báo cáo hóa học:" Research Article Systems of Generalized Quasivariational Inclusion Problems with Applications in LΓ-Spaces" doc

Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2011, Article ID 561573, 12 pages
doi:10.1155/2011/561573
Research Article
Systems of Generalized Quasivariational Inclusion
Problems with Applications in LΓ-Spaces
Ming-ge Yang,
1, 2
Jiu-ping Xu,
3
and Nan-jing Huang
1, 3
1
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2
Department of Mathematics, Luoyang Normal University, Luoyang, Henan 471022, China
3
College of Business and Management, Sichuan University, Chengdu, Sichuan 610064, China
Correspondence should be addressed to Nan-jing Huang,
Received 27 September 2010; Accepted 22 October 2010
Academic Editor: Yeol J. E. Cho
Copyright q 2011 Ming-ge Yang et al. 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.
We first prove that the product of a family of LΓ-spaces is also an LΓ-space. Then, by using a
Himmelberg type fixed point theorem in LΓ-spaces, we establish existence theorems of solutions
for systems of generalized quasivariational inclusion problems, systems of variational equations,
and systems of generalized quasiequilibrium problems in LΓ-spaces. Applications of the existence
theorem of solutions for systems of generalized quasiequilibrium problems to optimization
problems are given in LΓ-spaces.

1. Introduction
In 1979, Robinson 1 studied the following parametric variational inclusion problem: given
x ∈ R
n
,findy ∈ R
m
such that
0 ∈ g

x, y

 Q

x, y

, 1.1
where g : R
n
× R
m
→ R
p
is a single-valued function and Q : R
n
× R
m
 R
p
is a multivalued
map. It is known that 1.1 covers variational inequality problems and a vast of variational

system important in applications. Since then, various types of variational inclusion problems
have been extended and generalized by many authors see, e.g., 2–7 and the references
therein.
On the other hand, Tarafdar 8 generalized the classical Himmelberg fixed point
theorem 9 to locally H-convex uniform spaces or LC-spaces.Park10 generalized
the result of Tarafdar 8 to locally G-convex spaces or LG-spaces. Recently, Park 11
2 Fixed Point Theory and Applications
introduced the concept of abstract convex spaces which include H-spaces and G-convex
spaces as special cases. With this new concept, he can study the KKM theory and its
applications in abstract convex spaces. More recently, Park 12 introduced the concept of
LΓ-spaces which include LC-spaces and LG-spaces as special cases. He also established the
Himmelberg type fixed point theorem in LΓ-spaces. To see some related works, we refer to
13–21 and the references therein. However, to the best of our knowledge, there is no paper
dealing with systems of generalized quasivariational inclusion problems in LΓ-spaces.
Motivated and inspired by the works mentioned above, in this paper, we first prove
that the product of a family of LΓ-spaces is also an LΓ-space. Then, by using the Himmelberg
type fixed point theorem due to Park 12, we establish existence theorems of solutions for
systems of generalized quasivariational inclusion problems, systems of variational equations,
and systems of generalized quasiequilibrium problems in LΓ-spaces. Applications of the
existence theorem of solutions for systems of generalized quasiequilibrium problems to
optimization problems are given in LΓ-spaces.
2. Preliminaries
For a set X, X will denote the family of all nonempty finite subsets of X.IfA is a subset of
a topological space, we denote by intA and
A the interior and closure of A, respectively.
A multimap or simply a map T : X  Y isafunctionfromasetX into the power
set 2
Y
of Y ; that is, a function with the values Tx ⊂ Y for all x ∈ X. Given a map T : X  Y,
the map T

−
: Y  X defined by T
−
y{x ∈ X : y ∈ Tx} for all y ∈ Y, is called the lower
inverse of T. For any A ⊂ X, TA :

x∈A
Tx. For any B ⊂ Y , T
−
B : {x ∈ X : Tx∩B
/
 ∅}.
As usual, the set {x, y ∈ X × Y : y ∈ Tx}⊂X × Y is called the graph of T.
For topological spaces X and Y , a map T : X  Y is called
i closed if its graph GraphT is a closed subset of X × Y,
ii upper semicontinuous in short, u.s.c. if for any x ∈ X and any open set V in Y
with T x ⊂ V , there exists a neighborhood U of x such that Tx

 ⊂ V for all x

∈ U,
iii lower semicontinuous in short, l.s.c. if for any x ∈ X and any open set V in Y with
Tx∩V
/
 ∅, there exists a neighborhood U of x such that Tx

∩V
/
 ∅ for all x


∈ U,
iv continuous if T is both u.s.c. and l.s.c.,
v compact if TX is contained in a compact subset of Y.
Lemma 2.1 see 22. Let X and Y be topological spaces, T : X  Y be a map. Then, T is l.s.c. at
x ∈ X if and only i f for any y ∈ Tx and for any net {x
α
} in X converging to x, there exists a net
{y
α
} in Y such that y
α
∈ Tx
α
 for each α and y
α
converges to y.
Lemma 2.2 see 23. Let X and Y be Hausdorff topological spaces and T : X  Y be a map.
i If T is an u.s.c. map with closed values, then T is closed.
ii If Y is a compact space and T is closed, then T is u.s.c.
iii If X is compact and T is an u.s.c. map with compact values, then TX is compact.
In what follows, we introduce the concept of abstract convex spaces and map classes
R, RC and RO having certain KKM properties. For more details and discussions, we refer
the reader to 11, 12, 24.
Fixed Point Theory and Applications 3
Definition 2.3 see 11. An abstract convex space E, D; Γ consists of a topological space E,
a nonempty set D, and a map Γ : D  E with nonempty values. We denote Γ
A
:ΓA for
A ∈D.
In the case E  D,letE; Γ :E, E; Γ. It is obvious that any vector space E is an

abstract convex space with Γco, where co denotes the convex hull in vector spaces. In
particular, R;co is an abstract convex space.
Let E, D; Γ be an abstract convex space. For any D

⊂ D,theΓ-convex hull of D

is
denoted and defined by
co
Γ
D

:


Γ
A
| A ∈

D


⊂ E, 2.1
co is reserved for the convex hull in vector spaces.AsubsetX of E is called a Γ-convex
subset of E, D; Γ relative to D

if for any N ∈D

, we have Γ
N

⊂ X;thatis,co
Γ
D

⊂ X.
This means that X, D

; Γ|
D


 itself is an abstract convex space called a subspace of E, D; Γ.
When D ⊂ E, the space is denoted by E ⊃ D; Γ. In such case, a subset X of E is said to be
Γ-convex if co
Γ
X ∩ D ⊂ X; in other words, X is Γ-convex relative to D

 X ∩ D. When
E; Γ  R;co, Γ-convex subsets reduce to ordinary convex subsets.
Let E, D; Γ be an abstract convex space and Z a set. For a map F : E  Z with
nonempty values, if a map G : D  Z satisfies
F

Γ
A

⊂ G

A


, ∀A ∈

D

, 2.2
then G is called a KKM map with respect to F. A KKM map G : D  E is a KKM map with
respect to the identity map 1
E
. A map F : E  Z is said to have the KKM property and called
a R-map if, for any KKM map G : D  Z with respect to F, the family {Gy}
y∈D
has the
finite intersection property. We denote
R

E, Z

:

F : E  Z | F is a R-map

. 2.3
Similarly, when Z is a topological space, a RC-map is defined for closed-valued maps
G,andaRO-map is defined for open-valued maps G. In this case, we have
R

E, Z

⊂ RC


E, Z

∩ RO

E, Z

. 2.4
Note that if Z is discrete, then three classes R, RC and RO are identical. Some authors use
the notation KKME, Z instead of RCE, Z.
Definition 2.4 see 24. For an abstract convex space E, D; Γ, the KKM principle is the
statement 1
E
∈ RCE, E ∩ ROE, E.
A KKM space is an abstract convex space satisfying the KKM principle.
4 Fixed Point Theory and Applications
Definition 2.5. Let Y; Γ be an abstract convex space, Z be a real t.v.s., and F : Y  Z a map.
Then,
i F is {0}-quasiconvex-like if for any {y
1
,y
2
, ,y
n
}∈Y and any y ∈
Γ{y
1
,y
2
, ,y
n

} there exists j ∈{1, 2, ,n} such that Fy ⊂ Fy
j
,
ii F is {0}-quasiconvex if for any {y
1
,y
2
, ,y
n
}∈Y and any y ∈ Γ{y
1
,y
2
, ,y
n
}
there exists j ∈{1, 2, ,n} such that Fy
j
 ⊂ Fy.
Remark 2.6. If Y is a nonempty convex subset of a t.v.s. with Γco, then Definition 2.5 i and
ii reduce to Definition 2.4 iii and vi  in Lin 5, respectively.
Definition 2.7 see 25. A uniformity for a set X is a nonempty family U of subsets of X × X
satisfying the following conditions:
i each member of U contains the diagonal Δ,
ii for each U ∈U, U
−1
∈U,
iii for each U ∈U, there exists V ∈Usuch that V ◦ V ⊂ U,
iv if U ∈U, V ∈U, then U ∩ V ∈U,
v if U ∈Uand U ⊂ V ⊂ X × X, then V ∈U.

The pair X, U is called a uniform space. Every member in U is called an entourage.
For any x ∈ X and any U ∈U, we define Ux : {y ∈ X : x, y ∈ U}. The uniformity U is
called separating if

{U ⊂ X × X : U ∈U}Δ. The uniform space 
X, U is Hausdorff if and
only if U is separating. For more details about uniform spaces, we refer the reader to Kelley
25.
Definition 2.8 see 12. An abstract convex uniform space E, D; Γ; B is an abstract convex
space with a basis B of a uniformity of E.
Definition 2.9 see 12. An abstract convex uniform space E ⊃ D; Γ; B is called an LΓ-space
if
i D is dense in E,and
ii for each U ∈Band each Γ-convex subset A ⊂ E,theset{x ∈ E : A ∩ Ux
/
 ∅} is
Γ-convex.
Lemma 2.10 see 12, Corollary 4.5.
Let E ⊃ D; Γ; B be a Hausdorff KKM LΓ-space and T :
E  E a compact u.s.c. map with nonempty closed Γ-convex values. Then, T has a fixed point.
Lemma 2.11 see 24, Lemma 8.1. Let {E
i
,D
i
; Γ
i
}
i∈I
be any family of abstract convex spaces.
Let E :


i∈I
E
i
and D :

i∈I
D
i
. For each i ∈ I,letπ
i
: D → D
i
be the projection. For each
A ∈D, define ΓA :

i∈I
Γ
i
π
i
A. Then, E, D; Γ is an abstract convex space.
Lemma 2.12. Let I be any index set. For each i ∈ I,letX
i
; Γ
i
; B
i
 be an LΓ-space. If one defines
X :


i∈I
X
i
, ΓA :

i∈I
Γ
i
π
i
A for each A ∈X and B : {

n
j1
U
j
: U
j
∈S,j 
1, 2, ,n and n ∈ N},whereS : {{x, y ∈ X × X : x
i
,y
i
 ∈ U
i
} : i ∈ I,U
i
∈B
i

}. Then,
X; Γ; B is also an LΓ-space.
Fixed Point Theory and Applications 5
Proof. By Lemma 2.11, X; Γ is an abstract convex space. It is easy to check that S is a subbase
of the product uniformity of X. Since B is the basis generated by S,weobtainthatB is a basis
of the product uniformity, and the associated uniform topology on X.
Now, we prove that for each U ∈Band each Γ-convex subset A ⊂ X,theset{x ∈ X :
A ∩ Ux
/
 ∅} is Γ-convex. Firstly, we show that for each i ∈ I, π
i
A is a Γ
i
-convex subset
of X
i
. For any N
i
∈π
i
A, we can find some N ∈A with π
i
NN
i
. Since A is a Γ-
convex subset of X, we have ΓN ⊂ A. It follows that Γ
i
π
i
N  Γ

i
N
i
 ⊂ π
i
A. Thus, we
have shown that π
i
A is a Γ
i
-convex subset of X
i
. Secondly, we show that the set {x ∈ X :
A∩Ux
/
 ∅} is Γ-convex. Since each U
j
∈Shas the form U
j
 {x, y ∈ X×X : x
i
j
,y
i
j
 ∈ U
i
j
}
for some i

j
∈ I and U
i
j
∈B
i
j
, we have that
U

x



y ∈ X :

x, y

∈ U


⎧
⎨
⎩
y ∈ X:

x, y

∈
n


j1
U
j
⎫
⎬
⎭


y ∈ X :

x
i
j
,y
i
j

∈ U
i
j
∀ j  1, 2, ,n



y ∈ X : y
i
j
∈ U
i

j

x
i
j

∀ j  1, 2, ,n



i∈I\
{
i
j
:j1,2, ,n
}
X
i
×
n

j1
U
i
j

x
i
j


,
2.5
{
x ∈ X : A ∩ U

x

/
 ∅
}

⎧
⎨
⎩
x ∈ X : A ∩
⎛
⎝

i∈I\{i
j
:j1,2, ,n}
X
i
×
n

j1
U
i
j


x
i
j

⎞
⎠
/
 ∅
⎫
⎬
⎭

⎧
⎨
⎩
x ∈ X :

i∈I\{i
j
:j1,2, ,n}

π
i

A

∩ X
i


×
n

j1

π
i
j

A

∩ U
i
j

x
i
j

/
 ∅
⎫
⎬
⎭

⎧
⎨
⎩
x ∈ X :
n


j1

π
i
j

A

∩ U
i
j

x
i
j

/
 ∅
⎫
⎬
⎭

n

j1

x ∈ X : π
i
j


A

∩ U
i
j

x
i
j

/
 ∅


n

j1
⎛
⎝

i∈I\{i
j
}
X
i
×

x
i

j
∈ X
i
j
: π
i
j

A

∩ U
i
j

x
i
j

/
 ∅

⎞
⎠
.
2.6
By the definition of LΓ-spaces, we obtain that for each j ∈{1, 2, ,n},theset{x
i
j
∈ X
i

j
:
π
i
j
A ∩ U
i
j
x
i
j

/
 ∅} is Γ
i
j
-convex. It follows from 2.6 that the set {x ∈ X : A ∩ Ux
/
 ∅} is
a Γ-convex subset of X. Therefore X; Γ; B is an LΓ-space. This completes the proof.
Remark 2.13. Lemma 2.12 generalizes 26, Theorem 2.2 from locally FC-uniform spaces to
LΓ-spaces. The proof of Lemma 2.12 is different with the proof of 26, Theorem 2.2.
6 Fixed Point Theory and Applications
3. Existence Theorems of Solutions for Systems of Generalized
Quasivariational Inclusion Problems
Let I be any index set. For each i ∈ I,letZ
i
be a topological vector space, X
i
; Γ

1
i
; B
1
i
 be an LΓ-
space, and Y
i
; Γ
2
i
; B
2
i
 be an LΓ-space with 1
Y
i
∈ RCY
i
,Y
i
.LetX 

i∈I
X
i
, Y 

i∈I
Y

i
and
X ×Y ; Γ; B be the product LΓ-space as defined in Lemma 2.12. Furthermore, we assume that
X × Y; Γ; B is a KKM space. Throughout this paper, we use these notations unless otherwise
specified, and assume that all topological spaces are Hausdorff.
The following theorem is the main result of this paper.
Theorem 3.1. For each i ∈ I, suppose that
i A
i
: X × Y  X
i
is a compact u.s.c. map with nonempty closed Γ
1
i
-convex values,
ii T
i
: X  Y
i
is a compact continuous map with nonempty closed Γ
2
i
-convex values,
iii G
i
: X × Y
i
× Y
i
 Z

i
is a closed map with nonempty values,
iv for each x, v
i
 ∈ X × Y
i
, y
i
 G
i
x, y
i
,v
i
 is {0}-quasiconvex; for each x, y
i
 ∈ X × Y
i
,
v
i
 G
i
x, y
i
,v
i
 is {0}-quasiconvex-like and 0 ∈ G
i
x, y

i
,y
i
.
Then, there exists 
x, y ∈ X × Y with x x
i

i∈I
and y y
i

i∈I
such that for each i ∈ I, x
i
∈
A
i
x, y, y
i
∈ T
i
x and 0 ∈ G
i
x, y
i
,v
i
 for all v
i

∈ T
i
x.
Proof. For each i ∈ I, define H
i
: X  T
i
X by
H
i

x



y
i
∈ T
i

x

:0∈ G
i

x, y
i
,v
i


∀ v
i
∈ T
i

x


, ∀x ∈ X. 3.1
Then, H
i
x is nonempty for each x ∈ X. Indeed, fix any i ∈ I and x ∈ X, define Q
x
i
: T
i
x 
T
i
x by
Q
x
i

v
i



y

i
∈ T
i

x

:0∈ G
i

x, y
i
,v
i

, ∀v
i
∈ T
i

x

. 3.2
First, we show that Q
x
i
is a KKM map w.r.t. 1
T
i
x
. Suppose to the contrary that there exists a

finite subset {v
1
i
,v
2
i
, ,v
n
i
}⊂T
i
x such that Γ
2
i
{v
1
i
,v
2
i
, ,v
n
i
}
/
⊂

n
k1
Q

x
i
v
k
i
. Hence, there
exists
v
i
∈ Γ
2
i
{v
1
i
,v
2
i
, ,v
n
i
} satisfying v
i
/
∈ Q
x
i
v
k
i

 for all k  1, 2, ,n. Since T
i
x is Γ
2
i
-
convex, we have
v
i
∈ Γ
2
i
{v
1
i
,v
2
i
, ,v
n
i
} ⊂ T
i
x.Byv
i
/
∈ Q
x
i
v

k
i
 for all k  1, 2, ,n,we
know that 0
/
∈ G
i
x, v
i
,v
k
i
 for all k  1, 2, ,n. Since v
i
 G
i
x, v
i
,v
i
 is {0}-quasiconvex-
like, there exists 1 ≤ j ≤ n such that
0 ∈ G
i

x,
v
i
, v
i


⊂ G
i

x,
v
i
,v
j
i

. 3.3
This leads to a contradiction. Therefore, Q
x
i
is a KKM map w.r.t. 1
T
i
x
. Next, we show that
Q
x
i
v
i
 is closed for each v
i
∈ T
i
x. Indeed, if y

i
∈ Q
x
i
v
i
, then there exists a net {y
α
i
}
α∈Λ
in Q
x
i
v
i
 such that y
α
i
→ y
i
. For each α ∈ Λ, we have y
α
i
∈ T
i
x and 0 ∈ G
i
x, y
α

i
,v
i
.By
condition ii, T
i
x is closed, and hence y
i
∈ T
i
x. By condition iii, G
i
is closed, and hence
0 ∈ G
i
x, y
i
,v
i
. It follows that y
i
∈ Q
x
i
v
i
. Therefore, Q
x
i
v

i
 is closed. Since 1
Y
i
∈ RCY
i
,Y
i

and T
i
x is Γ
2
i
-convex, we have that 1
T
i
x
∈ RCT
i
x,T
i
x. Having that T
i
is compact, we
can deduce that

v
i
∈T

i
x
Q
x
i
v
i

/
 ∅.ThatisH
i
x is nonempty.
Fixed Point Theory and Applications 7
H
i
is closed for each i ∈ I. Indeed, if x, y
i
 ∈ GraphH
i
, then there exists a
net {x
α
,y
α
i
}
α∈Λ
in GraphH
i
 such that x

α
,y
α
i
 → x, y
i
. One has y
α
i
∈ T
i
x
α
 and
0 ∈ G
i
x
α
,y
α
i
,v
i
 for all v
i
∈ T
i
x
α
. By condition ii, T

i
is closed, and hence y
i
∈ T
i
x.
Let v
i
∈ T
i
x,sinceT
i
is l.s.c., there exists a net {v
α
i
} satisfying v
α
i
∈ T
i
x
α
 and v
α
i
→ v
i
.We
have 0 ∈ G
i

x
α
,y
α
i
,v
α
i
. Since G
i
is closed, we obtain 0 ∈ G
i
x, y
i
,v
i
. T hus, we have shown
that x, y
i
 ∈ GraphH
i
. Hence, H
i
is closed.
H
i
x is Γ
2
i
-convex for each i ∈ I and x ∈ X. Indeed, if {y

1
i
,y
2
i
, ,y
n
i
}∈H
i
x,
then we have that {y
1
i
,y
2
i
, ,y
n
i
}⊂T
i
x and 0 ∈ G
i
x, y
k
i
,v
i
 for all v

i
∈ T
i
x and all
k  1, 2, ,n. For any given
y
i
∈ Γ
2
i
{y
1
i
,y
2
i
, ,y
n
i
}, we have y
i
∈ T
i
x because T
i
x is Γ
2
i
-
convex. For each v

i
∈ T
i
x,sincey
i
 G
i
x, y
i
,v
i
 is {0}-quasiconvex, there exists 1 ≤ j ≤ n
such that
G
i

x, y
j
i
,v
i

⊂ G
i

x,
y
i
,v
i


. 3.4
Hence, 0 ∈ G
i
x, y
i
,v
i
 for all v
i
∈ T
i
x. It follows that y
i
∈ H
i
x and H
i
x is Γ
2
i
-convex.
Since H
i
X ⊂ T
i
X and T
i
X is compact. It follows from Lemma 2.2ii that H
i

is a
compact u.s.c. map for each i ∈ I. Define Q : X × Y  X × Y by
Q

x, y




i∈I
A
i

x, y


×


i∈I
H
i

x


, ∀

x, y


∈ X × Y.
3.5
It follows from the above discussions that for each i ∈ I, H
i
is a compact u.s.c. map with
nonempty closed Γ
2
i
-convex values. Thus, Q is a compact u.s.c. map with nonempty closed
Γ-convex values. By Lemma 2.10, there exists 
x, y ∈ X × Y such that x, y ∈ Qx, y.Thatis
there exists 
x, y ∈ X ×Y with x x
i

i∈I
and y y
i

i∈I
such that for each i ∈ I, x
i
∈ A
i
x, y,
y
i
∈ T
i
x and 0 ∈ G

i
x, y
i
,v
i
 for all v
i
∈ T
i
x. This completes the proof.
For the special case of Theorem 3.1, we have the following corollary which is actually
an existence theorem of solutions for variational equations.
Corollary 3.2. For each i ∈ I, suppose that conditions (i) and (ii) in Theorem 3.1 hold. Moreover,
iii
1
G
i
: X × Y
i
× Y
i
→ Z
i
is a continuous mapping;
iv
1
for each x, v
i
 ∈ X × Y
i

, y
i
→ G
i
x, y
i
,v
i
 is {0}-quasiconvex; for each x, y
i
 ∈ X × Y
i
,
v
i
→ G
i
x, y
i
,v
i
 is also {0}-quasiconvex and G
i
x, y
i
,y
i
0.
Then, there exists 
x, y ∈ X × Y with x x

i

i∈I
and y y
i

i∈I
such that for each i ∈ I, x
i
∈
A
i
x, y, y
i
∈ T
i
x and G
i
x, y
i
,v
i
0 for all v
i
∈ T
i
x.
Theorem 3.3. For each i ∈ I, suppose that conditions (i) and (ii) in Theorem 3.1 hold. Moreover,
iii
2

H
i
: X  Z
i
is a closed map with nonempty values and Q
i
: X × Y
i
× Y
i
 Z
i
is an u.s.c.
map with nonempty compact values;
iv
2
for each x, v
i
 ∈ X × Y
i
, y
i
 Q
i
x, y
i
,v
i
 is {0}-quasiconvex; for each x, y
i

 ∈ X × Y
i
,
v
i
 Q
i
x, y
i
,v
i
 is {0}-quasiconvex-like and 0 ∈ H
i
xQ
i
x, y
i
,y
i
.
Then, there exists 
x, y ∈ X × Y with x x
i

i∈I
and y y
i

i∈I
such that for each i ∈ I, x

i
∈
A
i
x, y, y
i
∈ T
i
x and 0 ∈ H
i
xQ
i
x, y
i
,v
i
 for all v
i
∈ T
i
x.
8 Fixed Point Theory and Applications
Proof. For each i ∈ I, define G
i
: X × Y
i
× Y
i
 Z
i

by
G
i

x, y
i
,v
i

 H
i

x

 Q
i

x, y
i
,v
i

, ∀

x, y
i
,v
i

∈ X × Y

i
× Y
i
. 3.6
Obviously, G
i
has nonempty values. Now, we show that G
i
is closed. Indeed, if
x, y
i
,v
i
,z
i
 ∈ GraphG
i
, then there exists a net {x
α
,y
α
i
,v
α
i
,z
α
i
}
α∈Λ

in GraphG
i
 such that
x
α
,y
α
i
,v
α
i
,z
α
i
 → x, y
i
,v
i
,z
i
. Since
z
α
i
∈ G
i

x
α
,y

α
i
,v
α
i

 H
i

x
α

 Q
i

x
α
,y
α
i
,v
α
i

, 3.7
there exist u
α
i
∈ H
i

x
α
 and w
α
i
∈ Q
i
x
α
,y
α
i
,v
α
i
 such that z
α
i
 u
α
i
 w
α
i
.Let
K 
{
x
α
: α ∈ Λ

}
∪
{
x
}
,L
i


y
α
i
: α ∈ Λ

∪

y
i

,M
i


v
α
i
: α ∈ Λ

∪
{

v
i
}
. 3.8
Then K is a compact subset of X, L
i
and M
i
are compact subsets of Y
i
. By condition iii
2
and
Lemma 2.2iii, Q
i
K×L
i
×M
i
 is a compact subset of Z
i
. Thus, we can assume that w
α
i
→ w
i
.
By condition iii
2
, Q

i
is closed, and hence w
i
∈ Q
i
x, y
i
,v
i
. Since z
α
i
− w
α
i
 u
α
i
∈ H
i
x
α
 and
H
i
is closed, we have z
i
− w
i
∈ H

i
x. Letting u
i
 z
i
− w
i
, it follows that
z
i
 u
i
 w
i
∈ H
i

x

 Q
i

x, y
i
,v
i

 G
i


x, y
i
,v
i

, 3.9
and so G
i
is closed.
By the above discussions, we know that condition iii of Theorem 3.1 is satisfied. It is
easy to check that condition iv of Theorem 3.1 is also satisfied. By Theorem 3.1, there exists

x, y ∈ X × Y with x x
i

i∈I
and y y
i

i∈I
such that for each i ∈ I, x
i
∈ A
i
x, y, y
i
∈ T
i
x
and

0 ∈ G
i

x, y
i
,v
i

 H
i

x

 Q
i

x, y
i
,v
i

, 3.10
for all v
i
∈ T
i
x. This completes the proof.
For the special case of Theorem 3.3, we have the following corollary which is actually
an existence theorem of solutions for variational equations.
Corollary 3.4. For each i ∈ I, suppose that conditions (i) and (ii) in Theorem 3.1 hold. Moreover,

iii
3
H
i
: X → Z
i
is a continuous map and Q
i
: X × Y
i
× Y
i
→ Z
i
is a continuous map;
iv
3
for each x, v
i
 ∈ X × Y
i
, y
i
→ Q
i
x, y
i
,v
i
 is {0}-quasiconvex; for each x, y

i
 ∈ X × Y
i
,
v
i
→ Q
i
x, y
i
,v
i
 is also {0}-quasiconvex and H
i
xQ
i
x, y
i
,y
i
0.
Then, there exists 
x, y ∈ X × Y with x x
i

i∈I
and y y
i

i∈I

such that for each i ∈ I, x
i
∈
A
i
x, y, y
i
∈ T
i
x and H
i
xQ
i
x, y
i
,v
i
0 for all v
i
∈ T
i
x.
From Theorem 3.3, we establish the following corollary which is actually an existence
theorem of solutions for systems of generalized vector quasiequilibrium problems.
Fixed Point Theory and Applications 9
Corollary 3.5. For each i ∈ I, suppose that conditions (i) and (ii) in Theorem 3.1 hold. Moreover,
iii
4
C
i

: X  Z
i
is a closed map with nonempty values and Q
i
: X × Y
i
× Y
i
 Z
i
is an u.s.c.
map with nonempty compact values;
iv
4
for each x, v
i
 ∈ X × Y
i
, y
i
 Q
i
x, y
i
,v
i
 is {0}-quasiconvex; for each x, y
i
 ∈ X × Y
i

,
v
i
 Q
i
x, y
i
,v
i
 is {0}-quasiconvex-like and Q
i
x, y
i
,y
i
 ∩ C
i
x
/
 ∅.
Then, there exists 
x, y ∈ X × Y with x x
i

i∈I
and y y
i

i∈I
such that for each i ∈ I, x

i
∈
A
i
x, y, y
i
∈ T
i
x, and Q
i
x, y
i
,v
i
 ∩ C
i
x
/
 ∅ for all v
i
∈ T
i
x.
Proof. Define H
i
: X  Z
i
by H
i
x−C

i
x for all x ∈ X. Since C
i
is a closed map with
nonempty values, we have that H
i
is a closed map with nonempty values. All the conditions
of Theorem 3.3 are satisfied. The conclusion of Corollary 3.5 follows from Theorem 3.3.This
completes the proof.
4. Applications to Optimization Problems
Let Z be a real topological vector space, D a proper convex cone in Z.Apointy ∈ A is called
a vector minimal point of A if for any y ∈ A, y −
y
/
∈−D \{0}. The set of vector minimal point
of A is denoted by Min
D
A.
Lemma 4.1 see 27. Let Z be a Hausdorff t.v.s., D be a closed convex cone in Z.IfA is a nonempty
compact subset of Z,thenMin
D
A
/
 ∅.
Theorem 4.2. For each i ∈ I, suppose that conditions (i), (ii) in Theorem 3.1 and conditions (iii)
4
,
(iv)
4
in Corollary 3.5 hold. Furthermore, let h : X × Y  Z be an u.s.c. map with nonempty compact

values, where Z is a real t.v.s. ordered by a proper closed convex cone in Z. Then, there exists a solution
to:
Min
x,y
h

x, y

, 4.1
where x x
i

i∈I
and y y
i

i∈I
such that for each i ∈ I, x
i
∈ A
i
x, y, y
i
∈ T
i
x, and Q
i
x, y
i
,v

i
∩
C
i
x
/
 ∅ for all v
i
∈ T
i
x.
Proof. By Corollary 3.5, there exists 
x, y ∈ X × Y with x x
i

i∈I
and y y
i

i∈I
such that
for each i ∈ I,
x
i
∈ A
i
x, y, y
i
∈ T
i

x and Q
i
x, y
i
,v
i
 ∩ C
i
x
/
 ∅ for all v
i
∈ T
i
x. For each
i ∈ I,let
M
i


x, y

∈ X × Y : x
i
∈ A
i

x, y

,y

i
∈ T
i

x

,
Q
i

x, y
i
,v
i

∩ C
i

x

/
 ∅∀v
i
∈ T
i

x


,

4.2
and M 

i∈I
M
i
. Then x, y ∈ M and M
/
 ∅.WeshowthatM
i
is closed for each i ∈ I.
Indeed, if x, y ∈
M
i
, then there exists a net {x
α
,y
α
}
α∈Λ
in M
i
such that x
α
,y
α
 → x, y.
For each α ∈ Λ, x
α
,y

α
 ∈ M
i
implies that
x
α
i
∈ A
i

x
α
,y
α

,y
α
i
∈ T
i

x
α

,Q
i

x
α
,y

α
i
,v
i

∩ C
i

x
α

/
 ∅∀v
i
∈ T
i

x
α

. 4.3
10 Fixed Point Theory and Applications
By the closedness of A
i
and T
i
, we have that x
i
∈ A
i

x, y and y
i
∈ T
i
x. Now, we prove
that Q
i
x, y
i
,v
i
 ∩ C
i
x
/
 ∅ for all v
i
∈ T
i
x. For any v
i
∈ T
i
x,sinceT
i
is l.s.c., there exists a
net {v
α
i
}

α∈Λ
satisfying v
α
i
∈ T
i
x
α
 and v
α
i
→ v
i
.Letu
α
i
∈ Q
i
x
α
,y
α
i
,v
α
i
 ∩ C
i
x
α

. Since Q
i
is
u.s.c. with nonempty compact values, we can assume that u
α
i
→ u
i
∈ Z
i
. By the closedness
of Q
i
and C
i
, we have that u
i
∈ Q
i
x, y
i
,v
i
 ∩ C
i
x.Thus,Q
i
x, y
i
,v

i
 ∩ C
i
x
/
 ∅. It follows
that M
i
is closed. Hence, M is closed. Note that M ⊂

i∈I
A
i
X × Y  ×

i∈I
T
i
X.Weknow
that M is a nonempty compact subset of X × Y . It follows from Lemma 2.2iii that hM is a
nonempty compact subset of Z.ByLemma 4.1,Min
D
hM
/
 ∅. That is there exists a solution
of the problem: Min
x,y
hx, y where x, y ∈ M. This completes the proof.
Theorem 4.3. For each i ∈ I, suppose that X
i

is compact and condition (ii) in Theorem 3.1 holds.
Moreover,
iii
5
Q
i
: X × Y
i
× Y
i
→ R is a continuous function;
iv
5
for each x, v
i
 ∈ X × Y
i
, y
i
→ Q
i
x, y
i
,v
i
 is {0}-quasiconvex; for each x, y
i
 ∈ X × Y
i
,

v
i
→ Q
i
x, y
i
,v
i
 is also {0}-quasiconvex and Q
i
x, y
i
,y
i
 ≥ 0.
Furthermore, let h : X × Y → R is a l.s.c. function. Then there exists a solution to:
min
x,y
h

x, y

, 4.4
where x x
i

i∈I
and y y
i


i∈I
such that for each i ∈ I, y
i
∈ T
i
x and Q
i
x, y
i
,v
i
 ≥ 0 for all
v
i
∈ T
i
x.
Proof. For each i ∈ I, define A
i
: X × Y  X
i
and C
i
: X  R by
A
i

x, y

 X

i
, ∀

x, y

∈ X × Y,
C
i

x



0, ∞

, ∀x ∈ X,
4.5
respectively. It is easy to check that all the conditions of Corollary 3.5 are satisfied. For each
i ∈ I, define
M
i


x, y

∈ X × Y : y
i
∈ T
i


x

,Q
i

x, y
i
,v
i

≥ 0 ∀v
i
∈ T
i

x


, 4.6
and M 

i∈I
M
i
. Then, by Corollary 3.5, there exists x, y ∈ M and hence M
/
 ∅. Arguing as
Theorem 4.2, we can prove that M is a nonempty compact subset of X × Y . Hence there exists
a solution to the problem min
x,y

hx, y where x, y ∈ M. This completes the proof.
Remark 4.4. Theorem 4.3 generalizes 28, Corollary 3.5 from locally convex topological
vector spaces to LΓ-spaces.
Theorem 4.5. For each i ∈ I, suppose that X
i
is compact and condition (ii) in Theorem 3.1 holds.
Moreover,
iii
6
F
i
: X × Y
i
→ R is a continuous function;
iv
6
for each x ∈ X, y
i
→ F
i
x, y
i
 is {0}-quasiconvex.
Fixed Point Theory and Applications 11
Furthermore, let h : X × Y → R be a l.s.c. function. Then, there exists a solution to the problem:
min
x,y
h

x, y


, 4.7
where x x
i

i∈I
and y y
i

i∈I
such that for each i ∈ I, y
i
is the solution of the problem
min
v
i
∈T
i
x
F
i
x, v
i
.
Proof. For each i ∈ I, define Q
i
: X × Y
i
× Y
i

→ R by
Q
i

x, y
i
,v
i

 F
i

x, v
i

− F
i

x, y
i

, ∀

x, y
i
,v
i

∈ X × Y
i

× Y
i
. 4.8
It is easy to check that all the conditions of Theorem 4.3 are satisfied. Theorem 4.5 follows
immediately from Theorem 4.3. This completes the proof.
Acknowledgments
This work was supported by the Key Program of NSFC Grant no. 70831005 and the Open
Fund PLN0904 of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation
Southwest Petroleum University.
References
1 S. M. Robinson, “Generalized equations and their solutions. I. Basic theory,” Mathematical
Programming Study, vol. 10, pp. 128–141, 1979.
2 S. S. Chang, “Set-valued variational inclusions in Banach spaces,” Journal of Mathematical Analysis and
Applications, vol. 248, no. 2, pp. 438–454, 2000.
3 X. P. Ding, “Perturbed proximal point algorithms for generalized quasivariational inclusions,” Journal
of Mathematical Analysis and Applications, vol. 210, no. 1, pp. 88–101, 1997.
4 N J. Huang, “A new class of generalized set-valued implicit variational inclusions in Banach spaces
with an application,” Computers & Mathematics with Applications, vol. 41, no. 7-8, pp. 937–943, 2001.
5 L J. Lin, “Systems of generalized quasivariational inclusions problems with applications to
variational analysis and optimization problems,” Journal of Global Optimization, vol. 38, no. 1, pp. 21–
39, 2007.
6 L J. Lin, “Variational inclusions problems with applications to Ekeland’s variational principle, fixed
point and optimization problems,” Journal of Global Optimization, vol. 39, no. 4, pp. 509–527, 2007.
7 L J. Lin, “Systems of variational inclusion problems and differential inclusion problems with
applications,” Journal of Global Optimization, vol. 44, no. 4, pp. 579–591, 2009.
8 E. Tarafdar, “Fixed point theorems in locally H-convex uniform spaces,” Nonlinear Analysis, vol. 29,
no. 9, pp. 971–978, 1997.
9 C. J. Himmelberg, “Fixed points of compact multifunctions,” Journal of Mathematical Analysis and
Applications, vol. 38, pp. 205–207, 1972.
10 S. Park, “Fixed point theorems in locally G-convex spaces,” Nonlinear Analysis, vol. 48, no. 6, pp. 869–

879, 2002.
11 S. Park, “On generalizations of the KKM principle on abstract convex spaces,” Nonlinear Analysis
Forum, vol. 11, no. 1, pp. 67–77, 2006.
12 S. Park, “Fixed point theory of multimaps in abstract convex uniform spaces,” Nonlinear Analysis,vol.
71, no. 7-8, pp. 2468–2480, 2009.
13 L. Deng and M. G. Yang, “Coincidence theorems with applications to minimax inequalities, section
theorem, best approximation and multiobjective games in topological spaces,” Acta Mathematica Sinica
(English Series), vol. 22, no. 6, pp. 1809–1818, 2006.
14
 L. Deng and M. G. Yang, “Weakly R-KKM mappings-intersection theorems and minimax inequalities
in topological spaces,” Applied Mathematics and Mechanics, vol. 28, no. 1, pp. 103–109, 2007.
12 Fixed Point Theory and Applications
15 S. Park, “Several episodes in recent studies on the KKM theory,” Nonlinear Analysis Forum, vol. 15, pp.
13–26, 2010.
16 S. Park, “Generalized convex spaces, L-spaces, and FC-spaces,” Journal of Global Optimization, vol. 45,
no. 2, pp. 203–210, 2009.
17 S. Park, “Remarks on fixed points, maximal elements, and equilibria of economies in abstract convex
spaces,” Taiwanese Journal of Mathematics, vol. 12, no. 6, pp. 1365–1383, 2008.
18 S. Park, “Remarks on the partial KKM principle,” Nonlinear Analysis Forum, vol. 14, pp. 51–62, 20009.
19 S. Park, “Comments on recent studies on abstract convex spaces,” Nonlinear Analysis Forum, vol. 13,
pp. 1–17, 2008.
20 S. Park, “The rise and decline of generalized convex spaces,” Nonlinear Analysis Forum, vol. 15, pp.
1–12, 2010.
21 M G. Yang, J P. Xu, N J. Huang, and S J. Yu, “Minimax theorems for vector-valued mappings in
abstract convex spaces,” Taiwanese Journal of Mathematics, vol. 14, no. 2, pp. 719–732, 2010.
22 N. X. Tan, “Quasi-variational inequalities in topological linear locally convex Hausdorff spaces,”
Mathematische Nachrichten, vol. 122, pp. 231–224, 1985.
23 J. P. Aubin and A. Cellina, Differential Inclusion, Springer, Berlin, Germany, 1994.
24 S. Park, “Equilibrium existence theorems in KKM spaces,” Nonlinear Analysis, vol. 69, no. 12, pp.
4352–5364, 2008.

25 J. L. Kelley, General Topology, vol. 2, Springer, New York, NY, USA, 1975.
26 X. P. Ding, “The generalized game and the system of generalized vector quasi-equilibrium problems
in locally FC-uniform spaces,” Nonlinear Analysis, vol. 68, no. 4, pp. 1028–1036, 2008.
27 D. C. Luc, Theory of Vector Optimization, vol. 319 of Lecture Notes in Economics and Mathematical Systems,
Springer, Berlin, Germany, 1989.
28
 L J. Lin, “Mathematical programming with system of equilibrium constraints,” Journal of Global
Optimization, vol. 37, no. 2, pp. 275–286, 2007.