# Fixed Points and Existence Theorems of Maximal Elements with Applications inFC-Spaces

Authors: Rong-Hua He
Journal: ISRN Applied Mathematics
 (2011)

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## Abstract

We present some fixed point theorems and existence theorems of maximal elements inFC-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.


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## Body

## 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 inFC-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].LetY be a nonempty set. We denote by 2Y 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 Δn be the standard n-dimensional simplex with vertices {e0,…,en}. If J is a nonempty subset of {0,1,…,n}, we denote by ΔJ the convex hull of the vertices {ej:j∈J}.LetX and Y be two sets, and, T:X→2Y 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 spacesX 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 (1.1)ccl⁡A=⋂{B⊂X:A⊂BandBis compactly closed inX},cint⁡A=⋃{B⊂X:B⊂AandBis compactly open inX}.It is easy to see thatccl(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=clK(A⋂K) and cintA⋂K=intK(A⋂K), where clK(A⋂K) (resp., intK(A⋂K)) denotes the closure (resp., interior) of A⋂K in K.A set-valued mappingT:X→2Y 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,φN) is said to be an FC-space if X is a topological space, and for each N={x0,…,xn}∈〈X〉, where some elements in N may be the same, there exists a continuous mapping φN:Δn→X. A subset D of (X,φN) is said to be an FC-subspace of X if for each N={x0,…,xn}∈〈X〉 and for each {xi0,…,xik}⊂N⋂D, φN(Δk)⊂D, where Δk=co({eij: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
LetX be a convex subset of a topological vector space, let Z and, V be nonempty sets and let F:X×Z→2V,C:X→2V 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) relationρ on 2V and looking for a point x0∈X such that F(x0,z)ρC(x0) for all z∈Z.Denote byρc the complementary relation of ρ, that is, for any A,B⊂V, exactly one of the following assertions AρB,AρcB 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: (1.2)inf⁡x∈Xh(x,x)≤sup⁡x∈Xinf⁡z∈Zf(x,z)+sup⁡z∈Zinf⁡x∈Xg(x,z),
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,φN) and (Z,φM*) be two FC-spaces. A set-valued mapping T:X→2Z is said to be weakly-FC valued if for each A={x0,…,xn}∈〈X〉,x∈φN(Δn) and B={z0,…,zn}∈〈Z〉 with zi∈T(xi)(i=0,…,n), we have T(x)⋂φM*(Δn′)≠∅, where Δn′ denotes the standard n-dimensional simplex corresponding, B={z0,…,zn}∈〈Z〉.Lemma 2.2 (see [10]).
LetX and Y be two topological spaces, and let G:X→2Y be a set-valued mapping. Then the following statements are equivalent:(i)
G-1:Y→2X 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,φN) be an FC-space, K be a nonempty compact subset of X and let S,Q:X→2X be set-valued mappings satisfying the following conditions:(i)
for eachx∈X and N∈〈S(x)〉,φN(Δk)⊆Q(x);(ii)
X=⋃y∈XcintS-1(y);(iii)
there exists a compactFC-subspace LN of X containing N such that(2.1)LN∖K⊆⋃{cint⁡S-1(y):y∈LN}.
Then there exists x¯∈X such that x¯∈Q(x¯).Proof.
SinceK⊆⋃y∈XcintS-1(y) and K is a nonempty compact subset of X, there exists a finite set N={y1,…,yn}∈〈X〉 such that
(2.2)K=⋃i=1ncint⁡S-1(yi).
By condition (iii), there exists a compact FC-subspace LN of X containing N such that
(2.3)LN∖K⊆⋃{cint⁡S-1(y):y∈LN}.
It follows from (2.2) and N⊂LN that
(2.4)LN⋂K⊆⋃{cint⁡S-1(y):y∈N}⊆⋃{cint⁡S-1(y):y∈LN}.
By (2.3) and (2.4), we have
(2.5)LN⊆⋃{cint⁡S-1(y):y∈LN}.
Since LN is compact, there exists a finite set M={z0,…,zm}∈〈LN〉 such that
(2.6)LN=⋃i=0m(cint⁡S-1(zi)⋂LN).
Since LN is also compact FC-subspace of X, there exists a continuous mapping φM:Δm→X such that
(2.7)φM(ΔJ)⊂LN,∀J⊂{0,…,m}.
We may assume that {ψi}i=0m is the continuous partition of unity subordinated to the open covering {cintS-1(zi)⋂LN}i=0m such that(1)
for eachi=0,…,m,ψi:LN→[0,1] is continuous;(2)
for eachi=0,…,m and x∈LN,(2.8)ψi(x)≠0⟹x∈cint⁡S-1(zi)⋂LN⊂S-1(zi)⟹(zi)∈S(x),(3)
for eachx∈LN,∑i=0mψi(x)=1.
Define a mappingψ:LN→Δm by
(2.9)ψ(x)=∑i=0mψi(x)ei,∀x∈LN.
Hence ψ is continuous, and for each x∈LN,
(2.10)ψ(x)=∑j∈J(x)ψj(x)ej∈ΔJ(x),∀x∈LN,
where J(x)={j∈{0,…,m}:ψj(x)≠0}. It follows from (2.8) that {zi:i∈J(x)}⊂S(x). By condition (i)
(2.11)φM∘ψ(x)∈φM(ΔJ(x))⊆Q(x),N⊂LN⊂X, and (2.7), we have, ψ∘φM:Δm→2Δm has a fixed point z∈Δm, that is, z=ψ∘φM(z). Let x¯=φM(z). Then
(2.12)x¯=φM(z)=φM∘(ψ∘φM(z))=φM∘ψ(x¯)⊂Q(x¯).
This completes the proof.Lemma 2.4.
Let(X,φN) and (Z,φM*) be two FC-spaces. let P,T:X→2Z 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.
Letx∈X,{y1,…,yn}∈〈X〉 with P(x)⋂T(yi)≠∅ for each i∈{1,…,n}, and y∈φN(Δn). For each i∈{1,…,n}, let zi∈P(x)⋂T(yi). It follows from T is weakly-FC valued that T(y)⋂φM*(Δn′)≠∅. Since x∈X,P(x) is FC-subspace of Z, we have ∅≠T(y)⋂φM*(Δn′)⊂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,φN) be an FC-space, let Z be a nonempty set, and, let P,T:X→2Z,Q:X→2X be three set-valued mappings satisfying the following conditions:(i)
for eachx∈X,Q(x) is FC-subspace of X;(ii)
for eachx∈X,{y∈X:P(x)⋂T(y)≠∅}⊆Q(x);(iii)
X=⋃z∈T(X)cintP-1(z);(iv)
there exists a compact subsetK of X such that for each N∈〈X〉, there exists a compact FC-subspace LN of X containing N such that(2.13)LN∖K⊆⋃y∈LNcint⁡{x∈X:P(x)⋂T(y)≠∅}.
Then there exists x¯∈X such that x¯∈Q(x¯).Proof.
Define a mappingS:X→2X by
(2.14)S(x)={y∈X:P(x)⋂T(y)≠∅}.
It is easy to see that for a family {Ai}i∈I of subsets of a topological space, ⋃i∈IcintAi⊆cint(⋃i∈IAi). Thus, we have
(2.15)⋃y∈Xcint⁡S-1(y)=⋃y∈Xcint⁡(⋃z∈T(y)P-1(z))⊇⋃y∈X⋃z∈T(y)cint⁡P-1(z)=⋃z∈T(X)cint⁡P-1(z).
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 φN(Δk)⊆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, whenQ(x)={y∈X:P(x)⋂T(y)≠∅}, Theorem 2.5 reduces to the following result.Corollary 2.6.
Let(X,φN) be an FC-space, let Z be a nonempty set, and let P,T:X→2Z be two set-valued mappings satisfying the following conditions:(i)
X=⋃z∈T(X)cintP-1(z);(ii)
for eachx∈X,Q(x) is FC-subspace of X;(iii)
there exists a compact subsetK of X such that for each N∈〈X〉, there exists a compact FC-subspace LN of X containing N such that(2.16)LN∖K⊆⋃y∈LNcint⁡{x∈X:P(x)⋂T(y)≠∅}.
Then there exists x¯∈X such that P(x¯)⋂T(x¯)≠∅.Remark 2.7.
Theorem2.3 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,φN) be an FC-space, Z be a nonempty set and P,T:X→2Z be two set-valued mappings satisfying the following conditions:(i)
P-1 is transfer compactly open valued;(ii)
for eachx∈X,{y∈X:P(x)⋂T(y)≠∅} is FC-subspace of X;(iii)
there exists a compact subsetK of X such that for each N∈〈X〉, there exists a compact FC-subspace LN of X containing N such that(2.17)LN∖K⊆⋃y∈LNcint⁡{x∈X:P(x)⋂T(y)≠∅},(iv)
for eachx∈X,P(x)⋂T(x)=∅.
Then at least one of the following assertions holds:(a)
there existsx¯∈X such that P(x¯)=∅,(b)
there existsz¯∈Z such that T-1(z¯)=∅.Proof.
By means of contradiction, suppose that for eachx∈X,P(x)≠∅, and for each z∈Z,T-1(z)≠∅ (that is, T(X)=Z). Then, by Lemma 2.2, we have
(2.18)X=⋃z∈Zcint⁡P-1(z)=⋃z∈T(X)cint⁡P-1(z).
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 Corollary2.8, if one assumes further that T(X)=Z, then there exists x¯∈X such that P(x¯)=∅.Corollary 2.10.
Let(X,φN) be an FC-space, and let P:X→2X be a set-valued mapping satisfying the following conditions:(i)
P-1 is transfer compactly open valued;(ii)
for eachx∈X,P(x) is FC-subspace of X;(iii)
for eachx∈X,x∉P(x);(iv)
there exists a compact subsetK of X such that for each N∈〈X〉, there exists a compact FC-subspace LN of X containing N such that(2.19)LN∖K⊆⋃y∈LNcint⁡P-1(y).
Then there exists x¯∈X such that P(x¯)=∅.Proof.
Define a mappingT:X→2X by T(x)={x} for all x∈X. Then Corollary 2.10 follows from Corollary 2.8.Remark 2.11.
Corollary2.8 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,φN) and (Z,φM*) be two FC-spaces and let P,T:X→2Z 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 Lemma2.4 and Corollary 2.6, we have that the conclusion of Corollary 2.12 holds.Corollary 2.13.
Let(X,φN) and (Z,φM*) be two FC-spaces and P,T:X→2Z 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 existsx¯∈X such that P(x¯)=∅;(b)
there existsz¯∈Z such that T-1(z¯)=∅.Proof.
By Lemma2.4 and Corollary 2.8, we have that the conclusion of Corollary 2.13 holds.Remark 2.14.
Corollary2.12 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,φN) be an FC-space, let Z and V be nonempty sets, let ρ be a relation on 2V, and let F:X×Z→2V,C:X→2V 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,φN) 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.
LetX 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,φN) be an FC-space, let Z and V be nonempty sets, and let ρ be a relation on 2V. Let F:X×Z→2V,C:X→2V, and T:X→2Z be set-valued mappings satisfying the following conditions:(i)
T(X)=Z;(ii)
F is (C,ρc)-transfer compactly continuous in the first variable;(iii)
there exists a compact subsetK of X such that for each N∈〈X〉, there exists a compact FC-subspace LN of X containing N such that(3.1)LN⋂⋂y∈LNccl⁡{x∈X:F(x,z)ρC(x),∀z∈T(y)}⊆K,(iv)
for eachx∈X, the set {y∈X:F(x,z)ρcC(x),for somez∈T(y)} is FC-subspace of X;(v)
for allx∈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.
LetP:X×Z→2Z be the set-valued mapping defined by
(3.2)P(x)={z∈Z:F(x,z)ρcC(x)}.
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∈LN∖K. Then, by condition (iii) there exists y∈LN such that
(3.3)x∈X∖ccl⁡{x′∈X:F(x′,z)ρC(x′),∀z∈T(y)}=cint⁡{x′∈X:F(x′,z)ρcC(x′),for somez∈T(y)}=cint⁡{x′∈X:P(x′)⋂T(y)≠∅}.
Thus all the conditions of Theorem2.5 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.
Theorem3.4 generalizes Theorem 6 in [2] in the following several aspects: (a) from G-convex space to FC-space without linear structure; (b) from (C,ρc)-transfer continuous to (C,ρc)-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,φN) and (Z,φM*) be two FC-spaces, let V be nonempty set, and let ρ be a relation on 2V. Let F:X×Z→2V,C:X→2V, and T:X→2Z be set-valued mappings satisfying the following conditions: P,T:X→2Z 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)ρcC(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 Lemma2.4 and  Corollary 2.12, we have that the conclusion of Corollary 3.6 holds.Remark 3.7.
Corollary3.6 generalizes Theorem 7 in [2] in the following several aspects: (a) from G-convex space to FC-space without linear structure; (b) from (C,ρc)-transfer continuous to (C,ρc)-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)ρcC(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,φN) 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 eachx∈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 eachx,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 existsx¯∈X such that f(z,x¯)≥α for all z∈Z;(b)
there existsz¯∈Z such that g(z¯,x)≥β for all x∈X.Proof.
LetP,T:X→2Z,Q:X→2X be the set-valued mappings defined by
(3.4)P(x)={z∈Z:f(z,x)<α},T(x)={z∈Z:g(z,x)<β},Q(x)={y∈X:h(x,y)<γ}
for all x∈X. By means of contradiction, suppose that both assertions (a) and (b) would be false; that is(1)
for eachx∈X, there exists z∈Z such that f(z,x)<α,(2)
for eachz∈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.
Theorem3.8 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 Theorem3.8, we may obtain the following minimax inequality.Corollary 3.10.
Let(X,φN) 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 eachx,y∈X,z∈Z, h(x,y)≤f(z,x)+g(z,y).Then
(3.5)inf⁡x∈Xh(x,x)≤sup⁡x∈Xinf⁡z∈Zf(z,x)+sup⁡z∈Zinf⁡x∈Xg(z,x),
with the convention ∞+(-∞)=∞.Proof.
We may suppose that(3.6)inf⁡x∈Xh(x,x)>-∞,sup⁡x∈Xinf⁡z∈Zf(z,x)<∞,sup⁡z∈Zinf⁡x∈Xg(z,x)<∞.
By means of contradiction, suppose that
(3.7)inf⁡x∈Xh(x,x)>sup⁡x∈Xinf⁡z∈Zf(z,x)+sup⁡z∈Zinf⁡x∈Xg(z,x)
and choose α,β,γ∈R such that
(3.8)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.
It is easy to  see that functionsf,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, then(3.9)α≤inf⁡z∈Zf(z,x¯)≤sup⁡x∈Xinf⁡z∈Zf(z,x),
a contradiction.If (b) happens, then(3.10)sup⁡z∈Zinf⁡x∈Xg(z,x)≥inf⁡x∈Xg(z¯,x)≥β,
a contradiction again.
This completes the proof.Remark 3.11.
Corollary3.10 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.



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*Source: 673591-2011-07-06.xml*