Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 247236 | https://doi.org/10.1155/2011/247236

Wei-Shih Du, "Applications of an HIDS Theorem to the Existence of Fixed Point, Abstract Equilibria and Optimization Problems", Abstract and Applied Analysis, vol. 2011, Article ID 247236, 17 pages, 2011. https://doi.org/10.1155/2011/247236

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

Academic Editor: Josip E. PečariΔ‡
Received16 Jun 2011
Accepted18 Jul 2011
Published29 Sep 2011

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 π‘“βˆΆπ‘‹Γ—π‘‹β†’β„ be a function with 𝑓(π‘₯,π‘₯)β‰₯0 for all π‘₯βˆˆπ‘‹. Then the scalar equilibrium problem in the sense of Blum and Oettli [1] is to find π‘₯βˆˆπ‘‹ such that 𝑓(π‘₯,𝑦)β‰₯0 for all π‘¦βˆˆπ‘‹. 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 [1–4] and references therein.

The famous Ekeland's variational principle (EVP, for short) [5–7] 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 [8–23] 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, 16–18, 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 π‘–βˆˆπΌ, let π‘Œπ‘– be a nonempty closed convex subset of a Hausdorff topological vector space (t.v.s., for short) 𝑉𝑖, π»π‘–βŠ†π‘Œπ‘–, βˆπ‘Œ=π‘–βˆˆπΌπ‘Œπ‘–, π΄π‘–βˆΆπ‘ŒβŠΈπ‘Œπ‘–, and let π‘‡π‘–βˆΆπ‘ŒβŠΈπ‘Œπ‘– be multivalued maps. Hybrid inclusion and disclusion systems (HIDS) are defined as follows:(HIDS)Find𝑣=(𝑣)π‘–βˆˆπΌβˆˆπ‘Œsuchthatπ‘£π‘–βˆˆπ»π‘–,π‘¦π‘–βˆ‰π΄π‘–(𝑣),βˆ€π‘¦π‘–βˆˆπ‘‡π‘–(𝑣),βˆ€π‘–βˆˆπΌ.(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 π‘’βˆˆπ‘‹ be given. For each π‘–βˆˆπΌ, let π‘ˆπ‘– and 𝑍𝑖 be real t.v.s. with zero vector πœƒπ‘ˆπ‘– and πœƒπ‘π‘–, respectively.

Example 1.1. For each π‘–βˆˆπΌ, let πΉπ‘–βˆΆπ‘‹Γ—π‘Œπ‘–β†’β„ and πΊπ‘–βˆΆπ‘ŒΓ—π‘Œπ‘–β†’β„ be functions. If 𝐻𝑖 and 𝐴𝑖 are defined as follows: 𝐻𝑖=ξ€½π‘¦π‘–βˆˆπ‘Œπ‘–βˆΆπΉπ‘–ξ€·π‘’,𝑦𝑖,𝐴≀0𝑖(𝑧𝑦)=π‘–βˆˆπ‘Œπ‘–βˆΆπΊπ‘–ξ€·π‘¦,𝑧𝑖,≀0(1.2) then HIDS will reduce to the following system of hybrid scalar equilibrium problem (P1):
(P1) Find 𝑣=(𝑣)π‘–βˆˆπΌβˆˆπ‘Œ such that 𝐹𝑖(𝑒,𝑣𝑖)≀0 and 𝐺𝑖(𝑣,𝑦𝑖)>0 for all π‘¦π‘–βˆˆπ‘‡π‘–(𝑣) and for all π‘–βˆˆπΌ.

Example 1.2. For each π‘–βˆˆπΌ, let πΉπ‘–βˆΆπ‘‹Γ—π‘Œπ‘–βŠΈπ‘ˆπ‘– and πΊπ‘–βˆΆπ‘ŒΓ—π‘Œπ‘–βŠΈπ‘π‘– be multivalued maps with nonempty values, and let 𝐢𝑖 and 𝐷𝑖 be nonempty subsets of π‘ˆπ‘– and 𝑍𝑖, respectively. If 𝐻𝑖 and 𝐴𝑖 are defined as follows: 𝐻𝑖=ξ€½π‘¦π‘–βˆˆπ‘Œπ‘–βˆΆπΉπ‘–ξ€·π‘’,π‘¦π‘–ξ€Έβˆ©ξ€·βˆ’πΆπ‘–β§΅ξ€½πœƒπ‘ˆπ‘–ξ€Ύ,𝐴=βˆ…π‘–ξ€½π‘§(𝑦)=π‘–βˆˆπ‘Œπ‘–βˆΆπΊπ‘–ξ€·π‘¦,π‘§π‘–ξ€Έβˆ©ξ€·βˆ’π·π‘–β§΅ξ€½πœƒπ‘π‘–ξ€Ύ,ξ€Ύξ€Έ=βˆ…(1.3) then HIDS will reduce to the following problem (P2), which is an abstract equilibrium problem:
(P2) Find 𝑣=(𝑣)π‘–βˆˆπΌβˆˆπ‘Œ such that 𝐹𝑖(𝑒,𝑣𝑖)∩(βˆ’πΆπ‘–β§΅{πœƒπ‘ˆπ‘–})=βˆ… and 𝐺𝑖(𝑣,𝑦𝑖)∩(βˆ’π·π‘–β§΅{πœƒπ‘π‘–})β‰ βˆ… for all π‘¦π‘–βˆˆπ‘‡π‘–(𝑣) and for all π‘–βˆˆπΌ.

Example 1.3. For each π‘–βˆˆπΌ, let πΉπ‘–βˆΆπ‘Œπ‘–βŠΈπ‘ˆπ‘– and πΊπ‘–βˆΆπ‘ŒΓ—π‘Œπ‘–βŠΈπ‘π‘– be multivalued maps. If 𝐻𝑖 and 𝐴𝑖 are defined as follows: 𝐻𝑖=ξ€½π‘¦π‘–βˆˆπ‘Œπ‘–βˆΆπ‘¦π‘–βˆˆπΉπ‘–ξ€·π‘¦π‘–,𝐴𝑖(𝑧𝑦)=π‘–βˆˆπ‘Œπ‘–βˆΆπ‘¦βˆ‰πΊπ‘–ξ€·π‘¦,𝑧𝑖,ξ€Έξ€Ύ(1.4) then HIDS will reduce to the following fixed point problem (P3):
(P3) Find 𝑣=(𝑣)π‘–βˆˆπΌβˆˆπ‘Œ such that π‘£π‘–βˆˆπΉπ‘–(𝑣𝑖) and π‘£βˆˆπΊπ‘–(𝑣,𝑦𝑖) for all π‘¦π‘–βˆˆπ‘‡π‘–(𝑣) for all π‘–βˆˆπΌ.

Example 1.4. For each π‘–βˆˆπΌ, let πΉπ‘–βˆΆπ‘‹Γ—π‘Œπ‘–βŠΈπ‘ˆπ‘– and πΊπ‘–βˆΆπ‘ŒΓ—π‘Œπ‘–βŠΈπ‘π‘– be multivalued maps with nonempty values. If 𝐻𝑖 and 𝐴𝑖 are defined as follows: 𝐻𝑖=ξ€½π‘¦π‘–βˆˆπ‘Œπ‘–βˆΆπœƒπ‘ˆπ‘–βˆ‰πΉπ‘–ξ€·π‘’,𝑦𝑖,𝐴𝑖𝑧(𝑦)=π‘–βˆˆπ‘Œπ‘–βˆΆπœƒπ‘π‘–βˆ‰πΊπ‘–ξ€·π‘¦,𝑧𝑖,ξ€Έξ€Ύ(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)π‘–βˆˆπΌβˆˆπ‘Œ such that πœƒπ‘ˆπ‘–βˆ‰πΉπ‘–(𝑒,𝑣𝑖) and πœƒπ‘π‘–βˆˆπΊπ‘–(𝑣,𝑦𝑖) for all π‘¦π‘–βˆˆπ‘‡π‘–(𝑣) and for all π‘–βˆˆπΌ.

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 𝐡 be nonempty sets. A multivalued map π‘‡βˆΆπ΄βŠΈπ΅ is a function from 𝐴 to the power set 2𝐡 of 𝐡. We denote ⋃𝑇(𝐴)={𝑇(π‘₯)∢π‘₯∈𝐴} and let π‘‡βˆ’βˆΆπ΅βŠΈπ΄ be defined by the condition that π‘₯βˆˆπ‘‡βˆ’(𝑦) if and only if π‘¦βˆˆπ‘‡(π‘₯). We recall that a point π‘₯∈𝐴 is a maximal element of π‘‡βˆΆπ΄β†’2𝐡 if 𝑇(π‘₯)=βˆ…. Let 𝑋 be a linear space with zero vector πœƒ. A nonempty subset 𝐢 of 𝑋 is called a convex cone if 𝐢+πΆβŠ†πΆ and πœ†πΆβŠ†πΆ for all πœ†β‰₯0. A cone 𝐢 in 𝑋 is pointed if 𝐢∩(βˆ’πΆ)={πœƒ}. Let 𝑍 be a real t.v.s., and let 𝐷 be a proper convex cone in 𝑍, and π΄βŠ†π‘. A point π‘¦βˆˆπ΄ is called a vectorial minimal point of 𝐴 with respect to 𝐷 if for any π‘¦βˆˆπ΄, π‘¦βˆ’π‘¦βˆ‰βˆ’π·β§΅{πœƒ}. 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 π‘Œ be linear spaces, and let 𝐢 be a proper convex cone in π‘Œ. A map π‘“βˆΆπ‘‹βŠΈπ‘Œ is called 𝐢-convex if for any π‘₯1,π‘₯2βˆˆπ‘‹ and πœ†βˆˆ[0,1], one has ξ€·π‘₯πœ†π‘“1ξ€Έ+ξ€·π‘₯(1βˆ’πœ†)𝑓2ξ€Έξ€·βˆ’π‘“πœ†π‘₯1+(1βˆ’πœ†)π‘₯2ξ€ΈβŠ†πΆ.(2.1)
Clearly, if 𝑓1 and 𝑓2 are 𝐢-convex and 𝛼β‰₯0, then 𝛼𝑓1 and 𝑓1+𝑓2 are 𝐢-convex.

Definition 2.2. Let 𝑋 be a nonempty convex subset of a vector space 𝐸, let π‘Œ be a nonempty convex subset of a vector space 𝑉, and let 𝑍 be a real t.v.s. Let πΉβˆΆπ‘‹Γ—π‘ŒβŠΈπ‘ and πΆβˆΆπ‘‹βŠΈπ‘ be multivalued maps such that for each π‘₯βˆˆπ‘‹, 𝐢(π‘₯) is a nonempty closed convex cone. For each fixed π‘₯βˆˆπ‘‹, π‘¦βŠΈπΉ(π‘₯,𝑦) is called 𝐢(π‘₯)-quasiconvex if for any 𝑦1, 𝑦2βˆˆπ‘Œ and πœ†βˆˆ[0,1], one has either 𝐹π‘₯,𝑦1ξ€Έξ€·βŠ†πΉπ‘₯,πœ†π‘¦1+(1βˆ’πœ†)𝑦2ξ€Έ+𝐢(π‘₯),(2.2) or 𝐹π‘₯,𝑦2ξ€Έξ€·βŠ†πΉπ‘₯,πœ†π‘¦1+(1βˆ’πœ†)𝑦2ξ€Έ+𝐢(π‘₯).(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 π‘Œ be a t.v.s. with zero vector πœƒ, and let 𝐢≠{πœƒ} be a pointed convex cone in π‘Œ. A map π‘“βˆΆπ‘‹β†’π‘Œ is said to be (i)vectorial lower semicontinuous with respect to 𝐢 (𝐢-v.l.s.c., for short) at π‘₯0βˆˆπ‘‹ if for any π‘ŽβˆˆπΆβ§΅{πœƒ}, there exists an open neighborhood 𝑁(π‘₯0) of π‘₯0 such that 𝑓(𝑦)βˆ’π‘“(π‘₯0)+π‘ŽβˆˆπΆβ§΅{πœƒ} for all π‘¦βˆˆπ‘(π‘₯0),(ii)vectorial upper semicontinuous with respect to 𝐢 (𝐢-v.u.s.c., for short) at π‘₯0βˆˆπ‘‹ if for any π‘ŽβˆˆπΆβ§΅{πœƒ}, there exists an open neighborhood 𝑁(π‘₯0) of π‘₯0 such that 𝑓(π‘₯0)βˆ’π‘“(𝑦)+π‘ŽβˆˆπΆβ§΅{πœƒ} for all π‘¦βˆˆπ‘(π‘₯0).
The function 𝑓 is called   𝐢-v.l.s.c. (resp., 𝐢-v.u.s.c.) on 𝑋 if 𝑓 is 𝐢-v.l.s.c. (resp., 𝐢-v.u.s.c.) at every point of 𝑋.

Proposition 2.4. Let 𝑋 be a t.s., let π‘Œ be a t.v.s with zero vector πœƒ, and let 𝐢≠{πœƒ} be a pointed convex cone in π‘Œ. Let 𝑓, π‘”βˆΆπ‘‹β†’π‘Œ be maps and 𝛾β‰₯0. Then the following hold: (a)𝑓 is 𝐢-v.u.s.c. (resp., 𝐢-v.l.s.c.) on π‘‹β‡”βˆ’π‘“ is 𝐢-v.l.s.c. (resp., 𝐢-v.u.s.c.) on 𝑋; (b)if 𝑓 and 𝑔 are 𝐢-v.u.s.c. (resp., 𝐢-v.l.s.c.) on 𝑋, then 𝛾𝑓 and 𝑓+𝑔 are 𝐢-v.u.s.c. (resp., 𝐢-v.l.s.c.) on 𝑋; (c)if 𝑓 is 𝐢-v.u.s.c. on 𝑋, then(i){π‘₯βˆˆπ‘‹βˆΆπœ†βˆ’π‘“(π‘₯)∈𝐢⧡{πœƒ}} is open in 𝑋 for all πœ†βˆˆπ‘Œ,(ii){π‘₯βˆˆπ‘‹βˆΆπœ†βˆ’π‘“(π‘₯)βˆ‰πΆβ§΅{πœƒ}} is closed in 𝑋 for all πœ†βˆˆπ‘Œ; (d)if 𝑓 is 𝐢-v.l.s.c. on 𝑋, then(iii){π‘₯βˆˆπ‘‹βˆΆπ‘“(π‘₯)βˆ’πœ†βˆˆπΆβ§΅{πœƒ}} is open in 𝑋 for all πœ†βˆˆπ‘Œ,(iv){π‘₯βˆˆπ‘‹βˆΆπ‘“(π‘₯)βˆ’πœ†βˆ‰πΆβ§΅{πœƒ}} is closed in 𝑋 for all πœ†βˆˆπ‘Œ.

Proof. Clearly, (a) and (b) hold from definition. To prove (c), it suffices to show (i). Suppose that 𝑓 is 𝐢-v.u.s.c. on 𝑋. Let πœ†βˆˆπ‘Œ and π‘₯0∈{π‘₯βˆˆπ‘‹βˆΆπœ†βˆ’π‘“(π‘₯)∈𝐢⧡{πœƒ}}. Then π›ΌβˆΆ=πœ†βˆ’π‘“(π‘₯0)∈𝐢⧡{πœƒ}. Since 𝑓 is 𝐢-v.u.s.c. at π‘₯0, there exists an open neighborhood 𝑁(π‘₯0) of π‘₯0 such that ξ€·π‘₯πœ†βˆ’π‘“(𝑦)=𝑓0ξ€Έβˆ’π‘“(𝑦)+π›ΌβˆˆπΆβ§΅{πœƒ}(2.4) for all π‘¦βˆˆπ‘(π‘₯0). Hence {π‘₯βˆˆπ‘‹βˆΆπœ†βˆ’π‘“(π‘₯)∈𝐢⧡{πœƒ}} 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 π›Όβˆˆπ‘‹ with 𝛼≠0𝑋, where 0𝑋 is the origin of 𝑋. Let π‘“βˆΆπ‘‹β†’πΏπ›Ό={π›Ύπ›ΌβˆΆπ›Ύβˆˆβ„} be a map. Hence in conclusion (c) (resp., (d)) of Proposition 2.4, we have 𝑓isC-v.u.s.c.onπ‘‹βŸΊ(i)⟺(ii)(resp.,𝑓is𝐢-v.l.s.c.onπ‘‹βŸΊ(iii)⟺(iv)).(2.5) In particular, if 𝑋=ℝ (the set of real numbers) and 𝐢=[0,∞), then the 𝐢-v.l.s.c. (resp. 𝐢-v.u.s.c.) function π‘“βˆΆπ‘‹β†’β„ is l.s.c. (resp. u.s.c.) in usual.
The concept of 𝐢-vectorial β„“-function and 𝐢-vectorial quasi-distance on topological spaces are introduced as follows.

Definition 2.6. Let 𝑋 and 𝐸 be t.v.s., let πœƒ be the zero vector of 𝐸, and let 𝐢≠{πœƒ}, a pointed convex cone in 𝐸. A map π‘βˆΆπ‘‹Γ—π‘‹β†’πΈ is called (a)a 𝐢-vectorial β„“-function (ℓ𝐢-function, for short) if the following are satisfied: (VL1)𝑝(π‘₯,π‘₯)∈𝐢 for all π‘₯βˆˆπ‘‹;(VL2) for any π‘₯βˆˆπ‘‹, 𝑝(π‘₯,β‹…) is 𝐢-convex;(VL3) for any π‘¦βˆˆπ‘‹, 𝑝(β‹…,𝑦) is 𝐢-v.u.s.c. (b)a 𝐢-vectorial quasi-distance if the following are satisfied:(VQD1)𝑝(π‘₯,π‘₯)∈𝐢 for all π‘₯βˆˆπ‘‹;(VQD2)𝑝(π‘₯,𝑦)+𝑝(𝑦,𝑧)βˆˆπ‘(π‘₯,𝑧)+𝐢 for any π‘₯,𝑦,π‘§βˆˆπ‘‹;(VQD3) for any π‘₯βˆˆπ‘‹, 𝑝(π‘₯,β‹…) is 𝐢-convex and 𝐢-v.l.s.c.;(VQD4) for any π‘¦βˆˆπ‘‹, 𝑝(β‹…,𝑦) is 𝐢-v.u.s.c.
If 𝐸=(βˆ’βˆž,∞] and letting 𝐢=[0,∞) be in (a) and (b), then the function π‘βˆΆπ‘‹Γ—π‘‹β†’(βˆ’βˆž,∞] is called a β„“-function and quasi-distance, respectively, introduced by Lin and Du [19]. For examples and results of β„“-function and quasi-distance, one can see [19].

Remark 2.7. (a) Obviously, a 𝐢-vectorial quasi-distance is a ℓ𝐢-function, but the reverse is not true;
(b) if 𝑝1 and 𝑝2 are 𝐢-vectorial quasi-distances (resp., ℓ𝐢-functions) and 𝛼β‰₯0, then 𝛼𝑝1 and 𝑝1+𝑝2 are 𝐢-vectorial quasi-distances (resp., ℓ𝐢-functions);
(c) if π‘“βˆΆπ‘‹β†’πΈ is a 𝐢-v.l.s.c. and 𝐢-convex function, then the function π‘βˆΆπ‘‹Γ—π‘‹β†’πΈ defined by 𝑝(π‘₯,𝑦)=𝑓(𝑦)βˆ’π‘“(π‘₯) is a 𝐢-vectorial quasi-distance.

Lemma 2.8 . (see [24, 25]). Let 𝑋 and π‘Œ be the Hausdorff topological spaces, and let π‘‡βˆΆπ‘‹βŠΈπ‘Œ be a multivalued map. Then 𝑇 is l.s.c. at π‘₯βˆˆπ‘‹ if and only if for any π‘¦βˆˆπ‘‡(π‘₯) and for any net {π‘₯𝛼} in 𝑋 converging to π‘₯, there exists a subnet {π‘₯πœ™(πœ†)}πœ†βˆˆΞ› of {π‘₯𝛼} and a net {π‘¦πœ†}πœ†βˆˆΞ› with π‘¦πœ†β†’π‘¦ such that π‘¦πœ†βˆˆπ‘‡(π‘₯πœ™(πœ†)) for all πœ†βˆˆΞ›.

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 π‘–βˆˆπΌ, let π‘Œπ‘– be a nonempty closed convex subset of a Hausdorff t.v.s. 𝑉𝑖. Let 𝑋 be a nonempty subset of a topological space 𝐸, βˆπ‘Œ=π‘–βˆˆπΌπ‘Œπ‘– and π‘’βˆˆπ‘‹. For each π‘–βˆˆπΌ, let 𝐻𝑖 be a nonempty closed subset of π‘Œπ‘–, let π΄π‘–βˆΆπ‘ŒβŠΈπ‘Œπ‘– be a multivalued map, and let π‘‡π‘–βˆΆπ‘ŒβŠΈπ‘Œπ‘– be a multivalued map with nonempty values. For each π‘–βˆˆπΌ, suppose that the following conditions are satisfied:(i)for each 𝑦=(𝑦𝑖)π‘–βˆˆπΌβˆˆπ‘Œ,π‘¦π‘–βˆ‰π΄π‘–(𝑦);(ii)for each π‘¦βˆˆπ‘Œ, co𝑇𝑖(𝑦)βŠ†π»π‘– and 𝐴𝑖(𝑦) is convex;(iii)for each π‘§π‘–βˆˆπ‘Œπ‘–, π‘‡βˆ’π‘–(𝑧𝑖) and π΄βˆ’π‘–(𝑧𝑖) are open in π‘Œ;(iv)there exist a nonempty compact subset 𝐾 of π‘Œ and a nonempty compact convex subset 𝑀𝑖 of π‘Œπ‘– for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘Œβ§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘—βˆ©π‘‡π‘—(𝑦)βˆ©π΄π‘—(𝑦).
Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘Œ such that for each π‘–βˆˆπΌ, π‘£π‘–βˆˆπ»π‘– and π‘¦π‘–βˆ‰π΄π‘–(𝑣) for all π‘¦π‘–βˆˆπ‘‡π‘–(𝑣).

Example 3.2. Let 𝑋 and π‘Œ be Hausdorff t.v.s., let π‘ˆ be a real t.v.s. with its zero vector πœƒ, and π‘’βˆˆπ‘‹.
(a) Let πΉβˆΆπ‘‹Γ—π‘ŒβŠΈπ‘ˆ be a multivalued map with nonempty values such that there exists 𝑀=𝑀(𝑒)βˆˆπ‘Œ such that πœƒβˆˆπΉ(𝑒,𝑀) and the map π‘¦βŠΈπΉ(𝑒,𝑦) is closed. Then it is easy to see that 𝐻={π‘¦βˆˆπ‘ŒβˆΆπœƒβˆˆπΉ(𝑒,𝑦)} is a nonempty closed subset of π‘Œ.
(b) Let πΊβˆΆπ‘ŒΓ—π‘ŒβŠΈπ‘ˆ be a multivalued map with nonempty values and π‘Š be a nonempty open set in π‘ˆ. Suppose that(i) for each π‘¦βˆˆπ‘Œ,πœƒβˆ‰πΊ(𝑦,𝑦)+π‘Š,(ii) for each π‘¦βˆˆπ‘Œ,𝐺(𝑦,β‹…) is {πœƒ}-quasiconvex and for each π‘§βˆˆπ‘Œ, 𝐺(β‹…,𝑧) is l.s.c.
Let π΄βˆΆπ‘ŒβŠΈπ‘Œ be defined by 𝐴(𝑦)={π‘§βˆˆπ‘ŒβˆΆπœƒβˆˆπΊ(𝑦,𝑧)+π‘Š}.(3.1)
Then π‘¦βˆ‰π΄(𝑦) for each π‘¦βˆˆπ‘Œ. We claim that π΄βˆ’(𝑧) is open in π‘Œ for each π‘§βˆˆπ‘Œ. Indeed, let π‘§βˆˆπ‘Œ be given, and let π‘¦βˆˆcl(π‘Œβ§΅π΄βˆ’(𝑧)). Then there exists a net {𝑦𝛼}π›ΌβˆˆΞ› in π‘Œβ§΅π΄βˆ’(𝑧) such that 𝑦𝛼→𝑦. Thus we have πœƒβˆ‰πΊ(𝑦𝛼,𝑧)+π‘Š or 𝐺(𝑦𝛼,𝑧)βŠ†π‘ˆβ§΅π‘Š. Clearly, π‘¦βˆˆπ‘Œ. For any π‘€βˆˆπΊ(𝑦,𝑧), since 𝐺(β‹…,𝑧) is l.s.c. at 𝑦 and 𝑦𝛼→𝑦, by Lemma 2.8, there exists a net {𝑀𝛼} with 𝑀𝛼→𝑀 such that π‘€π›ΌβˆˆπΊ(𝑦𝛼,𝑧)βŠ†π‘ˆβ§΅π‘Š. Since π‘ˆβ§΅π‘Š is closed, π‘€βˆˆπ‘ˆβ§΅π‘Š. So 𝐺(𝑦,𝑧)βŠ†π‘ˆβ§΅π‘Š. It implies cl(π‘Œβ§΅π΄βˆ’(𝑧))=π‘Œβ§΅π΄βˆ’(𝑧), and hence π΄βˆ’(𝑧) is open in π‘Œ. Next, we show that for each π‘¦βˆˆπ‘Œ, 𝐴(𝑦) is convex. Let π‘Ž, π‘βˆˆπ΄(𝑦). Then πœƒβˆˆπΊ(𝑦,π‘Ž)+π‘Š and πœƒβˆˆπΊ(𝑦,𝑏)+π‘Š. For any πœ†βˆˆ[0,1], let π‘’πœ†βˆΆ=πœ†π‘Ž+(1βˆ’πœ†)π‘βˆˆπ‘Œ. Suppose to the contrary that there exists πœ†0∈(0,1) such that πœƒβˆ‰πΊ(𝑦,π‘’πœ†0)+π‘Š. By the {πœƒ}-quasiconvexity of 𝐺(𝑦,β‹…), either ξ€·πœƒβˆˆπΊ(𝑦,π‘Ž)+π‘ŠβŠ†πΊπ‘¦,π‘’πœ†0ξ€Έ+π‘Š(3.2) or ξ€·πœƒβˆˆπΊ(𝑦,𝑏)+π‘ŠβŠ†πΊπ‘¦,π‘’πœ†0ξ€Έ+π‘Š,(3.3) which leads to a contradiction. Hence for each π‘¦βˆˆπ‘Œ, 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 π‘–βˆˆπΌ, let 𝑋𝑖 be a nonempty Hausdorff t.v.s., and let 𝐢𝑖≠{πœƒ} be a pointed convex cone in a t.v.s. 𝐸 with zero vector πœƒ. Let βˆπ‘‹=π‘–βˆˆπΌπ‘‹π‘–. For each π‘–βˆˆπΌ, let 𝑝𝑖, π‘žπ‘–βˆΆπ‘‹π‘–Γ—π‘‹π‘–β†’πΈ be 𝐢𝑖-vectorial quasi-distances and let π‘‡π‘–βˆΆπ‘‹βŠΈπ‘‹π‘– be a multivalued map with nonempty values. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=π‘žπ‘–(𝑒𝑖,𝑒𝑖)=πœƒ for all π‘–βˆˆπΌ. For each π‘–βˆˆπΌ, suppose that the following conditions are satisfied:(i)for each π‘¦βˆˆπ‘‹, co𝑇𝑖(𝑦)βŠ†{π‘₯π‘–βˆˆπ‘‹π‘–βˆΆπ‘žπ‘–(𝑒𝑖,π‘₯𝑖)βˆ‰πΆπ‘–β§΅{πœƒ}} and π‘‡βˆ’π‘–(𝑧𝑖) is open for all π‘§π‘–βˆˆπ‘‹π‘–;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘—βˆ©π‘‡π‘—(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)βˆˆβˆ’πΆπ‘—β§΅{πœƒ}.
Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)π‘žπ‘–(𝑒𝑖,𝑣𝑖)βˆ‰πΆπ‘–β§΅{πœƒ},(b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆπ‘‡π‘–(𝑣).

Proof. For each π‘–βˆˆπΌ, define πΉπ‘–βˆΆπ‘‹Γ—π‘‹π‘–βŠΈπΈ and πΊπ‘–βˆΆπ‘‹Γ—π‘‹Γ—π‘‹π‘–βŠΈπΈ by 𝐹𝑖π‘₯,𝑦𝑖=βˆ’π‘žπ‘–ξ€·π‘₯𝑖,𝑦𝑖+𝐢𝑖⧡{πœƒ},βˆ€π‘₯,π‘¦π‘–ξ€Έβˆˆπ‘‹Γ—π‘‹π‘–,(3.4)𝐺𝑑,π‘₯,𝑦𝑖=𝑝𝑖π‘₯𝑖,𝑦𝑖+𝐢𝑖⧡{πœƒ},βˆ€π‘‘,π‘₯,π‘¦π‘–ξ€Έβˆˆπ‘‹Γ—π‘‹Γ—π‘‹π‘–,(3.5) respectively. Thus, for each π‘–βˆˆπΌ, we let 𝐻𝑖π‘₯∢=π‘–βˆˆπ‘‹π‘–βˆΆπœƒβˆ‰πΉπ‘–ξ€·π‘’,π‘₯𝑖=ξ€½π‘₯ξ€Έξ€Ύπ‘–βˆˆπ‘‹π‘–βˆΆπ‘žπ‘–ξ€·π‘’π‘–,π‘₯π‘–ξ€Έβˆ‰πΆπ‘–β§΅ξ€Ύ{πœƒ},(3.6) and let π΄π‘–βˆΆπ‘‹βŠΈπ‘‹π‘– be defined by 𝐴𝑖𝑦(π‘₯)=π‘–βˆˆπ‘‹π‘–βˆΆπœƒβˆˆπΊπ‘–ξ€·π‘’,π‘₯,𝑦𝑖=ξ€½π‘¦ξ€Έξ€Ύπ‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–ξ€·π‘₯𝑖,π‘¦π‘–ξ€Έβˆˆβˆ’πΆπ‘–ξ€Ύβ§΅{πœƒ},βˆ€π‘₯=(π‘₯𝑖)π‘–βˆˆπΌβˆˆπ‘‹.(3.7) Clearly, for each π‘–βˆˆπΌ, πœƒβˆ‰πΊπ‘–(𝑒,π‘₯,π‘₯𝑖) for all π‘₯=(π‘₯𝑖)π‘–βˆˆπΌβˆˆπ‘‹, and hence π‘₯π‘–βˆ‰π΄π‘–(π‘₯) for all π‘₯=(π‘₯𝑖)π‘–βˆˆπΌβˆˆπ‘‹. By the 𝐢𝑖-vectorial lower semicontinuity of π‘žπ‘–(𝑒𝑖,β‹…), 𝐻𝑖 is a nonempty closed subset of 𝑋𝑖. By (i), for each π‘₯βˆˆπ‘‹, co𝑇𝑖(π‘₯)βŠ†π»π‘–. For each π‘¦π‘–βˆˆπ‘‹π‘–, by the vectorial upper semicontinuity of 𝑝𝑖(β‹…,𝑦𝑖), the set {π‘₯π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(π‘₯𝑖,𝑦𝑖)βˆˆβˆ’πΆπ‘–β§΅{πœƒ}} is open in 𝑋𝑖, and hence π΄βˆ’π‘–(𝑦𝑖)={π‘₯π‘–βˆˆπ‘‹π‘–βˆΆπ‘žπ‘–(π‘₯𝑖,𝑦𝑖)βˆˆβˆ’πΆπ‘–βˆβ§΅{πœƒ}}×𝑗≠𝑖𝑋𝑗 is open in 𝑋. For each π‘₯βˆˆπ‘‹, 𝐴𝑖(π‘₯) is convex from the 𝐢𝑖-convexity of 𝑝𝑖(π‘₯𝑖,β‹…) for all π‘–βˆˆπΌ. By (ii), there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘—βˆ©π‘‡π‘—(𝑦)βˆ©π΄π‘—(𝑦). Therefore all the conditions of Theorem 3.1 are satisfied. Applying Theorem 3.1, there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that: (1)π‘žπ‘–(𝑒𝑖,𝑣𝑖)βˆ‰πΆπ‘–β§΅{πœƒ},(2)𝑝𝑖(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆπ‘‡π‘–(𝑣).

Theorem 3.4. Let 𝐼, 𝑋𝑖, 𝑋, 𝐢𝑖, 𝐸, πœƒ, 𝑇𝑖, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=πœƒ for all π‘–βˆˆπΌ. For each π‘–βˆˆπΌ, suppose that the following conditions are satisfied:(i)for each π‘¦βˆˆπ‘‹, co𝑇𝑖(𝑦)βŠ†{π‘₯π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,π‘₯𝑖)βˆ‰πΆπ‘–β§΅{πœƒ}} and π‘‡βˆ’π‘–(𝑧𝑖) is open for all π‘§π‘–βˆˆπ‘‹π‘–;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘—βˆ©π‘‡π‘—(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)βˆˆβˆ’πΆπ‘—β§΅{πœƒ}.
Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)𝑝𝑖(𝑒𝑖,𝑣𝑖)βˆ‰πΆπ‘–β§΅{πœƒ},(b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆπ‘‡π‘–(𝑣)βˆͺ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,𝑧𝑖)βˆˆπΆπ‘–β§΅{πœƒ}}.

Proof. For each π‘–βˆˆπΌ, let π‘žπ‘–βˆΆπ‘‹π‘–Γ—π‘‹π‘–β†’πΈ be defined by π‘žπ‘–=𝑝𝑖. By Theorem 3.3, there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, we have (1)𝑝𝑖(𝑒𝑖,𝑣𝑖)βˆ‰πΆπ‘–β§΅{πœƒ},(2)𝑝𝑖(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆπ‘‡π‘–(𝑣).
We claim that for each π‘–βˆˆπΌ, 𝑝𝑖(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,𝑧𝑖)βˆˆπΆπ‘–β§΅{πœƒ}}. For each π‘₯π‘–βˆˆ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,𝑧𝑖)βˆˆπΆπ‘–β§΅{πœƒ}}, we have 𝑝𝑖(𝑒𝑖,π‘₯𝑖)βˆˆπΆπ‘–β§΅{πœƒ}. If 𝑝𝑖(𝑣𝑖,π‘₯𝑖)βˆˆβˆ’πΆπ‘–β§΅{πœƒ}, then, by (VQD2) (in Definition 2.6(b)), we obtain 𝑝𝑖𝑒𝑖,π‘£π‘–ξ€Έβˆˆπ‘π‘–ξ€·π‘’π‘–,π‘₯π‘–ξ€Έβˆ’π‘π‘–ξ€·π‘£π‘–,π‘₯𝑖+πΆπ‘–βŠ†πΆπ‘–β§΅{πœƒ},(3.8) which contradict with (1). Hence 𝑝𝑖(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆπ‘‡π‘–(𝑣)βˆͺ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,𝑧𝑖)βˆˆπΆπ‘–β§΅{πœƒ}}, and the proof is completed.

Remark 3.5. (i) Let 𝑋 be a nonempty Hausdorff t.v.s., and let 𝐢≠{πœƒ} be a nonempty pointed convex cone in a t.v.s. 𝐸 with zero vector πœƒ. If a map π‘βˆΆπ‘‹Γ—π‘‹β†’πΈ satisfies that for each π‘₯βˆˆπ‘‹, 𝑝(π‘₯,β‹…) is 𝐢-quasiconvex, then for each π‘’βˆˆπ‘‹ with 𝑝(𝑒,𝑒)=πœƒ, π‘Šπ‘’βˆΆ={π‘₯βˆˆπ‘‹βˆΆπ‘(𝑒,π‘₯)βˆ‰πΆβ§΅{πœƒ}} is a nonempty convex subset in 𝑋.
(ii) In Theorem 3.3, if for each π‘–βˆˆπΌ, π‘Šπ‘–βˆΆ={π‘₯π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,π‘₯𝑖)βˆ‰πΆπ‘–β§΅{πœƒ}} is convex and 𝑇𝑖(π‘₯)=π‘Šπ‘– for all π‘₯βˆˆπ‘‹, then conclusion (b) can be replaced with conclusion (b)ξ…ž, where
(b)ξ…žπ‘π‘–(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ},forallπ‘₯π‘–βˆˆπ‘‹π‘–.
Since the sum of two 𝐢𝑖-vectorial quasi-distances is also a 𝐢𝑖-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, 𝐢𝑖, 𝐸, πœƒ, 𝑇𝑖, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=πœƒ for all π‘–βˆˆπΌ. For each π‘–βˆˆπΌ, let π‘“π‘–βˆΆπ‘‹π‘–β†’πΈ be a 𝐢𝑖-v.l.s.c. and 𝐢𝑖-convex function and suppose that the following conditions are satisfied:(i)for each π‘¦βˆˆπ‘‹, co𝑇𝑖(𝑦)βŠ†{π‘₯π‘–βˆˆπ‘‹π‘–βˆΆπ‘“π‘–(π‘₯𝑖)βˆ’π‘“π‘–(𝑒𝑖)βˆ‰πΆπ‘–β§΅{πœƒ}} and π‘‡βˆ’π‘–(𝑧𝑖) is open for all π‘§π‘–βˆˆπ‘‹π‘–;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘—βˆ©π‘‡π‘—(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑗)βˆ’π‘“π‘—(𝑦𝑗)βˆˆβˆ’πΆπ‘—β§΅{πœƒ}.
Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)𝑓𝑖(𝑣𝑖)βˆ’π‘“π‘–(𝑒𝑖)βˆ‰πΆπ‘–β§΅{πœƒ}, (b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)+𝑓𝑖(π‘₯𝑖)βˆ’π‘“π‘–(𝑣𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆπ‘‡π‘–(𝑣).

In Theorem 3.6, if 𝐸=(βˆ’βˆž,∞] and 𝐢𝑖=[0,∞) for all π‘–βˆˆπΌ, 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 π‘–βˆˆπΌ, let π‘“π‘–βˆΆπ‘‹π‘–β†’(βˆ’βˆž,∞] be a l.s.c. and convex function and let π‘π‘–βˆΆπ‘‹π‘–Γ—π‘‹π‘–β†’(βˆ’βˆž,∞] be a quasi-distance. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=0 for all π‘–βˆˆπΌ. Suppose that the following conditions are satisfied: (i)for each π‘¦βˆˆπ‘‹, co𝑇𝑖(𝑦)βŠ†{π‘₯π‘–βˆˆπ‘‹π‘–βˆΆπ‘“π‘–(π‘₯𝑖)≀𝑓𝑖(𝑒𝑖)} and π‘‡βˆ’π‘–(𝑧𝑖) is open for all π‘§π‘–βˆˆπ‘‹π‘–;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘—βˆ©π‘‡π‘—(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)<𝑓𝑗(𝑦𝑗)βˆ’π‘“π‘—(𝑧𝑗).
Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)𝑓𝑖(𝑣𝑖)≀𝑓𝑖(𝑒𝑖), (b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)β‰₯𝑓𝑖(𝑣𝑖)βˆ’π‘“π‘–(π‘₯𝑖) for all π‘₯π‘–βˆˆπ‘‡π‘–(𝑣).

Theorem 3.8. Let 𝐼, 𝑋𝑖, 𝑋, 𝐢𝑖, 𝐸, πœƒ, 𝑇𝑖, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=πœƒ for all π‘–βˆˆπΌ. For each π‘–βˆˆπΌ, let π‘“π‘–βˆΆπ‘‹π‘–β†’πΈ be a 𝐢𝑖-v.l.s.c. and 𝐢𝑖-convex function and suppose that the following conditions are satisfied: (i)for each π‘¦βˆˆπ‘‹, co𝑇𝑖(𝑦)βŠ†{π‘₯π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,π‘₯𝑖)+𝑓𝑖(π‘₯𝑖)βˆ’π‘“π‘–(𝑒𝑖)βˆ‰πΆπ‘–β§΅{πœƒ}} and π‘‡βˆ’π‘–(𝑧𝑖) is open for all π‘§π‘–βˆˆπ‘‹π‘–;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘—βˆ©π‘‡π‘—(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑗)βˆ’π‘“π‘—(𝑦𝑗)βˆˆβˆ’πΆπ‘—β§΅{πœƒ}.
Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)𝑝𝑖(𝑒𝑖,𝑣𝑖)+𝑓𝑖(𝑣𝑖)βˆ’π‘“π‘–(𝑒𝑖)βˆ‰πΆπ‘–β§΅{πœƒ}, (b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)+𝑓𝑖(π‘₯𝑖)βˆ’π‘“π‘–(𝑣𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆπ‘‡π‘–(𝑣)βˆͺ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,𝑧𝑖)+𝑓𝑖(𝑧𝑖)βˆ’π‘“π‘–(𝑒𝑖)βˆˆπΆπ‘–β§΅{πœƒ}}.

Corollary 3.9. Let 𝐼, 𝑋𝑖, 𝑋, and 𝑇𝑖 be the same as in Theorem 3.3. For each π‘–βˆˆπΌ, let π‘“π‘–βˆΆπ‘‹π‘–β†’(βˆ’βˆž,∞] be a l.s.c. and convex function and π‘π‘–βˆΆπ‘‹π‘–Γ—π‘‹π‘–β†’(βˆ’βˆž,∞] be a quasi-distance. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=0 for all π‘–βˆˆπΌ. Suppose that the following conditions are satisfied: (i)for each π‘¦βˆˆπ‘‹, co𝑇𝑖(𝑦)βŠ†{π‘₯π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,π‘₯𝑖)≀𝑓𝑖(𝑒𝑖)βˆ’π‘“π‘–(π‘₯𝑖)} and π‘‡βˆ’π‘–(𝑧𝑖) is open for all π‘§π‘–βˆˆπ‘‹π‘–;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘—βˆ©π‘‡π‘—(𝑦) such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)<𝑓𝑗(𝑦𝑗)βˆ’π‘“π‘—(𝑧𝑗).
Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)𝑝𝑖(𝑒𝑖,𝑣𝑖)≀𝑓𝑖(𝑒𝑖)βˆ’π‘“π‘–(𝑣𝑖), (b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)β‰₯𝑓𝑖(𝑣𝑖)βˆ’π‘“π‘–(xi) for all π‘₯π‘–βˆˆπ‘‡π‘–(𝑣)βˆͺ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,𝑧𝑖)>𝑓𝑖(𝑒𝑖)βˆ’π‘“π‘–(𝑧𝑖)}.

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

Theorem 3.10. Let 𝐼, 𝑋𝑖, 𝑋, 𝐢𝑖, 𝐸, πœƒ, 𝑒, 𝑝𝑖, and π‘žπ‘– 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 π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘— such that π‘žπ‘—(𝑒𝑗,𝑧𝑗)βˆˆβˆ’πΆπ‘— and 𝑝𝑗(𝑦𝑗,𝑧𝑗)βˆˆβˆ’πΆπ‘—β§΅{πœƒ}. Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)π‘žπ‘–(𝑒𝑖,𝑣𝑖)βˆ‰πΆπ‘–β§΅{πœƒ}, (b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘žπ‘–(𝑒𝑖,𝑧𝑖)βˆˆβˆ’πΆπ‘–}.

Proof. For each π‘–βˆˆπΌ, let 𝐻𝑖 and multivalued maps 𝐹𝑖, 𝐺𝑖, and 𝐴𝑖 be defined as in the proof of Theorem 3.3. For each π‘–βˆˆπΌ, define π‘‡π‘–βˆΆπ‘‹βŠΈπ‘‹π‘– by 𝑇𝑖(π‘₯)=πΏπ‘–ξ€½π‘§βˆΆ=π‘–βˆˆπ‘‹π‘–βˆΆπ‘žπ‘–ξ€·π‘’π‘–,π‘§π‘–ξ€Έβˆˆβˆ’πΆπ‘–ξ€Ύ,βˆ€π‘₯βˆˆπ‘‹βŸΊπ‘‡βˆ’π‘–ξ€·π‘§π‘–ξ€Έ=𝑋ifπ‘§π‘–βˆˆπΏπ‘–,βˆ…ifπ‘§π‘–βˆˆπ‘‹π‘–β§΅πΏπ‘–.(3.9) Clearly, for each π‘–βˆˆπΌ, π‘‡βˆ’π‘–(𝑧𝑖) is open in 𝑋 for all π‘§π‘–βˆˆπ‘‹π‘–. By the 𝐢𝑖-convexity of π‘žπ‘–(𝑒𝑖,β‹…), 𝐿𝑖 is a nonempty convex subset of 𝑋𝑖 for all π‘–βˆˆπΌ. Since 𝐿𝑖 is convex in 𝑋𝑖 and πΏπ‘–βŠ†π»π‘–, we have co𝑇𝑖(π‘₯)βŠ†π»π‘– for all π‘–βˆˆπΌ. By our hypothesis, there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘—βˆ©π‘‡π‘—(𝑦)βˆ©π΄π‘—(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 β€œfor each π‘¦βˆˆπ‘Œ, co𝑇𝑖(𝑦)βŠ†{π‘₯π‘–βˆˆπ‘‹π‘–βˆΆπ‘žπ‘–(𝑒𝑖,π‘₯𝑖)βˆ‰πΆπ‘–β§΅{πœƒ}}” are not assumed. So Theorems 3.10 and 3.3 are different.

Theorem 3.12. Let 𝐼, 𝑋𝑖, 𝑋, 𝐢𝑖, 𝐸, πœƒ, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=πœƒ for all π‘–βˆˆπΌ. Suppose that there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘— such that 𝑝𝑗(𝑒𝑗,𝑧𝑗)βˆˆβˆ’πΆπ‘— and 𝑝𝑗(𝑦𝑗,𝑧𝑗)βˆˆβˆ’πΆπ‘—β§΅{πœƒ}. Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)𝑝𝑖(𝑒𝑖,𝑣𝑖)βˆ‰πΆπ‘–β§΅{πœƒ}, (b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,𝑧𝑖)βˆˆβˆ’πΆπ‘–}βˆͺ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,𝑧𝑖)βˆˆπΆπ‘–β§΅{πœƒ}}.

Proof. For each π‘–βˆˆπΌ, let π‘žπ‘–=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, 𝐢𝑖, 𝐸, πœƒ, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=πœƒ for all π‘–βˆˆπΌ. For each π‘–βˆˆπΌ, let π‘“π‘–βˆΆπ‘‹π‘–β†’πΈ be a 𝐢𝑖-v.l.s.c. and 𝐢𝑖-convex function. Suppose that there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘— such that 𝑓𝑗(𝑧𝑗)βˆ’π‘“π‘—(𝑒𝑗)βˆˆβˆ’πΆπ‘— and 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑗)βˆ’π‘“π‘—(𝑦𝑗)βˆˆβˆ’πΆπ‘—β§΅{πœƒ}. Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)𝑓𝑖(𝑣𝑖)βˆ’π‘“π‘–(𝑒𝑖)βˆ‰πΆπ‘–β§΅{πœƒ},(b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)+𝑓𝑖(π‘₯𝑖)βˆ’π‘“π‘–(𝑣𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘“π‘–(𝑧𝑖)βˆ’π‘“π‘–(𝑒𝑖)βˆˆβˆ’πΆπ‘–}.

Corollary 3.14. Let 𝐼, 𝑋𝑖, and 𝑋 be the same as in Theorem 3.3. For each π‘–βˆˆπΌ, let π‘“π‘–βˆΆπ‘‹π‘–β†’(βˆ’βˆž,∞] be a l.s.c. and convex function and π‘π‘–βˆΆπ‘‹π‘–Γ—π‘‹π‘–β†’(βˆ’βˆž,∞] be a quasi-distance. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=0 for all π‘–βˆˆπΌ. Suppose that there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘— such that 𝑓𝑗(𝑧𝑗)≀𝑓𝑗(𝑒𝑗) and 𝑝𝑗(𝑦𝑗,𝑧𝑗)<𝑓𝑗(𝑦𝑗)βˆ’π‘“π‘—(𝑧𝑗). Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)𝑓𝑖(𝑣𝑖)≀𝑓𝑖(𝑒𝑖), (b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)β‰₯𝑓𝑖(𝑣𝑖)βˆ’π‘“π‘–(π‘₯𝑖) for all π‘₯π‘–βˆˆ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘“π‘–(𝑧𝑖)≀𝑓𝑖(𝑒𝑖)}.

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

Theorem 3.16. Let 𝐼, 𝑋𝑖, 𝑋, 𝐢𝑖, 𝐸, πœƒ, and 𝑝𝑖 be the same as in Theorem 3.3. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=πœƒ for all π‘–βˆˆπΌ. For each π‘–βˆˆπΌ, let π‘“π‘–βˆΆπ‘‹π‘–β†’πΈ be a 𝐢𝑖-v.l.s.c. and 𝐢𝑖-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 π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘— such that 𝑝𝑗(𝑒𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑗)βˆ’π‘“π‘—(𝑒𝑗)βˆˆβˆ’πΆπ‘— and 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑗)βˆ’π‘“π‘—(𝑦𝑗)βˆˆβˆ’πΆπ‘—β§΅{πœƒ}. Then for each 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=πœƒ for all π‘–βˆˆπΌ, there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)𝑝𝑖(𝑒𝑖,𝑣𝑖)+𝑓𝑖(𝑣𝑖)βˆ’π‘“π‘–(𝑒𝑖)βˆ‰πΆπ‘–β§΅{πœƒ}, (b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)+𝑓𝑖(π‘₯𝑖)βˆ’π‘“π‘–(𝑣𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,𝑧𝑖)+𝑓𝑖(𝑧𝑖)βˆ’π‘“π‘–(𝑒𝑖)βˆˆβˆ’πΆπ‘–}βˆͺ{π‘§π‘–βˆˆπ‘‹π‘–βˆΆπ‘π‘–(𝑒𝑖,𝑧𝑖)+𝑓𝑖(𝑧𝑖)βˆ’π‘“π‘–(𝑒𝑖)βˆˆπΆπ‘–β§΅{πœƒ}}.
In Theorem 3.16, if 𝐸=(βˆ’βˆž,∞] and 𝐢𝑖=[0,∞) for all π‘–βˆˆπΌ, 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 π‘–βˆˆπΌ, let π‘“π‘–βˆΆπ‘‹π‘–β†’πΈ be a l.s.c. and convex function and let π‘π‘–βˆΆπ‘‹π‘–Γ—π‘‹π‘–β†’(βˆ’βˆž,∞] be a quasi-distance. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=0 for all π‘–βˆˆπΌ. Suppose that there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘— such that 𝑝𝑗(𝑒𝑗,𝑧𝑗)≀𝑓𝑗(𝑒𝑗)βˆ’π‘“π‘—(𝑧𝑗) and 𝑝𝑗(𝑦𝑗,𝑧𝑗)<𝑓𝑗(𝑦𝑗)βˆ’π‘“π‘—(𝑧𝑗). Then for each 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆπ‘‹ with 𝑝𝑖(𝑒𝑖,𝑒𝑖)=0 for all π‘–βˆˆπΌ, there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has (a)𝑝𝑖(𝑒𝑖,𝑣𝑖)≀𝑓𝑖(𝑒𝑖)βˆ’π‘“π‘–(𝑣𝑖), (b)𝑝𝑖(𝑣𝑖,π‘₯𝑖)β‰₯𝑓𝑖(𝑣𝑖)βˆ’π‘“π‘–(π‘₯𝑖) for all π‘₯π‘–βˆˆπ‘‹π‘–.

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 𝐢-vectorial β„“-function in a Hausdorff t.v.s.

Theorem 4.1. Let 𝐼 be any index set. For each π‘–βˆˆπΌ, let 𝑋𝑖 be a nonempty Hausdorff t.v.s. and 𝐢𝑖≠{πœƒ} be a nonempty pointed convex cone in a t.v.s. 𝐸 with zero vector πœƒ. Let βˆπ‘‹=π‘–βˆˆπΌπ‘‹π‘–. For each π‘–βˆˆπΌ, let 𝑝𝑖, π‘žπ‘–βˆΆπ‘‹π‘–Γ—π‘‹π‘–β†’πΈ be ℓ𝐢𝑖-function. Suppose that (i)β„’={π‘₯=(π‘₯𝑖)π‘–βˆˆπΌβˆˆπ‘‹βˆΆπ‘π‘–(π‘₯𝑖,π‘₯𝑖)=π‘žπ‘–(π‘₯𝑖,π‘₯𝑖)=πœƒforallπ‘–βˆˆπΌ}β‰ βˆ…, (ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘— such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)+π‘žπ‘—(𝑦𝑗,𝑧𝑗)βˆˆβˆ’πΆπ‘–β§΅{πœƒ}.
Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has 𝑝𝑖(𝑣𝑖,π‘₯𝑖)+π‘žπ‘–(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆπ‘‹π‘–.

Proof. Let 𝑒=(𝑒𝑖)π‘–βˆˆπΌβˆˆβ„’ be given. For each π‘–βˆˆπΌ, we define πΉπ‘–βˆΆπ‘‹Γ—π‘‹π‘–βŠΈπΈ, πΊπ‘–βˆΆπ‘‹Γ—π‘‹Γ—π‘‹π‘–βŠΈπΈ and π΄π‘–βˆΆπ‘‹βŠΈπ‘‹π‘– by 𝐹𝑖π‘₯,𝑦𝑖=𝐢𝑖⧡{πœƒ},βˆ€π‘₯,π‘¦π‘–ξ€Έβˆˆπ‘‹Γ—π‘‹π‘–,𝐺𝑑,π‘₯,𝑦𝑖=𝑝𝑖π‘₯𝑖,𝑦𝑖+π‘žπ‘–ξ€·π‘₯𝑖,𝑦𝑖+𝐢𝑖⧡{πœƒ},βˆ€π‘‘,π‘₯,π‘¦π‘–ξ€Έβˆˆπ‘‹Γ—π‘‹Γ—π‘‹π‘–,𝐴(4.1)𝑖(𝑦π‘₯)=π‘–βˆˆπ‘‹π‘–βˆΆπœƒβˆˆπΊπ‘–ξ€·π‘’,π‘₯,𝑦𝑖π‘₯ξ€Έξ€Ύ,βˆ€π‘₯=π‘–ξ€Έπ‘–βˆˆπΌβˆˆπ‘‹,(4.2) respectively. Clearly, 𝐻𝑖π‘₯∢=π‘–βˆˆπ‘‹π‘–βˆΆπœƒβˆ‰πΉπ‘–ξ€·π‘’,π‘₯𝑖=𝑋,βˆ€π‘–βˆˆπΌ.(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)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, we have 𝑝𝑖(𝑣𝑖,π‘₯𝑖)+π‘žπ‘–(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆπ‘‹π‘–.

An existence theorem of system of generalized vector equilibrium problem is immediate from Theorem 4.1 if π‘žπ‘–β‰‘0 (the zero map) for all π‘–βˆˆπΌ.

Theorem 4.2. Let 𝐼 be any index set. For each π‘–βˆˆπΌ, let 𝑋𝑖 be a nonempty Hausdorff t.v.s., and let 𝐢𝑖≠{πœƒ} be a nonempty pointed convex cone in a t.v.s. 𝐸 with zero vector πœƒ. Let βˆπ‘‹=π‘–βˆˆπΌπ‘‹π‘–. For each π‘–βˆˆπΌ, let π‘π‘–βˆΆπ‘‹π‘–Γ—π‘‹π‘–β†’πΈ be a ℓ𝐢𝑖-function. Suppose that (i)β„‹={π‘₯=(π‘₯𝑖)π‘–βˆˆπΌβˆˆπ‘‹βˆΆπ‘π‘–(π‘₯𝑖,π‘₯𝑖)=πœƒforallπ‘–βˆˆπΌ}β‰ βˆ…;(ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘— such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)βˆˆβˆ’πΆπ‘–β§΅{πœƒ}.
Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has 𝑝𝑖(𝑣𝑖,π‘₯𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆπ‘‹π‘–.

Remark 4.3. (a) Notice that Theorems 4.1 and 4.2 are indeed equivalent since the sum of two ℓ𝐢𝑖-functions is also a ℓ𝐢𝑖-function.
(b) In Theorem 4.1, the maps 𝑝𝑖 and π‘žπ‘– are only assumed to be ℓ𝐢𝑖-functions which need not be 𝐢𝑖-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, 𝐢𝑖, 𝐸, and πœƒ be the same as in Theorem 4.1. For each π‘–βˆˆπΌ, let π‘π‘–βˆΆπ‘‹π‘–Γ—π‘‹π‘–β†’πΈ be a ℓ𝐢𝑖-function and let π‘“π‘–βˆΆπ‘‹π‘–β†’πΈ be a 𝐢𝑖-v.l.s.c. and 𝐢𝑖-convex function. Suppose that (i)β„‹={π‘₯=(π‘₯𝑖)π‘–βˆˆπΌβˆˆπ‘‹βˆΆπ‘π‘–(π‘₯𝑖,π‘₯𝑖)=πœƒforallπ‘–βˆˆπΌ}β‰ βˆ…, (ii)there exist a nonempty compact subset 𝐾 of 𝑋 and a nonempty compact convex subset 𝑀𝑖 of 𝑋𝑖 for each π‘–βˆˆπΌ such that for each π‘¦βˆˆπ‘‹β§΅πΎ there exist π‘—βˆˆπΌ and π‘§π‘—βˆˆπ‘€π‘— such that 𝑝𝑗(𝑦𝑗,𝑧𝑗)+𝑓𝑗(𝑧𝑖)βˆ’π‘“π‘—(𝑦𝑖)βˆˆβˆ’πΆπ‘—β§΅{πœƒ}.
Then there exists 𝑣=(𝑣𝑖)π‘–βˆˆπΌβˆˆπ‘‹ such that for each π‘–βˆˆπΌ, one has 𝑝𝑖(𝑣𝑖,π‘₯𝑖)+𝑓𝑖(π‘₯𝑖)βˆ’π‘“π‘–(𝑣𝑖)βˆ‰βˆ’πΆπ‘–β§΅{πœƒ} for all π‘₯π‘–βˆˆπ‘‹π‘–.

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,