Bull. Korean Math. Soc. 50 (2013), No. 1, pp. 25–36
http://dx.doi.org/10.4134/BKMS.2013.50.1.025
NONEMPTY INTERSECTION THEOREMS
AND SYSTEM OF GENERALIZED VECTOR
EQUILIBRIUM PROBLEMS IN F C-SPACES
Rong-Hua He and Hong-Xu Li
Abstract. By using some existence theorems of maximal elements for
a family of set-valued mappings involving a better admissible set-valued
mapping under noncompact setting of F C-spaces, we present some nonempty intersection theorems for a family {Gi }i∈I in product F C-spaces.
Then, as applications, some new existence theorems of equilibrium for a
system of generalized vector equilibrium problems are proved in product
F C-spaces. Our results improve and generalize some recent results.
1. Introduction and preliminaries
The vector variational inequality problem (in short, V V IP ) was first introduced by Giannessi [17] in finite dimensional Euclidean spaces. Since then,
the V V IP has been generalized and applied by many authors in various directions. Inspired and motivated by applications of the V V IP with set-valued
mappings, various generalized vector variational inequality problem (in short,
GV V IP ) and generalized vector equilibrium problem (in short, GV EP ) have
become important developed directions of the classical V V IP (see, for example,
[1, 3, 4, 6, 15, 22] and the book edited by F. Giannessi [18], and the references
therein).
Recently, Ding [12] introduced and studied a system of generalized vector
equilibrium problems in the product space of G-convex spaces. Let X be a
topological space and I be any index set. Let {Di }i∈I , {EiQ
}i∈I , {Yi }i∈I and
{Zi }i∈I be four families of topological spaces, and let Y = i∈I Yi . For each
i ∈ I, let Ti : X → 2Di , Si : X → 2Ei , Ci : X → 2Zi and ηi : Di × Ei × Yi → 2Zi
be set-valued mappings. A system of generalized vector equilibrium problems
Received September 1, 2010; Revised March 19, 2012.
2010 Mathematics Subject Classification. 49J40, 49J53.
Key words and phrases. maximal element, nonempty intersection theorem, system of
generalized vector equilibrium problems, product F C-space.
This work is supported by the National Natural Science Foundation of China (11171046),
the National Natural Science Foundation of China (11071042) and the Scientific Research
Foundation of CUIT under Grant (KYTZ201114).
c
2013
The Korean Mathematical Society
25
26
RONG-HUA HE AND HONG-XU LI
(in short, SGV EP ) is to find x̂ ∈ X such that for each i ∈ I,
(1)
∀yi ∈ Yi , ∃dˆi ∈ Ti (x̂), eˆi ∈ Si (x̂) such that ηi (dˆi , eˆi , yi ) 6⊆ Ci (x̂).
Motivated and inspired by the above research works, we shall present some
nonempty intersection theorems for a family {Gi }i∈I in a product F C-spaces,
which are generalizations of G-convex spaces, by applying an existence theorems of maximal elements obtained by He and Zhang [20]. As applications,
several new existence theorems of equilibrium points for the SGV EP (1) are
established in product F C-space. Our results improve and generalize the corresponding results in [2, 12, 14, 19, 23].
Let X and Y be two nonempty sets. We denote by 2Y and hXi the family
of all subsets of Y and the family of all nonempty finite subsets of X, respectively. Let ∆n denote the standard n-dimensional simplex with the vertices
{e0 , . . . , en }. If J is a nonempty subset of {0, 1, . . . , n}, we shall denote by △J
the convex hull of the vertices {ej : j ∈ J}. For topological spaces X and Y ,
a subset A of X is said to be compactly open
T (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 [5]) are defined
respectively by
\
cclA = {B ⊂ X : A ⊂ B and B is compactly closed in X},
[
cintA = {B ⊂ X : B ⊂ A and B is compactly open in X}.
A set-valued mapping T : X → 2Y is said to be transfer compactly closed
valued on X (see [5]) if for each x ∈ X and y 6∈ T (x), there exists x′ ∈ X such
that y 6∈ cclT (x′ ). T is said to be transfer compactly open valued on X if for
each x ∈ X and y ∈ T (x), there exists x′ ∈ X such that y ∈ cintT (x′ ).
First, we recall the following concepts (see [13, 20]).
Definition 1.1. (X, ϕN ) is said to be an F C-space if X is a topological space
and for each N = {x0 , . . . , xn } ∈ hXi, where some elements in N may be same,
there exists a continuous mapping ϕN : ∆n → X. A subset D of X is said
to be an F C-subspace
T of X if for each N = {x0 , . . . , xn } ∈ hXi and for each
{xi0 , . . . , xik } ⊂ N D, ϕN (∆k ) ⊂ D, where ∆k = co({eij : j = 0, . . . , k}).
Definition 1.2. An F C-space (X, ϕN ) is said to be a CF C-space if for each
N ∈ hY i, there exists a compact F C-subspace LN of Y containing N .
Now, we give some lemmas which will be useful in the proof of our main
results (see [13, 20]).
Lemma 1.1.
Q set. For each i ∈ I, let (Yi , ϕNi ) be an F CQ Let I be any index
space, Y = i∈I Yi and ϕN = i∈I ϕNi . Then (Y, ϕN ) is also an F C-space.
The following notion was introduced by He and Zhang [20]. Let X be a
topological space and IQbe any index set. For each i ∈ I, let (Yi , ϕNi )i∈I be
an F C-space and Y = i∈I Yi such that (Y, ϕN ) is an F C-space defined as in
NONEMPTY INTERSECTION THEOREMS
27
Lemma 1.1. Let F ∈ B(Y, X) and Ai : X → 2Yi , i ∈ I be set-valued mappings.
For each i ∈ I,
(1) Ai : X → 2Yi is said to be a generalized GB -mapping if
(a) for each N = {y0 , . . . , yn } ∈ hY i and {yi0 , . . . , yik } ⊂ N ,
F (ϕN (∆k ))
\


k
\
j=0

 = ∅,
cintA−1
i (πi (yij ))
where πi is the projection of Y onto Yi and ∆k = co({eij : j =
0, . . . , k});
(b) A−1
i (yi ) = {x ∈ X : yi ∈ Ai (x)} is transfer compactly open in Yi
for each yi ∈ Yi .
(2) Ax,i : X → 2Yi is said to be a generalized GB -majorant of Ai at
x ∈ X if Ax,i is a generalized GB -mapping and there exists an open
neighborhood N (x) of x in X such that Ai (z) ⊂ Ax,i (z) for all z ∈
N (x).
(3) Ai is said to be a generalized GB -majorized if for each x ∈ X with
Ai (x) 6= ∅, there exists a generalized GB -majorant Ax,i of Ai at x, and
T
−1
for any N ∈ h{x ∈ X : Ai (x) 6= ∅}i, the mapping x∈N Ax,i
is transfer
compactly open in Yi .
(4) {Ai }i∈I is said to be a family of generalized GB -mappings (resp., GB majorant mappings) if for each i ∈ I, Ai : X → 2Yi is a generalized
GB -mapping (resp., GB -majorant mapping).
Lemma 1.2. Let X be a topological space, K be a nonempty compact subset
of X Q
and I be any index set. For each i ∈ I, let (Yi , ϕNi ) be an F C-space and
Y = i∈I Yi such that (Y, ϕN ) is an F C-space defined as in Lemma 1.1. Let
F ∈ B(Y, X) and for each i ∈ I, Ai : X → 2Yi be a generalized GB -mapping
such that the following condition holds:
(P) For each i ∈ I and Ni ∈ hYi i, there exists a compact F C-subspace LNi
of YiTcontaining Ni and for each x ∈ X\K, there exists i ∈ I satisfying
LNi cintAi (x) 6= ∅.
Then there exists x̂ ∈ K such that Ai (x̂) = ∅ for each i ∈ I.
Lemma 1.3. Let X be a topological space, and let Q
I be any index set. For each
i ∈ I, let (Yi , ϕNi ) be an CF C-space, and let Y = i∈I Yi . Let F ∈ B(Y, X) be
a compact mapping such that the following two conditions hold for each i ∈ I:
Yi
(i) A
Si : X → 2 be a generalized
S GB -majorized mapping;
(ii) i∈I {x ∈ X : Ai (x) 6= ∅} = i∈I cint{x ∈ X : Ai (x) 6= ∅}.
Then there exists x̂ ∈ X such that Ai (x̂) = ∅ for each i ∈ I.
28
RONG-HUA HE AND HONG-XU LI
2. Nonempty intersection theorems
Definition 2.1. Let D, E, Y and Z be nonempty sets and X be a topological
space. Let F : D × E × Y → 2Z and C : X → 2Z be two set-valued mappings. F (d, e, y) is said to be transfer compactly upper semicontinuous in x
with respect to C if for any nonempty compact subset K of X and any x ∈ K,
{(d, e, y) ∈ D × E × Y : F (d, e, y) ⊆ C(x)} =
6 ∅ implies that there exist a
relatively open neighborhood N (x) of x in K and (d′ , e′ , y ′ ) ∈ D × E × Y such
that F (d′ , e′ , y ′ ) ⊆ C(z) for all z ∈ N (x).
We first give the following nonempty intersection theorem which is in fact
equivalent to Lemma 1.2.
Theorem 2.1. Let X be a topological space, K be a nonempty compact subset
of X Q
and I be any index set. For each i ∈ I, let (Yi , ϕNi ) be an F C-space and
Y = i∈I Yi such that (Y, ϕN ) is an F C-space defined as in Lemma 1.1. Let
F ∈ B(Y, X) and Gi : Yi → 2X be such that the following conditions hold:
(i) For each yi ∈ Yi , Gi (yi ) is transfer compactly closed in Yi .
(ii) For each N = {y0 , . . . , yn } ∈ hY i and {yr0 , . . . , yrk } ⊂ N ,


k
\ \
cint(X\Gi (πi (yrj ))) = ∅,
F (ϕN (∆k )) 
j=0
where πi is the projection of Y onto Yi and ∆k = co({erj : j =
0, . . . , k}).
(iii) For each i ∈ I and Ni ∈ hYi i, there exists a compact F C-subspace LNi
of YiTcontaining Ni and for each x ∈ X\K, there exists i ∈ I satisfying
LNi cint(Yi \G−1
i (x)) 6= ∅.
Then we have


\ \ \

K
Gi (yi ) 6= ∅.
i∈I yi ∈Yi
Proof. Lemma 1.2 ⇒ Theorem 2.1. For each i ∈ I, define a set-valued mapping
Ai : X → 2Yi by
Ai (x) = Yi \G−1
∀x ∈ X.
i (x),
Then for each yi ∈ Yi , we have
−1
A−1
i (yi ) = {x ∈ X : yi ∈ Ai (x)} = {x ∈ X : yi ∈ Yi \Gi (x)}
= {x ∈ X : x 6∈ Gi (yi )} = X\Gi (yi ).
A−1
i (yi )
Hence
is transfer compactly open in X by condition (i). The condition (ii) of Theorem 2.1 implies that for each N = {y0 , . . . , yn } ∈ hY i and
{yr0 , . . . , yrk } ⊂ N ,


k
\ \
 = ∅,
cintA−1
F (ϕN (∆k )) 
i (πi (yrj ))
j=0
NONEMPTY INTERSECTION THEOREMS
29
where πi is the projection of Y onto Yi and ∆k = co({erj : j = 0, . . . , k}). So
for each i ∈ I, Ai : X → 2Yi is a generalized GB -mapping. The condition (iii)
of Theorem 2.1 implies that the condition (P) of Lemma 1.2 holds. Then by
Lemma 1.2, there exists x̂ ∈ K such that Ai (x̂) = ∅ for each i ∈ I. Therefore,
−1
we have Ai (x̂) = Yi \G−1
i (x̂) = ∅, and then Yi = Gi (x̂) for i ∈ I. This implies
that
\ \
x̂ ∈
Gi (yi ).
i∈I yi ∈Yi
Hence we obtain
K
\


\ \
i∈I yi ∈Yi

Gi (yi ) 6= ∅.
Theorem 2.1 ⇒ Lemma 1.2. For each i ∈ I, define a set-valued mapping
Gi : Yi → 2X by
Gi (yi ) = X\A−1
i (yi ), ∀yi ∈ Yi .
Since for each i ∈ I, Ai : X → 2Yi is a generalized GB -mapping, conditions (i)
and (ii) of Theorem 2.1 are satisfied. Condition (P) of Lemma 1.2 implies that
condition (iii) of Theorem 2.1 holds. Consequently, all conditions of Theorem
2.1 are satisfied, and then we have


\ \ \

K
Gi (yi ) 6= ∅.
i∈I yi ∈Yi
TT T
Taking any x̂ ∈ K ( i∈I yi ∈Yi Gi (yi )) 6= ∅, we have x̂ ∈ K and
x̂ ∈ Gi (yi ) = X\A−1
i (yi ) for yi ∈ Yi , i ∈ I,
which implies
yi 6∈ Ai (x̂)
for yi ∈ Yi , i ∈ I.
So Ai (x̂) = ∅ for all i ∈ I. This completes the proof.
Remark 2.1. Since Theorem 2.1 is an equivalent form of Lemma 1.2, Theorem
2.1 generalizes Theorem 2 and Theorem 4 of Park and Kim [23]. Moreover,
Theorem 2.1 improves and generalizes Theorem 2.1 of Ding [12] in the following ways: (a) The setting spaces are generalized from “G-convex spaces” to
“F C-spaces” without linear structure; (b) The assumption for mapping Gi is
generalized from “compactly closed” to “transfer compactly closed”; (c) Conditions (ii) and (iii) of Theorem 2.1 are weaker than conditions (ii) and (iii) of
Theorem 2.1 in [12].
Especially, if X = Y in Theorem 2.1, we have the following result, which
improves Theorem 3.2 in Ding [14].
Corollary 2.1. Let Q
I be any index set. For each i ∈ I, let (Xi , ϕNi ) be an
F C-space and X = i∈I Xi such that (X, ϕN ) is an F C-space defined as in
30
RONG-HUA HE AND HONG-XU LI
Lemma 1.1. Let K be a nonempty compact subset of X and Gi : Xi → 2X be
such that the following conditions hold:
(i) For each xi ∈ Xi , Gi (xi ) is transfer compactly closed in Xi .
(ii) For each N = {x0 , . . . , xn } ∈ hXi and {xr0 , . . . , xrk } ⊂ N ,


k
\ \
cint(X\Gi (πi (xrj ))) = ∅,
ϕN (∆k ) 
j=0
where πi is the projection of X onto Xi and ∆k = co({erj : j =
0, . . . , k}).
(iii) For each i ∈ I and Ni ∈ hXi i, there exists a compact F C-subspace LNi
of XT
i containing Ni and for each x ∈ X\K, there exists i ∈ I satisfying
LNi cint(Xi \G−1
i (x)) 6= ∅.
Then we have
!
\ \ \
K
Gi (xi ) 6= ∅.
i∈I xi ∈Xi
Q
Proof. For each i ∈ I, let Yi = Xi , X = Y = i∈I Xi and F be the identity
mapping on X. Then all the conditions of Theorem 2.1 are satisfied and the
proof is completed.
If Xi in Corollary 2.1 is a compact F C-space for each I, we have the following
result with simpler form.
Corollary 2.2. Let I be any
Q index set. For each i ∈ I, let (Xi , ϕNi ) be a
compact F C-space and X = i∈I Xi such that (X, ϕN ) is an F C-space defined
as in Lemma 1.1. Let Gi : Xi → 2X be such that the following conditions hold
for each i ∈ I:
(i) for each xi ∈ Xi , Gi (xi ) is transfer compactly closed in Xi ;
(ii) for each N = {x0 , . . . , xn } ∈ hXi and {xr0 , . . . , xrk } ⊂ N ,
TT
ϕN (∆k ) ( kj=0 cint(X\Gi (πi (xrj )))) = ∅, where πi is the projection
of X onto Xi and ∆k = co({erj : j = 0, . . . , k}).
Then we have
\ \
Gi (xi ) 6= ∅.
i∈I xi ∈Xi
Q
Proof. For each i ∈ I and Ni ∈ hXi i, let LNi = Xi and let K = i∈I Xi .
Then K = X is compact. Clearly the condition (iii) of Corollary 2.1 is satisfied
trivially. The conclusion of Corollary 2.2 holds from Corollary 2.1.
By Corollary 2.1, we also have the following result, which improves Theorem
3.3 of Ding [14].
Corollary 2.3. Let Q
I be any index set. For each i ∈ I, let (Xi , ϕNi ) be an
F C-space and X = i∈I Xi such that (X, ϕN ) is an F C-space defined as in
NONEMPTY INTERSECTION THEOREMS
31
Lemma 1.1, and let K be a nonempty compact subset of X. Let Gi : Xi → 2X
be such that the following conditions hold for each i ∈ I:
(i) For each xi ∈ Xi , Gi (xi ) is transfer compactly closed in Xi .
(ii) For each x ∈ X, Xi \G−1
i (x) is an F C-subspace of Xi .
(iii) For each x ∈ X, πi (x) ∈ G−1
i (x).
(iv) For each i ∈ I and Ni ∈ hXi i, there exists a compact F C-subspace LNi
of XT
i containing Ni and for each x ∈ X\K, there exists i ∈ I satisfying
LNi cint(Xi \G−1
i (x)) 6= ∅.
Then we have
!
\ \ \
Gi (xi ) 6= ∅.
K
i∈I xi ∈Xi
Proof. We first show that conditions (ii) and (iii) imply conditions (ii) of Corollary 2.1 holds. If it is false, then there exist N = {x0 , . . . , xn } ∈ hXi and
M = {xr0 , . . . , xrk } ⊂ N such that


k
\ \
cint(X\Gi (πi (xrj ))) 6= ∅.
ϕN (∆k ) 
j=0
Q
Hence there exists x̂ ∈ ϕN (∆k ) = i∈I ϕNi (∆k ) where Ni = πi (N ) such
that x̂ 6∈ Gi (πi (x)) for all x ∈ M . So we have πi (M ) ⊆ Xi \Gi−1 (x̂). Since
Xi \G−1
i (x̂) is an F C-subspace of Xi by (ii), we have πi (ϕN (∆k )) ⊂ Xi \
−1
G−1
(x̂).
So πi (x̂) ∈ Xi \G−1
i
i (x̂) and then πi (x̂) 6∈ Gi (x̂), which contradicts the
condition (iii). This show that the conditions (ii) and (iii) imply the condition
(ii) of Corollary 2.1 holds. Corollary 2.3 follows from Corollary 2.1.
As a direct consequence of Corollary 2.3, we have the following result, which
generalizes and improves Theorem 2.1 of Guillerme [19].
Corollary 2.4. Let I be any
Q index set. For each i ∈ I, let (Xi , ϕNi ) be a
compact F C-space and X = i∈I Xi such that (X, ϕN ) is an F C-space defined
as in Lemma 1.1. Let Gi : Xi → 2X be such that the following conditions hold
for each i ∈ I:
(iii) For each xi ∈ Xi , Gi (xi ) is transfer compactly closed in Xi .
(iii) For each x ∈ X, Xi \G−1
i (x) is an F C-subspace of Xi .
(iii) For each x ∈ X, πi (x) ∈ G−1
i (x).
Then we have
\ \
Gi (xi ) 6= ∅.
i∈I xi ∈Xi
3. System of generalized vector equilibrium problems
By using Theorem 2.1, we establish some new equilibrium existence results
for SGV EP (1) in this section.
32
RONG-HUA HE AND HONG-XU LI
Theorem 3.1. Let X be a topological space, K be a nonempty compact subset
{Di }i∈I , {Ei }i∈I
of X, I be any index set, (Yi , ϕNi ) be a family of F C-spaces, Q
and {Zi }i∈I be the families of topological spaces, and Y = i∈I Yi such that
(Y, ϕN ) is an F C-space defined as in Lemma 1.1. Let F ∈ B(Y, X), and for
each i ∈ I, Ti : X → 2Di , Si : X → 2Ei , Ci : X → 2Zi and ηi : Di × Ei × Yi →
2Zi be set valued mappings such that the following conditions hold for each
i ∈ I:
(i) ηi (di , ei , yi ) is transfer compactly upper semicontinuous in x with respect to Ci .
(ii) For each N = {y0 , . . . , yn } ∈ hY i and N1 = {yr0 , . . . , yrk } ⊂ N , and
for each x ∈ F (ϕN (∆k )), there exists di ∈ Ti (x), ei ∈ Si (x) and y ∈ N1
such that ηi (di , ei , πi (y)) 6⊆ Ci (x).
(iii) For each Ni ∈ hYi i, there exists a compact F C-subspace LNi of Yi
containing Ni , and for each x ∈ X\K, there exist i ∈ I and yi ∈ LNi
satisfying
yi 6∈ ccl{ui ∈ Yi : ∃di ∈ Ti (x), ei ∈ Si (x) such that ηi (di , ei , ui ) 6⊆ Ci (x)}.
Then there exists x̂ ∈ K satisfying that, for each i ∈ I and yi ∈ Yi , there exist
dˆi ∈ Ti (x̂) and eˆi ∈ Si (x̂) such that ηi (dˆi , eˆi , yi ) 6⊆ Ci (x̂).
Proof. For each i ∈ I, defined set-valued mappings Ti , Hi : Yi → 2X by
Ti (yi ) = {x ∈ X : ∃di ∈ Ti (x) and ei ∈ Si (x) such that ηi (di , ei , yi ) 6⊆ Ci (x)},
Hi (yi ) = X\Ti (yi )
= {x ∈ X : ∃di ∈ Ti (x) and ei ∈ Si (x) such that ηi (di , ei , yi ) ⊆ Ci (x)}.
Notice that for each i ∈ I, ηi (di , ei , yi ) is transfer compactly upper semicontinuous in x withTrespect to Ci . Then for any nonempty compact subset K of X,
if x ∈ Hi (yi ) K, we have x ∈ K and there exist di ∈ Ti (x) and ei ∈ Si (x)
such that ηi (di , ei , yi ) ⊆ Ci (x), i.e.,
{(di , ei , yi ) ∈ Di × Ei × Yi : ηi (di , ei , yi ) ⊆ Ci (x)} =
6 ∅.
By condition (i), there exists a relatively open neighborhood N (x) of x in K
and (d′i , e′i , yi′ ) ∈ Di × Ei × Yi such that ηi (d′i , e′i , yi′ ) ⊆ Ci (z) for all z ∈ N (x).
Then we have
\
x ∈ N (x) ⊆ intK (K {z ∈ X : ηi (d′i , e′i , yi′ ) ⊆ Ci (z)})
\
\
= intK (Hi (yi′ ) K) = cint(Hi (yi′ )) K.
This implies that Hi is transfer compactly open in Yi . Thus for each i ∈ I and
yi ∈ Yi , Ti (yi ) is transfer compactly closed in Yi . So condition (i) of Theorem
2.1 is satisfied.
NONEMPTY INTERSECTION THEOREMS
33
From the condition (ii) it follows that for each N = {y0 , . . . , yn } ∈ hY i and
N1 = {yr0 , . . . , yrk } ⊂ N , we have
[
[
F (ϕN (∆k )) ⊆
Ti (πi (y)) ⊂
cclTi (πi (y)),
y∈N1
y∈N1
and then
F (ϕN (∆k ))
\
= F (ϕN (∆k ))


\
y∈N1
\



(X\cclTi(πi (y)))
\
y∈N1

cint(X\Ti (πi (y))) = ∅.
This yields that condition (ii) of Theorem 2.1 is satisfied.
For each i ∈ I and x ∈ X, we have
Ti−1 (x) = {yi ∈ Yi : ∃di ∈ Ti (x), ei ∈ Si (x) such that ηi (di , ei , yi ) 6⊆ Ci (x)},
and hence
Yi \Ti−1 (x) = {yi ∈ Yi : ηi (di , ei , yi ) ⊆ Ci (x), ∀di ∈ Ti (x), ei ∈ Si (x)}.
By condition (iii), for each Ni ∈ hYi i, there exists a compact F C-subspace LNi
of Yi containing Ni , and for each x ∈ X\K, there exist i ∈ I and yi ∈ LNi
satisfying
yi ∈ Yi \cclTi−1(x) = cint(Yi \Ti−1 (x)).
So we have
\
cint(Yi \Ti−1 (x)) 6= ∅.
LNi
Then condition (iii) of Theorem 2.1 is satisfied. Now by Theorem 2.1 we have


\ \ \

K
Ti (yi ) 6= ∅.
i∈I yi ∈Yi
TT T
Taking any x̂ ∈ K ( i∈I yi ∈Yi Ti (yi )), we obtain that x̂ ∈ K, and for each
i ∈ I and yi ∈ Yi there exist dˆi ∈ Ti (x̂) and eˆi ∈ Si (x̂) such that ηi (dˆi , eˆi , yi ) 6⊆
Ci (x̂). This means that x̂ is an equilibrium point of the SGV EP (1). The
proof is complete.
Corollary 3.1. Let I be any index set, (Xi , ϕN ′ )i∈I and (Yi , ϕNi )i∈I be two
i
families
of
F
C-spaces.
Let
{Z
}
be
a
family
of
topological spaces and X =
i
i∈I
Q
Q
′
Y
such
that
(X,
ϕ
),
(Y,
ϕ
)
are
F C-spaces defined as in
X
,
Y
=
N
i
N
i∈I i
i∈I
Lemma 1.1. Let F : Y → X be a continuous single-valued mapping and for
each i ∈ I, Ci : X → 2Zi , fi : X × Yi → 2Zi be set valued mappings such that
the following conditions holds:
(i) For each i ∈ I, fi (x, yi ) is transfer compactly upper semicontinuous in
x with respect to Ci .
34
RONG-HUA HE AND HONG-XU LI
(ii) For each N = {y0 , . . . , yn } ∈ hY i and N1 = {yr0 , . . . , yrk } ⊂ N , and
for each x ∈ F (ϕN (∆k )), there exists y ∈ N1 such that fi (x, πi (y)) 6⊆
Ci (x).
Q
(iii) There exists a nonempty compact subset Ki of Xi . Let K = i∈I Ki .
For each Ni ∈ hYi i, there exists a compact F C-subspace LNi of Yi
containing Ni , and for each x ∈ X\K, there exist i ∈ I and yi ∈ LNi
satisfying
yi 6∈ ccl{ui ∈ Yi : fi (x, ui ) 6⊆ Ci (x)}.
Then there exists x̂ ∈ K such that for each i ∈ I and yi ∈ Yi we have fi (x̂, yi ) 6⊆
Ci (x̂).
Q
Proof. Clearly K = i∈I Ki is a nonempty compact subset of X. For each
Q
i ∈ I, let Di = Xi and Ei = X i = j∈I,j6=i Xj . Write x = (xi , xi ) and
fi (x, yi) = ηi (xi , xi , yi) for all x ∈ X and yi ∈ Yi . Let Ti (x) = πi (x) = xi and
Si (x) = π i (x) = xi for each i ∈ I and x ∈ X where πi and π i are the projection
mappings from X onto Xi and X i , respectively. It is easy to check that all
conditions of Theorem 3.1 are satisfied. By Theorem 3.1, there exists x̂ ∈ K
such that for each i ∈ I,
fi (x̂, yi ) = ηi (x̂i , x̂i , yi) 6⊆ Ci (x̂), ∀yi ∈ Yi .
This completes the proof.
Remark 3.1. (I) Theorem 3.1 generalizes Theorem 3.1 in [12] in several aspects:
(a) The setting spaces are generalized from “family of G-convex spaces” to
“family of F C-space” without linear structure; (b) Conditions (i), (ii) and
(iii) of Theorem 3.1 in [12] are replaced by condition (i) of Theorem 3.1; (c)
conditions (ii) and (iii) of Theorem 3.1 are weaker than conditions (iv) and (v)
of Theorem 3.1 in [12].
(II) Corollary 3.1 generalizes Theorem 3.2 in [12] in the aspects similar as
those of Theorem 3.1 to Theorem 3.1 in [12]. Moreover, Corollary 3.1 improves
Theorem 2.1 and Theorem 2.2 in [2] from topological vector spaces to F C-space
under weaker assumptions.
When I be a singleton in Corollary 3.1, we obtain the following result.
Corollary 3.2. Let I be any index set, (X, ϕN ′ ) and (Y, ϕN ) be two F Cspaces and Z be a topological space, F : Y → X be a continuous single-valued
mapping, C : X → 2Z and f : X × Y → 2Z be set-valued mappings such that
(i) For each f (x, y) is transfer compactly upper semicontinuous in x with
respect to C.
(ii) For each N = {y0 , . . . , yn } ∈ hY i, N1 = {yr0 , . . . , yrk } ⊂ N and x ∈
F (ϕN (∆k )), there exists y ∈ N1 such that f (x, y) 6⊆ C(x).
(iii) There exists a nonempty compact subset K of X. For each N ∈ hY i,
there exists a compact F C-subspace LN of Y containing N , and for
NONEMPTY INTERSECTION THEOREMS
35
each x ∈ X\K, there exists y ∈ LN satisfying
y 6∈ ccl{u ∈ Y : f (x, u) 6⊆ C(x)}.
Then there exists x̂ ∈ X such that f (x̂, y) 6⊆ C(x̂) for y ∈ Y .
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Rong-Hua He
Department of Mathematics
Chengdu University of Information Technology
Chengdu, Sichuan 610103, P. R. China
E-mail address: [email protected]
Hong-Xu Li
Department of Mathematics
Sichuan University
Chengdu, Sichuan 610064, P. R. China
E-mail address: [email protected]