Volume 2011, Article ID 945413,13pages doi:10.1155/2011/945413
Research Article
Solving the Set Equilibrium Problems
Yen-Cherng Lin and Hsin-Jung Chen
Department of Occupational Safety and Health, China Medical University, Taichung 40421, Taiwan
Correspondence should be addressed to Yen-Cherng Lin,[email protected]
Received 17 September 2010; Accepted 21 November 2010
Academic Editor: Qamrul Hasan Ansari
Copyrightq2011 Y.-C. Lin and H.-J. Chen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the weak solutions and strong solutions of set equilibrium problems in real Hausdorff topological vector space settings. Several new results of existence for the weak solutions and strong solutions of set equilibrium problems are derived. The new results extend and modify various existence theorems for similar problems.
1. Introduction and Preliminaries
LetX,Y,Zbe arbitrary real Hausdorfftopological vector spaces, letKbe a nonempty closed
convex set ofX, and let C ⊂ Y be a proper closed convex and pointed cone with apex at
the origin and intC /∅, that is,Cis proper closed with intC /∅and satisfies the following conditions:
1λC⊆C, for allλ >0; 2CC⊆C;
3C∩−C {0}.
LettingA,Bbe two sets ofY, we can define relations “≤C” and “/≤C” as follows:
1A≤CB⇔B−A⊂C; 2A/≤CB⇔B−A /⊂C.
Similarly, we can define the relations “≤intC” and “/≤intC” if we replace the setCby intC.
The trimappingf : Z×K×K → 2Y and mappingT : K → 2Z are given. The set
equilibrium problemSEPIis to find anx∈Ksuch that
for ally ∈K and for somes∈ Tx. Such solution is called a weak solution forSEPI. We note that1.1is equivalent to the following one:
fs, x, y/⊂ −intC 1.2
for ally∈Kand for somes∈Tx.
For the case whensdoes not depend ony, that is, to find anx∈Kwith somes∈Tx such that
fs, x, y/≤intC{0} 1.3
for ally∈K, we will call this solution a strong solution ofSEPI. We also note that1.3is equivalent to the following one:
fs, x, y/⊂ −intC 1.2
for ally∈K.
We note that iffis a vector-valued function and the mappings → fs, x, yis constant for eachx, y ∈K, thenSEPI reduces to the vector equilibrium problemVEP, which is to findx∈Ksuch that
fx, y/≤intC0 1.4
for ally∈K. Existence of a solution of this problem is investigated by Ansari et al.1,2. Iff is a vector-valued function andZ LX, Y which is denoted the space of all continuous linear mappings fromXtoY andfs, x, y s, y−x, wheres, ydenotes the evaluation of the linear mappingsaty, thenSEPI reduces toGVVIP: to findx∈Kand
s∈Txsuch that
s, y−x/≤intC0 1.5
for ally∈K. It has been studied by Chen and Craven3.
If we considerF : K → K,Z LX, Y,A : LX, Y → LX, Y, andfs, x, y As, y−x Fy−Fx, wheres, ydenotes the evaluation of the linear mappingsaty, thenSEPIreduces to theGVVIPwhich is discussed by Huang and Fang4and Zeng and Yao5: to find a vectorx∈Kands∈Txsuch that
As, y−xFy−Fx/≤intC0, ∀y∈K. 1.6
IfZ LX, Y,T :K → LX, Yis a single-valued mapping,fs, x, y Tx, y−x, then SEPI reduces to theweakvector variational inequalities problem which is considered by Fang and Huang6, Chiang and Yao7, and Chiang8as follows: to find a vectorx∈K such that
for ally ∈K. The vector variational inequalities problem was first introduced by Giannessi 9in finite-dimensional Euclidean space.
Summing up the above arguments, they show that for a suitable choice of the mapping
Tand the spacesX,Y, andZ, we can obtain a number of known classes of vector equilibrium problems, vector variational inequalities, and implicit generalized variational inequalities. It is also well known that variational inequality and its variants enable us to study many important problems arising in mathematical, mechanics, operations research, engineering sciences, and so forth.
In this paper we aim to derive some solvabilities for the set equilibrium problems. We also study some results of existence for the weak solutions and strong solutions of set equilibrium problems. LetK be a nonempty subset of a topological vector spaceX. A
set-valued function Φ from K into the family of subsets of X is a KKM mapping if for any
nonempty finite setA⊂K, the convex hull ofAis contained inx∈AΦx. Let us first recall the following results.
Fan’s Lemmasee10. LetKbe a nonempty subset of Hausdorfftopological vector spaceX. Let
G:K → 2Xbe a KKM mapping such that for anyy∈K,Gyis closed andGy∗is compact for somey∗∈K. Then there existsx∗ ∈Ksuch thatx∗∈Gyfor ally∈K.
Definition 1.1see11. LetΩbe a vector space, letΣbe a topological vector space, letKbe
a nonempty convex subset ofΩ, and letC⊂ Σbe a proper closed convex and pointed cone
with apex at the origin and intC /∅, andϕ:K → 2Σis said to be
1C-convexiftϕx1 1−tϕx2 ⊂ ϕtx1 1−tx2 Cfor everyx1, x2 ∈ Kand
t∈0,1;
2naturally quasi -C-convex if ϕtx1 1 −tx2 ⊂ co{ϕx1∪ϕx2} −C for every
x1, x2∈Kandt∈0,1.
The following definition can also be found in11.
Definition 1.2. LetY be a Hausdorfftopological vector space, letC ⊂ Y be a proper closed
convex and pointed cone with apex at the origin and intC /∅, and let A be a nonempty
subset ofY. Then
1a pointz∈Ais called a minimal pointofAifA∩z−C {z}; MinAis the set of all minimal points ofA;
2a pointz∈Ais called a maximal pointofAifA∩zC {z}; MaxAis the set of all maximal points ofA;
3a pointz∈Ais called a weakly minimal pointofAifA∩z−intC ∅; MinwAis the set of all weakly minimal points ofA;
4a pointz∈Ais called a weakly maximal pointofAifA∩zintC ∅; MaxwAis the set of all weakly maximal points ofA.
Definition 1.3. LetX,Y be two topological spaces. A mappingT :X → 2Y is said to be
2lower semicontinuous if for everyx ∈ X and every open neighborhood Vy of everyy∈Tx, there exists a neighborhoodWxofxsuch thatTu∩Vy/∅for allu∈Wx;
3continuous if it is both upper semicontinuous and lower semicontinuous.
We note thatT is lower semicontinuous atx0if for any net{xν} ⊂ X,xν → x0,y0 ∈Tx0
implies that there exists net yν ∈ Txν such that yν → y0. For other net-terminology
properties about these two mappings, one can refer to12.
Lemma 1.4 see 13. Let X,Y, and Z be real topological vector spaces, and let K and C be nonempty subsets of X and Y, respectively. Let F : K×C → 2Z,S : K → 2C be set-valued mappings. If bothF and Sare upper semicontinuous with nonempty compact values, then the set-valued mappingG:K → 2Zdefined by
Gx
y∈Sx
Fx, yFx, Sx, ∀x∈K 1.8
is upper semicontinuous with nonempty compact values.
By using similar technique of 11, Proposition 2.1, we can deduce the following lemma that slight-generalized the original one.
Lemma 1.5. LetL,Kbe two Hausdorfftopological vector spaces, and letL,Kbe nonempty compact convex subsets ofLandK, respectively. LetG:L×K → 2Ê
be continuous mapping with nonempty compact valued onL×K; the mappings → −Gs, xis naturally quasiÊ-convex onLfor each
x∈K, and the mappingx → Gs, xisÊ-convex onKfor eachs∈L. Assume that for eachx∈K,
there existssx∈Lsuch that
MinGsx, x≥ÊMin
x∈K
Maxw
s∈L
Gs, x. 1.9
Then, one has
Min
x∈K
Maxw
s∈L
Gs, x Max s∈L
Minw
x∈K
Gs, x. 1.10
2. Existence Theorems for Set Equilibrium Problems
Now, we state and show our main results of solvabilities for set equilibrium problems.
Theorem 2.1. LetX,Y,Zbe real Hausdorfftopological vector spaces, letKbe a nonempty closed convex subset ofX, and letC⊂Y be a proper closed convex and pointed cone with apex at the origin andintC /∅. Given mappingsf :Z×K×K → 2Y,T :K → 2Z, andν:K×K → 2Y, suppose that
1{0}≤Cνx, xfor allx∈K;
2for eachx∈K, there is ans∈Txsuch that for ally∈K,
3for eachx∈K, the set{y∈K :{0}/≤Cνx, y}is convex;
4there is a nonempty compact convex subsetDofK, such that for everyx∈K\D, there is ay∈Dsuch that for alls∈Tx,
fs, x, y≤intC{0}, 2.2
5for eachy∈K, the set{x∈K :fs, x, y≤intC{0}for alls∈Tx}is open inK.
Then there exists anx∈Kwhich is a weak solution of (SEP)I. That is, there is anx∈Ksuch that
fs, x, y/≤intC{0} 2.3
for ally∈Kand for somes∈Tx.
Proof. DefineΩ:K → 2Dby
Ωyx∈D:fs, x, y/≤intC{0}for somes∈Tx 2.4
for ally ∈K. From condition5we know that for eachy ∈K, the setΩyis closed inK, and hence it is compact inDbecause of the compactness ofD.
Next, we claim that the family{Ωy:y∈K}has the finite intersection property, and
then the whole intersection y∈KΩy is nonempty and any element in the intersection
y∈KΩyis a solution ofSEPI, for any given nonempty finite subsetN ofK. LetDN co{D∪N}, the convex hull ofD∪N. ThenDNis a compact convex subset ofK. Define the mappingsS, R:DN → 2DN, respectively, by
Syx∈DN:fs, x, y/≤intC{0}for somes∈Tx,
Ryx∈DN :{0}≤Cνx, y,
2.5
for eachy∈DN. From conditions1and2, we have
{0}≤Cνy, y ∀y∈DN, 2.6
and for eachy∈K, there is ans∈Tysuch that
νy, y−fs, y, y≤C{0}. 2.7
Hence{0}≤Cfs, y, y, and theny∈Syfor ally∈DN.
We can easily see thatShas closed values inDN. Since, for eachy∈DN,Ωy Sy∩
lemma if we claim thatSis a KKM mapping. Indeed, ifSis not a KKM mapping, neither isR sinceRy⊂Syfor eachy∈DN. Then there is a nonempty finite subsetMofDNsuch that
coM /⊂
u∈M
Ru. 2.8
Thus there is an elementu∈coM⊂DNsuch thatu /∈Rufor allu∈M, that is,{0}/≤Cνu, u for allu∈M. By3, we have
u∈coM⊂y∈K:{0}/≤Cνu, y, 2.9
and hence{0}/≤Cνu, uwhich contradicts 2.6. Hence R is a KKM mapping, and so is S. Therefore, there exists anx∈Kwhich is a solution ofSEPI. This completes the proof.
Theorem 2.2. LetX,Y,Zbe real Hausdorfftopological vector spaces, letKbe a nonempty closed convex subset ofX, and letC⊂Y be a proper closed convex and pointed cone with apex at the origin andintC /∅. Let the mappingf : Z×K×K → 2Y be such that for eachy ∈ K, the mappings
s, x → fs, x, yandT :K → 2Zare upper semicontinuous with nonempty compact values and
ν :K×K → 2Y. Suppose that conditions (1)–(4) ofTheorem 2.1hold. Then there exists anx∈K which is a solution of (SEP)I. That is, there is anx∈Ksuch that
fs, x, y/≤intC{0} 2.10
for ally∈Kand for somes∈Tx.
Proof. For any fixedy∈K, we define the mappingG:K → 2Y by
Gx
s∈Tx
fs, x, y 2.11
for alls∈ Zandx∈ K. Since the mappingss, x → fs, x, yandT :K → 2Zare upper
semicontinuous with nonempty compact values, byLemma 1.4, we know that G is upper
semicontinuous onKwith nonempty compact values. Hence, for eachy∈K, the set
x∈K:fs, x, y≤intC{0} ∀s∈Tx
{x∈K:Gx⊂−intC} 2.12
is open in K. Then all conditions of Theorem 2.1 hold. From Theorem 2.1, SEPI has a
solution.
In order to discuss the results of existence for the strong solution of SEPI, we introduce the condition . It is obviously fulfilled that ifY Ê,fis single-valued function.
Theorem 2.3. Under the framework ofTheorem 2.2, one has a weak solutionxof (SEP)I withs ∈
Tx. In addition, ifY Ê, C Ê, andK is compact,Txis convex, the mapping s, x →
is naturally quasiÊ-convex onTxfor eachx∈K, and the mappingx → fs, x, xisÊ-convex
onKfor eachs∈Tx. Assuming that for eachx∈K, there existstx∈Txsuch that
Minftx, x, x≥CMin
x∈K
Maxw
s∈Tx
fs, x, x,
thenxis a strong solution of (SEP)I; that is, there existss∈Txsuch that
fs, x, x/≤intC{0} 2.13
for allx∈K. Furthermore, the set of all strong solutions of (SEP)Iis compact.
Proof. FromTheorem 2.2, we know thatx∈Ksuch that1.1holds for allx∈Kand for some
s∈Tx. Then we have Minx∈KMaxs∈Txfs, x, x≥C0.
From condition and the convexity of Tx, Lemma 1.5 tells us that
Maxs∈TxMinwx∈Kfs, x, x≥C0. Then there is ans ∈ Txsuch that Minwx∈Kfs, x,
x≥C0. Thus for allρ∈x∈Kfs, x, x, we haveρ≥C0. Hence there existss∈Txsuch that
fs, x, x/≤intC{0} 2.14
for allx∈K. Such anxis a strong solution ofSEPI.
Finally, to see that the solution set ofSEPI is compact, it is sufficient to show that the solution set is closed due to the coercivity condition4ofTheorem 2.2. To this end, letΓ denote the solution set ofSEPI. Suppose that net{xα} ⊂ Γwhich converges to somep. Fix anyy∈K. For eachα, there is ansα∈Txαsuch that
fsα, xα, y≤/intC{0}. 2.15
SinceT is upper semicontinuous with compact values and the set{xα} ∪ {p}is compact, it follows thatT{xα} ∪ {p}is compact. Therefore without loss of generality, we may assume that the sequence{sα}converges to some s. Then s ∈ Tp and fsα, xα, y/⊂ −intC. Let Ω {s, x∈z∈KTz×K :fs, x, y⊂ −intC}. Since the mappings, x → fs, x, yis upper semicontinuous with nonempty compact values, the setΩis open inz∈KTz×K. Hencez∈KTz×K\Ωis closed inz∈KTz×K. By the factssα, xα∈z∈KTz×K\Ω andsα, xα→αs, p, we haves, p∈z∈KTz×K\Ω. This implies thatfs, p, y/⊂ −intC. We then obtain
fs, p, y/≤intC{0}. 2.16
Hencep∈ΓandΓis closed.
We would like to point out that condition is fulfilled if we takeY Êandfis a
single-valued function. The following is a concrete example for both Theorems2.1and2.3.
Example 2.4. Let X Y Ê, Z LX, Y,K 1,2,C Ê, andD 1,2. ChooseT :
TK×K×K → 2Y is defined byfs, x, y {aδxy−x : δ ∈ 0,1}, wherex ∈K,
s∈Txwithsax, for somea∈1,2,y∈K, andν:K×K → 2Y is defined by
νx, y
⎧ ⎨ ⎩
xy−x, y≥x,
3xy−x, y≤x.
2.17
Then all conditions of Theorems 2.1 and 2.3 are satisfied. By Theorems 2.1 and 2.3,
respectively, theSEPInot only has a weak solution, but also has a strong solution. A simple geometric discussion tells us thatx1 is a strong solution forSEPI.
Corollary 2.5. Under the framework ofTheorem 2.1, one has a weak solutionxof (SEP)I withs ∈
Tx. In addition, ifY ÊandCÊ,Kis compact,Txis convex,Ê-convex onTxfor each
x∈Kand the mappingx → fs, x, xisÊ-convex onKfor eachs∈Tx,f:Z×K×K → 2 Y
such thats, x → fs, x, yis continuous with nonempty compact values for each y ∈ K, and
T :K → 2Zis upper semicontinuous with nonempty compact values. Assume that condition holds, thenxis a strong solution of (SEP)I; that is, there existss∈Txsuch that
fs, x, x/≤intC{0} 2.18
for allx∈K. Furthermore, the set of all strong solutions of (SEP)Iis compact.
Theorem 2.6. Let X, Y, Z, K,C, T, f be as in Theorem 2.1. Assume that the mapping y →
fs, x, yisC-convex onKfor eachx∈Kands∈Txsuch that
1for eachx∈K, there is ans∈Txsuch thatfs, x, x/≤intC{0};
2there is a nonempty compact convex subsetDofK, such that for everyx∈K\D, there is ay∈Dsuch that for alls∈Tx,
fs, x, y≤intC{0}, 2.19
3for eachy∈K, the set{x∈K :fs, x, y≤intC{0}for alls∈Tx}is open inK.
Then there is anx∈Kwhich is a weak solution of (SEP)I.
Proof. For any given nonempty finite subsetN ofK. LettingDN coD∪N, thenDNis a nonempty compact convex subset ofK. DefineS:DN → 2DN as in the proof ofTheorem 2.1, and for eachy∈K, let
Ωyx∈D:fs, x, y/≤intC{0}for somes∈Tx. 2.20
We note that for eachx ∈ DN,Sxis nonempty and closed sincex ∈ Sxby conditions
1 and 3. For each y ∈ K,Ωy is compact in D. Next, we claim that the mappingS
is a KKM mapping. Indeed, if not, there is a nonempty finite subset M of DN, such that
coM /⊂x∈MSx. Then there is anx∗∈coM⊂DNsuch that
fs, x∗, x≤
for allx∈Mands∈Tx∗. Since the mapping
x−→fs, x∗, x 2.22
isC-convex onDN, we can deduce that
fs, x∗, x∗≤
intC{0} 2.23
for all s ∈ Tx∗. This contradicts condition 1. Therefore, S is a KKM mapping, and by Fan’s lemma, we havex∈DNSx/∅. Note that for anyu∈x∈DNSx, we haveu∈Dby condition2. Hence, we have
y∈N
Ωy
y∈N
Sy∩D /∅, 2.24
for each nonempty finite subset N of K. Therefore, the whole intersection y∈KΩy is nonempty. Letx∈y∈KΩy. Thenxis a solution ofSEPI.
Corollary 2.7. Let X, Y, Z, K, C,T, f be as in Theorem 2.1. Assume that the mappingy →
fs, x, y is C-convex onK for eachx ∈ K and s ∈ Tx, f : Z ×K×K → 2Y such that
s, x → fs, x, yis continuous with nonempty compact values for eachy∈K, andT :K → 2Z
is upper semicontinuous with nonempty compact values. Suppose that
1for eachx∈K, there is ans∈Txsuch thatfs, x, x/≤intC{0};
2there is a nonempty compact convex subsetDofK, such that for everyx∈K\D, there is ay∈Dsuch that for alls∈Tx,
fs, x, y≤intC{0}. 2.25
Then there is anx∈Kwhich is a weak solution of (SEP)I.
Proof. Using the technique of the proof inTheorem 2.2and applyingTheorem 2.6, we have the conclusion.
The following result is another existence theorem for the strong solutions ofSEPI. We need to combineTheorem 2.6and use the technique of the proof inTheorem 2.3.
Theorem 2.8. Under the framework ofTheorem 2.6, on has a weak solutionxof (SEP)Iwiths∈Tx.
In addition, ifY ÊandCÊ,Kis compact,Txis convex and the mappings → −fs, x, x
is naturally quasi C-convex onTxfor eachx ∈ K,f : Z×K×K → 2Y such thats, x →
fs, x, yis continuous with nonempty compact values for eachy ∈K, andT :K → 2Zis upper semicontinuous with nonempty compact values. Assuming that condition holds, thenxis a strong solution of (SEP)I; that is, there existss∈Txsuch that
for allx∈K. Furthermore, the set of all strong solutions of (SEP)Iis compact.
Using the technique of the proof inTheorem 2.3, we have the following result.
Corollary 2.9. Under the framework of Corollary 2.7, one has a weak solution x of (SEP)I with
s ∈ Tx. In addition, if Y Ê and C Ê, K is compact, Tx is convex, and the mapping
s → −fs, x, xis naturally quasi C-convex on Txfor eachx ∈ K. Assuming that condition holds, thenxis a strong solution of (SEP)I; that is, there existss∈Txsuch that
fs, x, x/≤intC{0} 2.27
for allx∈K. Furthermore, the set of all strong solutions of (SEP)Iis compact.
Next, we discuss the existence results of the strong solutions forSEPIwith the setK without compactness setting from Theorems2.10to2.14below.
Theorem 2.10. Letting X be a finite-dimensional real Banach space, under the framework of
Theorem 2.1, one has a weak solutionxof (SEP)I withs∈Tx. In addition, ifY ÊandCÊ,
Txis convex,fs, x, x {0}for alls ∈Txand for allx∈K, the mappingy → fs, x, yis
C-convex onKfor eachx∈K ands∈ Txand the mappings → −fs, x, xis naturally quasi
C-convex onTxfor eachx∈K,f :Z×K×K → 2Y such thats, x → fs, x, yis continuous for eachy∈K, andT :K → 2Zis upper semicontinuous with nonempty compact values. Assume that for somer >x, such that for eachx∈Kr, there is atx ∈Txsuch that the condition
Minftx, x, x≥CMin
x∈Kr
Maxw
s∈Tx
fs, x, x
is satisfied, whereKr . B0, r∩K. Thenxis a strong solution of (SEP)I; that is, there existss∈Tx
such that
fs, x, x/≤intC{0} 2.28
for allx∈K. Furthermore, the set of all strong solutions of (SEP)Iis compact.
Proof. Let us chooser >xsuch that condition holds. LettingB0, r {x∈X :x ≤
r}, then the setKr is nonempty and compact inX. We replaceKbyKr inTheorem 2.3; all conditions ofTheorem 2.3hold. Hence byTheorem 2.3, we haves∈Txsuch that
fs, x, z/≤intC{0} 2.29
for allz ∈ Kr. For anyx ∈K, choose t ∈ 0,1small enough such that1−txtx ∈ Kr. Puttingz 1−txtxin2.29, we have
We note that
fs, x,1−txtx≤C1−tfs, x, x tfs, x, x tfs, x, x 2.31
which implies that
fs, x, x/≤intC{0} 2.32
for allx∈K. This completely proves the theorem.
Corollary 2.11. Letting X be a finite-dimensional real Banach space, under the framework of
Theorem 2.2, one has a weak solutionxof (SEP)I withs∈Tx. In addition, ifY ÊandCÊ,
Txis convex,fs, x, x {0}for alls ∈Txand for allx∈K, the mappingy → fs, x, yis
C-convex onK for eachx∈Kands∈Tx, and the mappings → −fs, x, xis naturally quasi
C-convex onTxfor eachx∈K. Assume that for somer >x, condition holds. Thenxis a strong solution of (SEP)I; that is, there existss∈Txsuch that
fs, x, x/≤intC{0} 2.33
for all x∈K. Furthermore, the set of all strong solutions of (SEP)Iis compact.
Using a similar argument to that of the proof in Theorem 2.10 and combining
Theorem 2.6and Corollary 2.7, respectively, we have the following two results of existence for the strong solution ofSEPI.
Theorem 2.12. LetXbe a finite-dimensional real Banach space, under the framework ofTheorem 2.6, one has a weak solutionxof (SEP)Iwiths∈Tx. In addition, ifY ÊandCÊ,Txis convex,
fs, x, x {0}for alls∈Txand for allx∈K, the mappings → −fs, x, xis naturally quasi
C-convex onTxfor eachx∈K,f :Z×K×K → 2Y such thats, x → fs, x, yis continuous for eachy∈K, andT :K → 2Zis upper semicontinuous with nonempty compact values. Assume that for somer >x, condition holds. Thenxis a strong solution of (SEP)I; that is, there exists
s∈Txsuch that
fs, x, x/≤intC{0} 2.34
for allx∈K. Furthermore, the set of all strong solutions of (SEP)Iis compact.
In order to illustrate Theorems2.10and2.12more precisely, we provide the following concrete example.
is defined byfs, x, y {sxy−x:s∈Tx}, wherex∈K,y∈K, andν:K×K → 2Y is defined by
νx, y
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x2y−x
2
, y≥x,
x2y−x, y≤x.
2.35
We claim that condition holds. Indeed, We know that the weak solutionx1. For each
x ∈ Kr 1, r∩1,2, if we choose anytx ∈ T1, then Minftx, x, x Min{txx−1}
{txx− 1} and Minx∈KrMaxw
s∈Txfs, x, x Minx∈1,r∩1,2Maxw{sx− 1 : s ∈
1/2,1} Minx∈1,r∩1,2{x−1} Min0, r −1 ∩0,1 {0}. Hence condition and all other conditions of Theorems2.10and2.12are satisfied. By Theorems2.10and2.12, respectively, theSEPI not only has a weak solution, but also has a strong solution. We can see thatx1 is a strong solution forSEPI.
Theorem 2.14. Letting X be a finite-dimensional real Banach space, under the framework of
Corollary 2.7, one has a weak solutionxof (SEP)I withs∈Tx. In addition, ifY ÊandCÊ,
Txis convex,fs, x, x {0}for alls∈Txand for allx∈K, and the mappings → −fs, x, x is naturally quasiC-convex onTxfor eachx∈K. Assume that for somer > x, condition holds. Thenxis a strong solution of (SEP)I; that is, there existss∈Txsuch that
fs, x, x/≤intC{0} 2.36
for allx∈K. Furthermore, the set of all strong solutions of (SEP)Iis compact.
We would like to point out an open question naturally arising fromTheorem 2.3: is
Theorem 2.3extendable to the case of Y Ê
p or more general spaces, such as Hausdorff
topological vector spaces?
Acknowledgments
The authors would like to thank the referees whose remarks helped improving the paper. This work was partially supported by Grant no. 98-Edu-Project7-B-55 of Ministry of Education of
TaiwanRepublic of Chinaand Grant no. NSC98-2115-M-039-001- of the National Science
Council of TaiwanRepublic of Chinathat are gratefully acknowledged.
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