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 noncom
Trang 1Volume 2009, Article ID 905605, 12 pages
doi:10.1155/2009/905605
Research Article
1 Department of Mathematics, Chengdu University of Information Technology,
Chengdu, Sichuan 610103, China
2 Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Rong-Hua He,ywlcd@cuit.edu.cn
Received 30 March 2009; Accepted 1 September 2009
Recommended by Nikolaos Papageorgiou
LetI be a finite or infinite index set, let X be a topological space, and let Y i , ϕ N ii∈Ibe a family of
FC-spaces For each i ∈ I, let A i:X → 2 Y ibe 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
Copyrightq 2009 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 12 and 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 A i : X → 2 Y i 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 mappings in 2, 13; H-majorized mappings in 14 Some new existence theorems of maximal elements for generalized GB-majorized mappings are proved under noncompact setting of FC-spaces Our results improve and generalize the corresponding results in
2,4,14–16
Trang 22 Preliminaries
LetX and Y 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 of A Let Δ n denote the standardn-dimensional simplex with
the vertices{e0, , e n } If J is a nonempty subset of {0, 1, , n}, we will denote by Δ J the convex hull of the vertices{e j:j ∈ J}.
LetX and Y be two sets, and let T : X → 2 Y be a set-valued mapping We will use the following notations in the sequel:
i Tx {y ∈ Y : y ∈ Tx},
ii TA x∈A Tx,
iii T−1y {x ∈ X : y ∈ Tx}.
For topological spacesX and Y, a subset A of X is said to be compactly open resp.,
compactly closed if for each nonempty compact subset K of X, A∩K is open resp., closed in
K The compact closure of A and the compact interior of A see 17 are defined, respectively, by
cclA
B ⊂ X : A ⊂ B, B is compactly closed in X,
cintA
B ⊂ X : B ⊂ A, B is compactly open in X.
2.1
It is easy to see that cclX \ A X \ cint A, int A ⊂ cint A ⊂ A, A ⊂ ccl A ⊂ clA, A is compactly openresp., compactly closed in X if and only if A cint A resp., A ccl A For
each nonempty compact subsetK of X, ccl AK cl K AK and cint AK int K AK,
wherecl K AK resp., int K AK denotes the closure resp., interior of AK in K A
set-valued mappingT : X → 2 Y is transfer compactly open valued onX see 17 if for eachx ∈ X and y ∈ Tx, there exists x∈ X such that y ∈ cint Tx Let A i i 1, , m be
transfer compactly open valued, thenm
i1cintA i cintm i1 A i 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 class BY, X of better admissible mapping were
introduced by Ding in8 Note that the class BY, X of better admissible mapping includes many important classes of mappings, for example, the classBY, X in 18, U k
c Y, X in 19
and so on as proper subclasses Now we introduce the following definition
Definition 2.1 An FC-space Y, ϕ N is said to be an CFC-space if for each N ∈ Y, there
exists a compactFC-subspace L NofY containing N.
Y, ϕ N be a G-convex space, let the notion of CG-convex space was introduced by
Ding in4
and ϕ N i∈I ϕ N i Then Y, ϕ N is also an FC-space.
LetX be a topological space, and let I be any index set For each i ∈ I, let Y i , ϕ N ii∈I
be anFC-space, and let Y i∈I Y isuch thatY, ϕ N is an FC-space defined as inLemma 2.2
Trang 3LetF ∈ BY, X and for each i ∈ I, let A i:X → 2 Y ibe a set-valued mapping For eachi ∈ I,
1 A i:X → 2 Y i is said to be a generalizedGB-mapping if
a for each N {y 0, , y n } ∈ Y and {y i0 , , y i k } ⊂ N, Fϕ NΔk
k j0cintA−1
i π i y i j ∅, where π i is the projection of Y onto Y i and
Δk co{e i j :j 0, , k};
b A−1
i y i {x ∈ X : y i ∈ A i x} is transfer compactly open in Y ifor eachy i ∈ Y i;
2 A x,i : X → 2 Y i is said to be a generalized GB-majorant ofA i atx ∈ X if A x,i is a generalized GB-mapping and there exists an open neighborhoodNx of x in X
such thatA i z ⊂ A x,i z for all z ∈ Nx;
3 A i is said to be a generalizedGB-majorized if for eachx ∈ X with A i x / ∅, there
exists a generalized GB-majorant A x,i of A i at x, and for any N ∈ {x ∈ X :
A i x / ∅}, the mappingx∈N A−1
x,iis transfer compactly open inY i;
4 A i is said to be a generalized GB-majorized if for each x ∈ X, there exists a
generalizedGB-majorantA x,iofA iatx.
Then{A i}i∈I is said to be a family of generalized GB-mappingsresp., GB-majorant mappings if for each i ∈ I, Ai :X → 2 Y i is a generalizedGB-mappingresp., GB-majorant mapping
If for eachi ∈ I, let Y i , ϕ N i be a G-convex space, a family of GB-mappingsresp., GB -majorant mappings were introduced by Ding in 4 Clearly, each family of generalized GB -mappings must be a family of generalizedGB-majorant mappings IfF S is a single-valued
mapping andA i x is an FC-subspace of Y ifor eachx ∈ X, then condition y i /∈ A i Sy for
eachy ∈ Y implies that condition a in 1 holds Indeed, if a is false, then there exist N
{y0, , y n } ∈ Y, {y i0 , , y i k } ⊆ N, and y ∈ ϕ NΔk such that Fy Sy ∈k
j0 A−1
i π i y i j and henceπ i y i j ∈ A i Sy for each j 0, , k It follows from y ∈ ϕ NΔk that π i y ∈
ϕ N iΔk where N i π i N It follows from A i Sy being an FC-subspace of Y ithatπ i y ∈
ϕ N iΔk ⊂ A i Sy which contradicts condition y i /∈ A i Sy for each y ∈ Y Hence each
LS-mappingresp., LS-majorant mapping introduced by Deguire et al see 2, 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
let G : X → 2 Y be a set-valued mapping such that Gx / ∅ for each x ∈ K Then the following conditions are equivalent:
1 G have the compactly local intersection property;
2 for each y ∈ Y, there exists an open subset O y of X (which may be empty) such that
O y
K ⊂ G−1y and K y∈Y O y
K;
3 there exists a set-valued mapping F : X → 2 Y such that for each y ∈ Y, F−1y is open or
empty in X, F−1yK ⊂ G−1y, ∀y ∈ Y, and K y∈Y F−1yK;
Trang 44 for each x ∈ K, there exists y ∈ Y such that x ∈ cint G −1yK and K
y∈Y cint G−1yK y∈Y G−1yK;
5 G−1:Y → 2 X is transfer compactly open valued on Y.
A : X → 2 Y such that
i for each N {y0, , y n } ∈ Y and for each {y i0 , , y i k } ⊆ N,
F ϕ NΔk
⎛
⎝k
j0
cintA−1 y i j
⎞
⎠ ∅, 3.1
ii A−1:Y → 2 X is transfer compactly open valued;
iii there 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 K y∈Y0 cint A−1y c is empty or compact in X, where cint A−1y c denotes the complement of cint A−1y.
Then there exists a point x ∈ X such that Ax ∅.
be an FC-space, F ∈ BY, X and A : X → 2 Y be a generalized GB-mapping such that
i for each N {y0, , y n } ∈ Y, there exists a compact FC-subspace L N of Y containing
N such that for each x ∈ X \ K, L NcintAx / ∅.
Then there exists a point x ∈ K such that Ax ∅.
Proof Suppose that Ax / ∅ for each x ∈ X Since A is a generalized GB-mapping,A−1 is transfer compactly open valued ByLemma 3.1, we have
K
y∈Y
cintA−1
y K. 3.2
SinceK is compact, there exists a finite set N {y o , , y n } ∈ Y such that
K n
i0
cintA−1
y i K. 3.3
By conditioni and F ∈ BY, X, there exists a compact FC-subspace L NofY containing N
andFL N is compact in X, and hence we have
FL N
y∈L N
cintA−1
y FL N. 3.4
By using similar argument as in the proof ofLemma 3.2, we can show that there exists x ∈ X
such thatAx ∅ Condition i implies that x must be in K This completes the proof.
Trang 5Remark 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
A : X → 2 Y be a generalized GB-majorized mapping such that
i there exists a paracompact subset E of X such that {x ∈ X : Ax / ∅} ⊂ E;
ii there 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∈Y0 cint A−1y c is empty or compact.
Then there exists a point x ∈ X such that Ax ∅.
Proof Suppose that Ax / ∅ for each x ∈ X Since A is a generalized GB-majorized, for each
x ∈ X, there exists an open neighborhood Nx of x in X and a generalized GB-mapping
A x:X → 2 Y such that
a Az ⊂ A x z for each z ∈ Nx,
b for each N {y 0, , y n } ∈ Y and {y i0 , , y i k } ⊆ N, Fϕ NΔk
k
j0cintA−1
x y i j ∅,
c A−1
x is transfer compactly open inY,
d for any N ∈ {x ∈ X : Ax / ∅}, the mappingx∈N A−1
x is transfer compactly open
inX.
SinceAx / ∅ for each x ∈ X, it follows from condition i that X {x ∈ X : Ax / ∅} E is
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 each x ∈ X, Ox ⊂ Nx
sinceX is normal For each x ∈ X, define a mapping B x:X → 2 Y by
B x z
⎧
⎨
⎩
A x z, if z ∈ Ox,
Then for eachy ∈ Y, we have
B−1
x
y z ∈ Ox : y ∈ A x z
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−1
x y is transfer compactly open in Y by c.
Now define a mappingB : X → 2 Y by
Bz
x∈X
B x z, ∀z ∈ X. 3.7
Trang 6We claim thatB is a generalized GB-mapping andAz ⊂ Bz for each z ∈ X Indeed, for
any nonempty compact subsetC of X and each y ∈ Y with B−1y ∩ C / ∅, we may take any
fixedu ∈ B−1y ∩ C Since {Ox : x ∈ X} is locally finite, there exists an open neighborhood
V uofu in X such that {x ∈ X : V u ∩ Ox / ∅} {x1, , x n } is a finite set If x /∈ {x1, , x n}, then ∅ V u ∩ Ox V u ∩ Ox, and hence B x z Y for all z ∈ V u which implies that
Bz x∈X B x z n
i1 B x i z for all z ∈ V u It follows that for eachy ∈ Y,
B−1
y z ∈ X : y ∈ Bz⊃z ∈ V u:y ∈ Bz
z ∈ V u:y ∈n
i1
B x i z
V u
n i1
B−1
x i
y
. 3.8
For any nonempty compact subsetC of X and each y ∈ Y, if v ∈ V u∩ n
i1 B−1
x i yC ⊂
B−1yC Since V uis open inX, it follows from d that there exists y∈ Y such that
v ∈ V ucint
n
i1
B−1
x i
y
C cint
V un
i1
B−1
x i
y
C
cint B−1
y C.
3.9
This proves thatB−1:Y → 2 Xis transfer compactly open valued inY.
On the other hand, for eachN {y0, , y n } ∈ Y and N1 {y i0 , , y i k } ⊆ N, if t ∈
k
j0cintB−1y i j , then N1 ⊂ cint Bt Since there exists x0 ∈ X such that t ∈ Ox0 and N1 ⊂ cintBt ⊂ cint B x0 t cint A x0 t, we have t ∈k j0cintA−1
x0 y i j , and hence t /∈ Fϕ NΔk by
b Hence we have
F ϕ NΔk
⎛
⎝k
j0
cintB−1 y i j
⎞
⎠ ∅ 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, if y /∈ Bz, then there exists an x0 ∈ X such that y /∈ B x0 z A x0 z
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, , y n } ∈ Y, there exists a compact FC-subspace L NofY containing Y0∪ N such that
the setK y∈Y0 cint A−1y cis empty or compact Note thatAz ⊂ Bz for each z ∈ X
impliescint B−1y c ⊂ cint A−1y cfor eachy ∈ Y Hence K y∈Y0 cint B−1y c ⊂ K
andKis empty or compact ByLemma 3.2, there exists a pointx ∈ X such that Bx ∅, and
henceAx ∅ which contradicts the assumption that Ax / ∅ for each x ∈ X Therefore,
there existsx ∈ X such that Ax ∅.
Trang 7Theorem 3.6 Let X be a topological space, let K be a nonempty compact subset of X and Y, ϕ N be
an FC-space Let F ∈ BY, X and A : X → 2 Y be a generalized GB-majorized mapping such that
i there exists a paracompact subset E of X such that {x ∈ X : Ax / ∅} ⊂ E;
ii for each N {y0, , y n } ∈ Y, there exists a compact FC-subspace L N of Y containing
N such that for each x ∈ X \ K, L NcintAx / ∅.
Then there exists x ∈ K such that Ax ∅.
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 generalizedGB-mappingB : X → 2 Y such that
Ax ⊂ Bx for each x ∈ X It follows from condition ii that for each x ∈ X \ K, L N ∩ cintBx / ∅ ByTheorem 3.3, there existsx ∈ K such that Bx ∅, and hence Ax ∅
which contradicts the assumption thatAx / ∅ for each x ∈ X Therefore, there exists x ∈ X
such thatAx ∅ Condition ii implies x ∈ K This completes the proof.
Remark 3.7. Theorem 3.5generalizes4, Theorem 2.3 in 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 13 condition ii ofTheorem 3.5
is weaker than conditionii of 4, Theorem 2.3 If X is compact, condition i is 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 2.3 in the following ways: Section 21 from CH-convex space to FC-space without linear structure;Section 22 from
H-majorized correspondences to generalized GB-majorized mapping;Section 23 condition
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 in 15, Theorem 5.3, and Ding and Yuan in16, Theorem 2.3 in several aspects
BY, X and A : X → 2 Y be a generalized GB-majorized mapping Then there exists a point x ∈ X
such that Ax ∅.
Proof The conclusion ofCorollary 3.8follows fromTheorem 3.6withE K X.
compact mapping and A : X → 2 Y be a generalized GB-majorized mapping Then there exists a point
x ∈ X such that Ax ∅.
Proof Since F is a compact mapping, there exists a compact subset X0ofX such that FY ⊂
X0 The mappingA| X0:X0 → 2Ybe the restriction ofA to X0 It is easy to see thatA| X0is also generalizedGB-majorized ByCorollary 3.8, there exists x ∈ X0such thatA| X0 x Ax
∅
Remark 3.10. Corollary 3.8generalizes Deguire et al.2, Theorem 1 in the following ways:
1.1 from a convex subset of Hausdorff topological vector space to an FC-space without
linear structure;1.2 from a LS-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 generalizes2, Theorem 2 and 4, Corollary 2.4 in several aspects
Trang 8Theorem 3.11 Let X be a topological space, and let I be any index set For each i ∈ I, let Y i , ϕ N i
be an FC-space, and let Y i∈I Y i such that Y, ϕ N is an FC-space defined as in Lemma 2.2 Let
F ∈ BY, X such that for each i ∈ I,
i let A i:X → 2 Y i be a generalized GB-majorized mapping;
iii∈I {x ∈ X : A i x / ∅} i∈Icint{x ∈ X : Ai x / ∅};
iii there exists a paracompact subset E i of X such that {x ∈ X : A i x / ∅} ⊂ E i ;
iv there 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 sety∈Y0ccl{x ∈ X : ∃i ∈
Ix, π i y /∈ A i x} is empty or compact, where Ix {i ∈ I : A i x / ∅}.
Then there exists x ∈ X such that A i x ∅ for each i ∈ I.
Proof For each x ∈ X, Ix {i ∈ I : A i x / ∅} Define A : X → 2 Y by
Ax
⎧
⎪
⎪
i∈Ix
π−1
i A i x, if Ix / ∅,
3.11
Then for eachx ∈ X, Ax / ∅ if and only if Ix / ∅ Let x ∈ X with Ax / ∅, then there exists
j0∈ Ix such that A j0 x / ∅ By condition ii, there exists i0∈ Ix such that x ∈ cint{x ∈ X :
A i0 x / ∅} Since A i0is generalizedGB-majorized, there exist an open neighborhoodNx of
x in X and a generalized GB-majorantA x,i0ofA i0atx such that
a A i0 z ⊂ A x,i0 z for all z ∈ Nx,
b for each N {y0, , y n } ∈ Y and {y r0 , , y r k } ⊂ N,
F ϕ NΔk
⎛
⎝k
j0
cintA−1
x,i0 π i0 y r j
⎞
⎠ ∅, 3.12
c A−1
x,i0 :Y i → 2Xis transfer compactly open inY i,
d for each N ∈ {x ∈ X : A i0 x / ∅}, the mappingx∈N A−1
x,i0is transfer compactly open inY i
Without loss of generality, we can assume that Nx ⊂ cint{x ∈ X : A i0 x / ∅} Hence,
A i0 z / ∅ for each z ∈ Nx Define B x,i0:X → 2 Y by
B x,i0 z π−1
i0 A x,i0 z, ∀z ∈ X. 3.13
We claim thatB x,i0is a generalizedGB-majorant ofA at x Indeed, we have
a for each z ∈ Nx, Az i∈Iz π−1
i A i z ⊂ π−1
i0 A i0 z ⊂ π−1
i0 A x,i0 z
B x,i0 z,
Trang 9b for each N {y 0, , y n } ∈ Y and M {y r0 , , y r k } ⊂ N, if u ∈
k
j0cintB−1
x,i0 π i0 y r j , then M ⊂ cint B x,i0 u It is easy to see that π i0 M ⊂
cintπ i0 B x,i0 u, so that π i0 M ⊂ cint A x,i0 u, i.e., u ∈ k
j0cintA−1
x,i0 π i0 y r j and henceu /∈ Fϕ NΔk by b It follows that
F ϕ NΔk
⎛
⎝k
j0
cintB−1
x,i0 π i0 y r j
⎞
⎠ ∅, 3.14
c for each y ∈ Y, we have that
B−1
x,i0
y A−1
x,i0
π i0
is transfer compactly open inY by c.
HenceB x,i0is a generalizedGB-majorant ofA at x.
For eachN ∈ {x ∈ X : A i0 x / ∅} and y ∈ Y, by 3.15, we have
x∈N
B−1
x,i0
y
x∈N
A−1
x,i0
π i0
It follows fromd thatx∈N B−1
x,i0is transfer compactly open inY.
HenceA : X → 2 Y is generalizedGB-majorized By conditioniii, we have
{x ∈ X : Ax / ∅} ⊂ {x ∈ X : A i0 x / ∅} ⊂ E i0 3.17
By conditioniv, there exists a nonempty set Y0 ⊂ Y and for each N {y0, , y n } ∈ Y,
there exists a compactFC-subspace L N ofY containing Y0
N By the definition of A, for
eachy ∈ Y0, we have
A−1
y x ∈ X : y ∈ Ax
⎧
⎨
⎩x ∈ X:y ∈
i∈Ix
π−1
i A i x
⎫
⎬
⎭
⎧
⎨
⎩x ∈ X:π i
y ∈
i∈Ix
A i x
⎫
⎬
⎭.
3.18
It follows from condition iv that K y∈Y0 cint A−1y c y∈Y0ccl{x ∈ X : ∃i ∈
Ix, π i y /∈ A i x} is empty or compact and hence all conditions ofTheorem 3.5are satisfied
By Theorem 3.5, there exists x ∈ X such that Ax ∅ which implies Ix ∅, that is,
A i x ∅ for each i ∈ I.
Trang 10Theorem 3.12 Let X be a topological space, and let I be any index set For each i ∈ I, let Y i , ϕ N i be
an CFC-space, and let Y i∈I Y i Let F ∈ By, x be a compact mapping such that for each i ∈ I,
i let A i:X → 2 Y i be a generalized GB-majorized mapping;
iii∈I {x ∈ X : A i x / ∅} i∈Icint{x ∈ X : Ai x / ∅}.
Then there exists x ∈ X such that A i x ∅ for each i ∈ I.
Proof Since for each i ∈ I, let Y i , ϕ N i be an CFC-space, then for each N i ∈ Y i, there exists
a compactFC-subspace L N i ofY i containingN i LetL N i∈I L N i andN i∈I N i ∈ Y,
thenL Nis a compactFC-subspace of Y for each N ∈ Y, L Nis a compactFC-subspace of Y
containingN Hence Y, ϕ N is also an CFC-space.
For eachx ∈ X, Ix {i ∈ I : A i x / ∅} Define A : X → 2 Y
Ax
⎧
⎪
⎪
i∈Ix
π−1
i A i x, if Ix / ∅,
Then for eachx ∈ X, Ax / ∅ if and only if Ix / ∅ 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 exists x ∈ X such that Ax ∅, and so Ix ∅ Hence, we have
A i x ∅ for each i ∈ I.
Theorem 3.13 Let X be a topological space, let K be a nonempty compact subset of X, and let I be
any index set For each i ∈ I, let Y i , ϕ N i be an FC-space, and let Y i∈I Y i such that Y, ϕ N is
an FC-space defined as in Lemma 2.2 Let F ∈ BY, X such that for each i ∈ I, A i :X → 2 Y i be a generalized GB-mapping such that
i for each i ∈ I and N i ∈ Y i , there exists a compact FC-subspace L N i of Y i containing N i and for each x ∈ X \ K, there exists i ∈ I satisfying L N i
cintA i x / ∅.
Then there exists x ∈ K such that A i x ∅ for each i ∈ I.
Proof Suppose that the conclusion is not true, then for each x ∈ K, there exists i ∈ I such that
A i x / ∅ Since A iis a generalizedGB-mapping,A−1
i is transfer compactly open valued By Lemma 3.1, we have
K ⊂
i∈I
y i ∈Y i
cintA−1
i
y i 3.20
SinceK is compact, there exists a finite set J ⊂ I such that for each j ∈ J, there exists N j
{y1
j , y2
j , , y m j
j } ⊂ Y jwithK ⊂j∈Jm j
k1 cint A−1
j y k
j It follows that for each x ∈ K, there
exists aj ∈ J ⊂ I such that N j
cintA j x / ∅ We may take any fixed y0 y0
ii∈I ∈ Y For
eachi ∈ I\J, let N i {y0
i } By condition i, for each i ∈ I, there exists a compact FC-subspace
L N iofY icontainingN iand for eachx ∈ X \K, there exists i ∈ I satisfying L N i
cintA i x / ∅.
Hence for eachx ∈ X, there exists i ∈ I such that L N i
cintA i x / ∅ Let L Ni∈I L N i, then
L Nis a compactFC-subspace of Y and hence it is also a compact CFC-space Let X0 FL N,