Volume 2011, Article ID 381218,19pages doi:10.1155/2011/381218
Research Article
Existence and Stability of Solutions for Implicit
Multivalued Vector Equilibrium Problems
Sanhua Wang
1and Qiuying Li
21Department of Mathematics, Nanchang University, Nanchang, Jiangxi 330031, China 2College of Science and Technology, Nanchang University, Nanchang, Jiangxi 330029, China
Correspondence should be addressed to Sanhua Wang,wsh [email protected]
Received 31 October 2010; Accepted 14 January 2011
Academic Editor: M. Furi
Copyrightq2011 S. Wang and Q. Li. 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.
A class of implicit multivalued vector equilibrium problems is studied. By using the generalized Fan-Browder fixed point theorem, some existence results of solutions for the implicit multivalued vector equilibrium problems are obtained under some suitable assumptions. Moreover, a stability result of solutions for the implicit multivalued vector equilibrium problems is derived. These results extend and unify some recent results for implicit vector equilibrium problems, multivalued vector variational inequality problems, and vector variational inequality problems.
1. Introduction
LetXbe a Hausdorfftopological vector space,Ka nonempty subset ofX, andf:K×K → R
a function. Then, the scalar equilibrium problem consists in findingx∈Ksuch that
fx, y≥0, ∀y∈K. 1.1
This problem provides a unifying framework for many important problems, such as, optimization problems, variational inequality problems, complementary problems, minimal inequality problems, and fixed point problems, and has been widely applied to study the problems arising in economics, mechanics, and engineering sciencesee1. In recent years,
lots of existence results concerning equilibrium problems and variational inequality problems have been established by many authors in different ways. For details, we refer the reader to 1–26and the references therein.
a given map. Recently, Ansari et al.2studied the following vector equilibrium problems to findx∈Ksuch that
fx, y∈ −/ intC, ∀y∈K, 1.2
or to findx∈Ksuch that
fx, y∈C, ∀y∈K. 1.3
In case that the mapfis multivalued, Ansari et al.2also studied the following multivalued vector equilibrium problemfor short, MVEPto findx∈Ksuch that
fx, y/⊆ −intC, ∀y∈K, 1.4
or to findx∈Ksuch that
fx, y⊆C, ∀y∈K. 1.5
By using an abstract monotonicity condition, they gave some existence theorems of solutions for MVEP.
Very recently, in3,4, the authors studied the following implicit vector equilibrium
problemsfor short, IVEP: to findx∈Ksuch that
fgx, y∈ −/ intCx, ∀y∈K, 1.6
whereg :K → Kis a vector-valued map andC:K → 2Y is a multivalued map such that,
for allx∈K,Cxis a closed convex cone inY with intCx/∅. By using the famous FKKM
theorem and section theorem, they gave some existence results of solutions for IVEP.
Inspired and motivated by the research work mentioned above, in this paper, we consider a class of implicit multivalued vector equilibrium problems and introduce the concepts of Cx −h-pseudomonotonicity and V −h-hemicontinuity for multivalued maps.
By using the fixed point theorem of Chowdhury and Tan27, we obtain some existence
results of solutions for the implicit multivalued vector equilibrium problems in the setting of topological vector spaces. Furthermore, we derive a stability result of solutions for the implicit multivalued vector equilibrium problems. These results extend and unify some recent results for implicit vector equilibrium problems, multivalued vector variational inequality problems, and vector variational inequality problems.
2. Preliminaries
Throughout this paper, unless otherwise specified, we suppose that X, Y, and Z are topological vector spaces, andK ⊆ X andD ⊆ Z are nonempty subsets. We also suppose thatC:X → 2Y is a multivalued map such that, for anyx∈X,Cxis a proper, closed, and
In this paper, we consider the following implicit multivalued vector equilibrium problemfor short, IMVEPto findx∈Ksuch that
∀y∈K, ∃u∈Tx:Fgx, u, hx, y/⊆ −intCx. 2.1
We call thisxa solution for IMVEP. Some special cases of IMVEP.
1IfFis a single-valued map, then IMVEP reduces to the problem of findingx ∈K
such that
∀y∈K, ∃u∈Tx:Fgx, u, hx, y∈ −/ intCx, 2.2
which has been studied in5.
2IfF is a single-valued map,gx, v gx, hx, y y, then IMVEP reduces to IVEP.
3IfF is a single-valued map,gx, v x, hx, y y, then IMVEP reduces to the problem of findingx∈Ksuch that
Fx, y∈ −/ intCx, ∀y∈K, 2.3
which has been studied in6.
4Ifgx, v x, hx, y y,Cx CCis a closed convex cone inY with intC /∅,
then IMVEP reduces to MVEP.
Definition 2.1see28. LetXandY be two topological spaces. A multivalued mapT :X →
2Y is said to be
iupper semicontinuousfor short, u.s.c.atx0 ∈Xif, for each open setV inY with
Tx0⊆V, there exists an open neighborhoodUx0ofx0such thatTx⊆Vfor all
x∈Ux0;
iilower semicontinuousfor short, l.s.c.atx0 ∈Xif, for each open setV inY with
Tx0∩V /∅, there exists an open neighborhoodUx0ofx0such thatTx∩V /∅ for allx∈Ux0;
iiiclosed if the graph GrT {x, y∈X×Y :y∈Tx}is a closed subset ofX×Y. ivcompact-valued if, for eachx∈X,Txis a nonempty compact subset ofY.
Definition 2.2. LetX andY be topological vector spaces,K a nonempty convex subset ofX
andCa nonempty convex cone ofY. A multivalued mapF:X → 2Y is said to beC-convex
if, for anyx, y∈Xandt∈0,1, one has
Ftx 1−ty⊆tFx 1−tFy−C. 2.4
Definition 2.3. LetX,Y, andZbe topological vector spaces,Ka nonempty convex subset of
x ∈ X,Cxis a proper, closed, and convex cone inY with intCx/∅. Given two
vector-valued mapsg:K×D → X,h:K×K → X, and two multivalued mapsF :X×X → 2Y, T :K → 2D. Then,Tis said to be
iCx−h-pseudomonotone with respect toFandgonKif, for anyx, y∈Kand any
u∈Tx,v∈Ty, one has
Fgx, u, hx, y/⊆ −intCx impliesFgy, v, hx, y/⊆ −intCx; 2.5
iiweaklyCx−h-pseudomonotone with respect toFandgonK if, for anyx, y ∈K
and anyu∈Tx, one has
Fgx, u, hx, y/⊆ −intCx impliesFgy, v, hx, y/⊆ −intCxfor somev∈Ty; 2.6
iiiV −h-hemicontinuous with respect toFandg onKif, for anyx, y∈K,t∈0,1, ytxty−xand anyvt∈Tyt, there existsu∈Txsuch that, for any open set V withFgx, u, hx, y⊆V, there existst0∈0,1such that
∀t∈0, t0, Fgyt, vt, hx, y⊆V. 2.7
Remark 2.4. The above V −h-hemicontinuity for multivalued map is a generalization ofV -hemicontinuity for continuous linear operator.
Example 2.5. LetXY ZR,KD 0,1, andCx Rfor allx∈X. Letg:K×D → X,h:K×K → X,T :K → 2DandF:X×X → 2Y be defined as follows:
gx, u −1xu, ∀x, u∈K×D,
hx, yy−x, ∀x, y∈K×K,
Tx 0, x, ∀x∈K,
Fx, y
⎧ ⎨ ⎩
x, y, if x≥y,
y, x, otherwise. ∀
x, y∈X×X.
2.8
Then, T isCx−h-pseudomonotone with respect to F and g onK. Moreover, T isV −h -hemicontinuous with respect toFandgonK.
Proof. Firstly, we show thatT isCx−h-pseudomonotone with respect toFandgonK.
Indeed, for anyx, y∈0,1,u∈Tx 0, x, andv∈Ty 0, y, it is obvious that
−1xu≤ −x≤y−x. If
then,y−x≥0, that is,y≥x. It follows that
Fgy, v, hx, yF−1yv, y−x−1yv, y−x/⊆−∞,0. 2.10
HenceTisCx−h-pseudomonotone with respect toFandgonK.
Secondly, we show thatT isV−h-hemicontinuous with respect toFandgonK. Indeed, let x, y ∈ 0,1. Taking u x ∈ Tx, for any open set V with
Fgx, u, hx, y⊆V, that is,
Fgx, u, hx, yF−1xu, y−xF−12x, y−x−12x, y−x⊆V,
2.11
there existsr >0 such that
−12x−2r, y−x⊆V. 2.12
Letytxty−x, for allt∈0,1. Clearly,yt≥0.
Ify≤x, then for anyt0∈0,1and anyvt∈Tyt 0, yt, we have
0≤yt≤x. 2.13
And so
Fgyt, vt
, hx, yF−1ytvt
, y−x
−1ytvt, y−x
⊆−12yt
, y−x
⊆−12x, y−x
⊆V.
2.14
Ify > x, takingt0min{r/y−x,1/2} ∈0,1, then for anyt∈0, t0andvt∈Tyt, we have
yt≤yt0,
−1ytvt
≤ −1yt
−1−x−ty−x
y−xtx−1ty−1 ≤y−x1−0−1
≤y−x.
It follows that
Fgyt, vt, hx, yF−1ytvt, y−x
−1ytvt
, y−x
⊆−12yt
, y−x
⊆−12yt0
, y−x
−12xt0
y−x, y−x
⊆−12xr, y−x
⊆V.
2.16
Thus,TisV−h-hemicontinuous with respect toFandgonK.
Lemma 2.6see29. LetXandY be two topological spaces andF :X → 2Y a multivalued map.
iF is closed if and only if for any net{xα} ⊆X withxα → xand any net{yα}such that
yα∈Fxαwithyα → y, one hasy∈Fx.
iiIf F is compact valued, then F is u.s.c. atx ∈ X if and only if for any net {xα} ⊆ X
with xα → xand any net{yα}withyα ∈ Fxα, there existsy ∈ Fxand a subnet
{yβ} ⊆ {yα}such thatyβ → y.
The following lemma, which is a generalized form of Fan-Browder fixed piont theorem 30,31, is very important to establish our existence results of solutions for IMVEP.
Lemma 2.7 see27. Let K be a nonempty convex subset of a topological vector space X and
F, G:K → 2Kbe two multivalued maps such that
ifor anyx∈K,Fx⊆Gx; iifor anyx∈K,Gxis convex;
iiifor anyy∈K,F−1yis compactly open (i.e.,F−1y∩Lis open inLfor each nonempty compact subsetLofK);
ivthere exists a nonempty, closed, and compact subsetA⊆Kandy∈Asuch thatK\A⊆ G−1y;
vfor anyx∈K,Fx/∅.
Then, there existsx0∈Ksuch thatx0∈Gx0.
Lemma 2.8see28. LetXandY be two Hausdorfftopological vector spaces andT :X → 2Y a multivalued map. IfT is closed andTXis compact, thenT is u.s.c., whereTX x∈XTxand
Adenotes the closure of the setA.
Lemma 2.9see32. LetEbe a metric space andA, An∈En1,2, . . .be compact subsets. If,
3. Existence of Solutions for IMVEP
In this section, we will apply the generalized Fan-Browder fixed point theorem to establish some existence results of solutions for IMVEP. First of all, we have the following lemma.
Lemma 3.1. LetX,Y, andZbe topological vector spaces, andKa nonempty convex subset ofX, and
Da nonempty subset ofZ. LetC:X → 2Y be a multivalued map such that, for anyx∈X,Cxis a
proper, closed, and convex cone inYwithintCx/∅. Given two vector-valued mapsg:K×D → X,
h:K×K → X, and two multivalued mapsF:X×X → 2Y,T :K → 2D. Consider the following
problems.
IFindx∈Ksuch that,∀y∈K,∃u∈Tx:Fgx, u, hx, y/⊆ −intCx; IIFindx∈Ksuch that,∀y∈K,∃v∈Ty:Fgy, v, hx, y/⊆ −intCx; IIIFindx∈Ksuch that,∀y∈K,∀v∈Ty:Fgy, v, hx, y/⊆ −intCx.
Then,
iProblem (I) implies Problem (II) ifT is weaklyCx−h-pseudomonotone with respect toF
andgonK, moreover, implies Problem (III) ifT isCx−h-pseudomonotone with respect to
FandgonK;
iiProblem (II) implies Problem (I) ifT isV −h-hemicontinuous with respect toF and g
onK and, for anyx, y ∈ K and anyv ∈ Ty,Fgy, v, hx,·isCx-convex and
Fgy, v, hx, x⊆ −Cx; iiiProblem (III) implies Problem (II).
Proof. iIt follows from the weaklyCx−h-pseudomonotone with respect toF andgonK
andCx−h-pseudomonotone with respect toFandgonK, respectively.
iiLetxbe a solution ofII. Then,∀y∈K,∃v∈Tysuch that
Fgy, v, hx, y/⊆ −intCx. 3.1
Letyβxβy−x, for allβ∈0,1. SinceKis convex, we haveyβ∈K. Then, there exists vβ∈Tyβsuch that
Fgyβ, vβ, hx, yβ/⊆ −intCx, ∀β. 3.2
Since for anyx, y∈Kand anyv∈Ty,Fgy, v, hx,·isCx-convex, we have
Fgyβ, vβ
, hx, yβ
⊆βFgyβ, vβ
, hx, y1−βFgyβ, vβ
, hx, x−Cx. 3.3
Notice that for allx, y∈Kand anyv∈Ty,Fgy, v, hx, x⊆ −Cx. Then, we can obtain
that
Fgyβ, vβ
Indeed, suppose to the contrary, thatFgyβ, vβ, hx, y⊆ −intCxfor someβ, then
Fgyβ, vβ
, hx, yβ
⊆βFgyβ, vβ
, hx, y1−βFgyβ, vβ
, hx, x−Cx
⊆ −intCx−Cx−Cx
⊆ −intCx,
3.5
which contradicts3.2, and so3.4holds.
We claim that there existsu∈Txsuch that
Fgx, u, hx, y/⊆ −intCx, 3.6
and soxis a solution of ProblemI.
In fact, if it is not the case, then we have, for any u ∈ Tx, Fgx, u, hx, y ⊆
−intCx. And then it follows from the fact thatTisV−h-hemicontinuous with respect toF
andgonKthat there existsu∈Txandβ0∈0,1such thatFgx, u, hx, y⊆ −intCx and
∀β∈0, β0
, Fgyβ, vβ
, hx, y⊆ −intCx. 3.7
3.7contradicts3.4, and so3.6holds. iiiis obvious.
This completes the proof.
Now, we are ready to prove some existence theorems for IMVEP under suitable pseudomonotonicity assumptions.
Theorem 3.2. LetX,Y, andZbe topological vector spaces, andKa nonempty convex subset ofX
andD a nonempty subset ofZ. LetC : X → 2Y be a multivalued map such that, for anyx ∈ X, Cxis a proper, closed, and convex cone inY withintCx/∅. Given two mapsg :K×D → X,
h:K×K → X, and two multivalued mapsF :X×X → 2Y,T :K → 2D. Suppose the following
conditions are satisfied:
ihis continuous in the first variable;
iiT isCx−h-pseudomonotone andV−h-hemicontinuous with respect toFandgonK;
iiithe multivalued mapW:K → 2Y, defined byWx Y \ {−intCx}is closed;
ivFis u.s.c. and compact-valued, and it satisfies the following conditions:
afor any x, y ∈ K and v ∈ Ty, Fgy, v, hx,· is Cx-convex and
Fgy, v, hx, x⊆ −Cx;
bfor anyx∈K, there existsu∈Txsuch that0∈Fgx, u, hx, x;
vthere exists a nonempty, compact and closed subset A ⊆ K andy ∈ Asuch that, for all
x∈K\A, one has
Then, IMVEP is solvable, that is, there existsx∈Ksuch that
∀y∈K, ∃u∈Tx:Fgx, u, hx, y/⊆ −intCx. 3.9
Proof. Define two multivalued mapsH, G:K → 2Kas follows, for anyx∈K,
Hx y∈K:∃v∈Ty, Fgy, v, hx, y⊆ −intCx,
Gx y∈K:∀u∈Tx, Fgx, u, hx, y⊆ −intCx. 3.10
The proof is divided into the following steps. IFor allx∈K,Hx⊆Gx.
Indeed, lety∈Hx, then there existsv∈Tysuch that
Fgy, v, hx, y⊆ −intCx. 3.11
Ify /∈Gx, then there existsu∈Txsuch that
Fgx, u, hx, y/⊆ −intCx. 3.12
SinceT isCx−h-pseudomonotone with respect toFandgonK, then, by3.12, we have for
allv∈Ty,
Fgy, v, hx, y/⊆ −intCx, 3.13
which contradicts3.11. Thus,y∈Gx, and soHx⊆Gx.
IIFor allx∈K,Gxis convex.
In fact, for anyy1, y2 ∈ Gxandt ∈ 0,1, by the definition ofG, we have, for each
u∈Tx,
Fgx, u, hx, y1
⊆ −intCx,
Fgx, u, hx, y2
⊆ −intCx. 3.14
Letyt ty1 1−ty2. SinceKis convex, we haveyt ∈K. Noting thatFgx, u, hx,·is
Cx-convex, we have
Fgx, u, hx, yt
⊆tFgx, u, hx, y1
1−tFgx, u, hx, y2
−Cx
⊆ −intCx−intCx−Cx
⊆ −intCx.
3.15
Indeed, for any given compact subsetL ⊆ K, letP H−1y∩L. We will show that
P is open inLby proving thatPCis closed inL. Let{x
α} ⊆PCbe an arbitrary net such that xα → x0∈L. Then, for eachα,xα∈PC, that is,y /∈Hxα, thus, for anyv∈Ty,
Fgy, v, hxα, y/⊆ −intCxα, ∀α. 3.16
It follows that, for eachα, there existsωα ∈ Fgy, v, hxα, ysuch thatωα ∈ −/ intCxα,
that is,
ωα∈Wxα Y\ {−intCxα}, ∀α. 3.17
Sincehis continuous in the first variable andFis u.s.c. and compact-valued, it follows from
Lemma 2.6that there existsω0 ∈ Fgy, v, hx0, yand a subnet of{ωα}, we still denote
this subnet by{ωα}, such that ωα → ω0. Notice thatW is closed. We have ω0 ∈ Wx0
Y\ {−intCx0}. It follows thatω0/∈ −intCx0, and so
Fgy, v, hx0, y
/
⊆ −intCx0. 3.18
Then, by the arbitrary ofv, we havey /∈ Hx0, that is,x0 ∈/H−1y. Sincex0 ∈L, we know thatx0∈PC, and soPCis closed inL.
IVBy the assumptionv, there exists a nonempty, compact, and closed subsetA⊆K
andy∈Asuch that, for allx∈K\A, we have
∀u∈Tx, Fgx, u, hx, y⊆ −intCx. 3.19
This implies thaty∈Gx, that is,x∈G−1y. And thusK\A⊆G−1y. VGhas no fixed point inK.
Suppose that it is not the case, then there existsx0∈Ksuch thatx0∈Gx0, that is,
∀u∈Tx0, Fgx0, u, hx0, x0
⊆ −intCx0. 3.20
By the assumptioniv, we have 0∈Fgx0, u, hx0, x0for someu∈Tx0, and it follows that 0 ∈ −intCx0. This implies thatCx0is an absorbing set inY, which contradicts the assumption thatCx0is proper inY. Therefore,Ghas no fixed point.
SinceGhas no fixed point inK, it follows fromLemma 2.7that there existsx∈Ksuch thatHx ∅, that is,
∀y∈K, ∀v∈Ty, Fgy, v, hx, y/⊆ −intCx. 3.21
FromLemma 3.1, we havex∈Ksuch that
∀y∈K, ∃u∈Tx:Fgx, u, hx, y/⊆ −intCx. 3.22
Remark 3.3. Theorem 3.2is a multivalued extension of7, Theorem 3.
Remark 3.4. The conditionvofTheorem 3.2is satisfied automatically ifKis compact.
We now give an example to illustrateTheorem 3.2.
Example 3.5. Let X, Y, Z, K, D, C, g, h, T, andF be as in Example 2.5. We will show that all conditions ofTheorem 3.2are satisfied.
IIt follows fromExample 2.5that the conditioniiofTheorem 3.2is satisfied. And it is obvious thatFis compact valued andWis a closed mapping.
IIWe will show thatFis u.s.c. onX×X. Letx, y∈X, for any setV withFx, y⊆V.
1Ifxy, thenFx, y {x} ⊆V, and then there existsr >0 such thatx−r, xr⊆V. TakingUy Ux x−r, xr, then for anyx∈Uxandy∈Uy, we have
Fx, y
⎧ ⎨ ⎩
x, y⊆x−r, xr⊆V, if x≥y,
y, x⊆x−r, xr⊆V, otherwise. 3.23
2Ifx < y, thenFx, y x, y⊆V, and then there existsr : 0< r <y−x/2 such that
x−r, yr⊆V. 3.24
TakingUx x−r, xrandUy y−r, yr, we have
xr ≤xy−x
2
yx
2 y−
y−x
2 y−r. 3.25
It follows that for anyx∈Uxand anyy∈Uy, then we havex≤y. Thus,
Fx, yx, y⊆x−r, yr⊆V. 3.26
3Ifx > y, the argument is similar to2. Hence,Fis u.s.c. onX×X.
IIIWe will show that for anyx, y∈Kandv∈Ty,Fgy, v, hx,·isCx-convex.
For anyx, y∈Kandv∈Ty. Letuttu1 1−tu2for eacht∈0,1andu1, u2∈K. Since
ut−xtu1 1−tu−2−x≥0−x−x,
it follows that
Fgy, v, hx, ui
F−1yv, ui−x
−1yv, ui−x
, i1,2. 3.28
Thus, we have
Fgy, v, hx, ut
F−1yv, ut−x
−1yv, ut−x
−1yv, tu1−x 1−tu2−x
t−1yv, u1−x
1−t−1yv, u2−x
tFgy, v, hx, u1 1−tFgy, v, hx, u2
⊆tFgy, v, hx, u1 1−tFgy, v, hx, u2−Cx.
3.29
This shows thatFgy, v, hx,·isCx-convex.
IVObviously, for anyx, y∈Kand anyy∈Ty,
Fgy, v, hx, xF−1yv, x−x
−1yv,0⊆ −R−Cx,
3.30
that is, for anyx, y∈Kandy∈Ty,Fgy, v, hx, x⊆ −Cx.
For anyx∈Kandu∈Tx, we have
Fgx, u, hx, xF−1xu, x−x −1xu,0, 3.31
thus, 0∈Fgx, u, hx, x.
By the above arguments, we know that all the conditions ofTheorem 3.2are satisfied. ByTheorem 3.2, IMVEP is solvable.
Indeed, letx0∈0,1, then for anyy∈0,1, there existsu0∈Txsuch that
Fgx, u, hx, yFg0,0, h0, yF−1, y−1, y/⊆ −intR. 3.32
Thus,xis a solution of IMVEP.
We now obtain an existence theorem for IMVEP for weaklyCx−h-pseudomonotone
maps with respect toFandgunder additional assumptions.
Theorem 3.6. LetX,Y,Z,K,D,C,g,h,F, andTbe as inTheorem 3.2. Assume that the conditions (iii)–(v) ofTheorem 3.2and the following conditions are satisfied:
igis continuous in the second variable andhis continuous in the first variable;
Then, IMVEP is solvable, that is, there existsx∈Ksuch that
∀y∈K, ∃u∈Tx:Fgx, u, hx, y/⊆ −intCx. 3.33
Proof. Define two multivalued mapsH, G:K → 2Kas follows, for anyx∈K,
Hx y∈K:∀v∈Ty, Fgy, v, hx, y⊆ −intCx,
Gx y∈K:∀u∈Tx, Fgx, u, hx, y⊆ −intCx. 3.34
By using the same arguments as in the proof I of Theorem 3.2 and weakly Cx − h -pseudomonotonicity with respect to F and g on K of T, we see that for any x ∈ K,
Hx⊆Gx.
We have already seen in the proof ofTheorem 3.2that for eachx∈K,Gxis convex and the multivalued mapGhas no fixed point. Moreover, there exists a nonempty, compact, and closed subsetA⊆Kandy∈Asuch thatK\A⊆G−1y.
Next, we will show that, for eachy∈K,H−1yis compactly open.
Indeed, for any given compact subsetL ⊆ K, letP H−1y∩L. We will show that
P is open inLby proving thatPCis closed inL. Let{x
α} ⊆PCbe an arbitrary net such that xα → x0 ∈ L. Then, for eachα,xα ∈ PC, that is,y /∈ Hxα. Thus, for eachα, there exists vα∈Tysuch that
Fgy, vα, hxα, y/⊆ −intCxα. 3.35
SinceTis compact valued, there exists a subnet{vβ}of{vα}such thatvβ → v0∈Ty. Then, by virtue of3.35, for eachβ, there existsωβ∈Fgy, vβ, hxβ, ysuch thatωβ∈ −/ intCxβ,
that is,
ωβ∈WxβY\−intCxβ, ∀β. 3.36
Sincegis continuous in the second variable andhis continuous in the first variable, we have
gy, vβ
−→gy, v0
, hxβ, y
−→hx0, y
. 3.37
In addition,Fis u.s.c. and compact valued, it follows fromLemma 2.6that there existω0 ∈
Fgy, v0, hx0, yand a subnet of{ωβ}, we still denote this subnet by{ωβ}, such thatωβ → ω0. Notice thatW is closed. We have ω0 ∈ Wx0 Y \ {−intCx0}. It follows thatω0 ∈/ −intCx0, and so
Fgy, v0
, hx0, y
/
⊆ −intCx0. 3.38
Thus,y /∈ Hx0, that is,x0 /∈ H−1y. Since x0 ∈ L, we know thatx0 ∈ PC, and soPC is closed inL.
Hence, as in the proof ofTheorem 3.2, there existsx∈Ksuch thatHx ∅, that is,
FromLemma 3.1, we havex∈Ksuch that
∀y∈K, ∃u∈Tx:Fgx, u, hx, y/⊆ −intCx. 3.40
This completes the proof.
Remark 3.7. Theorem 3.6is a multivalued extension of7, Theorem 4.
Next, we will prove an existence result for IMVEP without any kind of pseudomono-tonicity assumption.
Theorem 3.8. LetX,Y,Z,K,D,C,g,h,F, andTbe as inTheorem 3.2. Assume that the conditions (iii)–(v) ofTheorem 3.2and the following conditions are satisfied:
igis continuous in both variables andhis continuous in the first variable;
iiT is u.s.c. and compact-valued.
Then, IMVEP is solvable, that is, there existsx∈Ksuch that
∀y∈K, ∃u∈Tx:Fgx, u, hx, y/⊆ −intCx. 3.41
Proof. Define a multivalued mapG:K → 2Kas in the proof ofTheorem 3.2. As we have seen
in the proof inTheorem 3.2that, for eachx∈K,Gxis convex and the multivalued mapG
has no fixed point. Moreover, there exists a nonempty, compact, and closed subsetA⊆Kand
y∈Asuch thatK\A⊆G−1y.
Now, we have only to show that, for anyy∈K,G−1yis compactly open.
Indeed, for any given compact subsetL ⊆ K, letP G−1y∩L. We will show that
P is open inLby proving thatPCis closed inL. Let{x
α} ⊆PCbe an arbitrary net such that xα → x0 ∈ L. Then, for each α,xα ∈ PC, that is,y /∈ Gxα. Thus, for eachα, there exists uα∈Txαsuch that
Fgxα, uα, h
xα, y
/
⊆ −intCxα. 3.42
SinceT is u.s.c. and compact-valued, it follow fromLemma 2.6that there existsu0 ∈Tx0 and a subnet{uβ} ⊆ {uα}such thatuβ → u0. Then, by the assumptioni, we have
gxβ, uβ−→gx0, u0, h
xβ, y−→hx0, y
. 3.43
Furthermore, by virtue of3.42, for eachβ, there exists someωβ∈Fgxβ, uβ, hxβ, ysuch thatωβ/∈ −intCxβ. It follows that
ωβ∈WxβY\−intCxβ, ∀β. 3.44
Since F is u.s.c. and compact-valued, it follows from Lemma 2.6 that there exist ω0 ∈
ωβ → ω0. Notice thatW is closed. We haveω0 ∈ Wx0 Y \ {−intCx0}, that is,ω0 ∈/ −intCx0, and so
Fgx0, u0, h
x0, y
/
⊆ −intCx0. 3.45
Thus,y /∈Gx0, that is,x0 /∈G−1y. Sincex0∈L, we know thatx0∈PC, and soPCis closed inL.
Thus, as in the proof ofTheorem 3.2, there existsx∈Ksuch thatGx ∅, that is,
∀y∈K, ∃u∈Tx, Fgx, u, hx, y/⊆ −intCx. 3.46
This completes the proof.
Remark 3.9. From the proof of Theorem 3.8, we can see that, if D is compact, then the conditioniican be replaced by the following conditioniiT is closed.
4. Stability of Solution Sets for IMVEP
In this section, we discuss the stability of solutions for IMVEP.
Throughout this section, letX andZ be Banach spaces, and Y a topological vector space,K ⊆ X a nonempty compact convex subset andD ⊆ Z a nonempty subset, andC :
X → 2Y a multivalued map such that, for allx∈X,Cxis a proper, closed convex cone in Y with intCx/∅.
Let
Eg, h, T:g:K×D−→X is continuous in both variables;
h:K×K−→X is continuous in the first variable;
T :K−→2D is u.s.c.and compact valued.
4.1
LetA, Bbe two compact sets in a normed spaceU, · . Recall the Hausdorffmetric defined by
HA, B max
sup
a∈A
inf
b∈Ba−b,supb∈Bainf∈Aa−b
. 4.2
For any giveng, h, T,g, h, T∈E, let
ρg, h, T,g, h, T sup x,y∈K×D
gx, y−gx, y sup
x,y∈K×K
hx, y−hx, y
sup
x∈K
HTx, Tx,
4.3
Assume that the multivalued mapsFandWsatisfy all the conditions ofTheorem 3.2. Then, it follows fromTheorem 3.8that for anyg, h, T ∈ E, IMVEP has a solution, that is, there existsx∈Ksuch that
∀y∈K, ∃u∈Tx:Fgx, u, hx, y/⊆ −intCx. 4.4
Let
ϕg, h, Tx∈K:∀y∈K, ∃u∈Tx, Fgx, u, hx, y/⊆ −intCx. 4.5
Thenϕg, h, T/∅, which implies thatϕis a multivalued map fromEintoK.
Theorem 4.1. ϕ:E → 2Kis an upper semicontinuous map with nonempty compact values.
Proof. SinceK is compact, it follows fromLemma 2.8that we need only to show thatϕ is closed. Letgn, hn, Tn, xn∈Grϕandgn, hn, Tn, xn → g, h, T, x. We will show that
g, h, T, x∈Grϕ.
Indeed, for anyy∈Kandn, sincexn∈ϕgn, hn, Tn, there existsun∈Tnxnsuch that
Fgnxn, un, hnxn, y/⊆ −intCxn. 4.6
Notice thatTandTnare compact-valued. Then, for any open setOwithTx⊆O, there exists ε >0 such that
{u∈Z:du, Tx< ε} ⊆O, 4.7
wheredu, Tx infu∈Txu−u. Sinceρgn, hn, Tn,g, h, T → 0 andT is u.s.c., there existsn0such that, for alln≥n0,
sup
x∈K
HTx, Tnx< ε
2, 4.8
Txn⊆
u∈Z:du, Tx< ε
2
. 4.9
It follows from4.8that
HTxn, Tnxn< ε
2. 4.10
By4.10,4.9, and4.7, for alln≥n0, we have
Tnxn⊆
u∈Z:du, Txn< ε
2
⊆ {u∈Z:du, Tx< ε}
⊆O.
SinceTx⊆Oand for alln≥n0,un∈Tnxn⊆O, byLemma 2.9, there exists a subsequence {unk}of{un}such that
unk −→u∈Tx. 4.12
Sincegnk is continuous, hnk is continuous in the first variable, and gnk → g, hnk → h, we
have
gnkxnk, unk−→gx, u, hnk
xnk, y
−→hx, y. 4.13
Then, it follows from4.6that
Fgnkxnk, unk, hnk
xnk, y
/
⊆ −intCxnk. 4.14
Thus, there exist ωnk ∈ Fgnkxnk, unk, hnkxnk, y such that ωnk /∈ −intCxnk, that is,
ωnk ∈ Wxnk Y \ {−intCxnk}. Since F is u.s.c. and compact valued, there exist
ω∈Fgx, u, hx, yand a subsequence of{ωnk}, we still denote this subsequence by{ωnk},
such thatωnk → ω. Notice thatW is closed. We haveω ∈Wx Y\ {−intCx}, that is,
ω∈ −intCx, and so
Fgx, u, hx, y/⊆ −intCx. 4.15
This implies thatg, h, T, x∈Grϕ, and soϕis closed. This completes the proof.
5. Conclusions
In this paper, the existence and stability of solutions for a class of implicit multivalued vector equilibrium problems are studied. By using the generalized Fan-Browder fixed point theorem 27, some existence results of solutions for the implicit multivalued vector equilibrium
problems are obtained under some suitable assumptions. These results generalize and extend some corresponding results of Ansari et al. 7. Also, inSection 4 of this paper, a stability result of solutions for the implicit multivalued vector equilibrium problems is obtained. It is worth mentioning that, up till now, there is no paper to consider the stability of solutions for the implicit multivalued vector equilibrium problems. So, the stability result obtained in
Section 4of this paper is new and interesting.
Acknowledgments
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