Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2011, Article ID 936428, 15 pages doi:10.1155/2011/936428 Research Article Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium Problems with an Application Ren-You Zhong, 1 Nan-Jing Huang, 1 andYeolJeCho 2 1 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China 2 Department of Mathematics Education and the RINS, Gyeongsang National University, Chinju 660-701, Republic of Korea Correspondence should be addressed to Yeol Je Cho, yjcho@gsnu.ac.kr Received 25 October 2010; Accepted 19 January 2011 Academic Editor: K. Teo Copyright q 2011 Ren-You Zhong et al. 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. This paper is devoted to the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set are perturbed by different parameters. By using the properties of recession cones, several equivalent characterizations are given for the set-valued vector equilibrium problems to have nonempty and bounded solution sets. As an application, the stability of solution set for the set-valued vector equilibrium problem in a reflexive Banach space is also given. The results presented in this paper generalize and extend some known results in Fan and Zhong 2008, He 2007, and Zhong and Huang 2010. 1. Introduction Let X and Y be reflexive Banach spaces. Let K be a nonempty closed convex subset of X. Let F : K × K → 2 Y be a set-valued mapping with nonempty values. Let P be a closed convex pointed cone in Y with int P /  ∅. The cone P induces a partial ordering in Y , which was defined by y 1 ≤ P y 2 if and only if y 2 − y 1 ∈ P. We consider the following set-valued vector equilibrium problem, denoted by SVEPF, K, which consists in finding x ∈ K such that F  x, y  ∩  − int P   ∅, ∀y ∈ K. 1.1 2 Journal of Inequalities and Applications It is well known that 1.1 is closely related to the following dual set-valued vector equilibrium problem, denoted by DSVEPF, K, which consists in finding x ∈ K such that F  y, x  ⊂  −P  , ∀y ∈ K. 1.2 We denote the solution sets of SVEPF, K and DSVEPF, K by S and S D , respectively. Let Z 1 ,d 1  and Z 2 ,d 2  be two metric spaces. Suppose that a nonempty closed convex set L ⊂ X is perturbed by a parameter u, which varies over Z 1 ,d 1 ,thatis,L : Z 1 → 2 X is a set-valued mapping with nonempty closed convex values. Assume that a set-valued mapping F : X × X → 2 Y is perturbed by a parameter v, which varies over Z 2 ,d 2 ,thatis,F : X × X × Z 2 → 2 Y . We consider a parametric set-valued vector equilibrium problem, denoted by SVEPF·, ·,v,Lu, which consists i n finding x ∈ Lu such that F  x, y, v  ∩  − int P   ∅, ∀y ∈ L  u  . 1.3 Similarly, we consider the parameterized dual set-valued vector equilibrium problem, denoted by DSVEPF·, ·,v,Lu, which consists in finding x ∈ Lu such that F  y, x, v  ⊂  −P  , ∀y ∈ L  u  . 1.4 We denote the solution sets of SVEPF·, ·,v,Lu  and DSVEPF·, ·,v,Lu by Su, v and S D u, v, respectively. In 1980, Giannessi 1 extended classical variational inequalities to the case of vector-valued functions. Meanwhile, vector variational inequalities have been researched quite extensively see, e.g., 2. Inspired by the study of vector variational inequalities, more general equilibrium problems 3 have been extended to the case of vector-valued bifunctions, known as vector equilibrium problems. It is well known that the vector equilibrium problem provides a unified model of several problems, for example, vector optimization, vector variational inequality, vector complementarity problem, and vector saddle point problem see 4–9. In recent years, the vector equilibrium problem has been intensively studied by many authors see, e.g., 1–3, 10–26 and the references therein. Among many desirable properties of the solution sets for vector equilibrium problems, stability analysis of solution set is of considerable interest see, e.g, 27–33 and the references therein. Assuming that the barrier cone of K has nonempty interior, McLinden 34 presented a comprehensive study of the stability of the solution set of the variational inequality, when the mapping is a maximal monotone set-valued mapping. Adly 35, Adly et al. 36, and Addi et al. 37 discussed the stability of the solution set of a so-called semicoercive variational inequality. He 38 studied the stability of variational inequality problem with either the mapping or the constraint set perturbed in reflexive Banach spaces. Recently, Fan and Zhong 39 extended the corresponding results of He 38 to the case that the perturbation was imposed on the mapping and the constraint set simultaneously. Very recently, Zhong and Huang 40 studied the stability analysis for a class of Minty mixed variational inequalities in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. They got a stability result for the Minty mixed variational inequality with Φ-pseudomonotone mapping in a reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters, which generalized and extended some known results in 38, 39. Journal of Inequalities and Applications 3 Inspired and motivated by the works mentioned above, in this paper, we further study the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set are perturbed. We present several equivalent characterizations for the vector equilibrium problem to have nonempty and bounded solution set by using the properties of recession cones. As an application, we show the stability of the solution set for the set- valued vector equilibrium problem in a reflexive Banach space, when both the mapping and the constraint set are perturbed by different parameters. The results presented in this paper extend some corresponding results of Fan and Zhong 39,He38, Zhong and Huang 40 from the variational inequality to the vector equilibrium problem. The rest of the paper is organized as follows. In Section 2, we recall some concepts in convex analysis and present some basic results. In Section 3, we present several equivalent characterizations for the set-valued vector equilibrium problems to have nonempty and bounded solution sets. In Section 4, we give an application to the stability of the solution sets for the set-valued vector equilibrium problem. 2. Preliminaries In this section, we introduce some basic notations and preliminary results. Let X be a reflexive Banach space and K be a nonempty closed convex subset of X. The symbols “ → ”and“” are used to denote strong and weak convergence, respectively. The barrier cone of K, denoted by barrK, is defined by barr  K  :  x ∗ ∈ X ∗ :sup x∈K  x ∗ ,x  < ∞  . 2.1 The recession cone of K, denoted by K ∞ , is defined by K ∞ : { d ∈ X : ∃t n −→ 0, ∃x n ∈ K, t n x n d } . 2.2 It is known that for any given x 0 ∈ K, K ∞  { d ∈ X : x 0  λd ∈ K, ∀λ>0 } . 2.3 We give some basic properties of recession cones in the following result which will be used in the sequel. Let {K i } i∈I be any family of nonempty sets in X. Then   i∈I K i  ∞ ⊂  i∈I  K i  ∞ . 2.4 4 Journal of Inequalities and Applications If, in addition,  i∈I K i /  ∅ and each set K i is closed and convex, then we obtain an equality in the previous inclusion, that is,   i∈I K i  ∞   i∈I  K i  ∞ . 2.5 Let Φ : K → R ∪{∞} be a proper convex and lower semicontinuous function. The recession function Φ ∞ of Φ is defined by Φ ∞  x  : lim t → ∞ Φ  x 0  tx  − Φ  x 0  t , 2.6 where x 0 is any point in Dom Φ. Then it follows that Φ ∞  x  : lim t → ∞ Φ  tx  t . 2.7 The function Φ ∞ · turns out to be proper convex, lower semicontinuous and so weakly lower semicontinuous with the property that Φ  u  v  ≤ Φ  u  Φ ∞  v  , ∀u ∈ Dom Φ,v∈ X. 2.8 Definition 2.1. A set-valued mapping G : K → 2 Y is said to be i upper semicontinuous at x 0 ∈ K if, for any neighborhood NGx 0  of Gx 0 , there exists a neighborhood Nx 0  of x 0 such that G  x  ⊂N  G  x 0  , ∀x ∈N  x 0  ; 2.9 ii lower semicontinuous at x 0 ∈ K if, for any y 0 ∈ Gx 0  and any neighborhood Ny 0  of y 0 , there exists a neighborhood Nx 0  of x 0 such that G  x   N  y 0  /  ∅, ∀x ∈N  x 0  . 2.10 We say G is continuous at x 0 if it is both upper and lower semicontinuous at x 0 ,and we say G is continuous on K if it is both upper and lower semicontinuous at every point of K. It is evident that G is lower semicontinuous at x 0 ∈ K if and only if, for any sequence {x n } with x n → x 0 and y 0 ∈ Gx 0 , there exists a sequence {y n } with y n ∈ Gx n  such that y n → y 0 . Definition 2.2. A set-valued mapping G : K → 2 Y is said to be weakly lower semicontinuous at x 0 ∈ K if, for any y 0 ∈ Gx 0  and for any sequence {x n }∈K with x n x 0 , there exists a sequence y n ∈ Gx n  such that y n → y 0 . Journal of Inequalities and Applications 5 We say G is weakly lower semicontinuous on K if it is weakly lower semicontinuous at every point of K.ByDefinition 2.2, we know that a weakly lower semicontinuous mapping is lower semicontinuous. Definition 2.3. A set-valued mapping G : K → 2 Y is said to be i upper P-convex on K if for any x 1 and x 2 ∈ K, t ∈ 0, 1, tG  x 1    1 − t  G  x 2  ⊂ G  tx 1   1 − t  x 2   P; 2.11 ii lower P-convex on K if for any x 1 and x 2 ∈ K, t ∈ 0, 1, G  tx 1   1 − t  x 2  ⊂ tG  x 1    1 − t  G  x 2  − P. 2.12 We say that G is P-convex if G is both upper P-convex and lower P-convex. Definition 2.4. Let {A n } be a sequence of sets in X. We define ω-lim sup n →∞ A n : { x ∈ X : ∃ { n k } ,x n k ∈ A n k such that x n k x } . 2.13 Lemma 2.5 see 36. Let K be a nonempty closed convex subset of X with intbarrK /  ∅.Then there exists no sequence {x n }⊂K such that x n →∞and x n /x n   0. Lemma 2.6 see 39. Let K be a nonempty closed convex subset of X with intbarrK /  ∅.Then there exists no sequence {d n }⊂K ∞ with each d n   1 such that d n  0. Lemma 2.7 see 39. Let Z, d be a metric space and u 0 ∈ Z be a given point. Let L : Z → 2 X be a set-valued mapping with nonempty values and let L be upper semicontinuous at u 0 . Then there exists a neighborhood U of u 0 such that Lu ∞ ⊂ Lu 0  ∞ for all u ∈ U. Lemma 2.8 see 41. Let K be a nonempty convex subset of a Hausdorff topological vector space E and G : K → 2 E be a set-valued mapping from K into E satisfying the following properties: i G is a KKM mapping, that is, for every finite subset A of K, coA ⊂  x∈A Gx; ii Gx is closed in E for every x ∈ K; iii Gx 0  is compact in E for some x 0 ∈ K. Then  x∈K Gx /  ∅. 3. Boundedness and Nonemptiness of Solution Sets In this section, we present several equivalent characterizations for the set-valued vector equilibrium problem to have nonempty and bounded solution set. First of all, we give some assumptions which will be used for next theorems. 6 Journal of Inequalities and Applications Let K be a nonempty convex and closed subset of X. Assume that F : K × K → 2 Y is a set-valued mapping satisfying the following conditions: f 0  for each x ∈ K, Fx, x0; f 1  for each x, y ∈ K, Fx, y ∩ − int P ∅ implies that Fy, x ⊂ −P; f 2  for each x ∈ K, Fx, · is P-convex on K; f 3  for each x ∈ K, Fx, · is weakly lower semicontinuous on K; f 4  for each x, y ∈ K,theset{ξ ∈ x, y : Fξ, y  − int P∅} is closed, here x, y stands for the closed line segment joining x and y. Remark 3.1. If F  x, y    Ax, y − x  Φ  y  − Φ  x  , ∀x, y ∈ K, 3.1 where A:K → 2 X ∗ is a set-valued mapping, Φ : K → R  {∞} is a proper, convex, lower semicontinuous function and P  R  , then condition f 1  reduces to the following Φ-pseudomonotonicity assumption which was used in 40. See 40 , Definition 2.2iii of 40: for all x, x ∗ , y,y ∗  in the graphA,  x ∗ ,y− x  Φ  y  − Φ  x  ≥ 0 ⇒  y ∗ ,y− x  Φ  y  − Φ  x  ≥ 0. 3.2 Remark 3.2. If, for each y ∈ K, the mapping F·,y is lower semicontinuous in K, then condition f 4  is fulfilled. Indeed, for each x, y ∈ K and for any sequence {ξ n }⊂{ξ ∈ x, y : Fξ, y  − int P∅} with ξ n → ξ 0 , we have ξ 0 ∈ x, y and Fξ 0 ,y  − int P∅.By the lower semicontinuity of F·,y, for any z ∈ Fξ 0 ,y, there exists z n ∈ Fξ n ,y such that z n → z. Since Fξ n ,y  − int P∅, we have z n ∈ Y \ − int P and so z ∈ Y \ − int P by the closedness of Y \ − int P. This implies that Fξ 0 ,y  − int P∅ and the set {ξ ∈ x, y : Fξ, y  − int P∅} is closed. The following example shows that conditions f 0 –f 4  can be satisfied. Example 3.3. Let X  R, Y  R 2 , P  R 2  and K 1, 2.Let F  x, y    y − x,  1, 1  x   y − x   , ∀x, y ∈ K. 3.3 It is obvious that f 0  holds. Since for each x, y ∈ K, Fx, · and F·,y are lower semicontinuous on K,byRemark 3.2, we known that conditions f 3  and f 4  hold. For each x, y ∈ K,ifFx, y ∩ −R 2  ∅, then we have y − x ≥ 0. This implies that F  y, x    x − y,  1, 1  y  x − y   ⊂  −R 2   3.4 and so f 1  holds. Moreover, for each x ∈ K, y 1 ,y 2 ∈ K and t 1 ,t 2 ∈ 0, 1 with t 1  t 2  1, it is easy to verify that F  x, t 1 y 1  t 2 y 2   t 1 F  x, y 1   t 2 F  x, y 2  3.5 Journal of Inequalities and Applications 7 which shows that Fx, · is R 2  -convex on K and so f 2  holds. Thus, F satisfies all conditions f 0 –f 4 . Theorem 3.4. Let K be a nonempty closed convex subset of X and F : K × K → 2 Y be a set-valued mapping satisfying assumptions f 0 -f 4 .ThenS  S D . Proof. From the assumption f 1 , it is easy to see that S ⊂ S D . We now prove that S D ⊂ S.Let x ∈ S D . Then for all y ∈ K, Fy, x ⊂ −P.Setx t  x  ty − x, where t ∈ 0, 1. Clearly, x t ∈ K. From the upper P -convexity of Fx, ·, we have  1 − t  F  x t ,x   tF  x t ,y  ⊂ F  x t ,x t   P. 3.6 Since Fx t ,x ⊂ −P,weobtain tF  x t ,y  ⊂−  1 − t  F  x t ,x   0  P ⊂ P  P ⊂ P. 3.7 This implies that Fx t ,y ⊂ P and so Fx t ,y ∩ − int P∅. Letting t → 0  , by assumption f 4 , we have Fx, y ∩ − int P∅.Thus,x ∈ S and S D ⊂ S. This completes the proof. Theorem 3.5. Let K be a nonempty closed convex subset of X and F : K × K → 2 Y be a set-valued mapping satisfying assumptions f 0 –f 4 . If the solution set S is nonempty, then S ∞  S D ∞  R 1 :  y∈K  d ∈ K ∞ : F  y, y  λd  ⊂  −P  , ∀λ>0  . 3.8 Proof. From the proof of Theorem 3.4, we know that S  S D   x ∈ K : F  y, x  ⊂  −P  , ∀y ∈ K    y∈K  x ∈ K : F  y, x  ⊂  −P   . 3.9 Let S y  {x ∈ X : Fy, x ⊂ −P}. Then S  S D   y∈K K ∩ S y . By the assumptions f 2  and f 3 , we know that the set S y is nonempty closed and convex. It follows from 2.5 and Theorem 3.4 that S ∞  S D ∞  ⎛ ⎝  y∈K K ∩ S y ⎞ ⎠ ∞   y∈K  K ∩ S  y  ∞   y∈K K ∞ ∩  S  y  ∞   y∈K  d ∈ K ∞ : d ∈  S  y  ∞    y∈K  d ∈ K ∞ : y  λd ∈ S  y  , ∀λ>0    y∈K  d ∈ K ∞ : F  y, y  λd  ⊂−P, ∀λ>0  . 3.10 Then this completes the proof. 8 Journal of Inequalities and Applications Remark 3.6. If F  y, x   Ay, x − y Φ  x  − Φ  y  , ∀x, y ∈ K, 3.11 where A : K → 2 X ∗ is a set-valued mapping, Φ : K → R  {∞} is a proper, convex, lower semicontinuous function and P  R  , then it follows from 3.8 and 2.8 that S D ∞   y∈K  d ∈ K ∞ : F  y, y  λd  ⊂  −P  , ∀λ>0   K ∞ ∩  d ∈ X : y ∗ ,y λd − y Φ  y  λd  − Φ  y  ≤ 0, ∀y ∈ K, y ∗ ∈ A  y  , ∀λ>0   K ∞ ∩  d ∈ X : y ∗ ,d Φ ∞  d  ≤ 0, ∀y ∗ ∈ A  K   . 3.12 Thus,weknowthatTheorem 3.5 is a generalization of 40, Theorem 3.1. Moreover, by 40, Remark 3.1, Theorem 3.5 is also a generalization of 38, Lemma 3.1. Theorem 3.7. Let K be a nonempty closed convex subset of X and F : K × K → 2 Y be a set- valued mapping satisfying assumptions f 0 –f 4 . Suppose that intbarrK /  ∅. Then the following statements are equivalent: i the solution set of SVEPF, K is nonempty and bounded; ii the solution set of DSVEPF, K is nonempty and bounded; iii R 1   y∈K {d ∈ K ∞ : Fy, y  λd ⊂ −P, ∀λ>0}  {0}; iv there exists a bounded set C ⊂ K such that for every x ∈ K \ C, there exists some y ∈ C such that Fy, x / ⊂−P. Proof. The implications i⇔ii and ii⇒iii follow immediately from Theorems 3.4 and 3.5 and the definition of recession cone. Now we prove that iii implies iv.Ifiv does not hold, then there exists a sequence {x n }⊂K such that for each n, x n ≥n and Fy, x n  ⊂ −P for every y ∈ K with y≤n. Without loss of generality, we may assume that d n  x n /x n  weakly converges to d. Then d ∈ K ∞ by the definition of the recession cone. Since intbarrK /  ∅,byLemma 2.5,weknow that d /  0. Let y ∈ K and λ>0 be any fixed points. For n sufficiently large, by the lower P-convexity of Fy, ·, F  y,  1 − λ  x n   y  λ  x n  x n  ⊂  1 − λ  x n   F  y, y   λ  x n  F  y, x n  − P ⊂ 0 − P − P ⊂−P. 3.13 Since  1 − λ  x n   y  λ  x n  x n y λd 3.14 and Fy,· is weakly lower semicontinuous, we know that Fy, y  λd ⊂−P and so d ∈ R 1 . However, it contradicts the assumption that R 1  {0}.Thusiv holds. Journal of Inequalities and Applications 9 Since i and ii are equivalent, it remains to prove that iv implies ii.LetG : K → 2 K be a set-valued mapping defined by G  y  :  x ∈ K : F  y, x  ⊂  −P   , ∀y ∈ K. 3.15 We first prove that Gy is a closed subset of K. Indeed, for any x n ∈ Gy with x n → x 0 , we have Fy, x n  ⊂ −P . It follows from the weakly lower semicontinuity of Fy, · that Fy, x 0  ⊂ −P. This shows that x 0 ∈ Gy and so Gy is closed. We next prove that G is a KKM mapping from K to K. Suppose to the contrary that there exist t 1 ,t 2 , ,t n ∈ 0, 1 with t 1  t 2  ··· t n  1, y 1 ,y 2 , ,y n ∈ K and y  t 1 y 1  t 2 y 2  ··· t n y n ∈ co{y 1 ,y 2 , ,y n } such that y/∈∪ i∈{1,2, ,n} Gy i . Then F  y i , y  / ⊂  −P  ,i 1, 2, ,n. 3.16 By assumption f 1 , we have F  y, y i  ∩  − int P  /  ∅,i 1, 2, ,n. 3.17 It follows from the upper P -convexity of F y, · that t 1 F  y, y 1   t 2 F  y, y 2   ··· t n F  y, y n  ⊂ F  y, y   P ⊂ P, 3.18 which is a contradiction with 3.17. Thus we know that G is a KKM mapping. We may assume that C is a bounded closed convex set otherwise, consider the closed convex hull of C instead of C.Let{y 1 , ,y m } be finite number of points in K and let M : coC ∪{y 1 , ,y m }. Then the reflexivity of the space X yields that M is weakly compact convex. Consider the set-valued mapping G  defined by G  y : Gy ∩ M for all y ∈ M. Then each G  y is a weakly compact convex subset of M and G  is a KKM mapping. We claim that ∅ /   y∈M G   y  ⊂ C. 3.19 Indeed, by Lemma 2.8, intersection in 3.19 is nonempty. Moreover, if there exists some x 0 ∈  y∈M G  y but x 0 /∈ C, t hen by iv, we have Fy, x 0  / ⊂−P for some y ∈ C.Thus,x 0 /∈ Gy and so x 0 /∈ G  y, which is a contradiction to the choice of x 0 . Let z ∈  y∈M G  y. Then z ∈ C by 3.19 and so z ∈  m i1 Gy i  ∩ C. This shows that the collection {Gy ∩ C : y ∈ K} has finite intersection property. For each y ∈ K, it follows from the weak compactness of Gy ∩ C that  y∈K Gy ∩ C is nonempty, which coincides with the solution set of DSVEPF, K. Remark 3.8. Theorem 3.7 establishes the necessary and sufficient conditions for the vector equilibrium problem to have nonempty and bounded solution sets. If F  y, x   Ay, x − y Φ  x  − Φ  y  , ∀x, y ∈ K, 3.20 10 Journal of Inequalities and Applications where A : K → 2 X ∗ is a set-valued mapping, Φ : K → R  {∞} is a proper, convex, lower semicontinuous function and P  R  , then problem 1.2 reduces to the following Minty mixed variational inequality: finding x ∈ K such that  y ∗ ,y− x  Φ  y  − Φ  x  ≥ 0, ∀y ∈ K, y ∗ ∈ A  y  , 3.21 which was considered by Zhong and Huang 40. Therefore, Theorem 3.7 is a generalization of 40, Theorem 3.2. Moreover, by 40, Remark 3.2, Theorem 3.7 is also a generalization of Theorem 3.4 due to He 38. Remark 3.9. By using a asymptotic analysis methods, many authors studied the necessary and sufficient conditions for the nonemptiness and boundedness of the solution sets to variational inequalities, optimization problems, and equilibrium problems, we refer the reader to references 42–49 for more details. 4. An Application As an application, in this section, we will establish the stability of solution set for the set- valued vector equilibrium problem when the mapping and the constraint set are perturbed by different parameters. Let Z 1 ,d 1  and Z 2 ,d 2  be two metric spaces. F : X × X × Z 2 → 2 Y is a set-valued mapping satisfying the following assumptions: f  0  for each u ∈ Z 1 , v ∈ Z 2 , x ∈ Lu, Fx, x, v0; f  1  for each u ∈ Z 1 , v ∈ Z 2 , x, y ∈ Lu, Fx, y, v∩− int P∅ implies that Fy, x, v ⊂ −P; f  2  for each u ∈ Z 1 , v ∈ Z 2 , x ∈ Lu, Fx, ·,v is P-convex on Lu; f  3  for each u ∈ Z 1 ,v ∈ Z 2 , x, y ∈ Lu and z ∈ Fx, y, v, for any sequences {x n }, {y n } and {v n } with x n → x, y n yand v n → v, there exists a sequence {z n } with z n ∈ Fx n ,y n ,v n  such that z n → z. The following Theorem 4.1 plays an important role in proving our results. Theorem 4.1. 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Corporation Journal of Inequalities and Applications Volume 2011, Article ID 936428, 15 pages doi:10.1155/2011/936428 Research Article Boundedness and Nonemptiness of Solution Sets for Set-Valued Vector Equilibrium. to the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set. study the characterizations of the boundedness and nonemptiness of solution sets for set-valued vector equilibrium problems in reflexive Banach spaces, when both the mapping and the constraint set