Volume 2010, Article ID 968271,13pages doi:10.1155/2010/968271
Research Article
Existence of Solutions for
η
-Generalized Vector
Variational-Like Inequalities
Xi Li,
1Jong Kyu Kim,
2and Nan-Jing Huang
1, 31Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2Department of Mathematics, Kyungnam University, Masan, Kyungnam 631-701, South Korea 3Science & Technology Finance and Mathematical Finance Key Laboratory of Sichuan Province,
Sichuan University, Sichuan 610064, China
Correspondence should be addressed to Nan-Jing Huang,nanjinghuang@hotmail.com
Received 22 October 2009; Accepted 9 December 2009
Academic Editor: Yeol Je Cho
Copyrightq2010 Xi Li et al. 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 introduce and study a class ofη-generalized vector variational-like inequalities and a class ofη -generalized strong vector variational-like inequalities in the setting of Hausdorfftopological vector spaces. An equivalence result concerned with two classes ofη-generalized vector variational-like inequalities is proved under suitable conditions. By using FKKM theorem, some new existence results of solutions for theη-generalized vector variational-like inequalities andη-generalized strong vector variational-like inequalities are obtained under some suitable conditions.
1. Introduction
Vector variational inequality was first introduced and studied by Giannessi1in the setting of finite-dimensional Euclidean spaces. Since then, the theory with applications for vector variational inequalities, vector complementarity problems, vector equilibrium problems, and vector optimization problems have been studied and generalized by many authorssee, e.g.,
2–15and the references therein.
Recently, Yu et al. 16considered a more general form of weak vector variational inequalities and proved some new results on the existence of solutions of the new class of weak vector variational inequalities in the setting of Hausdorfftopological vector spaces.
Very recently, Ahmad and Khan 17 introduced and considered weak vector variational-like inequalities with η-generally convex mapping and gave some existence results.
In 2008, Lee et al. 20 introduced a new class of strong vector variational-type inequalities in Banach spaces. They obtained the existence theorems of solutions for the inequalities without monotonicity in Banach spaces by using Brouwer fixed point theorem and Browder fixed point theorem.
Motivated and inspired by the work mentioned above, in this paper we introduce and study a class ofη-generalized vector variational-like inequalities and a class ofη-generalized strong vector variational-like inequalities in the setting of Hausdorff topological vector spaces. We first show an equivalence theorem concerned with two classes ofη-generalized vector variational-like inequalities under suitable conditions. By using FKKM theorem, we prove some new existence results of solutions for theη-generalized vector variational-like inequalities andη-generalized strong vector variational-like inequalities under some suitable conditions. The results presented in this paper improve and generalize some known results due to Ahmad and Khan17, Lee et al. 20, and Yu et al. 16.
2. Preliminaries
LetXandY be two real Hausdorfftopological vector spaces,K⊂Xa nonempty, closed, and convex subset, andC⊂Y a closed, convex, and pointed cone with apex at the origin. Recall that the Hausdorfftopological vector space Y is said to an ordered Hausdorfftopological vector space denoted byY, Cif ordering relations are defined inY as follows:
∀x, y∈Y, x≤y⇐⇒y−x∈C,
∀x, y∈Y, x/≤y⇐⇒y−x /∈C. 2.1
If the interior intCis nonempty, then the weak ordering relations inY are defined as follows:
∀x, y∈Y, x < y⇐⇒y−x∈intC,
∀x, y∈Y, x/<y⇐⇒y−x /∈intC. 2.2
Let LX, Y be the space of all continuous linear maps from X to Y and T : X →
LX, Y. We denote the value ofl ∈ LX, Yonx ∈ X byl, x. Throughout this paper, we assume thatCx : x ∈ K is a family of closed, convex, and pointed cones ofY such that intCx/∅for allx∈K,ηis a mapping fromK×KintoX, andfis a mapping fromK×K intoY.
In this paper, we consider the following two kinds of vector variational inequalities:
η-Generalized Vector Variational-Like Inequalityfor short,η-GVVLI: for eachz ∈K andλ∈0,1, findx∈Ksuch that
Tλx 1−λz, ηy, xfy, x/∈ −intCx, ∀y∈K, 2.3
η-Generalized Strong Vector Variational-Like Inequalityfor short,η-GSVVLI: for each
z∈Kandλ∈0,1, findx∈Ksuch that
η-GVVLI and η-GSVVLI encompass many models of variational inequalities. For example, the following problems are the special cases ofη-GVVLI andη-GSVVLI.
1Iffy, x 0 andCx Cfor allx, y∈K, thenη-GVVLI reduces to findingx∈K, such that for eachz∈K,λ∈0,1,
Tλx 1−λz, ηy, x/∈ −intC, ∀y∈K, 2.5
which is introduced and studied by Ahmad and Khan17. In addition, ifηy, x y−xfor eachx, y∈K, thenη-GVVLI reduces to the following model studied by Yu et al.16.
Findx∈Ksuch that for eachz∈K,λ∈0,1,
Tλx 1−λz, y−x/∈ −intC, ∀y∈K. 2.6
2Ifλ1 andCx Cfor allx∈K, thenη-GSVVLI is equivalent to the following vector variational inequality problem introduced and studied by Lee et al.20.
Findx∈Ksatisfying
Tx, ηy, xfy, x/∈ −C\ {0}, ∀y∈K. 2.7
For our main results, we need the following definitions and lemmas.
Definition 2.1. Let T : K → LX, Y and η : K ×K → K be two mappings and C
x∈KCx/∅.T is said to beη-monotone inCif and only if
Tx−Ty, ηx, y∈C, ∀x, y∈K. 2.8
Definition 2.2. LetT :K → LX, Yandη :K×K → Kbe two mappings. We say thatT is
η-hemicontinuous if, for any givenx, y, z∈Kandλ∈0,1, the mappingt→ Tλx 1−
ty−x 1−λz, ηy, xis continuous at 0.
Definition 2.3. A multivalued mapping A : X → 2Y is said to be upper semicontinuous onX if, for all x ∈ X and for each open setG in Y with Ax ⊂ G, there exists an open neighbourhoodOxofx∈Xsuch thatAx⊂Gfor allx∈Ox.
Lemma 2.4see21. LetY, Cbe an ordered topological vector space with a closed, pointed, and convex coneCwithintC /∅. Then for anyy, z∈Y, we have
1y−z∈intCandy /∈intCimplyz /∈intC; 2y−z∈Candy /∈intCimplyz /∈intC;
3y−z∈ −intCandy /∈ −intCimplyz /∈ −intC; 4y−z∈ −Candy /∈ −intCimplyz /∈ −intC.
Lemma 2.6see23. LetXbe a Hausdorfftopological space,A1, A2, . . . , Anbe nonempty compact
convex subsets ofX. Thenconvni1Aiis compact.
Lemma 2.7 see 24. Let X and Y be two topological spaces. If A : X → 2Y is upper semicontinuous with closed values, thenAis closed.
3. Main Results
Theorem 3.1. LetXbe a Hausdorfftopological linear space,K ⊂Xa nonempty, closed, and convex subset, andY, Cxan ordered topological vector space with intCx/∅ for allx ∈ K. Letη :
K×K → Xandf:K×K → Xbe affine mappings such thatηx, x fx, x 0for eachx∈K. LetT :K → LX, Ybe anη-hemicontinuous mapping. IfCx∈KCx/∅andTisη-monotone inC,then for eachz∈K,λ∈0,1, the following statements are equivalent
ifindx0∈K, such thatTzx0, ηy, x0fy, x0/∈ −intCx0, for ally∈K;
iifindx0∈K, such thatTzy, ηy, x0fy, x0/∈ −intCx0, for ally∈K,
whereTzis defined byTzx Tλx 1−λzfor allx∈K.
Proof. Suppose thatiholds. We can findx0∈K, such that
Tzx0, ηy, x0
fy, x0
/
∈ −intCx0, ∀y∈K. 3.1
SinceT isη-monotone, for eachx, y∈K,we have
Tλy 1−λz−Tλx 1−λz, ηλy 1−λz, λx 1−λz∈C. 3.2
On the other hand, we knowηis affine andηx, x 0. It follows that
Tzy−Tzx, ηy, x
1 λ
Tλy 1−λz−Tλx 1−λz, ηλy 1−λz, λx 1−λz∈C.
3.3
HenceTzis alsoη-monotone. That is
Tzx0, ηy, x0
−Tzy, ηy, x0
∈ −C, ∀y∈K. 3.4
SinceCx∈KCx, for ally∈K, we obtain
Tzx0, ηy, x0
fy, x0
−Tzy, ηy, x0
−fy, x0
∈ −C⊂ −Cx0. 3.5
ByLemma 2.4,
Tzy, ηy, x0
fy, x0
/∈ −intCx0, ∀y∈K, 3.6
Conversely, suppose thatiiholds. Then there existsx0∈Ksuch that
Tzy, ηy, x0
fy, x0
/∈ −intCx0, ∀y∈K. 3.7
For eachy∈K,t∈0,1, we letytty 1−tx0. Obviously,yt∈K. It follows that
Tzyt, ηyt, x0
fyt, x0
/
∈ −intCx0. 3.8
Sincefandηare affine andηx0, x0 fx0, x0 0, we have
Tλty 1−tx0
1−λz, tηy, x0
tfy, x0
/
∈ −intCx0. 3.9
That is
Tλx0t
y−x0
1−λz, ηy, x0
fy, x0
/
∈ −intCx0. 3.10
Considering theη-hemicontinuity ofT and lettingt → 0, we have
Tzx0, ηy, x0
fy, x0
/
∈ −intCx0, ∀y∈K. 3.11
This completes the proof.
Remark 3.2. If Cx Cand fy, x 0 for all x, y ∈ K, thenTheorem 3.1is reduced to Lemma 5 of17.
LetKbe a closed convex subset of a topological linear spaceX,and{Cx:x∈K}a family of closed, convex, and pointed cones of a topological spaceY such that intCx/∅for allx∈K. Throughout this paper, we define a set-valued mappingC:K → 2Y as follows:
Cx Y \ {−intCx}, ∀x∈K. 3.12
Theorem 3.3. LetX be a Hausdorfftopological linear space,K ⊂ X a nonempty, closed, compact, and convex subset, andY, Cxan ordered topological vector space withintCx/∅for allx∈K. Letη : K×K → X andf : K×K → X be affine mappings such thatηx, x fx, x 0
for eachx∈K. LetT :K → LX, Ybe anη-hemicontinuous mapping. Assume that the following conditions are satisfied
iCx∈KCx/∅andT isη-monotone inC;
iiC:K → 2Y is an upper semicontinuous set-valued mapping.
Then for eachz∈K,λ∈0,1, there existx0∈Ksuch that
Tλx0 1−λz, η
y, x0
fy, x0
/
Proof. For eachy∈K, we denoteTzx Tλx 1−λz,and define
F1
yx∈K:Tzx, ηy, xfy, x/∈ −intCx ,
F2
yx∈K:Tzy, ηy, xfy, x/∈ −intCx . 3.14
ThenF1yandF2yare nonempty sincey∈F1yandy∈F2y. The proof is divided into the following three steps.
IFirst, we prove the following conclusion:F1 is a KKM mapping. Indeed, assume
thatF1is not a KKM mapping; then there existu1, u2, . . . , um ∈K, t1 ≥ 0, t2 ≥0, . . . , tm ≥ 0 withmi1ti 1 andwmi1tiuisuch that
w /∈m
i1
F1ui, i1,2, . . . , m. 3.15
That is,
∀i1,2, . . . , m, Tzw, ηui, wfui, w∈ −intCw. 3.16
Sinceηandfare affine, we have
Tzw, ηw, wfw, w
Tzw, η
m
i1 tiui, w
f
m
i1 tiui, w
m
i1
tiTzw, ηui, wfui, w∈ −intCw.
3.17
On the other hand, we know ηw, w fw, w 0. Then we have 0
Tzw, ηw, wfw, w ∈ −intCw. It is impossible and soF1 : K → 2K is a KKM
mapping.
IIFurther, we prove that
y∈K
F1
y
y∈K
F2
y. 3.18
In fact, ifx∈F1y, thenTzx, ηy, xfy, x/∈intCx.From the proof ofTheorem 3.1, we know thatTzisη-monotone inCz. It follows that
Tzy−Tzx, ηy, x∈C, 3.19
and so
ByLemma 2.4, we have
Tzy, ηy, xfy, x/∈ −intCx, 3.21
and sox∈F2yfor eachy∈K. That is,F1y⊂F2yand so
y∈K
F1
y⊂
y∈K
F2
y. 3.22
Conversely, suppose thatx∈y∈KF2y.Then
Tzy, ηy, xfy, x/∈ −intCx, ∀y∈K. 3.23
It follows fromTheorem 3.1that
Tzx, ηy, xfy, x/∈ −intCx, ∀y∈K. 3.24
That is,x∈y∈KF1yand so
y∈K
F2
y⊂
y∈K
F1
y, 3.25
which implies that
y∈K
F1
y
y∈K
F2
y. 3.26
IIILast, we prove thaty∈KF2y/∅.Indeed, sinceF1is a KKM mapping, we know
that, for any finite set{y1, y2, . . . , yn} ⊂K,one has
convy1, y2, . . . , yn ⊂ n
i1 F1
yi⊂ n
i1 F2
yi. 3.27
This shows thatF2is also a KKM mapping.
Now, we prove thatF2yis closed for ally∈K. Assume that there exists a net{xα} ⊂
F2ywithxα → x∈K. Then
Tzy, ηy, xαfy, xα/∈ −intCxα. 3.28
Using the definition ofC, we have
Sinceηandfare continuous, it follows that
Tzy, ηy, xαfy, xα−→Tzy, ηy, xfy, x. 3.30
SinceCis upper semicontinuous mapping with close values, byLemma 2.7, we know thatC is closed, and so
Tzy, ηy, xfy, x∈Cx. 3.31
This implies that
Tzy, ηy, xfy, x/∈ −intCx, 3.32
and so F2y is closed. Considering the compactness of K and closeness of F2y ⊂ K, we know thatF2yis compact. ByLemma 2.5, we havey∈KF2y/∅,and it follows that
y∈KF1y/∅, that is, for eachz∈Kandλ∈0,1,there existsx0∈Ksuch that
Tλx0 1−λz, η
y, x0
fy, x0
/
∈ −intCx0, ∀y∈K. 3.33
Thus,η-GVVLI is solvable. This completes the proof.
Remark 3.4. The conditioniiinTheorem 3.3can be found in several papers see, e.g.,25,
26.
Remark 3.5. IfCx Candfy, x 0 for allx, y ∈ KinTheorem 3.3, then conditionii holds and conditioniis equivalent to theη-monotonicity ofT. Thus, it is easy to see that
Theorem 3.3is a generalization of17, Theorem 6.
In the above theorem, K is compact. In the following theorem, under some suitable conditions, we prove a new existence result of solutions forη-GVVLI without the compactness ofK.
Theorem 3.6. LetXbe a Hausdorfftopological linear space,K ⊂Xa nonempty, closed, and convex subset, and Y, Cx be an ordered topological vector space with intCx/∅ for all x ∈ K. Let η : K×K → X and f : K×K → X be affine mappings such thatηx, x fx, x 0for eachx ∈ K. LetT : K → LX, Ybe anη-hemicontinuous mapping. Assume that the following conditions are satisfied:
iC⊂x∈KCx/∅andT isη-monotone inC;
iiC:K → 2Y is an upper semicontinuous set-valued mapping;
iiithere exists a nonempty compact and convex subsetDofKand for eachz∈K,λ∈0,1,
x∈K\D, there existy0 ∈Dsuch that
Tλy0 1−λz
, ηy0, x
fy0, x
∈ −intCy0
Then for eachz∈K,λ∈0,1, there existx0∈Dsuch that
Tλx0 1−λz, η
y, x0
fy, x0
/
∈ −intCx0, ∀y∈K. 3.35
Proof. ByTheorem 3.1, we know that the solution set of the problemiiinTheorem 3.1is equivalent to the solution set of following variational inequality: findx∈K, such that
Tλy 1−λz, ηy, xfy, x/∈ −intCx, ∀y∈K. 3.36
For eachz∈Kandλ∈0,1,we denoteTzx Tλx 1−λz.LetG:K → 2Dbe defined as follows:
Gyx∈D:Tzy, ηy, xfy, x/∈ −intCx , ∀y∈K. 3.37
Obviously, for eachy∈K,
Gyx∈K:Tzy, ηy, xfy, x/∈ −intCx ∩D. 3.38
Using the proof ofTheorem 3.3, we obtain thatGyis a closed subset ofD. Considering the compactness ofDand closedness ofGy, we know thatGyis compact.
Now we prove that for any finite set {y1, y2, . . . , yn} ⊂ K, one has ni1Gyi/∅.
Let Yn ni1{yi}. Since Y is a real Hausdorff topological vector space, for each yi ∈
{y1, y2, . . . , yn},{yi}is compact and convex. LetNconvD∪Yn. ByLemma 2.6, we know
thatNis a compact and convex subset ofK. LetF1, F2:N → 2Nbe defined as follows:
F1
yx∈N:Tzx, ηy, xfy, x/∈ −intCx , ∀y∈N;
F2
yx∈N:Tzy, ηy, xfy, x/∈ −intCx , ∀y∈N.
3.39
Using the proof ofTheorem 3.3, we obtain
y∈N
F1
y
y∈N
F2
y/∅, 3.40
and so there existsy0∈
y∈NF2y.
Next we prove thaty0 ∈ D. In fact, ify0 ∈ K\D,then the assumption implies that
there existsu∈Dsuch that
Tλu 1−λz, ηu, y0
fu, y0
∈ −intCu, 3.41
which contradictsy0∈F2uand soy0∈D.
Considering the compactness ofGyfor eachy∈K, we know that there existsx0∈Dsuch
thatx0 ∈
y∈KGy/∅.Therefore, the solution set ofη-GVVLI is nonempty. This completes the proof.
In the following, we prove the solvability ofη-GSVVLI under some suitable conditions by using FKKM theorem.
Theorem 3.7. LetXbe a Hausdorfftopological linear space,K ⊂Xa nonempty, closed, and convex set, and Y, Cxan ordered Hausdorff topological vector space withintCx/∅ for allx ∈ K. Assume that for eachy ∈K, x → ηx, yandx → fx, yare affine,ηx, y ηy, x 0,and fx, y fy, x 0for allx∈K. LetT :K → LX, Ybe a mapping such that
ifor eachz, y ∈ K,λ ∈0,1,the set{x∈ K :Tλx 1−λz, ηy, xfy, x ∈
−Cx\ {0}}is open inK;
iithere exists a nonempty compact and convex subsetDofKand for eachz∈K,λ∈0,1,
x∈K\D,there existsu∈Dsuch that
Tλx 1−λz, ηu, xfy, x∈ −Cx\ {0}. 3.42
Then for eachz∈K,λ∈0,1,there existsx0∈Ksuch that
Tλx0 1−λz, η
y, x0
fy, x0
/
∈ −Cx0\ {0}, ∀y∈K. 3.43
Proof. For eachz∈Kandλ∈0,1,we denoteTzx Tλx 1−λz. LetG:K → 2Dbe defined as follows:
Gyx∈D:Tzx, ηy, xfy, x/∈ −Cx\ {0} , ∀y∈K. 3.44
Obviously, for eachy∈K,
Gyx∈K:Tzx, ηy, xfy, x/∈ −Cx\ {0} ∩D. 3.45
SinceGyis a closed subset ofD, considering the compactness ofDand closedness ofGy, we know thatGyis compact.
Now we prove that for any finite set {y1, y2, . . . , yn} ⊂ K, one has ni1Gyi/∅.
Let Yn ni1{yi}. Since Y is a real Hausdorff topological vector space, for each yi ∈ {y1, y2, . . . , yn},{yi}is compact and convex. LetNconvD∪Yn. ByLemma 2.6, we know thatNis a compact and convex subset ofK.
LetF:N → 2Nbe defined as follows:
We claim thatFis a KKM mapping. Indeed, assume thatFis not a KKM mapping. Then there existu1, u2, . . . , um∈K,t1≥0, t2≥0, . . . , tm≥0 withmi1ti1 andw
m
i1tiuisuch that
w /∈m
i1
Fui, i1,2, . . . , m. 3.47
That is,
∀i1,2, . . . , m Tzw, ηui, wfui, w∈ −Cw\ {0}. 3.48
Sinceηandfare affine, we have
Tzw, ηw, wfw, w
Tzw, η
m
i1 tiui, w
f
m
i1 tiui, w
m
i1
tiTzw, ηui, wfui, w∈ −Cw\ {0}.
3.49
On the other hand, we knowηw, w fw, w 0,and so
0Tzw, ηw, wfw, w∈ −Cw\ {0}, 3.50
which is impossible. Therefore,F:N → 2Nis a KKM mapping.
SinceFyis a closed subset ofN, it follows thatFyis compact. ByLemma 2.5, we have
y∈N
Fy/∅. 3.51
Thus, there existsy0∈y∈NFy.
Next we prove thaty0 ∈D. In fact, ify0 ∈N\D,then the conditioniiimplies that
there existsu∈Dsuch that
Tλy0 1−λz
, ηu, y0
fu, y0
∈ −Cy0
\ {0}, 3.52
which contradictsy0∈Fuand soy0∈D.
Since{y1, y2, . . . , yn} ⊂NandGyi Fyi∩Dfor eachyi ∈ {y1, y2, . . . , yn}, it follows that y0 ∈
n
i1Gyi. Thus, for any finite set {y1, y2, . . . , yn} ⊂ K, we have ni1Gyi/∅.
Considering the compactness ofGyfor each y ∈ K, it is easy to know that there exists
x0∈Dsuch thatx0 ∈y∈KGy/∅.Therefore, for eachz∈K,λ∈0,1,there existsx0 ∈K
such that
Tλx0 1−λz, η
y, x0
fy, x0
/
∈ −Cx0\ {0}, ∀y∈K. 3.53
Remark 3.8. IfKis compact,Cx C, andλ1, thenTheorem 3.7is reduced to Theorem 2.1 in20.
Acknowledgments
The authors greatly appreciate the editor and the anonymous referees for their useful comments and suggestions. This work was supported by the Key Program of NSFCGrant no. 70831005, the Kyungnam University Research Fund 2009, and the Open FundPLN0904 of State Key Laboratory of Oil and Gas Reservoir Geology and Exploitation Southwest Petroleum University.
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