(ProQuest: ... denotes non-US-ASCII text omitted.)
Ming-ge Yang 1, 3 and Jiu-ping Xu 2 and Nan-jing Huang 1, 2
Recommended by Yeol J. E. Cho
1, Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2, College of Business and Management, Sichuan University, Chengdu, Sichuan 610064, China
3, Department of Mathematics, Luoyang Normal University, Luoyang, Henan 471022, China
Received 27 September 2010; Accepted 22 October 2010
1. Introduction
In 1979, Robinson [1] studied the following parametric variational inclusion problem: given x∈...n , find y∈...m such that [figure omitted; refer to PDF] where g:...n ×...m [arrow right]...p is a single-valued function and Q:...n ×...m ......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] 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 . If A 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 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∈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 if for any y∈T(x) and for any net {xα } in X converging to x , there exists a net {yα } in Y such that yα ∈T(xα ) for each α and yα 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 [real] , [real]... and [real]... having certain KKM properties. For more details and discussions, we refer the reader to [11, 12, 24].
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 ΓA :=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, (...;co ) is an abstract convex space.
Let (E,D;Γ) be an abstract convex space. For any D'⊂D , the Γ -convex hull of D' is denoted and defined by [figure omitted; refer to PDF] (co is reserved for the convex hull in vector spaces). A subset X of E is called a Γ -convex subset of (E,D;Γ) relative to D' if for any N∈...D'... , we have ΓN ⊂X ; that is, coΓ 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 coΓ (X∩D)⊂X ; in other words, X is Γ -convex relative to D'=X∩D . When (E;Γ)=(...;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 [figure omitted; refer to PDF] 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 a [real] -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 [figure omitted; refer to PDF]
Similarly, when Z is a topological space, a [real]... -map is defined for closed-valued maps G , and a [real]... -map is defined for open-valued maps G . In this case, we have [figure omitted; refer to PDF] Note that if Z is discrete, then three classes [real] , [real]... and [real]... are identical. Some authors use the notation KKM(E,Z) instead of [real]...(E,Z) .
Definition 2.4 . (see [24]).
For an abstract convex space (E,D;Γ) , the KKM principle is the statement 1E ∈[real]...(E,E)∩[real]...(E,E) .
A KKM space is an abstract convex space satisfying the KKM principle.
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 {y1 ,y2 ,...,yn }∈...Y... and any y¯∈Γ({y1 ,y2 ,...,yn }) there exists j∈{1,2,...,n} such that F(y¯)⊂F(yj ) ,
(ii) : F is {0} -quasiconvex if for any {y1 ,y2 ,...,yn }∈...Y... and any y¯∈Γ({y1 ,y2 ,...,yn }) there exists j∈{1,2,...,n} such that F(yj )⊂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∈U such that V[composite function]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 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;[Bernoulli]) is an abstract convex space with a basis [Bernoulli] of a uniformity of E .
Definition 2.9 . (see [12]).
An abstract convex uniform space (E⊃D;Γ;[Bernoulli]) is called an LΓ -space if
(i) : D is dense in E , and
(ii) : for each U∈[Bernoulli] and each Γ -convex subset A⊂E , the set {x∈E:A∩U[x]≠∅} is Γ -convex.
Lemma 2.10 . (see [12, Corollary 4.5 ]).
Let (E⊃D;Γ;[Bernoulli]) 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 {(Ei ,Di ;Γi )}i∈I be any family of abstract convex spaces. Let E:=∏i∈IEi and D:=∏i∈IDi . For each i∈I , let πi :D[arrow right]Di be the projection. For each A∈...D... , define Γ(A):=∏i∈IΓi (πi (A)) . Then, (E,D;Γ) is an abstract convex space.
Lemma 2.12.
Let I be any index set. For each i∈I , let (Xi ;Γi ;[Bernoulli]i ) be an LΓ -space. If one defines X:=∏i∈IXi , Γ(A):=∏i∈IΓi (πi (A)) for each A∈...X... and [Bernoulli]:={...j=1nUj :Uj ∈...AE;,j=1,2,...,n and n∈...} , where ...AE;:={{(x,y)∈X×X:(xi ,yi )∈Ui }:i∈I,Ui ∈[Bernoulli]i } . Then, (X;Γ;[Bernoulli]) is also an LΓ -space.
Proof.
By Lemma 2.11, (X;Γ) is an abstract convex space. It is easy to check that ...AE; is a subbase of the product uniformity of X . Since [Bernoulli] is the basis generated by ...AE; , we obtain that [Bernoulli] is a basis of the product uniformity, and the associated uniform topology on X .
Now, we prove that for each U∈[Bernoulli] and each Γ -convex subset A⊂X , the set {x∈X:A∩U[x]≠∅} is Γ -convex. Firstly, we show that for each i∈I , πi (A) is a Γi -convex subset of Xi . For any Ni ∈...πi (A)... , we can find some N∈...A... with πi (N)=Ni . Since A is a Γ -convex subset of X , we have Γ(N)⊂A . It follows that Γi (πi (N))=Γi (Ni )⊂πi (A) . Thus, we have shown that πi (A) is a Γi -convex subset of Xi . Secondly, we show that the set {x∈X:A∩U[x]≠∅} is Γ -convex. Since each Uj ∈...AE; has the form Uj ={(x,y)∈X×X:(xij ,yij )∈Uij } for some ij ∈I and Uij ∈[Bernoulli]ij , we have that [figure omitted; refer to PDF] [figure omitted; refer to PDF] By the definition of LΓ -spaces, we obtain that for each j∈{1,2,...,n} , the set {xij ∈Xij :πij (A)∩Uij [xij ]≠∅} is Γij -convex. It follows from (2.6) that the set {x∈X:A∩U[x]≠∅} is a Γ -convex subset of X . Therefore (X;Γ;[Bernoulli]) 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 ].
3. Existence Theorems of Solutions for Systems of Generalized Quasivariational Inclusion Problems
Let I be any index set. For each i∈I , let Zi be a topological vector space, (Xi ;Γi1 ;[Bernoulli]i1 ) be an LΓ -space, and (Yi ;Γi2 ;[Bernoulli]i2 ) be an LΓ -space with 1Yi ∈[real]...(Yi ,Yi ) . Let X=∏i∈IXi , Y=∏i∈IYi and (X×Y;Γ;[Bernoulli]) be the product LΓ -space as defined in Lemma 2.12. Furthermore, we assume that (X×Y;Γ;[Bernoulli]) 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) Ai :X×Y...Xi is a compact u.s.c. map with nonempty closed Γi1 -convex values,
(ii) Ti :X...Yi is a compact continuous map with nonempty closed Γi2 -convex values,
(iii): Gi :X×Yi ×Yi ...Zi is a closed map with nonempty values,
(iv) for each (x,vi )∈X×Yi , yi ...Gi (x,yi ,vi ) is {0} -quasiconvex; for each (x,yi )∈X×Yi , vi ...Gi (x,yi ,vi ) is {0} -quasiconvex-like and 0∈Gi (x,yi ,yi ) .
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 ∈Ai (x¯,y¯) , y¯i ∈Ti (x¯) and 0∈Gi (x¯,y¯i ,vi ) for all vi ∈Ti (x¯) .
Proof.
For each i∈I , define Hi :X...Ti (X) by [figure omitted; refer to PDF] Then, Hi (x) is nonempty for each x∈X . Indeed, fix any i∈I and x∈X , define Qix :Ti (x)...Ti (x) by [figure omitted; refer to PDF] First, we show that Qix is a KKM map w.r.t. 1Ti (x) . Suppose to the contrary that there exists a finite subset {vi1 ,vi2 ,...,vin } ⊂Ti (x) such that Γi2 ({vi1 ,vi2 ,...,vin })⊄...k=1nQix (vik ) . Hence, there exists v¯i ∈Γi2 ({vi1 ,vi2 ,...,vin }) satisfying v¯i ∉Qix (vik ) for all k=1,2,...,n . Since Ti (x) is Γi2 -convex, we have v¯i ∈Γi2 ({vi1 ,vi2 ,...,vin })⊂Ti (x) . By v¯i ∉Qix (vik ) for all k=1,2,...,n , we know that 0∉Gi (x,v¯i ,vik ) for all k=1,2,...,n . Since vi ...Gi (x,v¯i ,vi ) is {0} -quasiconvex-like, there exists 1≤j≤n such that [figure omitted; refer to PDF] This leads to a contradiction. Therefore, Qix is a KKM map w.r.t. 1Ti (x) . Next, we show that Qix (vi ) is closed for each vi ∈Ti (x) . Indeed, if yi ∈Qix (vi )¯ , then there exists a net {yiα}α∈Λ in Qix (vi ) such that yiα [arrow right]yi . For each α∈Λ , we have yiα ∈Ti (x) and 0∈Gi (x,yiα ,vi ) . By condition (ii), Ti (x) is closed, and hence yi ∈Ti (x) . By condition (iii), Gi is closed, and hence 0∈Gi (x,yi ,vi ) . It follows that yi ∈Qix (vi ) . Therefore, Qix (vi ) is closed. Since 1Yi ∈[real]...(Yi ,Yi ) and Ti (x) is Γi2 -convex, we have that 1Ti (x) ∈[real]...(Ti (x),Ti (x)) . Having that Ti is compact, we can deduce that ...vi ∈Ti (x)Qix (vi )≠∅ . That is Hi (x) is nonempty.
Hi is closed for each i∈I . Indeed, if (x,yi )∈Graph(Hi )¯ , then there exists a net {(xα ,yiα )}α∈Λ in Graph(Hi ) such that (xα ,yiα )[arrow right](x,yi ) . One has yiα ∈Ti (xα ) and 0∈Gi (xα ,yiα ,vi ) for all vi ∈Ti (xα ) . By condition (ii), Ti is closed, and hence yi ∈Ti (x) . Let vi ∈Ti (x) , since Ti is l.s.c., there exists a net {viα } satisfying viα ∈Ti (xα ) and viα [arrow right]vi . We have 0∈Gi (xα ,yiα ,viα ) . Since Gi is closed, we obtain 0∈Gi (x,yi ,vi ) . Thus, we have shown that (x,yi )∈Graph(Hi ) . Hence, Hi is closed.
Hi (x) is Γi2 -convex for each i∈I and x∈X . Indeed, if {yi1 ,yi2 ,...,yin }∈...Hi (x)... , then we have that {yi1 ,yi2 ,...,yin }⊂Ti (x) and 0∈Gi (x,yik ,vi ) for all vi ∈Ti (x) and all k=1,2,...,n . For any given y¯i ∈Γi2 ({yi1 ,yi2 ,...,yin }) , we have y¯i ∈Ti (x) because Ti (x) is Γi2 -convex. For each vi ∈Ti (x) , since yi ...Gi (x,yi ,vi ) is {0} -quasiconvex, there exists 1≤j≤n such that [figure omitted; refer to PDF] Hence, 0∈Gi (x,y¯i ,vi ) for all vi ∈Ti (x) . It follows that y¯i ∈Hi (x) and Hi (x) is Γi2 -convex.
Since Hi (X)⊂Ti (X)¯ and Ti (X)¯ is compact. It follows from Lemma 2.2(ii) that Hi is a compact u.s.c. map for each i∈I . Define Q:X×Y...X×Y by [figure omitted; refer to PDF] It follows from the above discussions that for each i∈I , Hi is a compact u.s.c. map with nonempty closed Γi2 -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¯) . That is 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 ∈Ai (x¯,y¯) , y¯i ∈Ti (x¯) and 0∈Gi (x¯,y¯i ,vi ) for all vi ∈Ti (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 : Gi :X×Yi ×Yi [arrow right]Zi is a continuous mapping;
(iv)1 : for each (x,vi )∈X×Yi , yi [arrow right]Gi (x,yi ,vi ) is {0} -quasiconvex; for each (x,yi )∈X×Yi , vi [arrow right]Gi (x,yi ,vi ) is also {0} -quasiconvex and Gi (x,yi ,yi )=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 ∈Ai (x¯,y¯) , y¯i ∈Ti (x¯) and Gi (x¯,y¯i ,vi )=0 for all vi ∈Ti (x¯) .
Theorem 3.3.
For each i∈I , suppose that conditions (i) and (ii) in Theorem 3.1 hold. Moreover,
(iii)2 : Hi :X...Zi is a closed map with nonempty values and Qi :X×Yi ×Yi ...Zi is an u.s.c. map with nonempty compact values;
(iv)2 : for each (x,vi )∈X×Yi , yi ...Qi (x,yi ,vi ) is {0} -quasiconvex; for each (x,yi )∈X×Yi , vi ...Qi (x,yi ,vi ) is {0} -quasiconvex-like and 0∈Hi (x)+Qi (x,yi ,yi ) .
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 ∈Ai (x¯,y¯) , y¯i ∈Ti (x¯) and 0∈Hi (x¯)+Qi (x¯,y¯i ,vi ) for all vi ∈Ti (x¯) .
Proof.
For each i∈I , define Gi :X×Yi ×Yi ...Zi by [figure omitted; refer to PDF] Obviously, Gi has nonempty values. Now, we show that Gi is closed. Indeed, if (x,yi ,vi ,zi )∈Graph(Gi )¯ , then there exists a net {(xα ,yiα ,viα ,ziα )}α∈Λ in Graph(Gi ) such that (xα ,yiα ,viα ,ziα )[arrow right](x,yi ,vi ,zi ) . Since [figure omitted; refer to PDF] there exist uiα ∈Hi (xα ) and wiα ∈Qi (xα ,yiα ,viα ) such that ziα =uiα +wiα . Let [figure omitted; refer to PDF] Then K is a compact subset of X , Li and Mi are compact subsets of Yi . By condition (iii)2 and Lemma 2.2(iii), Qi (K×Li ×Mi ) is a compact subset of Zi . Thus, we can assume that wiα [arrow right]wi . By condition (iii)2 , Qi is closed, and hence wi ∈Qi (x,yi ,vi ) . Since ziα -wiα =uiα ∈Hi (xα ) and Hi is closed, we have zi -wi ∈Hi (x) . Letting ui =zi -wi , it follows that [figure omitted; refer to PDF] and so Gi 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 ∈Ai (x¯,y¯) , y¯i ∈Ti (x¯) and [figure omitted; refer to PDF] for all vi ∈Ti (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 : Hi :X[arrow right]Zi is a continuous map and Qi :X×Yi ×Yi [arrow right]Zi is a continuous map;
(iv)3 : for each (x,vi )∈X×Yi , yi [arrow right]Qi (x,yi ,vi ) is {0} -quasiconvex; for each (x,yi )∈X×Yi , vi [arrow right]Qi (x,yi ,vi ) is also {0} -quasiconvex and Hi (x)+Qi (x,yi ,yi )=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 ∈Ai (x¯,y¯) , y¯i ∈Ti (x¯) and Hi (x¯)+Qi (x¯,y¯i ,vi )=0 for all vi ∈Ti (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.
Corollary 3.5.
For each i∈I , suppose that conditions (i) and (ii) in Theorem 3.1 hold. Moreover,
(iii)4 : Ci :X...Zi is a closed map with nonempty values and Qi :X×Yi ×Yi ...Zi is an u.s.c. map with nonempty compact values;
(iv)4 : for each (x,vi )∈X×Yi , yi ...Qi (x,yi ,vi ) is {0} -quasiconvex; for each (x,yi )∈X×Yi , vi ...Qi (x,yi ,vi ) is {0} -quasiconvex-like and Qi (x,yi ,yi )∩Ci (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 ∈Ai (x¯,y¯) , y¯i ∈Ti (x¯) , and Qi (x¯,y¯i ,vi )∩Ci (x¯)≠∅ for all vi ∈Ti (x¯) .
Proof.
Define Hi :X...Zi by Hi (x)=-Ci (x) for all x∈X . Since Ci is a closed map with nonempty values, we have that Hi 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 . 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 MinD 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)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: [figure omitted; refer to PDF] where x=(xi)i∈I and y=(yi)i∈I such that for each i∈I , xi ∈Ai (x,y) , yi ∈Ti (x) , and Qi (x,yi ,vi )∩Ci (x)≠∅ for all vi ∈Ti (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 ∈Ai (x¯,y¯) , y¯i ∈Ti (x¯) and Qi (x¯,y¯i ,vi )∩Ci (x¯)≠∅ for all vi ∈Ti (x¯) . For each i∈I , let [figure omitted; refer to PDF] and M=...i∈IMi . Then (x¯,y¯)∈M and M≠∅ . We show that Mi is closed for each i∈I . Indeed, if (x,y)∈Mi ¯ , then there exists a net {(xα ,yα )}α∈Λ in Mi such that (xα ,yα )[arrow right](x,y) . For each α∈Λ , (xα ,yα )∈Mi implies that [figure omitted; refer to PDF] By the closedness of Ai and Ti , we have that xi ∈Ai (x,y) and yi ∈Ti (x) . Now, we prove that Qi (x,yi ,vi )∩Ci (x)≠∅ for all vi ∈Ti (x) . For any vi ∈Ti (x) , since Ti is l.s.c., there exists a net {viα}α∈Λ satisfying viα ∈Ti (xα ) and viα [arrow right]vi . Let uiα ∈Qi (xα ,yiα ,viα )∩Ci (xα ) . Since Qi is u.s.c. with nonempty compact values, we can assume that uiα [arrow right]ui ∈Zi . By the closedness of Qi and Ci , we have that ui ∈Qi (x,yi ,vi )∩Ci (x) . Thus, Qi (x,yi ,vi )∩Ci (x)≠∅ . It follows that Mi is closed. Hence, M is closed. Note that M⊂∏i∈IAi (X×Y)×∏i∈ITi (X) . We know 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 . By Lemma 4.1, MinD 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 Xi is compact and condition (ii) in Theorem 3.1 holds. Moreover,
(iii)5 : Qi :X×Yi ×Yi [arrow right]... is a continuous function;
(iv)5 : for each (x,vi )∈X×Yi , yi [arrow right]Qi (x,yi ,vi ) is {0} -quasiconvex; for each (x,yi )∈X×Yi , vi [arrow right]Qi (x,yi ,vi ) is also {0} -quasiconvex and Qi (x,yi ,yi )≥0 .
Furthermore, let h:X×Y[arrow right]... is a l.s.c. function. Then there exists a solution to: [figure omitted; refer to PDF] where x=(xi)i∈I and y=(yi)i∈I such that for each i∈I , yi ∈Ti (x) and Qi (x,yi ,vi )≥0 for all vi ∈Ti (x) .
Proof.
For each i∈I , define Ai :X×Y...Xi and Ci :X...... by [figure omitted; refer to PDF] respectively. It is easy to check that all the conditions of Corollary 3.5 are satisfied. For each i∈I , define [figure omitted; refer to PDF] and M=...i∈IMi . 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 Xi is compact and condition (ii) in Theorem 3.1 holds. Moreover,
(iii)6 : Fi :X×Yi [arrow right]... is a continuous function;
(iv)6 : for each x∈X , yi [arrow right]Fi (x,yi ) is {0} -quasiconvex.
Furthermore, let h:X×Y[arrow right]... be a l.s.c. function. Then, there exists a solution to the problem: [figure omitted; refer to PDF] where x=(xi)i∈I and y=(yi)i∈I such that for each i∈I , yi is the solution of the problem min vi ∈Ti (x)Fi (x,vi ) .
Proof.
For each i∈I , define Qi :X×Yi ×Yi [arrow right]... by [figure omitted; refer to PDF] 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).
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Abstract
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.
You have requested "on-the-fly" machine translation of selected content from our databases. This functionality is provided solely for your convenience and is in no way intended to replace human translation. Show full disclaimer
Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer