# Some New Coincidence Theorems in Product GFC-Spaces with Applications

Authors: Jianrong Zhao
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We first propose a new concept of GFC-subspace. Using this notion, we obtain a new continuous selection theorem. As a consequence, we establish some new collective fixed point theorems and coincidence theorems in product GFC-spaces. Finally, we give some applications of our theorems.


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## Body

## 1. Introduction and Preliminaries

Recently, Ding [1, 2], X. P. Ding and T. M. Ding [3], Tang et al. [4], Fang and Huang [5] established some collective fixed point theorems, coincidence theorems, and KKM-type theorems for the families of set-valued mappings, which are defined on FC-spaces or product FC-spaces without any convexity structure.Very recently, Khanh and Quan [6], Khanh et al. [7] defined a new notion by GFC-space which is a generalization of FC-space and proved some continuous selections theorems, collectively fixed point theorems, coincidence theorems, and KKM-type theorems in this new GFC-space.Motivated and inspired by the work mentioned above, we will give some new collective fixed point and coincidence theorems in product GFC-spaces in the present paper. For this purpose, we first propose a new concept of GFC-subspace. Using this notion, we obtain a new continuous selection theorem. As a consequence, we establish some new collective fixed point theorems and coincidence theorems in product GFC-spaces. Finally, we give some applications of our theorems.For our purpose, first, we present some known definitions and preliminary results. For a nonempty setA
,
〈
A
〉 denotes the family of all nonempty finite subsets and 2
A denotes the family of all subsets of A. We denote the standard n-simplex with vertices {
e
i
}
i
=
0
n by Δ
n. The following notion was introduced by Khanh and Quan [6].Definition 1 (see [6]).
(a) A generalized finitely continuous topological space or a GFC-space(
X
,
Y
,
{
φ
N
}
) consists of a topological space X and a nonempty set Y such that for each finite subset N
=
{
y
0
,
y
1
,
…
,
y
n
}
∈
〈
Y
〉, one has a continuous mapping φ
N
:
Δ
n
→
X.
(b) LetA
,
B
⊂
Y and S
:
Y
→
2
X be given. Then B is called an S-subset of Y with respect to A if for any N
=
{
y
0
,
y
1
,
…
,
y
n
}
∈
〈
Y
〉 and for any {
y
i
0
,
…
,
y
i
k
}
⊂
A
∩
N, one has φ
N
(
Δ
k
)
⊂
S
(
B
), where Δ
k
=
co
⁡
(
{
e
i
0
,
…
,
e
i
k
}
).Now we give a new concept as follows.Definition 2.
Assume that(
X
,
Y
,
{
φ
N
}
) is a GFC-space, C
⊂
Y, and D
⊂
X. Then D is said to be a GFC-subspace of X with respect to C if for each N
=
{
y
0
,
y
1
,
…
,
y
n
}
∈
〈
Y
〉 and for any {
y
i
0
,
…
,
y
i
k
}
⊂
C
∩
N, one has φ
N
(
Δ
k
)
⊂
D, where Δ
k
=
co
⁡
(
{
e
i
0
,
…
,
e
i
k
}
).Remark 3.
By the Definition2, we know that (
D
,
C
,
{
φ
N
}
) is also a GFC-space. Clearly, if B is an S-subset of Y with respect to A, then S
(
B
) is a GFC-subspace of X with respect to A. In addition, if X
=
Y, then a GFC-subspace D of X with respect to C coincides with an FC-subspace D of X with respect to C (see [1]).Assume that{
D
i
}
i
∈
I is a family of GFC-subspace of X with respect to C and ⋂
i
∈
I
‍
D
i
≠
∅, where I is an index set. It then follows from Definition 2 that ⋂
i
∈
I
‍
D
i is also a GFC-subspace of X with respect to C. For any given subset A of X, we define GFC-hull of A with respect to C by
(1)
GFC
(
A
,
C
)
=
∩
{
B
is a GFC-subspace of
X
with respect to
C
B
⊂
X
:
A
⊂
B
,
h
h
h
h
h
h
h
2
B
is a GFC-subspace of
X
with respect to
C
}
.We simply writeGFC
(
S
(
C
)
) instead of GFC
(
S
(
C
)
,
C
) when A
=
S
(
C
). We can easily show that GFC
(
A
,
C
) is also a GFC-subspace of X with respect to C.LetX be a topological space. A set A
⊂
X is called compactly closed (resp., compactly open) if for each nonempty compact set K
⊂
X such that A
∩
K is closed (resp., open) in K. And the compact interior and the compact closure of A (see [8]) are defined by
(2)
cint
⁡
(
A
)
=
∪
{
B
⊂
X
:
B
is compactly open in
X
,
B
⊂
A
}
,
ccl
⁡
(
A
)
=
∩
{
B
⊂
X
:
B
is
compactly
closed
in
X
,
B
⊂
A
}
.Clearly ifK is a nonempty compact subset of X, then we deriver that cint
⁡
(
A
)
∩
K
=
int
⁡
K
(
A
∩
K
)
,
ccl
⁡
(
A
)
∩
K
=
cl
⁡
K
(
A
∩
K
), and ccl
⁡
(
A
∖
K
)
=
X
∖
cint
⁡
(
A
).Assume thatY and Z are nonempty sets and X is a topological space. Define two set-valued mappings F
:
X
×
Y
→
2
Z and C
:
X
→
2
Z. F
(
x
,
y
) is called transfer compactly upper semicontinuous in x with respect to C (see [9]) if for any nonempty compact subset K of X and any x
∈
K, the set {
y
∈
Y
:
F
(
x
,
y
)
∈
C
(
x
)
}
≠
∅ means that there is a relatively open neighborhood N
(
x
) of x in K and a point y
′
∈
Y such that F
(
z
,
y
′
)
⊂
C
(
z
) for all z
∈
N
(
x
).A mappingT
:
X
→
2
Y is called transfer compactly open-valued (resp., transfer compactly closed-valued) on X (see [8]) if for each x
∈
X and each nonempty compact subset K of Y, y
∈
T
(
x
)
∩
K (resp., y
∉
T
(
x
)
∩
K) means that there is x
′
∈
X such that y
∈
int
⁡
K
(
T
(
x
′
)
∩
K
) (resp., y
∉
cl
⁡
K
(
T
(
x
′
)
∩
K
)).We need the following two results. The first statement was proved by Ding and Park [9].Lemma 4 (see [9]).
LetY and Z be two nonempty sets and X a topological space. Let F
:
X
×
Y
→
2
Z and C
:
X
→
2
Z be two set-valued mappings. Then F
(
x
,
y
) is transfer compactly upper semicontinuous in x with respect to C if and only if the mapping T
:
Y
→
2
X defined by T
(
y
)
=
{
x
∈
X
:
F
(
x
,
y
)
⊄
C
(
x
)
} is transfer compactly closed-valued on Y.The second statement is Lemma 1.1 of Ding [8].Lemma 5.
Assume thatX and Y are two topological spaces and F
:
X
→
2
Y is a set-valued mapping with nonempty values. Then the following statements are equivalent: (i)
F
-
1
:
Y
→
2
X is transfer compactly open-valued;
(ii)
for any compact subsetK of X and any x
∈
K, one has y
∈
Y satisfying that x
∈
cint
⁡
F
-
1
(
y
)
∩
K and K
=
⋃
y
∈
Y
‍
int
⁡
K
(
F
-
1
∩
K
).Throughout this paper, we always letI and J be any given index set. Now we give the following statement, which generalizes Lemma 1.1 of Ding [1].Lemma 6.
Suppose that(
X
i
,
Y
i
,
{
φ
N
i
}
) is a GFC-space for each i
∈
I. If X
=
∏
i
∈
I
‍
X
i
,
Y
=
∏
i
∈
I
‍
Y
i, and φ
N
=
∏
i
∈
I
‍
φ
N
i, then (
X
,
Y
,
{
φ
N
}
) is also a GFC-space.Proof.
Letπ
i
:
Y
→
Y
i be the projective mapping from Y to Y
i for each i
∈
I. For any given N
=
{
y
0
,
y
1
,
…
,
y
n
}
∈
〈
Y
〉, we denote N
i
=
π
i
(
N
)
=
{
π
i
(
y
0
)
,
…
,
π
i
(
y
n
)
}
∈
〈
Y
i
〉. Note that (
X
i
,
Y
i
,
{
φ
N
i
}
) is a GFC-space. Then we have a continuous mapping φ
N
i
:
Δ
n
→
X
i for each i
∈
I. So we may let a mapping φ
N
:
Δ
n
→
X by φ
N
(
α
)
=
∏
i
∈
I
‍
φ
N
i
(
α
), for any α
∈
Δ
n. It follows that φ
N is a continuous mapping, which means that (
X
,
Y
,
{
φ
N
}
) is a GFC-space. So Lemma 6 is proved.


## 2. Continuous Selection and Collective Fixed Points

Theorem 7.
Assume that(
X
,
Y
,
{
φ
N
}
) is a GFC-space and Z is a compact topological space. Suppose that F
:
Z
→
2
Y and G
:
Z
→
2
X are such that (i)
Z
=
⋃
y
∈
Y
‍
cint
⁡
F
-
1
(
y
);
(ii)
for any givenz
∈
Z, G
(
z
) is a GFC-subspace of X with respect to F
(
z
).Then one has a continuous selection g
:
Z
→
X of G satisfying g
=
φ
∘
ψ, where φ
:
Δ
n
→
X and ψ
:
Z
→
Δ
n are continuous for some n
∈
ℤ
+.Proof.
By condition (i), we know that there existsN
=
{
y
0
,
y
1
,
…
,
y
n
}
∈
〈
Y
〉 such that Z
=
⋃
i
=
0
n
‍
cint
⁡
F
-
1
(
y
i
) since Z is compact. Assume that {
ψ
i
}
i
=
0
n is the continuous partition of unity subordinated to the open covering {
cint
⁡
F
-
1
(
y
i
)
}
i
=
0
n; then for any i
∈
{
0,1
,
…
,
n
} and z
∈
Z, one has
(3)
ψ
i
(
z
)
≠
0
⟺
z
∈
cint
⁡
F
-
1
(
y
i
)
⊂
F
-
1
(
y
i
)
⟹
y
i
∈
F
(
z
)
.
Let ψ
:
Z
→
Δ
n be a mapping with ψ
(
z
)
=
∑
i
=
0
n
‍
ψ
i
(
z
)
e
i. Clearly ψ is continuous and for any z
∈
Z, one has ψ
(
z
)
=
∑
i
∈
J
(
x
)
‍
ψ
j
(
z
)
e
j where J
(
x
)
=
{
j
∈
{
0,1
,
…
,
n
}
:
ψ
i
(
z
)
≠
0
}. So by (3), we get {
y
j
:
j
∈
J
(
x
)
}
⊂
F
(
z
)
∩
N. It then follows from condition (ii) that for z
∈
Z, φ
N
(
Δ
J
(
x
)
)
⊂
G
(
z
). It is easy to see that g
(
z
)
=
φ
N
∘
ψ
(
z
)
∈
φ
N
(
Δ
J
(
x
)
)
⊂
G
(
z
), which implies that g
=
φ
N
∘
ψ is a continuous selection of G. The proof of Theorem 7 is completed.Remark 8.
Applying the definition of GFC-subspace, we extend Theorem 2.2 of Tarafdar [10], Proposition 1 of Browder [11], and Theorem 2.1 of Ding [2] to GFC-spaces without any convexity.Theorem 9.
Assume thatX
=
∏
i
∈
I
‍
X
i and (
X
i
,
Y
i
,
{
φ
N
i
}
) is a compact GFC-space for each i
∈
I. Suppose that F
i
:
X
→
2
Y
i and G
i
:
X
→
2
X
i are two set-valued mappings satisfying the following conditions: (i)
X
=
⋃
y
i
∈
Y
i
‍
cint
⁡
F
i
-
1
(
y
i
);
(ii)
for any givenx
∈
X, G
i
(
x
) is a GFC-subspace of X
i with respect to F
i
(
x
). Then one has a point x
^
=
(
x
^
i
)
i
∈
I
∈
X such that x
^
i
∈
G
i
(
x
^
) for i
∈
I.Proof.
For any giveni
∈
I, using Theorem 7, we obtain that there are continuous mappings φ
N
i
:
Δ
n
i
→
X
i and ψ
i
:
X
→
Δ
n
i such that g
i
=
φ
N
i
∘
ψ
i is a continuous selection of G
i for some positive integer n
i. Assume that E
i is the linear hull of the set {
e
0
,
e
1
,
…
,
e
n
i
} for each i
∈
I. Clearly E
i is a locally convex Hausdorff topological vector space as it is finite dimensional and Δ
n
i is a compact convex subset of E
i. Moreover, E
=
∏
i
∈
I
‍
E
i is a locally convex Hausdorff topological vector space and Δ
=
∏
i
∈
I
‍
Δ
n
i is a compact convex subset of E. Now let Φ
:
Δ
→
X and Ψ
:
X
→
Δ be two continuous mappings with
(4)
Φ
(
δ
)
=
∏
i
∈
I
‍
φ
N
i
(
P
i
(
δ
)
)
,
Ψ
(
x
)
=
∏
i
∈
I
‍
ψ
i
(
x
)
,
where P
i
:
Δ
→
Δ
n
i is the projection of Δ onto Δ
n
i for any given i
∈
I. It then follows from the Tychonoff fixed point theorem that the continuous mapping Ψ
∘
Φ
:
Δ
→
Δ has a fixed point δ
∈
Δ; that is, δ
=
Ψ
∘
Φ
(
δ
). Let x
^
=
Φ
(
δ
). It follows that
(5)
x
^
=
Ψ
∘
Φ
(
x
^
)
=
Φ
(
∏
i
∈
I
‍
ψ
i
(
x
^
)
)
=
∏
i
∈
I
‍
φ
N
i
(
P
i
(
∏
i
∈
I
‍
ψ
i
(
x
^
)
)
)
=
∏
i
∈
I
‍
φ
N
i
∘
ψ
i
(
x
^
)
.
This means that x
^
=
φ
N
i
∘
ψ
i
(
x
^
)
∈
G
i
(
x
^
) for each i
∈
I. So Theorem 9 is proved.Theorem 10.
Assume thatZ
i is a topological space and (
X
i
,
Y
i
,
{
φ
N
i
}
) is a GFC-space for each i
∈
I. Suppose that X
=
∏
i
∈
I
‍
X
i
,
S
i
:
Z
i
→
2
Y
i and T
i
:
Z
i
→
2
X
i and g
i
:
X
→
Z
i is a continuous mapping such that (i)
for any compact subsetD
i of Z
i, D
i
=
⋃
y
i
∈
Y
i
‍
(
cint
⁡
S
i
-
1
(
y
i
)
∩
D
i
);
(ii)
for anyz
i
∈
Z
i
,
T
i
(
z
i
) is a GFC-subspace of X
i with respect to S
i
(
z
i
);
(iii)
there is a nonempty subset ofY
i
0
⊂
Y
i such that the set B
i
=
⋂
y
i
∈
Y
i
0
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
c is empty or compact in Z
i, and for any N
i
∈
〈
Y
i
〉, there exists a compact GFC-subspace K
N
i of X
i with respect to L
N
i containing Y
i
0
∪
N
i.Then one has a point x
^
=
(
x
^
i
)
i
∈
I
∈
X such that x
^
i
∈
T
i
(
g
i
(
x
^
)
) for each i
∈
I.Proof.
For any giveni
∈
I, if B
i is a nonempty compact subset of Z
i, then by (i) one has
(6)
B
i
=
⋃
y
i
∈
Y
i
‍
(
cint
⁡
S
i
-
1
(
y
i
)
∩
B
i
)
⊂
⋃
y
i
∈
Y
i
‍
cint
⁡
S
i
-
1
(
y
i
)
.
Moreover, noticing thatB
i is compact, we can find that there is a finite set N
i
∈
〈
Y
i
〉 such that
(7)
B
i
=
⋂
y
i
∈
Y
i
0
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
c
⊂
⋃
y
i
∈
N
i
‍
cint
⁡
S
i
-
1
(
y
i
)
.
It then follows from (7) that
(8)
Z
i
=
⋃
y
i
∈
Y
i
0
‍
cint
⁡
S
i
-
1
(
y
i
)
∪
(
⋃
y
i
∈
N
i
‍
cint
⁡
S
i
-
1
(
y
i
)
)
.
IfB
i is empty in (iii), then we derive that
(9)
Z
i
=
Z
i
∖
B
i
=
Z
i
∖
⋂
y
i
∈
Y
i
0
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
c
=
⋃
y
i
∈
Y
i
0
‍
cint
⁡
S
i
-
1
(
y
i
)
.
From condition (iii), we know that there exists a compact GFC-subspaceK
N
i of X
i with respect to L
N
i containing Y
i
0
∪
N
i. So by (8) and (9), we get
(10)
Z
i
=
⋃
y
i
∈
L
N
i
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
.
Now letK
N
=
∏
i
∈
I
‍
K
N
i, L
N
=
∏
i
∈
I
‍
L
N
i, and φ
N
=
∏
i
∈
I
‍
φ
N
i. By condition (iii), we deduce that (
K
N
i
,
L
N
i
,
{
φ
N
i
}
) is a compact GFC-space. It then follows from Lemma 6 that (
K
N
,
L
N
,
{
φ
N
}
) is also a compact GFC-space. Let P
i
:
K
N
→
2
L
N
i and Q
i
:
K
N
→
2
K
N
i be two set-valued mappings with
(11)
P
i
(
x
)
=
S
i
(
g
i
(
x
)
)
∩
L
N
i
,
Q
i
(
x
)
=
T
i
(
g
i
(
x
)
)
∩
K
N
i
for
x
∈
K
N
.
In order to show that the conditions (i) and (ii) of Theorem9 hold, we only need to show that Q
i
(
x
) is a compact GFC-subspace of K
N
i with respect to P
i
(
x
), and K
N
=
⋃
y
i
∈
L
N
i
‍
cint
⁡
P
i
-
1
(
y
i
). By conditions (ii) and (iii), it is easy to see that Q
i
(
x
) is a compact GFC-subspace of K
N
i with respect to P
i
(
x
). It remains to show that K
N
=
⋃
y
i
∈
L
N
i
‍
cint
⁡
P
i
-
1
(
y
i
). On one hand, for each y
i
∈
L
N
i, we deduce that
(12)
P
i
-
1
(
y
i
)
=
{
x
∈
K
N
:
y
i
∈
P
i
(
x
)
}
=
{
x
∈
K
N
:
y
i
∈
S
i
(
g
i
(
x
)
)
∩
L
N
i
}
=
{
x
∈
K
N
:
y
i
∈
S
i
(
g
i
(
x
)
)
}
=
{
x
∈
K
N
:
x
∈
g
i
-
1
(
S
i
-
1
(
y
i
)
)
}
=
g
i
-
1
(
S
i
-
1
(
y
i
)
)
.
On the other hand, by (10), we obtain
(13)
g
i
(
K
N
)
⊂
Z
i
=
⋃
y
i
∈
L
N
i
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
.
It then follows form (12), (13), and the continuity of g
i that
(14)
K
N
⊂
g
i
-
1
(
⋃
y
i
∈
L
N
i
‍
cint
⁡
S
i
-
1
(
y
i
)
)
=
⋃
y
i
∈
L
N
i
‍
g
i
-
1
(
cint
⁡
S
i
-
1
(
y
i
)
)
=
⋃
y
i
∈
L
N
i
‍
(
cint
⁡
P
i
-
1
(
y
i
)
)
⊂
K
N
.
HenceK
N
=
⋃
y
i
∈
L
N
i
‍
cint
⁡
P
i
-
1
(
y
i
). Then Theorem 9 tells us that there exists a point x
^
∈
K
N
⊂
X such that x
^
i
∈
Q
i
(
x
^
)
=
T
i
(
g
i
(
x
^
)
)
∩
K
N
i
⊂
T
i
(
g
i
(
x
^
)
) for each i
∈
I. This completes the proof of Theorem 10.Theorem 11.
LetX
=
∏
i
∈
I
‍
X
i
,
Z
i be a topological space and (
X
i
,
Y
i
,
{
φ
N
i
}
) a GFC-space for each i
∈
I. Assume that S
i
:
Z
i
→
2
Y
i
,
T
i
:
Z
i
→
2
X
i, and R
i
:
Y
i
→
2
X
i are set-valued mappings and g
i
:
X
→
Z
i is a continuous mapping such that (i)
for any compact subsetD
i of Z
i, D
i
=
⋃
y
i
∈
Y
i
‍
(
cint
⁡
S
i
-
1
(
y
i
)
∩
D
i
);
(ii)
for anyz
i
∈
Z
i
,
G
F
C
(
R
i
(
S
i
(
z
i
)
)
)
⊂
T
i
(
z
i
);
(iii)
ifZ
i is not compact, then there is a nonempty subset of Y
i
0
⊂
Y
i such that a compact subset D
i of Z
i satisfying Z
i
∖
D
i
⊂
⋃
y
i
∈
Y
i
0
‍
cint
⁡
S
i
-
1
(
y
i
), and for any N
i
∈
〈
Y
i
〉, there exists a compact GFC-subspace K
N
i of X
i with respect to L
N
i containing Y
i
0
∪
N
i.Then we have a point x
^
=
(
x
^
i
)
i
∈
I
∈
X such that x
^
i
∈
T
i
(
g
i
(
x
^
)
) for each i
∈
I.Proof.
Using the similar argument of Theorem10, we only need to show that the conditions (ii) and (iii) of Theorem 10 are satisfied. By the definition of GFC
(
R
i
(
S
i
(
z
i
)
)
), we know that GFC
(
R
i
(
S
i
(
z
i
)
)
) is a GFC-subspace of X
i with respect to S
i
(
z
i
). It then follows from condition (ii) that T
i
(
z
i
) is a GFC-subspace of X
i with respect to S
i
(
z
i
), which means that the condition (i) of Theorem 10 holds. It remains to deal with condition (iii). If Z
i is noncompact, by (iii), we get
(15)
B
i
=
⋂
y
i
∈
Y
i
0
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
c
=
Y
i
∖
⋃
y
i
∈
Y
i
0
‍
cint
⁡
S
i
-
1
(
y
i
)
⊂
D
i
.
Clearly B
i is a closed subset of compact set D
i. If B
i is nonempty, then B
i is compact in Z
i. This means that the condition (iii) of Theorem 10 is satisfied. So the statement of Theorem 11 follows immediately from Theorem 10.Remark 12.
(a) LetV
j be a topological spaces for each j
∈
J. By Lemma 5, if using assumption (i) in Lemma 5 for assumption (i) in Theorems 9 and 10, then the statements of Theorems 9 and 10 are still true.
(b) The results of Theorems10 and 11 generalize Theorem 2.1 of Lan and Webb [12], Theorem 1 of Ansari and Yao [13], Theorem 2.2 of Ding and Park [14], Theorem 3.1 of Ding and Park [15], and Theorem 3.1 of Lin and Ansari [16] and Ding [2] to GFC-spaces.


## 3. Coincidence Theorems for Two Families of Set-Valued Mappings

Theorem 13.
Let(
X
i
,
Y
i
,
{
φ
N
i
}
) and (
U
j
,
V
j
,
{
φ
N
j
}
) be GFC-spaces for any i
∈
I and j
∈
J. Assume that X
=
∏
i
∈
I
‍
X
i and U
=
∏
i
∈
J
‍
U
j. Let S
i
:
U
→
2
Y
i
,
T
i
:
U
→
2
X
i
,
F
j
:
X
→
2
V
j, and G
j
:
X
→
2
U
j be set-valued mappings satisfying the following conditions: (i)
for any compact subsetA of X
,
A
=
⋃
v
j
∈
V
j
‍
(
cint
⁡
F
j
-
1
(
v
j
)
∩
A
);
(ii)
for anyx
∈
X, G
j
(
x
) is a GFC-subspace of U
j with respect to F
j
(
x
);
(iii)
there exists a nonempty subset ofY
i
0 of Y
i such that the set D
i
=
⋂
y
i
∈
Y
i
0
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
c is empty or compact in U and for each N
i
∈
〈
Y
i
〉, there is a compact GFC-subspace K
N
i of X
i with respect to L
N
i containing Y
i
0
∪
N
i;
(iv)
for any compact subsetB of U
,
B
=
⋃
y
i
∈
Y
i
‍
(
cint
⁡
S
i
-
1
(
y
i
)
∩
B
);
(v)
for anyu
∈
U, T
i
(
u
) is a GFC-subspace of X
i with respect to S
i
(
u
).Then one has x
^
=
(
x
^
i
)
i
∈
I
∈
X and u
^
=
(
u
^
j
)
j
∈
J
∈
U such that x
^
i
∈
T
i
(
u
^
) and u
^
j
∈
G
j
(
x
^
) for any i
∈
I and j
∈
J.Proof.
For any giveni
∈
I, if D
i is empty in (iii), it is easy to see that
(16)
U
=
U
∖
D
i
=
U
∖
⋂
y
i
∈
Y
i
0
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
c
=
⋃
y
i
∈
Y
i
0
‍
cint
⁡
S
i
-
1
(
y
i
)
.
IfD
i is nonempty compact set in U, by (iv), we know that
(17)
D
i
=
⋃
y
i
∈
Y
i
‍
(
cint
⁡
S
i
-
1
(
y
i
)
∩
D
i
)
⊂
⋃
y
i
∈
Y
i
‍
cint
⁡
S
i
-
1
(
y
i
)
.
Note thatD
i is compact. Then we can find a finite set N
i
∈
〈
Y
i
〉 such that
(18)
D
i
=
⋂
y
i
∈
Y
i
0
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
c
⊂
⋃
y
i
∈
N
i
‍
cint
⁡
S
i
-
1
(
y
i
)
.
It then follows from (16) and (18) that if either D
i is empty or compact in U, we get
(19)
U
=
⋃
y
i
∈
Y
i
0
‍
cint
⁡
S
i
-
1
(
y
i
)
∪
(
⋃
y
i
∈
N
i
‍
cint
⁡
S
i
-
1
(
y
i
)
)
.
Furthermore, by (iii), there exists a compact GFC-subspaceK
N
i of X
i with respect to L
N
i containing Y
i
0
∪
N
i. So by (19), we get
(20)
U
=
⋃
y
i
∈
L
N
i
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
.
Assume that K
N
=
∏
i
∈
I
‍
K
N
i
,
L
N
=
∏
i
∈
I
‍
L
N
i, and φ
N
=
∏
i
∈
I
‍
φ
N
i. It then follows from Lemma 6 that K
N is a compact GFC-subspace of X with respect to L
N. So (
K
N
,
L
N
,
{
φ
N
}
) is a compact GFC-space. We consider the restrictions F
j
∣
K
N and G
j
∣
K
N of F
j and G
j on K
N, respectively, for each j
∈
J. Then by condition (ii), we have G
j
∣
K
N
(
x
) as a GFC-subspace of U
j with respect to F
j
∣
K
N
(
x
) for any x
∈
X. From condition (i), we obtain
(21)
K
N
=
⋃
v
j
∈
V
j
‍
(
cint
⁡
F
j
-
1
(
v
j
)
∩
K
N
)
=
⋃
v
j
∈
V
j
‍
cint
⁡
F
j
-
1
∣
K
N
(
v
j
)
.
Moreover, by Theorem7, there exists a continuous selection g
j
:
K
N
→
U
j of G
j
∣
K
N for any j
∈
J. Let g
:
K
N
→
U be a mapping with g
(
x
)
=
∏
j
∈
J
‍
g
j
(
x
) for any x
∈
L
N. Clearly g is a continuous mapping. Define two set-valued mappings P
i
:
K
N
→
2
L
N
i and Q
i
:
K
N
→
2
K
N
i by
(22)
P
i
(
x
)
=
S
i
(
g
(
x
)
)
∩
L
N
i
,
Q
i
(
x
)
=
T
i
(
g
(
x
)
)
∩
K
N
i
.
SinceK
N
i is a compact GFC-subspace of X
i relative to L
N
i, by (v), we know that Q
i
(
x
) is also a compact GFC-subspace of X
i with respect to P
i
(
x
).
Now we claim thatK
N
=
⋃
y
i
∈
L
N
i
‍
cint
⁡
P
i
-
1
(
y
i
). For any given y
i
∈
L
N
i, we deduce that
(23)
P
i
-
1
(
y
i
)
=
{
x
∈
K
N
:
y
i
∈
P
i
(
x
)
}
=
{
x
∈
K
N
:
y
i
∈
S
i
(
g
(
x
)
)
∩
L
N
i
}
=
{
x
∈
K
N
:
y
i
∈
S
i
(
g
(
x
)
)
}
=
{
x
∈
K
N
:
x
∈
g
-
1
(
S
i
-
1
(
y
i
)
)
}
=
g
-
1
(
S
i
-
1
(
y
i
)
)
.
Then by (20) we obtain
(24)
g
(
K
N
)
⊂
U
=
⋃
y
i
∈
L
N
i
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
.
It then follows form (23), (24), and the continuity of g that
(25)
K
N
⊂
g
-
1
(
⋃
y
i
∈
L
N
i
‍
cint
⁡
S
i
-
1
(
y
i
)
)
=
⋃
y
i
∈
L
N
i
‍
g
-
1
(
cint
⁡
S
i
-
1
(
y
i
)
)
=
⋃
y
i
∈
L
N
i
‍
(
cint
⁡
P
i
-
1
(
y
i
)
)
⊂
K
N
.
HenceK
N
=
⋃
y
i
∈
L
N
i
‍
cint
⁡
P
i
-
1
(
y
i
). So the claim is proved.
Note thatQ
i
(
x
) is a compact GFC-subspace of X
i relative to P
i
(
x
). So by the above claim and Theorem 9, we know that there exists a point x
^
∈
K
N
⊂
X such that x
^
i
∈
Q
i
(
x
^
)
=
T
i
(
g
(
x
^
)
)
∩
K
N
i
⊂
T
i
(
g
(
x
^
)
) for any given i
∈
I. Assume that u
^
=
g
(
x
^
)
=
∏
j
∈
J
‍
g
j
(
x
^
). Thus we derive that there exist x
^
∈
X and u
^
∈
U such that x
^
i
∈
T
i
(
u
^
) and u
^
j
∈
G
j
(
x
^
) for any i
∈
I and j
∈
J, which implies that Theorem 13 is true.Theorem 14.
Let(
X
i
,
Y
i
,
{
φ
N
i
}
) and (
U
j
,
V
j
,
{
φ
N
j
}
) be GFC-spaces for any i
∈
I and j
∈
J. Assume that X
=
∏
i
∈
I
‍
X
i and U
=
∏
i
∈
J
‍
U
j. Let S
i
:
U
→
2
Y
i, T
i
:
U
→
2
X
i, R
i
:
Y
i
→
2
X
i, F
j
:
X
→
2
V
j, G
j
:
X
→
2
U
j, and H
j
:
V
j
→
2
U
j be set-valued mappings for each i
∈
I and j
∈
J. If the following conditions hold: (i)
for any compact subsetA of X, A
=
⋃
v
j
∈
V
j
‍
(
cint
⁡
F
j
-
1
(
v
j
)
∩
A
),
(ii)
for anyx
∈
X
,
G
F
C
(
H
j
(
F
j
(
x
)
)
)
⊂
G
j
(
x
),
(iii)
ifU is not compact, then there exists a nonempty subset of Y
i
0 of Y
i as well as a compact subset K of U such that U
∖
K
⊂
⋃
y
i
∈
Y
i
0
‍
cint
⁡
S
i
-
1
(
y
i
) and for each N
i
∈
〈
Y
i
〉, there is a compact GFC-subspace K
N
i of X
i with respect to L
N
i containing Y
i
0
∪
N
i,
(iv)
for any compact subsetB of U
,
B
=
⋃
v
j
∈
V
j
‍
(
cint
⁡
F
j
-
1
(
v
j
)
∩
B
),
(v)
for anyu
∈
U
,
G
F
C
(
R
i
(
S
i
(
u
)
)
)
⊂
T
i
(
u
),then there exist x
^
=
(
x
^
i
)
i
∈
I
∈
X and u
^
=
(
u
^
j
)
j
∈
J
∈
U such that x
^
i
∈
T
i
(
u
^
) and u
^
j
∈
G
j
(
x
^
) for any i
∈
I and j
∈
J.Proof.
By the definition of the GFC-hull with respect toF
j
(
x
) and condition (ii), it is easy to see that G
j
(
x
) is a GFC-subspace of U
j with respect to F
j
(
x
) for any x
∈
X and j
∈
J. Similarly, by (v), we obtain that T
i
(
u
) is a GFC-subspace of X
i with respect to S
i
(
u
) for any u
∈
U and i
∈
I. Then conditions (ii) and (v) of Theorem 13 hold. On the other hand, by (iii), if U is not compact, we know that
(26)
D
=
⋂
y
i
∈
Y
i
0
‍
(
cint
⁡
S
i
-
1
(
y
i
)
)
c
=
U
∖
⋃
y
i
∈
Y
i
0
‍
cint
⁡
S
i
-
1
(
y
i
)
⊂
K
.
If D is nonempty, then D is a closed subset of compact set K. This implies that D is compact in U. So condition (iii) tells us condition (iii) of Theorem 13 holds. Then the statement of Theorem 14 follows immediately from Theorem 13.Remark 15.
Theorems13 and 14 improve Theorem 9 of Yu and Lin [17], Theorem 3.3 of Lin and Ansari [16], and Theorems 3.1 and 3.2 of Ding [2] to GFC-spaces without any convexity structure.


## 4. Applications

In the current section, we will give some applications of our theorems.Theorem 16.
Let(
X
i
,
Y
i
,
{
φ
N
i
}
) be a GFC-space and Z
i a topological space for each i
∈
I. Assume that A
i and B
i are a subset of Y
i
×
Z
i and X
i
×
Z
i, respectively. Let X
=
∏
i
∈
I
‍
X
i and for each i
∈
I, let g
i
:
X
→
Z
i be a continuous mapping and R
i
:
Y
i
→
2
X
i a set-valued mapping satisfying the following conditions: (i)
for any compact subsetD
i of Z
i, D
i
=
⋃
y
i
∈
Y
i
‍
(
cint
⁡
{
z
i
∈
Z
i
:
(
y
i
,
z
i
)
∈
A
i
}
∩
D
i
);
(ii)
for anyz
i
∈
Z
i
,
G
F
C
(
R
i
(
{
y
i
∈
Y
i
:
(
y
i
,
z
i
)
∈
A
i
}
)
)
⊂
{
x
i
∈
X
i
:
(
x
i
,
z
i
)
∈
B
i
};
(iii)
ifZ
i is not compact, then there exists a nonempty subset of Y
i
0 of Y
i such that a compact subset D
i of Z
i such that Z
i
∖
D
i
⊂
⋃
y
i
∈
Y
i
0
‍
cint
⁡
{
z
i
∈
Z
i
:
(
y
i
,
z
i
)
∈
A
i
} and for each N
i
∈
〈
Y
i
〉, there is a compact GFC-subspace of X
i with respect to L
N
i containing Y
i
0
∪
N
i.Then we have a point x
^
=
(
x
^
i
)
i
∈
I
∈
X such that (
x
^
i
,
g
i
(
x
^
)
)
∈
B
i for each i
∈
I.Proof.
For any giveni
∈
I, we define two set-valued mappings S
i
:
Z
i
→
2
Y
i and T
i
:
Z
i
→
2
X
i by
(27)
S
i
(
z
i
)
=
{
y
i
∈
Y
i
:
(
y
i
,
z
i
)
∈
A
i
}
,
T
i
(
z
i
)
=
{
x
i
∈
X
i
:
(
x
i
,
z
i
)
∈
B
i
}
for all z
i
∈
Z
i. It then follows from assumptions (i)–(iii) and Theorem 11 that for any i
∈
I, there exists a point x
^
=
(
x
^
i
)
i
∈
I
∈
X such that x
^
i
∈
T
i
(
g
i
(
x
^
)
), which implies that (
x
^
i
,
g
i
(
x
^
)
)
∈
B
i. So Theorem 16 is true.Corollary 17.
For eachi
∈
I, let (
X
i
,
Y
i
,
{
φ
N
i
}
) be a GFC-space, R
i
:
Y
i
→
2
X
i a set-valued mapping, X
=
∏
i
∈
I
‍
X
i, and X
i
=
∏
j
∈
I
,
j
≠
i
‍
X
j. Assume that {
A
i
}
i
∈
I and {
B
i
}
i
∈
I are the families of subsets of Y
i
×
X
i and X
i
×
X
i, respectively. If the following conditions hold: (i)
for any compact subsetD
i of X
i, D
i
=
⋃
y
i
∈
Y
i
‍
(
cint
⁡
{
z
i
∈
Z
i
:
(
y
i
,
x
i
)
∈
A
i
}
∩
D
i
),
(ii)
for allx
i
∈
X
i
,
G
F
C
(
R
i
(
{
y
i
∈
Y
i
:
(
y
i
,
x
i
)
∈
A
i
}
)
)
⊂
{
x
i
∈
X
i
:
(
x
i
,
x
i
)
∈
B
i
},
(iii)
ifX
i is not compact, then there exists a nonempty subset of Y
i
0 of Y
i such that a compact subset D
i of X
i such that X
i
∖
D
i
⊂
⋃
y
i
∈
Y
i
0
‍
cint
⁡
{
x
i
∈
X
i
:
(
y
i
,
z
i
)
∈
A
i
} and for each N
i
∈
〈
Y
i
〉, there is a compact GFC-subspace of X
i relative to L
N
i containing Y
i
0
∪
N
i,then ⋂
i
∈
I
‍
B
i
≠
∅.Proof.
Define a mappingg
i
:
X
→
X
i as the projection of X onto X
i
=
∏
j
∈
I
,
j
≠
i
‍
X
j. Clearly g
i is a continuous mapping. Using Theorem 16 with Z
i
=
X
i and D
i
=
D
i, we find that there exists a point x
^
=
(
x
^
i
)
i
∈
I
∈
X such that (
x
^
i
,
g
i
(
x
^
)
)
∈
B
i for any i
∈
I, which implies that x
^
=
(
x
^
i
,
g
i
(
x
^
)
)
∈
⋂
i
∈
I
‍
B
i. Hence ⋂
i
∈
I
‍
B
i
≠
∅.In the rest of this section, for eachi
∈
I and j
∈
J, we always let Z
i and W
i be two nonempty set and (
X
i
,
Y
i
,
{
φ
N
i
}
) and (
U
j
,
V
j
,
{
φ
N
j
}
) be two GFC-spaces. Assume that X
=
∏
i
∈
I
‍
X
i and U
=
∏
i
∈
J
‍
U
j. For any i
∈
I and j
∈
J, let R
i
:
Y
i
→
2
X
i
,
H
j
:
V
j
→
2
U
j
,
C
j
:
X
→
2
Z
j
,
D
i
:
U
→
2
W
i
,
B
j
:
X
×
V
j
→
2
Z
j
,
A
j
:
X
×
U
j
→
2
Z
j
,
Q
i
:
Y
i
×
U
→
2
W
i, and P
i
:
X
i
×
U
→
2
W
i be set-valued mappings. Then we have the follows results.Theorem 18.
Suppose that the following conditions hold:(i)
for any compact subsetA of X, A
=
⋃
v
j
∈
V
j
‍
(
cint
⁡
{
x
∈
X
:
B
j
(
x
,
v
j
)
⊂
C
j
(
x
)
}
∩
A
);
(ii)
for anyx
∈
X
,
G
F
C
(
H
j
(
{
v
j
∈
V
j
:
B
j
(
x
,
v
j
)
⊂
C
j
(
x
)
}
)
)
⊂
{
u
j
∈
U
j
:
A
j
(
x
,
u
j
)
⊂
C
j
(
x
)
};
(iii)
ifU is not compact, then there exists a nonempty subset of Y
i
0 of Y
i such that a compact subset K of U such that U
∖
K
⊂
⋃
y
i
∈
Y
i
0
‍
cint
⁡
{
u
∈
U
:
Q
i
(
y
i
,
u
)
⊂
D
i
(
u
)
} and for each N
i
∈
〈
Y
i
〉, there is a compact GFC-subspace of X
i with respect to L
N
i containing Y
i
0
∪
N
i;
(iv)
for any compact subsetB of U, B
=
⋃
y
i
∈
Y
i
‍
(
cint
⁡
{
y
i
∈
Y
i
:
Q
i
(
y
i
,
u
)
⊂
D
i
(
u
)
}
∩
B
);
(v)
for anyu
∈
U
,
G
F
C
(
R
i
(
{
y
i
∈
Y
i
:
Q
i
(
y
i
,
u
)
⊂
D
i
(
u
)
}
)
)
⊂
{
x
i
∈
X
i
:
P
i
(
x
i
,
u
)
⊂
D
i
(
u
)
}.Then there exist (
x
^
,
u
^
)
∈
X
×
U and u
^
=
(
u
^
j
)
j
∈
J
∈
U such that A
j
(
x
^
,
u
^
j
)
⊂
C
j
(
x
^
) and P
i
(
x
^
i
,
u
^
)
⊂
D
i
(
u
^
) for each i
∈
I and j
∈
J.Proof.
Assume thatS
i
:
U
→
2
Y
i
,
T
i
:
U
→
2
X
i
,
F
j
:
X
→
2
V
j, and G
j
:
X
→
2
U
j are set-valued mappings as follows:
(28)
S
i
(
u
)
=
{
y
i
∈
Y
i
:
Q
i
(
y
i
,
u
)
⊂
D
i
(
u
)
}
,
T
i
(
u
)
=
{
x
i
∈
X
i
:
P
i
(
x
i
,
u
)
⊂
D
i
(
u
)
}
∀
u
∈
U
,
F
j
(
x
)
=
{
v
j
∈
V
j
:
B
j
(
x
,
v
j
)
⊂
C
j
(
x
)
}
,
(29)
G
j
(
x
)
=
{
u
j
∈
U
j
:
A
j
(
x
,
u
j
)
⊂
C
j
(
x
)
}
∀
x
∈
X
.
It then follows from Theorem 14 that there exist x
^
∈
X and u
^
∈
U such that x
^
i
∈
T
i
(
u
^
) and u
^
j
∈
G
j
(
x
^
) for any i
∈
I and j
∈
J. By the above definition, we obtain that A
j
(
x
^
,
u
^
j
)
⊂
C
j
(
x
^
) and P
i
(
x
^
i
,
u
^
)
⊂
D
i
(
u
^
) for any i
∈
I and j
∈
J. This completes the proof.Theorem 19.
For eachi
∈
I and j
∈
J, let Y
i and V
j be topological spaces. Suppose that the following conditions hold: (i)
for eachx
∈
X, the set {
v
j
∈
V
j
:
B
j
(
x
,
v
j
)
⊂
C
j
(
x
)
} is nonempty and B
j
(
x
,
v
j
) is transfer compactly upper semicontinuous in x with respect to C
j;
(ii)
for anyx
∈
X
,
G
F
C
(
H
j
(
{
v
j
∈
V
j
:
B
j
(
x
,
v
j
)
⊂
C
j
(
x
)
}
)
)
⊂
{
u
j
∈
U
j
:
A
j
(
x
,
u
j
)
⊂
C
j
(
x
)
};
(iii)
ifU is not compact, then there exists a nonempty subset of Y
i
0 of Y
i as well as a compact subset K of U such that U
∖
K
⊂
⋃
y
i
∈
Y
i
0
‍
cint
⁡
{
u
∈
U
:
Q
i
(
y
i
,
u
)
⊂
D
i
(
u
)
} and for each N
i
∈
〈
Y
i
〉, there is a compact GFC-subspace of X
i with respect to L
N
i containing Y
i
0
∪
N
i;
(iv)
for eachu
∈
U, the set {
y
i
∈
Y
i
:
Q
i
(
y
i
,
u
)
⊂
D
i
(
u
)
} is nonempty and Q
i
(
y
i
,
u
) is transfer compactly upper semicontinuous in u with respect to D
i;
(v)
for anyu
∈
U
,
G
F
C
(
R
i
(
{
y
i
∈
Y
i
:
Q
i
(
y
i
,
u
)
⊂
D
i
(
u
)
}
)
)
⊂
{
x
i
∈
X
i
:
P
i
(
x
i
,
u
)
⊂
D
i
(
u
)
}. Then there exist (
x
^
,
u
^
)
∈
X
×
U and u
^
=
(
u
^
j
)
j
∈
J
∈
U such that A
j
(
x
^
,
u
^
j
)
⊂
C
j
(
x
^
) and P
i
(
x
^
i
,
u
^
)
⊂
D
i
(
u
^
) for all i
∈
I and j
∈
J.Proof.
LetS
i
,
T
i
,
F
j, and G
j be set-valued mappings as defined in the proof of Theorem 18. Using the same argument of Theorem 18, we only need to show that the assumptions (i) and (iv) of Theorem 18 hold. From assumption (i) and Lemma 4, we know that for any x
∈
X
,
F
j
-
1 is transfer compactly open-valued. Thus Lemma 5 tells us that the assumption (i) of Theorem 18 holds. In a similar way, from (iv) and Lemma 4, we obtain that the assumption (iv) of Theorem 18 is satisfied. So the statement of Theorem 19 follows immediately from Theorem 18.



---

*Source: 392018-2014-05-06.xml*