Some Maximal Elements’ Theorems in FC -Spaces

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Volume 2009, Article ID 905605,12pages doi:10.1155/2009/905605

Research Article

Some Maximal Elements’ Theorems in FC -Spaces

Rong-Hua He

^{1, 2}

and Yong Zhang

¹

1Department 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, ϕNi_i∈Ibe a family of FC-spaces. For eachi∈I, letAi:X → 2^Yⁱbe 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 → 2^Yⁱ for each i ∈ I which involve a set-valued mapping F ∈ BY, X. The notion of generalized G_B-majorized mappings unifies and generalizes the corresponding notions of GB-majorized mappings in 4; LS-majorized mappings in 2, 13; H-majorized mappings in 14. Some new existence theorems of maximal elements for generalized G_B-majorized mappings are proved under noncompact setting of FC-spaces. Our results improve and generalize the corresponding results in 2,4,14–16.

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2. Preliminaries

LetX andY be two nonempty sets. We denote by 2^Y 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. LetΔ_n 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 → 2^Y 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

cclA

B⊂X :A⊂ B, Bis compactly closed inX , cintA

B⊂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, cclA

K cl_KA

Kand cintA

Kint_KA K, whereclKA

K resp., intKA

Kdenotes the closureresp., interiorofA

K inK. A set-valued mappingT : X → 2^Y is transfer compactly open valued onX see17if for eachx∈Xandy∈Tx, there existsx∈Xsuch thaty∈cintTx. LetA_i i1, . . . , mbe transfer compactly open valued, then_m

i1cintAi cint_m

i1Ai. 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,U^k_cY, Xin19 and so on as proper subclasses. Now we introduce the following definition.

Definition 2.1. AnFC-spaceY, ϕNis said to be anCFC-space if for eachN ∈ Y, there exists a compactFC-subspaceL_NofY containingN.

Y, ϕ_N 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, letY_i, ϕ_N_ibe anFC-space,Y

i∈IY_i andϕ_N

i∈Iϕ_N_i. ThenY, ϕ_Nis also anFC-space.

LetX be a topological space, and letIbe any index set. For eachi∈I, letYi, ϕNi_i∈I be anFC-space, and letY

i∈IY_isuch thatY, ϕ_Nis anFC-space defined as inLemma 2.2.

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LetF∈ BY, Xand for eachi∈I, letAi:X → 2^Yⁱbe a set-valued mapping. For eachi∈I, 1A_i:X → 2^Yⁱ is said to be a generalizedG_B-mapping if

afor each N {y0, . . . , yn} ∈ Y and {yi0, . . . , yik} ⊂ N, FϕNΔk _k

j0cintA⁻¹_i πiyij ∅, where πi is the projection of Y onto Yi and Δ_kco{e_i_j :j0, . . . , k};

bA⁻¹_i y_i {x∈X:y_i∈A_ix}is transfer compactly open inY_ifor eachy_i∈Y_i; 2A_x,i : X → 2^Yⁱ is said to be a generalized G_B-majorant ofA_i atx ∈ X ifA_x,i is a generalized GB-mapping and there exists an open neighborhoodNx ofxin X such thatA_iz⊂A_x,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 : A_ix/∅}, the mapping

x∈NA⁻¹_x,iis transfer compactly open inY_i;

4Ai is said to be a generalized GB-majorized if for each x ∈ X, there exists a generalizedG_B-majorantA_x,iofA_iatx.

Then{Ai}_i∈I is said to be a family of generalized GB-mappingsresp.,GB-majorant mappingsif for eachi∈I, Ai :X → 2^Yⁱ 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 generalizedG_B-majorant mappings. IfF Sis a single-valued mapping andA_ixis anFC-subspace ofY_ifor eachx∈X, then conditiony_i/∈A_iSyfor eachy∈Y implies that conditionain1holds. Indeed, ifais false, then there existN {y0, . . . , y_n} ∈ Y,{y_i₀, . . . , y_i_k} ⊆N, andy∈ϕ_NΔ_ksuch thatFy Sy∈_k

j0A⁻¹_i π_iy_i_j and henceπ_iy_i_j ∈ A_iSyfor each j 0, . . . , k. It follows fromy ∈ ϕ_NΔ_kthatπ_iy ∈ ϕNiΔkwhereNi πiN. It follows fromAiSybeing anFC-subspace ofYithatπiy∈ ϕ_N_iΔ_k ⊂ A_iSywhich contradicts condition y_i/∈A_iSyfor eachy ∈ Y. Hence each L_S-mappingresp.,L_S-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 → 2^Y 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 subsetO_y of X (which may be empty) such that Oy

K⊂G⁻¹yandK

y∈YOy K;

3there exists a set-valued mappingF:X → 2^Y such that for eachy∈Y, F⁻¹yis open or empty inX,F⁻¹yK⊂G⁻¹y,∀y∈Y,andK

y∈YF⁻¹yK;

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4for each x ∈ K, there exists y ∈ Y such that x ∈ cintG⁻¹y

K and K

y∈YcintG⁻¹y

K

y∈YG⁻¹y K;

5G⁻¹:Y → 2^Xis transfer compactly open valued onY.

Lemma 3.28. LetXbe a topological space, and letY, ϕNbe anFC-space,F ∈ BY, Xand A:X → 2^Y such that

ifor eachN{y0, . . . , y_n} ∈ Yand for each{y_i₀, . . . , y_i_k} ⊆N,

F ϕ_NΔ_k

⎛

⎝^k

j0

cintA⁻¹ y_i_j⎞

⎠∅, 3.1

iiA⁻¹:Y → 2^Xis transfer compactly open valued;

iiithere exists a nonempty setY0 ⊂Y and for eachN {y0, . . . , y_n} ∈ Y, there exists a compactFC-subspaceLN ofY containingY0∪N such thatK

y∈Y0cintA⁻¹y^cis empty or compact inX, wherecintA⁻¹y^cdenotes the complement of cintA⁻¹y.

Then there exists a pointx∈Xsuch thatAx ∅.

Theorem 3.3. LetXbe a topological space, letKbe a nonempty compact subset ofX, and letY, ϕ_N be anFC-space,F ∈ BY, XandA:X → 2^Y be a generalizedGB-mapping such that

ifor eachN{y0, . . . , yn} ∈ Y, there exists a compactFC-subspaceLNofY containing Nsuch that for eachx∈X\K, L_N

cintAx/∅.

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-subspaceL_NofY containingN andFLNis compact inX, and hence we have

FL_N

y∈LN

cintA⁻¹ y FL_N

. 3.4

By using similar argument as in the proof ofLemma 3.2, we can show that there existsx∈X such thatAx ∅. Conditioniimplies thatxmust be inK. This completes the proof.

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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 G_B-mappings to generalizedGB-mappings.

Theorem 3.5. LetX be a topological space, and letY, ϕNbe anFC-space. LetF ∈ BY, Xand A:X → 2^Y be a generalizedG_B-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, . . . , y_n} ∈ Y, there exists a compact FC-subspace L_N of Y containing Y0 ∪ N such that the set K

y∈Y0cintA⁻¹y^cis empty or compact.

Then there exists a pointx∈Xsuch thatAx ∅.

Proof. Suppose thatAx/∅for eachx∈X. SinceAis a generalizedG_B-majorized, for each x ∈ X, there exists an open neighborhoodNxofx inX and a generalized GB-mapping Ax:X → 2^Y such that

aAz⊂A_xzfor eachz∈Nx,

bfor each_k N {y0, . . . , yn} ∈ Y and {yi0, . . . , yik} ⊆ N, FϕNΔk

j0cintA⁻¹_x y_i_j ∅,

cA⁻¹_x is transfer compactly open inY,

dfor anyN∈ {x∈X :Ax/∅}, the mapping

x∈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 mappingB_x:X → 2^Y by

Bxz

⎧⎨

⎩

Axz, ifz∈Ox,

Y, ifz∈X\Ox. 3.5

Then for eachy∈Y, we have B_x⁻¹ y

z∈Ox:y∈Axz z∈X\Ox:y∈Y

A⁻¹_x y

Ox X\Ox

A⁻¹_x y X\Ox

Ox

X\Ox

A⁻¹_x y X\Ox.

3.6

HenceB_x⁻¹yis transfer compactly open inYbyc.

Now define a mappingB:X → 2^Y by

Bz

x∈X

Bxz, ∀z∈X. 3.7

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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 ∅ V_u ∩Ox V_u ∩Ox, and henceB_xz Y for allz ∈ V_u which implies that

Bz

x∈XB_xz _n

i1B_x_izfor allz∈V_u. It follows that for eachy∈Y,

B⁻¹ y

z∈X:y∈Bz

⊃

z∈Vu:y∈Bz

z∈Vu:y∈ⁿ

i1

Bxiz

Vu

_n

i1

B_x⁻¹_i y

. 3.8

For any nonempty compact subsetCofX and each y ∈ Y, ifv ∈ V_u∩_n

i1B⁻¹_x_iyC ⊂ B⁻¹y

C. SinceV_uis open inX, it follows fromdthat there existsy∈Y such that

v∈V_u cint

_n

i1

B⁻¹_x_i y

Ccint

V_uⁿ

i1

B_x⁻¹_i y C cintB⁻¹ y

C.

3.9

This proves thatB⁻¹:Y → 2^Xis 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⁻¹_x₀yij, and hencet /∈FϕNΔkby b. Hence we have

F ϕNΔk

⎛

⎝^k

j0

cintB⁻¹ yij

⎞

⎠∅ 3.10

for each N {y0, . . . , y_n} ∈ Y and N1 {y_i₀, . . . , y_i_k} ⊆ N. This shows that B is a generalizedGB-mapping.

For eachz ∈X, ify /∈Bz, then there exists anx0 ∈X such thaty /∈B_x₀z A_x₀z 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⁻¹y^cis empty or compact. Note thatAz⊂ Bzfor eachz∈ X impliescintB⁻¹y^c ⊂ cintA⁻¹y^cfor eachy ∈ Y. HenceK

y∈Y0cintB⁻¹y^c ⊂ K andKis empty or compact. ByLemma 3.2, there exists a pointx∈Xsuch thatBx ∅, and henceAx ∅which contradicts the assumption thatAx/∅for eachx ∈ X. Therefore, there existsx∈Xsuch thatAx ∅.

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Theorem 3.6. LetXbe a topological space, letKbe a nonempty compact subset ofXandY, ϕNbe anFC-space. LetF∈ BY, XandA:X → 2^Y be a generalizedG_B-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, L_N

cintAx/∅.

Then there existsx∈Ksuch thatAx ∅.

Proof. Suppose that Ax/∅for each x ∈ X. By using similar argument as in the proof of Theorem 3.5, we can show that there exists a generalizedG_B-mappingB:X → 2^Y 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 G_B-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, ϕ_N is a compact FC-space, then by letting K X Y L_N 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 generalizedG_B-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 → 2^Y be a generalizedG_B-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, ϕ_Nbe anCFC-space. LetF ∈ BY, Xbe a compact mapping andA:X → 2^Y 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 → 2^Ybe 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 aL_S-majorized mapping to a generalizedG_B-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 generalizes2, Theorem 2and4, Corollary 2.4in several aspects.

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Theorem 3.11. LetXbe a topological space, and letIbe any index set. For eachi∈I, letYi, ϕNi be anFC-space, and letY

i∈IY_isuch thatY, ϕ_Nis anFC-space defined as inLemma 2.2. Let F∈ BY, Xsuch that for eachi∈I,

iletA_i:X → 2^Yⁱ be a generalizedG_B-majorized mapping;

ii

i∈I{x∈X :A_ix/∅}

i∈Icint{x∈X:A_ix/∅};

iiithere exists a paracompact subsetE_iofXsuch that{x∈X:A_ix/∅} ⊂E_i;

ivthere exists a nonempty setY0 ⊂Y and for eachN {y0, . . . , y_n} ∈ Y, there exists a compactFC-subspaceLNofY containingY0

Nsuch that the set

y∈Y0ccl{x∈X:∃i∈ Ix, π_iy/∈A_ix}is empty or compact, whereIx {i∈I:A_ix/∅}.

Then there existsx∈Xsuch thatA_ix ∅for eachi∈I.

Proof. For eachx∈X,Ix {i∈I :A_ix/∅}. DefineA:X → 2^Y by

Ax

⎧⎪

⎨

⎪⎩

i∈Ix

π_i⁻¹Aix, 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: A_i₀x/∅}. SinceA_i₀is generalizedG_B-majorized, there exist an open neighborhoodNxof xinXand a generalizedG_B-majorantA_x,i₀ofA_i₀atxsuch that

aA_i₀z⊂A_x,i₀zfor allz∈Nx,

bfor eachN{y0, . . . , y_n} ∈ Yand{y_r₀, . . . , y_r_k} ⊂N,

F ϕ_NΔ_k

⎛

⎝^k

j0

cintA⁻¹_x,i₀ π_i0

y_r_j⎞

⎠∅, 3.12

cA⁻¹_x,i₀ :Y_i → 2^Xis transfer compactly open inY_i, dfor eachN ∈ {x∈X :Ai0x/∅}, the mapping

x∈NA⁻¹_x,i₀is transfer compactly open inYi.

Without loss of generality, we can assume that Nx ⊂ cint{x ∈ X : Ai0x/∅}. Hence, A_i₀z/∅for eachz∈Nx. DefineB_x,i₀:X → 2^Y by

Bx,i0z π_i⁻¹₀ Ax,i0z, ∀z∈X. 3.13

We claim thatBx,i0is a generalizedGB-majorant ofAatx. Indeed, we have afor each z ∈ Nx, Az

i∈Izπ_i⁻¹Aiz ⊂ π_i⁻¹₀ Ai0z ⊂ π_i⁻¹₀ Ax,i0z B_x,i₀z,

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bfor each_k N {y0, . . . , yn} ∈ Y and M {yr0, . . . , yrk} ⊂ N, if u ∈

j0cintB_x,i⁻¹₀π_i₀y_r_j, then M ⊂ cintB_x,i₀u. It is easy to see that π_i₀M ⊂ cintπ_i₀B_x,i₀u, so thatπ_i₀M⊂cintA_x,i₀u, i.e.,u∈_k

j0cintA⁻¹_x,i₀π_i₀y_r_jand henceu /∈FϕNΔkbyb. It follows that

F ϕ_NΔk

⎛

⎝^k

j0

cintB⁻¹_x,i₀ π_i₀

y_r_j⎞

⎠∅, 3.14

cfor eachy∈Y, we have that

B_x,i⁻¹₀ y

A⁻¹_x,i₀ πi0 y

3.15

is transfer compactly open inYbyc.

HenceB_x,i₀is a generalizedG_B-majorant ofAatx.

For eachN∈ {x∈X:A_i₀x/∅}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 fromdthat

x∈NB_x,i⁻¹₀is transfer compactly open inY.

HenceA:X → 2^Y is generalizedG_B-majorized. By conditioniii, we have

{x∈X:Ax/∅} ⊂ {x∈X:A_i₀x/∅} ⊂E_i₀. 3.17

By conditioniv, there exists a nonempty setY0 ⊂Y and for eachN {y0, . . . , yn} ∈ Y, there exists a compactFC-subspaceLN ofY containingY0

N. By the definition ofA, for eachy∈Y0, we have

A⁻¹ y

x∈X:y∈Ax

⎧⎨

⎩x∈X:y∈

i∈Ix

π_i⁻¹Aix

⎫⎬

⎭

⎧⎨

⎩x∈X:π_i y

∈

i∈Ix

A_ix

⎫⎬

⎭.

3.18

It follows from condition iv that K

y∈Y0cintA⁻¹y^c

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, A_ix ∅for eachi∈I.

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Theorem 3.12. LetXbe a topological space, and letIbe any index set. For eachi∈I, letYi, ϕNibe anCFC-space, and letY

i∈IY_i. LetF∈ By, xbe a compact mapping such that for eachi∈I, iletAi:X → 2^Yⁱ be a generalizedGB-majorized mapping;

ii

i∈I{x∈X :A_ix/∅}

i∈Icint{x∈X:A_ix/∅}.

Then there existsx∈Xsuch thatA_ix ∅for eachi∈I.

Proof. Since for eachi∈I, letY_i, ϕ_N_ibe anCFC-space, then for eachN_i ∈ Y_i, there exists a compactFC-subspaceLNi ofYi containingNi. LetLN

i∈ILNi andN

i∈INi ∈ Y, thenL_Nis a compactFC-subspace ofY for eachN∈ Y,L_Nis a compactFC-subspace ofY containingN. HenceY, ϕ_Nis also anCFC-space.

For eachx∈X,Ix {i∈I :Aix/∅}. DefineA:X → 2^Y

Ax

⎧⎪

⎨

⎪⎩

i∈Ix

π_i⁻¹Aix, 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 → 2^Y is a generalizedGB-majorized mapping.

By Corollary 3.9, there existsx ∈ X such thatAx ∅, and soIx ∅. Hence, we have A_ix ∅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,A_i :X → 2^Yⁱ be a generalizedGB-mapping such that

ifor eachi∈IandN_i∈ Y_i, there exists a compactFC-subspaceL_N_i ofY_icontainingN_i and for eachx∈X\K, there existsi∈IsatisfyingLNi

cintAix/∅.

Then there existsx∈Ksuch thatA_ix ∅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⁻¹_i is transfer compactly open valued. By Lemma 3.1, we have

K⊂

i∈I

yi∈Yi

cintA⁻¹_i yi

. 3.20

SinceK is compact, there exists a finite setJ ⊂ I such that for eachj ∈J, there existsN_j {y¹_j, y_j², . . . , y^m_j^j} ⊂Y_jwithK ⊂

j∈Jmj

k1cintA⁻¹_j y_j^k. It follows that for eachx∈K, there exists aj ∈J ⊂ Isuch thatNj

cintAjx/∅. We may take any fixedy⁰ y⁰_i_i∈I ∈Y. For eachi∈I\J, letN_i{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/∅. LetLN

i∈ILNi, then L_Nis a compactFC-subspace ofY and hence it is also a compactCFC-space. LetX0FL_N,

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thenX0is compact inX. DefineA_i:X0 → 2^L^Ni byA_ix LNi

Aix. For eachyi ∈LNi, we have

A_i₋₁ yi

x∈X0:yi ∈LNi

Aix X0

A⁻¹_i yi

. 3.21

SinceA⁻¹_i yi is transfer compactly open valued in Yi for eachi ∈ I and yi ∈ Yi, so that we claim thatA_i⁻¹y_iis transfer open valued inL_N_i. Noting that eachA_iis a generalized G_B-mapping, for eachM{y⁰, . . . , y^m} ∈ L_N ⊂ YandM1{y^r⁰, . . . , y^r^k} ⊂M, we have

F ϕ_MΔ_k

⎛

⎝^k

j0

cint A_i₋₁

π_i y^r^j⎞

⎠F ϕ_MΔ_k

⎛

⎝^k

j0

cint X0

A⁻¹_i π_i y^r^j⎞

⎠

⊂F ϕMΔk⎛

⎝^k

j0

cintA⁻¹_i πi y^r^j⎞

⎠∅, 3.22 whereΔkco{eij :j0, . . . , k}.

Hence for eachi∈I,A_iis a generalizedG_B-mapping and hence it is also a generalized G_B-majorized mapping. All conditions ofCorollary 3.8are satisfied. ByCorollary 3.8, there exists x ∈ X0 ⊂ X such that A_ix LNi

Aix ∅ for each i ∈ I, so we have LNi

cintAix⊂LNi

Aix A_ix ∅which contradicts the fact that for eachx∈X\K there exists i ∈ I such that L_N_i

cintA_ix/∅. 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 aG_B- 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.

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