IN G -CONVEX SPACES

(1)

PII. S0161171203211261 http://ijmms.hindawi.com

KKM THEOREM WITH APPLICATIONS TO LOWER AND UPPER BOUNDS EQUILIBRIUM PROBLEM

IN G -CONVEX SPACES

M. FAKHAR and J. ZAFARANI Received 9 November 2002

We give some new versions of KKM theorem for generalized convex spaces. As an application, we answer a question posed by Isac et al. (1999) for the lower and upper bounds equilibrium problem.

2000 Mathematics Subject Classiﬁcation: 47J20, 47H10, 54C60, 49J35.

1. Introduction. In [5], Isac et al. raised the following open problem which is closely related to the equilibrium problem. Given a closed nonempty subset Kin a locally convex semireﬂexive topological space, a mappingf:K×K→R, and two real numbersα,β, whereα≤β, it is interesting to know under what conditions there exists an ¯x∈Ksuch that

α≤f (x,y)¯ ≤β, ∀y∈K. (1.1) First, Li [8] gave some answers to the open problem (1.1) by introducing and using the concept of extremal subsets. Then Chadli et al. [1] gave some answers to this open problem by a method diﬀerent from that Li used. Our goal in this paper is to derive some more results in answering this problem inG-convex spaces. In fact, we will derive some results of problem (1.1) for bifunctions that are deﬁned onX×X, for whichXis aG-convex space.

LetX be nonempty set. We denote by 2^X the family of all subsets ofX, by Ᏺ^(X)the family of all nonempty ﬁnite subsets ofX, and by|A|the cardinality ofA∈Ᏺ^(X).

Let Y be a nonempty set and let X be a topological space. If F :Y →2^X is a multivalued map, then we say thatF is transfer closed-valued if, for any (y,x)∈Y×X withx∈F(y), there existsy∈Y such thatx∈clF(y); see Tian [14]. It is clear that this deﬁnition is equivalent to saying that

y∈YF(y)=

y∈YclF(y). IfB⊆YandA⊆X, then we say thatF:B→2^Ais transfer closed- valued if the multivalued mapy→F(y)∩Ais transfer closed-valued. In the case whenX=Y andA=B, we say thatF is transfer closed-valued onA.

Letfbe a bifunction onX×Y, thenf is calledλ-transfer lower semicontin uous (l.s.c.) on the ﬁrst variable onXif, for each(x,y)∈X×Ywithf (x,y) > λ, there existy∈Yand a neighborhoodU(x)ofxinXsuch thatf (z,y) > λfor

(2)

allz∈U(x). The bifunctionf is said to beλ-transfer upper semicontinuous (u.s.c.) on the ﬁrst variable onXif−f isλ-transfer l.s.c. on the ﬁrst variable.

If f is deﬁned on Y×X, then λ-transfer l.s.c. (u.s.c.) bifunction on second variable on X is deﬁned by a similar method. It is easily seen that an l.s.c (u.s.c.) bifunction isλ-transfer l.s.c (u.s.c.) bifunction for eachλ.

A generalized convex space orG-convex space was ﬁrst introduced by Park and Kim [12], and more recently, it has been generalized by Park [10]. AG- convex space(X,D;Γ)consists of a topological spaceX, a nonempty setD, and a multivalued mapΓ:Ᏺ^(D)→2^X\{∅}such that, for eachA∈Ᏺ^(D)^with the cardinality|A| =n+1, there exists a continuous functionΦA:∆n→Γ(A) such that eachJ∈Ᏺ(A)impliesΦA(∆J)⊂Γ(J), for which ifA= {a0,a1,...,an} andJ= {a_i₀,a_i₁,...,a_i_j}, then∆J=co{e_i₀,...,e_i_j}. WhenD=X, we will write (X;Γ)in place of(X,X;Γ). If(X,D;Γ)is aG-convex space,D⊆X, andK⊂X, thenKisG-convex if for eachA∈Ᏺ(D),A⊂KimpliesΓ(A)⊂K. TheG-convex hull ofKdenoted byG-coK is the set

{B⊂X:Bis aG-convex subset ofX containingK}.

Notice thatG-convex spaces contain most of the well-know spaces such as topological vector spaces, convex spaces, generalizedH-spaces, L-spaces,C- spaces, and hyperconvex metric spaces (see [10,11,12,13] and the references therein).

Let(X,D;Γ)be aG-convex space, then the multivalued mappingF:D→2^Xis called a KKM map if, for each ﬁnite subsetAofD, we haveΓ(A)⊆F(A); see Park and Lee [13]. IfxclF(x)is a KKM map, then we say that clFis a KKM map.

2. Main results. The KKM theorem is a very important tool in the study of the equilibrium problem. To solve problem (1.1) onG-convex spaces, we ﬁrst give some reﬁned versions of the KKM theorem. The following KKM theorem, due to Park and Lee [13, Theorem 1], is essential for obtaining our main results.

Theorem 2.1. Let(X,D;Γ)be a G-convex space and let F :D→2^X be a multimap such that

(1) F has closed (resp., open) values, (2) F is a KKM map.

Then{F(z):z∈D}has the ﬁnite intersection property. More precisely, for each N∈Ᏺ(D),Γ(N)∩(

z∈NF(z)≠∅). Further, if (3)

z∈MclF(z)is compact for someM∈Ᏺ^{(D), then}_z∈D^clF(z)≠∅. As a consequence of the above theorem, we obtain the following result which is a reﬁnement of [3, Theorem 1.1] and [7, Theorem 3.3].

Theorem2.2. Let(X,D;Γ)be aG-convex space such that, for eachA,B∈ Ᏺ(D)with A⊆B, Γ(A)⊆Γ(B). Suppose that F :D→2^X\ {∅}and G:D→ 2^X\{∅}are two multivalued maps such that

(1) F(x)⊆G(x)for allx∈D, (2) F is a KKM map,

(3)

KKM THEOREM WITH APPLICATIONS TO LOWER AND UPPER 3269 (3) for someM∈Ᏺ(D),

x∈MclF(x)is compact,

(4) for eachA∈Ᏺ(D)withM⊆A,G:A→2^Γ^(A)is transfer closed-valued, (5) for eachA∈Ᏺ^(D)^with^M⊆A,

cl





x∈A

G(x)



=

x∈A

G(x). (2.1)

Then

x∈DG(x)≠∅.

Proof. LetA∈Ᏺ(D) withM⊆A. Consider a multivalued mapFA:A→ 2^Γ(A)\{∅}deﬁned byF_A(x):=cl_Γ(A)(F(x)∩Γ(A))for allx∈A. ThenF_A(x)is closed inΓ(A). AlsoFA is a KKM map. In fact, ifB∈Ᏺ(A), thenΓ(B)⊆Γ(A) and Γ(B)⊆

x∈BF(x), thusΓ(B)⊆(

x∈BF(x))∩Γ(A)⊆

x∈BFA(x). So, by Theorem 2.1, we have

x∈A

F_A(x)≠∅. (2.2)

Let{Ai:i∈I}be the family of all ﬁnite subsets of Dcontaining the setM, partially ordered by⊆. Now, for eachi∈I, letXi=Γ(Ai). By (2.2),

x∈Ai

clX_i

F(x)∩Xi ≠∅, for eachi∈I. (2.3)

Take anyxi∈

x∈AiclX_i(F(x)∩Xi). For eachi∈I, letYi= {xj:j≥i, j∈I}. Clearly, we have that {Yi: i∈I}has ﬁnite intersection property, and Yi ⊆

x∈MclF(x), for alli∈I. Hence, by condition (3), clY_iis compact. Therefore i∈IclY_i≠∅. Choose any ¯x∈

i∈IclY_i. Also, for anyi,j∈I withj ≥i, we have

xj∈

x∈Aj

clX_j

F(x)∩Xj ⊆

x∈Aj

clX_j

G(x)∩Xj

=

x∈Aj

G(x)∩Xj ⊆

x∈Ai

G(x)∩Xj

⊆

x∈Ai

G(x).

(2.4)

Therefore,Y_i⊆

x∈AiG(x). Now, for anyx∈D, there existsi0∈I such that x∈Ai₀. It follows that

x¯∈clY_i₀⊆cl





z∈Ai0

G(z)



=

z∈Ai0

G(z)⊆G(x). (2.5)

Then ¯x∈G(x)for allx∈X, and the proof is completed.

ByTheorem 2.1and the fact that

x∈DG(x)=

x∈DclG(x), whenGis transfer closed-valued, we can obtain the following result.

(4)

Theorem2.3. Let(X,D;Γ)be aG-convex space. Suppose thatF:D→2^X\{∅}

andG:D→2^X\{∅}are two multivalued maps such that (1) F(x)⊆G(x)for allx∈D,

(2) clF is a KKM map, (3) for someM∈Ᏺ(D),

x∈MclF(x)is compact, (4) Gis transfer closed-valued.

Then

x∈DG(x)≠∅.

The following examples show that Theorems2.2and2.3are diﬀerent.

Example2.4. Assume thatX=RandD=N. If we deﬁneΓ(A)=co(A+1)for everyA∈Ᏺ(D), then(X,D;Γ)is aG-convex space andΓ(A)≠G- coA. Suppose thatF:D→2^Xis deﬁned as

F(x)=











{1,2}∪

(−∞,0)∩Q

ifx=1, (1,+∞) ifx=2, R ifx≠1,2.

(2.6)

By takingM= {1,2}andF=G, all the conditions ofTheorem 2.2are satisﬁed and

x∈DF(x)= {2}, but

x∈DclF(x)= {1,2}. Therefore,F is not transfer closed-valued and so we cannot applyTheorem 2.3.

The following example is a modiﬁed form of [14, Example 1].

Example2.5. IfX=[0,1],D=Q∩X, andΓ(A)=[minA,1], for everyA∈ Ᏺ(D), then(X,D;Γ)is aG-convex space. Suppose thatF:D→2^Xis deﬁned by F(x)=[x,1]∩Q. IfF=G, then all the conditions ofTheorem 2.3are satisﬁed.

ButFis not KKM map and moreover forA= {0,0.5}, conditions (4) and (5) are not satisﬁed.

By a method similar to that of the proof ofTheorem 2.2, we can obtain the following result which is an improvement of [2, Lemma 2] and [6, Lemma 3.1]

onG-convex spaces.

Theorem2.6. Let(X;Γ)be aG-convex space and letG-coAbe closed for eachA∈Ᏺ(X). Suppose thatF :X→2^X\ {∅}andG:X→2^X\ {∅}are two multivalued maps such that

(1) F(x)⊆G(x)for allx∈X, (2) F is aKKMmap,

(3) for someM∈Ᏺ^(X),_x∈M^clF(x)is compact,

(4) for eachA∈Ᏺ^(X)^with^M⊆A,Gis transfer closed-valued onG-coA, (5) for eachA∈Ᏺ(X)withM⊆A,

cl





x∈G-coA

G(x)



∩G-coA=





x∈G-coA

G(x)



∩G-coA. (2.7)

Then

x∈XG(x)≠∅.

(5)

KKM THEOREM WITH APPLICATIONS TO LOWER AND UPPER 3271 Remark2.7. (a) If, inTheorem 2.3,Xis Hausdorﬀ andX=D, then condition (3) can be replaced by the following condition:

(3) there exists a compact subset K of X such that, for each N ∈Ᏺ^(X), there exists a nonempty compactG-convex subset LN ofX such that

x∈LNclF(x)⊆K.

(b) If, inTheorem 2.6, for eachA∈Ᏺ^(X),^{G- coA}is compact, then, instead of conditions (3) and (4) we can assume that

(3) there existsM∈Ᏺ(X)such that cl(

x∈MF(x))is compact,

(4) for eachA∈Ᏺ^(X)^with^M⊆A,F is transfer closed-valued onG- coA.

Then the conclusion ofTheorem 2.6holds. In this case, we obtain a reﬁnement of Lemma 2.3 of Ding and Tarafdar [4]. Also condition (3) ofTheorem 2.6can be replaced by the following condition:

(3) there existsM∈Ᏺ(X)such that cl(

x∈MG(x))is compact.

(c)Example 2.4shows that, in general,Γ(A)≠G- coA. Therefore,Theorem 2.6 has its own applications.

Now, byTheorem 2.2, we obtain the following result, which gives an answer to problem (1.1).

Theorem2.8. Let(X,D;Γ)be aG-convex space such that for eachA,B∈ Ᏺ^(D)withA⊆B,Γ(A)⊆Γ(B). Suppose thatf andgare two real bifunctions deﬁned onX×Dsuch that

(1) for each(x,y)∈X×D, ifα≤f (x,y)≤β, thenα≤g(x,y)≤β;

(2) for eachA∈Ᏺ^(D)andB⊆Awith∅≠B≠A, either (i) α≤inf_x∈Γ(A)max_y∈Bf (x,y)or

(ii) sup_x∈Γ(A)min_y∈A\Bf (x,y)≤β.

ForB=A, condition (i) holds, and forB= ∅, condition (ii) is satisﬁed;

(3) there exist a compact subsetKofXandM∈Ᏺ^(D)such that, for every x∈X\K, there are a pointy∈Mand a neighborhoodU(x)ofxsuch that for anyz∈U(x),f (z,y) < αorf (z,y) > β;

(4) for eachA∈Ᏺ^(D)^with^M⊆A,g:Γ(A)×A→Risα-transfer u.s.c. and β-transfer l.s.c. on the ﬁrst variable onΓ(A);

(5) for eachA∈Ᏺ(D)withM⊆A,x∈X and for each net(xλ)inXcon- verging tox, ifα≤g(x_λ,y)≤βfor ally∈A, thenα≤g(x,y)≤β.

Then there existsx¯∈Xsuch thatα≤g(x,y)¯ ≤βfor ally∈D.

Proof. Assume thatF,G:D→2^X are deﬁned by F(y)=

x∈X:α≤f (x,y)≤β ,

G(y)=x∈X:α≤g(x,y)≤β. (2.8)

By condition (1),F(y)⊆G(y)for ally∈D. Condition (2) implies thatF is a KKM map, because if there existsA∈Ᏺ(D)such thatΓ(A)

y∈AF(y), then there is a point ˆx∈Γ(A)such thatf (x,y) < αˆ orf (x,y) > β, for allˆ y∈A.

LetB= {y∈A:f (ˆx,y) < α}, thenB=Aor∅, or∅≠B≠A. In the case when

(6)

B=AorB= ∅, we have max_y∈Af (x,y) < αˆ or min_y∈Af (ˆx,y) > β. If∅≠ B≠A, then max_y∈Bf (x,y) < αˆ and min_y∈A\Bf (x,y) > βˆ which contradicts condition (2). Also, by condition (3) we have

y∈MclF(y)⊆K. Now, we show that condition (4) implies thatG:A→2^Γ(A)is transfer closed-valued for each A∈Ᏺ(D)withM⊆A. Let(x,y)be a point inΓ(A)×Aandx∈Γ(A)∩G(y).

Theng(x,y) < αorg(x,y) > β. Ifg(x,y) < α, then there existy∈Aand a neighborhoodU(x)ofxinΓ(A)such thatg(z,y) < αfor allz∈U(x). Thus, x∈cl_Γ(A)(Γ(A)∩G(y)). Similarly, we can prove the case wheng(x,y) > β.

Moreover ifx∈cl(

y∈AG(y)), then there exists a net(x_λ)in

y∈AG(y)such thatxλ→x. Therefore,α≤g(xλ,y)≤βfor ally∈A, and by condition (5), we haveα≤g(x,y)≤β. Hencex∈

y∈AG(y)and so, byTheorem 2.2, we have

y∈DG(y)≠∅.

Remark2.9. (a) If inTheorem 2.8instead of condition (4) we assume the following condition:

(4) gisα-transfer u.s.c. andβ-transfer l.s.c. on the ﬁrst variable onX, then, byTheorem 2.3and without condition (5), we can obtain another answer for problem (1.1). In the above case, if X=D and X is Hausdorﬀ, then by Remark 2.7(a), condition (3) can be replaced by the following condition:

(3) there exists a compact subset K ofX such that, for every N∈Ᏺ(X) there is a nonempty compactG-convex subset LN of X such that for everyx∈X\K, there are a pointy∈L_N and a neighborhoodU(x)of xsuch that for anyz∈U(x)we havef (z,y) < αorf (z,y) > β.

(b) If inTheorem 2.8X=DandG- coAis compact for anyA∈Ᏺ(X), then we can concludeTheorem 2.8by replacing conditions (3), (4), and (5) by the following conditions:

(3) there exist a compact subsetKofXandM∈Ᏺ(X)such that, for every x∈X\K, there is a pointy∈Msuch thatf (x,y) < αorf (x,y) > β;

(4) for each A∈Ᏺ^(X)^with ^M⊆A,f :G- coA×G- coA→Ris α-transfer u.s.c. andβ-transfer l.s.c. on the ﬁrst variable onG- coA;

(5) for each A∈Ᏺ^(X)withM ⊆A,x,y ∈G- coA, and for each net(x_λ) in X converging to x, if α≤g(x_λ,z)≤β for allz∈Γ({x,y}), then α≤g(x,y)≤β.

(c) In part (a), ifX is a nonempty convex subset of a Hausdorﬀ topological vector space, then we can obtain a reﬁnement of [1, Theorem 2.3] and [8, Theorem 3.1].

Theorem2.10. Let(X;Γ)be a HausdorﬀG-convex space, for any ﬁnite sub- set A ofX, and letG-coA be compact. Suppose thatf, g1, and g2 are real bifunctions onX×Xsatisfying the following conditions:

(1) g1(x,x)≥αandg2(x,x)≤β, for allx∈X;

(2) for every x ∈X and for every A∈ Ᏺ^(X) ^if^A ⊆ {y ∈X :f (x,y) <

αorf (x,y) > β},Γ(A)⊆ {y∈X:g1(x,y) < αorg2(x,y) > β}; (3) there exist compact subsetKofXandM∈Ᏺ^(X)such that the set{y∈

M:f (x,y) < αorf (x,y) > β}is nonempty for eachx∈X\K;

(7)

KKM THEOREM WITH APPLICATIONS TO LOWER AND UPPER 3273 (4) for eachA∈Ᏺ(X)withM⊆A, f :G-coA×G-coA→Ris α-transfer

u.s.c. andβ-transfer l.s.c. on the ﬁrst variable onG-coA;

(5) for eachA∈Ᏺ^(X)^with^M⊆A, x,y ∈G-coA, and for each net(x_λ) in X converging to x, if α≤ f (xλ,z)≤β for all z ∈Γ({x,y}), then α≤f (x,y)≤β.

Then there existsx¯∈Xsuch thatα≤f (x,y)¯ ≤βfor eachy∈X.

Proof. LetF:X→2^Xbe deﬁned by F(y)=

x∈X:α≤f (x,y)≤β

. (2.9)

First, we show thatF is a KKM map. Assume that there existsA∈Ᏺ(X)such that Γ(A)

y∈AF(y). Therefore,Γ(A)contains a point x0which is not in

y∈AF(y). Hence, by condition (2), we haveg1(x0,x0) < αorg2(x0,x0) > β.

This contradicts condition (1). Condition (3) implies that

y∈MF(y)⊆K. As in the proof ofTheorem 2.8, condition (4) implies condition (4) ofRemark 2.7, and condition (5) implies condition (5) ofTheorem 2.6. Therefore, byTheorem 2.6and part (b) ofRemark 2.7, we have

y∈XF(y)≠∅.

Remark2.11. If, inTheorem 2.10, instead of conditions (3) and (4), we have the following conditions:

(3) there exists a compact subset K of X such that for every N ∈Ᏺ^(X) there is a nonempty compactG-convex subset L_N of X such that for everyx∈X\Kthere are a pointy∈LNand a neighborhoodU(x)ofx such that for anyz∈U(x), we havef (z,y) < αorf (z,y) > β;

(4) fisα-transfer u.s.c. andβ-transfer l.s.c. on the ﬁrst variable onX.

Then, byRemark 2.7(a) and without condition (5) we can obtain a reﬁnement of [1, Theorem 2.2]. Also ifg₁and g₂ are identical and equal tof, then we obtain an improvement of [8, Theorem 3.1].

3. Some applications. In this section, we give some applications ofTheorem 2.8andRemark 2.9.

Theorem3.1. Let(X,D;Γ)be aG-convex space such that for eachA,B∈ Ᏺ^(D)withA⊆B,Γ(A)⊆Γ(B). Suppose thatf1andg1are two real bifunctions deﬁned onD×Xsuch that

(1) for each(y,x)∈D×X, iff1(y,x)≤c, theng1(y,x)≤c, (2) for eachA∈Ᏺ^(D),sup_x∈Γ(A)min_y∈Af1(y,x)≤c,

(3) there exist a compact subsetKofXandM∈Ᏺ^(D)such that, for every x∈X\K, there exist a pointy∈Mand a neighborhoodU(x)ofxsuch that for anyz∈U(x), f1(y,z) > c,

(4) for eachA∈Ᏺ^(D)^with^M⊆A,g1:A×Γ(A)→Risc-transfer l.s.c. on the second variable onΓ(A),

(5) for eachA∈Ᏺ(D)withM⊆Aand each net(xλ)inXconverging tox, ifg1(y,x_λ)≤cfor ally∈A, theng1(y,x)≤c.

Then there existsx¯∈Xsuch thatg1(y,x)¯ ≤cfor ally∈D.

(8)

Proof. Deﬁnef,g:X×D→Rbyf (x,y)=e^f¹^(y,x)andg(x,y)=e^g¹^(y,x). Ifα=0 andβ=e^c, then it is easy to see that all of the conditions ofTheorem 2.8are satisﬁed. Therefore, there is a point ¯x∈Xsuch that 0≤g(¯x,y)≤e^c for ally∈D, that is,g1(y,x)¯ ≤cfor ally∈D.

Corollary3.2. Let(X,D;Γ)be aG-convex space such that for eachA,B∈ Ᏺ^(D)^with^A⊆B,Γ(A)⊆Γ(B). Suppose thatϕandψare two real bifunctions deﬁned onX×Dsuch that

(1) for each(x,y)∈X×D, ifϕ(x,y)≥0, thenψ(x,y)≥0, (2) for eachA∈Ᏺ^(D),^infx∈Γ(A)max_y∈Aϕ(x,y)≥0,

(3) there exist a compact subsetKofXandM∈Ᏺ(D)such that for every x∈X\Kthere exist a pointy∈Mand a neighborhoodU(x)ofxsuch that for anyz∈U(x), ϕ(z,y) <0,

(4) for eachA∈Ᏺ(D)withM⊆A,ψ:Γ(A)×A→Ris0-transfer u.s.c. on the ﬁrst variable onΓ(A),

(5) for eachA∈Ᏺ^(D)^with^M⊆Aand each net(x_λ)inXconverging tox, ifψ(xλ,y)≥0for ally∈A, thenψ(x,y)≥0.

Then there existsx¯∈Xsuch thatψ(¯x,y)≥0for ally∈D.

Proof. It is enough inTheorem 3.1to setc=0,f1(y,x)= −ϕ(x,y), and g1(y,x)= −ψ(x,y).

If (X,Γ)is aG-convex space, theng:X→R isG-quasiconvex if{x∈X: g(x) < λ}isG-convex for eachλ∈R.

Remark3.3. If inCorollary 3.2X=D, for eachx∈X,yϕ(x,y)isG- quasiconvex, andϕ(x,x)≥0, then condition (2) ofCorollary 3.2is satisﬁed.

SoCorollary 3.2improves [9, Corollary 2].

If X=D, X is Hausdorﬀ space andG- coA is compact for anyA∈Ᏺ^(X), then instead of conditions (3), (4), and (5) ofTheorem 3.1we can suppose that (3) there exist a compact subsetKofXandM∈Ᏺ^(X)such that, for every

x∈X\K, there exists a pointy∈Msuch thatf1(y,x) > c;

(4) for each A∈ Ᏺ(X) with M ⊆A, f1 is c-transfer l.s.c. on the second variable onG- coA,

(5) for eachA∈Ᏺ^(X)^with^M⊆A,x,y∈G- coA, and each net(x_λ)inX converging tox, ifg1(z,xλ)≤cfor allz∈Γ({x,y}), theng1(y,x)≤c.

In the above case we obtain a reﬁnement of [2, Theorem 2], [6, Theorem 3.2], and [15, Theorems 2.2 and 2.3].

The following corollary improves [9, Corollary 3].

Corollary3.4. Let(X;Γ)be a HausdorﬀG-convex space and letG-coAbe compact for allA∈Ᏺ(X). Suppose thatY is a topological space,T:X→2^Y is a multivalued mapping having a continuous selectionf, andφ:X×Y×X→Ris a function such that

(9)

KKM THEOREM WITH APPLICATIONS TO LOWER AND UPPER 3275 (1) φ(x,y,z)isG-quasiconvex inz,

(2) φ(x,f (x),z)≥0for allx∈X,

(3) there exist a compact subsetKofXandM∈Ᏺ^(X)such that, for every x∈X\Kandy∈Y there exists a pointz∈Msuch thatφ(x,y,z) <0, (4) for eachA∈Ᏺ(X)withM⊆A,φ(x,y,z)is0-transfer u.s.c. in(x,y)on

G-coA,

(5) for eachA∈Ᏺ(X) withM ⊆A, x,z∈G-coA, and for each net (xλ) inXconverging tox, ifφ(xλ,f (xλ),z)≥0for allz∈Γ({x,z}), then φ(x,f (x),z)≥0.

Then there exist anx¯∈Xandy¯∈T (¯x)such thatφ(¯x,y,z)¯ ≥0for allz∈X.

Proof. Letϕ(z,x)=ψ(z,x)= −φ(x,f (x),z)for(x,z)∈X×X. Thenψ satisﬁes all of the requirements ofRemark 3.3. Therefore, byTheorem 3.1, we have the conclusion.

Acknowledgments. The authors thank the referee for his comments and helpful suggestions. Also they would like to express their sincere gratitude to Professor Sehie Park for providing them some of his recent research articles onG-convex spaces. The second author is partially supported by the Inter- universities Research Project No. 31303378-37.

References

[1] O. Chadli, Y. Chiang, and J. C. Yao,Equilibrium problems with lower and upper bounds, Appl. Math. Lett.15(2002), no. 3, 327–331.

[2] M. S. R. Chowdhury and K.-K. Tan,Generalization of Ky Fan’s minimax inequal- ity with applications to generalized variational inequalities for pseudo- monotone operators and ﬁxed point theorems, J. Math. Anal. Appl.204 (1996), no. 3, 910–929.

[3] M. S. R. Chowdhury, E. Tarafdar, and K.-K. Tan,Minimax inequalities onG-convex spaces with applications to generalized games, Nonlinear Anal.43(2001), no. 2, 253–275.

[4] X. P. Ding and E. Tarafdar,Generalized variational-like inequalities with pseu- domonotone set-valued mappings, Arch. Math. (Basel)74(2000), no. 4, 302–

313.

[5] G. Isac, V. M. Sehgal, and S. P. Singh,An alternate version of a variational inequal- ity, Indian J. Math.41(1999), no. 1, 25–31.

[6] E. M. Kalmoun,On Ky Fan’s minimax inequalities, mixed equilibrium problems and hemivariational inequalities, JIPAM. J. Inequal. Pure Appl. Math.2(2001), no. 1, 1–13.

[7] W. A. Kirk, B. Sims, and G. X.-Z. Yuan,The Knaster-Kuratowski and Mazurkiewicz theory in hyperconvex metric spaces and some of its applications, Nonlinear Anal.39(2000), no. 5, 611–627.

[8] J. Li,A lower and upper bounds version of a variational inequality, Appl. Math.

Lett.13(2000), no. 5, 47–51.

[9] L.-J. Lin and S. Park,On some generalized quasi-equilibrium problems, J. Math.

Anal. Appl.224(1998), no. 2, 167–181.

[10] S. Park,Elements of the KKM theory for generalized convex spaces, Korean J.

Comput. Appl. Math.7(2000), no. 1, 1–28.

(10)

[11] ,Fixed points of better admissible maps on generalized convex spaces, J.

Korean Math. Soc.37(2000), no. 6, 885–899.

[12] S. Park and H. Kim,Admissible classes of multifunctions on generalized convex spaces, Proc. College Natur. Sci. Seoul Nat. Univ.18(1993), 1–21.

[13] S. Park and W. Lee,A uniﬁed approach to generalized KKM maps in generalized convex spaces, J. Nonlinear Convex Anal.2(2001), no. 2, 157–166.

[14] G. Q. Tian,Generalizations of the FKKM theorem and the Ky Fan minimax inequal- ity, with applications to maximal elements, price equilibrium, and comple- mentarity, J. Math. Anal. Appl.170(1992), no. 2, 457–471.

[15] R. U. Verma,Role of generalized KKM type selections in a class of minimax in- equalities, Appl. Math. Lett.12(1999), no. 4, 71–74.

M. Fakhar: Department of Mathematics, University of Isfahan, Isfahan 81745-163, Iran

E-mail address:[email protected]

J. Zafarani: Department of Mathematics, University of Isfahan, Isfahan 81745-163, Iran

E-mail address:[email protected]