A Class of Fan-Browder Type Fixed-Point Theorem and Its Applications in Topological Space

Abstract

A fixed-point theorem is proved under noncompact setting of general topological spaces. By applying the fixed-point theorem, several new existence theorems of solutions for equilibrium problems are proved under noncompact setting of topological spaces. These theorems improve and generalize the corresponding results in related literature.

1. Introduction

Let 𝑋 and 𝑌 be nonempty sets, let 𝑇∶𝑋→𝑌 be a single-valued mapping, let 𝐴∶𝑋→2^𝑋 be a set-valued mapping, let 𝑓∶𝑋×𝑌→ℝ∪{±∞} and 𝜙∶𝑋×𝑋→ℝ∪{±∞} be functions. The quasi-equilibrium problem QEP(𝑇,𝐴,𝑓) is to find 𝑥∈𝑋 such that 𝑥∈𝐴(𝑥),𝑓(𝑥,𝑇(𝑥))≤𝑓(𝑦,𝑇(𝑥)),∀𝑦∈𝐴(𝑥). (1.1) The QEP(𝑇,𝐴,𝑓) was introduced and studied by Noor and Oettli [1]. Cubiotti [2] and Ding [3] proved some existence theorems of solutions for the QEP(𝑇,𝐴,𝑓) in finite-dimensional space ℝ^𝑛 and topological vector spaces, respectively.

The quasi-equilibrium problem QEP(𝐴,𝜙) is to find 𝑥∈𝑋 such that 𝑥∈𝐴(𝑥),𝜙(𝑦,𝑥)≥0,∀𝑦∈𝐴(𝑥). (1.2) The QEP(𝐴,𝜙) has been studied by many authors; for example, see [4–8] and others.

The QEP(𝑇,𝐴,𝑓) and QEP(𝐴,𝜙) include many optimization problems, Nash-type equilibrium problems, quasivariational inequality problems, quasi-complementary problems, and others as special cases; see [1–8] and the references therein.

In this paper, we first prove a new Fan-Browder-type fixed-point theorem under noncompact setting of general topological spaces. Next, by applying the fixed-point theorem, some new existence theorems of solutions for the QEP(𝑇,𝐴,𝑓) and QEP(𝐴,𝜙) are proved in noncompact setting of general topological spaces. These results include a number of important known results in the fields as special cases.

2. Preliminaries

For a set 𝑋, we will denote by 2^𝑋 and ⟨𝑋⟩ the family of all subsets of 𝑋 and the family of all nonempty finite subsets of 𝑋. For any 𝐴∈⟨𝑋⟩, let |𝐴| denote the cardinality of 𝐴. Let Δ_𝑛 denote the standard 𝑛-dimensional simplex with vertices {𝑒₀,𝑒₁,…,𝑒_𝑛}. If 𝐽 is a nonempty subset of {0,1,…,𝑛}, we denote by Δ_𝐽 the convex hull of the vertices {𝑒_𝑗∶𝑗∈𝐽}. A set-valued mapping 𝐹∶𝑋→2^𝑌 is a function from a set 𝑋 into the power set 2^𝑌 of 𝑌, that is, a function with the value 𝐹(𝑥)⊂𝑌 for each 𝑥∈𝑋 and the fiber 𝐹⁻¹(𝑦)={𝑥∈𝑋∶𝑦∈𝐹(𝑥)} for each 𝑦∈𝑌. For 𝐴⊂𝑋, let 𝐹(𝐴)=⋃{𝐹(𝑥)∶𝑥∈𝐴}.

A subset 𝐴 of 𝑋 is said to be compactly open (resp., compactly closed) in 𝑋 if, for any nonempty compact subset 𝐾 of 𝑋, 𝐴∩𝐾 is open (resp., closed) in 𝐾. The following notions were introduced by Ding [9]. For any given nonempty subset 𝐴 of 𝑋, we define the compact closure and the compact interior of 𝐴, denoted by ccl(𝐴) and cint(𝐴), asccl(𝐴)={𝐵⊂𝑋∶𝐴⊂𝐵and𝐵iscompactlyclosedin𝑋},cint(𝐴)={𝐵⊂𝑋∶𝐵⊂𝐴and𝐵iscompactlyopenin𝑋}. (2.1) If 𝐴 is a subset of a vector space, then co(𝐴) denotes the convex hull of 𝐴.

If 𝑋 and 𝑌 are two topological spaces and 𝐺∶𝑋→2^𝑌 is a set-valued mapping, then 𝐺 is said to be transfer compactly open-valued (resp., transfer compactly closed-valued) on 𝑋 if, for each 𝑥∈𝑋 and for each compact subset 𝐾 of 𝑌 with 𝐺(𝑥)∩𝐾≠∅, 𝑦∈𝐺(𝑥)∩𝐾 (resp., 𝑦∉𝐺(𝑥)∩𝐾) implies that there exists a point 𝑥^′∈𝑋 such that 𝑦∈int_𝐾(𝐺(𝑥^′)∩𝐾) (resp., 𝑦∉cl_𝐾(𝐺(𝑥^′)∩𝐾)). A set-valued mapping 𝐺∶𝑋→2^𝑌 is said to have the compactly local intersection property on 𝑋 if, for each nonempty compact subset 𝐾 of 𝑋 and for each 𝑥∈𝑋 with 𝐺(𝑥)≠∅, there exists an open neighborhood 𝑁(𝑥) of 𝑥 in 𝑋 such that ⋂_{𝑧∈𝑁(𝑥)∩𝐾}𝐺(𝑧)≠∅; see [10]. A multivalued map 𝐺(𝑥) is said to be transfer open-valued [11], if for any 𝑥∈𝑋, 𝑦=𝑔(𝑥) there exists a 𝑍⊂𝑋 such that 𝑦∈int_𝑌(𝑍).

In [12], Deng and Xia introduced the following concept which is crucial to the study of KKM theory in general topological spaces.

Let 𝑋 be a nonempty set and let 𝑌 be a topological space. 𝑊∶𝑋→2^𝑌 is said to be a generalized relatively KKM (𝑅-KKM) mapping if, for each 𝐴∈⟨𝑋⟩ with |𝐴|=𝑛+1, there exists a continuous mapping 𝜑_𝐴∶Δ_𝑛→𝑌 such that, for each 𝐽∈⟨𝐴⟩, 𝜑_𝐴(Δ_𝐽)⊂𝑊(𝐽), (2.2) where Δ_𝐽 is the standard subsimplex of Δ_𝑛 corresponding to 𝐽.

Throughout this paper, all topological spaces are assumed to be Hausdorff.

In order to prove our main theorems, we need the following results.

Lemma 2.1. Let 𝑋 and 𝑌 be two topological spaces and 𝐺∶𝑋→2^𝑌 a set-valued mapping with nonempty values. Then the following conditions are equivalent:(I)𝐺 has the compactly local intersection property,(II)for each nonempty compact subset 𝐾 of 𝑋 and for each 𝑦∈𝑌, there exists an open subset 𝑂_𝑦 of 𝑋 (which may be empty) such that 𝑂_𝑦∩𝐾⊂𝐺⁻¹(𝑦) and 𝐾=⋃_𝑦∈𝑌(𝑂_𝑦∩𝐾),(III)for each nonempty compact subset 𝐾 of 𝑋, there exists a set-valued mapping 𝐹∶𝑋→2^𝑌 such that for each 𝑦∈𝑌, 𝐹⁻¹(𝑦) is open or empty in 𝑋 and 𝐹⁻¹(𝑦)∩𝐾⊂𝐺⁻¹(𝑦) for each 𝑦∈𝑌, and 𝐾=⋃_𝑦∈𝑌(𝐹⁻¹(𝑦)∩𝐾),(IV)for each nonempty compact subset 𝐾 of 𝑋 and for each 𝑥∈𝐾, there exists 𝑦∈𝑌 such that 𝑥∈cint𝐺⁻¹(𝑦)∩𝐾 and 𝐾=⋃_𝑦∈𝑌(cint𝐺⁻¹(𝑦)∩𝐾)=⋃_𝑦∈𝑌(𝐺⁻¹(𝑦)∩𝐾),(V)𝐺⁻¹∶𝑌→2^𝑋 is transfer compactly open-valued on 𝑋,(VI)𝑋=⋃_𝑦∈𝑌cint𝐺⁻¹(𝑦),(VII)for each 𝑦∈𝑌, 𝐺⁻¹(𝑦)={𝑥∈𝑋;𝑦∈𝐺(𝑥)} contains a relatively open subset 𝑂_𝑦 of 𝑌 (𝑂_𝑦 could be empty set for some 𝑦) such that ⋃_𝑦∈𝑌𝑂_𝑦=𝑌,(VIII)let 𝑆,𝑇∶𝑘→2^𝑘 be two multivalued maps, co(𝐺(𝑥))⊂𝑇(𝑥) and 𝐺(𝑥) is nonempty, and 𝐺⁻¹ is open in 𝑋.

Proof. By Lemma 1.1 of Ding in [10], (I), (II), (III), (IV), and (V) are equivalent. By Lemma 2.2 of Lin and Ansari in [13], (V) and (VI) are equivalent, and by Ansari in [14], (VI), (VII), and (VIII) are equivalent. This completes our proof.

Remark 2.2. Lemma 2.1 includes Lemma 1.1 of Ding in [10] and Lemma 2.2 of Ansari in [13] as special cases.

Lemma 2.3 (see [15]). Let 𝑋 and 𝑌 be topological spaces, let 𝐷 be a nonempty closed subset of 𝑋, and let Φ,Ψ∶𝑋→2^𝑌 be two set-valued mappings such that Φ(𝑥)⊂Ψ(𝑥) for each 𝑥∈𝑋. Suppose that Φ⁻¹,Ψ⁻¹∶𝑌→2^𝑋 are both transfer compactly open-valued on 𝑌. Then the mapping 𝐺∶𝑋→2^𝑌 defined by 𝐺(𝑥)=Φ(𝑥)if𝑥∈𝐷,Ψ(𝑥)if𝑥∈𝑋⧵𝐷 (2.3) is such that 𝐺⁻¹∶𝑌→2^𝑋 is also transfer compactly open-valued on 𝑌.

3. Fan-Browder Type Fixed-Point Theorem

Theorem 3.1. Let 𝑋 be a topological space, let 𝐾 be a nonempty compact subset of 𝑋, and let 𝐺∶𝑋→2^𝑋 be such that(i)𝐺 has nonempty values and satisfies one of the conditions (I)–(VIII) in Lemma 2.1,(ii)either,(a) for each 𝑁∈ℱ(𝑋), there is a nonempty compact subset 𝐿_𝑁 of 𝑋 containing 𝑁 such that there is a generalized 𝑅-KKM mapping 𝑊∶𝐿_𝑁→2^𝐿𝑁 satisfying, for each 𝑥∈𝑋, the fact that𝑀∈⟨𝐺(𝑥)∩𝐿_𝑁⟩ implies that 𝑊(𝑀)⊂𝐺(𝑥), having, moreover, that 𝐿_𝑁⧵𝐾⊂⋃_{𝑦∈𝐿𝑁}cint𝐺⁻¹(𝑦), or(b) if 𝑋 is a nonempty convex set of a topological vector space, for each 𝑁∈𝐹(𝑋), there is a nonempty compact subset 𝐿_𝑁 of 𝑋 containing 𝑁 such that there is a mapping 𝑊∶𝐿_𝑁→2^𝐿 satisfying, for each 𝐷⊂𝑁, the fact that 𝑊(𝐷) is convex and, for each 𝑥∈𝑋, the fact that 𝑀∈⟨𝐺(𝑥)∩𝐿_𝑁⟩ implies that 𝑊(𝑀)⊂𝐺(𝑥).Then there exists a point 𝑥₀∈𝑋 such that 𝑥₀∈𝐺(𝑥₀).

Proof. By (i) and Lemma 2.1, we have 𝑋=⋃_𝑦∈𝑋cint𝐺⁻¹(𝑦). Since 𝐾 is a nonempty compact subset of 𝑋, there exists a finite set 𝑁∈ℱ(𝑋) such that 𝐾=_𝑦∈𝑁(cint𝐺⁻¹(𝑦)∩𝐾). (3.1) For the 𝑁, consider the compact subset 𝐿_𝑁⊂𝑋 in condition (ii) satisfying 𝐿_𝑁⧵𝐾⊂_{𝑦∈𝐿𝑁}cint𝐺⁻¹(𝑦). (3.2) By (3.1), we have 𝐿_𝑁∩𝐾⊂_𝑦∈𝑁(cint𝐺⁻¹(𝑦)∩𝐿_𝑁). (3.3) Noting that 𝑁⊂𝐿_𝑁, it follows from (3.2) and (3.3) that 𝐿_𝑁=⋃_{𝑦∈𝐿𝑁}(cint𝐺⁻¹(𝑦)∩𝐿_𝑁). Since 𝐿_𝑁 is compact, there exists a finite set {𝑦₀,…,𝑦_𝑚}⊂𝐿_𝑁 such that 𝐿_𝑁=^𝑚_𝑖=0(cint𝐺⁻¹(𝑦_𝑖)∩𝐿_𝑁). (3.4) Since 𝑊∶𝐿_𝑁→2^𝐿𝑁 is a generalized 𝑅-KKM mapping, there exists a continuous mapping 𝑓∶Δ_𝑚→𝐿_𝑁 such that 𝑓(Δ_𝐽)⊂𝑊(𝐽) for each ∅≠𝐽⊂{𝑦₀,…,𝑦_𝑚}, where Δ_𝐽 is the face of Δ_𝑚 corresponding to 𝐽⊂{𝑦₀,…,𝑦_𝑚}. Let {𝜑_𝑖}^𝑚_𝑖=0 be a continuous partition of unity subordinated to the open covering {cint𝐺⁻¹(𝑦_𝑖)∩𝐿_𝑁}^𝑚_𝑖=0, that is, for each 𝑖∈{0,1,…,𝑚}, 𝜑_𝑖∶𝐿_𝑁→[0,1] is continuous, and {𝑥∈𝐿_𝑁∶𝜑_𝑖(𝑥)≠0}⊂cint𝐺⁻¹(𝑦_𝑖)∩𝐿_𝑁⊂𝐺⁻¹(𝑦_𝑖) (3.5) such that ∑^𝑚_𝑖=0𝜑_𝑖(𝑥)=1 for all 𝑥∈𝐿_𝑁. Define 𝜑∶𝐿_𝑁→Δ_𝑚 by 𝜑(𝑥)=(𝜑₀(𝑥),𝜑₁(𝑥),…,𝜑_𝑚(𝑥)),∀𝑥∈𝐿_𝑁. (3.6) Then 𝜑(𝑥)∈Δ_𝐽(𝑥) for all 𝑥∈𝐿_𝑁, where 𝐽(𝑥)={𝑦_𝑗=𝑦₀,𝑦₁,…,𝑦_𝑚∶𝜑_𝑗(𝑥)≠0}. Therefore, we have 𝑓(𝜑(𝑥))∈𝑓(Δ_𝐽(𝑥))⊂𝑊(𝐽(𝑥))⊂𝐺(𝑥),∀𝑥∈𝐿_𝑁. (3.7) It is easy to see that 𝜑∘𝑓∶Δ_𝑚→Δ_𝑚 is continuous. By Browder's fixed-point theorem, there exists 𝑧∈Δ_𝑚 such that 𝑧=(𝜑∘𝑓)(𝑧). Let 𝑥=𝑓(𝑧), then we have 𝑥=𝑓(𝑧)=𝑓(𝜑(𝑧))⊂𝐺(𝑥). This completes our proof.

Remark 3.2. Theorem 3.1 generalized Theorem 2.1 of Ding in [15] by dropping all the contractible conditions. Theorem 3.1 is also a noncompact variant of [8, Theorem 2] in general topological spaces.

4. Equilibrium Existence of 𝑄𝐸𝑃(𝑇,𝐴,𝑓) and 𝑄𝐸𝑃(𝐴,𝑓)

First, we prove the following equilibrium existence theorems of QEP(𝑇,𝐴,𝑓).

Theorem 4.1. Let 𝑋 be a topological space, let 𝐾 be a nonempty compact subset of 𝑋, and let 𝑌 be a nonempty set. Let 𝑇∶𝑋→𝑌, 𝐴∶𝑋→2^𝑋, and 𝑓∶𝑋×𝑌→ℝ∪{±∞} be such that(i)𝐴 has nonempty values on 𝑋 and satisfies one of the conditions (I)–(VIII) in Lemma 2.1,(ii)the set 𝐷={𝑥∈𝑋∶𝑥∈𝐴(𝑥)} is nonempty closed in 𝑋,(iii)the mapping 𝐵⁻¹∶𝑋→2^𝑋 is transfer compactly open-valued, where 𝐵∶𝑋→2^𝑋 is defined by 𝐵(𝑥)={𝑦∈𝐴(𝑥)∶𝑓(x,𝑇(𝑥))−𝑓(𝑦,𝑇(𝑥))>0}, (4.1)(iv)for each 𝑁∈ℱ(𝑋), there is a nonempty compact subset 𝐿_𝑁 of 𝑋 containing 𝑁 such that there is a generalized 𝑅-KKM mapping 𝑊∶𝐿_𝑁→2^𝐿𝑁 satisfying, for each 𝑥∈𝐷, the fact that 𝑀∈⟨𝐵(𝑥)∩𝐿_𝑁⟩ implies that 𝑊(𝑀)⊂𝐵(𝑥) and, for each 𝑥∈𝑋⧵𝐷, the fact that 𝑀∈⟨𝐴(𝑥)∩𝐿_𝑁⟩ implies that 𝑊(𝑀)⊂𝐴(𝑥). Moreover, for each 𝑥∈𝐿_𝑁⧵𝐾, if 𝑥∈𝑋⧵𝐷, then there exists 𝑦∈𝐿_𝑁 such that 𝑥∈cint𝐴⁻¹(𝑦); if 𝑥∈𝐷, then there exists 𝑦∈𝐿_𝑁 such that 𝑥∈cint𝐵⁻¹(𝑦).
Then there exists 𝑥∈𝑋 such that𝑥∈𝐴(𝑥),𝑓(𝑥,𝑇(𝑥))≤𝑓(𝑦,𝑇(𝑥)),∀𝑦∈𝐴(𝑥), (4.2) that is, 𝑥 is a solution of the QEP(𝑇,𝐴,𝑓).

Proof. Define a mapping 𝐺∶𝑋→2^𝑋 by 𝐺(𝑥)=𝐵(𝑥)if𝑥∈𝐷,𝐴(𝑥)if𝑥∈𝑋⧵𝐷. (4.3) From conditions (i), (iii) and Lemma 2.1, it follows that the mappings 𝐵⁻¹,𝐴⁻¹∶𝑋→2^𝑋 are both transfer compactly open-valued on 𝑋. Note that 𝐵(𝑥)⊂𝐴(𝑥) for each 𝑥∈𝑋 and 𝐷 is nonempty closed by (ii); by Lemma 2.3, 𝐺⁻¹∶𝑋→2^𝑋 is also transfer compactly open-valued on 𝑋. From condition (iv), it follows that, for each 𝑁∈ℱ(𝑋) and for each 𝑥∈𝑋, 𝑀∈⟨𝐺(𝑥)∩𝐿_𝑁⟩ implies that 𝑊(𝑀)⊂𝐺(𝑥). Moreover, 𝐿_𝑁⧵𝐾⊂⋃_{𝑦∈𝐿𝑁}cint𝐺⁻¹(𝑦). Hence, condition (ii) of Theorem 3.1 holds. Now assume that, for each 𝑥∈𝐷, 𝐵(𝑥)≠∅. Then for each 𝑥∈𝑋, 𝐺(𝑥)≠∅ by (i). It is easy to see that all conditions of Theorem 3.1 are satisfied. By Theorem 3.1, there exists 𝑥∈𝑋 such that 𝑥∈𝐺(𝑥). By the definition of 𝐷 and 𝐺, we must have {𝑥∈𝑋∶𝑥∈𝐺(𝑥)}⊂𝐷. It follows that 𝑥∈𝐵(𝑥)∩𝐷. In particular, we obtain 𝑓(𝑥,𝑇(𝑥))−𝑓(𝑥,𝑇(𝑥))>0 which is impossible. Therefore, there exists 𝑥∈𝐷 such that 𝐵(𝑥)=∅, that is, 𝑥∈𝐴(𝑥) and 𝑓(𝑥,𝑇(𝑥))≤𝑓(𝑦,𝑇(𝑥)) for all 𝑦∈𝐴(𝑥). This completes the proof.

Remark 4.2. Theorem 4.1 generalized Theorem 3.1 of Ding in [15] by dropping all the contractible conditions. Theorem 4.1 also improved and generalized [3, Theorem 2.1] and [2, Theorem 4.2] from topological vector spaces to general noncompact topological spaces without linear structure under much weaker assumptions.

Theorem 4.3. Let 𝑋 and 𝑌 be two topological spaces and let 𝐾 be a nonempty compact subset of 𝑋. Let 𝑇∶𝑋→𝑌, 𝐴∶𝑋→2^𝑋, and 𝑓∶𝑋×𝑌→ℝ∪{±∞} be such that(i)𝐴 has nonempty values on 𝑋 such that 𝐴⁻¹∶𝑋→2^𝑋 is compactly open-valued,(ii)the set 𝐷={𝑥∈𝑋∶𝑥∈𝐴(𝑥)} is nonempty closed in 𝑋,(iii)𝑇 and 𝑓 are continuous,(iv)for each 𝑁∈ℱ(𝑋), there is a nonempty compact subset 𝐿_𝑁 of 𝑋 containing 𝑁 such that there is a generalized 𝑅-KKM mapping 𝑊∶𝐿_𝑁→2^𝐿𝑁 satisfying, for each 𝑥∈𝐷, the fact that 𝑀∈⟨𝐴(𝑥)∩𝑃(𝑥)∩𝐿_𝑁⟩ implies that 𝑊(𝑀)⊂𝐴(𝑥)∩𝑃(𝑥), where 𝑃∶𝑋→2^𝑋 is defined by 𝑃(𝑥)={𝑦∈𝑋∶𝑓(𝑥,𝑇(𝑥))−𝑓(𝑦,𝑇(𝑥))>0}, and, for each 𝑥∈𝑋⧵𝐷, the fact that 𝑀∈⟨𝐴(𝑥)∩𝐿_𝑁⟩ implies that 𝑊(𝑀)⊂𝐴(𝑥). Moreover, for each 𝑥∈𝐿_𝑁⧵𝐾, if 𝑥∈𝑋⧵𝐷, then 𝐴(𝑥)∩𝐿_𝑁≠∅; if 𝑥∈𝐷, then there exists 𝑦∈𝐴(𝑥)∩𝐿_𝑁 satisfying 𝑓(𝑦,𝑇(𝑥))<𝑓(𝑥,𝑇(𝑥)).Then there exists 𝑥∈𝑋 such that 𝑥∈𝐴(𝑥),𝑓(𝑥,𝑇(𝑥))≤𝑓(𝑦,𝑇(𝑥)),∀𝑦∈𝐴(𝑥), (4.4) that is, 𝑥 is a solution of the QEP(𝑇,𝐴,𝑓).

Proof. Since 𝑇 and 𝑓 are both continuous, we have that, for each 𝑦∈𝑋, 𝑃⁻¹(𝑦) = {𝑥∈𝑋∶𝑓(𝑥,𝑇(𝑥))−𝑓(𝑦,𝑇(𝑥))>0} is open in 𝑋, and hence, 𝑃⁻¹∶𝑋→2^𝑋 has compactly open values. It follows that the mapping 𝐵∶𝑋→2^𝑋 defined by 𝐵(𝑥)=𝐴(𝑥)∩𝑃(𝑥) is such that the mapping 𝐵⁻¹=(𝐴∩𝑃)⁻¹=𝐴⁻¹∩𝑃⁻¹ also has compactly open values on 𝑋, and hence, it is also transfer compactly open-valued on 𝑋. By condition (iv), for each 𝑥∈𝐿_𝑁⧵𝐾, if 𝑥∈𝑋⧵𝐷, we have 𝐴(𝑥)∩𝐿_𝑁≠∅, and hence, there exists 𝑦∈𝐿_𝑁 such that 𝑥∈𝐴⁻¹(𝑦)=cint𝐴⁻¹(𝑦); if 𝑥∈𝐷, we have 𝑦∈𝐿_𝑁 and 𝑥∈𝐴⁻¹(𝑦)∩𝑃⁻¹(𝑦)=cint(𝐴⁻¹(𝑦)∩𝑃⁻¹(𝑦)). Hence, condition (iv) implies that condition (iv) of Theorem 4.1 is satisfied. It is easy to see that all conditions of Theorem 4.1 are satisfied. By Theorem 4.1, the conclusion of Theorem 4.3 holds. This completes the proof.

Second, we prove the following equilibrium existence theorem of QEP(𝐴,𝑓).

Theorem 4.4. Let 𝑋 be a topological space, let 𝐾 be a nonempty compact subset of 𝑋. Let 𝐴∶𝑋→2^𝑋 and 𝑓∶𝑋×𝑌→ℝ∪{±∞} be such that(i)𝐴 has nonempty values on 𝑋 such that 𝐴⁻¹∶𝑋→2^𝑋 has compactly open values and the mapping cl𝐴∶𝑋→2^𝑋 defined by (cl𝐴)(𝑥)=cl𝐴(𝑥) (the closure of 𝐴(𝑥)) is upper semicontinuous,(ii)𝑓 is a continuous function,(iii)for each 𝑁∈ℱ(𝑋), there is a nonempty compact subset 𝐿_𝑁 of 𝑋 containing 𝑁 such that there is a generalized 𝑅-KKM mapping 𝑊∶𝐿_𝑁→2^𝐿𝑁 satisfying, for each 𝑥∈𝐷, the fact that 𝑀∈⟨𝐴(𝑥)∩𝑃(𝑥)∩𝐿_𝑁⟩ implies that 𝑊(𝑀)⊂𝐴(𝑥)∩𝑃(𝑥), where 𝐷={𝑥∈𝑋∶𝑥∈cl𝐴(𝑥)} and 𝑃(𝑥)={𝑦∈𝑋∶𝑓(𝑥,𝑥)−𝑓(𝑦,𝑥)>0}, and for each 𝑥∈𝑋⧵𝐷, the fact that 𝑀∈⟨𝐴(𝑥)∩𝐿_𝑁⟩ implies that 𝑊(𝑀)⊂𝐴(𝑥). Moreover, for each 𝑥∈𝐿_𝑁⧵𝐾, if 𝑥∈𝑋⧵𝐷, then 𝐴(𝑥)∩𝐿_𝑁≠∅; if 𝑥∈𝐷, then there exists 𝑦∈𝐴(𝑥)∩𝐿_𝑁 satisfying 𝑓(𝑦,𝑥)<𝑓(𝑥,𝑥).Then there exists 𝑥∈𝑋 such that 𝑥∈𝐴(𝑥),𝑓(𝑥,𝑥)≤𝑓(𝑦,𝑥),∀𝑦∈𝐴(𝑥). (4.5) If one we further assumes that 𝑓(𝑥,𝑥)≥0 for all 𝑥∈𝑋, then one has that 𝑥∈𝐴(𝑥),𝑓(𝑦,𝑥)≥0,∀𝑦∈𝐴(𝑥), (4.6) that is, 𝑥 is a solution of the QEP(𝐴,𝑓).

Proof. Since cl𝐴∶𝑋→2^𝑋 is upper semicontinuous with closed values, the set 𝐷={𝑥∈𝑋∶𝑥∈cl𝐴(𝑥)} must be closed in 𝑋. By letting 𝑌=𝑋, with 𝑇 being the identity mapping and 𝐷 being in place of 𝐷, it is easy to see that all conditions of Theorem 4.3 are satisfied. By Theorem 4.3, there exists 𝑥∈𝑋 such that 𝑥∈𝐴(𝑥) and 𝑓(𝑥,𝑥)≤𝑓(𝑦,𝑥), for all 𝑦∈𝐴(𝑥). If 𝑓(𝑥,𝑥)≥0 for all 𝑥∈𝑋, then we must have 𝑥∈𝐴(𝑥) and 𝑓(𝑦,𝑥)≥0, for all 𝑦∈𝐴(𝑥), that is, 𝑥 is a solution of the QEP(𝐴,𝑓).

Remark 4.5. Theorem 4.4 generalized Theorem 3.3 of Ding in [15] by dropping all the contractible conditions. Theorem 4.4 is also a noncompact variant of [4, Theorem 6.4.21], [5, Theorem 3.1], Theorem 2.1 of [6, 7], and [8, Theorem 4] in general topological spaces.

From Theorem 4.4, we obtain the following existence result for generalized quasi-equilibrium problems.

Theorem 4.6. Let 𝑋 and 𝑌 be two topological spaces and let 𝐾 be a nonempty compact subset of 𝑋. Let 𝑇∶𝑋→2^𝑌 have a continuous selection 𝑔∶X→𝑌. Let 𝐴∶𝑋→2^𝑋 and 𝜙∶𝑋×𝑌×𝑋→ℝ∪{±∞} be such that(i)𝐴 satisfies condition (i) of Theorem 4.4,(ii)𝜙 is a continuous function,(iii)for each 𝑁∈ℱ(𝑋), there is a nonempty compact subset 𝐿_𝑁 of 𝑋 containing 𝑁 such that there is a generalized 𝑅-KKM mapping 𝑊∶𝐿_𝑁→2^𝐿𝑁 satisfying, for each 𝑥∈𝐷, the fact that 𝑀∈⟨𝐴(𝑥)∩𝑃(𝑥)∩𝐿_𝑁⟩ implies that 𝑊(𝑀)⊂𝐴(𝑥)∩𝑃(𝑥), where 𝐷={𝑥∈𝑋∶𝑥∈cl𝐴(𝑥)} and 𝑃(𝑥)={𝑧∈𝑋∶𝜙(𝑥,𝑔(𝑥),𝑥)−𝜙(𝑥,𝑔(𝑥),𝑧)>0}, and, for each 𝑥∈𝑋⧵𝐷, the fact that 𝑀∈⟨𝐴(𝑥)∩𝐿_𝑁⟩ implies that 𝑊(M)⊂𝐴(𝑥). Moreover, for each 𝑥∈𝐿_𝑁⧵𝐾, if 𝑥∈𝑋⧵𝐷, then 𝐴(𝑥)∩𝐿_𝑁≠∅; if 𝑥∈𝐷, then there exists 𝑦∈𝐴(𝑥)∩𝐿_𝑁 satisfying 𝜙(𝑥,𝑔(𝑥),𝑦)<𝜙(𝑥,𝑔(𝑥),𝑥).Then there exists 𝑥∈𝑋 and 𝑦=𝑔(𝑥)∈𝑇(𝑥) such that 𝑥∈𝐴(𝑥),𝜙(𝑥,𝑦,𝑥)≤𝜙(𝑥,𝑦,𝑧),∀𝑧∈𝐴(𝑥). (4.7) If one further assumes that 𝜙(𝑥,𝑔(𝑥),𝑥)≥0 for all 𝑥∈𝑋, then one has that 𝑥∈𝐴(𝑥),𝜙(𝑥,𝑦,𝑧)≥0,∀𝑧∈𝐴(𝑥). (4.8)

Proof. Define 𝑓∶𝑋×𝑋→ℝ∪{±∞} by 𝑓(𝑧,𝑥)=𝜙(𝑥,𝑔(𝑥),𝑧),∀(𝑧,𝑥)∈𝑋×𝑋. (4.9) Then the conclusion of Theorem 4.6 holds from Theorem 4.4.

Remark 4.7. Theorem 4.6 generalized Theorem 3.4 of Ding in [15] by dropping all the contractible conditions. Theorem 4.6 is a noncompact variant of [8, Corollary 5] and [16, Theorem 3.1] in general topological spaces.

Acknowledgment

The authors would like to express their thanks to the Editor-in-Chief and the anonymous referees for their valuable comments and suggestions on a previous draft, which resulted in the present version of the paper.