On Generalized Implicit Vector Equilibrium Problems in Topological Ordered Spaces

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Volume 2009, Article ID 513408,9pages doi:10.1155/2009/513408

Research Article

On Generalized Implicit Vector Equilibrium

Problems in Topological Ordered Spaces

Qun Luo

Department of Mathematics, Zhaoqing University, Zhaoqing, Guangdong 526061, China

Correspondence should be addressed to Qun Luo,[email protected]

Received 4 June 2009; Revised 29 August 2009; Accepted 30 September 2009

Recommended by Nanjing Jing Huang

We discuss three classes of generalized implicit vector equilibrium problems in topological ordered spaces. Under some conditions, we prove three new existence theorems of solutions for the generalized implicit vector equilibrium problems in topological ordered spaces by using the Fan-Browder fixed point theorem.

Copyrightq2009 Qun Luo. 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 and Preliminaries

It is well known that the vector equilibrium problem is closely related to vector variational inequality, vector optimization problem, and many otherssee, e.g.,1–6and the references therein.

Recently, a large of generalized vector equilibrium problems have been studied in diﬀerent conditions by many authors and a lot of results concerned with the existence of solutions and properties of solutions have been given in finite and infinite dimensional spaces

see7and the references therein.

The main purpose of this paper is to extend some known results for vector equilibrium problems to topological ordered spaces see 8. We discuss three classes of generalized implicit vector equilibrium problems in topological ordered spaces. Under some conditions, we prove three new existence theorems of solutions for the generalized implicit vector equilibrium problems in topological ordered spaces by using the Fan-Browder fixed point theorem.

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casex≤x, the setx, x {y∈X:x≤y≤x}is called an order interval. Now assume that

X,≤is a semilattice andA⊆Xis a nonempty finite subset. Thus, the set

ΔA a∈A

a,supA 1.1

is well defined and it has the following properties:

aA⊆ΔA,

bifA⊆A, thenΔA⊆ΔA.

A subsetE ⊆ X is said to beΔ-convex if, for any nonempty finite subsetA ⊆ E, we haveΔA⊆E.

For anyD⊂X, FDdenotes the family of all finite subsets ofDand

ΔD A∈FDΔ

A. 1.2

LetXbe a topological semilattice,K⊂Xa nonemptyΔ-convex subset,Ya Hausdorﬀ topological vector space. Assume thatA : K → 2K,g : K → 2K,f : K×K → 2Y, and

C :K → 2Y such that, for anyx∈ K,Cxis a closed, pointed, and convex cone inY and intCx/∅.

In this paper, we consider the following three classes generalized implicit vector equilibrium problems:

1 weak generalized implicit vector equilibrium problemWGIVEP: findx∗ ∈ K, such thatx∗∈Ax∗and for anyy∈Ax∗, there exists anu∈gx∗such that

fu, y∩intCx∗ ∅; 1.3

2 strong generalized implicit vector equilibrium problem SGIVEP: find x∗ ∈ K such thatx∗∈Ax∗and

fu, y∩intCx∗ ∅, ∀y∈Ax∗,∀u∈gx∗. 1.4

3uniform generalized implicit vector equilibrium problemUGIVEP: findx∗ ∈K such thatx∗∈Ax∗and there existsu∗∈gx∗such that

fu∗_{, y}_∩_C_x∗ _∅_, _∀_y_∈_A_x∗_. _1.5

Definition 1.1. LetXandY be two topological spaces.

1A mappingF :X → 2Y is called upper semicontinuoususcatx0 ∈Xif, for any

neighborhoodNFx0ofFx0, there exists a neighborhoodNx0ofx0such that

Fx⊂NFx0, ∀x∈Nx0. 1.6

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2A mappingF :X → 2Y is called lower semicontinuouslscatx0 ∈X if, for any

net{xα}inX such thatxα → x0and for anyy0 ∈ Fx0, there existsyα ∈ Fxαsuch that yα → y0.Fis called lsc onXif it is lsc at each point ofX.

3A mappingF : X → 2Y is called complement pseudo-upper semicontinuousc p-uscatx0∈Xif, fory /∈Fxαwithxα → x0, we havey /∈Fx0.Fis called c p-usc onXif

Fis c p-usc at each pointx0ofX.

Remark 1.2. 1By 9, if F is usc with closed values, then for any net{xα} inX such that

xα → x0 and for any net{yα} in Y with yα ∈ Fxα such that yα → y0 in Y, we have

y0∈Fx0.

2IfY\Fis usc with closed values, thenFis c p-usc, where

Y\Fx Y \Fx, ∀x∈X. 1.7

Lemma 1.3see9. LetXandY be two topological spaces. LetXbe compact andF:X → 2Y be usc such thatFxis compact for eachx∈X. ThenFX _x_∈_XFxis compact.

Definition 1.4. Let X be a topological semilattice or a Δ-convex subset of a topological semilattice, letY be a Hausdorﬀtopological vector space, and letC⊂Y be a closed, pointed, and convex cone with intC /∅.

1A mappingF :X → 2Y is called aΔ-convex mappingor aΔ-concave mapping with respect to Cif, for any nonempty finite subset D {x1, x2, . . . , xn} ⊆ X,x ∈ ΔD, zi∈Fxi,ti≥0 withi1,2, . . . , nandn_i₁ti1, there existsz∈Fxsuch that

n

i1

tizi−z∈C

orz−

n

i1

tizi∈C

. 1.8

2A mappingF:X → 2Y is called to haveΔ-inheritance if, for any nonempty finite subsetD {x1, x2, . . . , xn}ofX,x0 ∈ΔD,K ⊂XwithK∩Fxi ∅andi1,2, . . . , n, we

haveK∩Fx0 ∅.

3LetQ⊆X. A mappingG:X×X → 2Yis calledQ-Δ-convexorQ-Δ-concavewith respect toCin second argument if, for any nonempty finite subsetD {x1, x2, . . . , xn} ⊆ X, x∈ΔD,{u0, u1, u2, . . . , un} ⊆Q,zi∈Gui, xi,ti≥0 withi1,2, . . . , nandni1ti1, there

existsz∈Gu0, xsuch that

n

i1

tizi−z∈C

orz−

n

i1

tizi∈C

. 1.9

Remark 1.5. IfG:X×X → 2Y is aQ-Δ-convex mappingor aQ-Δ-concave mappingwith respect toCin second argument, then for anyu∈X,Gu,·:X → 2Yis aΔ-convex mapping

or aΔ-concave mappingwith respect toC.

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2. Existence Theorems

Theorem 2.1. LetKbe a nonempty compactΔ-convex subset of a topological semilattice with path-connected intervalsM, letY be a Hausdorﬀtopological vector space. LetA:K → 2Kbe a mapping with nonempty Δ-convex values, and letg : K → 2K andf : K×K → 2Y be mappings and

C:K → 2Y be a mapping such that, for eachx∈K,Cxis a closed, pointed, and convex cone inY withintCx/∅. Assume that

1For anyy∈K,A−1_y_{is open;} 2B{x∈K:x∈Ax}is closed; 3gis usc with compact values;

4fu, x∩intCx ∅for anyx∈Kandu∈gx;

5for anyx∈Kandu∈gx,y → fu, yisΔ-concave with respect toCx; 6for anyy∈K,x → fx, yis lsc;

7W :K → 2Y is usc, whereWx Y \intCxfor eachx∈K.

Then there exists anx∗ ∈ K such thatx∗ ∈ Ax∗and for anyy ∈ Ax∗, there exists an

u∗_∈_g_x∗_{such that}

fu∗_{, y}_∩_int_C_x∗ _∅_. _2.1

Furthermore, the solution set of (WGIVEP) is closed, and hence is compact.

Proof. DefineP :K → 2Kby

Px y∈K:∀u∈gx, fu, y∩intCx/∅, ∀x∈K. 2.2

We first prove that for anyy∈K,P−1_y_{is open, that is,}

K\P−1_y_x_∈_K_:_{x /}_∈_P−1_y

x∈K:y /∈Px

x∈K:∃u∈gx, fu, y∩intCx ∅

2.3

is closed. Let a net{xα} ⊂K\P−1yandxα → x0 ∈K. Then there existsuα ∈gxαsuch

thatfuα, y∩intCxα ∅,fuα, y ⊂ Wxα, for anyα. By3andLemma 1.3, we know that

gK x∈K

gx 2.4

is compact and so{uα} ⊂gKhas a cluster pointu0 ∈gK. We may assume thatuα → u0

and thus,u0∈gx0. For anyv0∈fu0, y, by6, there existsvα∈fuα, ysuch thatvα → v0

and sovα ∈fuα, y⊂Wxα. It follows from7thatv0 ∈Wx0 Y \intCx0and hence

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Suppose that there exists anx0 ∈ K such that Px0 is notΔ-convex, that is, there

exist y1, y2, . . . , yn ∈ Px0 such that Δy1, y2, . . . , yn/⊂Px0. Hence, there exists y0 ∈ Δy1, y2, . . . , yn,y0/∈Px0, that is, there existsu0 ∈ gx0such thatfu0, y0∩intCx0 ∅. For each i 1,2, . . . , n, yi ∈ Px0, take u u0, fu0, yi ∩ intCx0/∅. Let vi ∈

fu0, yi∩intCx0,i 1,2, . . . , n. For anyti ≥ 0, i 1,2, . . . , n,andni1ti 1, we have n

i1tivi∈intCx0. By5, there existsv∈fu0, y0such that

v− n

i1

tivi∈Cx0. 2.5

Sincev /∈intCx0, we know that

v−

v− n

i1

tivi

n

i1

tivi/∈intCx0, 2.6

which is a contradiction. Therefore, for anyx∈K,PxisΔ-convex. By1andLemma 1.6,B /∅. DefineS:K → 2Kby

Sx

⎧ ⎨ ⎩

Ax∩Px, if x∈B,

Ax, if x∈K\B. 2.7

ThenSxisΔ-convex for eachx∈K. It follows from1and2that

S−1_y_A−1_y_∩_P−1_y_∪_K_\_B_∩_A−1_y_, _∀_y_∈_K _2.8

is open.

Suppose that for allx∈K,Sxis nonempty. Then, byLemma 1.6Shas a fixed point, that is, there existsx0 ∈K, such thatx0 ∈Sx0. Ifx0 ∈B, thenx0 ∈Sx0 Ax0∩Px0,

hencex0 ∈ Px0, for allu ∈gx0,fu, x0∩intCx0/∅which contradicts to assumption 4; Ifx0 ∈K\B, thenx0 ∈Sx0 Ax0, hencex0∈Bwhich contradicts withx0 ∈K\B.

Therefore, there existsx∗ ∈ K, such thatSx∗ ∅. SinceAxis nonempty for anyx ∈K, thenx∗ ∈B,Sx∗ Ax∗∩Px∗ ∅, that is,x∗ ∈Ax∗and for anyy∈Ax∗,y /∈Px∗. Therefore,x∗∈Ax∗and for anyy∈Ax∗, there exists anu∗∈gx∗such that

fu∗_{, y}_∩_int_C_x∗ _∅_. _2.9

LetTdenote the solution set ofWGIVEPand{xα} ⊂T withxα → x0∈K. We show

thatx0∈T, that is,x0∈Ax0, and for ally∈Ax0, there existsu0∈gx0such that

fu0, y

∩intCx0 ∅. 2.10

In fact, it follows from2thatx0 ∈ Ax0. For anyy ∈ Ax0,x0 ∈ A−1y. By 1, there

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existsα0such that for anyα≥α0,xα ∈Nx0⊂A−1y. Thus,y∈Axαand so there exists

uα∈gxαsuch thatfuα, y∩intCxα ∅, that is,fuα, y⊂Wxα. SincegK x∈Kgx

is compact,{uα} ⊂gKhas a cluster pointu0 ∈gK. We may assume thatuα → u0. From 3, we have u0 ∈ gx0. By6, for any q ∈ fu0, y, there existsqα ∈ fuα, y such that

qα → q. It follows from7thatq∈Wx0, that is,fu0, y∩intCx0 ∅. Thus,T is closed,

and hence is compact. This completes the proof.

Theorem 2.2. LetKbe a nonempty compactΔ-convex subset of a topological semilattice with path-connected intervals M, let Y be a Hausdorﬀ topological vector space. Let A : K → 2K be with nonemptyΔ-convex values, and letg :K → 2Kandf :K×K → 2Y be mappings andC:K → 2Y be a mapping such that, for eachx ∈ K,Cxis a closed, pointed, and convex cone inY with

intCx/∅. Assume that

1For anyy∈K,A−1_y_{is open;} 2B{x∈K:x∈Ax}is closed; 3gis lsc;

4for allx∈Kandu∈gx,fu, x∩intCx ∅;

5for allx∈K,fisgx-Δ-concave with respect toCxin second argument; 6for ally∈K,x → fx, yis lsc;

7W :K → 2Y is usc, whereWx Y \intCxfor allx∈K.

Then there exists anx∗∈Ksuch thatx∗∈Ax∗and

fu, y∩intCx∗ ∅, ∀y∈Ax∗, ∀u∈gx∗. 2.11

Furthermore, the solution set of (SGIVEP) is closed, and hence is compact.

Proof. DefineP :K → 2Kby

Px y∈K:∃u∈gx, fu, y∩intCx/∅, ∀x∈K. 2.12

Then the proof is similar to that ofTheorem 2.1and so we omit it.

Theorem 2.3. LetKbe a nonempty compactΔ-convex subset of a topological semilattice with path-connected intervalsMand letY be a Hausdorﬀtopological vector space. LetA :K → 2Kbe with nonemptyΔ-convex values, letg :K → 2Kandf :K×K → 2Y be mappings, and letC:K → 2Y be a mapping such that, for eachx ∈ K,Cxis a closed, pointed, and convex cone inY with

intCx/∅. Assume that

1Ais usc with compact values;

2For anyx∈K,gxis nonemptyΔ-convex; 3gis c p-usc onX;

4fis usc with nonempty compact values; 5for allx, u∈K,fu, x∩Cx ∅;

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7ChasΔ-inheritance; 8Cis c p-usc onX.

Then there exists anx∗∈Ksuch thatx∗∈Ax∗and there existsu∗∈gx∗such that

fu∗_{, y}_∩_C_x∗ _∅_, _∀_y_∈_A_x∗_. _2.13

Proof. DefineF:K×K → 2K×Kby

Fx, u Bx, u×gx, ∀x, u∈K×K, 2.14

where

Bx, u y∈Ax:fu, z∩Cy∅, ∀z∈Ax. 2.15

The proof is divided into the following five steps.

IFor anyx, u∈K×K,Bx, uis nonempty.

If it is false, then there exists x, u ∈ K ×K such that Bx, u ∅, that is, for any

y∈Ax, there existsz∈Axsuch thatfu, z∩Cy/∅. Let

Gyz∈Ax:fu, z∩Cy/∅. 2.16

ThenG:Ax → 2Axis nonempty values. If there existsy0∈Axsuch thatGy0is notΔ

-convex, then there existk1, k2, . . . , kn∈Gy0such thatΔ{k1, k2, . . . , kn}/⊂Gy0, that is, there

existsq∈Δ{k1, k2, . . . , kn}withq /∈Gy0. Thus,fu, q∩Cy0 ∅. For eachi1,2, . . . , n,

ki ∈Gy0, takevi ∈ fu, ki∩Cy0. For anyti ≥ 0,i 1,2, . . . , nandni1ti 1, we have _n

i1tivi∈Cy0. By6, there existsv∈fu, qsuch that

v− n

i1

tivi∈Cy0

. 2.17

Sincefu, q∩Cy0 ∅,v /∈Cy0. Hence,

v−

v− n

i1

tivi

n

i1

tivi/∈Cy0

, 2.18

which is a contradiction. Thus, for anyy∈Ax,Gyis nonemptyΔ-convex. For anyz∈Ax,

G−1_z _y_∈_A_x_:_z_∈_G_y_y_∈_A_x_:_f_{u, z}_∩_C_y_/_∅_. _2.19

It follows from8that

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is closed and so G−1_z _{is open. Since} _A_x _{is nonempty compact and} _Δ_{-convex, by}

Lemma 1.6G has a fixed point. Thus, there exists y∗ ∈ Ax such that y∗ ∈ Gy∗, that is, fu, y∗∩Cy∗/∅ which contradicts with Assumption 5. Hence Bx, u/∅ for any

x, u∈K×K.

IIFor anyx, u∈K×K,Bx, uisΔ-convex. If it is false, then there existsx, u∈

K×Ksuch thatBx, uis notΔ-convex, that is, there existy1, y2, . . . , yn∈Bx, usuch that

Δy1, y2, . . . , yn/⊂Bx, u. 2.21

Thus, there existsy0 ∈ Δ{y1, y2, . . . , yn}such thaty0∈/Bx, u. Then y1, y2, . . . , yn ∈ Ax

and for allz∈Ax,fu, z∩Cyi ∅,i1,2, . . . , n. SinceAxisΔ-convex,y0 ∈Ax. By 7,fu, z∩Cy0 ∅. Sincey0/∈Bx, u, there existsz0∈Axsuch that

fu, z0∩C

y0

/

∅, 2.22

which is a contradiction. Therefore, for anyx, u∈K×K,Bx, uisΔ-convex.

IIIFx, uis nonemptyΔ-convex for anyx, u ∈ K×K. By stepsIandII, the conclusion follows directly from2.

IVFor anyy, v∈K×K,

F−1_{y, v}_{x, u}_∈_K_×_K_:_{y, v}_∈_F_{x, u}

x, u∈K×K:y∈Bx, uandv∈gx 2.23

is open. In fact, we only need to show that

K×K\F−1_{y, v}_{x, u}_∈_K_×_K_:_{y, v}_/_∈_B_{x, u}_×_g_x

x, u∈K×K:y /∈Bx, u∪x, u∈K×K:v /∈gx 2.24

is closed. Let a net{xα, uα} ⊂ {x, u∈K×K:y /∈Bx, u}andxα, uα → x0, u0∈K×K.

Ify /∈Ax0, theny /∈Bx0, u0and hence

x0, u0∈

x, u∈K×K:y /∈Bx, u. 2.25

Ify∈Ax0and there existszα∈Axαsuch that

fuα, zα∩Cy/∅. 2.26

Takepα ∈fuα, zα∩Cy. By1andLemma 1.3,AK _x_∈_KAxis compact and hence

{zα} ⊂AKhas a cluster pointz0∈AK. We may assume thatzα → z0and soz0∈Ax0.

Similarly, by4,{pα}has a cluster pointp0. We assume thatpα → p0and hencep0∈fu0, z0.

SinceCyclosed,p0 ∈Cy. Thus,fu0, z0∩Cy/∅,y /∈Bx0, u0,andx0, u0∈ {x, u∈

K×K:y /∈Bx, u}. Hence,{x, u∈K×K:y /∈Bx, u}is closed. Let a net

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andxα, uα → x0, u0∈K×K, thenv /∈gxα. By3, we havev /∈gx0, and hence{x, u∈

K×K:v /∈gx}is closed. Thus,K×K\F−1_{y, v}_{is closed and so}_F−1_{y, v}_{is open.} VThe UGIVEP has a solution. ByLemma 1.6,Fhas a fixed point. Thus, there exists

x∗_{, u}∗_∈_K_×_K_{such that}_x∗_{, u}∗_∈_F_x∗_{, u}∗ _B_x∗_{, u}∗_×_g_x∗_{, that is,}_x∗_∈_K_and_u∗_∈_g_x∗

such thatx∗∈Ax∗and

fu∗_{, y}_∩_C_x∗ _∅_, _∀_y_∈_A_x∗_. _2.28

This completes the proof.

Acknowledgments

This research was supported by the Natural Science Foundations of Guangdong Province

9251064101000015. The author is grateful to the referees for the valuable comments and suggestions.

References

1 J. Li, N.-J. Huang, and J. K. Kim, “On implicit vector equilibrium problems,”Journal of Mathematical Analysis and Applications, vol. 283, no. 2, pp. 501–512, 2003.

2 I. V. Konnov and J. C. Yao, “Existence of solutions for generalized vector equilibrium problems,”

Journal of Mathematical Analysis and Applications, vol. 233, no. 1, pp. 328–335, 1999.

3 L.-C. Ceng, S.-M. Guu, and J.-C. Yao, “On generalized implicit vector equilibrium problems in Banach spaces,”Computers & Mathematics with Applications, vol. 57, no. 10, pp. 1682–1691, 2009.

4 X. H. Chen, “Existence of solution for the implicit multi-valued vector equilibrium problem,”Journal of Applied Mathematics and Computing, vol. 30, no. 1-2, pp. 469–478, 2009.

5 N. J. Huang, J. Li, and H. B. Thompson, “Implicit vector equilibrium problems with applications,”

Mathematical and Computer Modelling, vol. 37, no. 12-13, pp. 1343–1356, 2003.

6 J. Li and N.-J. Huang, “Implicit vector equilibrium problems via nonlinear scalarisation,”Bulletin of the Australian Mathematical Society, vol. 72, no. 1, pp. 161–172, 2005.

7 G.-Y. Chen, X. X. Huang, and X. Q. Yang,Vector Optimization: Set-Valued and Variational Analysis, vol. 541 ofLecture Notes in Economics and Mathematical Systems, Springer, Berlin, Germany, 2005.

8 C. D. Horvath and J. V. Llinares Ciscar, “Maximal elements and fixed points for binary relations on topological ordered spaces,”Journal of Mathematical Economics, vol. 25, no. 3, pp. 291–306, 1996.

9 J.-P. Aubin and I. Ekeland,Applied Nonlinear Analysis, Pure and Applied Mathematics, John Wiley & Sons, New York, NY, USA, 1984.