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Volume 2008, Article ID 581849, 15 pagesdoi:10.1155/2008/581849 Research Article Connectedness and Compactness of Weak Efficient Solutions for Set-Valued Vector Equilibrium Problems Bin

Volume 2008, Article ID 581849, 15 pages

doi:10.1155/2008/581849

Research Article

Connectedness and Compactness of

Weak Efficient Solutions for Set-Valued

Vector Equilibrium Problems

Bin Chen, Xun-Hua Gong, and Shu-Min Yuan

Department of Mathematics, Nanchang University, Nanchang 330047, China

Correspondence should be addressed to Xun-Hua Gong,xunhuagong@gmail.com

Received 1 November 2007; Revised 17 July 2008; Accepted 5 September 2008

Recommended by C E Chidume

We study the set-valued vector equilibrium problems and the set-valued vector Hartman-Stampacchia variational inequalities We prove the existence of solutions of the two problems

In addition, we prove the connectedness and the compactness of solutions of the two problems in normed linear space

Copyrightq 2008 Bin Chen 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

1 Introduction

We know that one of the important problems of vector variational inequalities and vector equilibrium problems is to study the topological properties of the set of solutions Among its topological properties, the connectedness and the compactness are of interest Recently, Lee et al.1 and Cheng 2 have studied the connectedness of weak efficient solutions set for single-valued vector variational inequalities in finite dimensional Euclidean space Gong

3 5 has studied the connectedness of the various solutions set for single-valued vector equilibrium problem in infinite dimension space The set-valued vector equilibrium problem was introduced by Ansari et al.6 Since then, Ansari and Yao 7, Konnov and Yao 8, Fu

9, Hou et al 10, Tan 11, Peng et al 12, Ansari and Flores-Baz´an 13, Lin et al 14 and Long et al.15 have studied the existence of solutions for set-valued vector equilibrium and set-valued vector variational inequalities problems However, the connectedness and the compactness of the set of solutions to the set-valued vector equilibrium problem remained unstudied In this paper, we study the existence, connectedness, and the compactness of the weak efficient solutions set for set-valued vector equilibrium problems and the set-valued vector Hartman-Stampacchia variational inequalities in normed linear space

2 Preliminaries

Throughout this paper, letX, Y be two normed linear spaces, let A be a nonempty subset of

X, let F : A × A → 2 Y be a set-valued map, and letC be a closed convex pointed cone in Y.

We consider the following set-valued vector equilibrium problemSVEP: find x ∈ A,

such that

Definition 2.1 Let int C / ∅ A vector x ∈ A satisfying

is called a weak efficient solution to the SVEP Denote by Vw A, F the set of all weak efficient

solutions to the SVEP

LetY∗be the topological dual space ofY Let

be the dual cone ofC.

Definition 2.2 Let f ∈ C∗\ {0} A vector x ∈ A is called an f-efficient solution to the SVEP if

wherefFx, y ≥ 0 means that fz ≥ 0, for all z ∈ Fx, y Denote by V f A, F the set of all f-efficient solutions to the SVEP.

Definition 2.3 Let A be a nonempty convex subset in X A set-valued map F : A × A → 2 Y

is called to beC-convex in its second variable if, for each fixed x ∈ A, for every y1, y2 ∈ A,

t ∈ 0, 1, the following property holds:

tFx, y1  1 − tFx, y2 ⊂ Fx, ty1 1 − ty2  C. 2.5

Definition 2.4 Let A be a nonempty convex subset in X A set-valued map F : A × A → 2 Y

is called to beC-concave in its first variable if, for each fixed y ∈ A, for every x1, x2 ∈ A,

t ∈ 0, 1, the following property holds:

Ftx1 1 − tx2, y ⊂ tFx1, y  1 − tFx2, y  C. 2.6

Definition 2.5 Let A be a nonempty subset of X Let T : A → 2 LX,Y be a set-valued map, where LX, Y is the space of all bounded linear operators from X into Y let LX, Y be

equipped with operator norm topology Set Tx, y  {s, y : s ∈ Tx}, x, y ∈ A

i Let A be a convex subset of X T is said to be v-hemicontinuous if, for every pair of

pointsx, y ∈ A, the set-valued map

Jα : Tαy  1 − αx, y − x, α ∈ 0, 1, 2.7

is lower semicontinuous at 0

ii Let f ∈ C∗\ {0} T is said to be f-pseudomonotone on A if, for every pair of points

x, y ∈ A, fs, y − x ≥ 0, for all s ∈ Tx, then fs , y − x ≥ 0, for all s ∈ Ty.

The definition ofv-hemicontinuity was introduced by Lin et al 14

Definition 2.6 Let X be a Hausdorff topological vector space and let K ⊂ X be a nonempty

set.G : K → 2 Xis called to be a KKM map if for any finite set{x1, , x n } ⊂ K the relation

co{x1, , x n} ⊂n

i1

holds, where co{x1, , x n } denoted the convex hull of {x1, , x n}

For the definition of the upper semicontinuity and lower semicontinuity, see16 The following FKKM theorem plays a crucial role in this paper

Lemma 2.7 Let X be a Hausdorff topological vector space Let K be a nonempty convex subset of X,

and let G : K → 2 K be a KKM map If for each x ∈ K, Gx is closed in X, and if there exists a point

x0∈ K such that Gx0 is compact, thenx∈K Gx / ∅.

By definition, we can get the following lemma

Lemma 2.8 Let A be a nonempty convex subset of X Let F : A × A → 2 Y be a set-valued map, and let C ⊂ Y be a closed convex pointed cone Moreover, suppose that Fx, y is C-convex in its second variable Then, for each x ∈ A, Fx, A  C is convex.

3 Scalarization

In this section, we extend a result in3 to set-valued map

Theorem 3.1 Suppose that int C / ∅, and that Fx, A  C is a convex set for each x ∈ A Then

V w A, F  

f∈C∗ \{0}

Proof It is clear that

V w A, F ⊃ 

f∈C∗ \{0}

Now we prove that

V w A, F ⊂ 

f∈C∗ \{0}

Letx ∈ V w A, F By definition, Fx, y ∩ −int C  ∅, for all y ∈ A Thus

AsC is a convex pointed cone, we have

By assumption,Fx, A  C is a convex set By the separation theorem of convex sets, there

exist somef ∈ Y∗\ {0}, such that

inf{fFx, y  c : y ∈ A, c ∈ C} ≥ sup{f−c : c ∈ C} 3.6

By3.6, we obtain that f ∈ C∗\ {0} and

Therefore,x ∈ V f A, F Hence V w A, F ⊂f∈C∗ \{0}V f A, F Thus we have

V w A, F  

f∈C∗ \{0}

4 Existence of the weak efficient solutions

Theorem 4.1 Let A be a nonempty closed convex subset of X and let C ⊂ Y be a closed convex

pointed cone with int C / ∅ Let F : A × A → 2 Y be a set-valued map with Fx, x ⊂ C for all x ∈ A Suppose that for each y ∈ A, F·, y is lower semicontinuous on A, and that Fx, y is C-convex

in its second variable If there exists a nonempty compact subset D of A, and y ∈ D, such that Fx, y  ∩ −int C / ∅, for all x ∈ A \ D, then, for any f ∈ C∗\ {0}, V f A, F / ∅, V f A, F ⊂ D,

V w A, F / ∅, and V w A, F ⊂ D.

Proof Let f ∈ C∗\ {0} Define the set-valued map G : A → 2 Aby

Gy  {x ∈ A : fFx, y ≥ 0}, y ∈ A. 4.1

By assumption, y ∈ Gy, for all y ∈ A, so Gy / ∅ We claim that G is a KKM map.

Suppose to the contrary that there exists a finite subset{y1, , y n } of A, and there exists

x ∈ co{y1, , y n } such that x /∈n i1 Gy i  Then x n i1 t i y ifor somet i ≥ 0, 1 ≤ i ≤ n, with

n

i1 t i  1, and x /∈ Gy i , for all i  1, , n Then there exist z i ∈ Fx, y i, such that

AsFx, y is C-convex in its second invariable, we can get that

t1Fx, y1  t2Fx, y2  · · ·  t n Fx, y n  ⊂ Fx, x  C. 4.3

By4.3, we know that there exist z ∈ Fx, x, c ∈ C, such that

Hencefz  c  ft1z1  t2z2 · · · t n z n  By assumption, we have fz  c ≥ 0 By 4.2, however, we haveft1z1 t2z2 · · · t n z n  < 0 This is a contradiction Thus G is a KKM map.

Now we show that for eachy ∈ A, Gy is closed For any sequence, {x n } ⊂ Gy and x n → x0 BecauseA is a closed set, we have x0 ∈ A By assumption, for each y ∈ A, F·, y is lower

semicontinuous onA, then by 16, for each fixed y ∈ A, and for each z0 ∈ Fx0, y, there

existz n ∈ Fx n , y, such that z n → z0 Because{x n } ⊂ Gy, we have

Thusfz n  ≥ 0 By the continuity of f and z n → z0, we havefz0 ≥ 0 By the arbitrariness

ofz0 ∈ Fx0, y, we have fFx0, y ≥ 0, that is, x0 ∈ Gy Hence Gy is closed By the

assumption, we haveGy  ⊂ D, and Gy  is closed Since D is compact, Gy  is compact

ByLemma 2.7, we have

y∈A Gy / ∅ Thus there exists x ∈y∈A Gy This means that

Therefore, x ∈ V f A, F Next we show that V f A, F ⊂ D If x ∈ V f A, F, then x ∈



y∈A Gy ⊂ Gy  ⊂ D It follows from V f A, F ⊂ V w A, F that V w A, F / ∅, and by

Theorem 3.1, we haveV w A, F ⊂ D.

Theorem 4.2 Let A be a nonempty closed convex subset of X and let C ⊂ Y be a closed convex

pointed cone with int C / ∅ Let f ∈ C∗\ {0} Assume that T : A → 2 LX,Y is a v-hemicontinuous, f-pseudomonotone mapping Moreover, assume that the set-valued map F : A × A → 2 Y defined by Fx, y  Tx, y − x is C-convex in its second variable If there exists a nonempty compact subset D

of A, and y ∈ D, such that Tx, y − x ∩ −int C / ∅, for all x ∈ A \ D, then V f A, F / ∅ and

V f A, F ⊂ D.

Proof Let f ∈ C∗\ {0} Define the set-valued maps E, G : A → 2 Aby

Ey  {x ∈ A : fs, y − x ≥ 0 ∀s ∈ Tx}, y ∈ A, Gy  {x ∈ A : fs, y − x ≥ 0 ∀s ∈ Ty}, y ∈ A, 4.7

respectively As for eachy ∈ A, we have y ∈ Ey, then Ey / ∅ The proof of the theorem is

divided into four steps

I E is a KKM map on A.

Suppose to the contrary that there exists a finite subset{y1, , y n } of A, and there

existsx ∈ co{y1, , y n } such that x /∈n

i1 Ey i  Then x ∈ A, x  n

i1 t i y i for somet i ≥ 0,

1≤ i ≤ n, withn i1 t i  1, and x /∈ Ey i , for all i  1, , n Then there exist s i ∈ T x such that

fs i , y i − x < 0 for each i  1, 2, , n. 4.8 SinceFx, y is C-convex in its second variable, we have

t1T x, y1− x  · · ·  t n T x, y n − x ⊂ T x, x − x  C  C. 4.9 Letz i  s i , y i − x, for each i  1, 2, , n By 4.9, we know there exists c ∈ C, such that

Asf ∈ C∗\ {0}, we have

While by4.8, we have ft1z1 t2z2 · · ·  t n z n  < 0 This is a contraction Hence E is a KKM

map onA.

II Ey ⊂ Gy for all y ∈ A and G is a KKM map.

By thef-pseudomonotonicity of T, for each y ∈ A, we have Ey ⊂ Gy Since E is a

KKM map, so isG.

IIIy∈A Gy / ∅.

Now we show that for eachy ∈ A, Gy is closed Let {x n } be a sequence in Gy such

thatx n converges tox By the closedness of A, we have x ∈ A Since {x n } ⊂ Gy, then for

eachs ∈ Ty, we have

Asx n → x, and the continuity of f, then for each s ∈ Ty, we have

Consequently,x ∈ Gy Hence Gy is closed By the assumption, we have Gy  ⊂ D Then Gy  is compact since D is compact By step II, we know G is a KKM map ByLemma 2.7,



y∈A Gy / ∅.

IVy∈A Gy y∈A Ey.

Because Ey ⊂ Gy, we have y∈A Gy ⊃ y∈A Ey Now let us show that



y∈A Gy ⊂y∈A Ey Let x ∈y∈A Gy For each y ∈ A, and each s ∈ Ty, we have

For anys ∈ T x and for each fixed y ∈ A, define the set-valued mapping J : 0, 1 → 2 Y by

Jα  Tαy  1 − αx, y − x α ∈ 0, 1. 4.15

We pick a sequence{α n } ⊂ 0, 1 such that α n → 0 and set x n  α n y  1 − α n x Since A is a

convex set,x n ∈ A for each n It is clear that x n → x Let w  s, y−x We have w ∈ J0 Since

T is v-hemicontinuous, Jα is lower semicontinuous at 0 By 16, there exist w n ∈ Jα n 

Tα n y  1 − α n x, y − x, such that w n → w As w n ∈ Jα n , there exist s n ∈ Tx n such that

w n  s n , y − x By w n → w, we have s n , y − x → s, y − x By 4.14, we have

α n fs n , y − x  fs n , α n y  1 − α n x − x ≥ 0. 4.16

Sinceα n > 0, fs n , y − x ≥ 0 Hence fs, y − x ≥ 0 since f is continuous and w n → w.

Therefore, for anys ∈ T x and for each y ∈ A, we have fs, y − x ≥ 0 Hence x ∈y∈A Ey.

Thus

y∈A Ey y∈A Gy / ∅ This means that there exists x ∈ A, for each s ∈ Tx, we have fs, y − x ≥ 0, for all y ∈ A It follows that x ∈ V f A, F, thus V f A, F / ∅ By the proof of

Theorem 4.1, we knowV f A, F ⊂ D Since V f A, F ⊂ V w A, F, we have V w A, F / ∅ The

proof of the theorem is completed

5 Connectedness and compactness of the solutions set

In this section, we discuss the connectedness and the compactness of the weak efficient solutions set for set-valued vector equilibrium problems and the set-valued vector Hartman-Stampacchia variational inequalities in normed linear space

Theorem 5.1 Let A be a nonempty closed convex subset of X, let C ⊂ Y be a closed convex pointed

cone with int C / ∅, and let F : A×A → 2 Y be a set-valued map Assume that the following conditions are satisfied:

i for each y ∈ A, F·, y is lower semicontinuous on A;

ii Fx, y is C-concave in its first variable and C-convex in its second variable;

iii Fx, x ⊂ C, for all x ∈ A;

iv {Fx, y : x, y ∈ A} is a bounded subset in Y;

v there exists a nonempty compact convex subset D of A, and y ∈ D, such that Fx, y  ∩

−int C / ∅, for all x ∈ A \ D.

Then V w A, F is a nonempty connected compact set.

Proof We define the set-valued map H : C∗\ {0} → 2Dby

Hf  V f A, F, f ∈ C∗\ {0}. 5.1

ByTheorem 4.1, for eachf ∈ C∗\ {0}, we have Hf / ∅, hence V w A, F / ∅ and V w A, F ⊂

D It is clear that C∗\ {0} is convex, so it is a connected set Now we prove that, for each

f ∈ C∗\ {0}, Hf is a connected set Let x1, x2∈ Hf, we have x1, x2∈ D and

BecauseFx, y is C-concave in its first variable, for each fixed y ∈ A, and for above x1, x2 ∈

D, and t ∈ 0, 1, we have tx1 1 − tx2∈ D since D is convex, and

Ftx1 1 − tx2, y ⊂ tFx1, y  1 − tFx2, y  C. 5.3

Hence for eachy ∈ A, z ∈ Ftx1 1 − tx2, y, there exist z1 ∈ Fx1, y, z2 ∈ Fx2, y, and

c ∈ C, such that z  tz1 1 − tz2 c As f ∈ C∗\ {0} and by 5.2, we have

fz  tfz1  1 − tfz2  fc ≥ 0. 5.4 Thus

fFtx1 1 − tx2, y ≥ 0 ∀y ∈ A, 5.5

that istx1 1 − tx2∈ Hf So Hf is convex, therefore it is a connected set.

Now we show thatHf is upper semicontinuous on C∗\ {0} Since D is a nonempty

compact set, by16, we only need to prove that H is closed Let the sequence {f n , x n} ⊂ GraphH and fn , x n  → f0, x0, where {f n } converge to f0 with respect to the norm topology Asf n , x n  ∈ GraphH, we have

that is,f n Fx n , y ≥ 0, for all y ∈ A As x n → x0andD is compact, we have x0∈ D Since for

eachy ∈ A, F·, y is lower semicontinuous on A, for each fixed y ∈ A, and each z0∈ Fx0, y,

there existz n ∈ Fx n , y, such that z n → z0 Fromf n Fx n , y ≥ 0, we have

By the continuity off0andz n → z0, we have

LetQ  {Fx, y : x, y ∈ A} By assumption, Q is a bounded set in Y, then there exist some

M > 0, such that for each z ∈ Q, we have z ≤ M For any ε > 0, because f n − f0→ 0 with respect to norm topology, there existsn0 ∈ N, and when n ≥ n0, we havef n − f0 < ε.

Therefore, there existsn0∈ N, and when n ≥ n0, we have

|f n z n  − f0z n |  |f n − f0z n | ≤ f n − f0 z n ≤ Mε. 5.9 Hence

lim

n → ∞ f n z n  − f0z n   0. 5.10

Consequently, by5.8, 5.10, we have

lim

n → ∞ f n z n  lim

n → ∞ f n z n  − f0z n   f0z n

 lim

n → ∞ f n z n  − f0z n  lim

n → ∞ f0z n

 f0z0.

5.11

By5.7, we have f0z0 ≥ 0 So for any y ∈ A and for each z0∈ Fx0, y, we have f0z0 ≥ 0 Hence

This means that

Hence the graph of H is closed Therefore, H is a closed map By 16, H is upper

semicontinuous onC∗\{0} Because Fx, y is C-convex in its second variable, byLemma 2.8, for eachx ∈ A, Fx, A  C is convex It follows fromTheorem 3.1that

V w A, F  

f∈C∗ \{0}

Thus by17, Theorem 3.1 Vw A, F is a connected set.

Now, we show that V w A, F is a compact set We first show that V w A, F is a

closed set Let {x n } ⊂ V w A, F with x n → x0 Since D is compact, x0 ∈ D We claim that

x0∈ V w A, F Suppose to the contrary that x0/∈ V w A, F, then there exist some y0 ∈ A such

that

Thus there existsz0∈ Fx0, y0 such that

Hence−int C is a neighborhood of z0 SinceF·, y0 is lower semicontinuous at x0, there exists some neighborhoodUx0 of x0such that

Fx, y0 ∩ −int C / ∅ ∀x ∈ Ux0 ∩ A. 5.17 Sincex n → x0, there exist somen0, and when n ≥ n0, we havex n ∈ Ux0 ∩ A By 5.17,

This contradicts{x n } ⊂ V w A, F Thus x0 ∈ V w A, F This means that V w A, F is a closed

set SinceD is compact and V w A, F ⊂ D, V w A, F is compact.

Theorem 5.2 Let A be a nonempty closed convex subset of X, and let C ⊂ Y be a closed convex

pointed cone with int C / ∅ Assume that for each f ∈ C∗\{0}, T : A → 2 LX,Y is a v-hemicontinuous, f-pseudomonotone mapping Moreover, assume that the set-valued map F : A × A → 2 Y defined by Fx, y  Tx, y − x is C-convex in its second variable, and the set {Fx, y : x, y ∈ A} is a bounded set in Y If there exists a nonempty compact convex subset D of A, and y ∈ D, such that

Tx, y − x ∩ −int C / ∅, for all x ∈ A \ D, then V w A, F is a nonempty connected set.

Proof We define the set-valued map H : C∗\ {0} → 2Dby

Hf  V f A, F for each f ∈ C∗\ {0}. 5.19

ByTheorem 4.2, for eachf ∈ C∗\{0}, we have Hf  V f A, F / ∅ and V f A, F ⊂ D Hence

V w A, F / ∅ and V w A, F ⊂ D Clearly, C∗\ {0} is a convex set, hence it is a connected set Define the set-valued mapsE, G : A → 2 Aby

Ey  {x ∈ A : fs, y − x ≥ 0, ∀s ∈ Tx}, y ∈ A, Gy  {x ∈ A : fs, y − x ≥ 0, ∀s ∈ Ty}, y ∈ A, 5.20

respectively Now we prove that for eachf ∈ C∗\ {0}, Hf is a connected set Let x1, x2 ∈

Hf  V f A, F, then x1, x2∈y∈A Ey By the proof ofTheorem 4.2, we have

y∈A Gy 



y∈A Ey, so x1, x2 ∈y∈A Gy Hence for i  1, 2, and for each y ∈ A, s ∈ Ty, we have

Then, for eachy ∈ A, s ∈ Ty, and t ∈ 0, 1, we have tx1 1 − tx2∈ D since D is convex and

fs, y − tx1 1 − tx2 ≥ 0. 5.22

Hencetx1 1 − tx2 ∈ y∈A Gy  y∈A Ey Thus tx1 1 − tx2 ∈ Hf Consequently,

for eachf ∈ C∗\ {0}, Hf is a convex set Therefore, it is a connected set The following is

to prove thatH is upper semicontinuous on C∗\ {0} Since D is a nonempty compact set, by

16 we only need to show that H is a closed map Let sequence {f n , x n } ⊂ GraphH and

f n , x n  → f0, x0, where {f n } converges to f0 with respect to the norm topology ofY∗ As

f n , x n  ∈ GraphH, we have

Then, for eachs ∈ Tx n, we have that

f n s , y − x n  ≥ 0 ∀y ∈ A. 5.24

Consequently, by5.8, 5.10, we have

lim

n... x2 ∈y∈A Gy Hence for i  1, 2, and for each y ∈ A, s ∈ Ty, we have

Then, for eachy ∈ A, s ∈ Ty, and t ∈ 0, 1, we have tx1... somen0, and when n ≥ n0, we havex n ∈ Ux0 ∩ A By 5.17,