Generalized Vector Equilibrium-Like Problems Without

Hindawi Publishing CorporationJournal of Inequalities and ApplicationsVolume 2007, Article ID 61794, 13 pagesdoi:10.1155/2007/61794

Research ArticleGeneralized Vector Equilibrium-Like Problems withoutPseudomonotonicity in Banach Spaces

Lu-Chuan Ceng, Sy-Ming Guu, and Jen-Chih Yao

Received 10 January 2007; Accepted 21 March 2007

Recommended by Donal O’Regan

Let X and Y be real Banach spaces, D a nonempty closed convex subset of X , and C :D→ 2Y a multifunction such that for each u ∈ D, C(u) is a proper, closed and convexcone with intC(u) �=∅, where intC(u) denotes the interior of C(u). Given the mappingsT : D→ 2L(X ,Y), A : L(X ,Y)→ L(X ,Y), f : L(X ,Y)×D×D→ Y , and h : D→ Y , we studythe generalized vector equilibrium-like problem: find u0 ∈ D such that f (As0,u0,v) +h(v)−h(u0) �∈ − intC(u0) for all v ∈D for some s0 ∈ Tu0. By using the KKM techniqueand the well-known Nadler result, we prove some existence theorems of solutions forthis class of generalized vector equilibrium-like problems. Furthermore, these existencetheorems can be applied to derive some existence results of solutions for the generalizedvector variational-like inequalities. It is worth pointing out that there are no assumptionsof pseudomonotonicity in our existence results.

Copyright © 2007 Lu-Chuan Ceng et al. This is an open access article distributed underthe Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

1. Introduction

In 1980, Giannessi [1] first introduced and studied the vector variational inequality ina finite-dimensional Euclidean space, which is a vector-valued version of the variationalinequality of Hartman and Stampacchia. Subsequently, many authors investigated vectorvariational inequalities in abstract spaces, and extended vector variational inequalities tovector equilibrium problems, which include as special cases various problems, for exam-ple, vector complementarity problems, vector optimization problems, abstract economi-cal equilibria, and saddle-point problems (see, e.g., [1–17]).

In 1999, B.-S. Lee and G.-M. Lee [12] first established a vector version of Minty’slemma (see [18]) by using Nadler’s result [19]. They considered vector variational-like

2 Journal of Inequalities and Applications

inequalities for multifunctions under pseudomonotonicity and hemicontinuity condi-tions. Recently, Khan and Salahuddin [5] also established a vector version of Minty’slemma and applied it to obtain an existence theorem for a class of vector variational-like inequalities for compact-valued multifunctions under similar pseudomonotonicitycondition and similar hemicontinuity condition.

On the other hand, as a natural generalization of the vector equilibrium problem, thegeneralized vector equilibrium problem includes as special cases various problems, for ex-ample, generalized vector variational inequality problem, generalized vector variational-like inequality problem, generalized vector complementarity problem and vector equi-librium problem. Inspired by early results in this field, many authors have consideredand studied the generalized vector equilibrium problem, that is, the vector equilibriumproblem for multifunctions; see, for example, [6, 8, 13–15, 17].

In this paper, let X and Y be two real Banach spaces and D a nonempty closed con-vex subset of X . Let C : D→ 2Y be a multifunction such that for each u ∈ D, C(u) is aproper, closed, and convex cone with intC(u) �= ∅, where intC(u) denotes the interiorof C(u). For convenience, we let P = ⋂u∈DC(u). Given the mappings T : D → 2L(X ,Y),A : L(X ,Y)→ L(X ,Y), f : L(X ,Y)×D×D→ Y , and h : D→ Y , we consider the general-ized vector equilibrium-like problem as follows,

find u0 ∈D such that f(As0,u0,v

)+h(v)−h

(u0) �∈ − intC

(u0),

∀v ∈D for some s0 ∈ Tu0.(1.1)

In particular, if we put f (z,x, y)= 〈z,η(y,x)〉 for all (z,x, y)∈ L(X ,Y)×D×D, whereη : D × D → X , then the above problem reduces to the following generalized vectorvariational-like inequality problem:

find u0 ∈D such that⟨As0,η

(v,u0

)⟩+h(v)−h

(u0) �∈ − intC

(u0),

∀v ∈D for some s0 ∈ Tu0.(1.2)

By using the KKM technique [20] and the Nadler’s result [19], we prove some exis-tence theorems of solutions for this class of generalized vector equilibrium-like problems.Furthermore, these existence theorems can be applied to derive some existence results ofsolutions for the generalized vector variational-like inequalities. It is worth pointing outthat there are no assumptions of pseudomonotonicity in our existence results.

2. Preliminaries

In this section, we recall some notations, definitions and results, which are essential forour main results.

Definition 2.1 (see [11]). Let D be a nonempty subset of a vector space X . Then a mul-tifunction T : D→ 2X is called a KKM-map where 2X denotes the collection of all non-empty subsets of X , if for each nonempty finite subset {u1,u2, . . . ,un} of D, co{u1,u2, . . . ,un} ⊂

⋃ni=1Tui, where co{u1,u2, . . . ,un} denotes the convex hull of {u1,u2, . . . ,un}.

Lu-Chuan Ceng et al. 3

Lemma 2.2 (Fan’s lemma [20]). Let D be an arbitrary set in a Huasdorff topological vectorspace X . Let T : D→ 2X be a KKM-map such that Tu is closed for all u∈D and is compactfor at least one u∈D. Then

⋂u∈D Tu �= ∅.

Lemma 2.3 (Nadler’s theorem [19]). Let (X ,‖ · ‖) be a normed vector space and H theHausdorff metric on the collection CB(X) of all closed and bounded subsets of X , induced bya metric d in terms of d(x, y)= ‖x− y‖, which is defined by

H(A,B)=max(

supu∈A

infv∈B‖u− v‖, sup

v∈Binfu∈A

‖u− v‖)

, (2.1)

for A and B in CB(X). If A and B are any two members in CB(X), then for each ε > 0 andeach u∈A, there exists v ∈ B such that

‖u− v‖ ≤ (1 + ε)H(A,B). (2.2)

In particular, if A and B are any two compact subsets in X , then for each u∈ A, there existsv ∈ B such that

‖u− v‖ ≤H(A,B). (2.3)

Lemma 2.4 (see [16]). Let Y be a topological vector space with a pointed, closed and convexcone C such that intC �= ∅. Then for all x, y,z ∈ Y ,

(i) x− y ∈− intC and x �∈ − intC⇒ y �∈ − intC;(ii) x+ y ∈−C and x+ z �∈ − intC⇒ z− y �∈ −intC;

(iii) x+ z− y �∈ − intC and −y ∈−C⇒ x+ z �∈ − intC;(iv) x+ y �∈ − intC and y− z ∈−C⇒ x+ z �∈ − intC.

Definition 2.5 (see [13]). A multifunction T : D→ 2Y is called P-convex if, for all u,v ∈Dand λ∈ (0,1),

T(λu+ (1− λ)v

)⊆ λTu+ (1− λ)Tv−P. (2.4)

Similarly, one can define the P-convexity of single-valued mappings.

Definition 2.6. Let T : D→ 2Y . The graph of T , denoted by Gr(T), is the following set:

Gr(T)= {(x, y) : y ∈ Tx}. (2.5)

Definition 2.7 (see [16]). Let f : D×D→ Y be a vector-valued bifunction. Then f (x, y)is said to be hemicontinuous with respect to y if for any given x ∈D,

limλ→0+

f(x,λy1 + (1− λ)y2

)= f(x, y2

)(2.6)

for all y1, y2 ∈D.

Throughout the rest of this paper, by “→” and “⇀” we denote the strong convergenceand weak convergence, respectively.


3. Main results

In this section, we will present two theorems for the existence results to the generalizedvector equilibrium-like problem.

Theorem 3.1. Let X and Y be real Banach spaces, D a nonempty convex subset of X , and{C(u) : u ∈ D} a family of closed proper convex solid cones of Y such that for each u ∈D, C(u) �= Y . Let W : D→ 2Y be a multifunction, defined by W(u)= Y \ (− intC(u)), suchthat the graph Gr(W) is weakly closed in X ×Y . Suppose that the following conditions hold:

(i) for each u,v ∈ D, f (Atλ,vλ,vλ) ∈ C(u), for all tλ ∈ Tvλ, and f (Atλ,vλ,u) + f (Atλ,u,vλ)= 0, where vλ := u+ λ(v−u), λ∈ (0,1);

(ii) f (z,·,v),h(·) : D→ Y are weakly continuous for each (z,v)∈ L(X ,Y)×D;(iii) f (z,v,·) +h(·) is P-convex on D for each (z,v)∈ L(X ,Y)×D;(iv) there exists a bifunction p : D×D→ Y with the following properties:

(a) for each u,v ∈ D, p(u,v) �∈ − intC(u) implies f (At,u,v) + h(v) − h(u) �∈− intC(u), for all t ∈ Tv,

(b) for each finite subset �⊆D and each u∈ co�, v �→ p(u,v) is P-convex,(c) for each v ∈D, p(v,v) �∈ intC(v);(d) there exist a weakly compact convex subset K ⊆ D and v0 ∈ K such that p(u,

v0)∈− intC(u) for all u∈D \K .Then there exists a solution u0 ∈D such that

f(At,v,u0

)+h(u0)−h(v) �∈ intC

(u0)

(3.1)

for all v ∈D and t ∈ Tv.Moreover, suppose additionally that L(X ,Y) is reflexive and T : D→ 2L(X ,Y) is a

multifunction which takes bounded, closed, and convex values in L(X ,Y) and satisfiesthe following conditions:

(v) for each net {λ} ⊂ (0,1) such that λ→ 0+,

tλ s0,

tλ ∈ Tvλ

}

=⇒ f(Atλ,vλ,v

)− f(As0,vλ,v

)0, (3.2)

where vλ := u+ λ(v−u) for (u,v)∈D×D;(vi) for each u,v ∈D,

H(T(u+ λ(v−u)

),T(u)

)−→ 0 as λ−→ 0+, (3.3)

where H is the Hausdorff metric defined on CB(L(X ,Y)).Then there exists a solution u0 ∈D such that for some s0 ∈ Tu0,

f(As0,u0,v

)+h(v)−h

(u0) �∈ − intC

(u0) ∀v ∈D. (3.4)

Proof. For each v ∈D, we define G : D→ 2K by

G(v)= {u∈ K : f (At,u,v) +h(v)−h(u) �∈ − intC(u), ∀t ∈ Tv}

, ∀v ∈D. (3.5)


Firstly, we claim that G(v) is weakly closed for each v ∈D. Indeed, let {un} ⊆ G(v) besuch that un⇀ u0 ∈ K as n→∞. Since un ∈G(v) for all n, we have that

f(At,un,v

)+h(v)−h

(un) �∈ − intC

(un), ∀t ∈ Tv. (3.6)

Since from condition (ii) it follows that f (At,·,v)− h(·) : D→ Y is weakly continuous,we have

f(At,un,v

)+h(v)−h

(un)

f(At,u0,v

)+h(v)−h

(u0). (3.7)

Note that the graph Gr(W) is weakly closed inX ×Y . Hence, we have f (At,u0,v) +h(v)−h(u0) ∈ Y \ (− intC(u0)), that is, f (At,u0,v) + h(v)− h(u0) �∈ − intC(u0). This showsthat u0 ∈ G(v). Thus, G(v) is weakly closed. Since every element u0 ∈

⋂v∈DG(v) is a

solution of (3.1), we have to prove that

⋂

v∈DG(v) �= ∅. (3.8)

Since K is weakly compact, it is sufficient to show that the family {G(v)}v∈D has the finiteintersection property.

Let {v1,v2, . . . ,vm} be a finite subset of D. We claim that

m⋂

j=1

G(vj) �= ∅. (3.9)

Indeed, note that

V := co{v1,v2, . . . ,vm

}(3.10)

is a compact convex subset of D and also a weakly compact convex subset of D. We definea multifunction F : V → 2V as

F(v)= {u∈V : p(u,v) �∈ − intC(u)}

, ∀v ∈V. (3.11)

By (iv)(c), F(v) is nonempty for each v ∈V .Now we assert that F is a KKM-map.Indeed, suppose to the contrary that there exists a finite subset {y1, y2, . . . , yn} ⊆V and

scalars αi ≥ 0, i= 1,2, . . . ,n, with∑n

i=1αi = 1, such that

n∑

i=1

αiyi �∈n⋃

i=1

F(yi). (3.12)

Then, we have

p

( n∑

i=1

αiyi, yi

)

∈− intC

( n∑

i=1

αiyi

)

, ∀i. (3.13)


By (iv)(b), we have

p

( n∑

i=1

αiyi,n∑

i=1

αiyi

)

∈n∑

i=1

αip

( n∑

i=1

αiyi, yi

)

−P

⊆n∑

i=1

αi

(

− intC

( n∑

i=1

αiyi

))

−C

( n∑

i=1

αiyi

)

⊆− intC

( n∑

i=1

αiyi

)

−C

( n∑

i=1

αiyi

)

=− intC

( n∑

i=1

αiyi

)

,

(3.14)

a contradiction to condition (iv)(c). Hence, F is a KKM-map. From condition (iv)(a), wehave that

F(v)⊆G(v), ∀v ∈V. (3.15)

Observe that, for each v ∈ V , the closure clV (F(v)) of F(v) in V is closed in V , andtherefore is compact also. By Lemma 2.2,

⋂

v∈VclV(F(v)

) �= ∅. (3.16)

We can choose

u∈⋂

v∈VclV(F(v)

)(3.17)

and note that v0 ∈ K and F(v0)⊆ K by (iv)(d). Thus,

u∈ clV(F(v0))⊆ clD

(F(v0))= clK

(F(v0))⊆ K. (3.18)

Moreover, it is easy to see that for each v ∈V ,{u∈V : f (At,u,v) +h(v)−h(u) �∈ − intC(u), ∀t ∈ Tv

}(3.19)

is weakly closed. Since

u∈m⋂

j=1

clV(F(vj))

(3.20)

and since, for each j = 1,2, . . . ,m,

clV(F(vj))= clV

({u∈V : p

(u,vj

) �∈ − intC(u)})

⊆ clV({u∈V : f

(At,u,vj

)+h(vj)−h(u) �∈ − intC(u), ∀t ∈ Tvj

})

⊆ {u∈V : f(At,u,vj

)+h(vj)−h(u) �∈ − intC(u), ∀t ∈ Tvj

},

(3.21)


we have

f(At,u,vj

)+h(vj)−h(u) �∈ − intC(u), ∀t ∈ Tvj (3.22)

for all j = 1,2, . . . ,m, and hence,

u∈m⋂

j=1

G(vj). (3.23)

Therefore, {G(v)}v∈D has the finite intersection property and so

⋂

v∈DG(v) �= ∅, (3.24)

that is, there exists u0 ∈ K ⊆D such that

f(At,u0,v

)+h(v)−h

(u0) �∈ − intC

(u0)

(3.25)

for all v ∈D and t ∈ Tv.On the other hand, for any arbitrary v ∈ D, letting vλ = λv + (1− λ)u0, 0 < λ < 1, we

have vλ ∈D by the convexity of D. Hence, for all tλ ∈ Tvλ

f(Atλ,u0,vλ

)+h(vλ)−h

(u0) �∈ − intC

(u0). (3.26)

Since the operator

u �−→ f (z,u,v)−h(u) (3.27)

is P-convex for each (z,v)∈ L(X ,Y)×D, so from condition (i) we have

f(Atλ,vλ,vλ

)+h(vλ)−h

(vλ)= f

(Atλ,vλ,λv+ (1− λ)u0

)+h(λv+ (1− λ)u0

)−h(vλ)

∈ λ[f(Atλ,vλ,v

)+h(v)−h

(vλ)]

+ (1− λ)[f(Atλ,vλ,u0

)+h(u0)−h

(vλ)]−P

⊆ λ[f(Atλ,vλ,v

)+h(v)−h

(vλ)]

+ (1− λ)[f(Atλ,vλ,u0

)+h(u0)−h

(vλ)]−C

(u0)

⊆ λ[f(Atλ,vλ,v

)+h(v)−h

(vλ)]

− (1− λ)[f(Atλ,u0,vλ

)+h(vλ)−h

(u0)]−C

(u0).

(3.28)

Hence,

f(Atλ,vλ,v

)+h(v)−h

(vλ) �∈ − intC

(u0). (3.29)


Indeed, suppose to the contrary that

f(Atλ,vλ,v

)+h(v)−h

(vλ)∈− intC

(u0). (3.30)

Since − intC(u0) is a convex cone,

λ[f(Atλ,vλ,v

)+h(v)−h

(vλ)]∈− intC

(u0). (3.31)

Since condition (i) implies that

f(Atλ,vλ,vλ

)∈ C(u0), (3.32)

so from (3.28) we derive

(1− λ)[f(Atλ,u0,vλ

)+h(vλ)−h

(u0)]

∈ λ[f(Atλ,vλ,v

)+h(v)−h

(vλ)]− f

(Atλ,vλ,vλ

)−C(u0)

⊆− intC(u0)−C

(u0)−C

(u0)

⊆− intC(u0)−C

(u0)

=− intC(u0).

(3.33)

Thus,

f(Atλ,u0,vλ

)+h(vλ)−h

(u0)∈− intC

(u0), (3.34)

which contradicts (3.26). Consequently

f(Atλ,vλ,v

)+h(v)−h

(vλ) �∈ − intC

(u0). (3.35)

Since Tvλ and Tu0 are bounded closed subsets in L(X ,Y), by Lemma 2.3 for each tλ ∈ Tvλwe can find an sλ ∈ Tu0 such that

∥∥tλ− sλ

∥∥≤ (1 + λ)H

(Tvλ,Tu0

). (3.36)

Since L(X ,Y) is reflexive and Tu0 is a bounded, closed, and convex subset in L(X ,Y), Tu0

is a weakly compact subset in L(X ,Y). Hence, without loss of generality we may assumethat sλ⇀ s0 ∈ Tu0 as λ→ 0+. Moreover, for each φ ∈ (L(X ,Y))∗ we have

∣∣φ(tλ− s0

)∣∣≤ ∣∣φ(tλ− sλ

)∣∣+

∣∣φ(sλ− s0

)∣∣

≤ ‖φ‖∥∥tλ− sλ∥∥+

∣∣φ(sλ− s0

)∣∣

≤ ‖φ‖(1 + λ)H(Tvλ,Tu0

)+∣∣φ(sλ− s0

)∣∣.

(3.37)

Since H(Tvλ,Tu0)→ 0 as λ→ 0+, so tλ⇀ s0. Thus, according to condition (v) we have

∥∥ f(Atλ,vλ,v

)− f(As0,vλ,v

)∥∥−→ 0 as λ−→ 0+. (3.38)


Since h : D→ Y is weakly continuous and f (z,·,v) : D→ Y is continuous for each (z,v)∈L(X ,Y)×D, we deduce from (v) that

f(Atλ,vλ,v

)+h(v)−h

(vλ)− f

(As0,u0,v

)−h(v) +h(u0)

= f(Atλ,vλ,v

)− f(As0,u0,v

)− (h(vλ)−h

(u0))

= f(Atλ,vλ,v

)− f(As0,vλ,v

)+ f(As0,vλ,v

)− f(As0,u0,v

)

−(h(vλ)−h

(u0))

0 as λ−→ 0+,

(3.39)

that is,

f(Atλ,vλ,v

)+h(v)−h

(vλ)

f(As0,u0,v

)+h(v)−h

(u0)

as λ−→ 0+. (3.40)

Therefore, it follows from (3.29) and the weak closedness of Y \ (− intC(u0)) that

f(As0,u0,v

)+h(v)−h

(u0) �∈ − intC

(u0)

(3.41)

for all v ∈D.This completes the proof. �

Theorem 3.2. Let X and Y be real Banach spaces, let D be a nonempty convex subset ofX , and {C(u) : u ∈ D} a family of closed proper convex solid cones of Y such that for eachu∈D, C(u) �= Y . Let W : D→ 2Y be a multifunction, defined by W(u)= Y \ (− intC(u)),such that the graph Gr(W) is weakly closed in X ×Y . Suppose that the following conditionshold:

(i) for each u,v ∈ D, f (Atλ,vλ,vλ) ∈ C(u), for all tλ ∈ Tvλ, and f (Atλ,vλ,u) + f (Atλ,u,vλ)= 0, where vλ := u+ λ(v−u), λ∈ (0,1);

(ii) f (z,·,v),h(·) : D→ Y are weakly continuous for each (z,v)∈ L(X ,Y)×D;(iii) f (z,v,·) +h(·) is P-convex on D for each (z,v)∈ L(X ,Y)×D;(iv) there exists a bifunction q : D×D→ Y such that

(a) q(u,u) �∈ − intC(u), for all u∈D,(b) q(u,v)− f (At,u,v)∈−C(u), for all u,v ∈D, t ∈ Tv,(c) {v ∈D : q(u,v) +h(v)−h(u)∈− intC(u)} is convex for each u∈D;

(v) there exists a weakly compact convex subset K ⊆D such that for each u∈D \K thereexists v0 ∈D satisfying

f (At,u,v) +h(v)−h(u)∈− intC(u), ∀t ∈ Tv. (3.42)

Then there exists a solution u0 ∈D such that

f(At,v,u0

)+h(u0)−h(v) �∈ intC

(u0)

(3.43)

for all v ∈D and t ∈ Tv.Moreover, suppose additionally that L(X ,Y) is reflexive and T : D→ 2L(X ,Y) is a

multifunction which takes bounded, closed, and convex values in L(X ,Y) and satisfiesthe following conditions:


(vi) for each net {λ} ⊂ (0,1) such that λ→ 0+,

tλ s0,

tλ ∈ Tvλ

}

=⇒ f(Atλ,vλ,v

)− f(As0,vλ,v

)0, (3.44)

where vλ := u+ λ(v−u) for (u,v)∈D×D;(vii) for each u,v ∈D,

H(T(u+ λ(v−u)

),T(u)

)−→ 0 as λ−→ 0+, (3.45)

where H is the Hausdorff metric defined on CB(L(X ,Y)).Then there exists a solution u0 ∈D such that for some s0 ∈ Tu0,

f(As0,u0,v

)+h(v)−h

(u0) �∈ − intC

(u0) ∀v ∈D. (3.46)

Proof. Define

G(v)= {u∈ K : f (At,u,v) +h(v)−h(u) �∈ − intC(u), ∀t ∈ Tv}

, ∀v ∈D. (3.47)

Following the same proof as in Theorem 3.1, we can prove that G(v) is weakly closed foreach v ∈ D. We now claim that

⋂v∈DG(v) �= ∅. Indeed, since K is weakly compact, it

is sufficient to show that the family {G(v)}v∈D has the finite intersection property. Let{v1,v2, . . . ,vn} be a finite subset of D and set B = co{K ∪ {v1,v2, . . . ,vn}}. Then B is aweakly compact and convex subset of D.

We define two vector multifunctions F1,F2 : B→ 2B as follows:

F1(v)= {u∈ B : f (At,u,v) +h(v)−h(u) �∈ − intC(u), ∀t ∈ Tv}

, ∀v ∈ B,

F2(v)= {u∈ B : q(u,v) +h(v)−h(u) �∈ − intC(u)}

, ∀v ∈ B.(3.48)

From condition (iv)(a), (b), we have

q(v,v) +h(v)−h(v) �∈ − intC(v), ∀v ∈ B,

q(v,v)− f (At,v,v)∈−C(v), ∀t ∈ Tv.(3.49)

Now Lemma 2.4(ii) guarantees that

f (At,v,v) +h(v)−h(v) �∈ − intC(v), ∀t ∈ Tv, (3.50)

and so F1(v) is nonempty. Since F1(v) is a weakly closed subset of the weakly compactsubset B, we know that F1(v) is weakly compact.

Next we claim that F2 is a KKM-map. Indeed, suppose that there exists a finite subset{u1,u2, . . . ,un} of B and αi ≥ 0, i= 1,2, . . . ,n with

∑ni=1αi = 1, such that

u=n∑

i=1

αiui ∈n⋃

j=1

F2(uj). (3.51)


Then

q(u,uj

)+h(uj)−h(u)∈− intC(u), j = 1,2, . . . ,n. (3.52)

From condition (iv)(c), we derive

q(u, u)= q(u, u) +h(u)−h(u)∈− intC(u), (3.53)

which contradicts condition (iv)(a). Thus, F2 is a KKM-map. From condition (iv)(b) andLemma 2.4(ii), we have F2v ⊆ F1(v), for all v ∈ B. Indeed, if u ∈ F2(v), then q(u,v) +h(v)−h(u) �∈ − intC(u). By condition (iv)(b), we have

q(u,v)− f (At,u,v)∈−C(u), ∀t ∈ Tv. (3.54)

Consequently, it follows from Lemma 2.4(ii) that

f (At,u,v) +h(v)−h(u) �∈ − intC(u), ∀t ∈ Tv, (3.55)

that is, u∈ F1(v). This shows that F1 is also a KKM-map. According to Lemma 2.2, thereexists u∈ B such that u∈ F1(v) for all v ∈ B; that is, there exists u∈ B such that

f (At,u,v) +h(v)−h(u) �∈ − intC(u), ∀v ∈ B, t ∈ Tv. (3.56)

By condition (v), we get u ∈ K and u ∈ G(vi), i = 1,2, . . . ,n. Hence, {G(v)}v∈D has thefinite intersection property and moreover,

⋂

v∈DG(v) �= ∅, (3.57)

that is, there exists u0 ∈ K ⊆D such that

f(At,u0,v

)+h(v)−h

(u0) �∈ − intC

(u0)

(3.58)

for all v ∈D and t ∈ Tv.For the remainder of the proof, we can derive the conclusion of Theorem 3.2 by fol-

lowing the same proof as in Theorem 3.1. �

Remark 3.3. The above existence theorems can be applied to deriving some existenceresults of solutions for the generalized vector variational-like inequalities. Here we omitthem. It is worth pointing out that there are no assumptions of pseudomonotonicity inour existence results.

Acknowledgments

The research of the first author was partially supported by the Teaching and ResearchAward Fund for Outstanding Young Teachers in Higher Education Institutions of MOE,


China, and the Dawn Program Foundation in Shanghai. The research of the second au-thor was partially supported by NSC 95-2221-E-155-049. The research of the third authorwas partially supported by NSC 95-2221-E-110-078.

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Lu-Chuan Ceng: Department of Mathematics, Shanghai Normal University, Shanghai 200234, ChinaEmail address: [email protected]

Sy-Ming Guu: Department of Business Administration, College of Management,Yuan-Ze University, Chung-Li City 330, Taoyuan Hsien, TaiwanEmail address: [email protected]

Jen-Chih Yao: Department of Applied Mathematics, National Sun Yat-Sen University,Kaohsiung 804, TaiwanEmail address: [email protected]

mailto:[email protected]



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