Class 𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍), General 𝐾𝐾𝑀 Type Theorems, and Their Applications in Topological Vector Space

Abstract

The class 𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍) and generalized 𝐾𝐾𝑀 mapping are introduced, and some generalized 𝐾𝐾𝑀 theorems are proved. As applications, Ky Fan’s matching theorem and Fan-Browder fixed-point theorem are extended, and some existence theorems of solutions for the generalized vector equilibrium problems are established under noncompact setting, which improve and generalize some known results.

1. Introduction and Preliminaries

In 1929, Knaster, Kurnatoaski, and Mazurkiewicz proved the well-known 𝐾𝐾𝑀 theorem on 𝑛-simplex. In 1961, Fan [1] generalized the 𝐾𝐾𝑀 theorem from Euclid space to infinite dimensional topological vector spaces by introducing 𝐾𝐾𝑀 mapping. In 1989, Park [2] introduced 𝑠-𝐾𝐾𝑀 mapping which is a generalized form of the 𝐾𝐾𝑀 mapping and obtained some new 𝐾𝐾𝑀 theorems. In 1991, Chang and Zhang [3] improved fundamentally Ky Fan’s 𝐾𝐾𝑀 mapping that makes 𝐾𝐾𝑀 theory have great development. Since then, many results related to 𝐾𝐾𝑀 principle were obtained and applied universally in the fields of nonlinear analysis (see [4–17]).

Let 𝑋 and 𝑍 be topological spaces, let ⟨𝑋⟩ and 2^𝑋 denote the nonempty finite subset of 𝑋 and the set of the nonempty subsets of 𝑋, respectively, let𝐴⊂𝐵⊂𝑋,  int_𝐵𝐴 denote the interior of 𝐴 in 𝐵, and let cl_𝐵𝐴 be the closure of 𝐴 in 𝐵 (when 𝐵=𝑋, int_𝐵𝐴=int𝐴, and cl_𝐵𝐴=cl𝐴). 𝐴 is said to be compactly closed (resp., compactly open) in 𝑋 if, for every nonempty compact subset 𝐾 of 𝑋,  𝐴∩𝐾 is closed (resp., open) in 𝐾. The compact closure of 𝐴 and the compact interior of 𝐴 (see [10]) are defined, respectively, by =ccl𝐴𝐵⊂𝑋∶𝐴⊂𝐵and𝐵iscompactlyclosedin𝑋,=cint𝐴𝐵⊂𝑋∶𝐵⊂𝐴and𝐵iscompactlyopenin𝑋. (1) It is easy to see that ccl(𝑋⧵𝐴)=𝑋⧵cint𝐴,  int𝐴⊂cint𝐴⊂𝐴,  𝐴⊂ccl𝐴⊂cl𝐴. For every nonempty subset 𝐾 of 𝑋, the subset ccl𝐴∩𝐾 is closed in 𝐾 and cint𝐴∩𝐾 is open in 𝐾. The multivalued mapping 𝐹∶𝑋→2^𝑍 is said to be transfer compactly open valued (resp., transfer compactly closed valued) on 𝑋, if, for every 𝑥∈𝑋 and for each nonempty compact subset 𝐾 of 𝑍, 𝑧∈𝐹(𝑥)∩𝐾 (resp., 𝑧∉𝐹(𝑥)∩𝐾) implies that there exists 𝑥∈𝑋 such that 𝑧∈int_𝐾(𝑇(𝑥)∩𝐾) (resp., 𝑧∉cl_𝐾(𝑇(𝑥)∩𝐾)) (see [10]). The mappings 𝐹^𝑐∶𝑋→2^𝑍 and 𝐹⁻¹∶𝑍→2^𝑋 are defined as 𝐹^𝑐(𝑥)=𝑍⧵𝐹(𝑥)={𝑧∈𝑍∶𝑧∉𝐹(𝑥)} and 𝐹⁻¹(𝑧)={𝑥∈𝑋∶𝑧∈𝐹(𝑥)}, respectively.

Recently, many scholars (see [18–21]) not only studied further the 𝐾𝐾𝑀 theorem involving 𝐾𝐾𝑀 mapping, but also established some new 𝐾𝐾𝑀 theorem, fixed-point theorems, and coincidence theorems and utilized them to research the existence of solution to generalized vector equilibria, which makes the 𝐾𝐾𝑀 theory more perfect and rich.

In this paper, we first introduce the new generalized class 𝐾𝐾𝑀(𝑋,𝑌,𝑍) consisting of all multifunctions 𝑇∶𝑌→2^𝑍 that have the generalized 𝐾𝐾𝑀 property and prove some 𝐾𝐾𝑀 theorems for 𝔄-𝐾𝐾𝑀 mapping. Applying these 𝐾𝐾𝑀 theorems, the Ky Fan matching theorems and the Fan-Browder fixed-point theorems are generalized. Finally, we establish some new existence theorems of solutions for generalized vector equilibrium problems under noncompact setting. These theorems improve and generalize many known results in the literature.

Definition 1. Let 𝑋 be a nonempty set, 𝑌 a nonempty convex subset of a linear space, and 𝑍 a topological space, and let 𝑆∶𝑋→2^𝑌,  𝑇∶𝑌→2^𝑍, and 𝐹∶𝑋→2^𝑍 be three multivalued mappings. 𝐹 is said to be a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇 if, for any {𝑥₀,…,𝑥_𝑛}∈⟨𝑋⟩, there exists 𝑦_𝑖∈𝑆(𝑥_𝑖)(𝑖=0,1,…,𝑛), such that, for any {𝑦_𝑖0,…,𝑦_𝑖𝑘}⊂{𝑦₀,…,𝑦_𝑛}, one has 𝑇co𝑦_𝑖0,…,𝑦_𝑖𝑘⊂^𝑘_𝑗=0𝐹𝑥_𝑖𝑗. (2)

The multivalued mapping 𝑇∶𝑌→2^𝑍 is said to have the 𝔄-𝐾𝐾𝑀 property, if, for any 𝔄-KKM mapping 𝐹 with respect to 𝑇, the family {𝐹(𝑥)∶𝑥∈𝑋} has the finite intersection property. Let the set {𝑇∶𝑇hasthe𝔄-𝐾𝐾𝑀property}be denoted by 𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍).

Remark 2. 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇 is strictly weaker than the generalized 𝑆-𝐾𝐾𝑀 mapping with respect to 𝑇 in [5]. It is easy to see that Definition 1 is not the degenerate form of Definition 1 in [11] and Definition 3 in [6] and not a special case of Definition 1 in [6] and Definition 2 in [7].

Example 3. Let 𝑋=𝑌=𝑍=𝑅,  𝑇,𝑆,𝐹∶𝑅→2^𝑅 be three mappings defined as follows: 𝑇(𝑥)=[)(𝑥)=[),𝑆(𝑥)=[].𝑥,+∞,𝐹𝑥−1,+∞𝑥−2,𝑥+2 (3)

Then 𝐹 is 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇. In fact, for any {𝑥₀,…,𝑥_𝑛}∈⟨𝑅⟩, take 𝑦_𝑖=𝑥_𝑖∈𝑆(𝑥_𝑖). For any {𝑦_𝑖0,…,𝑦_𝑖𝑘}⊂{𝑦₀,…,𝑦_𝑛}, it is easy to know that co𝑦_𝑖0,…,𝑦_𝑖𝑘=min𝑦_𝑖0,…,𝑦_𝑖𝑘,max𝑦_𝑖0,…,𝑦_𝑖𝑘. (4) Therefore, by the definition of 𝑇 and 𝐹, we have 𝑇co𝑦_𝑖0,…,𝑦_𝑖𝑘=min𝑦_𝑖0,…,𝑦_𝑖𝑘=𝑥,+∞min_𝑖0,…,𝑥_𝑖𝑘,+∞_𝑘𝑗=0,𝐹𝑥_𝑖𝑗=min𝑥_𝑖0,…,𝑥_𝑖𝑘−1,+∞. (5) It follows that 𝑇co𝑦_𝑖0,…,𝑦_𝑖𝑘⊂^𝑘_𝑗=0𝐹𝑥_𝑖𝑗=^𝑘_𝑗=0𝐹𝑥_𝑖𝑗. (6) However, for {𝑥₀,…,𝑥_𝑛}∈⟨𝑅⟩,  𝑆({𝑥₀,…,𝑥_𝑛})=⋃^𝑛_𝑖=0[𝑥_𝑖−2,𝑥_𝑖+2],  𝑇(co(𝑆(𝑥₀,…,𝑥_𝑛))=[min{𝑥_𝑖−2},+∞). Hence 𝑇co𝑆𝑥₀,…,𝑥⊆_𝑛^/𝑛_𝑖=0𝐹𝑥_𝑖. (7) Hence, 𝐹 is not a generalized 𝑆-𝐾𝐾𝑀 mapping with respect to 𝑇.

Lemma 4 (see [12]). Let 𝑋 and 𝑍 be topological spaces and 𝐹∶𝑋→2^𝑍 a set-valued mapping with 𝐹(𝑋)=𝑍. Then the following conditions are equivalent:(i)𝐹is transfer compactly open valued,(ii)for each compact subset 𝐾 of 𝑍 and for each 𝑧∈𝐾, there exists 𝑥∈𝑋 such that 𝑧∈cint𝐹(𝑥)⋂𝐾 and ⋃_𝑥∈𝑋(𝐹(𝑥)∩𝐾)=⋃_𝑥∈𝑋(cint𝐹(𝑥)∩𝐾).

Lemma 5 (see [13]). Let 𝑋 and 𝑍 be topological spaces and 𝐹∶𝑋→2^𝑍 a set-valued mapping with 𝑋≠𝐹⁻¹(𝑧) for each 𝑧∈𝑍. Then the following conditions are equivalent:(i)𝐹 is transfer compactly closed valued,(ii)the mapping 𝐹^𝑐∶𝑋→2^𝑌 defined by 𝐹^𝑐(𝑥)=𝑌⧵𝐹(𝑥) for each 𝑥∈𝑋 is transfer compactly open valued,(iii)for each compact subset 𝐾 of 𝑍, ⋃_𝑥∈𝑋(𝐹^𝑐(𝑥)∩𝐾)=⋃_𝑥∈𝑋(cint𝐹^𝑐(𝑥)∩𝐾),(iv)for each compact subset 𝐾 of 𝑍, ⋂_𝑥∈𝑋(𝐹(𝑥)∩𝐾)=⋂_𝑥∈𝑋(ccl𝐹(𝑥)∩𝐾).

Lemma 6. Let 𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍),  and let 𝑋₁ be a nonempty subset of 𝑋 and 𝑌₁ a nonempty convex subset of 𝑌,  𝑆₁=𝑆|_𝑋1; then 𝑇|_𝑌1∈𝔄₁-𝐾𝐾𝑀(𝑋₁,𝑌₁,𝑍).

Proof. Suppose that 𝐹₁∶𝑋₁→2^𝑍 is a 𝔄₁-𝐾𝐾𝑀 mapping with respect to 𝑇|_𝑌1. Then, for each {𝑥₀,…,𝑥_𝑛}∈⟨𝑋₁⟩, there exists 𝑦_𝑖∈𝑆₁(𝑥_𝑖)(𝑖=0,1,…,𝑛), such that, for any {𝑦_𝑖0,…,𝑦_𝑖𝑘}⊂{𝑦₀,…,𝑦_𝑛}, we have 𝑇|_𝑌1(co{𝑦_𝑖0,…,𝑦_𝑖𝑘})⊂⋃^𝑘_𝑗=0𝐹₁(𝑥_𝑖𝑗). We define a set-valued mapping 𝐹∶𝑋→2^𝑍 by 𝐹(𝑥)=𝐹₁(𝑥),if𝑥∈𝑋₁;𝑍,if𝑥∈𝑋⧵𝑋₁. (8) Obviously, 𝐹 is also a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇. Since 𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍), the family {𝐹(𝑥)∶𝑥∈𝑋} has the finite intersection property which implies that the family {𝐹₁(𝑥)∶𝑥∈𝑋₁} has the finite intersection property.

2. General 𝐾𝐾𝑀 Theorems

Theorem 7. Let 𝑋 be a topological space, 𝑌 a convex space, and 𝑍 a Hausdorff space. Suppose that ∶𝑋→2^𝑌,𝑇∶𝑌→2^𝑍, and 𝐹∶𝑋→2^𝑍 are three multifunctions satisfying the following:(1)𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍) such that 𝑇(co𝑆(𝑋)) is compact in 𝑍,(2)𝐹 is transfer compactly closed values,(3)𝐹 is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇.
Then 𝑇(co𝑆(𝑋))∩(⋂_𝑥∈𝑋𝐹(𝑥))≠∅.

Proof. Define 𝐹^∗∶𝑋→2^𝑍 by 𝐹^∗(𝑥)=𝑇(co𝑆(𝑋))ccl𝐹(𝑥)∀𝑥∈𝑋. (9) By (3), for any {𝑥₀,…,𝑥_𝑛}∈⟨𝑋⟩, there exists 𝑦_𝑖∈𝑆(𝑥_𝑖)(𝑖=0,1,…,𝑛), such that, for any {𝑦_𝑖0,…,𝑦_𝑖𝑘}⊂{𝑦₀,…,𝑦_𝑛}, we have 𝑇(co{𝑦_𝑖0,…,𝑦_𝑖𝑘})⊂⋃^𝑘_𝑗=0ccl𝐹(𝑥_𝑖𝑗). It follows from 𝑇(co{𝑦_𝑖0,…,𝑦_𝑖𝑘})⊂𝑇(co𝑆(𝑋)) that we have𝑇co𝑦_𝑖0,…,𝑦_𝑖𝑘⊂^𝑘_𝑗=0𝑇(co𝑆(𝑋))∩ccl𝐹𝑥_𝑖𝑗=^𝑘_𝑗=0𝐹^∗𝑥_𝑖𝑗⊂^𝑘_𝑗=0ccl𝐹^∗𝑥_𝑖𝑗. (10) This shows that 𝐹^∗ is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇, and so {𝐹^∗(𝑥)∶𝑥∈𝑋} has finite intersection property. Since 𝐹(𝑥) is transfer compactly closed by (1) and so ccl𝐹(𝑥) is compactly closed and 𝑇(co𝑆(𝑋)) is compact in 𝑍 by (1), consequently 𝐹^∗(𝑥) is closed in compact subset 𝑇(co𝑆(𝑋)) of Hausdorff space 𝑍. Therefore, 𝑇(co𝑆(𝑋))∩{ccl𝐹(𝑥)∶𝑥∈𝑋}=_𝑥∈𝑋𝐹^∗(𝑥)≠∅. (11) By Lemma 5, we have that 𝑇(co𝑆(𝑥)∩(⋂_𝑥∈𝑋𝐹(𝑥))=𝑇(co𝑆(𝑥)∩(⋂_𝑥∈𝑋ccl𝐹(𝑥))≠∅ holds.

Remark 8. Theorem 7 improves Theorem 4.3 of Chang et al. [5] in the following two aspects: (1) the generalized 𝑆-𝐾𝑀𝑀 mapping is generalized to 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇; (2) the compactly closed values property is replaced by the transfer compactly closed values property.

Theorem 9. Let 𝑋 be a topological space, 𝑌 a topological vector space, and 𝑍 a Hausdorff space. Suppose that 𝑆∶𝑋→2^𝑌,𝑇∶𝑌→2^𝑍, and 𝐹∶𝑋→2^𝑍 are three multivalued mappings satisfying the following:(1)𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍),(2)𝐹 is transfer compactly closed values,(3)𝐹 is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇,(4)for each compact subset 𝑀₁ of 𝑋,  𝑇(𝑆(𝑀₁)) is compact in 𝑍, and for each convex subset 𝑀₂ of 𝑋, 𝑆(𝑀₂) is convex,(5)there exists a nonempty compact subset 𝐾 of 𝑍 such that, for each 𝑁∈⟨𝑋⟩, there exists compact convex subset 𝐿^𝑋_𝑁 of 𝑋 including 𝑁 such that 𝑇(𝑆(𝐿^𝑋_𝑁)∩(⋂_𝑥∈𝐿^𝑋_𝑁ccl𝐹(𝑥)⊂𝐾. Then ⋂_𝑥∈𝑋𝐹(𝑥)≠∅.

Proof. Suppose that the conclusion is not true; then ∩_𝑥∈𝑋𝐹(𝑥)=∅. Define 𝐺∶𝑋→2^𝑍 by 𝐺(𝑥)=𝑍⧵𝐹(𝑥) for each 𝑥∈𝑋; then 𝐺(𝑥) is nonempty for all 𝑥∈𝑋. From (2) and Lemma 5, it follows that 𝐺 is transfer compactly open mapping on 𝑋. Since 𝐾 is compact in 𝑍, by Lemma 4, we have 𝐾=∪_𝑥∈𝑋cint𝐺(𝑥)∩𝐾. Hence there exists 𝑁={𝑥₀,𝑥₁,…,𝑥_𝑛} such that 𝐾⊂∪^𝑛_𝑖=0cint𝐺(𝑥_𝑖)∩𝐾=∪^𝑛_𝑖=0cint𝐺(𝑥_𝑖). By (5), there exists compact convex subset 𝐿^𝑋_𝑁 of 𝑋 including 𝑁 such that 𝑇𝑆𝐿^𝑋_𝑁∩_{𝑥∈𝐿𝑋𝑁}ccl𝐹(𝑥)𝐾 (12) and 𝑇(𝑆(𝐿^𝑋_𝑁))⧵𝐾⊂⋃_{𝑥∈𝐿𝑋𝑁}cint𝐺(𝑥). Since {𝑥₀,𝑥₁,…,𝑥_𝑛}⊂𝐿^𝑋_𝑁, we have 𝐾⊂⋃^𝑛_𝑖=0cint𝐺(𝑥_𝑖)⊂⋃_{𝑥∈𝐿𝑋𝑁}cint𝐺(𝑥), so 𝑇(𝑆(𝐿^𝑋_𝑁))⊂⋃_{𝑥∈𝐿𝑋𝑁}cint𝐺(𝑥). Therefore, 𝑇𝑆𝐿^𝑋_𝑁∩_{𝑥∈𝐿𝑋𝑁}(𝑥)=ccl𝐹𝑇𝑆𝐿^𝑋_𝑁∩_{𝑥∈𝐿𝑋𝑁}((𝑥=ccl𝑍⧵𝐺))𝑇𝑆𝐿^𝑋_𝑁∩_{𝑥∈𝐿𝑋𝑁}(𝑍⧵cint𝐺(𝑥))=∅. (13) Since 𝑆(𝐿^𝑋_𝑁) is a convex subset of 𝑌, by Lemma 6, 𝑇|_𝑆(𝐿^𝑋_𝑁)∈𝔄-𝐾𝐾𝑀(𝐿^𝑋_𝑁,𝑆(𝐿^𝑋_𝑁),𝑍). Define set-valued mapping 𝐹^∗,𝐺^∗∶𝐿^𝑋_𝑁→2^{𝑇(𝑆(𝐿𝑋𝑁))} by 𝐹^∗(𝑥)=𝐹(𝑥)∩𝑇(𝑆(𝐿^𝑋_𝑁)) and 𝐺^∗(𝑥)=𝑇(𝑆(𝐿^𝑋_𝑁))⧵𝐹^∗(𝑥) for each 𝑥∈𝐿^𝑋_𝑁. Then we have 𝐺^∗(𝑥)=𝑇(𝑆(𝐿^𝑋_𝑁))⧵(𝐹(𝑥)∩𝑇(𝑆(𝐿^𝑋_𝑁)))=𝑇(𝑆(𝐿^𝑋_𝑁))∩(𝑍⧵𝐹(𝑥))=𝑇(𝑆(𝐿^𝑋_𝑁))∩𝐺(𝑥), and𝑇𝑆𝐿^𝑋_𝑁=_{𝑥∈𝐿𝑋𝑁}cint𝐺(𝑥)∩𝑇𝑆𝐿^𝑋_𝑁=_{𝑥∈𝐿𝑋𝑁}𝐺^∗(𝑥). (14) By Lemma 4, we have that 𝐺^∗ is transfer compactly open valued on 𝐿^𝑥_𝑁. Hence it follows from Lemma 5 that 𝐹^∗ is transfer compactly closed valued on 𝐿^𝑋_𝑁.
We claim that 𝐹^∗ is a 𝔄|_𝐿^𝑋_𝑁-𝐾𝐾𝑀 mapping with respect to 𝑇|_𝑆(𝐿^𝑋_𝑁). Since 𝐹 is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇, for any {𝑥₀,…,𝑥_𝑛}∈⟨𝐿^𝑋_𝑁⟩⊂⟨𝑋⟩, there exists 𝑦_𝑖∈𝑆(𝑥_𝑖)=𝑆(𝑥_𝑖)∩𝑆(𝐿^𝑋_𝑁)=𝑆|_𝐿^𝑋_𝑁(𝑥_𝑖)(𝑖∈{0,1,…,𝑛}) such that, for any {𝑦_𝑖0,…,𝑦_𝑖𝑘}⊂{𝑦₀,…,𝑦_𝑛}∈⟨𝑌⟩, 𝑇co𝑦_𝑖0,…,𝑦_𝑖𝑘⊂^𝑘_𝑗=0𝐹𝑥_𝑖𝑗. (15) Since 𝑆(𝐿^𝑋_𝑁) is convex in 𝑌 and 𝑦_𝑖𝑗∈𝑆(𝐿^𝑋_𝑁) for each 𝑗=0,1,…,𝑘, we have 𝑇|_𝑆(𝐿^𝑋_𝑁)co𝑦_𝑖0,…,𝑦_𝑖𝑘=𝑇co𝑦_𝑖0,…,𝑦_𝑖𝑘⊂^𝑘_𝑗=0𝐹𝑥_𝑖𝑗. (16) Hence 𝑇|_𝐿^𝑋_𝑁(co{𝑦_𝑖0,…,𝑦_𝑖𝑘})⊂⋃^𝑘_𝑗=0(𝐹(𝑥_𝑖𝑗)∩𝑇(𝑆(𝐿^𝑋_𝑁)))=∪^𝑘_𝑗=0𝐹^∗(𝑥_𝑖𝑗). Therefore, 𝐹^∗ is a 𝔄|_𝐿^𝑋_𝑁-𝐾𝐾𝑀 mapping with respect to 𝑇|_𝑆(𝐿^𝑋_𝑁). By Theorem 7, we have 𝑇(𝑆(𝐿^𝑋_𝑁))∩(⋂_𝑥∈𝐿^𝑋_𝑁ccl𝐹(𝑥))=𝑇(𝑆(𝐿^𝑋_𝑁))∩(⋂_𝑥∈𝐿^𝑋_𝑁𝐹(𝑥))=𝑇(𝑆(𝐿^𝑋_𝑁))∩(⋂_𝑥∈𝐿^𝑋_𝑁𝐹^∗(𝑥))≠∅, which is a contradiction. Therefore, ⋂_𝑥∈𝑋𝐹(𝑥)≠∅.

Remark 10. Theorem 9 generalizes Theorem 3.3 of Lin and Wan [14] in the following two aspects: (1) from transfer closed values to transfer compactly closed values; (2) from generalized 𝐾𝐾𝑀 mapping to 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇.

Theorem 11. Let 𝑋,𝑌 be two convex spaces and 𝑍 a Hausdorff space. Suppose that 𝑆∶𝑋→2^𝑌,𝑇∶𝑌→2^𝑍, and 𝐹∶𝑋→2^𝑍 are three multivalued mappings and 𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍) satisfying the following:(1)𝑆(𝐶) is a compact convex subset of 𝑌 if 𝐶 is a compact convex subset of 𝑋,(2)for any compact subset 𝑄 of 𝑌, 𝑇(𝑄) is compact in 𝑍,(3)𝐹 is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇 such that 𝐹 is transfer compactly closed,(4)there exist a nonempty compact convex subset 𝐿 of 𝑋 and a compact subset 𝐾 of 𝑍 such that ccl𝐹(_𝑥∈𝐿𝑥)⊆𝐾. (17)
Then 𝑇(co𝑆(𝑋))∩(⋂{𝐹(𝑥)∶𝑥∈𝑋})≠∅.

Proof. Assume that 𝑇(co𝑆(𝑥))∩(⋂{𝐹(𝑥)∶𝑥∈𝑋})=∅; then we have 𝑍=𝑇(co𝑆(𝑋))^𝑐∪(⋃{𝐹^𝑐(𝑥)∶𝑥∈𝑋}), and 𝐾⊂𝑇(co𝑆(𝑋))^𝑐∪(⋃{𝐹^𝑐(𝑥)∩𝐾∶𝑥∈𝑋}). It follows from Lemma 5 that 𝐹^𝑐(𝑥)∩𝐾∶𝑥∈𝑋=∪cint𝐹^𝑐(𝑥)∩𝐾∶𝑥∈𝑋. (18) Since 𝐹^𝑐 is transfer compactly open (by condition (3)), 𝐾⊂𝑇(co𝑆(𝑋))^𝑐∪(⋃{cint𝐹^𝑐(𝑥)∩𝐾∶𝑥∈𝑋})=(𝑇(co𝑆(𝑋))^𝑐∩𝐾)∪(⋃{cint𝐹^𝑐(𝑥)∩𝐾∶𝑥∈𝑋}), where cint𝐹^𝑐(𝑥)∩𝐾 is open in 𝐾 for each 𝑥∈𝑋 and (𝑇(co𝑆(𝑋))^𝑐∩𝐾) is open in 𝐾. Therefore, there exists a finite subset {𝑥₀,…,𝑥_𝑛} of 𝑋 such that 𝐾⊆𝑇(co𝑆(𝑋))^𝑐∩𝐾∪^𝑛_𝑖=0cint𝐹^𝑐𝑥_𝑖⊆∩𝐾𝑇(co𝑆(𝑋))^𝑐∪^𝑛_𝑖=0cint𝐹^𝑐𝑥_𝑖. (19) By (4), we have 𝐾^𝑐⊆_𝑥∈𝐿cint𝐹^𝑐(𝑥). (20) Let 𝑀=co(𝐿∪{𝑥₀,…,𝑥_𝑛}); then 𝑀 is a compact convex subset of 𝑋 such that 𝑍=𝑇(co𝑆(𝑥))^𝑐∪_𝑥∈𝑀cint𝐹^𝑐(𝑥); (21) that is, 𝑇(co𝑆(𝑥))∩_𝑥∈𝑀ccl𝐹(𝑥)=∅. (∗) Define 𝐹^∗∶𝑀→2^𝑍 by 𝐹^∗(𝑥)=𝑇𝑆(𝑀)∩ccl𝐹(𝑥). (22) Since 𝐹 is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇, for any {𝑥₀,𝑥₁,…,𝑥_𝑛}∈𝑀, there exists 𝑦_𝑖∈𝑆(𝑥_𝑖) such that, for any {𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘}⊂{𝑦₀,𝑦₁,…,𝑦_𝑛},𝑇(co{𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘}⊆∪^𝑘_𝑗=0𝐹(𝑥_𝑗). It follows that 𝑇co𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘⊆𝑇𝑆(𝑀)^𝑘_𝑗=0𝐹𝑥_𝑗⊂𝑇𝑆(𝑀)^𝑘_𝑗=0ccl𝐹𝑥_𝑗⊂^𝑘_𝑗=0𝐹^∗𝑥_𝑖𝑗, (23) which shows that 𝐹^∗ is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇 for the triple (𝑀,𝑌,𝑍). Since 𝑇∈ 𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍), it follows easily that 𝑇∈𝔄-𝐾𝐾𝑀(𝑀,𝑌,𝑍). Moreover, since 𝑆(𝑀) is compact convex, we have co𝑆(𝑀)⊆𝑆(𝑀), and 𝑇(co𝑆(𝑀))⊆𝑇(𝑆(𝑀)). Conditions (1) and (2) imply that 𝑇(𝑆(𝑀)) is compact in 𝑍. Then the compactness of 𝑇(𝑆(𝑀)) implies that 𝑇(𝑆(𝑀)) is compact in 𝑍. Applying Theorem 7 to 𝑆, 𝑇, and 𝐹^∗, we obtain that 𝑇(co𝑆(𝑀))∩𝐹^∗(𝑥)∶𝑥∈𝑀≠∅,𝑇(co𝑆(𝑋))∩{ccl𝐹(𝑥)∶𝑥∈𝑀}≠∅, (24) which contradicts (∗). This completes the proof.

Remark 12. Theorem 11 improves Theorem 5.1 of Chang et al. [5] in the following two aspects: (1) the generalized 𝑆-𝐾𝑀𝑀 mapping is generalized to 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇; (2) the compactly closed values property is replaced by the transfer compactly closed values property.

3. Matching Theorems and Fixed-Point Theorems

In order to apply the above theorem to show the fixed-point theorems, we first establish the following generalization of the Ky Fan’s matching theorem.

Theorem 13. Let 𝑋 be a nonempty set, 𝑌 a convex space, and 𝑍 a Hausdorff space. Suppose that 𝑆∶𝑋→2^𝑌,  𝑇∶𝑌→2^𝑍, and 𝐹∶𝑋→2^𝑍 are three multivalued mappings satisfying the following:(1)𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍) such that 𝑇(co𝑆(𝑋)) is compact in 𝑍,(2)𝐹 is transfer compactly open in 𝑍,(3)𝑇(co𝑆(𝑋))⊆𝐹(𝑋).
Then there exists {𝑥₀,𝑥₁,…,𝑥_𝑛}∈⟨𝑋⟩ such that 𝑇(co𝑆(𝑋))∩^𝑛_𝑖=0𝐹𝑥_𝑖≠∅. (25)

Proof. Assume that, for any {𝑥₀,𝑥₁,…,𝑥_𝑛}∈⟨𝑋⟩, 𝑇(co𝑆(𝑋))∩(⋂^𝑛_𝑖=0𝐹(𝑥_𝑖))=∅. Then 𝑇(co𝑆(𝑋))⊆∪^𝑛_𝑖=0𝐹^𝑐(𝑥_𝑖) for any {𝑥₀,𝑥₁,…,𝑥_𝑛}∈⟨𝑋⟩. Noting that 𝐹^𝑐 is transfer compactly closed, we have that the conditions of Theorem 7 are satisfied for the mappings 𝑆, 𝑇, and 𝐹^𝑐. Thus 𝑇(co𝑆(𝑋))∩𝐹^𝑐(𝑥)∶𝑥∈𝑋≠∅, (26) which implies that 𝑇(co𝑆(𝑋))⊆^/𝐹(𝑋); this contradicts with (3). So there exists a nonempty finite subset 𝑀⊂⟨𝑋⟩ such that 𝑇(co𝑆(𝑋))∩{𝐹(𝑥)∶𝑥∈𝑀}≠∅.

Remark 14. Theorem 13 implies that Theorem 7 holds and if not then 𝑇(co𝑆(𝑋))⊆∪_𝑥∈𝑋𝐹^𝑐(𝑥)=𝐺(𝑋), where 𝐺∶𝑋→2^𝑍 is defined by 𝐺(𝑥)=𝐹^𝑐(𝑥). Theorem 13 shows that there exists 𝑀∈𝑋 such that 𝑇(co𝑆(𝑀))∩{𝐺(𝑥)∶𝑥∈𝑀}≠∅, (27) which implies that 𝑇(co𝑆(𝑀)⊆^/𝐹(𝑀), contradicting the fact that 𝐹 is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇.

Theorem 15. Let 𝑋 be a nonempty subset of a compact convex space 𝑌 and 𝑍 a Hausdorff space. Suppose that 𝐴∶𝑋→2^𝑍 satisfies the following:(1)𝐴 is transfer compactly open in 𝑍,(2)𝐴(𝑋)=𝑍.
Then for any 𝑓∈𝒞(𝑌,𝑍) there exist a finite subset {𝑥₀,…,𝑥_𝑛} of 𝑋 and an 𝑥∈co{𝑥₀,…,𝑥_𝑛} such that 𝑓(𝑥)∈∩^𝑛_𝑖=0𝐴𝑥_𝑖.

Proof. Let 𝑆∶𝑋→2^𝑌be defined by 𝑆(𝑥)={𝑥} for 𝑥∈𝑋. Then 𝑓∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍). Since 𝑓(co(𝑆(𝑋))⊆𝑓(𝑌) and 𝑓(𝑌) is compact, 𝑓(co(𝑆(𝑋)) is compact in 𝑍. Furthermore, by (2) we have that 𝑓(co(𝑆(𝑋))⊆𝑓(𝑌)⊆𝐴(𝑋). (28) So all of the conditions of Theorem 13 are satisfied for the mappings 𝑆, 𝑓, and 𝐴. Thus, there exists a finite subset {𝑥₀,…,𝑥_𝑛} of 𝑋 such that 𝑓co𝑥₀,…,𝑥_𝑛∩^𝑛_𝑖=1𝐴𝑥_𝑖≠∅. (29)

Remark 16. If 𝐴(𝑥) is compactly open in 𝑍 for each 𝑥∈𝑋,  𝑋 is a subset of a convex subset 𝑌 of a topological vector space 𝐸, and 𝑓 is the inclusion mapping of 𝑋 into 𝑌, then Theorem 15 reduces to Lemma 1 by Fan in [15].

In the sequel, we give the famous Fan-Browder type fixed-point theorem. We first give the following conclusion.

Theorem 17. Suppose that 𝑋,𝑌 are two convex spaces and 𝑍 is a Hausdorff space. Assume that 𝑆∶𝑋→2^𝑌,  𝐺∶𝑍→2^𝑌, and 𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍) are three functions satisfying the following:(1)𝑆(𝐶) is a compact convex subset of 𝑌 if 𝐶 is a compact convex subset of 𝑋,(2)𝑇(𝑄) is compact in 𝑍 if 𝑄 is compact in 𝑌,(3)for any 𝑧∈𝑇(𝑆(𝑋)),  𝐺(𝑧) is a nonempty convex subset of 𝑌,(4)there exists a transfer compactly open values mapping 𝐻∶𝑋→2^𝑍 such that any 𝑥∈𝑋,  𝐻(𝑥)⊂𝐺⁻¹(𝑆(𝑥)), and ∪_𝑥∈𝑋𝐻(𝑥)=𝑍,(5)there exist a nonempty compact convex subset 𝐿 of 𝑋 and a compact subset 𝐾 of 𝑍 such that 𝑀=_𝑥∈𝐿𝐻^𝑐(𝑥)⊂𝐾. (30)
Then there exists a finite subset {𝑥₀,𝑥₁,…,𝑥_𝑛} of 𝑋; for any 𝑦_𝑖∈𝑆(𝑥_𝑖)(𝑖=0,1,…,𝑛) there exist {𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘}⊂{𝑦₀,𝑦₁,…,𝑦_𝑛},𝑧∈𝑇(co{𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘}) such that, for any 𝑗=0,1,…,𝑘, there exists 𝑦^′_𝑖𝑗∈𝑆(𝑥_𝑖𝑗) such that co{𝑦^′_𝑖0,𝑦^′_𝑖1,…,𝑦^′_𝑖𝑘}⊂𝐺(𝑧).

Proof. Define 𝐹∶𝑋→2^𝑍 by 𝐹(𝑥)=𝐻^𝑐(𝑥) for 𝑥∈𝑋. Then 𝐹(𝑥) is transfer compactly closed in 𝑍.
(i) Suppose that 𝑀=∅. In this case, it is easy to know that 𝑇(co(𝑆(𝑋)))∩(⋂{𝐹(𝑥)∶𝑥∈𝑋})=∅. Then it follows from Theorem 11 that 𝐹 is not a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇; that is, there exists a finite subset {𝑥₀,𝑥₁,…,𝑥_𝑛} of 𝑋; for any 𝑦_𝑖=𝑆(𝑥_𝑖)(𝑖=0,1,…,𝑛), there exists {𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘}⊂{𝑦₀,𝑦₁,…,𝑦_𝑛} such that 𝑇(co{𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘})⊆^/⋃^𝑘_𝑗=0𝐹(𝑥_𝑖𝑗). Choose 𝑧∈𝑇(co{𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘}) such that 𝑧∉⋃^𝑘_𝑗=0𝐹(𝑥_𝑖𝑗). Then ∈𝐻(𝑥_𝑖𝑗)⊂𝐺⁻(𝑆(𝑥_𝑖𝑗)), and 𝑆(𝑥_𝑖𝑗)⋂𝐺(𝑧)≠∅ for any 𝑗=0,1,…,𝑘. Thus for any 𝑗=0,1,…,𝑘, there is 𝑦^′_𝑖𝑗∈𝑆(𝑥_𝑖𝑗) such that 𝑦^′_𝑖𝑗∈𝐺(𝑧). Since 𝐺(𝑧) is convex, we see that co{𝑦^′_𝑖0,𝑦^′_𝑖1,…,𝑦^′_𝑖𝑘}⊂𝐺(𝑧).
(ii) Suppose that 𝑀≠∅. Checking the proof of case (i), it suffices to show that 𝐹 is not a 𝔄-𝐾𝐾𝑀 mapping. On the contrary, assume that, for any {𝑥₀,𝑥₁,…,𝑥_𝑛}∈⟨𝑋⟩, there exists 𝑦_𝑖∈𝑆(𝑥_𝑖) such that, for any {𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘}⊂{𝑦₀,𝑦₁,…,𝑦_𝑛}, we have 𝑇(co({𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘}))⊂∪^𝑘_𝑗=0𝐹(𝑥_𝑖𝑗). Then 𝐹 is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇, and so by Theorem 11, we have𝑇(co(𝑆(𝑋)))∩{𝐹(𝑥)∶𝑥∈𝑋}≠∅. (31) In particular, ⋂_𝑥∈𝑋𝐻^𝑐(𝑥)≠∅; that is, _𝑥∈𝑋𝐻(𝑥)≠𝑍 (32) which contradicts the assumption that ⋃_𝑥∈𝑋𝐻(𝑥)=𝑍. This completes the proof.

Remark 18. Theorem 17 improves Corollary 5.2 of Chang et al. [5] in the following three aspects: (1) from generalized 𝑆-𝐾𝑀𝑀 mapping to 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇; (2) from compactly closed values to transfer compactly closed values; (3) from the single-valued mapping 𝑠 to the multivalued mapping 𝑆.

For Theorem 17, if 𝑆∶𝑋→2^𝑌 reduces to a single-value mapping 𝑠∶𝑋→𝑌, we have the following conclusion.

Theorem 19. Suppose that 𝑋, 𝑌 are two convex spaces and 𝑍 is a Hausdorff space. Assume that 𝑠∶𝑋→𝑌,𝐺∶𝑍→2^𝑌, and 𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍) are three mappings satisfying the following conditions:(1)𝑠(𝐶) is a compact convex subset of 𝑌 if 𝐶 is a compact convex subset of 𝑋,(2)𝑇(𝑄) is compact in 𝑍 if 𝑄 is compact in 𝑌,(3)for any 𝑧∈𝑇(𝑠(𝑋)), 𝐺(𝑧) is a nonempty convex subset of 𝑌,(4)there exists a transfer compactly open values mapping 𝐻∶𝑋→2^𝑍 such that any 𝑥∈𝑋,𝐻(𝑥)⊂𝐺⁻¹(𝑠(𝑥)), and ∪_𝑥∈𝑋𝐻(𝑥)=𝑍,(5)there exist a nonempty compact convex subset 𝐿 of 𝑋 and a compact subset 𝐾 of 𝑍 such that 𝑀=_𝑥∈𝐿𝐻^𝑐(𝑥)⊂𝐾. (33)
Then there exists a finite subset {𝑥₀,𝑥₁,…,𝑥_𝑛} of 𝑋,  𝑦_𝑖=𝑠(𝑥_𝑖)(𝑖=0,1,…,𝑛), such that, for any {𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘}⊂{𝑦₀,𝑦₁,…,𝑦_𝑛}, there exists  𝑧∈𝑇(co{𝑠(𝑥_𝑖0),𝑠(𝑥_𝑖1),…,𝑠(𝑥_𝑖𝑘)}) such that co{𝑠(𝑥_𝑖0),𝑠(𝑥_𝑖1),…,𝑠(𝑥_𝑖𝑘)}⊂𝐺(𝑧).

Proof. The proof is similar to Theorem 17.

Corollary 20. Suppose that 𝑋 is a compact convex space. Assume that 𝑠∶𝑋→𝑋,𝐺∶𝑋→2^𝑋, and 𝑇∈𝔖-𝐾𝐾𝑀(𝑋,𝑋,𝑋) are three functions satisfying the following:(1)𝑠(𝐶) is a compact convex subset of 𝑋 if 𝐶 is a compact convex subset of 𝑋,(2)𝑇(𝑄) is compact in 𝑋 if 𝑄 is compact in 𝑋,(3)for any 𝑧∈𝑇(𝑠(𝑋)),𝐺(𝑧) is a nonempty convex subset of 𝑋,(4)there exists a transfer compactly open values mapping 𝐻∶𝑋→2^𝑍 such that any 𝑥∈𝑋,𝐻(𝑥)⊂𝐺⁻¹(𝑠(𝑥)), and ∪_𝑥∈𝑋𝐻(𝑥)=𝑍.
Then there exists a finite subset 𝑥₀,𝑥₁,…,𝑥_𝑛 of 𝑋,𝑦_𝑖=𝑠(𝑥_𝑖)(𝑖=0,1,…,𝑛), such that, for any {𝑦_𝑖0,𝑦_𝑖1,…,𝑦_𝑖𝑘}⊂{𝑦₀,𝑦₁,…,𝑦_𝑛}, there exists  𝑧∈𝑇(co{𝑠(𝑥_𝑖0),𝑠(𝑥_𝑖1),…,𝑠(𝑥_𝑖𝑘)}) such that co{𝑠(𝑥_𝑖0),𝑠(𝑥_𝑖1),…,𝑠(𝑥_𝑖𝑘})⊂𝐺(𝑧).

Corollary 21. Let 𝑋 be a compact convex space. Suppose that G∶𝑋→2^𝑋 satisfies the following:(1)for any 𝑦∈𝑋,  𝐺(𝑦) is a nonempty convex subset of 𝑋,(2)𝐺⁻ is transfer open in 𝑋.
Then there is an 𝑥∈𝑋 such that 𝑥∈𝐺(𝑥).

Proof. Let 𝑠 and 𝑇 be the identity mapping id_𝑋; it follows from Corollary 20 that there exist a finite subset {𝑥₀,𝑥₁,…,𝑥_𝑛} of 𝑋 and 𝑧∈co{𝑥₀,𝑥₁,…,𝑥_𝑛} such that co{𝑥₀,𝑥₁,…,𝑥_𝑛}⊂𝐺(𝑧). Furthermore, 𝑧∈𝐺(𝑧). This completes the proof.

Remark 22. If 𝑋 is a nonempty compact convex subset of a topological vector space and for any 𝑥∈𝑋,  𝐺⁻(𝑥) is open in 𝑋, the above corollary is just the Fan-Browder type fixed-point theorem 1 in [16].

4. Generalized Vector Equilibrium Problems

In this section, we will introduce some definitions and conclusions and show the existence of solutions to the generalized vector equilibrium problems.

Definition 23. Let 𝑌, 𝑍 be two topological spaces and 𝑋 nonempty sets. Let 𝐻∶𝑍×𝑋→2^𝑌,  𝐶∶𝑍→2^𝑌,  𝑄∶𝑍→2^𝑊 be multivalued mapping. A generalized vector equilibrium problem (𝑋,𝑌,𝑍;𝐻,𝐶) is to find 𝑧∈𝑍 such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅,for all 𝑥∈𝑋. A generalized vector equilibrium problem (𝑋,𝑌,𝑍;𝑄,𝐻,𝐶) is to find 𝑧∈𝑍 such that, for each 𝑦∈𝑌, there exists 𝑤∈𝑄(𝑧) satisfying 𝜓(𝑧,𝑤,𝑦)∩𝐶(𝑧)≠∅.

Lemma 24. Let 𝑌 be a topological vector space and 𝑋,𝑍 nonempty sets. Let 𝑆∶𝑋→2^𝑌,  𝑇∶𝑌→2^𝑍,  𝐻∶𝑍×𝑋→2^𝑌,  𝐶∶𝑍→2^𝑌,  𝐺∶𝑍×𝑌→2^𝑋,  𝐷∶𝑍→2^𝑋. Suppose that the following conditions are satisfied:(1)for each 𝑦∈𝑌, there exists 𝑧∈𝑇(𝑦) such that 𝐺(𝑧,𝑦)∩𝐷(𝑧)=∅,(2)for each 𝑥∈𝑋,𝑦∈𝑌, there exists 𝑧∈𝑇(𝑦) such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)=∅ implies that 𝐺(𝑧,𝑦)∩𝐷(𝑧)≠∅ whenever 𝑦∈𝑆(𝑥) and 𝑧∈𝑇(𝑦),(3)for each 𝑦∈𝑌,  𝑅(𝑦)={𝑦∈𝑌∶𝐺(𝑧,𝑦)∩𝐷(𝑧)≠∅,∀𝑧∈𝑇(𝑦)} is convex.
Then 𝐹∶𝑋→2^𝑍defined by 𝐹(𝑥)={𝑧∈𝑍∶𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅} is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇.

Proof. If the conclusion does not hold, then there exists {𝑥₀,𝑥₁,…,𝑥_𝑛},  ∀𝑦_𝑖∈𝑆(𝑥_𝑖),∃{𝑦_𝑖0,…,𝑦_𝑖𝑘} such that 𝑇co𝑦_𝑖0,…,𝑦⊆_𝑖𝑘^/𝑘_𝑗=0𝐹𝑥_𝑖𝑗. (34) Therefore, there exist 𝑦∈co{𝑦_𝑖0,…,𝑦_𝑖𝑘} and 𝑧∈𝑇(𝑦) such that, for each 𝑗=0,…,𝑘,  𝑧∉𝐹₁(𝑥_𝑖𝑗). By the definition of 𝐹₁, we have 𝐻𝑧,𝑥_𝑖𝑗∩𝐶(𝑧)=∅,∀𝑗=0,…,𝑘. (35) By condition (2), for all 𝑧∈𝑇(𝑦_𝑖𝑗) and 𝑗=0,…,𝑘, we have 𝐺𝑧,𝑦_𝑖𝑗∩𝐷(𝑧)≠∅. (36) Thus 𝑦_𝑖𝑗∈𝑅(𝑦), and 𝑦∈co{𝑦_𝑖0,…,𝑦_𝑖𝑘}⊂𝑅(𝑦). Therefore, 𝐺(𝑧,𝑦)∩𝐷(𝑧)≠∅,∀𝑧∈𝑇(𝑦); this contradicts with (1). The proof is completed.

Lemma 25. Let 𝑌 be a topological vector space and 𝑋,𝑍 nonempty sets. Let 𝑆∶𝑋→2^𝑌,𝑇∶𝑌→2^𝑍,  𝐻∶𝑍×𝑋→2^𝑌,  𝐶∶𝑍→2^𝑌,  𝐺∶𝑍×𝑌→2^𝑋,  𝐷∶𝑍→2^𝑋. Suppose that the following conditions are satisfied:(1)for each 𝑦∈𝑌 and 𝑧∈𝑇(𝑦),  𝐺(𝑧,𝑦)∩𝐷(𝑧)≠∅,(2)for each 𝑥∈𝑋,  𝑦∈𝑌, there exists 𝑧∈𝑇(𝑦) such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)=∅ implies that 𝐺(𝑧,𝑦)∩𝐷(𝑧)=∅ for all 𝑦∈𝑆(𝑥) and some 𝑧∈𝑇(𝑦),(3)for each 𝑦∈𝑌,  𝑅(𝑦)={𝑦∈𝑌∶∃𝑧∈𝑇(𝑦) such that 𝐺(𝑧,𝑦)∩𝐷(𝑧)=∅} is convex.
Then 𝐹∶𝑋→2^𝑍 defined by 𝐹(𝑥)={𝑧∈𝑍∶𝐹(𝑧,𝑥)∩𝐶(𝑧)≠∅} is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇.

Proof. There exists {𝑥₀,𝑥₁,…,𝑥_𝑛},  ∀𝑦_𝑖∈𝑆(𝑥_𝑖),∃{𝑦_𝑖0,…,𝑦_𝑖𝑘} such that 𝑇co𝑦_𝑖0,…,𝑦⊆_𝑖𝑘^/𝑘_𝑗=0𝐹𝑥_𝑖𝑗. (37) Therefore, there exist 𝑦∈co{𝑦_𝑖0,…,𝑦_𝑖𝑘} and 𝑧∈𝑇(𝑦) such that, for each 𝑗=0,…,𝑘,  𝑧∉𝐹₁(𝑥_𝑖𝑗). By the Definition of 𝐹, we have 𝐻𝑧,𝑥_𝑖𝑗∩𝐶(𝑧)=∅,∀𝑗=0,…,𝑘. (38) By the condition (2), there exists 𝑧∈𝑍 such that, for each 𝑗=0,…,𝑘, we have that 𝑧∈𝑇(𝑦_𝑖𝑗)  and 𝐺𝑧,𝑦_𝑖𝑗∩𝐷(𝑧)=∅. (39) Thus 𝑦_𝑖𝑗∈𝑅(𝑦), and 𝑦∈co{𝑦_𝑖0,…,𝑦_𝑖𝑘}⊂𝑅(𝑦). Therefore, 𝐺(𝑧,𝑦)∩𝐷(𝑧)=∅ for some 𝑧∈𝑇(𝑦); this contradicts with (i). The proof is completed.

Remark 26. To avoid the structure of the space, Lemmas 24 and 25 generalize Lemmas 2.4 and 2.5 of X. P. Ding and T. M. Ding [13] from the following two aspects: (1) from generalized 𝐾𝐾𝑀 mapping with respect to 𝑇 to 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇; (2) condition (2) in our results is obviously weaker than that in [13].

Definition 27 (see [13]). Let 𝑌 and 𝑍 be topology spaces and 𝑋 nonempty set. Let 𝐻∶𝑍×𝑋→2^𝑌 and 𝐶∶𝑍→2^𝑌 be set-valued mappings. 𝐻(𝑧,𝑥) is said to be a 𝐶-transfer compactly continuous mapping of the generalized vector equilibrium problem (𝑋,𝑌,𝑍;𝐻,𝐶) in first argument if, for any compact subset 𝐾 of 𝑍 and any 𝑧∈𝐾, there exists 𝑥∈𝑋 such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)=∅; then there is a point 𝑥^′∈𝑋 such that 𝑧∈cint{𝑥∈𝑍∶𝐻(𝑧,𝑥^′)∩𝐶(𝑧)=∅}.

Proposition 28 (see [13]). Let 𝑌 and 𝑍 be topological spaces and 𝑋 nonempty set. Let 𝐻∶𝑍×𝑋→2^𝑌 and 𝐶∶𝑍→2^𝑌 be set-valued mappings. 𝐻(𝑧,𝑥) is said to be a 𝐶-transfer compactly continuous mapping of the generalized vector equilibrium problem (𝑋,𝑌,𝑍;𝐻,𝐶) in first argument if and only if the mapping 𝐹∶𝑋→2^𝑍 defined by 𝐹(𝑥)={𝑧∈𝑍∶𝐻(𝑧,𝑥)∩𝐶(𝑧)}≠∅ is a transfer compactly closed-valued mapping.

Proposition 29 (see [13]). Let 𝑋,𝑌, and 𝑍 be topological spaces. Let 𝐻∶𝑍×𝑋→2^𝑌 and 𝐶∶𝑍→2^𝑌 be set-valued mappings such that(i)𝐶 has closed (resp., open) graph;(ii)for each 𝑥∈𝑋,𝐻(⋅,𝑥) is upper semicontinuous on each compact subset of 𝑍.
Then the mapping 𝐹∶𝑥→2^𝑍 defined by 𝐹(𝑥)={𝑧∈𝑍∶𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅} has compactly closed values.

Theorem 30. Let 𝑋 be a topological space, 𝑌 a topological vector space, and 𝑍 a Hausdorff space; let 𝑆∶𝑋→2^𝑌,  𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍),  𝐻∶𝑍×𝑋→2^𝑌,  𝐶∶𝑍→2^𝑌,𝐺∶𝑍×𝑌→2^𝑋,  𝐷∶𝑍→2^𝑋 be multivalued mappings. Suppose that the following conditions are satisfied:(1)𝐻(𝑧,𝑥) is a 𝐶-transfer compactly continuous mapping of the generalized vector equilibrium problem (𝑋,𝑌,𝑍;𝐻,𝐶),(2)for each 𝑦∈𝑌, there exists 𝑧∈𝑇(𝑦) such that 𝐺(𝑧,𝑦)∩𝐷(𝑧)=∅,(3)for each 𝑥,𝑦, there exists 𝑧∈𝑇(𝑦) such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)=∅ which implies that 𝐺(𝑧,𝑦)∩𝐷(𝑧)≠∅ whenever 𝑦∈𝑆(𝑥) and 𝑧∈𝑇(𝑦),(4)for each 𝑦∈𝑌, the set 𝑅(𝑦)={𝑦∈𝑌∶𝐺(𝑧,𝑦)∩𝐷(𝑧)≠∅,∀𝑧∈𝑇(𝑦)} is a convex subset of 𝑌,(5)for each compact subset 𝑀₁ of 𝑋,  𝑇(𝑆(𝑀₁)) is compact in 𝑍, and for each convex subset 𝑀₂ of 𝑋,  𝑆(𝑀₂) is convex,(6)setting 𝐹∶𝑋→2^𝑍 by 𝐹(𝑥)={𝑧∈𝑋∶𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅}, there exists a nonempty compact subset 𝐾 of 𝑍 such that, for each 𝑁∈⟨𝑋⟩, there exists compact convex subset 𝐿^𝑋_𝑁 of 𝑋 including 𝑁 such that 𝑇(𝑆(𝐿^𝑋_𝑁)∩(⋂_𝑥∈𝐿^𝑋_𝑁ccl𝐹₁(𝑥))⊂𝐾.
Then there exists 𝑧∈𝑍 such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅; that is, 𝑧 is a solution to the generalized vector equilibrium problem (𝑋,𝑌,𝑍;𝐻,𝐶).

Proof. By condition (1), 𝐹 is transfer compactly closed-valued mapping from Proposition 28. By conditions (2)–(4) and Lemma 24, we know that 𝐹 is a 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇. Conditions (5) and (6) imply that conditions (4) and (5) of Theorem 9 hold. From Theorem 9, it follows that ∩_𝑥∈𝑋𝐹(𝑥)≠∅ and 𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅ for all 𝑥∈𝑋.

Theorem 31. Let 𝑋 be a topological space, 𝑌 a topological vector space, and 𝑍 a Hausdorff space, and let 𝑆∶𝑋→2^𝑌,  𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍),  𝐻∶𝑍×𝑋→2^𝑌,  𝐶∶𝑍→2^𝑌,  𝐺∶𝑍×𝑌→2^𝑋,  𝐷∶𝑍→2^𝑋 be multivalued mappings. Suppose that the following conditions are satisfied:(1)𝐻(𝑧,𝑥) is a 𝐶-transfer compactly continuous mapping of the generalized vector equilibrium problem (𝑋,𝑌,𝑍;𝐻,𝐶),(2)for each 𝑦∈𝑌 and 𝑧∈𝑇(𝑦),  𝐺(𝑧,𝑦)∩𝐷(𝑧)≠∅,(3)for each 𝑥,𝑦, there exists 𝑧∈𝑇(𝑦) such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)=∅ implies that 𝐺(𝑧,𝑦)∩𝐷(𝑧)=∅ for all 𝑦∈𝑆(𝑥) and some 𝑧∈𝑇(𝑦),(4)for each 𝑦∈𝑌,  𝑅(𝑦)={𝑦∈𝑌∶∃𝑧∈𝑇(𝑦) such that 𝐺(𝑧,𝑦)∩𝐷(𝑧)}=∅ is convex,(5)for each compact subset 𝑀₁ of 𝑋,  𝑇(𝑆(𝑀₁)) is compact in 𝑍, and for each convex subset 𝑀₂ of 𝑋,  𝑆(𝑀₂) is convex,(6)setting 𝐹∶𝑋→2^𝑍 by 𝐹₁(𝑥)={𝑧∈𝑋∶𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅}, there exists a nonempty compact subset 𝐾 of 𝑍 such that, for each 𝑁∈⟨𝑋⟩, there exists compact convex subset 𝐿^𝑋_𝑁 of 𝑋 including 𝑁 such that 𝑇(𝑆(𝐿^𝑋_𝑁)∩(⋂_𝑥∈𝐿^𝑋_𝑁ccl𝐹(𝑥))⊂𝐾.
Then there exists 𝑥∈𝑋 such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅; that is, 𝑧 is a solution to the generalized vector equilibrium problem (𝑋,𝑌,𝑍;𝐻,𝐶).

Proof. By condition (1), 𝐹 is transfer compactly closed-valued mapping from Proposition 29. By conditions (2)–(4) and Lemma 25, we have that 𝐹 is a generalized 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇. Conditions (5) and (6) imply that conditions (4) and (5) of Theorem 9 hold. From Theorem 9, it follows that ∩_𝑥∈𝑋𝐹(𝑥)≠∅ and 𝐻(𝑧,𝑥)⊂𝐶(𝑧) for all 𝑥∈𝑋.

Theorem 32. Let 𝑋 be a topological space, 𝑌 a topological vector space, and 𝑍 a Hausdorff space, and let 𝑆∶𝑋→2^𝑌,  𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍),  𝐻∶𝑍×𝑋→2^𝑌,  𝐶∶𝑍→2^𝑌,  𝐺∶𝑍×𝑌→2^𝑋,  𝐷∶𝑍→2^𝑋 be multivalued mappings. Suppose that the following conditions are satisfied:(1)𝐶 has closed graph,(2)for each 𝑥∈𝑋,  𝐻(⋅,𝑥) is upper semicontinuous on each compact subset of 𝑍 with nonempty compact values on 𝑍,(3)for each 𝑦∈𝑌, there exists 𝑧∈𝑇(𝑦) such that 𝐺(𝑧,𝑦)∩𝐷(𝑧)=∅,(4)for each 𝑥,𝑦, there exists 𝑧∈𝑇(𝑦) such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)=∅ implies that 𝐺(𝑧,𝑦)∩𝐷(𝑧)≠∅ whenever 𝑦∈𝑆(𝑥) and 𝑧∈𝑇(𝑦),(5)for each 𝑦∈𝑌,  𝑅(𝑦)={𝑦∈𝑌∶∃𝑧∈𝑇(𝑦) such that 𝐺(𝑧,𝑦)∩𝐷(𝑧)=∅} is convex,(6)for each compact subset 𝑀₁ of 𝑋,  𝑇(𝑆(𝑀₁)) is compact in 𝑍, and for each convex subset 𝑀₂ of 𝑋,  𝑆(𝑀₂) is convex,(7)setting 𝐹∶𝑋→2^𝑍 by 𝐹(𝑥)={𝑧∈𝑋∶𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅}, there exists a nonempty compact subset 𝐾 of 𝑍 such that, for each 𝑁∈⟨𝑋⟩, there exists compact convex subset 𝐿^𝑋_𝑁 of 𝑋 including 𝑁 such that 𝑇(𝑆(𝐿^𝑋_𝑁)∩(⋂_𝑥∈𝐿^𝑋_𝑁ccl𝐹(𝑥))⊂𝐾.
Then there exists 𝑥∈𝑋 such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅; that is, 𝑧 is a solution to the generalized vector equilibrium problem (𝑋,𝑌,𝑍;𝐻,𝐶).

Proof. By condition (1) and Proposition 29, 𝐹 has compactly closed values and so it is a transfer compactly closed-valued mapping. Hence the conclusion of Theorem 32 holds from Theorem 30.

Theorem 33. Let 𝑋 be a topological space, 𝑌 a topological vector space, and 𝑍 a Hausdorff space, and let 𝑆∶𝑋→2^𝑌,  𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍),  𝐻∶𝑍×𝑋→2^𝑌,  𝐶∶𝑍→2^𝑌,  𝐺∶𝑍×𝑌→2^𝑋,  𝐷∶𝑍→2^𝑋 be multivalued mappings. Suppose that the following conditions are satisfied:(1)𝐶 has closed graph,(2)for each 𝑥∈𝑋,  𝐻(⋅,𝑥) is upper semicontinuous on each compact subset of 𝑍 with nonempty compact values on 𝑍,(3)for each 𝑦∈𝑌, there exists 𝑧∈𝑇(𝑦) such that 𝐺(𝑧,𝑦)∩𝐷(𝑧)≠∅,(4)for each 𝑥,𝑦, there exists 𝑧∈𝑇(𝑦) such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)=∅ implies that 𝐺(𝑧,𝑦)∩𝐷(𝑧)=∅ for all 𝑦∈𝑆(𝑥) and some 𝑧∈𝑇(𝑦),(5)for each 𝑦∈𝑌,  𝑅(𝑦)={𝑦∈𝑌∶∃𝑧∈𝑇(𝑦) such that 𝐺(𝑧,𝑦)∩𝐷(𝑧)}=∅ is convex,(6)for each compact subset 𝑀₁ of 𝑋,𝑇(𝑆(𝑀₁)) is compact in 𝑍, and for each convex subset 𝑀₂ of 𝑋,  𝑆(𝑀₂) is convex,(7)setting 𝐹∶𝑋→2^𝑍 by 𝐹(𝑥)={𝑧∈𝑋∶𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅}, there exists a nonempty compact subset 𝐾 of 𝑍 such that, for each 𝑁∈⟨𝑋⟩, there exists compact convex subset 𝐿^𝑋_𝑁 of 𝑋 including 𝑁 such that 𝑇(𝑆(𝐿^𝑋_𝑁)∩(⋂_𝑥∈𝐿^𝑋_𝑁ccl𝐹(𝑥))⊂𝐾.
Then there exists 𝑥∈𝑋 such that 𝐻(𝑧,𝑥)∩𝐶(𝑧)≠∅.

Proof. By condition (1) and Proposition 29, 𝐹 has compactly closed values and so it is a transfer compactly closed-valued mapping. Hence the conclusion of Theorem 33 holds from Theorem 31.

The following result is a simplicity version of Theorem 1 in [17].

Lemma 34. Let 𝑋, 𝑌, 𝑊, and 𝑍 be topological spaces. Let 𝜓∶𝑍×𝑊×𝑌→2^𝑋 and 𝑄∶𝑍→2^𝑊 be set-valued mappings.(1)If, for each fixed 𝑦∈𝑌,(𝑧,𝑤)→𝜓(𝑧,𝑤,𝑦) and 𝑄 are both lower semicontinuous, then the mapping 𝐻∶𝑍×𝑋→2^𝑌 defined by 𝐻(𝑧,𝑥)=⋃_{𝑤∈𝑄(𝑥)}𝜓(𝑧,𝑤,𝑦) satisfies that, for each 𝑥∈𝑋,𝑧→𝐻(𝑧,𝑥) is lower semicontinuous on 𝑍.(2)If, for each fixed 𝑦∈𝑌,  (𝑧,𝑤)→𝜓(𝑧,𝑤,𝑦) and 𝑄 are both upper semicontinuous with compact values, then, for each 𝑥∈𝑋,  𝑧→𝐻(𝑧,𝑥) is upper semicontinuous on 𝑍 with nonempty compact values.

Theorem 35. Let 𝑋 be a topological space, 𝑌 a topological vector space, and 𝑍 a Hausdorff space, and let 𝑆∶𝑋→2^𝑌,  𝑇∈𝔄-𝐾𝐾𝑀(𝑋,𝑌,𝑍),  𝐶∶𝑍→2^𝑌,  𝐷∶𝑍→2^𝑋,  𝜙∶𝑍×𝑊×𝑋→2^𝑌,  𝜓∶𝑍×𝑊×𝑌→2^𝑋,  𝑄∶𝑍→2^𝑊 be multivalued mappings. Suppose that the following conditions are satisfied:(1)𝐶 has closed graph,(2)for each 𝑦∈𝑌,  (𝑧,𝑤)→𝜓(𝑧,𝑤,𝑦) and 𝑄 are both upper semicontinuous compact values on 𝑍,(3)for each 𝑦∈𝑌, there exists 𝑧∈𝑇(𝑦) such that (⋃_{𝑤∈𝑄(𝑧)}𝜙(𝑧,𝑤,𝑦))∩𝐷(𝑧)≠∅,(4)for each 𝑥, 𝑦, there exists 𝑧∈𝑇(𝑦) such that (⋃_{𝑤∈𝑄(𝑧)}𝜙(𝑧,𝑤,𝑦))∩𝐶(𝑧)=∅ implies that (⋃_{𝑤∈𝑄(𝑧)}𝜓(𝑧,𝑤,𝑦))∩𝐷(𝑧)=∅ whenever 𝑦∈𝑆(𝑥) and 𝑧∈𝑇(𝑦),(5)for each 𝑦∈𝑌,  𝑅(𝑦)={𝑦∈𝑌∶⋃_{𝑤∈𝑄(𝑧)}𝜙(𝑧,𝑤,𝑦)∩𝐷(𝑧)≠∅,∀𝑧∈𝑇(𝑦)} is convex,(6)for each compact subset 𝑀₁ of 𝑋,  𝑇(𝑆(𝑀₁)) is compact in 𝑍, and for each convex subset 𝑀₂ of 𝑋,  𝑆(𝑀₂) is convex,(7)setting 𝐹∶𝑋→2^𝑍 by 𝐹(𝑥)={𝑧∈𝑋∶⋃_{𝑤∈𝑄(𝑧)}𝜓(𝑧,𝑤,𝑦)∩𝐶(𝑧)≠∅}, there exists a nonempty compact subset 𝐾 of 𝑍 such that, for each 𝑁∈⟨𝑋⟩, there exists compact convex subset 𝐿^𝑋_𝑁 of 𝑋 including 𝑁 such that 𝑇(𝑆(𝐿^𝑋_𝑁)∩(⋂_𝑥∈𝐿^𝑋_𝑁ccl𝐹(𝑥))⊂𝐾.
Then there exists 𝑧∈𝑍 such that, for each 𝑦∈𝑌, there exists 𝑤∈𝑄(𝑧) satisfying 𝜓(𝑧,𝑤,𝑦)∩𝐶(𝑧)≠∅; that is, 𝑧 is a solution to the generalized vector equilibrium problem (𝑋,𝑌,𝑍;𝑄,𝐻,𝐶).

Proof. Define set-valued mappings 𝐻∶𝑍×𝑋→2^𝑌,𝐺∶𝑍×𝑌→2^𝑋 by 𝐻(𝑧,𝑥)=⋃_{𝑤∈𝑄(𝑧)}𝜓(𝑧,𝑤,𝑦) and 𝐺(𝑧,𝑦)=⋃_{𝑤∈𝑄(𝑧)}𝜙(𝑧,𝑤,𝑦)) for each (𝑧,𝑥)∈𝑍×𝑋 and (𝑧,𝑦)∈𝑍×𝑌, respectively. By Lemma 34 and Theorem 32, the rest is similar to the proof of Theorem 4.7 in [13].

Remark 36. For the above results, Theorems 30 and 31 generalize Propositions 4.3 and 4.4 of Lin and Wan in [14] in the following two aspects: (1) from transfer closed values to transfer compactly closed values; (2) from generalized 𝐾𝐾𝑀 mapping to 𝔄-𝐾𝐾𝑀 mapping with respect to 𝑇. To avoid the structure of the space, condition (3) in our results is more general than that in Theorems 4.1, 4.2, 4.3, 4.5, and 4.7 of [13].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the referees’ suggestion for improving the paper. The research is supported by the Natural Science Foundation of Hunan Province (no. 2014JJ4044).