CUBO A Mathematical Journal

Vol.10, N
o
¯ 04, (73–83). December 2008

An Intersection Theorem and its Applications

Mircea Balaj

Department of Mathematics,

University of Oradea, Romania

email: mbalaj@uoradea.ro

and

Donal O’Regan

Department of Mathematics,

National University of Ireland, Galway, Ireland

email: donal.oregan@nuigalway.ie

ABSTRACT

In this paper we obtain a very general intersection theorem for the values of a map.

From this we derive existence theorems for two types of vectorial equilibrium problems,

an analytic alternative and a minimax inequality involving three real functions.

RESUMEN

En este art́ıculo obtenemos un teorema general de intersección para los valores de

una aplicación. A través de este resultado deducimos teoremas de existencia para dos

tipos de problemas de equilibrio vectoriales, una alternativa anaĺıtica y una desigualdad

minimax envolviendo tres funciones reales.

Key words and phrases: The better admissible class, fixed point, quasiconvex map, equilibrium

problem.

Math. Subj. Class.: 54C60, 49J35, 91B50.



74 Mircea Balaj and Donal O’Regan CUBO
10, 4 (2008)

1. Introduction and preliminaries

A multimap (or simply a map) T : X ⊸ Y is a function from a set X into the power set 2Y of

Y , that is a function with the values T (x) ⊆ Y for x ∈ X. To a map T : X ⊸ Y we associate

two other maps T c : X ⊸ Y and T − : Y ⊸ X defined by T c(x) = Y \ T (x), and respectively

T −(y) = {x ∈ X : y ∈ T (x)} The values of T − are called the fibers of T .

Let T : X ⊸ Y be a map. As usual the set {(x, y) ∈ X × Y : y ∈ T (x)} is called the graph of

T . For A ⊆ X, and B ⊆ Y let T (A) =
⋃

x∈A
T (x) and T −(B) = {x ∈ X : T (x) ∩ B 6= ∅}.

For topological spaces X and Y a map T : X ⊸ Y is said to be: upper semicontinuous (u.s.c.)

if for any closed set F ⊆ Y the set T −(F ) is closed in X; lower semicontinuous (l.s.c.) if for any

open set U ⊆ Y the set T −(U ) is open in X; compact if T (X) is contained in a compact subset of

Y ; closed if its graph is closed in X × Y .

The following lemma collects known facts about u.s.c. or l.s.c. maps (see for example [7] for

assertion (i), [16] for assertion (ii) and [9] for assertion (iii)).

Lemma 1 Let X and Y be topological spaces and T : X ⊸ Y be a map.

(i) If T has compact values, then it is u.s.c. if and only if for each x ∈ X, any net {xt} converging

to x and any net {yt} with yt ∈ T (xt) for all index t, there exists a subnet {yt′} of {yt} and

y ∈ T (x) such that {yt′} converges to y.

(ii) T is l.s.c. in x ∈ X if and only if for any y ∈ T (x) and any net {xt} converging to x, there

exists a net {yt} converging to y, with yt ∈ T (xt) for each t.

(iii) If Y is compact and T is closed, then T is u.s.c..

If X is a subset of a topological vector space we denote by coX and X the convex hull and

the closure of X respectively.

Let Y be a convex set in a topological vector space and X be a topological space. The better

admissible class B of mappings from Y into X (see [15]) is defined as follows:

T ∈ B(Y, X) ⇔ T : Y ⊸ X is a mapping such that for any nonempty finite subset A of Y

and any continuous mapping p : T (co A) → co A the composition p ◦ T|co A : co A ⊸ co A has a

fixed point.

The class B(Y, X) includes many important classes of mappings, such as U kc (Y, X) in [14],

KKM (Y, X) in [3] and A(Y, X) in [2], as proper subclasses.

Definition 1. Let X be a convex set in a vector space and Y a vector space. A mapping T : X ⊸ Y

is called:

(i) quasiconvex, if for every convex subset C of Y , T −(C) is a convex set;

(ii) convex, if for each x1, x2 ∈ X and λ ∈ (0, 1), λT (x1) + (1 − λ)T (x2) ⊆ T (λx1 + (1 − λ)x2);



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An Intersection Theorem and its Applications 75

(iii) concave, if for each x1, x2 ∈ X and λ ∈ (0, 1), T (λx1 + (1 − λ)x2) ⊆ λT (x1) + (1 − λ)T (x2).

Lemma 2 If a map T : X ⊸ Y is convex then it is quasiconvex.

Proof. Let C be a convex subset of Y , x1, x2 ∈ T
−
(C) and λ ∈ (0, 1). If y1 ∈ T (x1) ∩ C,

y2 ∈ T (x2) ∩ C, then

λy1 + (1 − λ)y2 ∈ (λT (x1) + (1 − λ)T (x2)) ∩ C ⊆ T (λx1 + (1 − λ)x2) ∩ C,

hence λx1 + (1 − λ)x2 ∈ T
−
(C).

Let us describe in short the contents on the next sections. We obtain first a very general

intersection theorem involving three maps, one of them from the class B. Two types of applications

of this result will be given in the last two sections.

The first one, offers existence theorems for the following types of vectorial equilibrium prob-

lems:

Let X be a topological space, Y be a convex set in a topological vector space, Z be a topological

vector space and V be nonempty set. Let F : Y × Z ⊸ V , C : Z ⊸ V and P : X ⊸ Z.

(I) Find x0 ∈ X such that F (y, z) ⊆ C(z) for each y ∈ Y and z ∈ P (x0);

and respectively,

(II) Find x0 ∈ X such that F (y, z) ∩ C(z) 6= ∅ for each y ∈ Y and z ∈ P (x0).

Finally, we obtain an analytic alternative and a minimax inequality involving three real func-

tions.

From now all (topological) vector spaces will be assumed real and all topological (vector)

spaces will be assumed Hausdorff.

2. An intersection theorem

Theorem 1. Let X be a topological space, Y be a convex set in a topological vector space and Z

be a nonempty set. Let P : X ⊸ Z, Q : Y ⊸ Z two maps satisfying the following conditions:

(i) for each y ∈ Y , {x ∈ X : P (x) ⊆ Q(y)} is closed;

(ii) P has convex values and Qc is quasiconvex;

(iii) there exists a compact mapping T ∈ B(Y, X) such that for each y ∈ Y , P (T (y)) ⊆ Q(y).

Then there exists x0 ∈ X such that P (x0) ⊆
⋂

y∈Y
Q(y).



76 Mircea Balaj and Donal O’Regan CUBO
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Proof. Let S : Y ⊸ X be the map defined by

S(y) = {x ∈ X : P (x) * Q(y)}.

Suppose that the conclusion of theorem is false. Then X =
⋃

y∈Y
S(y). Let X0 = T (Y ). Since

X0 is compact there exists a finite set A = {y1, y2, . . . , yn} ⊆ Y such that X0 =
⋃n

i=1
(S(yi) ∩ X0).

Let {α1, α2, . . . , αn} be a partition of unity on X0 subordinated to the cover {S(yi) ∩ X0 : 0 ≤

i ≤ n}. Recall that this means that















αi : X0 → [0, 1] is continuous, for each i ∈ {1, 2, . . . , n};

αi(x) > 0 ⇒ x ∈ S(yi);
∑n

i=1
αi(x) = 1 for each x ∈ X0.

Define f : T (co A) → co A by

f (x) =

n
∑

i=1

αi(x)yi for all x ∈ T (co A).

Since f is continuous and T ∈ B(Y, X), f ◦ T|A : coA ⊸ coA has a fixed point. Hence

there exists ỹ ∈ coA such that ỹ ∈ f (T (ỹ)). Then, for some x̃ ∈ T (ỹ) we have ỹ = f (x̃). Let

I = {i ∈ {1, . . . , n} : αi(x̃) > 0}. Then ỹ = f (x̃) ∈ co{yi : i ∈ I}. For each i ∈ I, x̃ ∈ S(yi), hence

P (x̃) ∩ Qc(yi) 6= ∅. By (ii) it follows that P (x̃) ∩ Q
c
(ỹ) 6= ∅, or equivalently, P (x̃) * Q(y). Since

x̃ ∈ T (ỹ), we get P (T (y)) * Q(y), which contradicts (iii).

Proposition 2. If Z is topological space, then condition (i) in Theorem 1 is fulfilled in any of the

following cases:

(i1) P has open fibers;

(i2) P is l.s.c. and Q has closed values;

Proof. If P has open values then for each y ∈ Y the set {x ∈ X : P (x) * Q(y)} =
⋃

z∈Qc(y)
P −(z)

is open, hence {x ∈ X : P (x) ⊆ Q(y)} = X \ {x ∈ X : P (x) * Q(y)} is closed.

By the definition of lower semicontinuity it follows that if (i2) holds then each set {x ∈ X :

P (x) ⊆ Q(y)} is closed.

3. Equilibrium Theorems

In [5], [6], [10-13], for a suitable choice of the sets Y, Z and V and of the maps F : Y × Z ⊸ V

and C : Z ⊸ V the authors study, all or part of the following problems:



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An Intersection Theorem and its Applications 77

(I) Find z0 ∈ Z such that F (y, z0) ⊆ C(z0) for all y ∈ Y ;

(II) Find z0 ∈ Z such that F (y, z0) ∩ C(z0) 6= ∅ for all y ∈ Y ;

(III) Find z0 ∈ Z such that F (y, z0) * C(z0) for all y ∈ Y ;

(IV) Find z0 ∈ Z such that F (y, z0) ∩ C(z0) = ∅ for all y ∈ Y .

Each existence result concerning problem (I) (respectively, (II)), yields an existence theorem

for problem (IV) (respectively, (III)), if we take into account the following equivalences: F (y, z) ⊆

C(z) ⇔ F (y, z) ∩ Cc(z) = ∅ and F (y, z) ∩ C(z) 6= ∅ ⇔ F (y, z) * Cc(z). For this reason we can fix

our attention on problems (I) and (II), only.

In this section we study equilibrium problems more general than (I) and (II):

Let X be a topological space, Y be a convex set in a topological vector space, Z be a topological

vector space and V be a nonempty set. Let F : Y × Z ⊸ V , C : Z ⊸ V and P : X ⊸ Z.

(V) Find x0 ∈ X such that F (y, z) ⊆ C(z) for each y ∈ Y and z ∈ P (x0);

and respectively,

(VI) Find x0 ∈ X such that F (y, z) ∩ C(z) 6= ∅ for each y ∈ Y and z ∈ P (x0).

Of course, when X = Z and P (z) = {z} for all z ∈ Z, problem (V) (respectively (VI)), reduces

to problem (I) (respectively (II)).

Theorem 3. Suppose that the maps F , C and P satisfy the following conditions:

(i) one of the following two requirements is fulfilled:

(i1) P has open fibers;

(i2) P is l.s.c., C is closed map and for each y ∈ Y , F (y, ·) is l.s.c.

(ii) F and Cc are convex maps, P has convex values;

(iii) there exists a compact mapping T ∈ B(Y, X) such that F (y, z) ⊆ C(z), for each y ∈ Y and

z ∈ P (T (y)).

Then there exists x0 ∈ X such that F (y, z) ⊆ C(z) for each y ∈ Y and z ∈ P (x0).

Proof. Let Q : Y ⊸ Z be the map defined by

Q(y) = {z ∈ Z : F (x, z) ⊆ C(z)}.

We prove that if (i2) holds, then Q has closed values. Let y ∈ Y and {zt}t∈∆ be a net in Q(y)

converging to z ∈ Z. If v ∈ F (y, z), since F (y, ·) is l.s.c., there exists a net {vt}t∈∆ converging to v

such that vt ∈ F (y, zt), for all t ∈ ∆. Since zt ∈ Q(y), vt ∈ F (y, zt) ⊆ C(zt). The map C is closed,

hence v ∈ C(z). Thus, F (y, z) ⊆ C(z), hence z ∈ Q(y). By Proposition 2, in both cases (i1) and

(i2), condition (i) in Theorem 1 is satisfied.



78 Mircea Balaj and Donal O’Regan CUBO
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We show next that the map Qc is convex. Let y1, y2 ∈ Y , λ ∈ (0.1) and z ∈ λQ
c
(y1) +

(1 − λ)Qc(y2). There exist z1, z2 ∈ Z such that z = λz1 + (1 − λ)z2 and v1, v2 ∈ V such that

vi ∈ F (yi, zi) ∩ C
c
(zi), for i = 1, 2. Since the maps F and C

c
are convex,

λv1 + (1 − λ)v2 ∈ λF (y1, z1) + (1 − λ)F (y2, z2) ⊆ F (λy1 + (1 − λ)y2, λz1 + (1 − λ)z2),

and similarly, λv1 + (1 − λ)v2 ∈ C
c
(λz1 + (1 − λ)z2). Thus, λv1 + (1 − λ)v2 ∈ F (λy1 + (1 −

λ)y2, z) ∩ C
c
(z), hence z ∈ Q(λy1 + (1 − λ)y2).

Hence Qc is convex and by Lemma 2, it is quasiconvex. It is clear that condition (iii) is equiva-

lent to the requirement similarly denoted in Theorem 1, hence all requirements of this theorem are

fulfilled. Consequently, there exists x0 ∈ X such that P (x0) ⊆
⋂

y∈Y
Q(y), that is, F (y, z) ⊆ C(z),

for each y ∈ Y and z ∈ P (x0).

Theorem 4. Suppose that the maps F , C and P satisfy the following conditions:

(i) one of the following two requirements is fulfilled:

(i1) P has open fibers;

(i2) P is l.s.c., C is u.s.c. with compact values and for each y ∈ Y , F (y, ·) is closed.

(ii) F is concave map, Cc is convex map and P has convex values;

(iii) there exists a compact mapping T ∈ B(Y, X) such that F (y, z) ∩ C(z) 6= ∅, for each y ∈ Y

and z ∈ P (T (y)).

Then there exists x0 ∈ X such that F (y, z) ∩ C(z) 6= ∅ for each y ∈ Y and z ∈ P (x0).

Proof. The proof is similar to that of Theorem 3. Let Q : Y ⊸ Z be the map defined by

Q(y) = {z ∈ Z : F (x, z) ∩ C(z) 6= ∅}.

We show first that if (i2) holds, then Q has closed values. Let y ∈ Y and {zt}t∈∆ be a net

in Q(y) converging to z ∈ Z. Then, for each t ∈ ∆, there exists vt ∈ F (yt, zt) ∩ C(zt). Since C is

u.s.c. with compact values, by Lemma 1 (i), there exist a subnet {vt′} of {vt} and v ∈ C(z) such

that vt′ → v. Since F (y, ·) is closed, v ∈ F (y, z). Therefore F (y, z) ∩ C(z) 6= ∅, hence z ∈ Q(y).

Let y1, y2 ∈ Y , λ ∈ (0.1) and z ∈ λQ
c
(y1) + (1 − λ)Q

c
(y2). There exist z1, z2 ∈ Z such that

z = λz1 + (1 − λ)z2 and F (y1, z1) ⊆ C
c
(z1), F (y2, z2) ⊆ C

c
(z2). By (ii) we infer that

F (λy1 + (1 − λ)y2, λz1 + (1 − λ)z2) ⊆ λF (y1, z1) + (1 − λ)F (y2, z2) ⊆ λC
c
(z1) + (1 − λ)C

(z2) ⊆

Cc(λz1 + (1 − λ)z2).

It follows that z ∈ Qc(λy1 + (1 − λ)y2), hence the map Q
c

is convex. The maps P and Q

satisfy all the requirements of Theorem 1 and the desired conclusion follows from this theorem.



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An Intersection Theorem and its Applications 79

4. Analytic alternative, minimax inequality

Definition 2. (see [1]). Let X and Y be convex sets in two vector spaces. We say that a function

q : Y × Z → R is (y, z)-quasiconvex if for any finite subset {(y1, z1), . . . , (yn, zn)} of Y × Z, and

each y ∈ co {y1, . . . , yn} there exists z ∈ co {z1, . . . zn} such that q(y, z) ≤ max1≤i≤n q(yi, zi).

It is clear that any function q : Y × Z → R quasiconvex on Y × Z is (y, z)-quasiconvex but

Example 2 in [1] shows that the converse is not true.

Definition 3. Let X and Z be topological spaces. A function p : X×Z → R is said to be marginally

upper semicontinuous in x (see [8]) if for every open subset U of Z the function x → infz∈U p(x, z)

is upper semicontinuous on X.

Any function upper semicontinuous in x is marginally upper semicontinuous in x but the

example given in [8], p.249 shows that the converse is not true.

Theorem 5. Let X be topological space, Y and Z be convex sets in topological vector spaces,

p : X × Z → R, q : Y × Z → R, t : X × Y → R be functions and α, β, λ be real numbers. Suppose

that the following conditions are satisfied:

(i) one of the following requirements is fulfilled:

(i1) for each z ∈ Z the set {x ∈ X : p(x, z) < α} is open;

(i2) p is marginally upper semicontinuous in x and for each y ∈ Y the set {z ∈ Z : q(y, z) ≥ β}

is closed;

(ii) for each x ∈ X the set {z ∈ Z : p(x, z) < α} is convex;

(iii) q is (y, z)-quasiconvex;

(iv) for x ∈ X, y ∈ Y and z ∈ Z the following implication holds: p(x, z) < α and q(y, z) < β ⇒

t(x, y) < λ;

(v) the map T : Y ⊸ X defined by T (y) = {x ∈ X : t(x, y) ≥ λ} is compact and belongs to the

class B(Y, X).

Then at least one of the following assertions holds:

(a) There exists x0 ∈ X such that p(x0, z) ≥ α, for all z ∈ Z.

(b) There exists z0 ∈ Z such that q(y, z0) ≥ β, for all y ∈ Y .

Proof. Define the maps P : X ⊸ Z, Q : Y ⊸ Z, T : X ⊸ Y by

P (x) = {z ∈ Z : p(x, z) < α}, Q(y) = {z ∈ Z : q(y, z) ≥ β}, and



80 Mircea Balaj and Donal O’Regan CUBO
10, 4 (2008)

T (y) = {x ∈ X : t(x, y) ≥ λ}.

If (i1) holds, then P has open fibers, If (i2) holds, then Q has closed values and we claim that P

is l.s.c. Indeed, since p is marginally upper semicontinuous in x, for each open U ⊆ Z the set

{x ∈ X : P (x) ∩ U 6= ∅} = {x ∈ X : infz∈U p(x, z) < α}

is open. Hence, according to Proposition 2, condition (i) in Theorem 1 holds.

Let C be a convex subset of Z, y1, y2 ∈ Q
c
(C) and y ∈ co{y1, y2}. Then there exist z1, z2 ∈ C

such that q(y1, z1) < β, q(y2, z2) < β. Since q is (y, z)-quasiconvex, there exists z ∈ co{z1, z2} ⊆ C

such that

q(y, z) ≤ max{q(y1, z1), q(y2, z2)] < β.

Thus y ∈ Qc(C), hence Q is quasiconvex. We prove that for each y ∈ Y , P (T (y)) ⊆ Q(y).

Suppose that for some y ∈ Y there exists x ∈ T (y) and z ∈ P (x) \ Q(y). By x ∈ T (y), we get

t(x, y) ≥ λ. On the other hand, since z ∈ P (x) \ Q(y), we have p(x, z) < α, q(y, z) < β and, by

(iv), we get t(x, y) < λ; a contradiction. Therefore the maps P, Q, T satisfy all the requirement

of Theorem 1. According to this theorem there exists x0 ∈ X such that P (x0) ⊆
⋂

y∈Y
Q(y).

Suppose that both assertions in the conclusion of theorem are false. This means that:

(a’) P (x) 6= ∅, for all x ∈ X;

(b’) for each z ∈ Z there exists y ∈ Y such that z /∈ Q(y).

The following contradiction completes the proof:

∅ 6= P (x0) ⊆
⋂

y∈Y
Q(y) = ∅.

Theorem 6. Let X be a topological compact space, Y and Z be two convex sets in topological

vector spaces and p : X × Z → R, q : Y × Z → R, t : X × Y → R functions. Suppose that the

following conditions are fulfilled:

(i) one of the following requirements is fulfilled:

(i1) p is u.s.c. in x;

(i2) p is marginally upper semicontinuous in x and q is u.s.c. in z;

(ii) p is quasiconvex in z;

(iii) q is (y, z)-quasiconvex;

(iv) for x ∈ X, y ∈ Y and z ∈ Z the following implication holds: t(x, y) ≤ p(x, z) + q(y, z);

(v) for each λ < infy∈Y supx∈X t(x, y) the map T : Y ⊸ X, defined by T (y) = {x ∈ X :

t(x, y) ≥ λ} belongs to the class B(Y, X).



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An Intersection Theorem and its Applications 81

Then,

infy∈Y supx∈X t(x, y) ≤ supx∈X infz∈Z p(x, z) + supz∈Zinfy∈Y q(y, z),

with the convention ∞ + (−∞) = ∞.

Proof. We may suppose that

inf
y∈Y

sup
x∈X

t(x, y) > −∞, supx∈X infz∈Z p(x, z) < ∞,

supz∈Zinfy∈Y q(y, z) < ∞.

By way of contradiction suppose that

inf
y∈Y

sup
x∈X

t(x, y) > supx∈X infz∈Z p(x, z) + supz∈Zinfy∈Y q(y, z)

and choose α, β, λ ∈ R such that supx∈Xinfz∈Z p(x, z) < α, supz∈Zinfy∈Y q(y, z) < β, λ <

infy∈Y supx∈X t(x, y), and α + β < λ.

We prove that condition (iv) in Theorem 5 is fulfilled. Let x ∈ X, y ∈ Y and z ∈ Z such

that p(x, z) < α and q(y, z) < β. Since α + β < λ, by condition (iv) in the theorem that must be

proved, we get t(x, y) ≤ p(x, z) + q(y, z) < α + β < λ.

It is easy to see that all the requirements of Theorem 5 are fulfilled. We prove that none of

assertions (a), (b) of the conclusion of Theorem 5 can take place.

If (a) happens, then

α ≤ infz∈Z p(x0, z) ≤ supx∈X infz∈Z p(x, z); a contradiction.

If (b) happens, then

β ≤ infy∈Y q(y, z0) ≤ supz∈Z infy∈Y q(y, z); a contradiction.

Corollary 7. Let X, Y and Z be convex subsets of three topological vector spaces, X being compact

and p : X × Z → R, q : Y × Z → R, t : X × Y → R three functions satisfying conditions (i), (ii),

(iii), (iv) of Theorem 6 and

(v’) t is upper semicontinuous on X × Y and for each y ∈ Y, t(., y) is quasiconcave on X.

Then, infy∈Y supx∈X t(x, y) ≤ supx∈Xinfz∈Z p(x, z) + supz∈Z infy∈Y q(y, z),

with the convention ∞ + (−∞) = ∞.

Proof. It suffices to prove that condition (v) in Theorem 6 is fulfilled. Obviously for each λ <

infy∈Y supx∈X t(x, y) the map T defined in condition (v) of Theorem 6 has nonempty values.

Moreover, by (v’) the values of T are convex. Since t is upper semicontinuous on X × Y the map

T is closed. Since X is compact, by Lemma 1, T is upper semicontinuous with compact values.

Consequently T is a Kakutani map. Since, K(Y, X) ⊂ B(Y, X), it follows that condition (v) from

Theorem 6 is satisfied.



82 Mircea Balaj and Donal O’Regan CUBO
10, 4 (2008)

The results obtained in this section generalize Theorems 19, 20 and Corollary 21 in [1], where

the corresponding map T , in each result, had the KKM property. Obviously, the condition T ∈

B(Y, X) is a weaker one.

Received: February 2008. Revised: March 2008.

References

[1] M. Balaj, Coincidence and maximal element theorems and their applications to generalized

equilibrium problems and minimax inequalities, Nonlinear Anal., 68 (2008), 3962–3971.

[2] H. Ben-El-Mechaiekh, S Chebbi, M. Florenzano and J.-V. Llinares, Abstract con-

vexity and fixed points J. Math. Anal. Appl. 222 (1998), 138–150.

[3] T.-H Chang and C.-L. Yen, KKM property and fixed point theorems, J. Math. Anal.

Appl., 203 (1996), 224–235.

[4] X.P. Ding, New H-KKM theorems and equilibria of generalized games, Indian J. Pure Appl.

Math., 27 (1996), 1057–1071.

[5] X.P. Ding and Y.J. Park, Fixed points and generalized vector equilibrium problems in

generalized convex spaces Indian J. Pure Appl. Math., 34 (2003), 973–990.

[6] X.P. Ding and J.Y. Park, Generalized vector equilibrium problems in generalized convex

spaces, J. Optim. Theory Appl., 120 (2004), 327–353.

[7] J.Y. Fu and A.H. Wan, Generalized vector equilibrium problems with set-valued mappings,

Math. Methods Oper. Res., 56 (2002), 259–268.

[8] G.H. Greco and M.P. Moschen, A minimax inequality for marginally semicontinuous

functions in Minimax Theory and Applications (B. Ricceri, S. Simons eds), Kluwer Acad.

Publ., Dordrecht, 1998, pp. 41–50.

[9] M. Lassonde, Fixed points for Kakutani factorizable multifunctions, J. Math. Anal. Appl.,

152 (1990), 46–60.

[10] L.J. Lin, Q.H. Qnsari and J.Y. Wu, Geometric properties and coincidence theorems with

applications to generalized vector equilibrium problems, J. Optim. Theory Appl., 117 (2003),

121–137.

[11] L.J. Lin and H.L. Chen, The study of KKM theorems with applications to vector equi-

librium problems and implicit vector variational inequalities problems, J. Global Optim., 32

(2005), 135–157.



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10, 4 (2008)

An Intersection Theorem and its Applications 83

[12] L.J. Lin, Z.T. Yu and G. Kassay, Existence of equilibria for multivalued mappings and

its application to vectorial equilibria, J. Optim. Theory Appl., 114 (2002), 189–208.

[13] L.J. Lin and W.P. Wan, KKM type theorems and coincidence theorems with applications

to the existence of equilibrium, J. Optim. Theory Appl., 123 (2004), 105–122.

[14] S. Park, Foundations of the KKM theory via coincidences of composites of upper semicon-

tinuous maps, J. Korean Math. Soc., 31 (1994), 493–519.

[15] S. Park, Fixed points of the better admissible multimaps, Math. Sci. Res. Hot-Line, 1(9)

(1997), 1–6.

[16] N.X. Tan and P.N. Tinh, On the existence of equilibrium points of vector functions, Nu-

merical Functional Analysis Optimiz., 19 (1998), 141–156.


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