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
Trang 1Volume 2011, Article ID 561573, 12 pages
doi:10.1155/2011/561573
Research Article
Systems of Generalized Quasivariational Inclusion
Ming-ge Yang,1, 2 Jiu-ping Xu,3 and Nan-jing Huang1, 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,nanjinghuang@hotmail.com
Received 27 September 2010; Accepted 22 October 2010
Academic Editor: Yeol J E Cho
Copyrightq 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, Robinson1 studied the following parametric variational inclusion problem: given
x ∈ R n , find y∈ Rmsuch that
0∈ gx, y
Qx, y
where g :Rn× Rm → Rp is a single-valued function and Q :Rn× Rm Rpis a multivalued map It is known that1.1 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
Trang 2introduced 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, Park12 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 Park12, 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 If A is a subset of
a topological space, we denote by intA and A the interior and closure of A, respectively.
A multimapor simply a map T : X Y is a function from a set X into the power
set 2Y 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 ∈ Tx} for all y ∈ Y, is called the lower inverse of T For any A ⊂ X, TA :x∈A Tx For any B ⊂ Y, T−B : {x ∈ X : Tx∩B / ∅}.
As usual, the set{x, y ∈ X × Y : y ∈ Tx} ⊂ 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 GraphT 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 Tx ⊂ V , there exists a neighborhood U of x such that Tx ⊂ 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 Tx∩V / ∅, there exists a neighborhood U of x such that Tx ∩V / ∅ for all x ∈ U,
iv continuous if T is both u.s.c and l.s.c.,
v compact if TX 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 if for any y ∈ Tx and for any net {x α } in X converging to x, there exists a net {y α } in Y such that y α ∈ Tx 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 TX 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 to11,12,24
Trang 3Definition 2.3see 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 : ΓA 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
LetE, D; Γ be an abstract convex space For any D ⊂ D, the Γ-convex hull of D is denoted and defined by
coΓD :Γ
A | A ∈D
co is reserved for the convex hull in vector spaces A subset X of E is called a Γ-convex
subset ofE, D; Γ relative to D if for any N ∈ D , we have ΓN ⊂ X; that is, coΓD ⊂ X.
This means thatX, 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 coΓX ∩ 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
FΓ A ⊂ GA, ∀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 1E A map F : E Z is said to have the KKM property and called
aR-map if, for any KKM map G : D Z with respect to F, the family {Gy} y∈D has the finite intersection property We denote
RE, Z :F : E Z | F is a R-map . 2.3
Similarly, when Z is a topological space, aRC-map is defined for closed-valued maps
G, and a RO-map is defined for open-valued maps G In this case, we have
RE, Z ⊂ RCE, Z ∩ ROE, Z. 2.4
Note that if Z is discrete, then three classesR, RC and RO are identical Some authors use the notation KKME, Z instead of RCE, Z
Definition 2.4 see 24 For an abstract convex space E, D; Γ, the KKM principle is the
statement 1E ∈ RCE, E ∩ ROE, E.
A KKM space is an abstract convex space satisfying the KKM principle
Trang 4Definition 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 {y1, y2, , y n } ∈ Y and any y ∈ Γ{y1, y2, , y n } there exists j ∈ {1, 2, , n} such that Fy ⊂ Fy j,
ii F is {0}-quasiconvex if for any {y1, y2, , y n } ∈ Y and any y ∈ Γ{y1, y2, , y n}
there exists j ∈ {1, 2, , n} such that Fy j ⊂ Fy.
Remark 2.6 If Y is a nonempty convex subset of a t.v.s withΓ co, thenDefinition 2.5i and
ii reduce toDefinition 2.4iii and vi in Lin 5, respectively
Definition 2.7see 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 ∈ U such that V ◦ V ⊂ U,
iv if U ∈ U, V ∈ U, then U ∩ V ∈ U,
v if U ∈ U and U ⊂ V ⊂ X × X, then V ∈ U.
The pairX, U is called a uniform space Every member in U is called an entourage For any x ∈ X and any U ∈ U, we define Ux : {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 ifU is separating For more details about uniform spaces, we refer the reader to Kelley
25
Definition 2.8see 12 An abstract convex uniform space E, D; Γ; B is an abstract convex
space with a basisB of a uniformity of E.
Definition 2.9see 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 ∈ B and each Γ-convex subset A ⊂ E, the set {x ∈ E : A ∩ Ux / ∅} 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;Γi}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, let π i : D → D i be the projection For each
A ∈ D, define ΓA : i∈IΓi 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;Γi;Bi be an LΓ-space If one defines
X : i∈I X i , ΓA : i∈IΓi π i A for each A ∈ X and B : { n j1 U j : U j ∈ S, j
1, 2, , n and n ∈ N}, where S : {{x, y ∈ X × X : x i , y i ∈ U i } : i ∈ I, U i ∈ Bi } Then,
X; Γ; B is also an LΓ-space.
Trang 5Proof ByLemma 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 SinceB is the basis generated by S, we obtain that B is a basis
of the product uniformity, and the associated uniform topology on X.
Now, we prove that for each U ∈ B and each Γ-convex subset A ⊂ X, the set {x ∈ X :
A ∩ Ux / ∅} 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 ΓN ⊂ 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∩Ux / ∅} is Γ-convex Since each U j ∈ S has 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∈ Bi j, we have that
Ux y ∈ X :x, y
∈ U
⎧
⎨
⎩y ∈ X :
x, y
∈n
j1
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 :j1,2, ,n}
X i×n
j1
U i j
x i j
,
2.5
{x ∈ X : A ∩ Ux / ∅}
⎧
⎨
⎩x ∈ X : A ∩
⎛
i∈I\{i j :j1,2, ,n}
X i×n
j1
U i j
x i j
⎞
⎠ / ∅
⎫
⎬
⎭
⎧
⎨
⎩x ∈ X :
i∈I\{i j :j1,2, ,n}
π i A ∩ X i ×n
j1
π i j A ∩ U i j
x i j
/ ∅
⎫
⎬
⎭
⎧
⎨
⎩x ∈ X :
n
j1
π i j A ∩ U i j
x i j
/ ∅
⎫
⎬
⎭
n
j1
x ∈ X : π i j A ∩ U i j
x i j
/ ∅
n
j1
⎛
⎝
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}, the set {x i j ∈ X i j :
π i j A ∩ U i j x i j / ∅} is Γ i j-convex It follows from2.6 that the set {x ∈ X : A ∩ Ux / ∅} is
aΓ-convex subset of X Therefore X; Γ; B is an LΓ-space This completes the proof.
Remark 2.13. Lemma 2.12 generalizes26, Theorem 2.2 from locally FC-uniform spaces to LΓ-spaces The proof ofLemma 2.12is different with the proof of 26, Theorem 2.2.
Trang 63 Existence Theorems of Solutions for Systems of Generalized
Quasivariational Inclusion Problems
Let I be any index set For each i ∈ I, let Z ibe a topological vector space,X i;Γ1
i;B1
i be an
LΓ-space, andY i;Γ2
i;B2
i be an LΓ-space with 1 Y i ∈ RCY i , Y i Let X i∈I X i , Y i∈I Y iand
X ×Y; Γ; B be the product LΓ-space as defined inLemma 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 closedΓ1
i -convex values,
ii T i : X Y i is a compact continuous map with nonempty closedΓ2
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 ii∈I and y y ii∈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, yi, v i
, ∀v i ∈ T i x. 3.2
First, we show that Q i xis a KKM map w.r.t 1T i x Suppose to the contrary that there exists a finite subset{v1
i , v2
i , , v n
i } ⊂ T i x such that Γ2
i {v1
i , v2
i , , v n
i }/⊂n k1 Q x
i v k
i Hence, there
exists v i ∈ Γ2
i {v1
i , v2
i , , v n
i } satisfying v i / ∈ Q x
i v k
i for all k 1, 2, , n Since T i x is Γ2
i
-convex, we have v i ∈ Γ2
i {v1
i , v2
i , , v n
i } ⊂ T i x By v 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 i j
This leads to a contradiction Therefore, Q xi is a KKM map w.r.t 1T 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 conditionii, T i x is closed, and hence y i ∈ T i x By condition iii, G iis 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 1Y i ∈ RCY i , Y i
and T i x is Γ2
i-convex, we have that 1T i x ∈ RCT i x, T i x Having that T iis compact, we can deduce that v ∈T x Q x
i v i / ∅ That is H i x is nonempty.
Trang 7H i is closed for each i ∈ I Indeed, if x, y i ∈ GraphH i, then there exists a net {x α , y α
i}α∈Λ in GraphHi 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, since T iis 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 iis closed, we obtain 0 ∈ G i x, y i , v i Thus, we have shown thatx, y i ∈ GraphH i Hence, H iis closed
H i x is Γ2
i -convex for each i ∈ I and x ∈ X Indeed, if {y1
i , y2
i , , y n
i } ∈ H i x,
then we have that {y1
i , y2
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∈ Γ2
i {y1
i , y2
i , , y n
i}, we have y i ∈ T i x because T i x is Γ2
i
-convex For each v i ∈ T i x, since y i G i x, y i , v i is {0}-quasiconvex, there exists 1 ≤ j ≤ n
such that
G i
x, y i j , v i
⊂ G i
x, y i , v i
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 Γ2
i-convex
Since H i X ⊂ T i X and T i X is compact It follows fromLemma 2.2ii that H iis 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 closedΓ2
i -convex values Thus, Q is a compact u.s.c map with nonempty closed
Γ-convex values ByLemma 2.10, there existsx, y ∈ X × Y such that x, y ∈ Qx, y That is
there existsx, y ∈ X×Y with x x ii∈I and y y ii∈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 ofTheorem 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,
iii1G i : X × Y i × Y i → Z i is a continuous mapping;
iv1for 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 ii∈I and y y ii∈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,
iii2H 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;
iv2for 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 ii∈I and y y ii∈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.
Trang 8Proof For each i ∈ I, define G i : X × Y i × Y i Z iby
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 ∈ GraphG i , then there exists a net {x α , y α
i , v α
i , z α
i}α∈Λin GraphGi 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
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 iy α
i : α∈ Λ ∪y i
, M iv 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 conditioniii2and
By conditioniii2, 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− wi ∈ 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
and so G iis closed
By the above discussions, we know that conditioniii ofTheorem 3.1is satisfied It is easy to check that conditioniv ofTheorem 3.1is also satisfied ByTheorem 3.1, there exists
x, y ∈ X × Y with x x ii∈I and y y ii∈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
for all v i ∈ T i x This completes the proof.
For the special case ofTheorem 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,
iii3H i : X → Z i is a continuous map and Q i : X × Y i × Y i → Z i is a continuous map;
iv3for 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 ii∈I and y y ii∈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.
theorem of solutions for systems of generalized vector quasiequilibrium problems
Trang 9Corollary 3.5 For each i ∈ I, suppose that conditions (i) and (ii) in Theorem 3.1 hold Moreover,
iii4C 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;
iv4for 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 ii∈I and y y ii∈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 iis a closed map with
nonempty values, we have that H iis a closed map with nonempty values All the conditions
completes the proof
4 Applications to Optimization Problems
Let Z be a real topological vector space, D a proper convex cone in Z A point y ∈ 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 If A is a nonempty compact subset of Z, then Min D A / ∅.
Theorem 4.2 For each i ∈ I, suppose that conditions (i), (ii) in Theorem 3.1 and conditions (iii)4, (iv)4in 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:
Minx,y h
x, y
where x x ii∈I and y y ii∈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 ByCorollary 3.5, there existsx, y ∈ X × Y with x x ii∈I and y y ii∈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 Thenx, y ∈ M and M / ∅ We show that M i is closed for each i ∈ I.
Indeed, ifx, y ∈ M i, then there exists a net{x α , y α}α∈Λ in M isuch thatx α , y α → x, y For each α ∈ Λ, x α , y α ∈ M iimplies 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
Trang 10By 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, since T iis l.s.c., there exists a net{v α
i}α∈Λ satisfying v i α ∈ T i x α and v α
i → v i Let u α i ∈ Q i x α , y α
i , v α
i ∩ C i x α Since Q iis
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 ∩ Cix 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 We know that M is a nonempty compact subset of X × Y It follows fromLemma 2.2iii that hM is a nonempty compact subset of Z ByLemma 4.1, MinD hM / ∅ That is there exists a solution
of the problem: Minx,y hx, 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,
iii5Q i : X × Y i × Y i → R is a continuous function;
iv5for 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:
minx,y h
x, y
where x x ii∈I and y y ii∈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 ofCorollary 3.5are satisfied For each
i ∈ I, define
M ix, 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, byCorollary 3.5, there existsx, y ∈ M and hence M / ∅ Arguing as
a solution to the problem minx,y hx, 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,
iii6F i : X × Y i → R is a continuous function;
iv6for each x ∈ X, y i → F i x, y i is {0}-quasiconvex.