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

Authors: Yi-An Chen; Yi-Ping Zhang
Journal: Abstract and Applied Analysis
 (2010)

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## 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.


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## Body

## 1. Introduction

LetX and Y be nonempty sets, let T:X→Y be a single-valued mapping, let A:X→2X 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 x̂∈X such that (1.1)x̂∈A(x̂),f(x̂,T(x̂))≤f(y,T(x̂)),∀y∈A(x̂).
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 ℝn and topological vector spaces, respectively.The quasi-equilibrium problemQEP(A,ϕ) is to find x̂∈X such that (1.2)x̂∈A(x̂),ϕ(y,x̂)≥0,∀y∈A(x̂).
The QEP(A,ϕ) has been studied by many authors; for example, see [4–8] and others.TheQEP(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 theQEP(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 setX, we will denote by 2X 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 Δn denote the standard n-dimensional simplex with vertices {e0,e1,…,en}. If J is a nonempty subset of {0,1,…,n}, we denote by ΔJ the convex hull of the vertices {ej:j∈J}. A set-valued mapping F:X→2Y is a function from a set X into the power set 2Y 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 subsetA 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), as(2.1)ccl(A)=⋂{B⊂X:A⊂BandBiscompactlyclosedinX},cint(A)=⋂{B⊂X:B⊂AandBiscompactlyopeninX}.
If A is a subset of a vector space, then co(A) denotes the convex hull of A.IfX and Y are two topological spaces and G:X→2Y 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∈intK(G(x′)∩K) (resp., y∉clK(G(x′)∩K)). A set-valued mapping G:X→2Y 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)∩KG(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∈intY(Z).In [12], Deng and Xia introduced the following concept which is crucial to the study of KKM theory in general topological spaces.LetX be a nonempty set and let Y be a topological space. W:X→2Y 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 φA:Δn→Y such that, for each J∈〈A〉, (2.2)φA(ΔJ)⊂W(J),
where ΔJ is the standard subsimplex of Δn 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.
LetX and Y be two topological spaces and G:X→2Y 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 subsetK of X and for each y∈Y, there exists an open subset Oy of X (which may be empty) such that Oy∩K⊂G-1(y) and K=⋃y∈Y(Oy∩K),(III)
for each nonempty compact subsetK of X, there exists a set-valued mapping F:X→2Y 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 subsetK 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→2X is transfer compactly open-valued on X,(VI)
X=⋃y∈YcintG-1(y),(VII)
for eachy∈Y, G-1(y)={x∈X;y∈G(x)} contains a relatively open subset Oy of Y (Oy could be empty set for some y) such that ⋃y∈YOy=Y,(VIII)
letS,T:k→2k be two multivalued maps, co(G(x))⊂T(x) and G(x) is nonempty, and G-1 is open in X.Proof.
By Lemma1.1 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.
Lemma2.1 includes Lemma 1.1 of Ding in [10] and Lemma 2.2 of Ansari in [13] as special cases.Lemma 2.3 (see [15]).
LetX and Y be topological spaces, let D be a nonempty closed subset of X, and let Φ,Ψ:X→2Y be two set-valued mappings such that Φ(x)⊂Ψ(x) for each x∈X. Suppose that Φ-1,Ψ-1:Y→2X are both transfer compactly open-valued on Y. Then the mapping G:X→2Y defined by
(2.3)G(x)={Φ(x)ifx∈D,Ψ(x)ifx∈X∖D
is such that G-1:Y→2X is also transfer compactly open-valued on Y.


## 3. Fan-Browder Type Fixed-Point Theorem

Theorem 3.1.
LetX be a topological  space, let K be a nonempty compact subset of X, and let G:X→2X be such that(i)
G has nonempty values and satisfies one of the conditions (I)–(VIII) in Lemma 2.1,(ii)
either,
(a)
for eachN∈ℱ(X), there is a nonempty compact subset LN of X containing N such that there is a generalized R-KKM mapping W:LN→2LN satisfying, for each x∈X, the fact thatM∈〈G(x)∩LN〉 implies that W(M)⊂G(x), having, moreover, that LN∖K⊂⋃y∈LNcintG-1(y),or(b)
ifX is a nonempty convex set of a topological vector space, for each N∈F(X), there is a nonempty compact subset LN of X containing N such that there is a mapping W:LN→2L satisfying, for each D⊂N, the fact that W(D) is convex and, for each x∈X, the fact that M∈〈G(x)∩LN〉 implies that W(M)⊂G(x).Then there exists a point x0∈X such that x0∈G(x0).Proof.
By (i) and Lemma2.1, 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
(3.1)K=⋃y∈N(cintG-1(y)∩K).
For the N, consider the compact subset LN⊂X in condition (ii) satisfying
(3.2)LN∖K⊂⋃y∈LNcintG-1(y).
By (3.1), we have
(3.3)LN∩K⊂⋃y∈N(cintG-1(y)∩LN).
Noting that N⊂LN, it follows from (3.2) and (3.3) that LN=⋃y∈LN(cintG-1(y)∩LN). Since LN is compact, there exists a finite set {y0,…,ym}⊂LN such that
(3.4)LN=⋃i=0m(cintG-1(yi)∩LN).
Since W:LN→2LN is a generalized R-KKM mapping, there exists a continuous mapping f:Δm→LN such that f(ΔJ)⊂W(J) for each ∅≠J⊂{y0,…,ym}, where ΔJ is the face of Δm corresponding to J⊂{y0,…,ym}. Let {φi}i=0m be a continuous partition of unity subordinated to the open covering {cintG-1(yi)∩LN}i=0m, that is, for each i∈{0,1,…,m}, φi:LN→[0,1] is continuous, and
(3.5){x∈LN:φi(x)≠0}⊂cintG-1(yi)∩LN⊂G-1(yi)
such that ∑i=0mφi(x)=1 for all x∈LN. Define φ:LN→Δm by
(3.6)φ(x)=(φ0(x),φ1(x),…,φm(x)),∀x∈LN.
Then φ(x)∈ΔJ(x) for all x∈LN, where J(x)={yj=y0,y1,…,ym:φj(x)≠0}. Therefore, we have
(3.7)f(φ(x))∈f(ΔJ(x))⊂W(J(x))⊂G(x),∀x∈LN.
It is easy to see that φ∘f:Δm→Δm is continuous. By Browder's fixed-point theorem, there exists z∈Δm such that z=(φ∘f)(z). Let x̂=f(z), then we have x̂=f(z)=f(φ(z))⊂G(x̂). This completes our proof.Remark 3.2.
Theorem3.1 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 ofQEP(T,A,f) and QEP(A,f)

First, we prove the following equilibrium existence theorems ofQEP(T,A,f).Theorem 4.1.
LetX 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→2X, 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 setD={x∈X:x∈A(x)} is nonempty closed in X,(iii)
the mappingB-1:X→2X is transfer compactly open-valued, where B:X→2X is defined by
(4.1)B(x)={y∈A(x):f(x,T(x))-f(y,T(x))>0},(iv)
for eachN∈ℱ(X), there is a nonempty compact subset LN of X containing N such that there is a generalized R-KKM mapping W:LN→2LN satisfying, for each x∈D, the fact that M∈〈B(x)∩LN〉 implies that W(M)⊂B(x) and, for each x∈X∖D, the fact that M∈〈A(x)∩LN〉 implies that W(M)⊂A(x). Moreover, for each x∈LN∖K, if x∈X∖D, then there exists y∈LN such that x∈cintA-1(y); if x∈D, then there exists y∈LN such that x∈cintB-1(y).
Then there existsx̂∈X such that(4.2)x̂∈A(x̂),f(x̂,T(x̂))≤f(y,T(x̂)),∀y∈A(x̂),
that is, x̂ is a solution of the QEP(T,A,f).Proof.
Define a mappingG:X→2X by
(4.3)G(x)={B(x)ifx∈D,A(x)ifx∈X∖D.
From conditions (i), (iii) and Lemma 2.1, it follows that the mappings B-1,A-1:X→2X 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→2X 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)∩LN〉 implies that W(M)⊂G(x). Moreover, LN∖K⊂⋃y∈LNcintG-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 x̂∈X such that x̂∈G(x̂). By the definition of D and G, we must have {x∈X:x∈G(x)}⊂D. It follows that x̂∈B(x̂)∩D. In particular, we obtain f(x̂,T(x̂))-f(x̂,T(x̂))>0 which is impossible. Therefore, there exists x̂∈D such that B(x̂)=∅, that is, x̂∈A(x̂) and f(x̂,T(x̂))≤f(y,T(x̂)) for all y∈A(x̂). This completes the proof.Remark 4.2.
Theorem4.1 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.
LetX and Y be two topological spaces and let K be a nonempty compact subset of X. Let T:X→Y, A:X→2X, and f:X×Y→ℝ∪{±∞} be such that(i)
A has nonempty values on X such that A-1:X→2X is compactly open-valued,(ii)
the setD={x∈X:x∈A(x)} is nonempty closed in X,(iii)
T and f are continuous,(iv)
for eachN∈ℱ(X), there is a nonempty compact subset LN of X containing N such that there is a generalized R-KKM mapping W:LN→2LN satisfying, for each x∈D, the fact that M∈〈A(x)∩P(x)∩LN〉 implies that W(M)⊂A(x)∩P(x), where P:X→2X 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)∩LN〉 implies that W(M)⊂A(x). Moreover, for each x∈LN∖K, if x∈X∖D, then A(x)∩LN≠∅; if x∈D, then there exists y∈A(x)∩LN satisfying f(y,T(x))<f(x,T(x)).Then there exists x̂∈X such that
(4.4)x̂∈A(x̂),f(x̂,T(x̂))≤f(y,T(x̂)),∀y∈A(x̂),
that is, x̂ is a solution of the QEP(T,A,f).Proof.
SinceT 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→2X has compactly open values. It follows that the mapping B:X→2X 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∈LN∖K, if x∈X∖D, we have A(x)∩LN≠∅, and hence, there exists y∈LN such that x∈A-1(y)=cintA-1(y); if x∈D, we have y∈LN 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 ofQEP(A,f).Theorem 4.4.
LetX be a topological space, let K be a nonempty compact subset of X. Let A:X→2X and f:X×Y→ℝ∪{±∞} be such that(i)
A has nonempty values on X such that A-1:X→2X has compactly open values and the mapping clA:X→2X defined by (clA)(x)=clA(x) (the closure of A(x)) is upper semicontinuous,(ii)
f is a continuous function,(iii)
for eachN∈ℱ(X), there is a nonempty compact subset LN of X containing N such that there is a generalized R-KKM mapping W:LN→2LN satisfying, for each x∈D¯, the fact that M∈〈A(x)∩P(x)∩LN〉 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)∩LN〉 implies that W(M)⊂A(x). Moreover, for each x∈LN∖K, if x∈X∖D¯, then A(x)∩LN≠∅; if x∈D¯, then there exists y∈A(x)∩LN satisfying f(y,x)<f(x,x).Then there exists x̂∈X such that
(4.5)x̂∈A(x̂),f(x̂,x̂)≤f(y,x̂),∀y∈A(x̂).
If one we further assumes that f(x,x)≥0 for all x∈X, then one has that
(4.6)x̂∈A(x̂),f(y,x̂)≥0,∀y∈A(x̂),
that is, x̂ is a solution of the QEP(A,f).Proof.
SinceclA:X→2X 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 x̂∈X such that x̂∈A(x̂) and f(x̂,x̂)≤f(y,x̂), for all y∈A(x̂). If f(x,x)≥0 for all x∈X, then we must have x̂∈A(x̂) and f(y,x̂)≥0, for all y∈A(x̂), that is, x̂ is a solution of the QEP(A,f).Remark 4.5.
Theorem4.4 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 Theorem4.4, we obtain the following existence result for generalized quasi-equilibrium problems.Theorem 4.6.
LetX and Y be two topological spaces and let K be a nonempty compact subset of X. Let T:X→2Y have a continuous selection g:X→Y. Let A:X→2X 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 eachN∈ℱ(X), there is a nonempty compact subset LN of X containing N such that there is a generalized R-KKM mapping W:LN→2LN satisfying, for each x∈D¯, the fact that M∈〈A(x)∩P(x)∩LN〉 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)∩LN〉 implies that W(M)⊂A(x). Moreover, for each x∈LN∖K, if x∈X∖D¯, then A(x)∩LN≠∅; if x∈D¯, then there exists y∈A(x)∩LN satisfying ϕ(x,g(x),y)<ϕ(x,g(x),x).Then there exists x̂∈X and ŷ=g(x̂)∈T(x̂) such that
(4.7)x̂∈A(x̂),ϕ(x̂,ŷ,x̂)≤ϕ(x̂,ŷ,z),∀z∈A(x̂).
If one further assumes that ϕ(x,g(x),x)≥0 for all x∈X, then one has that
(4.8)x̂∈A(x̂),ϕ(x̂,ŷ,z)≥0,∀z∈A(x̂).Proof.
Definef:X×X→ℝ∪{±∞} by
(4.9)f(z,x)=ϕ(x,g(x),z),∀(z,x)∈X×X.
Then the conclusion of Theorem 4.6 holds from Theorem 4.4.Remark 4.7.
Theorem4.6 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.



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