On the Existence Result for System of Generalized Strong Vector Quasiequilibrium Problems

Fixed Point Theory and Applications Volume 2011, Article ID 475121,9pages doi:10.1155/2011/475121

Research Article

On the Existence Result for System of Generalized Strong Vector Quasiequilibrium Problems

Somyot Plubtieng and Kanokwan Sitthithakerngkiet

Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Correspondence should be addressed to Somyot Plubtieng,[email protected] Received 3 December 2010; Accepted 12 January 2011

Academic Editor: Qamrul Hasan Ansari

Copyrightq2011 S. Plubtieng and K. Sitthithakerngkiet. 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 a new type of the system of generalized strong vector quasiequilibrium problems with set-valued mappings in real locally convex Hausdorfftopological vector spaces. We establish an existence theorem by using Kakutani-Fan-Glicksberg fixed-point theorem and discuss the closedness of strong solution set for the system of generalized strong vector quasiequilibrium problem. The results presented in the paper improve and extend the main results of Long et al.

2008.

1. Introduction

The equilibrium problem is a generalization of classical variational inequalities. This problem contains many important problems as special cases, for instance, optimization, Nash equilibrium, complementarity, and fixed-point problemssee1–3and the references therein. Recently, there has been an increasing interest in the study of vector equilibrium problems. Many results on existence of solutions for vector variational inequalities and vector equilibrium problems have been establishedsee, e.g.,4–16.

LetX andZbe real locally convex Hausdorffspace,K ⊂ X a nonempty subset and C⊂Zbe a closed convex pointed cone. LetF:K×K → 2^Zbe a given set-valued mapping.

Ansari et al.17introduced the following set-valued vector equilibrium problemsVEPsto findx∈Ksuch that

F x, y

/

⊆ −intC ∀y∈K, 1.1

or to findx∈Ksuch that

F x, y

⊂C ∀y∈K. 1.2

If intCis nonempty, andxsatisfies1.1, then we callxa weak efficient solution for VEP, and ifxsatisfies1.2, then we callxa strong solution for VEP. Moreover, they also proved an existence theorem for a strong vector equilibrium problem1.2 see17.

In 2000, Ansari et al. 5 introduced the system of vector equilibrium problems SVEPs, that is, a family of equilibrium problems for vector-valued bifunctions defined on a product set, with applications in vector optimization problems and Nash equilibrium problem 11 for vector-valued functions. The SVEP contains system of equilibrium problems, systems of vector variational inequalities, system of vector variational-like inequalities, system of optimization problems and the Nash equilibrium problem for vector- valued functions as special cases. But, by usingSVEP, we cannot establish the existence of a solution to the Debreu type equilibrium problem 7 for vector-valued functions which extends the classical concept of Nash equilibrium problem for a noncooperative game. Moreover, Ansari et al. 18 introduced the following concept of system of vector quasiequilibrium problems.

LetIbe any index set and for eachi∈I, letXibe a topological vector space. Consider a family of nonempty convex subsets{K_i}_i∈I withK_i ⊂ X_i. We denote by K

i∈IK_iand

X

i∈IXi. For eachi ∈ I, let Yi be a topological vector space and letCi : K → 2^Yi and Si :K → 2^Ki be multivalued mappings andFi:K×K → Yibe a bifunction. The system of vector quasiequilibrium problemsSVQEPs, that is, to findx∈Ksuch that for eachi∈I,

x_i∈S_ix:F_i x, y_i

/∈ −intC_ix ∀y_i∈S_ix. 1.3

IfSix Ki for allx ∈ K, thenSVQEPreduces toSVEP see5and if the index set I is singleton, thenSVQEPbecomes the vector quasiequilibrium problem. Many authors studied the existence of solutions for systems ofvectorquasiequilibrium problems, see, for example,19–23and references therein.

On the other hand, it is well known that a strong solution of vector equilibrium problem is an ideal solution, It is better than other solutions such as efficient solution, weak efficient solution, proper efficient solution and supper efficient solutionsee13. Thus, it is important to study the existence of strong solution and properties of the strong solution set.

In general, the ideal solutions do not exist.

Very recently, the generalized strong vector quasiequilibrium problemGSVQEPsis introduced by Long et al.16. LetX,Y, andZare real locally convex Hausdorfftopological vector spaces, K ⊂ X and D ⊂ Y are nonempty compact convex subsets, andC ⊂ Z is a nonempty closed convex cone. LetS:K → 2^K,T :K → 2^D, andF :K×D×K → 2^Zare three set-valued mappings. They considered the GSVQEP: findingx∈K, y∈Txsuch that x∈Sxand

F x, y, z

⊂C, ∀z∈Sx. 1.4

Moreover, they gave an existence theorem for a generalized strong vector quasiequilibrium problem without assuming that the dual of the ordering cone has a weak^∗compact base.

Motivated and inspired by research works mentioned above, in this paper, we introduce a different kind of systems of generalized strong vector quasiequilibrium problem without assuming that the dual of the ordering cone has a weak^∗ compact base. LetX,Y, and Zare real locally convex Hausdorfftopological vector spaces,K ⊂ X and D ⊂ Y are nonempty compact convex subsets, andC ⊂ Zis a nonempty closed convex cone. We also suppose thatS1, S2 :K → 2^K,T1, T2 :K → 2^DandF1, F2 :K×D×K → 2^Zare set-valued mappings. We consider the following system of generalized strong vector quasiequilibrium problemSGSVQEPs: findingx, u∈K×Kandv∈T1x,y∈T2usuch thatx∈S1x, u∈S2usatisfying

F1

x, y, z

⊂C ∀z∈S1x,

F2u, v, z⊂C ∀z∈S2u. 1.5 We call thisx, ua strong solution for theSGSVQEP.

At a quick glance, our required solution seems to be similar to such a thing of Ansari et al.5,18, in the case ofI {1,2}andK1 K2. In fact, however, the main different point comes from the independent choice of coordinate. In this paper, we establish an existence theorem of strong solution set for the system of generalized strong vector quasiequilibrium problem by using Kakutani-Fan-Glicksberg fixed-point theorem and discuss the closedness of the solution set. Moreover, we apply our result to obtain the result of Long et al.16.

2. Preliminaries

Throughout this paper,we suppose that X, Y, and Z are real locally convex Hausdorff topological vector spaces, K ⊂ X and D ⊂ Y are nonempty compact convex subsets, and C ⊂ Z is a nonempty closed convex cone. We also suppose that S1, S2 : K → 2^K, T1, T2:K → 2^D, andF1, F2:K×D×K → 2^Zare set-valued mappings.

Definition 2.1. LetXandY be two topological vector spaces andK a nonempty subset ofX and letF:K → 2^Y be a set-valued mapping.

iFis called upperC-continuous atx0∈Kif, for any neighbourhoodUof the origin inY, there is a neighbourhoodV ofx0such that, for allx∈V,

Fx⊂Fx0 U C. 2.1

iiFis called lowerC-continuous atx0 ∈Kif, for any neighbourhoodUof the origin inY, there is a neighbourhoodV ofx0such that, for allx∈V,

Fx0⊂Fx U−C. 2.2

Definition 2.2. LetXandYbe two topological vector spaces andKa nonempty convex subset ofX. A set-valued mappingF : K → 2^Y is said to be properlyC-quasiconvex if, for any x, y∈Kandt∈0,1, we have

eitherFx⊂F

tx 1−ty

C or F y

⊂F

tx 1−ty

C. 2.3

Definition 2.3. LetXandYbe two topological vector spaces, andT:X → 2^Y be a set-valued mapping.

iT is said to be upper semicontinuous at x ∈ X if, for any open setV containing Tx, there exists an open setUcontainingxsuch that, for allt∈U,Tt⊂V;T is said to be upper semicontinuous onXif it is upper semicontinuous at allx∈X.

iiTis said to be lower semicontinuous atx∈Xif, for any open setVwithTx∩V /∅, there exists an open setUcontainingxsuch that, for allt∈U,Tt∩V /∅;Tis said to be lower semicontinuous onXif it is lower semicontinuous at allx∈X.

iiiT is said to be continuous onXif, it is at the same time upper semicontinuous and lower semicontinuous onX.

ivTis said to be closed if the graph, GraphT, ofT, that is, GraphT {x, y:x∈X andy∈Tx}, is a closed set inX×Y.

Lemma 2.4 see 12. Let K be a nonempty compact subset of locally convex Hausdorff vector topology spaceE. IfS : K → 2^Kis upper semicontinuous and for any x ∈ K, Sxis nonempty, convex and closed, then there exists anx^∗∈Ksuch thatx^∗∈Sx^∗.

Lemma 2.5see24. LetXandY be two Hausdorfftopological vector spaces andT :X → 2^Y be a set-valued mapping. Then, the following properties hold:

iifT is closed and TX is compact, then T is upper semicontinuous, where TX

x∈XTxandEdenotes the closure of the setE,

iiifT is upper semicontinuous and for anyx∈X, Txis closed, thenTis closed,

iiiTis lower semicontinuous atx∈Xif and only if for anyy∈Txand any net{xα}, xα → x, there exists a net{yα}such thatyα∈Txαandyα → y.

3. Main Results

In this section, we apply Kakutani-Fan-Glicksberg fixed-point theorem to prove an existence theorem of strong solutions for the system of generalized strong vector quasiequilibrium problem. Moreover, we also prove the closedness of strong solution set for the system of generalized strong vector quasiequilibrium problem.

Theorem 3.1. For eachi {1,2}, letS_i : K → 2^Kbe continuous set-valued mappings such that for anyx ∈ K, S_ix are nonempty closed convex subsets ofK. LetT_i : K → 2^D be upper semi continuous set-valued mappings such that for anyx∈K, Tixare nonempty closed convex subsets ofDandFi:K×D×K → 2^Zbe set-valued mappings satisfy the following conditions:

ifor allx, y∈K×D, F_ix, y, S_ix⊂C,

iifor ally, z∈D×K, Fi·, y, zare properlyC-quasiconvex, iiiF_i·,·,·are upperC-continuous,

ivfor ally∈D, Fi·, y,·are lower−C-continuous.

Then, SGSVQEP has a solution. Moreover, the set of all strong solutions is closed.

Proof. For anyx, y∈K×D, define set-valued mappingsA, B:K×D → 2^Kby A

x, y

a∈S1x:F1

a, y, z

⊂C, ∀z∈S1x , B

x, y

b∈S2x:F2

b, y, z

⊂C, ∀z∈S2x

. 3.1

Step 1. Show thatAx, yandBx, yare nonempty.

For anyx∈K, we note thatS1xandS2xare nonempty. Thus, for anyx, y∈K×D, we haveAx, yandBx, yare nonempty.

Step 2. Show thatAx, yandBx, yare convex subsets ofK.

Leta1, a2 ∈ Ax, yand λ ∈ 0,1. Puta λa1 1−λa2. Sincea1, a2 ∈S1xand S1xis convex set, we havea∈S1x. Byii,F1·, y, zis properlyC-quasiconvex. Without loss of generality, we can assume that

F1

a1, y, z

⊂F1

λa1 1−λa2, y, z

C. 3.2

We claim thata∈Ax, y. In fact, ifa /∈Ax, y, then there existsz^∗∈S1xsuch that F1

a, y, z^∗

/⊆C. 3.3

It follows that F1

a1, y, z^∗

⊂F1

λa1 1−λa2, y, z^∗

C/⊆C C⊂C, 3.4

which contradicts toa1∈Ax, y. Thereforea∈Ax, yand henceAx, yis a convex subset ofK. Similarly, we haveBx, yis convex subset ofK.

Step 3. Show thatAx, yandBx, yare closed subsets ofK.

Let{a_α}be a sequence inAx, ysuch thata_α → a^∗. Thus, we havea_α ∈S1x. Since S1xis a closed subset ofK, it follows thata^∗∈S1x. By the lower semicontinuity ofS1and Lemma 2.5iii, for anyz^∗ ∈S1xand any net{xα} → x, there exists a net{zα}such that z_α∈S1x_αandz_α → z^∗. This implies that

F1

aα, y, zα

⊂C. 3.5

SinceF1·, y,·is lower−C-continuous, for any neighbourhoodUof the origin inZ, there is a subnet{a_β, z_β}of{a_α, z_α}such that

F1

a^∗, y, z^∗

⊂F1

aβ, y, zβ

U C. 3.6

From3.5and3.6, we have

F1

a^∗, y, z^∗

⊂U C. 3.7

We claim thatF1a^∗, y, z^∗⊂C. Assume that there existsp∈F1a^∗, y, z^∗andp /∈C. Thus, we note that 0/∈C−pandC−pis closed. HenceZ\C−pis open and 0∈Z\C−p. SinceZis a locally convex space, there exists a neighbourhoodU0of the origin such thatU0⊂Z\C−p is convex andU0 −U0. This implies that 0/∈U0 C−p, that is, p /∈U0 C, which is a contradiction. ThereforeF1a^∗, y, z^∗ ⊂ C. This mean thata^∗ ∈ Ax, yand so Ax, yis a closed subset ofK. Similarly, we haveBx, yis a closed subset ofK.

Step 4. Show thatAx, yandBx, yare upper semicontinuous.

Let{x_α, y_α : α ∈ I} ⊂ K×D be given such thatx_α, y_α → x, y ∈ K×D, and let a_α ∈ Ax_α, y_αsuch that a_α → a. Sincea_α ∈ S1x_αand S1 is upper semicontinuous, it follows byLemma 2.5ii that a ∈ S1x. We now claim that a ∈ Ax, y. Assume that a /∈Ax, y. Then, there existsz^∗ ∈S1xsuch that

F1

a, y, z^∗

/⊆C, 3.8

which implies that there is a neighbourhoodU0of the origin inZsuch that F1

a, y, z^∗

U0/⊆C. 3.9

SinceF1is upperC-continuous, for any neighbourhoodUof the origin inZ, there exists a neighbourhoodU1ofa, y, z^∗such that

F1

a, y, z

⊂F1

a, y, z^∗

U C, ∀ a,y, z

∈U1. 3.10

Without loss of generality, we can assume thatU0U. This implies that F1

a, y, z

⊂F1

a, y, z^∗

U0 C/⊆C C⊂C, ∀ a,y, z

∈U1. 3.11

Thus there isα0∈Isuch that F1

a_α, y_α, z_α

/⊆C, ∀α≥α0, 3.12

which contradicts toa_α∈Ax_α, y_α. Hencea∈Ax, yand, therefore,Ais a closed mapping.

SinceKis a compact set andAx, yis a closed subset ofK, we note thatAx, yis compact.

Then,Ax, yis also compact. Hence, byLemma 2.5i,Ax, yis an upper semicontinuous mapping. Similarly, we note thatBx, yis an upper semicontinuous mapping.

Step 5. Show that SGSVQEP has a solution.

Define the set-valued mappingHa:K×D → 2^K×DandGb :K×D → 2^K×Dby Ha

x, y

A x, y

, T1a

∀ x, y

∈K×D, G_b

x, y

B x, y

, T2b

∀ x, y

∈K×D. 3.13

Then,HaandGbare upper semicontinuous and, for allx, y∈K×D,Hax, y, andGbx, y are nonempty closed convex subsets ofK×D.

Define the set-valued mappingM:K×D×K×D → 2^K×D×K×Dby

M

x, y ,u, v

Hu

x, y

, Gxu, v , ∀

x, y ,u, v

∈K×D×K×D. 3.14 Then, M is also upper semicontinuous and, for all x, y,u, v ∈ K ×D×K ×D, Mx, y,u, vis a nonempty closed convex subset ofK×D×K×D. ByLemma 2.4, there exists a pointx, y,u, v∈K×D×K×Dsuch thatx, y,u, v∈Mx, y,u, v, that is

x, y

∈Hu x, y

, u, v∈Gxu, v. 3.15 This implies that x ∈ Ax, y, y ∈ T1u,u ∈ Bu, v, and v ∈ T2x. Then, there exists x, u∈K×Kandy∈T1u,v∈T2xsuch thatx∈S1x,u∈S2u,

F1

x, y, z

⊂C, ∀z∈S1x, F2u, v, z⊂C, ∀z∈S2u. 3.16

Hence SGSVQEP has a solution.

Step 6. Show that the set of solutions of SGSVQEP is closed.

Let{xα, uα:α∈I}be a net in the set of solutions of SGSVQEP such thatxα, uα→ x^∗, u^∗. By definition of the set of solutions of SGSVQEP, we note that there existv_α∈T1x_α, yα∈T2uα,xα∈S1xα, anduα∈S2uαsatisfying

F1

xα, yα, z

⊂C, ∀z∈S1xα, F2uα, vα, z⊂C, ∀z∈S2uα. 3.17 SinceS1 and S2 are continuous closed valued mappings, we obtain x^∗ ∈ S1x^∗and u^∗ ∈ S2u^∗. Letvα → v^∗andyα → y^∗. SinceT1 andT2are upper semicontinuous closed valued mappings, it follows byLemma 2.5iithatT1andT2are closed. Thus, we note thatv^∗∈T1x^∗ andy^∗∈T2u^∗. SinceF1·, y^∗,·andF2·, v^∗,·are lower−C-continuous, we have

F1

x^∗, y^∗, z

⊂C, ∀z∈S1x^∗, F2u^∗, v^∗, z⊂C, ∀z∈S2u^∗. 3.18 This means thatx^∗, u^∗belongs to the set of solutions of SGSVQEP. Hence the set of solutions of SGSVQEP is closed set. This completes the proof.

If we takeS S1 S2,F F1 F2, andT T1 T2. Then, fromTheorem 3.1, we derive the following result.

Corollary 3.2. LetS:K → 2^Kbe a continuous set-valued mapping such that for anyx∈K, Sx is nonempty closed convex subset ofK. LetT : K → 2^D be an upper semicontinuous set-valued mapping such that for anyx∈K, Txis a nonempty closed convex subset ofDandF:K×D×K → 2^Zbe set-valued mapping satisfy the following conditions:

ifor allx, y∈K×D, Fx, y, Sx⊂C,

iifor ally, z∈D×K, F·, y, zis properlyC-quasiconvex,

iiiF·,·,·is an upperC-continuous,

ivfor ally∈D, F·, y,·is a lower−C-continuous, vifx∈Sxandu∈SuthenTx Tu.

Then, GSVQEP has a solution. Moreover, the set of all solution of GSVQEP is closed.

Now we give an example to explain thatTheorem 3.1is applicable.

Example 3.3. LetX Y Z R,C 0, ∞, andK D 0,1. For eachx ∈ K, let S1x x,1,S2x 0, xandT1x 1−x,1,T2x x,1. We consider the set-valued mappingsF1, F2:K×D×K → 2^Zdefined by

F1

x, y, z

x−y z, ∞

∀ x, y, z

∈K×D×K, F2

x, y, z

y−x z, ∞

∀ x, y, z

∈K×D×K. 3.19 Then, it is easy to check that all of conditioni–ivinTheorem 3.1are satisfied. Hence, by Theorem 3.1, SGSVQEP has a solution. LetEbe the set of all strong solutions for SGSVQEP.

Then, we note that

E

x, u, y, v

∈K×K×T2u×T1x:x∈S1x, u∈S2usuch that F1

x, y, z

⊂C, ∀z∈S1x, F2u, v, z⊂C, ∀z∈S2u

1/3≤a≤0.5

{a} ×1−a,2a×0,1−a×1−a,1.

3.20

Acknowledgments

The authors would like to thank the referees for the insightful comments and suggestions. S.

Plubtieng the Thailand Research Fund for financial support under Grants no. BRG5280016.

Moreover, K. Sitthithakerngkiet would like to thanks the Office of the Higher Education Commission, Thailand for supporting by grant fund under Grant no. CHE-Ph.D-SW- RG/41/2550, Thailand.

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