Volume 2009, Article ID 905605,12pages doi:10.1155/2009/905605
Research Article
Some Maximal Elements’ Theorems in
FC
-Spaces
Rong-Hua He
1, 2and Yong Zhang
11Department of Mathematics, Chengdu University of Information Technology,
Chengdu, Sichuan 610103, China
2Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Rong-Hua He,[email protected]
Received 30 March 2009; Accepted 1 September 2009
Recommended by Nikolaos Papageorgiou
LetIbe a finite or infinite index set, letXbe a topological space, and letYi, ϕNii∈Ibe a family of
FC-spaces. For eachi∈I, letAi:X → 2Yibe a set-valued mapping. Some new existence theorems
of maximal elements for a set-valued mapping and a family of set-valued mappings involving a better admissible set-valued mapping are established under noncompact setting ofFC-spaces. Our results improve and generalize some recent results.
Copyrightq2009 R.-H. He and Y. Zhang. 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
It is well known that many existence theorems of maximal elements for various classes of set-valued mappings have been established in different spaces. For their applications to mathematical economies, generalized games, and other branches of mathematics, the reader may consult1–12and the references therein.
In most of the known existence results of maximal elements, the convexity assumptions play a crucial role which strictly restrict the applicable area of these results. In this paper, we will continue to study existence theorems of maximal elements in general topological spaces without convexity structure. We introduce a new class of generalized
GB-majorized mappings Ai : X → 2Yi for each i ∈ I which involve a set-valued mapping F ∈ BY, X. The notion of generalized GB-majorized mappings unifies and
generalizes the corresponding notions of GB-majorized mappings in 4; LS-majorized
2. Preliminaries
LetX andY be two nonempty sets. We denote by 2Y andX the family of all subsets of
Y and the family of all nonempty finite subsets of X, respectively. For each A ∈ X, we denote by|A|the cardinality ofA. LetΔn denote the standardn-dimensional simplex with the vertices{e0, . . . , en}. IfJ is a nonempty subset of{0,1, . . . , n},we will denote byΔJ the
convex hull of the vertices{ej:j ∈J}.
LetXandY be two sets, and letT :X → 2Y be a set-valued mapping. We will use the following notations in the sequel:
iTx {y∈Y :y∈Tx}, iiTA x∈ATx,
iiiT−1y {x∈X:y∈Tx}.
For topological spacesXandY, a subsetAofX is said to be compactly openresp., compactly closedif for each nonempty compact subsetKofX,A∩Kis openresp., closedin
K. The compact closure ofAand the compact interior ofAsee17are defined, respectively, by
cclAB⊂X :A⊂ B, Bis compactly closed inX,
cintAB⊂X :B⊂A, Bis compactly open inX.
2.1
It is easy to see that cclX \A X \cintA, intA ⊂ cintA ⊂ A, A ⊂ cclA ⊂ clA, Ais compactly openresp., compactly closedinXif and only ifAcintAresp.,AcclA. For each nonempty compact subsetKofX, cclAK clKAKand cintAKintKAK, whereclKAK resp., intKAKdenotes the closureresp., interiorofAK inK. A set-valued mappingT : X → 2Y is transfer compactly open valued onX see17if for eachx∈Xandy∈Tx, there existsx∈Xsuch thaty∈cintTx. LetAi i1, . . . , mbe transfer compactly open valued, thenmi1cintAi cintmi1Ai. It is clear that each transfer
open valued correspondence is transfer compactly open valued. The inverse is not true in general.
The definition ofFC-space and the classBY, Xof better admissible mapping were introduced by Ding in8. Note that the classBY, Xof better admissible mapping includes many important classes of mappings, for example, the classBY, Xin18,UkcY, Xin19 and so on as proper subclasses. Now we introduce the following definition.
Definition 2.1. An FC-spaceY, ϕNis said to be anCFC-space if for eachN ∈ Y, there exists a compactFC-subspaceLNofY containingN.
Y, ϕN be a G-convex space, let the notion ofCG-convex space was introduced by Ding in4.
Lemma 2.28. LetI be any index set. For eachi∈I, letYi, ϕNibe anFC-space,Y i∈IYi andϕN i∈IϕNi. ThenY, ϕNis also anFC-space.
LetF∈ BY, Xand for eachi∈I, letAi:X → 2Yibe a set-valued mapping. For eachi∈I,
1Ai:X → 2Yi is said to be a generalizedG
B-mapping if
afor each N {y0, . . . , yn} ∈ Y and {yi0, . . . , yik} ⊂ N, FϕNΔk kj0cintA−i1πiyij ∅, where πi is the projection of Y onto Yi and Δkco{eij :j0, . . . , k};
bA−1
i yi {x∈X:yi∈Aix}is transfer compactly open inYifor eachyi∈Yi;
2Ax,i : X → 2Yi is said to be a generalized G
B-majorant ofAi atx ∈ X ifAx,i is a generalized GB-mapping and there exists an open neighborhoodNx ofxin X
such thatAiz⊂Ax,izfor allz∈Nx;
3Ai is said to be a generalizedGB-majorized if for eachx ∈X withAix/∅, there exists a generalized GB-majorant Ax,i of Ai at x, and for any N ∈ {x ∈ X :
Aix/∅}, the mappingx∈NA−x,i1 is transfer compactly open inYi;
4Ai is said to be a generalized GB-majorized if for each x ∈ X, there exists a
generalizedGB-majorantAx,iofAiatx.
Then{Ai}i∈I is said to be a family of generalized GB-mappingsresp.,GB-majorant
mappingsif for eachi∈I, Ai :X → 2Yi is a generalizedGB-mappingresp.,GB-majorant
mapping.
If for eachi∈I, letYi, ϕNibe aG-convex space, a family ofGB-mappingsresp.,GB -majorant mappingswere introduced by Ding in4. Clearly, each family of generalizedGB
-mappings must be a family of generalizedGB-majorant mappings. IfF Sis a single-valued
mapping andAixis anFC-subspace ofYifor eachx∈X, then conditionyi/∈AiSyfor eachy∈Y implies that conditionain1holds. Indeed, ifais false, then there existN
{y0, . . . , yn} ∈ Y,{yi0, . . . , yik} ⊆N, andy∈ϕNΔksuch thatFy Sy∈ k
j0A−i1πiyij and henceπiyij ∈ AiSyfor each j 0, . . . , k. It follows fromy ∈ ϕNΔkthatπiy ∈ ϕNiΔkwhereNi πiN. It follows fromAiSybeing anFC-subspace ofYithatπiy∈ ϕNiΔk ⊂ AiSywhich contradicts condition yi/∈AiSyfor eachy ∈ Y. Hence each
LS-mappingresp.,LS-majorant mappingintroduced by Deguire et al.see2, page 934
must be a generalizedGB-mappingresp.,GB-majorant mapping. The inverse is not true in
general.
3. Maximal Elements
In order to obtain our main results, we need the following lemmas.
Lemma 3.13. LetXandY be topological spaces, letKbe a nonempty compact subset ofX,and letG : X → 2Y be a set-valued mapping such thatGx/∅ for eachx ∈ K. Then the following conditions are equivalent:
1Ghave the compactly local intersection property;
2for each y ∈ Y, there exists an open subsetOy of X (which may be empty) such that OyK⊂G−1yandKy∈YOyK;
3there exists a set-valued mappingF:X → 2Y such that for eachy∈Y, F−1yis open or
empty inX,F−1yK⊂G−1y,∀y∈Y,andK
4for each x ∈ K, there exists y ∈ Y such that x ∈ cintG−1yK and K
y∈YcintG−1yK y∈YG−1yK;
5G−1:Y → 2Xis transfer compactly open valued onY.
Lemma 3.28. LetXbe a topological space, and letY, ϕNbe anFC-space,F ∈ BY, Xand
A:X → 2Y such that
ifor eachN{y0, . . . , yn} ∈ Yand for each{yi0, . . . , yik} ⊆N,
F ϕNΔk ⎛ ⎝k
j0
cintA−1y
ij
⎞
⎠∅, 3.1
iiA−1:Y → 2Xis transfer compactly open valued;
iiithere exists a nonempty setY0 ⊂Y and for eachN {y0, . . . , yn} ∈ Y, there exists a
compactFC-subspaceLN ofY containingY0∪N such thatK
y∈Y0cintA
−1ycis
empty or compact inX, wherecintA−1ycdenotes the complement ofcintA−1y.
Then there exists a pointx∈Xsuch thatAx ∅.
Theorem 3.3. LetXbe a topological space, letKbe a nonempty compact subset ofX, and letY, ϕN be anFC-space,F ∈ BY, XandA:X → 2Y be a generalizedGB-mapping such that
ifor eachN{y0, . . . , yn} ∈ Y, there exists a compactFC-subspaceLNofY containing
Nsuch that for eachx∈X\K, LNcintAx/∅.
Then there exists a pointx∈Ksuch thatAx ∅.
Proof. Suppose thatAx/∅ for eachx ∈ X. SinceA is a generalizedGB-mapping,A−1 is
transfer compactly open valued. ByLemma 3.1, we have
K
y∈Y
cintA−1 y K. 3.2
SinceKis compact, there exists a finite setN{yo, . . . , yn} ∈ Ysuch that
Kn
i0
cintA−1 yi K
. 3.3
By conditioniandF ∈ BY, X, there exists a compactFC-subspaceLNofY containingN andFLNis compact inX, and hence we have
FLN
y∈LN
cintA−1 y FLN. 3.4
Remark 3.4. Theorem 3.3generalizes in 4, Theorem 2.2 in the following several aspects: a from G-convex space to FC-space without linear structure; b from GB-mappings to generalizedGB-mappings.
Theorem 3.5. LetX be a topological space, and letY, ϕNbe anFC-space. LetF ∈ BY, Xand
A:X → 2Y be a generalizedGB-majorized mapping such that
ithere exists a paracompact subsetEofXsuch that{x∈X :Ax/∅} ⊂E;
iithere exists a nonempty set Y0 ⊂ Y and for each N {y0, . . . , yn} ∈ Y, there
exists a compact FC-subspace LN of Y containing Y0 ∪ N such that the set K
y∈Y0cintA
−1ycis empty or compact.
Then there exists a pointx∈Xsuch thatAx ∅.
Proof. Suppose thatAx/∅for eachx∈X. SinceAis a generalizedGB-majorized, for each
x ∈ X, there exists an open neighborhoodNxofx inX and a generalized GB-mapping
Ax:X → 2Y such that
aAz⊂Axzfor eachz∈Nx,
bfor each N {y0, . . . , yn} ∈ Y and {yi0, . . . , yik} ⊆ N, FϕNΔk k
j0cintA−x1yij ∅, cA−1
x is transfer compactly open inY,
dfor anyN∈ {x∈X :Ax/∅}, the mappingx∈NA−1
x is transfer compactly open inX.
SinceAx/∅for eachx∈X, it follows from conditionithatX{x∈X :Ax/∅}Eis paracompact. By Dugundji in20, Theorem VIII.1.4, the open covering{Nx:x∈X}has an open precise locally finite refinement{Ox :x ∈ X}, and for eachx ∈ X, Ox ⊂ Nx sinceXis normal. For eachx∈X, define a mappingBx:X → 2Y by
Bxz ⎧ ⎨ ⎩
Axz, ifz∈Ox,
Y, ifz∈X\Ox. 3.5
Then for eachy∈Y, we have
B−1
x y
z∈Ox:y∈Axz z∈X\Ox:y∈Y
A−1
x y
Ox X\Ox
A−1
x y X\Ox
OxX\OxA−1
x y X\Ox.
3.6
HenceB−1
x yis transfer compactly open inYbyc. Now define a mappingB:X → 2Y by
Bz
x∈X
We claim thatBis a generalizedGB-mapping andAz ⊂ Bzfor eachz ∈X. Indeed, for
any nonempty compact subsetCofXand eachy∈Y withB−1y∩C /∅, we may take any
fixedu∈B−1y∩C. Since{Ox:x∈X}is locally finite, there exists an open neighborhood
VuofuinXsuch that{x∈X :Vu∩Ox/∅}{x1, . . . , xn}is a finite set. Ifx /∈ {x1, . . . , xn}, then ∅ Vu ∩Ox Vu ∩Ox, and henceBxz Y for allz ∈ Vu which implies that
Bz x∈XBxz ni1Bxizfor allz∈Vu. It follows that for eachy∈Y,
B−1 yz∈X:y∈Bz⊃z∈V
u:y∈Bz
z∈Vu:y∈ n
i1
Bxiz
Vu
n
i1
B−1
xi y
. 3.8
For any nonempty compact subsetCofX and each y ∈ Y, ifv ∈ Vu∩ni1B−xi1y
C ⊂
B−1yC. SinceV
uis open inX, it follows fromdthat there existsy∈Y such that
v∈Vu
cint
n
i1
B−1
xi y
Ccint
Vu
n
i1
B−1
xi y
C
cintB−1 yC.
3.9
This proves thatB−1:Y → 2Xis transfer compactly open valued inY.
On the other hand, for eachN{y0, . . . , yn} ∈ YandN1{yi0, . . . , yik} ⊆N, ift∈ k
j0cintB−1yij, thenN1 ⊂cintBt. Since there existsx0 ∈Xsuch thatt∈Ox0andN1 ⊂ cintBt⊂cintBx0t cintAx0t, we havet∈
k
j0cintA−x10yij, and hencet /∈FϕNΔkby b. Hence we have
F ϕNΔk ⎛ ⎝k
j0
cintB−1yij
⎞
⎠∅ 3.10
for each N {y0, . . . , yn} ∈ Y and N1 {yi0, . . . , yik} ⊆ N. This shows that B is a generalizedGB-mapping.
For eachz ∈X, ify /∈Bz, then there exists anx0 ∈X such thaty /∈Bx0z Ax0z and z ∈ Ox0 ⊂ Nx0. It follows from a that y /∈Az. Hence we have Az ⊂ Bz
for eachz ∈ X. By condition ii, there exists a nonempty set Y0 ⊂ Y and for each N
{y0, . . . , yn} ∈ Y, there exists a compactFC-subspaceLNofY containingY0∪Nsuch that
the setK y∈Y0cintA−1ycis empty or compact. Note thatAz⊂ Bzfor eachz∈ X
impliescintB−1yc ⊂ cintA−1ycfor eachy ∈ Y. HenceK
y∈Y0cintB
−1yc ⊂ K
Theorem 3.6. LetXbe a topological space, letKbe a nonempty compact subset ofXandY, ϕNbe
anFC-space. LetF∈ BY, XandA:X → 2Y be a generalizedGB-majorized mapping such that
ithere exists a paracompact subsetEofXsuch that{x∈X :Ax/∅} ⊂E;
iifor eachN{y0, . . . , yn} ∈ Y, there exists a compactFC-subspaceLNofY containing
Nsuch that for eachx∈X\K, LNcintAx/∅.
Then there existsx∈Ksuch thatAx ∅.
Proof. Suppose thatAx/∅for each x ∈ X. By using similar argument as in the proof of Theorem 3.5, we can show that there exists a generalizedGB-mappingB:X → 2Y such that
Ax ⊂ Bxfor each x ∈ X. It follows from conditioniithat for eachx ∈ X\K, LN ∩ cintBx/∅. ByTheorem 3.3, there existsx ∈ K such that Bx ∅, and hence Ax ∅ which contradicts the assumption thatAx/∅for eachx∈X. Therefore, there existsx∈X such thatAx ∅. Conditioniiimpliesx∈K. This completes the proof.
Remark 3.7. Theorem 3.5generalizes4, Theorem 2.3in several aspects:Section 11 from
G-convex space to FC-space without linear structure; Section 12 from a GB-majorized mapping to a generalizedGB-majorized mapping;Section 13conditioniiofTheorem 3.5
is weaker than conditioniiof 4, Theorem 2.3. IfX is compact, conditioniis satisfied trivially. If X Y, ϕN is a compact FC-space, then by letting K X Y LN for all N ∈ X, conditions i and ii are satisfied automatically. Theorem 3.6 unifies and generalizes Shen’s14, Theorem 2.1, Corollary 2.2 and Theorem 2.3in the following ways: Section 21fromCH-convex space toFC-space without linear structure;Section 22from
H-majorized correspondences to generalizedGB-majorized mapping;Section 23condition ii of Theorem 3.6 is weaker than that in the corresponding results of Shen in 14. Theorem 3.6also generalizes in4, Theorem 2.4, Ding in15, Theorem 5.3, and Ding and Yuan in16, Theorem 2.3in several aspects.
Corollary 3.8. LetX be a compact topological space, and letY, ϕNbe an CFC-space. Let F ∈ BY, XandA:X → 2Y be a generalizedGB-majorized mapping. Then there exists a pointx ∈X such thatAx ∅.
Proof. The conclusion ofCorollary 3.8follows fromTheorem 3.6withEKX.
Corollary 3.9. LetXbe a topological space, and letY, ϕNbe anCFC-space. LetF ∈ BY, Xbe a compact mapping andA:X → 2Y be a generalizedGB-majorized mapping. Then there exists a point
x∈Xsuch thatAx ∅.
Proof. SinceFis a compact mapping, there exists a compact subsetX0ofX such thatFY⊂
X0. The mappingA|X0:X0 → 2Ybe the restriction ofAtoX0. It is easy to see thatA|X0is also generalizedGB-majorized. ByCorollary 3.8, there existsx ∈X0such thatA|X0x Ax
∅.
Remark 3.10. Corollary 3.8generalizes Deguire et al.2, Theorem 1in the following ways: 1.1from a convex subset of Hausdorff topological vector space to an FC-space without linear structure;1.2from aLS-majorized mapping to a generalizedGB-majorized mapping. Corollary 3.8 also generalizes 4, Corollary 2.3 from CG-convex space to CFC-space and from a GB-majorized mapping to a generalized GB-majorized mapping.Corollary 3.9
Theorem 3.11. LetXbe a topological space, and letIbe any index set. For eachi∈I, letYi, ϕNi
be anFC-space, and letY i∈IYisuch thatY, ϕNis anFC-space defined as inLemma 2.2. Let F∈ BY, Xsuch that for eachi∈I,
iletAi:X → 2Yi be a generalizedG
B-majorized mapping;
iii∈I{x∈X :Aix/∅}i∈Icint{x∈X:Aix/∅};
iiithere exists a paracompact subsetEiofXsuch that{x∈X:Aix/∅} ⊂Ei;
ivthere exists a nonempty setY0 ⊂Y and for eachN {y0, . . . , yn} ∈ Y, there exists a
compactFC-subspaceLNofY containingY0Nsuch that the sety∈Y0ccl{x∈X:∃i∈
Ix, πiy/∈Aix}is empty or compact, whereIx {i∈I:Aix/∅}.
Then there existsx∈Xsuch thatAix ∅for eachi∈I.
Proof. For eachx∈X,Ix {i∈I :Aix/∅}. DefineA:X → 2Y by
Ax
⎧ ⎪ ⎨ ⎪ ⎩
i∈Ix
π−1
i Aix, ifIx/∅,
∅, ifIx ∅.
3.11
Then for eachx∈X,Ax/∅if and only ifIx/∅. Letx∈XwithAx/∅, then there exists
j0∈Ixsuch thatAj0x/∅. By conditionii, there existsi0∈Ixsuch thatx∈cint{x∈X:
Ai0x/∅}. SinceAi0is generalizedGB-majorized, there exist an open neighborhoodNxof
xinXand a generalizedGB-majorantAx,i0ofAi0atxsuch that
aAi0z⊂Ax,i0zfor allz∈Nx,
bfor eachN{y0, . . . , yn} ∈ Yand{yr0, . . . , yrk} ⊂N,
F ϕNΔk ⎛ ⎝k
j0
cintA−x,i1 0
πi0
yrj
⎞
⎠∅, 3.12
cA−1
x,i0 :Yi → 2
Xis transfer compactly open inY i,
dfor eachN ∈ {x∈X :Ai0x/∅}, the mapping
x∈NA−x,i10is transfer compactly open inYi.
Without loss of generality, we can assume that Nx ⊂ cint{x ∈ X : Ai0x/∅}. Hence,
Ai0z/∅for eachz∈Nx. DefineBx,i0:X → 2Y by
Bx,i0z π−
1
i0 Ax,i0z, ∀z∈X. 3.13
We claim thatBx,i0is a generalizedGB-majorant ofAatx. Indeed, we have
afor each z ∈ Nx, Az i∈Izπi−1Aiz ⊂ πi−01Ai0z ⊂ π−
1
i0 Ax,i0z
bfor each N {y0, . . . , yn} ∈ Y and M {yr0, . . . , yrk} ⊂ N, if u ∈ k
j0cintBx,i−10πi0yrj, then M ⊂ cintBx,i0u. It is easy to see that πi0M ⊂ cintπi0Bx,i0u, so thatπi0M⊂cintAx,i0u, i.e.,u∈kj0cintA−1
x,i0πi0yrjand henceu /∈FϕNΔkbyb. It follows that
F ϕNΔk ⎛ ⎝k
j0
cintB−x,i10πi0
yrj
⎞
⎠∅, 3.14
cfor eachy∈Y, we have that
B−1
x,i0 y
A−1
x,i0 πi0 y
3.15
is transfer compactly open inYbyc.
HenceBx,i0is a generalizedGB-majorant ofAatx.
For eachN∈ {x∈X:Ai0x/∅}andy∈Y, by3.15, we have
x∈N
B−1
x,i0 y
x∈N
A−1
x,i0 πi0 y
. 3.16
It follows fromdthatx∈NB−1
x,i0is transfer compactly open inY.
HenceA:X → 2Y is generalizedGB-majorized. By conditioniii, we have
{x∈X:Ax/∅} ⊂ {x∈X:Ai0x/∅} ⊂Ei0. 3.17
By conditioniv, there exists a nonempty setY0 ⊂Y and for eachN {y0, . . . , yn} ∈ Y,
there exists a compactFC-subspaceLN ofY containingY0N. By the definition ofA, for
eachy∈Y0, we have
A−1 yx∈X:y∈Ax
⎧ ⎨
⎩x∈X:y∈
i∈Ix
π−1
i Aix ⎫ ⎬ ⎭
⎧ ⎨
⎩x∈X:πi y∈
i∈Ix
Aix ⎫ ⎬ ⎭.
3.18
It follows from condition iv that K y∈Y0cintA−1yc
y∈Y0ccl{x ∈ X : ∃i ∈
Ix, πiy∈/Aix}is empty or compact and hence all conditions ofTheorem 3.5are satisfied. By Theorem 3.5, there exists x ∈ X such thatAx ∅ which impliesIx ∅, that is,
Theorem 3.12. LetXbe a topological space, and letIbe any index set. For eachi∈I, letYi, ϕNibe
anCFC-space, and letY i∈IYi. LetF∈ By, xbe a compact mapping such that for eachi∈I,
iletAi:X → 2Yi be a generalizedGB-majorized mapping;
iii∈I{x∈X :Aix/∅}i∈Icint{x∈X:Aix/∅}.
Then there existsx∈Xsuch thatAix ∅for eachi∈I.
Proof. Since for eachi∈I, letYi, ϕNibe anCFC-space, then for eachNi ∈ Yi, there exists a compactFC-subspaceLNi ofYi containingNi. LetLN
i∈ILNi andN
i∈INi ∈ Y, thenLNis a compactFC-subspace ofY for eachN∈ Y,LNis a compactFC-subspace ofY containingN. HenceY, ϕNis also anCFC-space.
For eachx∈X,Ix {i∈I :Aix/∅}. DefineA:X → 2Y
Ax
⎧ ⎪ ⎨ ⎪ ⎩
i∈Ix
π−1
i Aix, ifIx/∅,
∅, ifIx ∅.
3.19
Then for eachx∈X,Ax/∅if and only ifIx/∅. By using similar argument as in the proof ofTheorem 3.11, we can show thatA : X → 2Y is a generalizedGB-majorized mapping.
By Corollary 3.9, there existsx ∈ X such thatAx ∅, and soIx ∅. Hence, we have
Aix ∅for eachi∈I.
Theorem 3.13. LetXbe a topological space, letKbe a nonempty compact subset ofX,and letIbe any index set. For eachi∈I, letYi, ϕNibe anFC-space, and letY
i∈IYisuch thatY, ϕNis
anFC-space defined as inLemma 2.2. LetF ∈ BY, Xsuch that for eachi∈I,Ai :X → 2Yi be a
generalizedGB-mapping such that
ifor eachi∈IandNi∈ Yi, there exists a compactFC-subspaceLNi ofYicontainingNi and for eachx∈X\K, there existsi∈IsatisfyingLNi
cintAix/∅.
Then there existsx∈Ksuch thatAix ∅for eachi∈I.
Proof. Suppose that the conclusion is not true, then for eachx∈K, there existsi∈Isuch that
Aix/∅. SinceAiis a generalizedGB-mapping,A−i1 is transfer compactly open valued. By
Lemma 3.1, we have
K⊂ i∈I
yi∈Yi
cintA−i1 yi. 3.20
SinceK is compact, there exists a finite setJ ⊂ I such that for eachj ∈J, there existsNj {y1
j, yj2, . . . , ymjj} ⊂YjwithK ⊂j∈Jmkj1cintA−j1yjk. It follows that for eachx∈K, there
exists aj ∈J ⊂ Isuch thatNjcintAjx/∅. We may take any fixedy0 y0ii∈I ∈Y. For eachi∈I\J, letNi{y0
i}. By conditioni, for eachi∈I, there exists a compactFC-subspace
LNiofYicontainingNiand for eachx∈X\K, there existsi∈IsatisfyingLNi
cintAix/∅. Hence for eachx∈X, there existsi∈Isuch thatLNi
cintAix/∅. LetLNi∈ILNi, then
thenX0is compact inX. DefineAi:X0 → 2LNi byAix LNi
Aix. For eachyi ∈LNi, we have
A i
−1 y
i
x∈X0:yi ∈LNi
Aix
X0
A−1
i yi
. 3.21
SinceA−i1yi is transfer compactly open valued in Yi for eachi ∈ I and yi ∈ Yi, so that we claim thatAi−1yiis transfer open valued inLNi. Noting that eachAiis a generalized
GB-mapping, for eachM{y0, . . . , ym} ∈ LN ⊂ YandM1{yr0, . . . , yrk} ⊂M, we have
F ϕMΔk ⎛ ⎝k
j0
cint Ai−1 πi yrj ⎞
⎠F ϕMΔk ⎛ ⎝k
j0
cintX0
A−1
i πi yrj ⎞ ⎠
⊂F ϕMΔk ⎛ ⎝k
j0
cintA−i1 πi yrj ⎞ ⎠∅,
3.22
whereΔkco{eij :j0, . . . , k}.
Hence for eachi∈I,Aiis a generalizedGB-mapping and hence it is also a generalized
GB-majorized mapping. All conditions ofCorollary 3.8are satisfied. ByCorollary 3.8, there
exists x ∈ X0 ⊂ X such that Aix LNi
Aix ∅ for each i ∈ I, so we have
LNi
cintAix⊂LNi
Aix Aix ∅which contradicts the fact that for eachx∈X\K there exists i ∈ I such that LNicintAix/∅. Therefore, there exists x ∈ K such that
Aix ∅for eachi∈I.
Remark 3.14. Theorem 3.11 generalizes 4, Theorem 2.5 in several aspects. Theorem 3.12 improves 2, Theorem 3from convex subsets of topological vector spaces toCFC-spaces without linear structure and from a family of LS-majorized mappings to the family of
generalizedGB-majorized mappings. Theorem 3.13generalizes4, Theorem 2.6in several
aspects:1.1fromG-convex spaces toFC-spaces without linear structure;1.2from aGB -mapping to a generalized GB-mapping;1.3 conditioni ofTheorem 3.13is weaker than
condition iof 4, Theorem 2.6.Theorem 3.13 improves and generalizes2, Theorem 7 in the following ways:2.1from nonempty convex subsets of Hausdorfftopological vector spaces toFC-space without linear structure;2.2from the family ofLS-majorized mappings
to the family of generalizedGB-majorized mappings.
Acknowledgment
This work is supported by a Grant of the Natural Science Development Foundation of CUIT of Chinano. CSRF200709.
References
2 P. Deguire, K. K. Tan, and G. X.-Z. Yuan, “The study of maximal elements, fixed points forLS
-majorized mappings and their applications to minimax and variational inequalities in product topological spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 37, no. 7, pp. 933–951, 1999.
3 X.-P. Ding, “Maximal element principles on generalized convex spaces and their applications,” inSet Valued Mappings with Applications in Nonlinear Analysis, R. P. Argawal, Ed., vol. 4, pp. 149–174, Taylor & Francis, London, UK, 2002.
4 X.-P. Ding, “Maximal elements for GB-majorized mappings in product G-convex spaces and
applications—I,”Applied Mathematics and Mechanics, vol. 24, no. 6, pp. 583–594, 2003.
5 X.-P. Ding, “Maximal elements for GB-majorized mappings in product G-convex spaces and
applications—II,”Applied Mathematics and Mechanics, vol. 24, no. 9, pp. 899–905, 2003.
6 X.-P. Ding and F. Q. Xia, “Equilibria of nonparacompact generalized games with LFc-majorized
correspondence inG-convex spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 6, pp. 831–849, 2004.
7 X.-P. Ding, J.-C. Yao, and L.-J. Lin, “Solutions of system of generalized vector quasi-equilibrium problems in locallyG-convex uniform spaces,”Journal of Mathematical Analysis and Applications, vol. 298, no. 2, pp. 398–410, 2004.
8 X.-P. Ding, “Maximal element theorems in productFC-spaces and generalized games,”Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 29–42, 2005.
9 X.-P. Ding, “Maximal elements ofGKKM-majorized mappings in product FC-spaces and applications. I,”Nonlinear Analysis: Theory, Methods & Applications, vol. 67, no. 3, pp. 963–973, 2007.
10 W. K. Kim and K.-K. Tan, “New existence theorems of equilibria and applications,”Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 1, pp. 531–542, 2001.
11 L.-J. Lin, Z.-T. Yu, Q. H. Ansari, and L.-P. Lai, “Fixed point and maximal element theorems with applications to abstract economies and minimax inequalities,”Journal of Mathematical Analysis and Applications, vol. 284, no. 2, pp. 656–671, 2003.
12 S. P. Singh, E. Tarafdar, and B. Watson, “A generalized fixed point theorem and equilibrium point of an abstract economy,”Journal of Computational and Applied Mathematics, vol. 113, no. 1-2, pp. 65–71, 2000.
13 G. X.-Z. Yuan,KKM Theory and Applications in Nonlinear Analysis, vol. 218 ofMonographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1999.
14 Z.-F. Shen, “Maximal element theorems ofH-majorized correspondence and existence of equilibrium for abstract economies,”Journal of Mathematical Analysis and Applications, vol. 256, no. 1, pp. 67–79, 2001.
15 X.-P. Ding, “Fixed points, minimax inequalities and equilibria of noncompact abstract economies,” Taiwanese Journal of Mathematics, vol. 2, no. 1, pp. 25–55, 1998.
16 X.-P. Ding and G. X.-Z. Yuan, “The study of existence of equilibria for generalized games without lower semicontinuity in locally topological vector spaces,” Journal of Mathematical Analysis and Applications, vol. 227, no. 2, pp. 420–438, 1998.
17 C.-M. Chen and T.-H. Chang, “Some results for the family KKMX, Yand theΦ-mapping,”Journal of Mathematical Analysis and Applications, vol. 329, no. 1, pp. 92–101, 2007.
18 S. Park, “Coincidence theorems for the admissible multimaps on generalized convex spaces,”Journal of the Korean Mathematical Society, vol. 37, no. 4, pp. 885–899, 2000.
19 S. Park and H. Kim, “Foundations of the KKM theory on generalized convex spaces,”Journal of Mathematical Analysis and Applications, vol. 209, no. 2, pp. 551–571, 1997.
