Fixed Points and Existence Theorems of Maximal Elements with Applications in FC-Spaces

Abstract

We present some fixed point theorems and existence theorems of maximal elements in FC-space from which we derive several coincidence theorems. Applications of these results to generalized equilibrium problems and minimax theory will be given. Our results improve and generalize some recent results.

1. Introduction and Preliminaries

It is well known that many fixed point theorems, coincidence theorems, and maximal elements have been established in topological vector spaces, 𝐻-spaces, and 𝐺-convex spaces by many authors (see [1–11]). In most of the known existence results of maximal elements and fixed point theorems, the convexity assumptions play a crucial role which strictly restricts the applicable area of these results. In fact, what we meet may not have convexity structure. Hence it is quite reasonable and valuable to study fixed point theorems, coincidence theorems, and maximal elements in general topological spaces without convexity structure.

In this paper, we will continue to study some problems in nonlinear analysis in 𝐹𝐶-spaces. Some fixed point theorems and existence theorems of maximal elements are proved under noncompact setting of 𝐹𝐶-spaces. We will apply these results to give some coincidence theorems. More specifically, the application of these results to generalized equilibrium problems and minimax theory will be given in 𝐹𝐶-spaces. Our results improve and generalize the corresponding results in [1, 2, 12–14].

Let 𝑌 be a nonempty set. We denote by 2^𝑌 and ⟨𝑌⟩ the family of all subsets of 𝑌 and the family of all nonempty finite subsets of 𝑌, respectively. For each 𝐴∈⟨𝑋⟩, we denote by |𝐴| the cardinality of 𝐴. Let Δ_𝑛 be the standard n-dimensional simplex with vertices {𝑒₀,…,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}.

Let 𝑋 and 𝑌 be two sets, and, 𝑇∶𝑋→2^𝑌 be a set-valued mapping. We will use the following notations in the following material: (i)𝑇(𝑥)={𝑦∈𝑌∶𝑦∈𝑇(𝑥)},(ii)𝑇(𝐴)=⋃_𝑥∈𝐴𝑇(𝑥),(iii)𝑇⁻¹(𝑦)={𝑥∈𝑋∶𝑦∈𝑇(𝑥)},(iv)𝑇⁻¹(𝐵)={𝑥∈𝑋∶𝑇(𝑥)⋂𝐵≠∅}.

For topological spaces 𝑋 and 𝑌, a subset 𝐴 of 𝑋 is said to be compactly open (resp., compactly closed) if for each nonempty compact subset 𝐾 of 𝑋, 𝐴⋂𝐾 is open (resp., closed) in 𝐾. The compact closure of 𝐴 and the compact interior of 𝐴 (see [5]) are defined, respectively, by 𝑐𝑐cinl𝐴t𝐴=={𝐵{𝐵⊂⊂𝑋𝑋∶∶𝐴𝐵⊂⊂𝐵𝐴anandd𝐵𝐵isiscocommppacatcltylycloopseendiinn𝑋𝑋}}.,(1.1)

It is easy to see that 𝑐cl(𝑋⧵𝐴)=𝑋⧵𝑐int𝐴,int𝐴⊂𝑐int𝐴⊂𝐴,𝐴⊂𝑐cl𝐴⊂cl𝐴, 𝐴 is compactly open in 𝑋 if and only if 𝐴=𝑐int𝐴. For each nonempty compact subset 𝐾 of 𝑋, 𝑐cl𝐴⋂𝐾=cl_𝐾(𝐴⋂𝐾) and 𝑐int𝐴⋂𝐾=int_𝐾(𝐴⋂𝐾), where cl_𝐾(𝐴⋂𝐾) (resp., int_𝐾(𝐴⋂𝐾)) denotes the closure (resp., interior) of 𝐴⋂𝐾 in 𝐾.

A set-valued mapping 𝑇∶𝑋→2^𝑌 is said to be transfer compactly open valued on 𝑋 (see [5]) if for each 𝑥∈𝑋 and 𝑦∈𝑇(𝑥), there exists 𝑥′∈𝑋 such that 𝑦∈𝑐int𝑇(𝑥^). It is clear that each transfer open valued correspondence is transfer compactly open valued. The inverse is not true in general.

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

Definition 1.1 (see [7]). (𝑋,𝜑_𝑁) is said to be an 𝐹𝐶-space if 𝑋 is a topological space, and for each 𝑁={𝑥₀,…,𝑥_𝑛}∈⟨𝑋⟩, where some elements in 𝑁 may be the same, there exists a continuous mapping 𝜑_𝑁∶Δ_𝑛→𝑋. A subset 𝐷 of (𝑋,𝜑_𝑁) is said to be an 𝐹𝐶-subspace of 𝑋 if for each 𝑁={𝑥₀,…,𝑥_𝑛}∈⟨𝑋⟩ and for each {𝑥_𝑖0,…,𝑥_𝑖𝑘}⊂𝑁⋂𝐷, 𝜑_𝑁(Δ_𝑘)⊂𝐷, where Δ_𝑘=co({𝑒_𝑖𝑗∶𝑗=0,…,𝑘}) and co denotes convex hull.
In many recent papers (see [9, 11, 15]), one studies one or more of the following generalized equilibrium problems
Let 𝑋 be a convex subset of a topological vector space, let 𝑍 and, 𝑉 be nonempty sets and let 𝐹∶𝑋×𝑍→2^𝑉,𝐶∶𝑋→2^𝑉 be set-valued mappings. Find 𝑥₀∈𝑋 such that one of the following situations occurs:𝐹(𝑥₀,𝑧)⊆𝐶(𝑥₀) for all 𝑧∈𝑍;𝐹(𝑥₀,𝑧)⋂𝐶(𝑥₀)≠∅ for all 𝑧∈𝑍;𝐹(𝑥₀,𝑧)⊈𝐶(𝑥₀) for all 𝑧∈𝑍;𝐹(𝑥₀,𝑧)⋂𝐶(𝑥₀)=∅ for all 𝑧∈Z. 

In this paper, we will try a unified approach for all these problems considering a (binary) relation 𝜌 on 2^𝑉 and looking for a point 𝑥₀∈𝑋 such that 𝐹(𝑥₀,𝑧)𝜌𝐶(𝑥₀) for all 𝑧∈𝑍.

Denote by 𝜌^𝑐 the complementary relation of 𝜌, that is, for any 𝐴,𝐵⊂𝑉, exactly one of the following assertions 𝐴𝜌𝐵,𝐴𝜌^𝑐𝐵 holds.

Since Fan [16] and Liu [17] extended the von Neumann-Sion principle to obtain two-function minimax inequalities, many such results involving two or more functions have appeared in the literature. We also obtain a very general minimax inequality of the following type: inf_𝑥∈𝑋ℎ(𝑥,𝑥)≤sup_𝑥∈𝑋inf_𝑧∈𝑍𝑓(𝑥,𝑧)+sup_𝑧∈𝑍inf_𝑥∈𝑋𝑔(𝑥,𝑧),(1.2) which is, to the best of our knowledge, different from all the minimax inequalities known in the literature.

2. Fixed Points and Maximal Elements

In order to prove our main results, we need the following definition and lemmas.

Definition 2.1. Let (𝑋,𝜑_𝑁) and (𝑍,𝜑^∗𝑀) be two 𝐹𝐶-spaces. A set-valued mapping 𝑇∶𝑋→2^𝑍 is said to be weakly-𝐹𝐶 valued if for each 𝐴={𝑥₀,…,𝑥_𝑛}∈⟨𝑋⟩,𝑥∈𝜑_𝑁(Δ_𝑛) and 𝐵={𝑧₀,…,𝑧_𝑛}∈⟨𝑍⟩ with 𝑧_𝑖∈𝑇(𝑥_𝑖)(𝑖=0,…,𝑛), we have 𝑇(𝑥)⋂𝜑^∗𝑀(Δ^𝑛)≠∅, where Δ^𝑛 denotes the standard n-dimensional simplex corresponding, 𝐵={𝑧₀,…,𝑧_𝑛}∈⟨𝑍⟩.

Lemma 2.2 (see [10]). Let 𝑋 and 𝑌 be two topological spaces, and let 𝐺∶𝑋→2^𝑌 be a set-valued mapping. Then the following statements are equivalent: (i)𝐺⁻¹∶𝑌→2^𝑋 is transfer compactly open valued, and for all 𝑥∈𝑋,𝐺(𝑥) is nonempty;(ii)𝑋=⋃_𝑦∈𝑌𝑐int𝐺⁻¹(𝑦).

Theorem 2.3. Let (𝑋,𝜑_𝑁) be an 𝐹𝐶-space, 𝐾 be a nonempty compact subset of 𝑋 and let 𝑆,𝑄∶𝑋→2^𝑋 be set-valued mappings satisfying the following conditions: (i)for each 𝑥∈𝑋 and 𝑁∈⟨𝑆(𝑥)⟩,𝜑_𝑁(Δ_𝑘)⊆𝑄(𝑥);(ii)𝑋=⋃_𝑦∈𝑋𝑐int𝑆⁻¹(𝑦);(iii)there exists a compact 𝐹𝐶-subspace 𝐿_𝑁 of 𝑋 containing 𝑁 such that 𝐿_𝑁⧵𝐾⊆𝑐int𝑆⁻¹(𝑦)∶𝑦∈𝐿_𝑁.(2.1) Then there exists 𝑥∈𝑋 such that 𝑥∈𝑄(𝑥).

Proof. Since 𝐾⊆⋃_𝑦∈𝑋𝑐int𝑆⁻¹(𝑦) and 𝐾 is a nonempty compact subset of 𝑋, there exists a finite set 𝑁={𝑦₁,…,𝑦_𝑛}∈⟨𝑋⟩ such that 𝐾=^𝑛_𝑖=1𝑐int𝑆⁻¹𝑦_𝑖.(2.2) By condition (iii), there exists a compact 𝐹𝐶-subspace 𝐿_𝑁 of 𝑋 containing 𝑁 such that 𝐿_𝑁⧵𝐾⊆𝑐int𝑆⁻¹(𝑦)∶𝑦∈𝐿_𝑁.(2.3) It follows from (2.2) and 𝑁⊂𝐿_𝑁 that 𝐿_𝑁𝐾⊆𝑐int𝑆⁻¹(𝑦)∶𝑦∈𝑁⊆𝑐int𝑆⁻¹(𝑦)∶𝑦∈𝐿_𝑁.(2.4) By (2.3) and (2.4), we have 𝐿_𝑁⊆𝑐int𝑆⁻¹(𝑦)∶𝑦∈𝐿_𝑁.(2.5) Since 𝐿_𝑁 is compact, there exists a finite set 𝑀={𝑧₀,…,𝑧_𝑚}∈⟨𝐿_𝑁⟩ such that 𝐿_𝑁=^𝑚_𝑖=0𝑐int𝑆⁻¹𝑧_𝑖𝐿_𝑁.(2.6) Since 𝐿_𝑁 is also compact 𝐹𝐶-subspace of 𝑋, there exists a continuous mapping 𝜑_𝑀∶Δ_𝑚→𝑋 such that 𝜑_𝑀Δ_𝐽⊂𝐿_𝑁,∀𝐽⊂{0,…,𝑚}.(2.7) We may assume that {𝜓_𝑖}^𝑚𝑖=0 is the continuous partition of unity subordinated to the open covering {𝑐int𝑆⁻¹(𝑧_𝑖)⋂𝐿_𝑁}^𝑚𝑖=0 such that(1)for each 𝑖=0,…,𝑚,𝜓_𝑖∶𝐿_𝑁→[0,1] is continuous;(2)for each 𝑖=0,…,𝑚 and 𝑥∈𝐿_𝑁,𝜓_𝑖(𝑥)≠0⟹𝑥∈𝑐int𝑆⁻¹𝑧_𝑖𝐿_𝑁⊂𝑆⁻¹𝑧_𝑖⟹𝑧_𝑖∈𝑆(𝑥),(2.8)(3) for each 𝑥∈𝐿_𝑁,∑^𝑚𝑖=0𝜓_𝑖(𝑥)=1.
Define a mapping 𝜓∶𝐿_𝑁→Δ_𝑚 by 𝜓(𝑥)=^𝑚_𝑖=0𝜓_𝑖(𝑥)𝑒_𝑖,∀𝑥∈𝐿_𝑁.(2.9) Hence 𝜓 is continuous, and for each 𝑥∈𝐿_𝑁, 𝜓(𝑥)=_{𝑗∈𝐽(𝑥)}𝜓_𝑗(𝑥)𝑒_𝑗∈Δ_𝐽(𝑥),∀𝑥∈𝐿_𝑁,(2.10) where 𝐽(𝑥)={𝑗∈{0,…,𝑚}∶𝜓_𝑗(𝑥)≠0}. It follows from (2.8) that {𝑧_𝑖∶𝑖∈𝐽(𝑥)}⊂𝑆(𝑥). By condition (i) 𝜑_𝑀∘𝜓(𝑥)∈𝜑_𝑀Δ_𝐽(𝑥)⊆𝑄(𝑥),(2.11)𝑁⊂𝐿_𝑁⊂𝑋, and (2.7), we have, 𝜓∘𝜑_𝑀∶Δ_𝑚→2^Δ𝑚 has a fixed point 𝑧∈Δ_𝑚, that is, 𝑧=𝜓∘𝜑_𝑀(𝑧). Let 𝑥=𝜑_𝑀(𝑧). Then 𝑥=𝜑_𝑀(𝑧)=𝜑_𝑀∘𝜓∘𝜑_𝑀(𝑧)=𝜑_𝑀∘𝜓𝑥⊂𝑄𝑥.(2.12) This completes the proof.

Lemma 2.4. Let (𝑋,𝜑_𝑁) and (𝑍,𝜑^∗𝑀) be two 𝐹𝐶-spaces. let 𝑃,𝑇∶𝑋→2^𝑍 be set-valued mappings. If for each 𝑥∈𝑋,𝑃(𝑥) is 𝐹𝐶-subspace of 𝑍 and 𝑇 being weakly-𝐹𝐶 valued, then for each 𝑥∈𝑋,{𝑦∈𝑋∶𝑃(𝑥)⋂𝑇(𝑦)≠∅} is 𝐹𝐶-subspace of 𝑋.

Proof. Let 𝑥∈𝑋,{𝑦₁,…,𝑦_𝑛}∈⟨𝑋⟩ with 𝑃(𝑥)⋂𝑇(𝑦_𝑖)≠∅ for each 𝑖∈{1,…,𝑛}, and 𝑦∈𝜑_𝑁(Δ_𝑛). For each 𝑖∈{1,…,𝑛}, let 𝑧_𝑖∈𝑃(𝑥)⋂𝑇(𝑦_𝑖). It follows from 𝑇 is weakly-𝐹𝐶 valued that 𝑇(𝑦)⋂𝜑^∗𝑀(Δ^𝑛)≠∅. Since 𝑥∈𝑋,𝑃(𝑥) is 𝐹𝐶-subspace of 𝑍, we have ∅≠𝑇(𝑦)⋂𝜑^∗𝑀(Δ^𝑛)⊂𝑇(𝑦)⋂𝑃(𝑥). Hence, for each 𝑥∈𝑋,{𝑦∈𝑋∶𝑃(𝑥)⋂𝑇(𝑦)≠∅} is 𝐹𝐶-subspace of 𝑋. This completes the proof.

Theorem 2.5. Let (𝑋,𝜑_𝑁) be an 𝐹𝐶-space, let 𝑍 be a nonempty set, and, let 𝑃,𝑇∶𝑋→2^𝑍,𝑄∶𝑋→2^𝑋 be three set-valued mappings satisfying the following conditions: (i)for each 𝑥∈𝑋,𝑄(𝑥) is 𝐹𝐶-subspace of 𝑋;(ii)for each 𝑥∈𝑋,{𝑦∈𝑋∶𝑃(𝑥)⋂𝑇(𝑦)≠∅}⊆𝑄(𝑥);(iii)𝑋=⋃_{𝑧∈𝑇(𝑋)}𝑐int𝑃⁻¹(𝑧);(iv)there exists a compact subset 𝐾 of 𝑋 such that for each 𝑁∈⟨𝑋⟩, there exists a compact 𝐹𝐶-subspace 𝐿_𝑁 of 𝑋 containing 𝑁 such that 𝐿_𝑁⧵𝐾⊆_{𝑦∈𝐿𝑁}𝑐int𝑥∈𝑋∶𝑃(𝑥)𝑇(𝑦)≠∅.(2.13) Then there exists 𝑥∈𝑋 such that 𝑥∈𝑄(𝑥).

Proof. Define a mapping 𝑆∶𝑋→2^𝑋 by 𝑆(𝑥)=𝑦∈𝑋∶𝑃(𝑥)𝑇(𝑦)≠∅.(2.14) It is easy to see that for a family {𝐴_𝑖}_𝑖∈𝐼 of subsets of a topological space, ⋃_𝑖∈𝐼𝑐int𝐴_𝑖⊆𝑐int(⋃_𝑖∈𝐼𝐴_𝑖). Thus, we have _𝑦∈𝑋𝑐int𝑆⁻¹(𝑦)=_𝑦∈𝑋𝑐int_{𝑧∈𝑇(𝑦)}𝑃⁻¹(𝑧)⊇_{𝑦∈𝑋𝑧∈}_𝑇(𝑦)𝑐int𝑃⁻¹(𝑧)=_{𝑧∈𝑇(𝑋)}𝑐int𝑃⁻¹(𝑧).(2.15) By condition (iii), it follows that 𝑋=⋃_𝑦∈𝑋𝑐int𝑆⁻¹(𝑦). Clearly condition (iv) of Theorem 2.5 implies condition (iii) of Theorem 2.3. By condition (i) and (ii), for each 𝑥∈𝑋 and 𝑁∈⟨𝑆(𝑥)⟩, we have 𝜑_𝑁(Δ_𝑘)⊆𝑄(𝑥). Then all conditions of Theorem 2.3 are satisfied. By Theorem 2.3, there exists 𝑥∈𝑋 such that 𝑥∈𝑄(𝑥). This completes the proof.

Particularly, when 𝑄(𝑥)={𝑦∈𝑋∶𝑃(𝑥)⋂𝑇(𝑦)≠∅}, Theorem 2.5 reduces to the following result.

Corollary 2.6. Let (𝑋,𝜑_𝑁) be an 𝐹𝐶-space, let 𝑍 be a nonempty set, and let 𝑃,𝑇∶𝑋→2^𝑍 be two set-valued mappings satisfying the following conditions: (i)𝑋=⋃_{𝑧∈𝑇(𝑋)}𝑐int𝑃⁻¹(𝑧);(ii)for each 𝑥∈𝑋,𝑄(𝑥) is 𝐹𝐶-subspace of 𝑋;(iii)there exists a compact subset 𝐾 of 𝑋 such that for each 𝑁∈⟨𝑋⟩, there exists a compact 𝐹𝐶-subspace 𝐿_𝑁 of 𝑋 containing 𝑁 such that 𝐿_𝑁⧵𝐾⊆_{𝑦∈𝐿𝑁}𝑐int𝑥∈𝑋∶𝑃(𝑥)𝑇(𝑦)≠∅.(2.16) Then there exists 𝑥∈𝑋 such that 𝑃(𝑥)⋂𝑇(𝑥)≠∅.

Remark 2.7. Theorem 2.3 generalizes Theorem 4 in [14]. Theorem 2.5 generalizes Theorem 1 in [2]. Corollary 2.6 generalizes Theorem 8 in [12], Theorem 3.6 in [13], and Theorem 2 in [2].

Corollary 2.8. Let (𝑋,𝜑_𝑁) be an 𝐹𝐶-space, 𝑍 be a nonempty set and 𝑃,𝑇∶𝑋→2^𝑍 be two set-valued mappings satisfying the following conditions: (i)𝑃⁻¹ is transfer compactly open valued;(ii)for each 𝑥∈𝑋,{𝑦∈𝑋∶𝑃(𝑥)⋂𝑇(𝑦)≠∅} is 𝐹𝐶-subspace of 𝑋;(iii)there exists a compact subset 𝐾 of 𝑋 such that for each 𝑁∈⟨𝑋⟩, there exists a compact 𝐹𝐶-subspace 𝐿_𝑁 of 𝑋 containing 𝑁 such that 𝐿_𝑁⧵𝐾⊆_{𝑦∈𝐿𝑁}𝑐int𝑥∈𝑋∶𝑃(𝑥)𝑇(𝑦)≠∅,(2.17)(iv)for each 𝑥∈𝑋,𝑃(𝑥)⋂𝑇(𝑥)=∅.
Then at least one of the following assertions holds: (a)there exists 𝑥∈𝑋 such that 𝑃(𝑥)=∅,(b)there exists 𝑧∈𝑍 such that 𝑇⁻¹(𝑧)=∅.

Proof. By means of contradiction, suppose that for each 𝑥∈𝑋,𝑃(𝑥)≠∅, and for each 𝑧∈𝑍,𝑇⁻¹(𝑧)≠∅ (that is, 𝑇(𝑋)=𝑍). Then, by Lemma 2.2, we have 𝑋=_𝑧∈𝑍𝑐int𝑃⁻¹(𝑧)=_{𝑧∈𝑇(𝑋)}𝑐int𝑃⁻¹(𝑧).(2.18) Then all conditions of Corollary 2.6 are satisfied, and it follows that there exists 𝑥∈𝑋 such that 𝑃(𝑥)⋂𝑇(𝑥)≠∅, but this contradicts condition (iv). This completes the proof.

Corollary 2.9. In Corollary 2.8, if one assumes further that 𝑇(𝑋)=𝑍, then there exists 𝑥∈𝑋 such that 𝑃(𝑥)=∅.

Corollary 2.10. Let (𝑋,𝜑_𝑁) be an 𝐹𝐶-space, and let 𝑃∶𝑋→2^𝑋 be a set-valued mapping satisfying the following conditions: (i)𝑃⁻¹ is transfer compactly open valued;(ii)for each 𝑥∈𝑋,𝑃(𝑥) is 𝐹𝐶-subspace of 𝑋;(iii)for each 𝑥∈𝑋,𝑥∉𝑃(𝑥);(iv)there exists a compact subset 𝐾 of 𝑋 such that for each 𝑁∈⟨𝑋⟩, there exists a compact 𝐹𝐶-subspace 𝐿_𝑁 of 𝑋 containing 𝑁 such that 𝐿_𝑁⧵𝐾⊆_{𝑦∈𝐿𝑁}𝑐int𝑃⁻¹(𝑦).(2.19) Then there exists 𝑥∈𝑋 such that 𝑃(𝑥)=∅.

Proof. Define a mapping 𝑇∶𝑋→2^𝑋 by 𝑇(𝑥)={𝑥} for all 𝑥∈𝑋. Then Corollary 2.10 follows from Corollary 2.8.

Remark 2.11. Corollary 2.8 generalizes Theorem 3 in [2]. Corollary 2.9 and Corollary 2.10 improve Corollary 1 and Corollary 2 in [2], respectively. If the 𝐹𝐶-space 𝑋 is compact, condition (iii) in Theorem 2.3 is satisfied trivially. Consequently, condition (iv) of Theorem 2.5, condition (iv) of Corollary 2.10, condition (iii) of Corollary 2.6, condition (iii) of Corollary 2.8 and condition (iii) of Corollary 2.9 are all satisfied trivially.

Corollary 2.12. Let (𝑋,𝜑_𝑁) and (𝑍,𝜑^∗𝑀) be two 𝐹𝐶-spaces and let 𝑃,𝑇∶𝑋→2^𝑍 be two set-valued mappings satisfying conditions (i) and (iii) in Corollary 2.6. If for each 𝑥∈𝑋,𝑃(𝑥) is 𝐹𝐶-subspace of 𝑍 and 𝑇 is weakly-𝐹𝐶 valued, then there exists 𝑥∈𝑋 such that 𝑃(𝑥)⋂𝑇(𝑥)≠∅.

Proof. By Lemma 2.4 and Corollary 2.6, we have that the conclusion of Corollary 2.12 holds.

Corollary 2.13. Let (𝑋,𝜑_𝑁) and (𝑍,𝜑^∗𝑀) be two 𝐹𝐶-spaces and 𝑃,𝑇∶𝑋→2^𝑍 be set-valued mappings satisfying conditions (i), (iii) and (iv) in Corollary 2.8. If for each 𝑥∈𝑋,𝑃(𝑥) is 𝐹𝐶-subspace of 𝑍 and 𝑇 is weakly-𝐹𝐶 valued, then at least one of the following assertions holds: (a)there exists 𝑥∈𝑋 such that 𝑃(𝑥)=∅;(b)there exists 𝑧∈𝑍 such that 𝑇⁻¹(𝑧)=∅.

Proof. By Lemma 2.4 and Corollary 2.8, we have that the conclusion of Corollary 2.13 holds.

Remark 2.14. Corollary 2.12 generalizes Theorem 4 in [2]. Corollary 2.13 generalizes Theorem 5 in [2].

3. Generalized Equilibrium Theorems and Minimax Inequality

Inspired by Balaj and Lin [2], we have the following concept.

Definition 3.1. Let (𝑋,𝜑_𝑁) be an 𝐹𝐶-space, let 𝑍 and 𝑉 be nonempty sets, let 𝜌 be a relation on 2^𝑉, and let 𝐹∶𝑋×𝑍→2^𝑉,𝐶∶𝑋→2^𝑉 be set-valued mappings. We say that 𝐹 is (𝐶,𝜌)-transfer compactly continuous in the first variable if for any 𝑥∈𝑋,𝑧∈{𝑢∈𝑍∶𝐹(𝑥,𝑢)𝜌𝐶(𝑥)}, there exists 𝑥∈𝑋 such that 𝑧∈𝑐int{𝑢∈𝑍∶𝐹(𝑥,𝑢)𝜌𝐶(𝑥)}.

The following concept generalizes the corresponding notion in Ding [6, 8].

Definition 3.2. Let (𝑋,𝜑_𝑁) be an 𝐹𝐶-space and 𝑌 be topological space. For each 𝑦∈𝑌,𝑓∶𝑌×𝑋→𝑅 is said to be 𝐹𝐶-quasiconvex (resp., 𝐹𝐶-quasiconcave) in the first variable, if for each 𝜆∈𝑅, the set {𝑥∈𝑋∶𝑓(𝑦,𝑥)<𝜆} (resp., {𝑥∈𝑋∶𝑓(𝑥,𝑦)>𝜆}) is an 𝐹𝐶-subspace of 𝑋.

The following concept generalizes the corresponding of notion in Chen and Chang [5].

Definition 3.3. Let 𝑋 and 𝑌 be two topological spaces, and let 𝑓∶𝑌×𝑋→𝑅∪{−∞,+∞} be a function. Then 𝑓 is said to be 𝛼-transfer compactly lower semicontinuous (in short, 𝛼-transfer compactly l.s.c) in the first variable if for each 𝑦∈𝑌 with 𝑥∈{𝑢∈𝑋∶𝑓(𝑦,𝑢)>𝛼}, there exists 𝑦∈𝑌 such that 𝑥∈𝑐int{𝑢∈𝑋∶𝑓(𝑦,𝑢)>𝛼}. 𝑓 is said to be 𝛼-transfer compactly u.s.c in the first variable if and only if –𝑓 is 𝛼-transfer compactly l.s.c in the first variable. If 𝑓 is 𝛼-transfer compactly l.s.c in the first variable for each 𝛼∈𝑅, we say that 𝑓 is transfer compactly l.s.c in the first variable.

Theorem 3.4. Let (𝑋,𝜑_𝑁) be an 𝐹𝐶-space, let 𝑍 and 𝑉 be nonempty sets, and let 𝜌 be a relation on 2^𝑉. Let 𝐹∶𝑋×𝑍→2^𝑉,𝐶∶𝑋→2^𝑉, and 𝑇∶𝑋→2^𝑍 be set-valued mappings satisfying the following conditions: (i)𝑇(𝑋)=𝑍;(ii)𝐹 is (𝐶,𝜌^𝑐)-transfer compactly continuous in the first variable;(iii) there exists a compact subset 𝐾 of 𝑋 such that for each 𝑁∈⟨𝑋⟩, there exists a compact 𝐹𝐶-subspace 𝐿_𝑁 of 𝑋 containing 𝑁 such that 𝐿_𝑁_{𝑦∈𝐿𝑁}𝑐cl{𝑥∈𝑋∶𝐹(𝑥,𝑧)𝜌𝐶(𝑥),∀𝑧∈𝑇(𝑦)}⊆𝐾,(3.1)(iv) for each 𝑥∈𝑋, the set {𝑦∈𝑋∶𝐹(𝑥,𝑧)𝜌^𝑐𝐶(𝑥),forsome𝑧∈𝑇(𝑦)} is 𝐹𝐶-subspace of 𝑋;(v) for all 𝑥∈𝑋 and any 𝑧∈𝑇(𝑥),𝐹(𝑥,𝑧)𝜌𝐶(𝑥).Then there exists 𝑥∈𝑋 such that 𝐹(𝑥,𝑧)𝜌𝐶(𝑥) for all 𝑧∈𝑍.

Proof. Let 𝑃∶𝑋×𝑍→2^𝑍 be the set-valued mapping defined by 𝑃(𝑥)={𝑧∈𝑍∶𝐹(𝑥,𝑧)𝜌^𝑐𝐶(𝑥)}.(3.2) We prove that the mappings satisfy the conditions of Theorem 2.5. It is clear that condition (ii) is equivalent to the fact that the mapping 𝑃⁻¹ is transfer compactly open valued. By condition (iv), for each 𝑥∈𝑋 the set {𝑦∈𝑋∶𝑃(𝑥)⋂𝑇(𝑦)≠∅} is 𝐹𝐶-subspace of 𝑋. Let 𝑁∈⟨𝑋⟩ and for all 𝑥∈𝐿_𝑁⧵𝐾. Then, by condition (iii) there exists 𝑦∈𝐿_𝑁 such that 𝑥∈𝑋=⧵𝑐i𝑐nctl𝑥𝑥^_∈𝑋∶𝐹𝑥^,𝑧𝜌𝐶𝑥^,∀𝑧∈𝑇(𝑦)=𝑐int𝑥_∈𝑋∶𝐹𝑥^,𝑧𝜌^𝑐𝐶𝑥^,forsome𝑧∈𝑇(𝑦)∈𝑋∶𝑃𝑥^𝑇(𝑦)≠∅.(3.3)
Thus all the conditions of Theorem 2.5 are fulfilled, and by condition (i), 𝑇⁻¹(𝑧)≠∅ for all z∈𝑍. Hence, there exists 𝑥∈𝑋 such that 𝑃(𝑥)=∅, that is, 𝐹(𝑥,𝑧)𝜌𝐶(𝑥) for all 𝑧∈𝑍. This completes the proof.

Remark 3.5. Theorem 3.4 generalizes Theorem 6 in [2] in the following several aspects: (a) from 𝐺-convex space to 𝐹𝐶-space without linear structure; (b) from (𝐶,𝜌^𝑐)-transfer continuous to (𝐶,𝜌^𝑐)-transfer compactly continuous; (c) conditions (iii) and (iv) of Theorem 3.4 are weaker than conditions (iii) and (iv) of Theorem 6 in [2].

Corollary 3.6. Let (𝑋,𝜑_𝑁) and (𝑍,𝜑^∗𝑀) be two 𝐹𝐶-spaces, let 𝑉 be nonempty set, and let 𝜌 be a relation on 2^𝑉. Let 𝐹∶𝑋×𝑍→2^𝑉,𝐶∶𝑋→2^𝑉, and 𝑇∶𝑋→2^𝑍 be set-valued mappings satisfying the following conditions: 𝑃,𝑇∶𝑋→2^𝑍 are set-valued mappings satisfying conditions (i), (ii), (iii), and (v) in Theorem 3.4 and if for each 𝑥∈𝑋,{𝑧∈𝑍∶𝐹(𝑥,𝑧)𝜌^𝑐𝐶(𝑥)} is 𝐹𝐶-subspace of 𝑍 and 𝑇 is weakly-𝐹𝐶 valued, then there exists 𝑥∈𝑋 such that 𝐹(𝑥,𝑧)𝜌𝐶(𝑥) for all 𝑧∈𝑍.

Proof. By Lemma 2.4 and Corollary 2.12, we have that the conclusion of Corollary 3.6 holds.

Remark 3.7. Corollary 3.6 generalizes Theorem 7 in [2] in the following several aspects: (a) from 𝐺-convex space to 𝐹𝐶-space without linear structure; (b) from (𝐶,𝜌^𝑐)-transfer continuous to (𝐶,𝜌^𝑐)-transfer compactly continuous; (c) from 𝐺-weakly convex to weakly-𝐹𝐶 valued; (c) conditions (iii) and “for each 𝑥∈𝑋,{𝑧∈𝑍∶𝐹(𝑥,𝑧)𝜌^𝑐𝐶(𝑥)} is 𝐹𝐶-subspace of 𝑍” of Corollary 3.6 are weaker than conditions (iii) and (vii) of Theorem 7 in [2].

Theorem 3.8. Let (𝑋,𝜑_𝑁) be a compact 𝐹𝐶-space, let 𝑍 be nonempty set, let 𝑓,𝑔∶𝑍×𝑋→𝑅,ℎ∶𝑋×𝑋→𝑅 be three functions, and let 𝛼,𝛽,𝛾 be three real numbers satisfying the following conditions: (i)for each 𝑥∈𝑋, the set {𝑦∈𝑋∶ℎ(𝑥,𝑦)<𝛾} is 𝐹𝐶-subspace of 𝑋;(ii)𝑓 is 𝛼-transfer compactly u.s.c in the first variable;(iii)for each 𝑥,𝑦∈𝑋,𝑧∈𝑍, 𝑓(𝑧,𝑥)<𝛼 and 𝑔(𝑧,𝑦)<𝛽⇒ℎ(𝑥,𝑦)<𝛾;(iv)ℎ(𝑥,𝑥)≥𝛾 for all 𝑥∈𝑋.Then, at least one of the following assertions holds: (a)there exists 𝑥∈𝑋 such that 𝑓(𝑧,𝑥)≥𝛼 for all 𝑧∈𝑍;(b)there exists 𝑧∈𝑍 such that 𝑔(𝑧,𝑥)≥𝛽 for all 𝑥∈𝑋.

Proof. Let 𝑃,𝑇∶𝑋→2^𝑍,𝑄∶𝑋→2^𝑋 be the set-valued mappings defined by 𝑃𝑇𝑄(((𝑥𝑥𝑥)))==={{{𝑧𝑧𝑦∈∈∈𝑍𝑍𝑋∶∶∶𝑓𝑔ℎ(((𝑧𝑧𝑥,,,𝑥𝑥𝑦)))<<<𝛽𝛼𝛾}}},,(3.4) for all 𝑥∈𝑋. By means of contradiction, suppose that both assertions (a) and (b) would be false; that is(1)for each 𝑥∈𝑋, there exists 𝑧∈𝑍 such that 𝑓(𝑧,𝑥)<𝛼,(2)for each 𝑧∈𝑍, there exists 𝑥∈𝑋 such that 𝑔(𝑧,𝑥)<𝛽.
By (2), 𝑇(𝑋)=𝑍. Arbitrarily choose 𝑥∈𝑋. By (1) and condition (ii), it follows that 𝑥∈{𝑢∈𝑋∶𝑓(𝑧,𝑥)<𝛼}=𝑃⁻¹(𝑧), and there exists 𝑧∈𝑍 such that 𝑥∈𝑐int𝑃⁻¹(𝑧). Hence, we infer that 𝑋=⋃_𝑧∈𝑍𝑐int𝑃⁻¹(𝑧)=⋃_{𝑧∈𝑇(𝑋)}𝑐int𝑃⁻¹(𝑧). By condition (i), 𝑄(𝑥) is 𝐹C-subspace of 𝑋, and by condition (iii), {𝑦∈𝑋∶𝑃(𝑥)⋂𝑇(𝑦)≠∅}⊆𝑄(𝑥) for each 𝑥∈𝑋. Taking into account Remark 2.11, all conditions of Theorem 2.3 are satisfied, and it follows that there exists 𝑥∈𝑋 such that 𝑥∈𝑄(𝑥). Hence, we have ℎ(𝑥,𝑥)<𝛾, which contradicts condition (iv). This completes the proof.

Remark 3.9. Theorem 3.8 generalizes Theorem 12 in [2] in the following several aspects: (a) from 𝐺-convex space to 𝐹𝐶-space without linear structure; (b) from 𝛼-transfer u.s.c to 𝛼-transfer compactly u.s.c; (c) conditions (i) of Theorem 3.8 are weaker than conditions (i) of Theorem 12 in [2].

From Theorem 3.8, we may obtain the following minimax inequality.

Corollary 3.10. Let (𝑋,𝜑_𝑁) be a compact 𝐹𝐶-space, let 𝑍 be nonempty set, and let 𝑓,𝑔∶𝑍×𝑋→𝑅,ℎ∶𝑋×𝑋→𝑅 be three functions satisfying the following conditions: (i)ℎ(𝑥,⋅) is 𝐹𝐶-quasiconvex, for each 𝑥∈𝑋;(ii)𝑓 is transfer compactly u.s.c in the first variable;(iii)for each 𝑥,𝑦∈𝑋,𝑧∈𝑍, ℎ(𝑥,𝑦)≤𝑓(𝑧,𝑥)+𝑔(𝑧,𝑦).Then inf_𝑥∈𝑋ℎ(𝑥,𝑥)≤sup_𝑥∈𝑋inf_𝑧∈𝑍𝑓(𝑧,𝑥)+sup_𝑧∈𝑍inf_𝑥∈𝑋𝑔(𝑧,𝑥),(3.5) with the convention ∞+(−∞)=∞.

Proof. We may suppose that inf_𝑥∈𝑋ℎ(𝑥,𝑥)>−∞,sup_𝑥∈𝑋inf_𝑧∈𝑍𝑓(𝑧,𝑥)<∞,sup_𝑧∈𝑍inf_𝑥∈𝑋𝑔(𝑧,𝑥)<∞.(3.6) By means of contradiction, suppose that inf_𝑥∈𝑋ℎ(𝑥,𝑥)>sup_𝑥∈𝑋inf_𝑧∈𝑍𝑓(𝑧,𝑥)+sup_𝑧∈𝑍inf_𝑥∈𝑋𝑔(𝑧,𝑥)(3.7) and choose 𝛼,𝛽,𝛾∈𝑅 such that sup_𝑥∈𝑋inf_𝑧∈𝑍𝑓(𝑧,𝑥)<𝛼,sup_𝑧∈𝑍inf_𝑥∈𝑋𝑔(𝑧,𝑥)<𝛽,𝛾<inf_𝑥∈𝑋ℎ(𝑥,𝑥),𝛼+𝛽<𝛾.(3.8)
It is easy to see that functions 𝑓,𝑔, and ℎ satisfy all the conditions of Theorem 3.8. We prove that neither assertion (a) nor (b) of the conclusion of Theorem 3.8 can take place.If (a) happens, then𝛼≤inf_𝑧∈𝑍𝑓𝑧,𝑥≤sup_𝑥∈𝑋inf_𝑧∈𝑍𝑓(𝑧,𝑥),(3.9) a contradiction.If (b) happens, thensup_𝑧∈𝑍inf_𝑥∈𝑋𝑔(𝑧,𝑥)≥inf_𝑥∈𝑋𝑔𝑧,𝑥≥𝛽,(3.10) a contradiction again.
This completes the proof.

Remark 3.11. Corollary 3.10 generalizes Theorem 13 in [2] in the following several aspects: (a) from 𝐺-convex space to 𝐹𝐶-space without linear structure; (b) from 𝐺-quasiconvex to 𝐹𝐶-quasiconvex; (c) from transfer u.s.c to transfer compactly u.s.c.

Acknowledgment

This work is supported by the Scientific Research Foundation of CUIT under Grant KYTZ201114.