Volume 2009, Article ID 654370,7pages doi:10.1155/2009/654370
Research Article
Existence of Solutions to the System of
Generalized Implicit Vector Quasivariational
Inequality Problems
Zhi Lin
College of Science, Chongqing Jiaotong University, Chongqing 400074, China
Correspondence should be addressed to Zhi Lin,[email protected]
Received 31 March 2009; Revised 30 July 2009; Accepted 3 September 2009
Recommended by Yeol Je Cho
We study the system of generalized implicit vector quasivariational inequality problems and prove a new existence result of its solutions by Kakutani-Fan-Glicksberg’s fixed points theorem. As a special case, we also derive a new existence result of solutions to the generalized implicit vector quasivariational inequality problems.
Copyrightq2009 Zhi Lin. 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.
1. Introduction
The system of generalized implicit vector quasivariational inequality problems generalizes the generalized implicit vector quasivariational inequality problems, and the latter had been studied in 1–3. In this paper, we study the system of generalized implicit vector quasivariational inequality problems and prove a new existence result of its solutions by Kakutani-Fan-Glicksberg’s fixed points theorem. For other existence results with respect to the system of generalized implicit vector quasivariational inequality problems, we refer the reader to4–6and references therein.
Let I be an index set finite or infinite. For each i ∈ I, let Xi and Yi be two
Hausdorfftopological vector spaces, Ki a nonempty subset ofXi, andCi a closed, convex
and pointed cone of Yi with intCi/∅, where intCi denotes the interior of Ci. Denote that Ki
j∈I,j /iKj, Ki∈IKi Ki×Ki,X
i∈IXi. For eachx∈K, we can writex xi, xi.
For eachi∈I, letDibe a nonempty subset of the continuous linear operators spaceLXi, Yi
from Xi into Yi and let F : Di ×Ki×Ki −→ Yi,Gi : K −→ 2Ki,Ti : K −→ 2Di be three
set-valued maps, where 2Di and 2Ki denote the family of all nonempty subsets ofD
iandKi,
briefly, SGIVQIPis as follows: findx xi, xi∈Ksuch that for eachi∈I, xi∈Gix,and
∀yi∈Gix, ∃ui ∈Tix, Fi
ui, xi, yi
/
⊂ −intCi. 1.1
x xi, xi is said to be a solution of the SGIVQIP. An SGIVQIP is usually denoted by {Ki, Di, Gi, Ti, Fi}i∈I.
IfI is a singleton, then the SGIVQIP coincides with the generalized implicit vector quasivariational inequality problems briefly, GIVQIP. A GIVQIP is usually denoted by
{K, D, G, T, F}.
Throughout this paper, unless otherwise specified, assume that for eachi∈I,Kiis a
nonempty convex compact subset of a Banach spaceXi, Yiis a Hausdorfftopological vector
space, andCiis a closed, convex, and pointed cone ofYiwith intCi/∅, where intCidenotes
the interior ofCi.
2. Preliminaries
In this section, we introduce some useful notations and results.
Definition 2.1. LetXandYbe two topological spaces andKa nonempty convex subset ofX.
F:K−→2Y is a set-valued map.
1F is called upper semicontinuous atx0 ∈ K if, for any open setG ⊃ Fx0, there
exists an open neighborhoodUofx0inKsuch that for allx∈U,
G⊃Fx; 2.1
and upper semicontinuous onKif it is upper semicontinuous at every point ofK.
2Fis called lower semicontinuous atx0 ∈Kif, for any open setG∩Fx0/∅, there
exists an open neighborhoodUofx0inKsuch that for allx∈U,
G∩Fx/∅; 2.2
and lower semicontinuous onKif it is lower semicontinuous at every point ofK.
3F is called continuous atx0 ∈ K if, it is both upper semicontinuous and lower
semicontinuous atx0; and continuous onKif it is continuous at every point ofK.
Definition 2.2. LetXandYbe two topological vector spaces andKa nonempty convex subset ofX.AlsoF:K−→2Y is a set-valued map.
1F is called upperC-semicontinuous atx0 ∈Kif, for any open neighborhoodV of
the zero elementθinY, there exists an open neighborhoodUofx0inKsuch that,
for allx∈U,
Fx⊂Fx0 VC; 2.3
2F is called lowerC-semicontinuous atx0 ∈ Kif, for any open neighborhood V of
the zero elementθinY, there exists an open neighborhoodUofx0inKsuch that,
for allx∈U,
Fx∩Fx0 V C/∅; 2.4
and lowerC-semicontinuous onKif it is lowerC-semicontinuous at every point of
K.
3F is calledC-continuous atx0 ∈ Kif it is upper C-semicontinuous and lowerC
-semicontinuous at x0 ∈ K; andC-continuous onK if it isC-continuous at every
point ofK.
Definition 2.3. LetXandYbe two topological vector spaces andKa nonempty convex subset ofX. LetF:K → 2Y be a set-valued map.
1Fis calledC-convex if, for eachx1, x2∈K, t∈0,1,
Ftx1 1−tx2⊂tFx1 1−tFx2−C; 2.5
andC-concave if−FisC-convex.
2Fis calledC-quasiconvex-like if, for eachx1, x2∈K,t∈0,1,
eitherFtx1 1−tx2⊂Fx1−C or Ftx1 1−tx2⊂Fx2−C; 2.6
andC-quasiconcave-like if−FisC-quasiconvex-like.
Lemma 2.47, Theorem 1. LetKbe a nonempty paracompact subset of a Hausdorfftopological spaceX and,Zbe a nonempty subset of a Hausdorfftopological vector spaceY. Suppose thatS, T :
K→2Zbe two set-valued maps with following conditions:
1for eachx∈K,coSx⊂Tx;
2for eachy∈Z,S−1y {x∈K:y∈Sx}is open.
Then T has a continuous selection, that is, there is a continuous mapf:K→Zsuch thatfx∈Tx for eachx∈K.
3. Existence of Solutions to the SGIVQIP
Lemma 3.1. LetD, W, Xbe three Hausdorfftopological spaces,Za topological vector space, and C a closed, convex, and pointed cone of Z. LetT :W ×X → 2D andF :D×W ×W → 2Z be two set-valued maps. Assume thatw, x, y∈W×X×Wand
1T·,·is upper semicontinuous onW×Xwith nonempty and compact values;
2F·,·,·is upperC-semicontinuous onD×W×Wwith nonempty and compact values;
Then there exist open neighborhoodUwofwand open neighborhoodUxofx, and open neighborhoodUyof y such that{Fu, w, y : u ∈Tw, x} ⊂ −intCwheneverw ∈Uw, x∈Ux,y∈Uy.
Proof. By3and compactness ofFu, w, y, there exists an open neighborhoodVuof the zero elementθofZsuch thatFu, w, yVu⊂ −intC. By2, there exist open neighborhood
Ou ofu and open neighborhoodOuw ofw, open neighborhood Ouy of y such that Fu, w, y⊂Fu, w, yVu−C⊂ −intC−C⊂ −intCwheneveru∈Ou, w∈Ouw, y∈ Ouy. SinceTw, xis compact and∪u∈Tw,xOu⊃Tw, x, there exist finiteu1, u2, . . . , uM∈ Tw, xsuch thatMj1Ouj⊃Tw, x. Taking
Ow ∩Mj1Oujw, U
y∩Mj1Ouj
y. 3.1
Clearly,OwandUyare open neighborhood ofw andy, respectively. Thus for eachu∈ ∪M
j1Ouj, we haveFu, w, y⊂ −intCwheneverw ∈Ow,y ∈Uy.By1, there exist
open neighborhoodUwofwwithUw⊂ Owand open neighborhoodUxofxsuch thatTw, x⊂ ∪Mj1Oujwheneverw∈Uw, x∈Ux, which implies that
Fu, w, y: u∈Tw, x⊂ Fu, w, y: u∈ ∪Mj1O
uj⊂ −intC. 3.2
wheneverw∈Uw,x∈Ux,y∈Uy.
The proof is finished.
By Lemma3.1, we obtain the following result.
Theorem 3.2. Consider an SGIVQIP{Ki, Di, Gi, Ti, Fi}i∈I. For eachi∈I, assume that
1Gi·is continuous onKwith convex compact values and for eachx∈K,intGix/∅;
2Ti·is upper semicontinuous onKwith nonempty and compact values;
3Fi·,·,·is upperCi- semicontinuous onDi×Ki×Kiwith nonempty and compact values;
4for eachx∈Kand eachui∈Tix,Fiui, xi,·isCi- convex orCi- quasiconvex-like;
5for eachx∈Kand eachui ∈Tix, ifxi ∈int Gix, thenFiui, xi, xi/⊂−int Ci, wherexi is theithcomponent ofx.
Then the SGIVQIP has a solution, that is, there existsx xi, xi ∈ K such that for each i∈I, xi∈Gix,and
∀yi∈Gix, ∃ui ∈Tix, Fi
ui, xi, yi
/
Proof. For eachi∈I, define a set-valued mapSi:K → 2Ki∪ {∅}by
Six
yi∈Ki:Fi
ui, xi, yi
⊂ −intCi, ∀ui∈Tix
. 3.4
Step 1. We prove that the setJi {x ∈ K : Gix∩Six ∅}is closed. For any sequence xn∈J
i{x∈K:Gix∩Six ∅}withxn → x0, we have
∀yn
i ∈Gixn, ∃uin∈Tixn, Fi
uni, xni, yin
/
⊂ −intCi. 3.5
Ifx0/∈J
i, then there existsz0i ∈Gix0such that for eachui ∈Tix0, Fiui, xi0, z0i⊂ −intCi.
By Lemma3.1, there exist open neighborhood Ux0 ofx0 and open neighborhoodUz0 i
ofz0i, such that{Fui, xi, zi : ui ∈ Tx} ⊂ −intCi wheneverx ∈ Ux0, zi ∈ Uz0i. By
1, there existzn
i ∈ Gixnsuch thatzin → z0i n → ∞, which implies that there exists
a positive integer N such that xn ∈ Ux0, zn
i ∈ Uz0i whenevern > N. Thus we have Fiui, xn, zni⊂ −intCi, for allui∈Tixnwhenevern > N, a contradiction. This shows that Jiis closed,that is,Wi{x∈K:Gix∩Six/∅}is open.
Without loss of generality, assume thatWi/∅.
Define a set-valued mapPi:K→2Ki ∪ {∅}by
Pix intGix∩Six for eachx∈K. 3.6
Step 2. We prove that for eachx∈Wi, Pixis nonempty and convex.
For eachyi ∈Six, we haveFiui, xi, yi⊂ −intCi, for allui ∈Tix. By Lemma3.1,
there exists an open neighborhoodUyiofyisuch that{Fiui, xi, yi :ui ∈Tix} ⊂ −intCi
wheneveryi∈Uyi, which implies thatUyi⊂Six, that is,Sixis open. By4, it is easy
to verify thatSixis convex.
SinceGixis convex and intGix/∅, then for eachx ∈ Wi, Pixis nonempty and
convex.
Step 3. We prove thatPi|Wi has a continuous selectionfi:Wi→2 Ki.
For eachy0
i ∈ Pix, we haveyi0 ∈ intGixandy0i ∈ Six. Byy0i ∈ intGix, there
existsε0>0 such thatyi0ε0⊂intGix, whereyi0ε0{zi∈Ki:dizi, y0i< ε0}. SinceGix
is continuous with convex compact values, then there exists an open neighborhoodOxof
xsuch that
Gix⊂Gi
x1
2ε0, 3.7
wheneverx ∈ Ox, where Gix 1/2ε0 {zi ∈ Ki : dizi, Gix < 1/2ε0}. Thus
y0
i ε0 ⊂ intGix ⊂ Gix ⊂ Gix 1/2ε0 whenever x ∈ Ox, which implies that y0
i 1/2ε0 ⊂ Gixwheneverx ∈Ox, that is,y0i ∈intGixwheneverx∈ Ox. This
shows that the set{x ∈ K : y0i ∈ intGix}is open. Byyi0 ∈ Six, we haveFiui, xi, yi0 ⊂ −intCi, ∀ui∈Tix. By Lemma3.1, there exists an open neighborhoodOxofxsuch that
Fi
ui, xi, yi0
:ui∈Ti
wheneverx∈Ox, which implies thatOx⊂ {x∈K :y0i ∈Six}, that is,{x∈K :yi0 ∈ Six}is open. Hence, for eachyi∈Pix, the setPi−1yi {x∈K:yi∈intGix∩Six}is
open.
By Lemma2.4,Pi|Wihas a continuous selectionfi:Wi→2 Ki.
Step 4. We prove that the SGIVQIP has a solution.
For eachi∈I, define the set-valued mapHi:K→2Ki by
Hix
⎧ ⎨ ⎩
fix, if x∈Wi,
Gix, ifx∈Ji.
3.9
Note thatHixis upper semicontinuous whenx∈intJiandHixis upper semicontinuous
whenx∈Wi, and it is easy to verify thatHixis also upper semicontinuous whenx∈∂Ji,
where∂Ji denotes the boundary ofJi. Thus,Hixis upper semicontinuous with nonempty
convex compact values. By8, Theorem 7.1.15, the set-valued map H : K → 2K defined
byHx i∈IHixis closed with nonempty convex values. By Kakutani-Fan-Glicksberg’s
fixed points theoremsee9, pages 550,Hhas a fixed point, that is, there existsx∈Hx. The condition5implies that for eachi∈I,xi/∈intGix∩Six, that is,xi/fixfor each i∈I. Thus we have that for eachi∈I,xi∈Gix,and
∀yi∈Gix, ∃ui ∈Tix, Fi
ui, xi, yi
/
⊂ −intCi. 3.10
The proof is finished.
IfIis a singleton, we obtain the following existence result of solutions to the GIVQIP by Theorem3.2.
Corollary 3.3. Consider a GIVQIP{K, D, G, T, F}. Assume that
1G·is continuous onKwith convex compact values and for eachx∈K,int Gx/∅;
2T·is upper semicontinuous onKwith nonempty and compact values;
3F·,·,·is upperC-semicontinuous onD×K×Kwith nonempty and compact values;
4for eachx∈Kand eachu∈Tx,Fu, x,·isC-convex orC-quasiconvex-like;
5for eachx∈Kand eachu∈Tx, ifx∈intGx, thenFu, x, x/⊂ −intC.
Then the GIVQIP has a solution, that is, there existsx∈Ksuch thatx∈Gx,
∀y∈Gx, ∃u∈Tx, Fu, x, y/⊂ −int C. 3.11
Remark 3.4. Theorem3.2, Corollary3.3, and each corresponding result in literatures1–6do not include each other as a special case.
Acknowledgments
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