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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 247236, 17 pages
http://dx.doi.org/10.1155/2011/247236
Research Article

Applications of an HIDS Theorem to the Existence of Fixed Point, Abstract Equilibria and Optimization Problems

Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 824, Taiwan

Received 16 June 2011; Accepted 18 July 2011

Academic Editor: Josip E. Pečarić

Copyright © 2011 Wei-Shih Du. 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.

Abstract

By applying hybrid inclusion and disclusion systems (HIDS), we establish several vectorial variants of system of Ekeland's variational principle on topological vector spaces, some existence theorems of system of parametric vectorial quasi-equilibrium problem, and an existence theorem of system of the Stampacchia-type vectorial equilibrium problem. As an application, a vectorial minimization theorem is also given. Moreover, we discuss some equivalence relations between our vectorial variant of Ekeland's variational principle, common fixed point theorem, and maximal element theorem.

1. Introduction

Let 𝑋 be a nonempty subset of a topological space (t.s., for short), and let 𝑓𝑋×𝑋 be a function with 𝑓(𝑥,𝑥)0 for all 𝑥𝑋. Then the scalar equilibrium problem in the sense of Blum and Oettli [1] is to find 𝑥𝑋 such that 𝑓(𝑥,𝑦)0 for all 𝑦𝑋. The equilibrium problem was extensively investigated and generalized to the vectorial equilibrium problems for single-valued or multivalued maps and contains optimization problems, variational inequalities problems, saddle point problems, the Nash equilibrium problems, fixed point problems, complementary problems, bilevel problems, and semi-infinite problems as special cases and have some applications in mathematical program with equilibrium constraint; for detail one can refer to [14] and references therein.

The famous Ekeland's variational principle (EVP, for short) [57] is a forceful tool in various fields of applied mathematical analysis and nonlinear analysis. A number of generalizations in various different directions of these results for functions defined in metric (or quasimetric) spaces and more general in topological vector spaces have been investigated by several authors in the past; see [823] and references therein. It is wellknown that the original EVP is equivalent to Caristi's fixed point theorem, to Takahashi's nonconvex minimization theorem; and to the flower petal theorem for detail, see [14, 1618, 20] and references therein. EVPs were extended to the vector case by using scalarization method and were applied to the study of efficiency (or approximative efficiency) and others; see, for example, [3, 8, 9, 12, 13, 15, 23].

Recently, the author first studied the following mathematical model about hybrid inclusion and disclusion systems (HIDS, for short) [11]. Let 𝐼 be any index set. For each 𝑖𝐼, let 𝑌𝑖 be a nonempty closed convex subset of a Hausdorff topological vector space (t.v.s., for short) 𝑉𝑖, 𝐻𝑖𝑌𝑖, 𝑌=𝑖𝐼𝑌𝑖, 𝐴𝑖𝑌𝑌𝑖, and let 𝑇𝑖𝑌𝑌𝑖 be multivalued maps. Hybrid inclusion and disclusion systems (HIDS) are defined as follows:(HIDS)Find𝑣=(𝑣)𝑖𝐼𝑌suchthat𝑣𝑖𝐻𝑖,𝑦𝑖𝐴𝑖(𝑣),𝑦𝑖𝑇𝑖(𝑣),𝑖𝐼.(1.1)

In fact, HIDS contains several important problems as special cases. Let 𝑋 be a nonempty subset of a topological space 𝐸, and let 𝑢𝑋 be given. For each 𝑖𝐼, let 𝑈𝑖 and 𝑍𝑖 be real t.v.s. with zero vector 𝜃𝑈𝑖 and 𝜃𝑍𝑖, respectively.

Example 1.1. For each 𝑖𝐼, let 𝐹𝑖𝑋×𝑌𝑖 and 𝐺𝑖𝑌×𝑌𝑖 be functions. If 𝐻𝑖 and 𝐴𝑖 are defined as follows: 𝐻𝑖=𝑦𝑖𝑌𝑖𝐹𝑖𝑢,𝑦𝑖,𝐴0𝑖(𝑧𝑦)=𝑖𝑌𝑖𝐺𝑖𝑦,𝑧𝑖,0(1.2) then HIDS will reduce to the following system of hybrid scalar equilibrium problem (P1):
(P1) Find 𝑣=(𝑣)𝑖𝐼𝑌 such that 𝐹𝑖(𝑢,𝑣𝑖)0 and 𝐺𝑖(𝑣,𝑦𝑖)>0 for all 𝑦𝑖𝑇𝑖(𝑣) and for all 𝑖𝐼.

Example 1.2. For each 𝑖𝐼, let 𝐹𝑖𝑋×𝑌𝑖𝑈𝑖 and 𝐺𝑖𝑌×𝑌𝑖𝑍𝑖 be multivalued maps with nonempty values, and let 𝐶𝑖 and 𝐷𝑖 be nonempty subsets of 𝑈𝑖 and 𝑍𝑖, respectively. If 𝐻𝑖 and 𝐴𝑖 are defined as follows: 𝐻𝑖=𝑦𝑖𝑌𝑖𝐹𝑖𝑢,𝑦𝑖𝐶𝑖𝜃𝑈𝑖,𝐴=𝑖𝑧(𝑦)=𝑖𝑌𝑖𝐺𝑖𝑦,𝑧𝑖𝐷𝑖𝜃𝑍𝑖,=(1.3) then HIDS will reduce to the following problem (P2), which is an abstract equilibrium problem:
(P2) Find 𝑣=(𝑣)𝑖𝐼𝑌 such that 𝐹𝑖(𝑢,𝑣𝑖)(𝐶𝑖{𝜃𝑈𝑖})= and 𝐺𝑖(𝑣,𝑦𝑖)(𝐷𝑖{𝜃𝑍𝑖}) for all 𝑦𝑖𝑇𝑖(𝑣) and for all 𝑖𝐼.

Example 1.3. For each 𝑖𝐼, let 𝐹𝑖𝑌𝑖𝑈𝑖 and 𝐺𝑖𝑌×𝑌𝑖𝑍𝑖 be multivalued maps. If 𝐻𝑖 and 𝐴𝑖 are defined as follows: 𝐻𝑖=𝑦𝑖𝑌𝑖𝑦𝑖𝐹𝑖𝑦𝑖,𝐴𝑖(𝑧𝑦)=𝑖𝑌𝑖𝑦𝐺𝑖𝑦,𝑧𝑖,(1.4) then HIDS will reduce to the following fixed point problem (P3):
(P3) Find 𝑣=(𝑣)𝑖𝐼𝑌 such that 𝑣𝑖𝐹𝑖(𝑣𝑖) and 𝑣𝐺𝑖(𝑣,𝑦𝑖) for all 𝑦𝑖𝑇𝑖(𝑣) for all 𝑖𝐼.

Example 1.4. For each 𝑖𝐼, let 𝐹𝑖𝑋×𝑌𝑖𝑈𝑖 and 𝐺𝑖𝑌×𝑌𝑖𝑍𝑖 be multivalued maps with nonempty values. If 𝐻𝑖 and 𝐴𝑖 are defined as follows: 𝐻𝑖=𝑦𝑖𝑌𝑖𝜃𝑈𝑖𝐹𝑖𝑢,𝑦𝑖,𝐴𝑖𝑧(𝑦)=𝑖𝑌𝑖𝜃𝑍𝑖𝐺𝑖𝑦,𝑧𝑖,(1.5) then HIDS will reduce to the following system of mixed type of parametric variational inclusion and disclusion problem (P4):
(P4) Find 𝑣=(𝑣)𝑖𝐼𝑌 such that 𝜃𝑈𝑖𝐹𝑖(𝑢,𝑣𝑖) and 𝜃𝑍𝑖𝐺𝑖(𝑣,𝑦𝑖) for all 𝑦𝑖𝑇𝑖(𝑣) and for all 𝑖𝐼.

In this paper, we study the existence theorems of system of parametric vectorial quasi-equilibrium problems, vectorial variants of system of Ekeland's variational principle (VSEVP, for short), and the existence theorems of system of the Stampacchia-type vectorial equilibrium problem by using an HIDS theorem established by the author [11]. Our results improve and generalize some theorems in [19] to the vector case. Till now, to my knowledge, there are extremely few results about Stampacchia-type vectorial equilibrium problem in the literature. The existence of VSEVP and the Stampacchia-type vectorial equilibrium problem are established by applying the HIDS theorem without the use of any scalarization method. So our results are completely different from [3, 8, 9, 12, 13, 15, 23]. As an application, a vectorial minimization theorem is also proved. Moreover, we prove some equivalence relations between our VSEVP, common fixed point theorem, and maximal element theorem.

2. Preliminaries

Let 𝐴 and 𝐵 be nonempty sets. A multivalued map 𝑇𝐴𝐵 is a function from 𝐴 to the power set 2𝐵 of 𝐵. We denote 𝑇(𝐴)={𝑇(𝑥)𝑥𝐴} and let 𝑇𝐵𝐴 be defined by the condition that 𝑥𝑇(𝑦) if and only if 𝑦𝑇(𝑥). We recall that a point 𝑥𝐴 is a maximal element of 𝑇𝐴2𝐵 if 𝑇(𝑥)=. Let 𝑋 be a linear space with zero vector 𝜃. A nonempty subset 𝐶 of 𝑋 is called a convex cone if 𝐶+𝐶𝐶 and 𝜆𝐶𝐶 for all 𝜆0. A cone 𝐶 in 𝑋 is pointed if 𝐶(𝐶)={𝜃}. Let 𝑍 be a real t.v.s., and let 𝐷 be a proper convex cone in 𝑍, and 𝐴𝑍. A point 𝑦𝐴 is called a vectorial minimal point of 𝐴 with respect to 𝐷 if for any 𝑦𝐴, 𝑦𝑦𝐷{𝜃}. The set of vectorial minimal point of 𝐴 is denoted by Min𝐷𝐴. The convex hull of 𝐴 and the closure of 𝐴 are denoted by co𝐴 and cl𝐴, respectively.

Definition 2.1. Let 𝑋 and 𝑌 be linear spaces, and let 𝐶 be a proper convex cone in 𝑌. A map 𝑓𝑋𝑌 is called 𝐶-convex if for any 𝑥1,𝑥2𝑋 and 𝜆[0,1], one has 𝑥𝜆𝑓1+𝑥(1𝜆)𝑓2𝑓𝜆𝑥1+(1𝜆)𝑥2𝐶.(2.1)
Clearly, if 𝑓1 and 𝑓2 are 𝐶-convex and 𝛼0, then 𝛼𝑓1 and 𝑓1+𝑓2 are 𝐶-convex.

Definition 2.2. Let 𝑋 be a nonempty convex subset of a vector space 𝐸, let 𝑌 be a nonempty convex subset of a vector space 𝑉, and let 𝑍 be a real t.v.s. Let 𝐹𝑋×𝑌𝑍 and 𝐶𝑋𝑍 be multivalued maps such that for each 𝑥𝑋, 𝐶(𝑥) is a nonempty closed convex cone. For each fixed 𝑥𝑋, 𝑦𝐹(𝑥,𝑦) is called 𝐶(𝑥)-quasiconvex if for any 𝑦1, 𝑦2𝑌 and 𝜆[0,1], one has either 𝐹𝑥,𝑦1𝐹𝑥,𝜆𝑦1+(1𝜆)𝑦2+𝐶(𝑥),(2.2) or 𝐹𝑥,𝑦2𝐹𝑥,𝜆𝑦1+(1𝜆)𝑦2+𝐶(𝑥).(2.3)
Now, we define the concept of vectorial upper and lower semicontinuous on t.v.s.

Definition 2.3. Let 𝑋 be a t.s., let 𝑌 be a t.v.s. with zero vector 𝜃, and let 𝐶{𝜃} be a pointed convex cone in 𝑌. A map 𝑓𝑋𝑌 is said to be (i)vectorial lower semicontinuous with respect to 𝐶 (𝐶-v.l.s.c., for short) at 𝑥0𝑋 if for any 𝑎𝐶{𝜃}, there exists an open neighborhood 𝑁(𝑥0) of 𝑥0 such that 𝑓(𝑦)𝑓(𝑥0)+𝑎𝐶{𝜃} for all 𝑦𝑁(𝑥0),(ii)vectorial upper semicontinuous with respect to 𝐶 (𝐶-v.u.s.c., for short) at 𝑥0𝑋 if for any 𝑎𝐶{𝜃}, there exists an open neighborhood 𝑁(𝑥0) of 𝑥0 such that 𝑓(𝑥0)𝑓(𝑦)+𝑎𝐶{𝜃} for all 𝑦𝑁(𝑥0).
The function 𝑓 is called   𝐶-v.l.s.c. (resp., 𝐶-v.u.s.c.) on 𝑋 if 𝑓 is 𝐶-v.l.s.c. (resp., 𝐶-v.u.s.c.) at every point of 𝑋.

Proposition 2.4. Let 𝑋 be a t.s., let 𝑌 be a t.v.s with zero vector 𝜃, and let 𝐶{𝜃} be a pointed convex cone in 𝑌. Let 𝑓, 𝑔𝑋𝑌 be maps and 𝛾0. Then the following hold: (a)𝑓 is 𝐶-v.u.s.c. (resp., 𝐶-v.l.s.c.) on 𝑋𝑓 is 𝐶-v.l.s.c. (resp., 𝐶-v.u.s.c.) on 𝑋; (b)if 𝑓 and 𝑔 are 𝐶-v.u.s.c. (resp., 𝐶-v.l.s.c.) on 𝑋, then 𝛾𝑓 and 𝑓+𝑔 are 𝐶-v.u.s.c. (resp., 𝐶-v.l.s.c.) on 𝑋; (c)if 𝑓 is 𝐶-v.u.s.c. on 𝑋, then(i){𝑥𝑋𝜆𝑓(𝑥)𝐶{𝜃}} is open in 𝑋 for all 𝜆𝑌,(ii){𝑥𝑋𝜆𝑓(𝑥)𝐶{𝜃}} is closed in 𝑋 for all 𝜆𝑌; (d)if 𝑓 is 𝐶-v.l.s.c. on 𝑋, then(iii){𝑥𝑋𝑓(𝑥)𝜆𝐶{𝜃}} is open in 𝑋 for all 𝜆𝑌,(iv){𝑥𝑋𝑓(𝑥)𝜆𝐶{𝜃}} is closed in 𝑋 for all 𝜆𝑌.

Proof. Clearly, (a) and (b) hold from definition. To prove (c), it suffices to show (i). Suppose that 𝑓 is 𝐶-v.u.s.c. on 𝑋. Let 𝜆𝑌 and 𝑥0{𝑥𝑋𝜆𝑓(𝑥)𝐶{𝜃}}. Then 𝛼=𝜆𝑓(𝑥0)𝐶{𝜃}. Since 𝑓 is 𝐶-v.u.s.c. at 𝑥0, there exists an open neighborhood 𝑁(𝑥0) of 𝑥0 such that 𝑥𝜆𝑓(𝑦)=𝑓0𝑓(𝑦)+𝛼𝐶{𝜃}(2.4) for all 𝑦𝑁(𝑥0). Hence {𝑥𝑋𝜆𝑓(𝑥)𝐶{𝜃}} is an open set in 𝑋 and (i) is proved. Obviously, (ii) is immediate from (i). It is easy to see that conclusion (d) follows from (a) and (c).

Remark 2.5. Let 𝑋 be a t.v.s. and 𝛼𝑋 with 𝛼0𝑋, where 0𝑋 is the origin of 𝑋. Let 𝑓𝑋𝐿𝛼={𝛾𝛼𝛾} be a map. Hence in conclusion (c) (resp., (d)) of Proposition 2.4, we have 𝑓isC-v.u.s.c.on𝑋(i)(ii)(resp.,𝑓is𝐶-v.l.s.c.on𝑋(iii)(iv)).(2.5) In particular, if 𝑋= (the set of real numbers) and 𝐶=[0,), then the 𝐶-v.l.s.c. (resp. 𝐶-v.u.s.c.) function 𝑓𝑋 is l.s.c. (resp. u.s.c.) in usual.
The concept of 𝐶-vectorial -function and 𝐶-vectorial quasi-distance on topological spaces are introduced as follows.

Definition 2.6. Let 𝑋 and 𝐸 be t.v.s., let 𝜃 be the zero vector of 𝐸, and let 𝐶{𝜃}, a pointed convex cone in 𝐸. A map 𝑝𝑋×𝑋𝐸 is called (a)a 𝐶-vectorial -function (𝐶-function, for short) if the following are satisfied: (VL1)𝑝(𝑥,𝑥)𝐶 for all 𝑥𝑋;(VL2) for any 𝑥𝑋, 𝑝(𝑥,) is 𝐶-convex;(VL3) for any 𝑦𝑋, 𝑝(,𝑦) is 𝐶-v.u.s.c. (b)a 𝐶-vectorial quasi-distance if the following are satisfied:(VQD1)𝑝(𝑥,𝑥)𝐶 for all 𝑥𝑋;(VQD2)𝑝(𝑥,𝑦)+𝑝(𝑦,𝑧)𝑝(𝑥,𝑧)+𝐶 for any 𝑥,𝑦,𝑧𝑋;(VQD3) for any 𝑥𝑋, 𝑝(𝑥,) is 𝐶-convex and 𝐶-v.l.s.c.;(VQD4) for any 𝑦𝑋, 𝑝(,𝑦) is 𝐶-v.u.s.c.
If 𝐸=(,] and letting 𝐶=[0,) be in (a) and (b), then the function 𝑝𝑋×𝑋(,] is called a -function and quasi-distance, respectively, introduced by Lin and Du [19]. For examples and results of -function and quasi-distance, one can see [19].

Remark 2.7. (a) Obviously, a 𝐶-vectorial quasi-distance is a 𝐶-function, but the reverse is not true;
(b) if 𝑝1 and 𝑝2 are 𝐶-vectorial quasi-distances (resp., 𝐶-functions) and 𝛼0, then 𝛼𝑝1 and 𝑝1+𝑝2 are 𝐶-vectorial quasi-distances (resp., 𝐶-functions);
(c) if 𝑓𝑋𝐸 is a 𝐶-v.l.s.c. and 𝐶-convex function, then the function 𝑝𝑋×𝑋𝐸 defined by 𝑝(𝑥,𝑦)=𝑓(𝑦)𝑓(𝑥) is a 𝐶-vectorial quasi-distance.

Lemma 2.8 . (see [24, 25]). Let 𝑋 and 𝑌 be the Hausdorff topological spaces, and let 𝑇𝑋𝑌 be a multivalued map. Then 𝑇 is l.s.c. at 𝑥𝑋 if and only if for any 𝑦𝑇(𝑥) and for any net {𝑥𝛼} in 𝑋 converging to 𝑥, there exists a subnet {𝑥𝜙(𝜆)}𝜆Λ of {𝑥𝛼} and a net {𝑦𝜆}𝜆Λ with 𝑦𝜆𝑦 such that 𝑦𝜆𝑇(𝑥𝜙(𝜆)) for all 𝜆Λ.

3. The Existence of System of VSEVP and Abstract Equilibrium Problems

The following existence theorem for the solution of HIDS was established in [11].

Theorem 3.1 (HIDS theorem [11]). Let 𝐼 be any index set. For each 𝑖𝐼, let 𝑌𝑖 be a nonempty closed convex subset of a Hausdorff t.v.s. 𝑉𝑖. Let 𝑋 be a nonempty subset of a topological space 𝐸, 𝑌=𝑖𝐼𝑌𝑖 and 𝑢𝑋. For each 𝑖𝐼, let 𝐻𝑖 be a nonempty closed subset of 𝑌𝑖, let 𝐴𝑖𝑌𝑌𝑖 be a multivalued map, and let 𝑇𝑖𝑌𝑌𝑖 be a multivalued map with nonempty values. For each 𝑖𝐼, suppose that the following conditions are satisfied:(i)for each 𝑦=(𝑦𝑖)𝑖𝐼𝑌,𝑦𝑖𝐴𝑖(𝑦);(ii)for each 𝑦𝑌, co𝑇𝑖(𝑦)𝐻𝑖 and 𝐴𝑖(𝑦) is convex;(iii)for each 𝑧𝑖𝑌𝑖, 𝑇𝑖(𝑧𝑖) and 𝐴𝑖(𝑧𝑖) are open in 𝑌;(iv)there exist a nonempty compact subset 𝐾 of 𝑌 and a nonempty compact convex subset 𝑀𝑖 of 𝑌𝑖 for each 𝑖𝐼 such that for each 𝑦𝑌𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗𝑇𝑗(𝑦)𝐴𝑗(𝑦).
Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑌 such that for each 𝑖𝐼, 𝑣𝑖𝐻𝑖 and 𝑦𝑖𝐴𝑖(𝑣) for all 𝑦𝑖𝑇𝑖(𝑣).

Example 3.2. Let 𝑋 and 𝑌 be Hausdorff t.v.s., let 𝑈 be a real t.v.s. with its zero vector 𝜃, and 𝑢𝑋.
(a) Let 𝐹𝑋×𝑌𝑈 be a multivalued map with nonempty values such that there exists 𝑤=𝑤(𝑢)𝑌 such that 𝜃𝐹(𝑢,𝑤) and the map 𝑦𝐹(𝑢,𝑦) is closed. Then it is easy to see that 𝐻={𝑦𝑌𝜃𝐹(𝑢,𝑦)} is a nonempty closed subset of 𝑌.
(b) Let 𝐺𝑌×𝑌𝑈 be a multivalued map with nonempty values and 𝑊 be a nonempty open set in 𝑈. Suppose that(i) for each 𝑦𝑌,𝜃𝐺(𝑦,𝑦)+𝑊,(ii) for each 𝑦𝑌,𝐺(𝑦,) is {𝜃}-quasiconvex and for each 𝑧𝑌, 𝐺(,𝑧) is l.s.c.
Let 𝐴𝑌𝑌 be defined by 𝐴(𝑦)={𝑧𝑌𝜃𝐺(𝑦,𝑧)+𝑊}.(3.1)
Then 𝑦𝐴(𝑦) for each 𝑦𝑌. We claim that 𝐴(𝑧) is open in 𝑌 for each 𝑧𝑌. Indeed, let 𝑧𝑌 be given, and let 𝑦cl(𝑌𝐴(𝑧)). Then there exists a net {𝑦𝛼}𝛼Λ in 𝑌𝐴(𝑧) such that 𝑦𝛼𝑦. Thus we have 𝜃𝐺(𝑦𝛼,𝑧)+𝑊 or 𝐺(𝑦𝛼,𝑧)𝑈𝑊. Clearly, 𝑦𝑌. For any 𝑤𝐺(𝑦,𝑧), since 𝐺(,𝑧) is l.s.c. at 𝑦 and 𝑦𝛼𝑦, by Lemma 2.8, there exists a net {𝑤𝛼} with 𝑤𝛼𝑤 such that 𝑤𝛼𝐺(𝑦𝛼,𝑧)𝑈𝑊. Since 𝑈𝑊 is closed, 𝑤𝑈𝑊. So 𝐺(𝑦,𝑧)𝑈𝑊. It implies cl(𝑌𝐴(𝑧))=𝑌𝐴(𝑧), and hence 𝐴(𝑧) is open in 𝑌. Next, we show that for each 𝑦𝑌, 𝐴(𝑦) is convex. Let 𝑎, 𝑏𝐴(𝑦). Then 𝜃𝐺(𝑦,𝑎)+𝑊 and 𝜃𝐺(𝑦,𝑏)+𝑊. For any 𝜆[0,1], let 𝑒𝜆=𝜆𝑎+(1𝜆)𝑏𝑌. Suppose to the contrary that there exists 𝜆0(0,1) such that 𝜃𝐺(𝑦,𝑒𝜆0)+𝑊. By the {𝜃}-quasiconvexity of 𝐺(𝑦,), either 𝜃𝐺(𝑦,𝑎)+𝑊𝐺𝑦,𝑒𝜆0+𝑊(3.2) or 𝜃𝐺(𝑦,𝑏)+𝑊𝐺𝑦,𝑒𝜆0+𝑊,(3.3) which leads to a contradiction. Hence for each 𝑦𝑌, 𝐴(𝑦) is convex.
Applying Theorem 3.1, we establish the following existence theorem of system of parametric vectorial quasi-equilibrium problem.

Theorem 3.3. Let 𝐼 be any index set. For each 𝑖𝐼, let 𝑋𝑖 be a nonempty Hausdorff t.v.s., and let 𝐶𝑖{𝜃} be a pointed convex cone in a t.v.s. 𝐸 with zero vector 𝜃. Let 𝑋=𝑖𝐼𝑋𝑖. For each 𝑖𝐼, let 𝑝𝑖, 𝑞𝑖𝑋𝑖×𝑋𝑖𝐸 be 𝐶𝑖-vectorial quasi-distances and let 𝑇𝑖𝑋𝑋𝑖 be a multivalued map with nonempty values. Let 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=𝑞𝑖(𝑢𝑖,𝑢𝑖)=𝜃 for all 𝑖𝐼. For each 𝑖𝐼, suppose that the following conditions are satisfied:(i)for each 𝑦𝑋, co𝑇𝑖(𝑦){𝑥𝑖𝑋𝑖𝑞𝑖(𝑢𝑖,𝑥𝑖)𝐶𝑖{𝜃}} and 𝑇𝑖(𝑧𝑖) is open for all 𝑧𝑖𝑋𝑖;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗𝑇𝑗(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)𝐶𝑗{𝜃}.
Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑞𝑖(𝑢𝑖,𝑣𝑖)𝐶𝑖{𝜃},(b)𝑝𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑇𝑖(𝑣).

Proof. For each 𝑖𝐼, define 𝐹𝑖𝑋×𝑋𝑖𝐸 and 𝐺𝑖𝑋×𝑋×𝑋𝑖𝐸 by 𝐹𝑖𝑥,𝑦𝑖=𝑞𝑖𝑥𝑖,𝑦𝑖+𝐶𝑖{𝜃},𝑥,𝑦𝑖𝑋×𝑋𝑖,(3.4)𝐺𝑡,𝑥,𝑦𝑖=𝑝𝑖𝑥𝑖,𝑦𝑖+𝐶𝑖{𝜃},𝑡,𝑥,𝑦𝑖𝑋×𝑋×𝑋𝑖,(3.5) respectively. Thus, for each 𝑖𝐼, we let 𝐻𝑖𝑥=𝑖𝑋𝑖𝜃𝐹𝑖𝑢,𝑥𝑖=𝑥𝑖𝑋𝑖𝑞𝑖𝑢𝑖,𝑥𝑖𝐶𝑖{𝜃},(3.6) and let 𝐴𝑖𝑋𝑋𝑖 be defined by 𝐴𝑖𝑦(𝑥)=𝑖𝑋𝑖𝜃𝐺𝑖𝑢,𝑥,𝑦𝑖=𝑦𝑖𝑋𝑖𝑝𝑖𝑥𝑖,𝑦𝑖𝐶𝑖{𝜃},𝑥=(𝑥𝑖)𝑖𝐼𝑋.(3.7) Clearly, for each 𝑖𝐼, 𝜃𝐺𝑖(𝑢,𝑥,𝑥𝑖) for all 𝑥=(𝑥𝑖)𝑖𝐼𝑋, and hence 𝑥𝑖𝐴𝑖(𝑥) for all 𝑥=(𝑥𝑖)𝑖𝐼𝑋. By the 𝐶𝑖-vectorial lower semicontinuity of 𝑞𝑖(𝑢𝑖,), 𝐻𝑖 is a nonempty closed subset of 𝑋𝑖. By (i), for each 𝑥𝑋, co𝑇𝑖(𝑥)𝐻𝑖. For each 𝑦𝑖𝑋𝑖, by the vectorial upper semicontinuity of 𝑝𝑖(,𝑦𝑖), the set {𝑥𝑖𝑋𝑖𝑝𝑖(𝑥𝑖,𝑦𝑖)𝐶𝑖{𝜃}} is open in 𝑋𝑖, and hence 𝐴𝑖(𝑦𝑖)={𝑥𝑖𝑋𝑖𝑞𝑖(𝑥𝑖,𝑦𝑖)𝐶𝑖{𝜃}}×𝑗𝑖𝑋𝑗 is open in 𝑋. For each 𝑥𝑋, 𝐴𝑖(𝑥) is convex from the 𝐶𝑖-convexity of 𝑝𝑖(𝑥𝑖,) for all 𝑖𝐼. By (ii), there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗𝑇𝑗(𝑦)𝐴𝑗(𝑦). Therefore all the conditions of Theorem 3.1 are satisfied. Applying Theorem 3.1, there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that: (1)𝑞𝑖(𝑢𝑖,𝑣𝑖)𝐶𝑖{𝜃},(2)𝑝𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑇𝑖(𝑣).

Theorem 3.4. Let 𝐼, 𝑋𝑖, 𝑋, 𝐶𝑖, 𝐸, 𝜃, 𝑇𝑖, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=𝜃 for all 𝑖𝐼. For each 𝑖𝐼, suppose that the following conditions are satisfied:(i)for each 𝑦𝑋, co𝑇𝑖(𝑦){𝑥𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑥𝑖)𝐶𝑖{𝜃}} and 𝑇𝑖(𝑧𝑖) is open for all 𝑧𝑖𝑋𝑖;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗𝑇𝑗(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)𝐶𝑗{𝜃}.
Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑝𝑖(𝑢𝑖,𝑣𝑖)𝐶𝑖{𝜃},(b)𝑝𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑇𝑖(𝑣){𝑧𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑧𝑖)𝐶𝑖{𝜃}}.

Proof. For each 𝑖𝐼, let 𝑞𝑖𝑋𝑖×𝑋𝑖𝐸 be defined by 𝑞𝑖=𝑝𝑖. By Theorem 3.3, there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, we have (1)𝑝𝑖(𝑢𝑖,𝑣𝑖)𝐶𝑖{𝜃},(2)𝑝𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑇𝑖(𝑣).
We claim that for each 𝑖𝐼, 𝑝𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖{𝑧𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑧𝑖)𝐶𝑖{𝜃}}. For each 𝑥𝑖{𝑧𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑧𝑖)𝐶𝑖{𝜃}}, we have 𝑝𝑖(𝑢𝑖,𝑥𝑖)𝐶𝑖{𝜃}. If 𝑝𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃}, then, by (VQD2) (in Definition 2.6(b)), we obtain 𝑝𝑖𝑢𝑖,𝑣𝑖𝑝𝑖𝑢𝑖,𝑥𝑖𝑝𝑖𝑣𝑖,𝑥𝑖+𝐶𝑖𝐶𝑖{𝜃},(3.8) which contradict with (1). Hence 𝑝𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑇𝑖(𝑣){𝑧𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑧𝑖)𝐶𝑖{𝜃}}, and the proof is completed.

Remark 3.5. (i) Let 𝑋 be a nonempty Hausdorff t.v.s., and let 𝐶{𝜃} be a nonempty pointed convex cone in a t.v.s. 𝐸 with zero vector 𝜃. If a map 𝑝𝑋×𝑋𝐸 satisfies that for each 𝑥𝑋, 𝑝(𝑥,) is 𝐶-quasiconvex, then for each 𝑢𝑋 with 𝑝(𝑢,𝑢)=𝜃, 𝑊𝑢={𝑥𝑋𝑝(𝑢,𝑥)𝐶{𝜃}} is a nonempty convex subset in 𝑋.
(ii) In Theorem 3.3, if for each 𝑖𝐼, 𝑊𝑖={𝑥𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑥𝑖)𝐶𝑖{𝜃}} is convex and 𝑇𝑖(𝑥)=𝑊𝑖 for all 𝑥𝑋, then conclusion (b) can be replaced with conclusion (b), where
(b)𝑝𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃},forall𝑥𝑖𝑋𝑖.
Since the sum of two 𝐶𝑖-vectorial quasi-distances is also a 𝐶𝑖-vectorial quasi-distance, the following results related with system of VSEVP for vectorial quasi-distances in a Hausdorff t.v.s. are immediate from Theorem 3.3.

Theorem 3.6. Let 𝐼, 𝑋𝑖, 𝑋, 𝐶𝑖, 𝐸, 𝜃, 𝑇𝑖, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=𝜃 for all 𝑖𝐼. For each 𝑖𝐼, let 𝑓𝑖𝑋𝑖𝐸 be a 𝐶𝑖-v.l.s.c. and 𝐶𝑖-convex function and suppose that the following conditions are satisfied:(i)for each 𝑦𝑋, co𝑇𝑖(𝑦){𝑥𝑖𝑋𝑖𝑓𝑖(𝑥𝑖)𝑓𝑖(𝑢𝑖)𝐶𝑖{𝜃}} and 𝑇𝑖(𝑧𝑖) is open for all 𝑧𝑖𝑋𝑖;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗𝑇𝑗(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑗)𝑓𝑗(𝑦𝑗)𝐶𝑗{𝜃}.
Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑓𝑖(𝑣𝑖)𝑓𝑖(𝑢𝑖)𝐶𝑖{𝜃}, (b)𝑝𝑖(𝑣𝑖,𝑥𝑖)+𝑓𝑖(𝑥𝑖)𝑓𝑖(𝑣𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑇𝑖(𝑣).

In Theorem 3.6, if 𝐸=(,] and 𝐶𝑖=[0,) for all 𝑖𝐼, then we have the following result.

Corollary 3.7. Let 𝐼, 𝑋𝑖, 𝑋, and 𝑇𝑖 be the same as in Theorem 3.3. For each 𝑖𝐼, let 𝑓𝑖𝑋𝑖(,] be a l.s.c. and convex function and let 𝑝𝑖𝑋𝑖×𝑋𝑖(,] be a quasi-distance. Let 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=0 for all 𝑖𝐼. Suppose that the following conditions are satisfied: (i)for each 𝑦𝑋, co𝑇𝑖(𝑦){𝑥𝑖𝑋𝑖𝑓𝑖(𝑥𝑖)𝑓𝑖(𝑢𝑖)} and 𝑇𝑖(𝑧𝑖) is open for all 𝑧𝑖𝑋𝑖;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗𝑇𝑗(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)<𝑓𝑗(𝑦𝑗)𝑓𝑗(𝑧𝑗).
Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑓𝑖(𝑣𝑖)𝑓𝑖(𝑢𝑖), (b)𝑝𝑖(𝑣𝑖,𝑥𝑖)𝑓𝑖(𝑣𝑖)𝑓𝑖(𝑥𝑖) for all 𝑥𝑖𝑇𝑖(𝑣).

Theorem 3.8. Let 𝐼, 𝑋𝑖, 𝑋, 𝐶𝑖, 𝐸, 𝜃, 𝑇𝑖, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=𝜃 for all 𝑖𝐼. For each 𝑖𝐼, let 𝑓𝑖𝑋𝑖𝐸 be a 𝐶𝑖-v.l.s.c. and 𝐶𝑖-convex function and suppose that the following conditions are satisfied: (i)for each 𝑦𝑋, co𝑇𝑖(𝑦){𝑥𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑥𝑖)+𝑓𝑖(𝑥𝑖)𝑓𝑖(𝑢𝑖)𝐶𝑖{𝜃}} and 𝑇𝑖(𝑧𝑖) is open for all 𝑧𝑖𝑋𝑖;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗𝑇𝑗(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑗)𝑓𝑗(𝑦𝑗)𝐶𝑗{𝜃}.
Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑝𝑖(𝑢𝑖,𝑣𝑖)+𝑓𝑖(𝑣𝑖)𝑓𝑖(𝑢𝑖)𝐶𝑖{𝜃}, (b)𝑝𝑖(𝑣𝑖,𝑥𝑖)+𝑓𝑖(𝑥𝑖)𝑓𝑖(𝑣𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑇𝑖(𝑣){𝑧𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑧𝑖)+𝑓𝑖(𝑧𝑖)𝑓𝑖(𝑢𝑖)𝐶𝑖{𝜃}}.

Corollary 3.9. Let 𝐼, 𝑋𝑖, 𝑋, and 𝑇𝑖 be the same as in Theorem 3.3. For each 𝑖𝐼, let 𝑓𝑖𝑋𝑖(,] be a l.s.c. and convex function and 𝑝𝑖𝑋𝑖×𝑋𝑖(,] be a quasi-distance. Let 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=0 for all 𝑖𝐼. Suppose that the following conditions are satisfied: (i)for each 𝑦𝑋, co𝑇𝑖(𝑦){𝑥𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑥𝑖)𝑓𝑖(𝑢𝑖)𝑓𝑖(𝑥𝑖)} and 𝑇𝑖(𝑧𝑖) is open for all 𝑧𝑖𝑋𝑖;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗𝑇𝑗(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)<𝑓𝑗(𝑦𝑗)𝑓𝑗(𝑧𝑗).
Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑝𝑖(𝑢𝑖,𝑣𝑖)𝑓𝑖(𝑢𝑖)𝑓𝑖(𝑣𝑖), (b)𝑝𝑖(𝑣𝑖,𝑥𝑖)𝑓𝑖(𝑣𝑖)𝑓𝑖(xi) for all 𝑥𝑖𝑇𝑖(𝑣){𝑧𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑧𝑖)>𝑓𝑖(𝑢𝑖)𝑓𝑖(𝑧𝑖)}.

By using Theorem 3.1 again, we have the following result.

Theorem 3.10. Let 𝐼, 𝑋𝑖, 𝑋, 𝐶𝑖, 𝐸, 𝜃, 𝑢, 𝑝𝑖, and 𝑞𝑖 be the same as in Theorem 3.3. Suppose that there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗 such that 𝑞𝑗(𝑢𝑗,𝑧𝑗)𝐶𝑗 and 𝑝𝑗(𝑦𝑗,𝑧𝑗)𝐶𝑗{𝜃}. Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑞𝑖(𝑢𝑖,𝑣𝑖)𝐶𝑖{𝜃}, (b)𝑝𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖{𝑧𝑖𝑋𝑖𝑞𝑖(𝑢𝑖,𝑧𝑖)𝐶𝑖}.

Proof. For each 𝑖𝐼, let 𝐻𝑖 and multivalued maps 𝐹𝑖, 𝐺𝑖, and 𝐴𝑖 be defined as in the proof of Theorem 3.3. For each 𝑖𝐼, define 𝑇𝑖𝑋𝑋𝑖 by 𝑇𝑖(𝑥)=𝐿𝑖𝑧=𝑖𝑋𝑖𝑞𝑖𝑢𝑖,𝑧𝑖𝐶𝑖,𝑥𝑋𝑇𝑖𝑧𝑖=𝑋if𝑧𝑖𝐿𝑖,if𝑧𝑖𝑋𝑖𝐿𝑖.(3.9) Clearly, for each 𝑖𝐼, 𝑇𝑖(𝑧𝑖) is open in 𝑋 for all 𝑧𝑖𝑋𝑖. By the 𝐶𝑖-convexity of 𝑞𝑖(𝑢𝑖,), 𝐿𝑖 is a nonempty convex subset of 𝑋𝑖 for all 𝑖𝐼. Since 𝐿𝑖 is convex in 𝑋𝑖 and 𝐿𝑖𝐻𝑖, we have co𝑇𝑖(𝑥)𝐻𝑖 for all 𝑖𝐼. By our hypothesis, there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗𝑇𝑗(𝑦)𝐴𝑗(𝑦). Thus all the conditions of Theorem 3.1 are satisfied, and the conclusion follows from Theorem 3.1.

Remark 3.11. In Theorem 3.10, the multivalued map 𝑇𝑖 and the condition “for each 𝑦𝑌, co𝑇𝑖(𝑦){𝑥𝑖𝑋𝑖𝑞𝑖(𝑢𝑖,𝑥𝑖)𝐶𝑖{𝜃}}” are not assumed. So Theorems 3.10 and 3.3 are different.

Theorem 3.12. Let 𝐼, 𝑋𝑖, 𝑋, 𝐶𝑖, 𝐸, 𝜃, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=𝜃 for all 𝑖𝐼. Suppose that there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗 such that 𝑝𝑗(𝑢𝑗,𝑧𝑗)𝐶𝑗 and 𝑝𝑗(𝑦𝑗,𝑧𝑗)𝐶𝑗{𝜃}. Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑝𝑖(𝑢𝑖,𝑣𝑖)𝐶𝑖{𝜃}, (b)𝑝𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖{𝑧𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑧𝑖)𝐶𝑖}{𝑧𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑧𝑖)𝐶𝑖{𝜃}}.

Proof. For each 𝑖𝐼, let 𝑞𝑖=𝑝𝑖. Applying Theorem 3.10 and following the same argument as in the proof of Theorem 3.4, one can prove the theorem.

Theorem 3.13. Let 𝐼, 𝑋𝑖, 𝑋, 𝐶𝑖, 𝐸, 𝜃, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=𝜃 for all 𝑖𝐼. For each 𝑖𝐼, let 𝑓𝑖𝑋𝑖𝐸 be a 𝐶𝑖-v.l.s.c. and 𝐶𝑖-convex function. Suppose that there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗 such that 𝑓𝑗(𝑧𝑗)𝑓𝑗(𝑢𝑗)𝐶𝑗 and 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑗)𝑓𝑗(𝑦𝑗)𝐶𝑗{𝜃}. Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑓𝑖(𝑣𝑖)𝑓𝑖(𝑢𝑖)𝐶𝑖{𝜃},(b)𝑝𝑖(𝑣𝑖,𝑥𝑖)+𝑓𝑖(𝑥𝑖)𝑓𝑖(𝑣𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖{𝑧𝑖𝑋𝑖𝑓𝑖(𝑧𝑖)𝑓𝑖(𝑢𝑖)𝐶𝑖}.

Corollary 3.14. Let 𝐼, 𝑋𝑖, and 𝑋 be the same as in Theorem 3.3. For each 𝑖𝐼, let 𝑓𝑖𝑋𝑖(,] be a l.s.c. and convex function and 𝑝𝑖𝑋𝑖×𝑋𝑖(,] be a quasi-distance. Let 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=0 for all 𝑖𝐼. Suppose that there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗 such that 𝑓𝑗(𝑧𝑗)𝑓𝑗(𝑢𝑗) and 𝑝𝑗(𝑦𝑗,𝑧𝑗)<𝑓𝑗(𝑦𝑗)𝑓𝑗(𝑧𝑗). Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑓𝑖(𝑣𝑖)𝑓𝑖(𝑢𝑖), (b)𝑝𝑖(𝑣𝑖,𝑥𝑖)𝑓𝑖(𝑣𝑖)𝑓𝑖(𝑥𝑖) for all 𝑥𝑖{𝑧𝑖𝑋𝑖𝑓𝑖(𝑧𝑖)𝑓𝑖(𝑢𝑖)}.

Remark 3.15. [19, Theorem  4.3] is a special case of Corollary 3.14.

Theorem 3.16. Let 𝐼, 𝑋𝑖, 𝑋, 𝐶𝑖, 𝐸, 𝜃, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=𝜃 for all 𝑖𝐼. For each 𝑖𝐼, let 𝑓𝑖𝑋𝑖𝐸 be a 𝐶𝑖-v.l.s.c. and 𝐶𝑖-convex function, and suppose that there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗 such that 𝑝𝑗(𝑢𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑗)𝑓𝑗(𝑢𝑗)𝐶𝑗 and 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑗)𝑓𝑗(𝑦𝑗)𝐶𝑗{𝜃}. Then for each 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=𝜃 for all 𝑖𝐼, there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑝𝑖(𝑢𝑖,𝑣𝑖)+𝑓𝑖(𝑣𝑖)𝑓𝑖(𝑢𝑖)𝐶𝑖{𝜃}, (b)𝑝𝑖(𝑣𝑖,𝑥𝑖)+𝑓𝑖(𝑥𝑖)𝑓𝑖(𝑣𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖{𝑧𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑧𝑖)+𝑓𝑖(𝑧𝑖)𝑓𝑖(𝑢𝑖)𝐶𝑖}{𝑧𝑖𝑋𝑖𝑝𝑖(𝑢𝑖,𝑧𝑖)+𝑓𝑖(𝑧𝑖)𝑓𝑖(𝑢𝑖)𝐶𝑖{𝜃}}.
In Theorem 3.16, if 𝐸=(,] and 𝐶𝑖=[0,) for all 𝑖𝐼, then we have the following system of Lin and Du's variant of system of Ekeland's variational principle in t.v.s.

Corollary 3.17. Let 𝐼, 𝑋𝑖, and 𝑋 be the same as in Theorem 3.3. For each 𝑖𝐼, let 𝑓𝑖𝑋𝑖𝐸 be a l.s.c. and convex function and let 𝑝𝑖𝑋𝑖×𝑋𝑖(,] be a quasi-distance. Let 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=0 for all 𝑖𝐼. Suppose that there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗 such that 𝑝𝑗(𝑢𝑗,𝑧𝑗)𝑓𝑗(𝑢𝑗)𝑓𝑗(𝑧𝑗) and 𝑝𝑗(𝑦𝑗,𝑧𝑗)<𝑓𝑗(𝑦𝑗)𝑓𝑗(𝑧𝑗). Then for each 𝑢=(𝑢𝑖)𝑖𝐼𝑋 with 𝑝𝑖(𝑢𝑖,𝑢𝑖)=0 for all 𝑖𝐼, there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has (a)𝑝𝑖(𝑢𝑖,𝑣𝑖)𝑓𝑖(𝑢𝑖)𝑓𝑖(𝑣𝑖), (b)𝑝𝑖(𝑣𝑖,𝑥𝑖)𝑓𝑖(𝑣𝑖)𝑓𝑖(𝑥𝑖) for all 𝑥𝑖𝑋𝑖.

Remark 3.18. Corollary 3.17 generalizes [19, Theorem  4.1].

4. A Vectorial Minimization Theorem and Equivalent Formulations of VSEVP

Using Theorem 3.1 again, we also obtain an existence theorem of system of generalized vectorial equilibrium problem of the Stampacchia-type which can be regarded as a weak form of VSEVP for 𝐶-vectorial -function in a Hausdorff t.v.s.

Theorem 4.1. Let 𝐼 be any index set. For each 𝑖𝐼, let 𝑋𝑖 be a nonempty Hausdorff t.v.s. and 𝐶𝑖{𝜃} be a nonempty pointed convex cone in a t.v.s. 𝐸 with zero vector 𝜃. Let 𝑋=𝑖𝐼𝑋𝑖. For each 𝑖𝐼, let 𝑝𝑖, 𝑞𝑖𝑋𝑖×𝑋𝑖𝐸 be 𝐶𝑖-function. Suppose that (i)={𝑥=(𝑥𝑖)𝑖𝐼𝑋𝑝𝑖(𝑥𝑖,𝑥𝑖)=𝑞𝑖(𝑥𝑖,𝑥𝑖)=𝜃forall𝑖𝐼}, (ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗 such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑞𝑗(𝑦𝑗,𝑧𝑗)𝐶𝑖{𝜃}.
Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has 𝑝𝑖(𝑣𝑖,𝑥𝑖)+𝑞𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑋𝑖.

Proof. Let 𝑢=(𝑢𝑖)𝑖𝐼 be given. For each 𝑖𝐼, we define 𝐹𝑖𝑋×𝑋𝑖𝐸, 𝐺𝑖𝑋×𝑋×𝑋𝑖𝐸 and 𝐴𝑖𝑋𝑋𝑖 by 𝐹𝑖𝑥,𝑦𝑖=𝐶𝑖{𝜃},𝑥,𝑦𝑖𝑋×𝑋𝑖,𝐺𝑡,𝑥,𝑦𝑖=𝑝𝑖𝑥𝑖,𝑦𝑖+𝑞𝑖𝑥𝑖,𝑦𝑖+𝐶𝑖{𝜃},𝑡,𝑥,𝑦𝑖𝑋×𝑋×𝑋𝑖,𝐴(4.1)𝑖(𝑦𝑥)=𝑖𝑋𝑖𝜃𝐺𝑖𝑢,𝑥,𝑦𝑖𝑥,𝑥=𝑖𝑖𝐼𝑋,(4.2) respectively. Clearly, 𝐻𝑖𝑥=𝑖𝑋𝑖𝜃𝐹𝑖𝑢,𝑥𝑖=𝑋,𝑖𝐼.(4.3) Using the same argument in the proof of Theorem 3.3, one can verify that all the conditions of Theorem 3.1 are satisfied. By Theorem 3.1, there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, we have 𝑝𝑖(𝑣𝑖,𝑥𝑖)+𝑞𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑋𝑖.

An existence theorem of system of generalized vector equilibrium problem is immediate from Theorem 4.1 if 𝑞𝑖0 (the zero map) for all 𝑖𝐼.

Theorem 4.2. Let 𝐼 be any index set. For each 𝑖𝐼, let 𝑋𝑖 be a nonempty Hausdorff t.v.s., and let 𝐶𝑖{𝜃} be a nonempty pointed convex cone in a t.v.s. 𝐸 with zero vector 𝜃. Let 𝑋=𝑖𝐼𝑋𝑖. For each 𝑖𝐼, let 𝑝𝑖𝑋𝑖×𝑋𝑖𝐸 be a 𝐶𝑖-function. Suppose that (i)={𝑥=(𝑥𝑖)𝑖𝐼𝑋𝑝𝑖(𝑥𝑖,𝑥𝑖)=𝜃forall𝑖𝐼};(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗 such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)𝐶𝑖{𝜃}.
Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has 𝑝𝑖(𝑣𝑖,𝑥𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑋𝑖.

Remark 4.3. (a) Notice that Theorems 4.1 and 4.2 are indeed equivalent since the sum of two 𝐶𝑖-functions is also a 𝐶𝑖-function.
(b) In Theorem 4.1, the maps 𝑝𝑖 and 𝑞𝑖 are only assumed to be 𝐶𝑖-functions which need not be 𝐶𝑖-vectorial quasi-distances. So Theorem 4.1 is different from any theorem in Section 3 and is not a special case of any theorem in Section 3.

Theorem 4.4. Let 𝐼, 𝑋𝑖, 𝑋, 𝐶𝑖, 𝐸, and 𝜃 be the same as in Theorem 4.1. For each 𝑖𝐼, let 𝑝𝑖𝑋𝑖×𝑋𝑖𝐸 be a 𝐶𝑖-function and let 𝑓𝑖𝑋𝑖𝐸 be a 𝐶𝑖-v.l.s.c. and 𝐶𝑖-convex function. Suppose that (i)={𝑥=(𝑥𝑖)𝑖𝐼𝑋𝑝𝑖(𝑥𝑖,𝑥𝑖)=𝜃forall𝑖𝐼}, (ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗 such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑖)𝑓𝑗(𝑦𝑖)𝐶𝑗{𝜃}.
Then there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, one has 𝑝𝑖(𝑣𝑖,𝑥𝑖)+𝑓𝑖(𝑥𝑖)𝑓𝑖(𝑣𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑋𝑖.

Remark 4.5. [19, Theorem  4.2] is a special case of Theorem 4.4.
Applying Theorem 4.4, we obtain the following vectorial minimization theorem.

Theorem 4.6 (vectorial minimization theorem). Let 𝐼, 𝑋𝑖, 𝑋, 𝐶𝑖, 𝐸, and 𝜃 be the same as in Theorem 4.1. For each 𝑖𝐼, let 𝑝𝑖𝑋𝑖×𝑋𝑖𝐸 be a 𝐶𝑖-function and let 𝑓𝑖𝑋𝑖𝐸 be a 𝐶𝑖-v.l.s.c. and 𝐶𝑖-convex function. Suppose that (i)={𝑥=(𝑥𝑖)𝑖𝐼𝑋𝑝𝑖(𝑥𝑖,𝑥𝑖)=𝜃forall𝑖𝐼}, (ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗 such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑖)𝑓𝑗(𝑦𝑖)𝐶𝑗{𝜃},(iii)for 𝑖𝐼 and 𝑥𝑖𝑋𝑖 with 𝑓𝑖(𝑥𝑖)Min𝐶𝑖𝑓𝑖(𝑋𝑖), there exists 𝑦𝑖𝑋𝑖 with 𝑦𝑖𝑥𝑖 such that 𝑝𝑖(𝑥𝑖,𝑦𝑖)+𝑓𝑖(𝑦𝑖)𝑓𝑖(𝑥𝑖)𝐶𝑖{𝜃}.
Then there exists ̂𝑥=(̂𝑥𝑖)𝑖𝐼𝑋 such that 𝑓𝑖(̂𝑥𝑖)Min𝐶𝑖𝑓𝑖(Xi) for all 𝑖𝐼.

Proof. Applying Theorem 4.4, there exists 𝑣=(𝑣𝑖)𝑖𝐼𝑋 such that for each 𝑖𝐼, we have 𝑝𝑖(𝑣𝑖,𝑥𝑖)+𝑓𝑖(𝑥𝑖)𝑓𝑖(𝑣𝑖)𝐶𝑖{𝜃} for all 𝑥𝑖𝑋𝑖. We claim that 𝑓𝑖(𝑣𝑖)Min𝐶𝑖𝑓𝑖(𝑋𝑖) for all 𝑖𝐼. Suppose to the contrary that there exists 𝑖0𝐼 such that 𝑓𝑖0(𝑣𝑖0)Min𝐶𝑖0𝑓𝑖0(𝑋𝑖0). Then, by our assumption, there exists 𝑦𝑖0=𝑦𝑖0(𝑣𝑖0)𝑋𝑖0 with 𝑦𝑖0𝑣𝑖0 such that 𝑝𝑖0(𝑣𝑖0,𝑦𝑖0)+𝑓𝑖0(𝑦𝑖0)𝑓𝑖0(𝑣𝑖0)𝐶𝑖0{𝜃}, which leads to a contradiction. Therefore 𝑓𝑖(𝑣𝑖)Min𝐶𝑖𝑓𝑖(𝑋𝑖) for all 𝑖𝐼.

The following scalar minimization theorem follows from Theorem 4.6 immediately.

Corollary 4.7. Let 𝐼, 𝑋𝑖, and 𝑋 be the same as in Theorem 4.1. For each 𝑖𝐼, let 𝑓𝑖𝑋𝑖(,] be a l.s.c. and convex function and let 𝑝𝑖𝑋𝑖×𝑋𝑖(,] be a -function. Suppose that (i)𝒟={𝑥=(𝑥𝑖)𝑖𝐼𝑋𝑝𝑖(𝑥𝑖,𝑥𝑖)=0forall𝑖𝐼}, (ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each 𝑖𝐼 such that for each 𝑦𝑋𝐾 there exist 𝑗𝐼 and 𝑧𝑗𝑀𝑗 such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)<𝑓𝑗(𝑦𝑖)𝑓𝑗(𝑧𝑖),(iii)for any 𝑖𝐼 and 𝑥𝑖𝑋𝑖 with 𝑓𝑖(𝑥𝑖)>inf𝑧𝑖𝑋𝑖𝑓𝑖(𝑧𝑖) there exists 𝑦𝑖𝑋𝑖 with 𝑦𝑖𝑥𝑖 such that 𝑝𝑖(𝑥𝑖,𝑦𝑖)<𝑓𝑖(𝑥𝑖)𝑓𝑖(𝑦𝑖).
Then there exists ̂𝑥=(̂𝑥𝑖)𝑖𝐼𝑋 such that 𝑓𝑖(̂𝑥𝑖)=inf𝑧𝑖𝑋𝑖𝑓𝑖(𝑧𝑖) for all 𝑖𝐼.

Remark 4.8. (a) [19, Theorem  5.5] is a special case of Corollary 4.7.
(b) Theorems 4.4 and 4.6 are equivalent if they further add the condition “for each 𝑖𝐼, 𝑝𝑖(𝑥𝑖,𝑦𝑖)𝐶𝑖 for all 𝑥𝑖, 𝑦𝑖𝑋𝑖.” Indeed, it suffices to show that Theorem 4.6 implies Theorem 4.4. Suppose that for each 𝑥=(𝑥