉ → π a vector-valued mapping. For each (π₯, π§) ∈ πΈ × πΉ, define ππ₯π§ (π’) = π(π’, π₯, π§). The following assumptions are in operation: (a) π₯ σ³¨→ int πΆ(π₯) has a continuous select π : πΈ → π; (b) both πΆ and π are usc on πΈ, where π (π₯) = π \ int πΆ (π₯) , ∀V ∈ π (π₯) . (50) Proof. Denoting πΎ = πΈ × πΉ, we see that πΎ is compact. Μ : πΎ → πΈ and π : πΈ → 2πΎ as π Μ (π€) = Define π Μ (π₯, π§) = π(π§), for all π€ = (π₯, π§) ∈ πΎ and π(π₯) = {π₯} × π Μ is continuous on πΎ π(π₯), for all π₯ ∈ πΈ, respectively. Then π π(π€))-quasiconvex and for each π€ ∈ πΎ, ππ€ is generalized πΆ(Μ according to condition (d). Since π is strict and usc with compact values, so is π. Replacing πΉ, π, and π in Theorem 17 Μ , and π, respectively, we see that these conclusions are by πΎ, π true. (51) (c) π and π are continuous, π is strict and continuous, and π is strict and usc; (d) for each (π₯, π§) ∈ πΈ × πΉ, ππ₯π§ is πΆ(π(π§))-quasiconvex; (e) for each π₯ ∈ πΈ, both π(π₯) and π(π₯) are closed and convex. Then there exist π₯ ∈ π(π₯) and π§ ∈ π(π₯) such that π (π₯, π₯, π§) ∈ π€MinπΆ(π(π§)) π (π (π₯) , π₯, π§) . (52) Additionally, on the assumption that π(π₯, π₯, π§) ∈ πΆ(π(π§)) for each π₯ ∈ πΈ and π§ ∈ π(π₯), π (π (π₯) , π₯, π§) ∩ (− int πΆ (π (π§))) = 0. (53) 5. An Application of GVQEP1: A VVIP Let π be a real πΉ-space (π is called an πΉ-space [14], if it is a TVS such that its topology is induced by a complete invariant metric), π a real locally convex TVS, πΈ ⊂ π a nonempty subset and π· ⊂ π a proper, closed, and convex cone with int π· =ΜΈ 0 and let π : πΈ → 2πΏ(π,π) be strict and continuous with compact values. A VVIP is described as follows: to find π₯ ∈ πΈ such that β¨π (π₯) , π₯ − π₯β© ⊂ΜΈ − int π·, Furthermore, if π(π₯, π₯, π§) ⊂ πΆ(π(π§)) for each π₯ ∈ πΈ and π§ ∈ π(π₯), then π (V, π₯, π§) ⊂ΜΈ − int πΆ (π (π§)) , ∀π₯ ∈ πΈ; ∀π₯ ∈ πΈ, (54) β¨π (π§) , π₯β© = β {β¨π, π₯β© : π ∈ π (π§)} . (55) where The VVIP was investigated in [18, 19]. If π is single valued, then the VVIP becomes a vector variational inequality problem, which was discussed in [20–22]. Note that π is indeed locally convex. Assume that πΈ is compact and convex and β¨π(π₯), π₯β© is a singleton for each π₯ ∈ πΈ. Abstract and Applied Analysis 7 Acknowledgment Let π = π, 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). πΈ = πΉ, πΆ (π₯) = π·, ∀π₯ ∈ πΈ, π : πΈ σ³¨→ π be any continuous mapping, π (π₯, π§) = β¨π (π§) , π₯β©, π (π₯) = πΈ, (56) ∀ (π₯, π§) ∈ πΈ × πΈ, ∀π₯ ∈ πΈ, π (π₯) = {π₯} , ∀π₯ ∈ πΈ in Theorem 17. Then the GVQEP1 reduces to the VVIP. All the assumptions of Theorem 17 are verified as follows. (a) and (b) Since π· is a constant cone, π₯ σ³¨→ int π· has a continuous selection and both πΆ and π are usc. (c) Obviously, π is strict, π is continuous, π is strict and continuous, and π is strict and usc. In addition, π is continuous with compact values. In fact, for each fixed (π₯0 , π§0 ) ∈ πΈ × πΈ, letting π(π) = β¨π, π₯0 β©, we see that π is continuous with compact values. So π(π₯0 , π§0 ) = π(π(π§0 )) is compact by the compactness of π(π§0 ) and Lemma 3 (2). Moreover, take πΏ(π, π₯, π§) = β¨π, π₯β© and π»(π₯, π§) = π(π§). Obviously, π» is continuous with compact values. πΏ is continuous in view of the bilinearity of itself and [11, Theorem 2.17]. Thus π (π₯, π§) = β πΏ (π, π₯, π§) (57) π∈π»(π₯,π§) is continuous by [13, Theorem 1]. (d) For each π§ ∈ πΈ, π₯ σ³¨→ π(π₯, π§) is generalized π·quasiconvex by its linearity. (e) The assertion that π has compact values was verified in (c). Clearly, for each π₯ ∈ πΈ, both π(π₯) and π(π₯) are closed and convex. Thus there exist (π₯, π§) ∈ πΈ × πΈ, π₯ ∈ π(π₯) = πΈ, and π§ ∈ π(π₯) = {π₯} such that π (π₯, π₯) − π (π₯, π₯) ⊂ΜΈ − int π·, ∀π₯ ∈ πΈ, that is, β¨π (π₯) , π₯β© − β¨π (π₯) , π₯β© ⊂ΜΈ − int π·, ∀π₯ ∈ πΈ. (58) Since β¨π(π₯), π₯β© is a singleton for each π₯ ∈ πΈ, β¨π (π₯) , π₯ − π₯β© ⊂ΜΈ − int π·, ∀π₯ ∈ πΈ, (59) which implies that π₯ is a solution of VVIP. Incidentally, the set-valued mapping π satisfying the conditions above exists. For instance, let π = R2 , π = R, and πΈ = {π₯ = (π₯1 , π₯2 ) : π₯12 + π₯22 ≤ 1} ⊂ π. (60) Define π by π (π₯) = {(ππ₯2 , −ππ₯1 ) : |π| ≤ 1} , ∀π₯ = (π₯1 , π₯2 ) ∈ πΈ. (61) Clearly, π is strict and continuous with compact values. Moreover, β¨π(π₯), π₯β© = {0} for each π₯ ∈ πΈ. References [1] X. H. Gong, “Scalarization and optimality conditions for vector equilibrium problems,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 73, no. 11, pp. 3598–3612, 2010. [2] G. Y. Chen, X. Q. Yang, and H. 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