ON EXISTENCE OF A SOLUTION FOR THE SYSTEM OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS WITH UPPER SEMICONTINUOUS SET-VALUED MAPS

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1 ON EXISTENCE OF A SOLUTION FOR THE SYSTEM OF GENERALIZED VECTOR QUASI-EQUILIBRIUM PROBLEMS WITH UPPER SEMICONTINUOUS SET-VALUED MAPS JIANWEN PENG AND XINMIN YANG Received 28 September 2004 and in revised form 7 July 2005 We introduce a new model of the system of generalized vector quasi-equilibrium problems with upper semicontinuous set-valued maps and present several existence results of a solution for this system of generalized vector quasi-equilibrium problems and its special cases. The results in this paper extend and improve some results in the literature. 1. Introduction and preliminaries Throughout this paper, we use inta and CoA to denote the interior and the convex hull of a set A, respectively. Let I be an index set. For each i I, lety i, E i be two Hausdorff topological vector spaces. Consider a family of nonempty convex subsets {X i } i I with X i E i.letx = i I X i and E = i I E i. An element of the set X i = j I\i X i will be denoted by x i ;therefore, x X will be written as x = (x i,x i X i X i.foreachi I, letd i : X 2 Xi and F i : X X i 2 Yi be two set-valued maps, let C i : X 2 Yi be a set-valued map such that C i (x is a convex, pointed, and closed cone with intc i (x for all x X. Then the system of generalized vector quasi-equilibrium problems with set-valued maps (in short, SGVQEP is to find x = ( x i, x i inx such that for each i I, x i D i ( x, F i ( x, yi intci ( x, y i D i ( x. (1.1 Remark 1.1. In [32], we introduced and studied another type of system of generalized vector quasi-equilibrium problems with lower semicontinous set-valued maps, which is to find x = ( x i, x i inx such that for each i I, x i D i ( x, F i ( x, yi Yi \ ( intc i ( x, y i D i ( x. (1.2 It is apparent that this problem is different from the SGVQEP. The following problems are special cases of the SGVQEP. (1 For each i I and for all x X,ifD i (x X i, then the SGVQEP reduces to the system of generalized vector equilibrium problems with set-valued maps (in short, SGVEP Copyright 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005:15 ( DOI: /IJMMS

2 2410 System of generalized vector quasi-equilibrium problems which is to find x in X such that for each i I, F i ( x, yi intci ( x, y i X i. (1.3 This problem was introduced and studied by Ansari et al. in [20]. (2 For each i I, if the set-valued map F i is replaced by a vector-valued map ϕ i : X X i Y i, then the SGVQEP reduces to a system of vector quasi-equilibrium problems (in short, SVQEP which is to find x = ( x i, x i inx such that for each i I, x i D i ( x, ϕ i ( x, yi intci ( x, y i D i ( x. (1.4 For each i I, letϕ i : X Y i be a vector-valued map, and let f i (x, y i = ϕ i (x i, y i ϕ i (x, then the SVQEP is equivalent to the following Debreu-type equilibrium problem for vector-valued maps (in short, Debreu VEP which is to find x = ( x i, x i inx such that for each i I, x i D i ( x, ϕ i ( x i, y i ϕi ( x intc i ( x, y i D i ( x. (1.5 The SVQEP and the Debreu VEP were introduced and studied by Ansari et al. [21]. And if D i (x X i for each i I and for all x X, then the Debreu VEP becomes the Nash equilibrium problem for vector-valued maps in [19]. If Y i = R, C i (x ={y R y 0} for each i I and for all x X,andϕ i is a scalar realvalued function from X X i to R, then the SVQEP reduces to the model of generalized game in [22, page 286] and quasivariational inequalities in [23, page ]. (3 For each i I, letη i : X i X i E i be a single-valued map and T i : X 2 L(Ei,Yi a set-valued map, where L(E i,y i denotes the space of all continuous linear operators from E i into Y i.letf i (x, y i = T i (x,η i (y i,x i = v i T i(x v i,η i (y i,x i. Then the SGVQEP reduces to the system of generalized vector quasivariational-like inequality problems (in short, SGVQVLIP, which is to find x = ( x i, x i inx such that for each i I, x i D i ( x, y i D i ( x, v i T i ( x: v i,η i ( yi, x i / intci ( x. (1.6 For each i I, letη i (y i,x i = y i x i for all x i, y i X i, then the SGVQVLIP reduces to the system of generalized vector quasivariational inequality problems (in short, SGVQVIP which is to find x = ( x i, x i inx such that for each i I, x i D i ( x, y i D i ( x, v i T i ( x: v i, y i x i / intci ( x. (1.7 The SGVQVLIP and the SGVQVIP were introduced by Peng [18]. For each i I,ifD i (x X i for all x X, then the SGVQVLIP reduces to the system of generalized vector variational-like inequality problems (in short, SGVVLIP which is to find x = ( x i, x i inx such that for each i I, y i X i, v i T i ( x: v i,η i ( yi,x i / intci ( x. (1.8 For each i I, ifd i (x X i for all x X, andη i (y i,x i = y i x i for all x i, y i X i, then the SGVQVLIP reduces to the system of generalized vector variational inequality

3 problems (in short, SGVVIP which is to find x = ( x i, x i inx such that J. Peng and X. Yang 2411 y i X i, v i T i ( x: v i, y i x i / intci ( x. (1.9 The SGVVLIP and the SGVVIP were introduced by Ansari et al. in [20]. For each i I, ify i Y and C i (x C for all x X, wherec is a convex, closed, and pointed cone in Y with intc, then the SGVVIP reduces to another kind of system of generalized vector variational inequality problems (in short, II-SGVVIP which is to find x = ( x i, x i inx such that y i X i, v i T i ( x: v i, y i x i / intc. (1.10 This was studied by Allevi et al. [17]. If T i is a single-valued function, then the II-SGVVIP reduces to the system of vector variational inequality problems (in short, SVVIP, which is to find x = ( x i, x i inx such that Ti ( x, y i x i / intc, yi X i. (1.11 This was considered by Ansari et al. in [19]. Let Y = R and C = R + ={r R : r 0}. Foreachi I, ift i is replaced by f i : X R, then the SVVIP reduces to the system of scalar variational inequality problems which is to find x = ( x i, x i inx such that fi ( x, y i x i 0, yi X i. (1.12 This was considered in [16, 24, 25, 26]. (4 If the index set I is singleton, then the SGVQEP reduces to the generalized vector quasi-equilibrium problem (in short, GVQEP studied in [12, 15] which contains the generalized vector equilibrium problems studied in [27, 28, 29] as special cases, and the SGVVLIP reduces to the generalized vector variational-like inequality problem studied by Ding and Tarafdar [31]. In this paper, we present some existence results of a solution for the SGVQEP and its special cases with or without Φ-condensing maps. Now we introduce a new definition as follows. Definition 1.2. Let I be an index set. For each i I,letY i be a topological vector space and X i a nonempty convex subset of a Hausdorff topological vector space E i,andf i : X X i 2 Yi be a set-valued map, let C i : X 2 Yi be a set-valued map such that C i (x beaconvex and closed cone with intc i (x for all x X i. F i (x,z i issaidtobec i x -0-partially diagonal quasiconvex if for any finite set {y i1, y i2,..., y in } in X i,andforallx = (x i,x i X with x i Co{y i1, y i2,..., y in }, there exists some j (j = 1,2,...,nsuchthat F i ( x, yij intci (x. (1.13 Remark 1.3. (a If the formula (1.13 isreplacedbyf i (x, y ij Y i \ ( intc i (x, then Definition 1.2 becomes [32, Definition 1.1], which is called to be the second type C i x -0- partially diagonal quasiconvexity of F i in this paper.

4 2412 System of generalized vector quasi-equilibrium problems (b For each i I, if the set-valued map F i : X X i 2 Yi is replaced by a singlevalued map f i : X X i Y i and (1.13 isreplacedby f i (x, y ij / intc i (x, then the C i x -0-partially diagonal quasiconvexity of F i reduces to the C i x -0-partially diagonal quasiconvexity of f i.furthermore,ifc i (x = C i, then the C i x -0-partially diagonal quasiconvexity of f i reduces to the C i -0-partially diagonal quasiconvexity of f i.ify i = R and C i ={r R : r 0} for each i I, then the C i -0-partially diagonal quasiconvexity of f i reduces to [30,Definition3].Furthermore,letI ={1},then[30,Definition3]reducesto the γ-diagonal quasiconvexity in [3], here γ = 0. (c For each i I, lete i be a real normed space with dual space E i, X i E i, Z i = R. Let i denote the norm on E i.ifwedefineanormone as x = n x i i, x = ( x 1,x 2,...,x n E, (1.14 i=1 then it is easy to verify that is a norm on E. And hence E is also a real normed space. Let C i : X 2 Zi be defined as C i (x = [ x,+, for all x X, andlet[e 1,e 2 ] denote the line segment joining e 1 and e 2. Choosing p i E i, we define a set-valued map F : X X i 2 Zi as F ( x,z i = { u,zi x i : u [ 2 x z i i p i, x z i i p i ]}, ( x,zi X Xi, (1.15 Then, F is C i x -0-partially diagonal quasiconvex in the second argument. Otherwise, there exists finite set {z i1,z i2,...,z in } X i, and there is x X with x i = n j=1 α j z ij (α j 0, n j=1 α j = 1 such that for all j = 1,2,...,n, F(x,z ij intc i (x. Then for each j, for all λ j [0,1], we have λj ( 2 x z ij i p i + ( 1 λj ( x z ij i p i,zij x i < x 0. (1.16 It follows that pi,z ij x i > 0, j = 1,2,...,n. (1.17 Then we have n 0 < α j pi,z ij x i = pi,x i x i = 0, (1.18 j=1 a contradiction. Definition 1.4 [18]. For each i I, lete i, Y i be two topological vector spaces, X i anonempty and convex subset of E i, C i : X 2 Yi a set-valued map such that C i (x isaclosed, pointed, and convex cone for each x X. Letη i : X i X i E i be a single-valued map. T i : X 2 L(Ei,Yi is said to satisfy the generalized partial L-η i -condition if and only if for any finite set {y i1, y i2,..., y in } in X i,forall x = ( x i, x i with x i = n j=1 α j y ij,whereα j 0 and n j=1 α j = 1, there exists v i T i ( xsuchthat v i, n ( α j η i yij, x i / intc i ( x. (1.19 j=1

5 J. Peng and X. Yang 2413 Definition 1.5 [36]. Let X be a nonempty convex subset of a topological vector space E and P a closed, pointed, and convex cone in a topological vector space Y.LetF : X 2 Y be a set-valued map. Then F is said to be naturally quasiconvex on X,ifforanyx 1,x 2 X and λ [0,1], F ( λx 1 +(1 λx 2 Co { F ( x1 F ( x2 } P. (1.20 Definition 1.6 [8]. Let E be a Hausdorff topological space and L a lattice with least element, denoted by 0. A map Φ :2 E L is a measure of noncompactness provided that the following conditions hold for all M,N 2 E : (i Φ(M = 0ifandonlyifM is precompact (i.e., it is relatively compact; (ii Φ(convM = Φ(M; where convm denotes the convex closure of M; (iii Φ(M N = max{φ(m,φ(n}. Definition 1.7 [8]. Let Φ :2 E L be a measure of noncompactness on E and D E.Asetvalued map Q : D 2 E is called Φ-condensing provided that if M D with Φ(Q(M Φ(M, then M is relatively compact. It is clear that if Q : D 2 E is Φ-condensing and Q : D 2 E satisfies Q (x Q(x for all x D,thenQ is also Φ-condensing. In the next section, we will use the following particular form of a maximal element theorem for a family of set-valued maps due to Deguire et al. [33, Theorem 7] (also see [20, Theorem 1]. Lemma 1.8. Let I be any index set and {X i } i I a family of nonempty convex subsets where each X i is contained in a Hausdorff topological vector space E i. For each i I,letS i : X 2 Xi be a set-valued map such that (i S i (x is convex, (ii for each x X, x i / S i (x, (iii for each y i X i, S i 1 (y i is open in X, (iv there exist a nonempty compact subset N of X and a nonempty compact convex subset B i of X i for each i I such that for each x X \ N there exists i I satisfying S i (x B i. Then there exists x X such that S i ( x = for all i I. The following lemma is a particular form of [35,Corollary4]. Lemma 1.9 (maximal element theorem. Let I beanyindexsetand{x i } i I a family of nonempty, closed, and convex subsets where each X i is contained in a locally convex Hausdorff topological vector space E i. For each i I, lets i : X 2 Xi be a set-valued map such that (i for each x X, x i / CoS i (x, (ii for each y i X i, S 1 i (y i is open in X, (iii the set-valued map S : X 2 X defined as S(x = i I S i (x, forallx X, isφcondensing. Then there exists x X such that S i (x = for all i I. Lemma 1.10 [34]. Let X and Y be topological spaces and T : X 2 Y be an upper semicontinuous set-valued map with compact values. Suppose {x α } is a net in X such that x α x 0.

6 2414 System of generalized vector quasi-equilibrium problems If y α T(x α for each α, then there is a y 0 T(x 0 and a subset {y β } of {y α } such that y β y Existence results In this section, we present some existence results of a solution for the SGVQEP and its special cases without or with Φ-condensing maps. Theorem 2.1. Let I be any index set. For each i I,letY i be a topological vector space and X i a nonempty convex set in a Hausdorff topological vector space E i,letf i : X X i 2 Yi be a set-valued map, and let D i : X 2 Xi be a set-valued map such that D i (x is nonempty and convex for all x X, D 1 i (y i is open in X for all y i X i,andthesetw i ={x X : x i D i (x} is closed in X. LetC i : X 2 Yi be a set-valued map such that C i (x is a convex, pointed, and closed cone with intc i (x for all x X. Assume that the following conditions are satisfied: (i for each i I, F i is C i x -0-partially diagonal quasiconvex; (ii for each i I, for each y i X i, F i (, y i is upper semicontinuous on X with compact values; (iii for each i I, the set-valued map intc i has open graph in X Y i ; (iv there exist a nonempty and compact subset N of X and a nonempty, compact, and convex subset B i of X i for each i I such that for all x = (x i,x i X \ N, there exist i I and ȳ i B i, such that ȳ i D i (x and F i (x, ȳ i intc i (x. Then, the solution set of the SGVQEP is nonempty. Proof. For each i I, we define a set-valued map P i : X 2 Xi by P i (x = { y i X i : F i ( x, yi intci (x }, x = ( x i,x i X. (2.1 By hypothesis (i, we have x i / CoP i (xforeachi I and for all x X. To see this, suppose, by way of contradiction, that there exist i I and a point x = ( x i, x i X such that x i CoP i ( x. Then there exist finite points y i1, y i2,..., y in in X i,andλ j 0with n j=1 λ j = 1suchthat x i = n j=1 λ j y ij and y ij P i ( x forallj = 1,2,...,n. That is, F i ( x, y ij intc i ( x forj = 1,2,...,n, which contradicts the hypothesis that F i is C i x -0-partially diagonal quasiconvex. By hypothesis (ii, we can prove that for each i I, andforeachy i X i, the set P i 1 (y i ={x X : F i (x, y i intc i (x} is open, that is, P i has open lower sections. Let Q i (y i ={x X : F i (x, y i intc i (x} for all y i X i.wecanprovethatq i (y i ={x X : F i (x, y i intc i (x} =X \ P i 1 (y i isclosedforally i X i. In fact, consider a net x t X such that x t x X. Foreveryt there exists u t F i (x t, y i withu t / intc i (x t. Since F i (, y i is upper semicontinuous with compact values, by Lemma 1.10,itsuffices to find a subset {u tj } which converges to some u F i (x, y i ; then (x tj,u tj (x,u. It follows that u/ intc i (x since the graph of intc i ( is open, hence, x Q i (y i. So we have proved that P i has open lower sections. Then by [4, Lemma5],weknow that the set-valued map CoP i : X 2 Xi defined by CoP i (x = Co(P i (x, for all x X also has open lower sections. For each i I, we also define another set-valued map S i : X 2 Xi

7 J. Peng and X. Yang 2415 by D i (x CoP i (x if x W i, S i (x = D i (x if x/ W i. (2.2 Then, for each i I, it is clear that S i (x is convex for all x X and x i / S i (x. Since 1 i I, y i X i, S ( i y i = { x X : y i S i (x } = { x W i : y i D i (x CoP i (x } { x X \ W i : y i D i (x } = ( 1 W i D ( i y i CoP 1 ( [( y i X \ Wi D 1 ( ] y i = [( 1 W i D ( i y i 1 ( ( ] CoPi y i X \ Wi [( 1 W i D ( i y i 1 ( CoPi y i 1 ( ] Di y i = { X [( 1 D ( i y i 1 ( ( ]} [( CoPi y i X \ Wi 1 Wi D ( ( i y i 1 ( ] Di y i = [( 1 D ( i y i 1 ( ( ] CoPi y i X \ Wi 1 ( Di y i = ( D i 1 ( y i ( CoPi 1 ( y i (( X \ Wi ( Di 1 ( y i, (2.3 and D i 1 (y i, CoP i 1 (y i andx \ W i are open in X,wehaveS i 1 (y i openinx. By assumption (iv, we know that the condition (iv of Lemma 1.8 holds. Hence, by Lemma 1.8, there exists x X such that S i ( x =,foralli I. Sinceforalli I and for all x X, D i (xisnonempty,wehave x W i,andd i ( x CoP i ( x =,foralli I. This implies x i D i ( xandd i ( x P i ( x =,foralli I. Therefore, for all i I, x i D i ( x, F i ( x, yi intci ( x, y i D i ( x. (2.4 That is, the solution set of the SGVQEP is nonempty. This completes the proof. Corollary 2.2. If we replace, in Theorem 2.1, condition (i by the following conditions; (a for each i I,forallx X, y i F i (x, y i is natural P i -quasiconvex; (b for each i I,forallx X, F i (x,x i intc i (x; then the conclusion of Theorem 2.1 still holds, that is, the solution set of the SGVQEP is nonempty. Proof. For each i I, we define a set-valued map P i : X 2 Xi by P i (x = { y i X i : F i ( x, yi intci (x }, x = ( x i,x i X. (2.5 Then P i (x is convex for each i I and for all x X. To prove it, we fix arbitrary i I and x X.Lety i1, y i2 P i (xandλ [0,1], then we have F i ( x, yij intci (x, j = 1,2. (2.6

8 2416 System of generalized vector quasi-equilibrium problems By the convexity of intc i (x, we have Since F i is natural P i -quasiconvex, Co { F i ( x, yi1 Fi ( x, yi2 } intci (x. (2.7 F i ( x,λyi1 +(1 λy i2 Co { Fi ( x, yi1 Fi ( x, yi2 } Pi intc i (x P i intc i (x. (2.8 Hence, λy i1 +(1 λy i2 P i (xandsop i (xisconvex. We show that F i is C i x -0-partially diagonal quasiconvex for each i I. If not, then there exists a finite subset {y i1, y i2,..., y in } in X i, and a point x = (x i,x i X with x i Co{y i1, y i2,..., y in } such that F i ( x, yij intci (x, j = 1,2,...,n. (2.9 Since P i (x ={y i X i : F i (x, y i intc i (x} is a convex set, x i P i (x, that is, F i (x,x i intc i (x, which contradicts the condition (b. By Theorem 2.1, we know the conclusion of Corollary 2.2 holds. This completes the proof. Corollary 2.3. If we replace, in Theorem 2.1, condition (i by the following conditions: (a for each i I, F i is C i (x-quasiconvex-like; (b for each i I,forallx X, F i (x,x i intc i (x; then the conclusion of Theorem 2.1 still holds, that is, the solution set of the (SGVQEP is nonempty. Proof. For each i I, let the set-valued map P i : X 2 Xi be defined the same as that in the proof of Corollary 2.2. Then by the assumption (a and the proof of [20, Theorem 3], it is easy to see that for all i I and for all x X, P i (x is convex. The conclusion of Corollary 2.3 follows from the Corollary 2.2. This completes the proof. Remark 2.4. If D i (x = X i for all x X and for all i I, thenbycorollary 2.3, werecover [20,Theorem 3].Hence,Theorem 2.1 generalizes [20, Theorem 3] from the system of generalized vector equilibrium problems to the system of generalized vector quasiequilibrium problems with more general convex conditions. Remark 2.5. If F i is replaced by a single-valued map f i : X X i Y i,thenby[21,remark 5(1 andcorollary2.2], we can obtain [21, Theorem 2]. Hence, Theorem2.1 generalizes [21, Theorem 2] from single-valued case to set-valued case with more general convex conditions. Remark 2.6. For each i I, let the set-valued map F i be replaced by a single-valued map ϕ i : X X i Y i,andletc i (x = C i and D i (x = X i for all x X.ByTheorem 2.1,wehave the existence result of the SVEP as follows. Let I be any index set. For each i I,letY i be a topological vector space, X i anonempty compact convex set in a Hausdorff topological vector space E i,andϕ i : X X i Y i a single-valued map. Let C i Y i be a convex, pointed, and closed cone with intc i (x

9 J. Peng and X. Yang 2417 for all x X. Assume that the following conditions are satisfied (i for each i I, ϕ i is C i -0-partially diagonal quasiconvex; (ii for each i I,foreachy i X i, ϕ i (, y i iscontinuousonx. Then, there exists x = ( x i, x i inx such that for each i I, x i X i and ϕ i ( x, y i intc i, for all y i X i. That is, the solution set of the SVEP is nonempty. And if the condition (i of the above result is replaced by ϕ i (x,x i = 0forallx X and the map y i ϕ i (x, y i isc i -quasiconvex, then we recover [19, Theorem 2.1]. If ϕ i (x,x i = 0forallx X and the map y i ϕ i (x, y i isc i -quasiconvex, then ϕ i must be C i -0-partially diagonal quasiconvex. Hence, Theorem 2.1 generalizes [19, Theorem 2.1] in several aspects. Now we establish an existence result for a solution to the SGVQEP involving Φ- condensing maps. Theorem 2.7. Let I be any index set. For each i I,letY i be a topological vector space and X i a nonempty, closed, and convex set in a locally convex Hausdorff topological vector space E i.letf i : X X i 2 Yi be a set-valued map and D i : X 2 Xi a set-valued map such that for all x X, D i (x is a nonempty convex set, D i 1 (y i is open in X for all y i X i,andthe set W i ={x X : x i D i (x} is closed in X.LetC i : X 2 Yi be a set-valued map such that C i (x is a closed, pointed, and convex cone with intc i (x for all x X. Assume that the set-valued map D : X 2 X defined as D(x = i I D i (x,forallx X,isΦ-condensing and the conditions (i, (ii, and (iii of Theorem 2.1 hold. Then the solution set of SGVQEP is nonempty. Proof. In view of Lemma 1.9, itissufficient to show that the set-valued map S : X 2 X defined as S(x = i I S i (x forallx K is Φ-condensing, where S i s are the same as defined in the proof of Theorem 2.1. By the definition of S i, S i (x D i (x forallx X and for each i I, and therefore S(x D(xforallx X.SinceD is Φ-condensing, so is S. This completes the proof. Remark 2.8. Theorem 2.7 generalizes [21, Theorem 3] from single-valued case to setvalued case with more general convex conditions. Remark 2.9. If we replace, in Theorem 2.7, condition (i and (ii of Theorem 2.1,respectively, by the following conditions: (a for each i I, F i is the second-type C i x -0-partially diagonal quasiconvex; (b for each i I,foreachz i X i, F i (,z i is lower semicontinuous on X; then there exists x = ( x i, x i inx such that for each i I, This is [32, Theorem 2.1]. x i D i ( x, F i ( x,zi Yi \ ( intc i ( x, z i D i ( x. (2.10 We will prove the existence of solutions for the SGVQVLIP as follows. Corollary Let I be any index set. For each i I,letY i be a real Hausdorff topological vector space and X i a nonempty convex set in a real Hausdorff topological vector space E i.let the set-valued maps D i : X 2 Xi, C i : X 2 Yi,andthesetW i ={x X : x i D i (x} be as

10 2418 System of generalized vector quasi-equilibrium problems in Theorem 2.1,andletL(E i,y i be equipped with the σ-topology. Assume that the following conditions are satisfied: (i for each i I, T i : X 2 L(Ei,Yi is an upper semicontinuous set-valued map with nonempty compact values, η i : X i X i E i is continuous with respect to the second argument, such that T i satisfies the generalized partial L-η i -condition; (ii there exist a nonempty and compact subset N of X and a nonempty, compact, and convex subset B i of X i for each i I such that for all x = (x i,x i X \ N, there exist i I and ȳ i B i, such that ȳ i D i (x and v i,η i (ȳ i,x i intc i (x, forallv i T i (x. Then the SGVQVLI has a solution x X. Proof. For each i I, define a set-valued map P i : X 2 Xi by P i (x = { y i X i : ( T i (x,η i yi,x i intci (x } = { y i X i : ( v i,η i yi,x i intci (x, v i T i (x }, x X. (2.11 From the proof of [18, Theorem 3.1], we know that x i / Co(P i (x for all x = (x i,x i X and the set P 1 i (y i ={x X : T i (x,η i (y i,x i intc i (x} is open for each i I and for each y i X i. That is, P i has open lower sections in X. The remainder of the proof is same as that in the proof of Theorem 2.1. In view of Lemma 1.9 and the proof of Corollary 2.10, it is easy to obtain an existence result of a solution to SGVQVLI as follows. Corollary Let I be any index set. For each i I,letY i be a real Hausdorff topological vector space and X i a nonempty, closed, and convex set in a real locally convex Hausdorff topological vector space E i, let the set-valued maps D i : X 2 Xi, D : X 2 X, C i : X 2 Yi and the set W i ={x X : x i D i (x} be the same as those in Theorem 2.7. Assume that condition (iof Corollary 2.10 holds. Then the solution set of SGVQVLI is nonempty. Remark Let I be an index set and let I be countable. For each i I, lety i be a real Hausdorff topological vector space, let X i be a nonempty, compact, convex, and metrizable set in a real locally convex Hausdorff topological vector space E i,letd i : X 2 Xi be an upper semicontinuous set-valued mapping with nonempty convex closed values and open lower sections, let C i : X 2 Yi be a set-valued mapping such that C i (x isa closed pointed and convex cone with intc i (x for each x X, and the set-valued map M i = Y i \ ( intc i :X 2 Yi is upper semicontinuous, and let L(E i,y i beequipped with the σ-topology. Suppose that the condition (i of Corollary 2.10 satisfied, then the SGVQVLI has a solution x X. The above is [18, Theorem 3.1]. It is easy to see that Corollaries 2.10 and 2.11 generalize [18, Theorem 3.1] without compactness and metrizability of X i and with weaker conditions of D i. Corollaries 2.10 and 2.11 also generalize [20, Corollaries 2 and 3] and [19, Theorems 3.1 and 3.2] in several aspects. Remark By the above results, it is easy to obtain the existence results of a solution for the other special cases of the SGVQEP and they are omitted here.

11 Acknowledgments J. Peng and X. Yang 2419 This research was supported by the National Natural Science Foundation of China (Grant no , the Education Committee Project Research Foundation of Chongqing (Grant no , and the Natural Science Foundation of Chongqing (no The authors would like to express their thanks to the referees for their comments and suggestions that improved the presentation of this manuscript. References [1] J.-P. Aubin, Mathematical Methods of Game and Economic Theory, Studies in Mathematics and Its Applications, vol. 7, North-Holland, Amsterdam, [2] J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, Pure and Applied Mathematics (New York, John Wiley & Sons, New York, [3] J. X. Zhou and G. Y. Chen, Diagonal convexity conditions for problems in convex analysis and quasi-variational inequalities, J. Math. Anal. Appl. 132 (1988, no. 1, [4] G. Q. Tian and J. X. Zhou, Quasi-variational inequalities without the concavity assumption, J. Math. Anal. Appl. 172 (1993, no. 1, [5] G. X.-Z. Yuan and E. Tarafdar, Generalized quasi-variational inequalities and some applications, Nonlinear Anal. 29 (1997, no. 1, [6] L. J. Lin and S. Park, On some generalized quasi-equilibrium problems, J. Math. Anal. Appl. 224 (1998, no. 2, [7] S. Park, On minimax inequalities on spaces having certain contractible subsets, Bull. Austral. Math. Soc. 47 (1993, no. 1, [8] P.M.FitzpatrickandW.V.Petryshyn,Fixed point theorems for multivalued noncompact acyclic mappings, Pacific J. Math. 54 (1974, no. 2, [9] E.Michael, Continuous selections. I,Ann.ofMath.(263 (1956, [10] W. Shafer and H. Sonnenschein, Equilibrium in abstract economies without ordered preferences, J. Math. Econom. 2 (1975, no. 3, [11] U. Mosco, Implicit variational problems and quasi variational inequalities, Nonlinear Operators and the Calculus of Variations (Summer School, Univ. Libre Bruxelles, Brussels, 1975, Lecture Notes in Math., vol. 543, Springer, Berlin, 1976, pp [12] Q. H. Ansari and F. Flores-Bazán, Generalized vector quasi-equilibrium problems with applications, J. Math. Anal. Appl.277 (2003, no. 1, [13] J.-Y. Fu, Generalized vector quasi-equilibrium problems, Math.MethodsOper.Res.52 (2000, no. 1, [14] Y. H. Cheng, G. Y. Chen, and D. L. Zhu, Existence of solutions of strong vector quasi-equilibrium problems, OR Trans.11 (2001, no. 4, [15] J. W. Peng, Generalized vectorial quasi-equilibrium problem on W-space, J. Math. Res. Exposition 22 (2002, no. 4, [16] J.-S. Pang, Asymmetric variational inequality problems over product sets: applications and iterative methods, Math. Programming 31 (1985, no. 2, [17] E. Allevi, A. Gnudi, and I. V. Konnov, Generalized vector variational inequalities over product sets, Nonlinear Anal. 47 (2001, no. 1, [18] J. W. Peng, System of generalised set-valued quasi-variational-like inequalities, Bull. Austral. Math. Soc. 68 (2003, no. 3, [19] Q. H. Ansari, S. Schaible, and J. C. Yao, System of vector equilibrium problems and its applications, J. Optim. TheoryAppl.107 (2000, no. 3, [20], The system of generalized vector equilibrium problems with applications, J. GlobalOptim. 22 (2002, no. 1-4, 3 16.

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13 Mathematical Problems in Engineering Special Issue on Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios Call for Papers Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points. Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from Qualitative Theory of Differential Equations, allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers. This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems. Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment. Authors should follow the Mathematical Problems in Engineering manuscript format described at Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at mts.hindawi.com/ according to the following timetable: Guest Editors José Roberto Castilho Piqueira, Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, São Paulo, Brazil; piqueira@lac.usp.br Elbert E. Neher Macau, Laboratório Associado de Matemática Aplicada e Computação (LAC, Instituto Nacional de Pesquisas Espaciais (INPE, São Josè dos Campos, São Paulo, Brazil ; elbert@lac.inpe.br Celso Grebogi, Center for Applied Dynamics Research, King s College, University of Aberdeen, Aberdeen AB24 3UE, UK; grebogi@abdn.ac.uk Manuscript Due December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009 Hindawi Publishing Corporation