Volume 2007, Article ID 78696,13pages doi:10.1155/2007/78696
Research Article
Coincidence Theorems, Generalized Variational Inequality
Theorems, and Minimax Inequality Theorems for the Φ -Mapping on G-Convex Spaces
Chi-Ming Chen, Tong-Huei Chang, and Ya-Pei Liao
Received 14 December 2006; Revised 27 February 2007; Accepted 5 March 2007 Recommended by Simeon Reich
We establish some coincidence theorems, generalized variational inequality theorems, and minimax inequality theorems for the familyG-KKM(X,Y) and theΦ-mapping on G-convex spaces.
Copyright © 2007 Chi-Ming Chen et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and preliminaries
In 1929, Knaster et al. [1] had proved the well-known KKM theorem onn-simplex. In 1961, Fan [2] had generalized the KKM theorem in the infinite-dimensional topological vector space. Later, the KKM theorem and related topics, for example, matching theorem, fixed point theorem, coincidence theorem, variational inequalities, minimax inequalities, and so on had been presented in grand occasions. Recently, Chang and Yen [3] intro- duced the family, KKM(X,Y), and got some results about fixed point theorems, coinci- dence theorems, and some applications on this family. Later, Ansari et al. [4] and Lin and Chen [5] studied the coincidence theorems for two families of set-valued mappings, and they also gave some applications of the existence of minimax inequality and equilibrium problems. In this paper, we establish some coincidence theorems, generalized variational inequality theorems, and minimax inequality theorems for the familyG-KKM(X,Y) and theΦ-mapping onG-convex spaces.
LetXandY be two sets, and letT:X→2Ybe a set-valued mapping. We will use the following notations in the sequel:
(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=φ},
(v)T∗(y)= {x∈X:y /∈T(x)},
(vi) ifDis a nonempty subset ofX, thenDdenotes the class of all nonempty finite subsets ofD.
For the case thatX andY are two topological spaces, then T:X→2Y is said to be closed if its graphᏳT= {(x,y)∈X×Y:y∈T(x)}is closed.T is said to be compact if the imageT(X) ofXunderTis contained in a compact subset ofY.
LetX be a topological space. A subsetDof X is said to be compactly closed (resp., compactly open) inX if for any compact subsetK ofX, the setD∩K is closed (resp., closed) inK. Obviously,Dis compactly open inX if and only if its complementDc is compactly closed inX.
The compact closure ofDis defined by
ccl(D)= ∩B⊂X:D⊂B,Bis compactly closed inX, (1.1) and the compact interior ofDis defined by
cint(D)= ∪B⊂X:B⊂D,Bis compactly open inX. (1.2) Remark 1.1. It is easy to see that ccl(X\D)=X\cint(D),Dis compactly open inXif and only ifD=cint(D), and for each nonempty compact subsetK ofX, we have cint(D)∩ K=intK(D∩K), where intK(D∩K) denotes the interior ofD∩KinK.
Definition 1.2 [6,7]. LetXandY be two topological spaces, and letT:X→2Y.
(i)Tis said to be transfer compactly closed (resp., transfer closed) onXif for anyx∈ Xand anyy /∈T(x), there existsx∈Xsuch thaty /∈cclT(x) (resp.,y /∈clT(x)).
(ii)Tis said to be transfer compactly open (resp., transfer open) onXif for anyx∈X and anyy∈T(x), there existsx∈Xsuch thaty∈cintT(x) (resp.,y∈intT(x)).
(iii)T is said to have the compactly local intersection property on X if for each nonempty compact subsetK ofX and for eachx∈X withT(x)=φ, there ex- ists an open neighborhoodN(x) ofxinXsuch that∩z∈N(x)∩KT(z)=φ.
Remark 1.3. IfT:X→2Y is transfer compactly open (resp., transfer compactly closed) andYis compact, thenTis transfer open (resp., transfer closed).
We denote byΔnthe standardn-simplex with vectorse0,e1,...,en, whereeiis the (i+ 1)th unit vector inn+1.
A generalized convex space [8] or aG-convex space (X,D;Γ) consists of a topological spaceX, a nonempty subsetDofX, and a functionΓ:D →2Xwith nonempty values (in the sequal, we writeΓ(A) byΓAfor eachA∈ D) such that
(i) for eachA,B∈ D,A⊂Bimplies thatΓA⊂ΓB,
(ii) for eachA∈ Dwith|A| =n+ 1, there exists a continuous functionφA:Δn→ ΓAsuch thatJ∈ Aimplies thatφA(Δ|J|−1)⊂ΓJ, whereΔ|J|−1denotes the faces ofΔncorresponding toJ∈ A.
Particular forms ofG-convex spaces can be found in [8] and references therein. For a G-convex space (X,D;Γ) andK⊂X,
(i)KisG-convex if for eachA∈ D,A⊂KimpliesΓA⊂K,
(ii) the G-convex hull of K, denoted by G-Co(K), is the set ∩{B⊂X|Bis aG- convex subset ofXcontainingK}.
Definition 1.4 [9]. AG-convex spaceX is said to be a locallyG-convex space ifX is a uniform topological space with uniformityᐁwhich has an open baseᏺ= {Vi|i∈I}of symmetric encourages such that for eachV∈ᏺ, the setV[x]= {y∈X|(x,y)∈V}is a G-convex set, for eachx∈X.
Let (X,D;Γ) be aG-convex space which has a uniformityᐁandᐁhas an open sym- metric base familyᏺ. Then a nonempty subsetK ofX is said to be almostG-convex if for any finite subsetBofKand for anyV∈ᏺ, there is a mappinghB,V:B→Xsuch that x∈V[hB,V(x)] for allx∈BandG-Co(hB,V(B))⊂K. subset ofK. We call the mapping hB,V:B→XaG-convex-inducing mapping.
Remark 1.5. (i) In general, theG-convex-inducing mappinghB,Vis not unique. IfU⊂V, then it is clear that anyhB,Ucan be regarded as anhB,V.
(ii) It is clear that theG-convex set is almostG-convex, but the inverse is not true, for a counterexample.
LetE= 2 be the Euclidean topological space. Then the setB= {x=(x1,x2)∈E: x12/3+x2/32 <1}is aG-convex set, but the setB= {x=(x1,x2)∈E: 0< x2/31 +x22/3<1}is an almostG-convex set, not aG-convex set.
Applying Ding [10, Proposition 1] and Lin [11, Lemma 2.2], we have the following lemma.
Lemma 1.6. Let X andY be two topological spaces, and letF:X→2Y be a set-valued mapping. Then the following conditions are equivalent:
(i)Fhas the compactly local intersection property,
(ii) for each compact subsetKofXand for eachy∈Y, there exists an open subsetOyof Xsuch thatOy∩K⊂F−1(y) andK=
y∈Y(Oy∩K),
(iii) for any compact subsetK ofX, there exists a set-valued mappingP:X→2Y such thatP(x)⊂F(x) for eachx∈X,P−1(y) is open inXandP−1(y)∩K⊂F−1(y) for eachy∈Y andK=
y∈Y(P−1(y)∩K),
(iv) for each compact subsetK ofX and for each x∈K, there exists y∈Y such that x∈cintF−1(y)∩KandK=
y∈Y(cintF−1(y)∩K), (v)F−1is transfer compactly open valued onY,
(vi)X=
y∈YcintF−1(y).
Definition 1.7. LetY be a topological space and letXbe aG-convex space. A set-valued mappingT:Y→2Xis called aΦ-mapping if there exists a set-valued mappingF:Y→2X such that
(i) for eachy∈Y,A∈ F(y)implies thatG-Co(A)⊂T(y), (ii)Fsatisfies one of the conditions (i)–(vi) inLemma 1.6.
Moreover, the mappingFis called a companion mapping ofT.
Remark 1.8. IfT:Y→2Xis aΦ-mapping, then for each nonempty subsetY1ofY,T|Y1: Y1→2Xis also aΦ-mapping.
Let X be a G-convex space. A real-valued function f :X→ is said to be G- quasiconvex if for eachξ∈ , the set{x∈X: f(x)≤ξ}isG-convex, and f is said to beG-quasiconcave if−f isG-quasiconvex.
Definition 1.9. LetXbe a nonempty almostG-convex subset of aG-convex space. A real- valued function f :X→ is said to be almostG-quasiconvex if for eachξ∈ , the set {x∈X: f(x)≤ξ}is almostG-convex, and f is said to be almostG-quasiconcave if−f is almostG-quasiconvex.
Definition 1.10. LetXbe aG-convex space,Y a nonempty set, and let f,g:X×Y→ be two real-valued functions. For anyy∈Y,gis said to be f-G-quasiconcave inxif for eachA= {x1,x2,...,xn} ∈ X,
1min≤i≤nfxi,y≤g(x,y), ∀x∈G-Co(A). (1.3) Definition 1.11. LetX be a nonempty almostG-convex subset of aG-convex spaceE which has a uniformityᐁandᐁhas an open symmetric base familyᏺ,Y a nonempty set, and let f,g:X×Y→ be two real-valued functions. For anyy∈Y,gis said to be almost f-G-quasiconcave inxif for eachA= {x1,x2,...,xn} ∈ Xand for everyV∈ᏺ, there exists aG-convex-inducing mappinghA,V:A→Xsuch that
1min≤i≤nfxi,y≤g(x,y), ∀x∈G-CohA,V(A). (1.4) Remark 1.12. It is clear that if f(x,y)≤g(x,y) for each (x,y)∈X×Y, and if for each y∈Y, the mappingx→ f(x,y) is almostG-quasiconcave (G-quasiconcave), theng is almost f-G-quasiconcave inx(f-G-quasiconcave).
Definition 1.13. LetXbe aG-convex space,Y a topological space, and letT,F:X→2Y be two set-valued functions satisfying
TG-Co(A)⊂F(A) for anyA∈ X. (1.5) ThenFis called a generalizedG-KKM mapping with respect toT. If the set-valued func- tionT:X→2Y satisfies the requirement that for any generalized G-KKM mappingF with respect toT the family{F(x)|x∈X}has the finite intersection property, thenT is said to have the G-KKM property. The class G-KKM(X,Y) is defined to be the set {T:X→2Y|Thas theG-KKM property}.
We now generalize theG-KKM property on aG-convex space to theG-KKM∗prop- erty on an almostG-convex subset of aG-convex space.
Definition 1.14. LetX be a nonempty almostG-convex subset of aG-convex spaceE which has a uniformityᐁandᐁhas an open symmetric base familyᏺ, andY a topo- logical space. LetT,F:X→2Ybe two set-valued functions satisfying that for each finite subsetAofXand for anyV∈ᏺ, there exists aG-convex-inducing mappinghA,V:A→X such that
TG-CohA,V(A)⊂F(A). (1.6)
Then F is called a generalized G-KKM∗mapping with respect to T. If the set-valued functionT:X→2Ysatisfies the requirement that for any generalizedG-KKM∗mapping Fwith respect toTthe family{F(x)|x∈X}has the finite intersection property, thenT is said to have theG-KKM∗property. The classG-KKM∗(X,Y) is defined to be the set {T:X→2Y|Thas theG-KKM∗property}.
2. Coincidence theorems for theΦ-mapping and theG-KKM family
Throughout this paper, we assume that the setG-Co(A) is compact wheneverAis a com- pact set.
The following lemma will play important roles for this paper.
Lemma 2.1. LetY be a compact set,XaG-convex space. LetT:Y →2Xbe aΦ-mapping.
Then there exists a continuous function f :Y→X such that for eachy∈Y, f(y)∈T(y), that is,Thas a continuous selection.
Proof. SinceYis compact, there existsA={x0,x1,...,xn}⊂Xsuch thatY=n
i=0intF−1(xi).
SinceXis aG-convex space andA∈ X, there exists a continuous mappingφA:Δn→ Γ(A) such thatφA(Δ|J|−1)⊂ΓJfor eachJ∈ A.
Let{λi}ni=0be the partition of the unity subordinated to the cover{intF−1(xi)}ni=0ofY. Define a continuous mappingg:Y→Δnby
g(y)= n i=0
λi(y)ei=
i∈I(y)
λi(y)ei, for eachy∈Y, (2.1)
whereI(y)= {i∈ {0, 1, 2,...,n}:λi=0}. Note thati∈I(y) if and only ify∈F−1(xi), that is,xi∈F(y). SinceTis aΦ-mapping, we conclude thatφA◦g(y)∈φA(ΔI(y))⊂G-Co{xi: i∈I(y)} ⊂T(y), for eachy∈Y. This completes the proof.
LetXbe aG-convex space. A polytope inXis denoted byΔ=G-Co(A) for eachA∈ X. By the conception of theG-KKM(X,Y) family we immediately have the following proposition.
Proposition 2.2 [12]. LetX be aG-convex space, and letY andZ be two topological spaces. Then
(i)T∈G-KKM(X,Y) if and only ifT∈G-KKM(Δ,Y) for every polytopyΔinX, (ii) ifY is a normal space,Δa polytope inX, and ifT:X→2Ysatisfies the requirement
that f Thas a fixed point inΔfor all f ∈Ꮿ(Y,Δ), thenT∈G-KKM(Δ,Y).
FollowingLemma 2.1andProposition 2.2, we prove the following important lemma for this paper.
Lemma 2.3. LetXbe aG-convex space and letYbe a compactG-convex space. IfT:X→2Y is aΦ-mapping, thenT∈G-KKM(X,Y).
Proof. SinceT is a Φ-mapping, we have that for anyA∈ X, letΔ=G-Co(A), T|Δ: Δ→Yis also aΦ-mapping. SinceΔis compact and byLemma 2.1,T|Δhas a continuous selection function, that is, there is a continuous function f :Δ→Y such that for each
x∈Δ,f(x)∈T(x). So we conclude that f−1Thas a fixed point inΔ. ByProposition 2.2, T∈G-KKM(Δ,Y), and so we conclude thatT∈G-KKM(X,Y).
The following lemma is an extension of Chang et al. [13, Proposition 2.3].
Lemma 2.4. LetXbe a nonempty almostG-convex subset of aG-convex spaceEwhich has a uniformityᐁandᐁhas an open symmetric base familyᏺ, and letY,Zbe two topological spaces. IfT∈G-KKM∗(X,Y), then f T∈G-KKM∗(X,Z) for all f ∈Ꮿ(Y,Z).
Proof. LetF be a generalizedG-KKM∗ mapping with respect to f T such thatF(x) is closed for allx∈X, and letA∈ X. Then for anyV∈ᏺ, there exists aG-convex-induc- ing mappinghA,V :A→Xsuch that f T(G-Co(hA,V(A)))⊂F(A). SoT(G-Co(hA,V(A)))⊂ f−1F(A). Therefore, f−1F is a generalizedG-KKM∗mapping with respect toT. Since T∈KKM∗(X,Y) and f−1F(x) is closed for allx∈X, so the family{f−1F(x) :x∈X} has the finite intersection property, and so does the family{F(x) :x∈X}. Hence f T∈
G-KKM∗(X,Z).
Theorem 2.5. LetXbe a nonempty almostG-convex subset of a locallyG-convex spaceE, and letT∈G-KKM∗(X,X) be compact and closed. ThenThas a fixed point.
Proof. SinceEis a locallyG-convex space, there exists a uniform structureᐁ, letᏺ= {Vi|i∈I}be an open symmetric base family for the uniform structure ᐁsuch that for anyU∈ᏺ, the setU[x]= {y∈X|(x,y)∈U}isG-convex for eachx∈X, and let U∈ᏺ.
We now claim that for anyV ∈ᏺ, there exists xV ∈X such thatV[xV]∩T(xV)= φ. Suppose it is not the case, then there is aV ∈ᏺsuch that V[xV]∩T(xV)=φ, for allxV∈X. LetV1∈ᏺsuch thatV1◦V1⊂V. SinceT is compact, henceK=TX is a compact subset ofX. DefineF:X→2Xby
F(x)=K\V1[x] for eachx∈X. (2.2) We will show that
(1)F(x) is nonempty and closed for eachx∈X,
(2)Fis a generalizedG-KKM∗mapping with respect toT.
(1) is obvious. To prove (2), we use the contradiction. LetA= {x1,x2,...,xn} ∈ X. Sup- poseFis not a generalizedG-KKM∗mapping with respect toT. Then there existsV2∈ᏺ such that for anyG-convex-inducing mappinghA,V2:A→X, one hasT(G-Co(hA,V2(A))) F(A). LetV3∈ᏺsuch thatV3⊂V1∩V2. ThenT(G-Co(hA,V3(A)))F(A). So there exist μ∈G-Co(hA,V3(A)) andν∈T(μ) such thatν∈/n
i=1Fxi. From the definition ofF, it fol- lows thatν∈V1[xi] for eachi∈ {1, 2,...,n}. Hence,ν∈V1◦V3[hA,V3(xi)]⊂V[hA,V3(xi)]
for each i∈ {1, 2,...,n}, since X is almost G-convex. Thus, hA,V3(xi)∈V[ν], for each i∈ {1, 2,...,n}, and henceμ∈G-Co(hA,V3(A))⊂V[ν], that is,ν∈V[μ]. Therefore,ν∈ T(μ)∩V[μ]. This contradictsV[x]∩T(x)=φ, for allx∈X. Hence,F is a generalized G-KKM∗mapping with respect toT.
SinceT∈G-KKM∗(X,X), the family{F(x) :x∈X}has finite intersection property, and so we conclude thatx∈XF(x)=φ. Letη∈
x∈XF(x)⊂K⊂X. Thenη∈K\V1[x], for allx∈X. This implies thatη∈K\V1[η]. So we have reached a contradiction. Therefore, we have proved that for eachVi∈ᏺ, there is anxVi∈Xsuch thatV[xVi]∩T(xVi)=φ.
Let yVi∈Vi[xVi]∩T(xVi), then (xVi,yVi)∈ᏳT and (xVi,yVi)∈Vi. SinceT is compact, without loss of generality, we may assume that{yVi}i∈I converges to y0, that is, there exists V0∈ᏺ such that (yVj,y0)∈Vj for all Vj∈ᏺ withVj⊂V0. Let VU∈ᏺwith VU◦VU⊂Vj⊂V0, then we have (xVU,yVU)∈VU and (yVU,y0)∈VU, so (xVU,yVU)◦ (yVU,y0)=(xVU,y0)∈VU◦VU⊂Vj, that is,xVU→y0. The closedness ofTimplies that (y0,y0)∈ᏳT, that is,y0∈T(y0). This completes the proof.
Corollary 2.6. LetXbe a nonemptyG-convex subset of a locallyG-convex spaceE, and letT∈G-KKM(X,X) be compact and closed. ThenThas a fixed point.
We now establish the main coincidence theorem for theΦ-mapping and the family G-KKM(X,Y).
Theorem 2.7. LetXbe a nonemptyG-convex subset of a locallyG-convex spaceE, and let Y be a topological space. Assume that
(i)T∈G-KKM(X,Y) is compact and closed, (ii)F:Y→2XisΦ-mapping.
Then there exists (x,y)∈X×Ysuch thaty∈T(x) andx∈F(y).
Proof. SinceTis compact, we have thatK=T(X) is compact inY. By (ii), we have that F|Kis also aΦ-mapping. ByLemma 2.1,F|Khas a continuous selection f :K→X. So, by Lemma 2.4, we have f T∈KKM(X,X), and so byCorollary 2.6, there existsx∈Xsuch thatx∈f T(x)⊂FT(x), that is, there existsy∈T(x) such thatx∈F(y).
ApplyingLemma 2.3,Theorem 2.7, andCorollary 2.6, we immediately have the fol- lowing coincidence theorem for twoΦ-mappings.
Theorem 2.8. LetXbe a nonemptyG-convex subset of a locallyG-convex spaceE, andY a topological space. IfT:X→2Y,F:Y→2Xare twoΦ-mappings, and ifTis compact and closed, then there exists (x,y)∈X×Y such thaty∈T(x) andx∈F(y).
3. Generalized variational theorems and minimax inequality theorems
Lemma 3.1 [14]. LetXandY be two topological spaces, and letF:X→2Ybe a set-valued mapping. ThenFis transfer closed if and only ifx∈XF(x)=
x∈XF(x).
Definition 3.2 [15]. Let X andY be two topological spaces, and let f :X×Y → ∪ {−∞,∞}be a function. For someγ∈ , f(x,y) is said to beγ-transfer compactly lower semicontinuous inyif for eachy∈ {u∈Y :f(x,u)> γ}, there exists anx∈Xsuch that y∈cint{u∈Y:f(x,u)> γ}.f is said to beγ-transfer compactly upper semicontinuous inyif for eachy∈ {u∈Y:f(x,u)< γ}, there exists anx∈Xsuch thaty∈cint{u∈Y:
f(x,u)< γ}.
Definition 3.3. LetXandYbe two topological spaces, and let f :X×Y→ ∪ {−∞,∞}
be a function. Thenf is said to be transfer compactly lower semicontinuous (resp., trans- fer lower semicontinuous) inyif for eachy∈Y andγ∈ withy∈ {u∈Y :f(x,u)>
γ}, there exists anx∈X such that y∈cint{u∈Y :f(x,u)> γ}(resp., y∈int{u∈Y: f(x,u)> γ}).
f is said to be transfer compactly upper semicontinuous in yif−f is transfer com- pactly lower semicontinuous iny.
Lemma 3.4 [15]. LetXandYbe two topological spaces, and let f :X×Y→ ∪ {−∞,∞}
be a function. For someγ∈ , f :X×Y → is said to be γ-transfer compactly lower (resp., upper) semicontinuous inyif and only if the set-valued mappingF:X→2Ydefined byF(x)= {y∈Y : f(x,y)≤γ}(resp., F(x)= {y∈Y: f(x,y)≥γ}) for eachx∈X is transfer compactly closed.
Applying Lemmas3.1,3.4, andRemark 1.3, we immediately obtain the following the- orem.
Theorem 3.5. LetX be a nonempty almostG-convex subset of aG-convex spaceEwhich has a uniformityᐁandᐁhas an open symmetric base familyᏺ,Y a topological space, and letF∈G-KKM∗(X,Y) be compact. If f,g:X×Y → are two real-valued functions satisfying the following conditions:
(i) for eachx∈X, the mappingy→f(x,y) is transfer compactly lower semicontinuous onY,
(ii) for eachy∈Y,gis almost f-G-quasiconave inx, then for eachξ∈ , one of the following properties holds:
(1) there exists (x,y)∈ᏳFsuch that
g(x,y)> ξ, (3.1)
(2) or there existsy∈Ysuch that
f(x,y)≤ξ, ∀x∈X. (3.2)
Proof. Letξ∈ . SinceFis compact,F(X) is compact inY. DefineT,S:X→2Yby T(x)=
y∈F(X) :g(x,y)≤ξ, ∀x∈X, S(x)=
y∈F(X) : f(x,y)≤ξ, ∀x∈X. (3.3) Suppose the conclusion (1) is false. Then for each (x,y)∈ᏳF,g(x,y)≤ξ. This implies thatᏳF⊂ᏳT.
LetA= {x1,x2,...,xn} ∈ X. By the condition (ii), we claim that Sis a generalized G-KKM∗mapping with respect toT. If the above statement is not true, then there ex- istsV∈ᏺsuch that for anyG-convex-inducing mappinghA,V :A→X, one hasT(G- Co(hA,V(A)))S(A). So there existx0∈G-Co(hA,V(A)) andy0∈T(x0) such thaty0∈/ S(A). From the definitions ofTandS, it follows thatg(x0,y0)≤ξand f(xi,y0)> ξfor all i=1, 2,...,n. This contradicts the condition (ii). Therefore,Sis a generalizedG-KKM∗ mapping with respect toT, and so we get thatSis a generalizedG-KKM∗mapping with respect toF. SinceF∈G-KKM∗(X,Y), the family{S(x) :x∈X}has the finite intersec- tion property, and sinceS(x) is compact for eachx∈X, so we havex∈XS(x)=φ. From Lemmas3.1and3.4,Remark 1.3, and the condition (i), we have that∩x∈XS(x)=φ. Take y0∈
x∈XS(x), then f(x,y0)≤ξfor allx∈X.
Theorem 3.6. If all of the assumptions ofTheorem 3.5hold, then one immediately concludes the following inequality:
yinf∈Ysup
x∈Xf(x,y)≤ sup
(x,y)∈ᏳF
g(x,y). (3.4)
Proof. Letξ=sup(x,y)∈ᏳFg(x,y). Then the conclusion (1) ofTheorem 3.5is false. So there existsy0∈Ysuch that f(x,y0)≤ξfor allx∈X. This implies that supx∈X f(x,y0)≤ξ, ad so we have infy∈Ysupx∈X f(x,y)≤sup(x,y)∈ᏳFg(x,y).
Corollary 3.7. LetXbe aG-convex space,Ya topological space, and letF∈G-KKM(X,Y) be compact. If f,g:X×Y→ are two real-valued functions satisfying the following condi- tions:
(i) for eachx∈X, the mappingy→f(x,y) is transfer compactly lower semicontinuous onY,
(ii) for eachy∈Y,gis f-G-quasiconave inx, then for eachξ∈ , one of the following properties holds:
(1) there exists (x,y)∈ᏳFsuch that
g(x,y)> ξ, (3.5)
(2) or there existsy∈Ysuch that
f(x,y)≤ξ, ∀x∈X. (3.6)
Corollary 3.8. If all of the assumptions ofCorollary 3.7hold, then one immediately con- cludes the following inequality:
yinf∈Ysup
x∈Xf(x,y)≤ sup
(x,y)∈ᏳF
g(x,y). (3.7)
Proposition 3.9. LetX andY be twoG-convex spaces, and letT,F:X→2Y be two set- valued mappings. Then the following two statements are equivalent:
(i) for eachy∈Y, ifA∈ T∗(y), thenG-Co(A)⊂F∗(y).
(ii)Tis a generalizedG-KKM mapping with respect toF.
ApplyingProposition 3.9, we conclude the following variational theorems and mini- max inequality theorems for theΦ-mapping.
Theorem 3.10. LetX be a nonemptyG-convex space,Y a nonempty compactG-convex space, and letS,F:X→2Ybe two set-valued mappings satisfying the following conditions:
(i)Fis aΦ-mapping,
(ii)Sis transfer compactly closed valued onX, (iii) for eachy∈Y,F∗(y) isG-convex, (iv) for eachx∈X,F(x)⊂S(x).
Then there existsy∈Y such thatS∗(y)=φ.
Proof. ByLemma 2.3,F∈G-KKM(X,Y). By conditions (iii) and (iv), we have thatG- Co(S∗(y))⊂F∗(y) for each y∈Y. So, byProposition 3.9,S is a generalizedG-KKM
mapping with respect toF. Therefore, the family{S(x) :x∈X}has the finite intersection property. SinceY is compact,x∈XS(x)=φ. ByLemma 3.1, we havex∈XS(x)=φ. Let y∈
x∈XS(x). ThenS∗(y)=φ.
Theorem 3.11. LetXandY be twoG-convex spaces, and letS,T,G,H:X→2Y be four set-valued mappings satisfying the following conditions:
(i) for eachx∈X,T(x)⊂G(x)⊂H(x)⊂S(x), (ii) for eachy∈Y,H∗(y) isG-convex,
(iii) for eachx∈X,G(x) isG-convex,
(iv)T−1is transfer compactly open valued onY, (v)Sis transfer compactly closed valued onX.
Then one has the following two properties.
(1) IfY is compact, then there existsy∈Y such thatS∗(y)=φ.
(2) IfXis compact, then there existsx∈Xsuch thatT(x)=φ.
Proof. Case (1). SupposeY is compact. We defineF:X→2Yby
F(x)=G-CoT(x), for eachx∈X. (3.8)
ThenF is aΦ-mapping andF−1 is transfer compactly open valued on Y, and soF∈ G-KKM(X,Y). By conditions (i), (ii), and (iii), we haveG-Co(S∗(y))⊂H∗(y)⊂G∗(y)⊂ F∗(y) for eachy∈Y. Applying Proposition 3.9andTheorem 3.10, we could conclude that there existsy∈Ysuch thatS∗(y)=φ.
Case (2). SupposeX is compact. Conditions (i)–(v) are equivalent to the following statements:
(i) for eachy∈Y,S∗(y)⊂H∗(y)⊂G∗(y)⊂T∗(y), (ii) for eachy∈Y,H∗(y) isG-convex,
(iii) for eachx∈X, (G∗)∗(x) isG-convex, (iv)T∗is transfer compactly closed valued onY,
(v) (S∗)−1is transfer compactly open valued onX.
We now consider the four set-valued mappings S∗,H∗,G∗,T∗:Y →2X, then by the same process of the proof of Case (1), we also conclude that there existsx∈Xsuch that
T(x)=φ.
Theorem 3.12. LetXandYbe twoG-convex spaces, and letf,g,p,q:X×Y→ be four real-valued functions satisfying the following conditions:
(i) for each (x,y)∈X×Y,f(x,y)≤g(x,y)≤p(x,y)≤q(x,y), (ii) for eachy∈Y,x→g(x,y) isG-quasiconcave,
(iii) for eachx∈X,y→p(x,y) isG-quasiconvex,
(iv) for eachy∈Y,x→q(x,y) is transfer compactly upper semicontinuous, (v) for eachx∈X,y→f(x,y) is transfer compactly lower semicontinuous.
Then for anyλ∈ , one has the following two properties.
(1) IfY is compact, then there existsy∈Y such that f(x,y)≤λfor allx∈X.
(2) IfXis compact, then there existsx∈Xsuch thatq(x,y)≥λfor ally∈Y.
Proof. Letλ∈ . We defineS,T,G,H:X→2Yby T(x)=
y∈Y:q(x,y)< λ, G(x)=
y∈Y:p(x,y)< λ, H(x)=
y∈Y:g(x,y)≤λ,
S(x)=y∈Y:f(x,y)< λ for eachx∈X.
(3.9)
Then by condition (i),T(x)⊂G(x)⊂H(x)⊂S(x) for eachx∈X. Conditions (ii) and (iii) imply thatG(x) isG-convex for allx∈X andH∗(y) is G-convex for all y∈Y. Conditions (iv) and (v) imply thatT−1is transfer compactly open valued onY andSis transfer compactly closed valued onX. So all the conditions ofTheorem 3.10are satisfied.
Therefore, we have the following properties.
(1) IfY is compact, then there existsy∈Y such thatS∗(y)=φ, that is, there exists y∈Ysuch that f(x,y)≤λfor allx∈X.
(2) IfX is compact, then there existsx∈Xsuch thatT(x)=φ, that is, there exists
x∈Xsuch thatq(x,y)≥λfor ally∈Y.
FollowingTheorem 2.8, we also have the variational inequality theorem and minimax inequality theorem.
Theorem 3.13. LetXbe a nonemptyG-convex subset of a locallyG-convex spaceE, andY a compact topological space. If f,g,p,q:X×Y→ are four real-valued functions, anda, bare two real numbers, suppose the following conditions hold:
(i)g(x,y)≤f(x,y) andp(x,y)≤q(x,y) for allx∈X,y∈Y,
(ii) for eachx∈X,y→f(x,y) isG-quasiconcave onYand for eachy∈Y,x→p(x,y) isG-quasiconvex onX,
(iii) for eachy∈Y,x→g(x,y) is transfer compactly lower semicontinuous and for each x∈X,y→q(x,y) is transfer compactly upper semicontinuous inY,
(iv) f is upper semicontinuous onX×Y. Then one of the following statesment holds:
(1) there existsμ∈Xsuch thatg(μ,y)≤afor eachy∈Y, (2) there existsν∈Ysuch thatq(x,ν)≥bfor eachx∈X,
(3) there exists (μ,ν)∈X×Ysuch that f(μ,ν)≥aandp(μ,ν)< b.
Proof. LetS,T:X→2YandH,F:Y→2Xbe defined by Sx=
y∈Y:g(x,y)−a >0, for eachx∈X, Tx=
y∈Y: f(x,y)−a≥0, for eachx∈X, H y=
x∈X:q(x,y)−b <0, for eachy∈Y, F y=
x∈X:p(x,y)−b≤0, for eachy∈Y.
(3.10)
By the assumption (i), we have thatSx⊂Txfor eachx∈X, and by (ii),TxisG-convex for eachx∈X, and soG-Co(Sx)⊂Txfor eachx∈X. By the assumption (iii),S−1is transfer compactly open valued onY. Similarly, by (ii) and (iii), we haveG-Co(H y)⊂F yfor each y∈YandH−1is transfer compactly open valued onX.
Suppose that the conditions (1) and (2) are false. ThenSx=φfor eachx∈X and H y=φ for each y∈Y. So, we conclude that T is a Φ-mapping with a companion mappingS andF is aΦ-mapping with a companion mapping H. By the assumption (iv),T is closed. Hence, all of the assumptions ofTheorem 2.8hold, and so there exists (μ,ν)∈X×Ysuch thatν∈T(μ) andμ∈F(ν), that is, f(μ,ν)≥aandp(μ,ν)< b.
Theorem 3.14. LetX be a nonemptyG-convex subset of a locallyG-convex spaceE,Y a compact topological space. If f,g,p,q:X×Y → are four real-valued functions, anda,b are two real numbers, suppose the following conditions hold:
(i)g(x,y)≤f(x,y)≤p(x,y)≤q(x,y) for allx∈X,y∈Y,
(ii) for eachx∈X,y→f(x,y) isG-quasiconcave onYand for eachy∈Y,x→P(x,y) isG-quasiconvex onX,
(iii) for eachy∈Y,x→g(x,y) is transfer compactly lower semicontinuous and for each x∈X,y→q(x,y) is transfer compactly upper semicontinuous inY,
(iv) f is upper semicontinuous onX×Y. Then
xinf∈Xsup
y∈Yg(x,y)≤sup
y∈Yinf
x∈Xq(x,y). (3.11)
Proof. Letε >0 and let a=inf
x∈Xsup
y∈Yg(x,y)−ε, b=sup
y∈Yinf
x∈Xq(x,y) +ε. (3.12) Then for eachx∈X, there existsy∈Y such thatg(x,y)> a, and for each y∈Y, there existx∈Xsuch thatq(x,y)< b. Therefore, the conclusions (1) and (2) ofTheorem 3.13 are false. So there existμ∈Xandν∈Y such thatf(μ,ν)≥aandp(μ,ν)< b, that is
f(μ,ν)≥inf
x∈Xsup
y∈Yg(x,y)−ε, p(μ,ν)<sup
y∈Yinf
x∈Xq(x,y) +ε. (3.13) So by (i), we have
xinf∈Xsup
y∈Yg(x,y)−ε <sup
y∈Yinf
x∈Xq(x,y) +ε. (3.14)
Sinceεis an arbitrary positive number, by lettingε↓0, we get
xinf∈Xsup
y∈Yg(x,y)≤sup
y∈Yinf
x∈Xq(x,y). (3.15)
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Chi-Ming Chen: Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu 300, Taiwan
Email address:[email protected]
Tong-Huei Chang: Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu 300, Taiwan
Email address:[email protected]
Ya-Pei Liao: Department of Applied Mathematics, National Hsinchu University of Education, Hsinchu 300, Taiwan
Email address:yypp [email protected]
