DSpace at VNU: On topological existence theorems and applications to optimization-related problems

Math Meth Oper Res
DOI 10.1007/s00186-014-0462-0
ORIGINAL ARTICLE

On topological existence theorems and applications
to optimization-related problems
Phan Quoc Khanh · Lai Jiu Lin · Vo Si Trong Long

Received: 16 June 2013 / Accepted: 17 January 2014
© Springer-Verlag Berlin Heidelberg 2014

Abstract In this paper, we establish a continuous selection theorem and use it to
derive five equivalent results on the existence of fixed points, sectional points, maximal
elements, intersection points and solutions of variational relations, all in topological
settings without linear structures. Then, we study the solution existence of a number of
optimization-related problems as examples of applications of these results: quasivariational inclusions, Stampacchia-type vector equilibrium problems, Nash equilibria,
traffic networks, saddle points, constrained minimization, and abstract economies.
Keywords Continuous selections · Fixed points · Variational relations ·
Quasivariational inclusions · Nash equilibria · Traffic networks
Mathematics Subject Classification

47H10 · 90C47 · 90C48 · 90C99

P. Q. Khanh
Department of Mathematics, International University, Vietnam National University,
Hochiminh City, Vietnam
e-mail:
P. Q. Khanh
Federation University Australia, Ballarat, Victoria, Australia
L. J. Lin
Department of Mathematics, National Changhua University of Education,

Changhua, Taiwan
e-mail:
V. S. T. Long (B)
Department of Mathematics, Cao Thang College of Technology,
Hochiminh City, Vietnam
e-mail:

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1 Introduction
Existence of solutions takes a central place in the theory for any class of problems
and plays also a vital role in applications. Studies of the existence of solutions of a
problem are based on existence results for important points in nonlinear analysis like
fixed points, maximal points, intersection points, etc. During a long period in the past,
it was believed that such existence results needed both topological and linear/convex
structures. But, originated from Wu (1959) and Horvath (1991), two directions of dealing with pure topological existence theorems have been developed. The first approach
is based on replacing convexity assumptions by connectedness conditions, and the
second one on replacing a convex hull by an image of a simplex through a continuous map. Very recently, in Khanh and Quan (2013), a combination of the two ways
was discussed. This paper follows the idea of the second approach. Recently, this
idea was intensively developed in combination with the KKM theory (KKM means
Knaster–Kuratowski–Mazurkiewicz) to obtain pure topological existence theorems
and applications in the study of the existence solutions to optimization-related problems, (see, e.g., Ding 2005, 2007; Hai et al. 2009; Khanh et al. 2011; Khanh and Quan
2010; Khanh et al. 2009; Park 2008; Park and Kim 1996). Inspired by these results
and a definition in Chang and Zhang (1991), in this paper we propose a definition of
a general type of KKM mappings in terms of a GFC-space (defined in Khanh and
Quan (2010); Khanh et al. (2009)) and use it to establish equivalent topological sufficient conditions for the existence of many important points in nonlinear analysis and
apply these conditions to various optimization-related problems. Our results improve

or generalize a number of recent ones in the literature.
The outline of the paper is as follows. Section 2 contains definitions and preliminary
facts for our later use. In Sect. 3, we establish purely topological sufficient conditions
for the existence of important points in nonlinear analysis and prove the equivalence
of these conditions. Then, applications to investigating the solution existence for various optimization-related problems are presented. Section 4 is devoted to existence
theorems on product GFC-spaces and applications to problems concerning systems of
subproblems.
2 Preliminaries
Recall first some definitions for our later use. For a set X , by 2 X and X we denote
the family of all nonempty subsets, and the family of the nonempty finite subsets,
respectively, of X . N, Q, and R denote the set of the natural numbers, the set of rational
numbers, and that of the real numbers, respectively, and R = R ∪ {−∞, +∞}. If X
is a topological space and A ⊂ X , then intA signify the interior of A. Let X and Y
be nonempty sets. For F : Y → 2 X we define F − : X → 2Y and F ∗ : X → 2Y ,
respectively, by F − (x) = {y ∈ Y : x ∈ F(y)} and F ∗ (x) = Y \ F − (x). F − and F ∗
are called the inverse and dual, respectively, map of F. For x ∈ X , F − (x) is called
the inverse image, or the fiber, of F at x, F ∗ (x) is called the cofiber of F at x. n ,
n ∈ N, denotes the standard n-simplex, i.e., the simplex with vertices being the points
e0 = (1, 0, ..., 0), ..., en = (0, ..., 0, 1) of Rn+1 .

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Definition 2.1 (Khanh and Quan 2010) Let X be a topological space, Y a nonempty
set, and a family of continuous mappings ϕ : n → X, n ∈ N. A triple (X, Y, )
is said to be a generalized finitely continuous topological space (GFC-space in short)
if, for each finite subset N = {y0 , y1 , ..., yn } ∈ Y , there is ϕ N : n → X of the
family . If X is compact, (X, Y, ) is called a compact GFC-space. Later we also

use (X, Y, {ϕ N }) to denote (X, Y, ).
Observe that a GFC-space is equipped only with topological structures, without
linear or convex structures. The same notion was introduced in Park (2008) under
the name “ A -space” (independently and earlier than the GFC-space). These spaces
are generalizations of other topological structures as G-convex spaces Park and Kim
(1996), FC-spaces Ding (2005, 2007), etc, in order to study topics of existence, mainly
in optimization-related problems, without linear or convex structures. Note that if
Y = X , then (X, Y, ) (written as (X, )) collapses to a FC-space. Allowing to take
Y different from X may help to have a suitable family {ϕ N : N ∈ Y } in many
situations, see, e.g., Example 3.1 below.
Definition 2.2 (Hai et al. 2009; Khanh et al. 2011; Khanh and Quan 2010; Khanh
et al. 2009) Let (X, Y, ) be a GFC-space, Z a topological space, F : Y → Z and
T : X → 2 Z . F is said to be a KKM mapping with respect to T (T -KKM mapping
in short) if, for each N = {y0 , ..., yn } ∈ Y and each {yi0 , ..., yik } ⊂ N , one has
T (ϕ N ( k )) ⊂ ∪kj=0 F(yi j ), where ϕ N ∈ is corresponding to N and k is the face
of n formed by {ei0 , ..., eik }.
Definition 2.3 Let (X, Y, ) be a GFC-space, Z a topological space, A a nonempty
set, F : A → 2 Z , and T : X → 2 Z . F is said to be a general KKM mapping with
respect to T (g-T -KKM mapping in short) if, for each N A = {a0 , ..., an } ∈ A , there
exists N = {y0 , ..., yn } ∈ Y such that, for each {i 0 , ..., i k } ⊂ {0, ..., n}, one has
T (ϕ N ( k )) ⊂ kj=0 F(ai j ), where ϕ N ∈ is corresponding to N and k is the face
of n formed by {ei0 , ..., eik }.
Note that Definition 2.3 is a natural generalization of Definition 2.1 of Chang and
Zhang (1991), where X = Y = Z is a topological vector space, A is a convex subset
of another topological vector space, T is the identity map, and ϕ N (·) = co(·) (the
usual convex hull). Consequently, it also generalizes Definition 2.1 of Ansari et al.
(2000). We also see that every T -KKM mapping is a g-T -KKM when A = Y , but the
converse is not true as explained by the following example.
Example 2.1 Let X = Z = R and Y = Q. For each N = {y0 , ..., yn } ∈ Y let ϕ N
n

n
be defined by ϕ N (e) = i=0
λi yi for all e = i=0
λi ei ∈ n . Clearly, (X, Y, {ϕ N })
Z
is a GFC-space. Let F : Y → 2 be given by F(y) ≡ [0, +∞) and T be the identity
map. Let N = {−1}. Then,
T (ϕ N (

0 ))

= {−1} ⊂ F(−1) = [0, +∞).

Hence, F is not a T -KKM mapping. Now, for each N A = {a0 , ..., an } ∈ A = Y ,
we take N = {y0 , ..., yn } = {|a0 |, ..., |an |} ∈ Y , where | · | denotes absolute value.
It is easy to see that

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P. Q. Khanh et al.

T (ϕ N (

n ))

= [minN , maxN ] ⊂ [0, +∞) = F(ai ), ∀i ∈ {0, ..., n}.

This means that F is a g-T -KKM.
Lemma 2.1 Let (X, Y, {ϕ N }) be a GFC-space, Z a topological space, A a nonempty

set, H : Z → 2 A , and T : Z → 2 X . Then, the following statements are equivalent
(i) for each z ∈ Z and N A = {a0 , ..., an } ∈ A , there exists N = {y0 , y1 , ..., yn } ∈
Y such that, for each {ai0 , ..., aik } ⊂ N A ∩ H (z), one has ϕ N ( k ) ⊂ T (z),
where k is the simplex formed by {ei0 , ..., eik };
(ii) H ∗ is a g-T ∗ -KKM mapping.
Proof (i) ⇒ (ii). Suppose to the contrary that N A = {a0 , ..., an } ∈ A exists such
that, for each N = {y0 , y1 , ..., yn } ∈ Y , there exists {i 0 , ..., i k } ⊂ {0, ..., n},
k

T ∗ (ϕ N (

k ))

⊂

H ∗ (ai j ).

(1)

j=0

We choose N given in condition (i) associated with N A . By (1), there are x0 ∈
ϕ N ( k ) and z 0 ∈ T ∗ (x0 ) such that
k

/
z0 ∈

H ∗ (ai j ) =

j=0

k

Z \ H − (ai j ) = Z \

j=0

k

H − (ai j ).

j=0

It follows that {ai0 , ..., aik } ⊂ N A ∩ H (z 0 ). Hence, from (i) one has ϕ N ( k ) ⊂
/ Z \ T − (x0 ) = T ∗ (x0 ), a contradiction.
T (z 0 ). Hence, z 0 ∈ T − (x0 ), and so z 0 ∈
(ii) ⇒ (i). Suppose there exist z 0 ∈ Z and N A = {a0 , ..., an } ∈ A such that, for
each N = {y0 , ..., yn } ∈ Y , there exists {ai0 , ..., aik } ⊂ N A ∩ H (z 0 ) such that
ϕN (

k)

⊂ T (z 0 ).

(2)

Since H ∗ is a g-T ∗ -KKM mapping and N is arbitrary, one can take N associated
with N A such that
T ∗ (ϕ N (

k
k ))

⊂

H ∗ (ai j ).

(3)

j=0

Since {ai0 , ..., aik } ⊂ N A ∩ H (z 0 ), one has z 0 ∈
k

z0 ∈
/ Z\
j=0

By (2), there is x0 ∈ ϕ N (

k)

H − (ai j ) =

k

H − (ai j ), i.e.,

H ∗ (ai j ).

(4)

j=0

such that x0 ∈
/ T (z 0 ). This means that

z 0 ∈ T ∗ (x0 ) ⊂ T ∗ (ϕ N (
In view of (3), (5) contradicts (4).

123

k
j=0

k )).

(5)


On topological existence theorems and applications...

To end this section, we state our variational relation problem. For a set U and a
point x under consideration, we adopt the notations
α1 (x; U ) means ∀x ∈ U ; α2 (x; U ) means ∃x ∈ U.
Let X and Z be nonempty sets, S : X → 2 X and F : X × X → 2 Z have nonempty
values, and R(x, w, z) be a relation linking x ∈ X , w ∈ X and z ∈ Z . Our problem
is, for α ∈ {α1 , α2 },
(VRα ) find x¯ ∈ X such that , ∀w ∈ S(x),

¯ α(z, F(x,
¯ w)),
R(x,
¯ w, z) holds.
Note that the first variational relation problem was investigated in Khanh and Luc
(2008); Luc (2008). Developments have been obtained in some papers, (e.g., Balaj
and Lin 2010, 2011; Khanh and Long 2013; Khanh et al. 2011; Lin 2012; Luc et al.
2010).
3 Topological existence theorems and applications to optimization-related
problems
3.1 Topological existence theorems
In this subsection, we establish the existence of important objects in applied analysis in
pure topological settings of GFC-spaces. Let us begin with the existence of continuous
selections of set-valued maps. For a set-valued map T : X → 2 Z between two
topological spaces X and Z , recall that a (single-valued) continuous map t : X → Z
is called a continuous selection of T if t (x) ∈ T (x) for all x ∈ X .
Theorem 3.1 (continuous selections) Let Z be a compact topological space,
(X, Y, ) be a GFC-space, and T : Z → 2 X be a set-valued mapping with nonempty values. Assume that there are a nonempty set A and H : Z → 2 A such that the
following conditions hold
(i) H ∗ is a g-T ∗ -KKM mapping;
(ii) Z = a∈A int H − (a).
Then, T has a continuous selection of the form t = ϕ ◦ ψ for continuous maps
ϕ : n → X and ψ : Z → n , for some n ∈ N.
Proof Since Z is compact, by (ii), there exists N A = {a¯ 0 , ..., a¯ n } ∈ A such that
n
n
int H − (a¯i ). Then, there is a continuous partition of unity {ψi }i=0
of Z
Z = i=0
n

−1
associated with the finite open cover {int H (a¯ i )}i=0 . From (i) there exists N =
{y0 , ..., yn } ∈ Y associated with N A = {a¯ 0 , .., a¯ n }. Moreover, due to the GFC-space
structure, there is ϕ N : n → X corresponding to N . Now, we define the continuous
maps ψ : Z → n and t : Z → X , respectively, by
n

ψ(z) =

ψi (z)ei , t (z) = (ϕ N ◦ ψ)(z).
i=0

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P. Q. Khanh et al.

Suppose to the contrary that t is not a selection of T , i.e., there exists z 0 ∈ Z and
/ T (z 0 ), or equivalently, z 0 ∈ Z \ T − (x0 ) = T ∗ (x0 ).
t (z 0 ) = ϕ N (ψ(z 0 )) := x0 ∈
Furthermore, one has
n

ψ(z 0 ) =

ψi (z 0 )ei =

ψ j (z 0 )e j ∈

J (z 0 ) ,

j∈J (z 0 )

i=0

where J (z 0 ) := { j ∈ {0, ..., n} : ψ j (z 0 ) = 0}. Since H ∗ is a g-T ∗ -KKM mapping,
one has
z 0 ∈ T ∗ (x0 ) = T ∗ (ϕ N (ψ(z 0 ))) ⊂ T ∗ (ϕ N (

J (z 0 ) ))

H ∗ (a¯ i ).

⊂
i∈J (z 0 )

Hence, there exists j0 ∈ J (z 0 ), z 0 ∈ H ∗ (a¯ j0 ) = Z \ H − (a¯ j0 ), i.e.,
/ H − (a¯ j0 ).
z0 ∈

(6)

n ,
On the other hand, by the definition of J (z 0 ) and the partition {ψi }i=0

z 0 ∈ {z ∈ Z : ψ j0 (z) = 0} ⊂ int H − (a¯ j0 ) ⊂ H − (a¯ j0 ),
contradicting (6). Finally, putting ϕ = ϕ N we arrive at the conclusion.
Remark 1 Theorem 3.1 improves Theorem 2.1 of Khanh et al. (2011) since assumption (i) is weaker than the corresponding assumption (i) of that result. Consequently,
it also improves Theorem 2.1 of Ding (2007), Theorem 3.1 of Yannelis and Prabhakar
(1983), and Theorem 1 of Yu and Lin (2003). The next example gives a case where Theorem 3.1 is more convenient than Theorem 2.1 of Ding (2007) in terms of FC-spaces

and Theorem 2.1 of Khanh et al. (2011) with a GFC-space setting. Recall that for a
FC-space (X, ) and A, B ⊂ X , B is said to be a FC-subspace of X relative to A if,
for each N = {x0 , ..., xn } ∈ X and {xi0 , ..., xik } ⊂ N ∩ A, ϕ N ( k ) ⊂ B.
Example 3.1 Let Z = [0, 1], X = R, and F, T : Z → 2 X be defined by F(z) ≡ X ,
T (z) = [0, z]. For the continuous functions ϕ N : n → X defined by, for each
e ∈ n,
ϕ N (e) =

0,
if N ∈ N ,
√
2, otherwise

(we adopt that N contains also zero), (X, {ϕ N } N ∈ X ) is a FC-space. We show that
assumption (i) of Theorem 2.1 in Ding (2007) is not satisfied.
For N ∈
/ N , z ∈ Z , and
√
{xi0 , ..., xik } ⊂ N ∩ F(z) = N , one has ϕ N ( k ) = { 2} ⊂ T (z) = [0, z] ⊂ [0, 1].
This means that T (z) is not a FC-subspace of X relative to F(z), as that assumption
(i) requires. Now, take the GFC-space (X, Y, {ϕ N } N ∈ Y ) with Y = Q. We claim that
assumption (i) of Theorem 2.1 in Khanh et al. (2011) is not fulfilled, i.e., there does
not exist any map H : Z → 2Y such that, for each z ∈ Z , N = {y0 , .., yn } ∈ Y and

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On topological existence theorems and applications...

{yi0 , ..., yik } ⊂ N ∩ H (z), one has ϕ N ( k ) ⊂ T (z). Indeed, suppose to the contrary

that such a H exists. We take z¯ ∈ Z and N = { y¯0 , ..., y¯n } ∈ Y√ such that N ∩ H (¯z ) =
{ y¯i0 , ..., y¯ik } = ∅. Set N ∗ = N ∪ {0.5}. Then, ϕ N ∗ ( k ) = { 2} ⊂ T (¯z ) = [0, z¯ ], a
contradiction. To apply our Theorem 3.1, we take A = N, and H : Z → 2 A defined
by H (z) = F(z) ∩ N. For each z ∈ Z and N A = {a0 , ..., an } ∈ A = N , we
choose N ≡ N A ∈ Y to see that, for each {ai0 , ..., aik } ⊂ N A ∩ H (z) = N A ,
ϕ N ( k ) = {0} ⊂ T (z) = [0, z], i.e., (i) of Lemma 2.1, which is equivalent to (i)
of Theorem 3.1, is fulfilled. (ii) of Theorem 3.1 is satisfied because Z = H − (0) =
intH − (0). By Theorem 3.1, T has a continuous selection.
We now apply the above result on continuous selections to prove the following five
topological existence results. We will first demonstrate a result on fixed points, and
then show that it is equivalent to all the other four theorems.
Theorem 3.2 (fixed points) Let (X, Y, ) be a compact GFC-space and T : X → 2 X .
Assume that there are a nonempty set A and H : X → 2 A such that the following
conditions hold
(i) T has nonempty values and H ∗ is a g-T ∗ -KKM mapping;
(ii) X = a∈A int H − (a).
Then, T has a fixed point x¯ ∈ X , i.e., x¯ ∈ T (x).
¯
Proof According to Theorem 3.1, T has a continuous selection t = ϕ ◦ ψ, where
ϕ : n → X and ψ : X → n are continuous. Then, ψ ◦ ϕ : n → n is also
continuous. By virtue of the Tikhonov fixed-point theorem, there exists e¯ ∈ n such
that ψ ◦ ϕ(e)
¯ = e.
¯ Setting x¯ = ϕ(e),
¯ we have
x¯ = ϕ(ψ(x))
¯ = t (x)
¯ ∈ T (x).
¯
The proof is complete.

Remark 2 Theorem 3.2 sharpens Corollary 3.1 (ii1 ) of Khanh et al. (2011) since
assumption (i) is weaker than the corresponding assumption (i) of that result. Applied
to the special case where X = Y = A is a nonempty compact convex subset of
a topological vector space, T ≡ H , and ϕ N (·) = co(·), Theorem 3.2 generalizes
Theorem 1 of Browder (1968).
Theorem 3.3 (sectional points) Let (X, Y, ) be a compact GFC-space and M be a
subset of X × X . Assume that there are a nonempty set A and H : X → 2 A such that
the following conditions hold
(i) for each x ∈ X and N A = {a0 , ..., an } ∈ A , there exists N = {y0 , ..., yn } ∈ Y
/
such that, for each {ai0 , ..., aik } ⊂ N A ∩ H (x), ϕ N ( k ) ⊂ {w ∈ X : (x, w) ∈
M};
(ii) X = a∈A int H − (a);
(iii) (x, x) ∈ M for all x ∈ X .
Then, there exists x¯ ∈ X such that {x}
¯ × X ⊂ M.

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Observe that, for a similar reason as in Remark 2, Theorem 3.3 generalizes Lemma
4 of Fan (1961).
Theorem 3.4 (maximal elements) Let (X, Y, ) be a compact GFC-space and T :
X → 2 X . Assume that there are a nonempty set A and H : X → 2 A such that the
following conditions hold
(i) H ∗ is a g-T ∗ -KKM mapping;
(ii) w∈X T − (w) ⊂ a∈A int H − (a);
(iii) x ∈

/ T (x) for all x ∈ X .
Then, T has a maximal point x¯ ∈ X , i.e., T (x)
¯ = ∅.
Theorem 3.5 (intersection points) Let (X, Y, ) be a compact GFC-space and G :
X → 2 X . Assume that there are a nonempty set A and H : X → 2 A such that the
following conditions hold
(i) H ∗ is a g-G-KKM mapping;
(ii) x∈X X \ G(x) ⊂ a∈A int H − (a);
(iii) x ∈ G(x) for all x ∈ X .
Then,

x∈X

G(x) = ∅.

Theorem 3.6 (solutions of variational relations) Let (X, Y, ) be a compact GFCspace, Z a nonempty set, S : X → 2 X , F : X × X → 2 Z , and R(x, w, z) be a relation
linking x ∈ X , w ∈ X and z ∈ Z , and i ∈ {1, 2}. Assume that there are a nonempty
set A and H : X → 2 A such that the following conditions hold
(i) for each x ∈ X and N A = {a0 , ..., an } ∈ A , there exists N = {y0 , ..., yn } ∈ Y
such that, for each {ai0 , ..., aik } ⊂ N A ∩ H (x), one has ϕ N ( k ) ⊂ {w ∈ S(x) :
α3−i (z, F(x, w)), R(x, w, z) does not hold};
(ii) w∈X {x ∈ S − (w) : α3−i (z, F(x, w)), R(x, w, z) does not hold} ⊂
a∈A
int H − (a);
(iii) x ∈
/ {w ∈ S(x) : α3−i (z, F(x, w)), R(x, w, z) does not hold} for all x ∈ X .
Then, there exists a solution x¯ ∈ X of (VRαi ), i.e., ∀w ∈ S(x),
¯ αi (z, F(x,
¯ w)),
R(x,

¯ w, z) holds.
Now we will prove the equivalence of the above five theorems following the diagram
Theorem 3.2 ⇒ Theorem 3.4 ⇒ Theorem 3.6 ⇒ Theorem 3.2
Theorem 3.3

Theorem 3.5

Theorem 3.2 ⇒ Theorem 3.3. Let T (x) = {w ∈ X : (x, w) ∈
/ M} for x ∈ X . If
there is x¯ ∈ X such that T (x)
¯ = ∅, then {x}
¯ × X ⊂ M and we are done. Suppose
T (x) = ∅ for all x ∈ X and the conclusion of Theorem 3.3 is false. By (i) of Theorem
3.3 and Lemma 2.1, (i) of Theorem 3.2 is satisfied. Since the two assumptions (ii) are
¯ w) ∈
/ M},
the same, by Theorem 3.2, x¯ ∈ T (x)
¯ for some x¯ ∈ X , i.e., x¯ ∈ {w ∈ X : (x,
which contradicts (iii) of Theorem 3.3.

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On topological existence theorems and applications...

Theorem 3.3 ⇒ Theorem 3.2. Assume that all the assumptions of Theorem 3.2 are
fulfilled and set M = {(x, w) ∈ X × X : w ∈
/ T (x)}. Suppose to the contrary that
x ∈
/ T (x) for all x ∈ X . Then, (x, x) ∈ M for all x ∈ X , i.e., (iii) of Theorem 3.3

holds. According to Lemma 2.1, for each x ∈ X and N A = {a0 , ..., an } ∈ A , there
exists N = {y0 , ..., yn } ∈ Y such that, for each {ai0 , ..., aik } ⊂ N A ∩ H (x), one
/ M. It
has ϕ N ( k ) ⊂ T (x). Hence, for each w ∈ ϕ N ( k ), w ∈ T (x), i.e., (x, w) ∈
/ M}, i.e., (i) of Theorem 3.3 is satisfied.
follows that ϕ N ( k ) ⊂ {w ∈ X : (x, w) ∈
Since the two assumptions (ii) are the same, applying Theorem 3.3, one obtains x¯ ∈ X
such that {x}
¯ × X ⊂ M. It implies that w ∈
/ T (x)
¯ for all w ∈ X , contradicting the
assumption that T has the nonempty values.
Theorem 3.4 ⇒ Theorem 3.5. We set T (x) = X \ G − (x) for x ∈ X. Then,
−
T (x) = X \G(x) and T ∗ (x) = G(x). It is not hard to see that, under the assumptions
of Theorem 3.5, all assumptions of Theorem 3.4 are fulfilled. Therefore, there exists
¯ = ∅. Hence, x¯ ∈ x∈X G(x).
x¯ ∈ X such that T (x)
¯ = ∅, i.e., X \ G − (x)
Theorem 3.5 ⇒ Theorem 3.4. Under the assumptions of Theorem 3.4, let G(x) =
X \ T − (x) for x ∈ X . Then, assumptions (i) and (ii) of Theorem 3.4 clearly imply
the corresponding assumptions (i) and (ii) of Theorem 3.5. From (iii) of Theorem 3.4,
one has x ∈ X \ T − (x) = G(x) for all x ∈ X , i.e., (iii) of Theorem 3.5 is satisfied.
By Theorem 3.5, there exists x¯ ∈ x∈X G(x) = X \ x∈X T − (x). It follows that
¯ = ∅.
x¯ ∈
/ x∈X T − (x), that is, T (x)
Theorem 3.2 ⇒ Theorem 3.4. Suppose that T (x) = ∅ for each x ∈ X . Then, (i) of
Theorem 3.4 implies (i) of Theorem 3.2. Since X = w∈X T − (w), (ii) of Theorem
3.2 is satisfied along with (ii) of Theorem 3.4. By Theorem 3.2, T has a fixed point.

This contradicts (iii) of Theorem 3.4 and we are done.
Theorem 3.4 ⇒ Theorem 3.6. Let T : X → 2 X be defined by
T (x) = {w ∈ S(x) : α3−i (z, F(x, w)), R(x, w, z) does not hold}.
By (i) of Theorem 3.6 and Lemma 2.1, H ∗ is a g-T ∗ -KKM mapping, i.e., (i) of
Theorem 3.4 is fulfilled. It is not difficult to see that (ii) and (iii) of Theorem 3.6 imply
the corresponding (ii) and (iii) of Theorem 3.4. Applying this theorem, we have x¯ ∈ X
¯ w)), R(x,
¯ w, z) holds.
such that T (x)
¯ = ∅. Consequently, ∀w ∈ S(x),
¯ αi (z, F(x,
Theorem 3.6 ⇒ Theorem 3.2. Let the assumptions of Theorem 3.2 be satisfied.
We define two mappings S : X → 2 X , F : X × X → 2 Z and a relation R by, for
x, w ∈ X ,
S(x) ≡ X, F(x, w) = {z 0 } for an arbitrary z 0 ∈ Z ,
αi (z, F(x, w)), R(x, w, z) holds ⇔ w ∈
/ T (x).
Then, one has, for all x ∈ X ,
{w ∈ S(x) : α3−i (z, F(x, w)), R(x, w, z) does not hold} = T (x).
Suppose, for all x ∈ X , x ∈
/ T (x). Then, (iii) of Theorem 3.6 is fulfilled. Clearly,
by (ii) of Theorem 3.2, (ii) of Theorem 3.6 is fulfilled. Since H ∗ is a g-T ∗ -KKM
mapping, by Lemma 2.1, (i) of Theorem 3.6 holds. According to this theorem, x¯ ∈ X

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P. Q. Khanh et al.

exists such that, for all w ∈ S(x)

¯ = X, αi (z, F(x,
¯ w)), R(x,
¯ w, z) holds. This means
that w ∈
/ T (x)
¯ for all w ∈ X , contradicting the assumption (i) of Theorem 3.2 that T
has nonempty values.
Remark 3 The existence of the above-mentioned points has been obtained in a number
of contributions, to various extents of generality and relaxation of assumptions, see,
e.g., recent papers Hai et al. (2009); Khanh et al. (2011); Khanh and Quan (2010);
Khanh et al. (2009). Our assumption (i) here is of a very simple form and directly in
terms of a general KKM mapping (the map T or H ∗ ) with respect to another map.
Example 2.1 shows that being such a general KKM mapping is properly weaker than
being a usual KKM mapping following Definition 2.2, and hence assumption (i) is
weaker than the existing corresponding conditions.
3.2 Optimization-related problems
A. Quasivariational inclusion problems. Now we consider the following quasivariational inclusion problem. For any given sets U and V , we adopt the notations
r1 (U, V ) means U ∩ V = ∅; r2 (U, V ) means U ⊆ V ;
r3 (U, V ) means U ∩ V = ∅; r4 (U, V ) means U

V,

and the convention that r5 = r1 , r6 = r2 . Let X, Z , Z be nonempty sets, S : X → 2 X ,
F : X × X → 2 Z , G : X × Z → 2 Z and K : X × X × Z → 2 Z . For r ∈ {r1 , r2 , r3 , r4 }
and α ∈ {α1 , α2 }, we consider the following quasivariational inclusion problem
(QIPr α ) find x¯ ∈ X such that, ∀w ∈ S(x),
¯ α(z, F(x,
¯ w)),
r (K (x,
¯ w, z), G(x,

¯ z)).
This formulation was proposed in Hai et al. (2009) and has been used in some papers,
(e.g., Hai et al. 2009; Khanh and Long 2013; Khanh et al. 2011). It looks complicated,
but the used notations make it include much more particular cases with similar proofs
of the existence of solutions.
Theorem 3.7 For problem (QIPr j αi ), j = 1, ..., 4 and i = 1, 2, assume that there are
Y and such that (X, Y, ) is a compact GFC-space. Assume, further that there are
a nonempty set A and H : X → 2 A such that the following conditions hold
(i) for each x ∈ X and N A = {a0 , ..., an } ∈ A , there exists N = {y0 , ..., yn } ∈
Y such that, for each {ai0 , ..., aik } ⊂ N A ∩ H (x), ϕ N ( k ) ⊂ {w ∈ S(x) :
α3−i (z, F(x, w)), r j+2 (K (x, w, z), G(x, z))};
(ii) w∈X {x ∈ S − (w) : α3−i (z, F(x, w)), r j+2 (K (x, w, z), G(x, z))} ⊂ a∈A
int H − (a);
(iii) x ∈
/ {w ∈ S(x) : α3−i (z, F(x, w)), r j+2 (K (x, w, z), G(x, z))} for all x ∈ X .
Then, problem (QIPr j αi ) has a solution.

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On topological existence theorems and applications...

Proof Employ Theorem 3.6 with the relation R defined by
R(x, w, z) holds if and only if r j (F(x, w, z), G(x, z)).
Under the assumptions of Theorem 3.7, the assumptions of Theorem 3.6 are easily
seen to be satisfied. Hence, x¯ ∈ X exists such that, ∀w ∈ S(x),
¯ αi (z, F(x,
¯ w)),
R(x,
¯ w, z) holds. Consequently, x¯ is a solution of (QIPri α j ) in this case.

B. Stampacchia-type vector equilibrium problems. Since problem (QIPr j αi )
includes most of optimization-related problems, sufficient conditions for the existence
of their solutions can be derived directly from Theorem 3.7. Here, we mention only
some important problems as examples, also for the sake of comparison with several
recent existing results. First, we discuss a relatively general model of Stampacchia-type
vector equilibrium problems, and then apply the obtained result to other ones.
Let X, Z , S, F, and K be as for problem (QIPr α ). Let Z be a linear space and C :
X → 2 Z be nonempty-convex-cone-valued. For r ∈ {r1 , r2 , r3 , r4 } and α ∈ {α1 , α2 },
we consider the following Stampacchia-type vector equilibrium problem
(VEPr α ) find x¯ ∈ X such that, ∀w ∈ S(x),
¯ α(z, F(x,
¯ w)),
r (K (x,
¯ w, z), (−C(x)
¯ \ {0})).
Then, Theorem 3.7 becomes a sufficient condition for the existence of solutions to
problem (VEPr α ) with the simple replacement of G(x, z) by −C(x) \ {0}. Note that,
here we do not need Z to be equipped with a topology, and C to be closed-valued, as
assumed in Lin (2012). The following example provides, for problem (VEPr α ), a case
when Theorem 3.7 is applicable, while a recent existing result is not.
Example 3.2 Let X = Z = Z = R ∪ {±∞}, S : X → 2 X , F : X × X → 2 Z , K :
X × X × Z → 2 Z , and C : X :→ 2 Z be given by
S(x) =

{0},
if x = 1,
{1, 1 − x}, if x = 1,

F(x, w) ≡ {0}, K (x, w, z) = w2 (x 2 + 1), C(x) ≡ (−∞, 0].
Problem (VEPr4 α ) in this case is: find x¯ ∈ X such that, ∀w ∈ S(x),

¯
w 2 (x 2 + 1) ∈
/ (0, +∞).
To have a GFC-space, take Y = N and {ϕ N } defined by, for N ∈ Y and e ∈
ϕ N (e) =

n,

1, if 1 ∈
/ N,
0, otherwise.

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P. Q. Khanh et al.

Then, (X, Y, {ϕ N }) is a GFC-space. Next, we choose A = Y and H : X → 2 A defined
by
H (x) =

N, if x = 1,
∅, if x = 1.

To check assumption (i) of Theorem 3.7, consider x ∈ X and N A = {a0 , ..., an } ∈ A .
If x = 1 we take N = {a0 + 2, ..., an + 2} ∈ Y to see that, for each {ai0 , ..., aik } ⊂
N A ∩ H (x) = N A ,
ϕN (

k)

= {1} ⊂ {w ∈ S(x) : w 2 (x 2 + 1) ∈ (0, +∞)} = {1, 1 − x}.

If x = 1 then N A ∩ H (1) = ∅. Hence, assumption (i) of Theorem 3.7 is fulfilled.
Clearly, intH − (a) = int(X \ {1}) = X \ {1} for all a ∈ A, and so a∈A intH − (a)=X \
{1}. For (ii) we only need to show that
{x ∈ S − (w) : w 2 (x 2 + 1) ∈ (0, +∞)}.

1∈
/
w∈X

Suppose 1 ∈ S − (w)
¯ for some w¯ ∈ X such that w¯ 2 (12 + 1) ∈ (0, +∞). By the
/ (0, +∞), a contradiction.
definition of S, w¯ = 0. It implies that w¯ 2 (12 + 1) = 0 ∈
This means that assumption (ii) of Theorem 3.7 is satisfied. For each x ∈ X , it is easy
to see that
x∈
/

{w ∈ S(x) : w 2 (x 2 + 1) ∈ (0, +∞)} = {1, 1 − x}, if x = 1
if x = 1.
{w ∈ S(1) : w2 (12 + 1) ∈ (0, +∞)} = ∅,

Thus, (iii) of Theorem 3.7 is checked. We can also check directly that x¯ = 1 is a
solution. However, Theorem 4.1 of Lin (2012) does not work, since, for each x ∈ X ,
/ [0, +∞) for all w ∈ S(y), i.e., assumption
there is no y ∈ X such that w2 (x 2 + 1) ∈
(ii) of that theorem is not satisfied.

C. Nash equilibria. Let I = {1, ..., n} be a set of players and X 1 , ..., X n nonempty
sets. A n-person non-cooperative game is a 2n-tuple (X 1 , X 2 , ..., X n , f 1 , f 2 , ..., f n ),
where the ith player has the nonempty strategy set X i and the pay-off function f i :
X := i∈I X i → R. For a point x ∈ X , xiˆ stands for its projection onto X iˆ = j =i X j .
A point x¯ = (x¯1 , x¯2 , ..., x¯n ) ∈ X is said to be a Nash equilibrium point of if, for all
i ∈ I and wi ∈ X i ,
¯ ≥ f i (x¯iˆ , wi ).
f i (x)
n
: X × X → R by
(x, w) = i=1
( f i (x) − f (xiˆ , wi )). Then, x¯ is a
We define
Nash equilibrium point of if and only if x¯ is a solution of the equilibrium problem

(NEP) find x¯ ∈ X such that, for all w ∈ X,

123

(x,
¯ w) ≥ 0.


On topological existence theorems and applications...

Theorem 3.8 For the game , assume that there are Y and such that (X, Y, ) is a
compact GFC-space. Assume further that there are a nonempty set A and H : X → 2 A
such that the following conditions hold
(i) for x ∈ X and each N A = {a0 , ..., an } ∈ A , there exists N = {y0 , ..., yn } ∈ Y
such that, for each {ai0 , ..., aik } ⊂ N A ∩ H (x), one has ϕ N ( k ) ⊂ {w ∈ X :

(x, w) < 0};
(ii) w∈X {x ∈ X : (x, w) < 0} ⊂ a∈A int H − (a).
Then,

has a Nash equilibrium point.

Proof We simply apply Theorem 3.7 with r = r1 , S(x) ≡ X , Z = R, G(x) ≡
(x, w) (problem (NEP) does not include Z , α, and F).
(−∞, 0), and K (x, w) =
In this case, we note that (iii) of Theorem 3.7 is always satisfied.
Example 3.3 Consider a 2-person non-cooperative game with X 1 = X 2 = [0, 1] ∪
[2, 3], f 1 (x1 , x2 ) = 2x1 − 3x2 , and f 2 (x1 , x2 ) = x1 + 2x2 . This game is equivalent
to the equilibrium problem: find x¯ = (x¯1 , x¯2 ) ∈ X := X 1 × X 2 such that, for all
w = (w1 , w2 ) ∈ X ,
(x,
¯ w) = x¯1 + x¯2 − w1 − w2 ≥ 0.
Then, for Y = X and ϕ N defined by, for N ∈ Y and e ∈
ϕ N (e) =

n,

(3, 3), if N ∈ [2, 3] × [2, 3] ,
(0, 0) otherwise,

(X, Y, {ϕ N }) is a compact GFC-space. We choose A = Y and H : X → 2 A defined
by
H (x) =

A, if x = (3, 3),
∅, if x = (3, 3).

Consider x = (x1 , x2 ) ∈ X and N A = {a0 , ..., an } ∈ A . If x = (3, 3), we take
any N = {y0 , ..., yn } ∈ [2, 3] × [2, 3] ⊂ Y to see that, for each {ai0 , ..., aik } ⊂
N A ∩ H (x) = N A , one has
ϕN (

k)

= {(3, 3)} ⊂ {w ∈ X :

(x, w) < 0} = {w ∈ X : x1 + x2 − w1 − w2 < 0}.

If x = (3, 3), then N A ∩ H (3, 3) = ∅. Hence, assumption (i) of Theorem 3.8 is
fulfilled. Clearly, intH − (a) = int(X \ {(3, 3)}) = X \ {(3, 3)} for all a ∈ A, and so
−
a∈A intH (a)=X \ {(3, 3)}. (ii) of Theorem 3.8 is satisfied because
{x ∈ X :

(3, 3) ∈
/
w∈X

(x, w) < 0} =

{x ∈ X : x1 + x2 − w1 − w2 < 0}.
w∈X

Hence, all assumptions of Theorem 3.8 are checked. A direct computation gives that
the solution set is {(3, 3)}. However, trying with Corollary 3.5 of Hai et al. (2009), for
x = (0, 0), we see that the set

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P. Q. Khanh et al.

{y ∈ X :

(x, y) < 0} = ((0, 1] ∪ [2, 3]) × ((0, 1] ∪ [2, 3])

is not convex, and hence assumption (i) of that result is not satisfied.
D. Traffic networks. Let a network consist of nodes and links (or arcs). Let Q =
(Q 1 , ..., Q l ) be the set of pairs called O/D pairs, each of which consists of an origin
node and a destination one. Assume that P j , j = 1, ..., l, is the set of paths connecting
the pair Q j , and that P j includes r j ≥ 1 paths. Let m = r1 + ... + rl , and x =
(x1 , ..., xm ) denote a path (vector) flow. Assume that the constraint of the capacity of
paths is of the form
X = {x ∈ Rm : 0 ≤ xs ≤

s, s

= 1, ..., m}.

Let a vector cost T (x) = (T1 (x), ..., Tm (x)) be a multifunction of flow x. Let the travel
demand g j of the O/D pair Q j depend on equilibrium (vector) flow x¯ and denote the
travel vector demand by g = (g1 , ..., gl ). Denote the Kronecker numbers by
1, if s ∈ P j ,
0, if s ∈
/ Pj ,
φ = (φ js ), j = 1, ..., l, s = 1, ..., m.

φ js =

Then, the set of all feasible path flows is
S(x)
¯ = {x ∈ X : φx = g(x)}.
¯
For the case of multivalued costs, the following generalization of the Wardrop
equilibrium was proposed in Khanh and Luu (2004).
Definition 3.1 (i) A feasible path flow x¯ is said to be a weak equilibrium flow if, ∀Q j ,
¯
∀q, s ∈ P j , ∃z ∈ T (x),
[z q < z s ] ⇒ [x¯q =

q

or x¯s = 0],

for j = 1, ..., l and q, s = 1, ..., r j .
(ii) A feasible path flow x¯ is called a strong equilibrium flow if (i) is satisfied with
∃z ∈ T (x)
¯ replaced by ∀z ∈ T (x).
¯
In Khanh and Luu (2004), it is proved that a feasible path flow x¯ is a strong (weak)
equilibrium flow if and only if x¯ is a solution of the quasivariational inequality, with
α = α1 (α = α2 , respectively),
¯ such that, ∀w ∈ S(x),
¯ α(z, T (x)),
¯
(QVIα ) find x¯ ∈ S(x)

z, w − x¯ ≥ 0.
Theorem 3.9 For our traffic network problem, with α ∈ {α1 , α2 }, assume that there
are Y and such that (X, Y, ) is a compact GFC-space. Assume further that there
are a nonempty set A and H : X → 2 A such that the following conditions hold

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On topological existence theorems and applications...

(i) for each x ∈ X and N A = {a0 , ..., an } ∈ A , there exists N = {y0 , ..., yn } ∈ Y
such that, for each {ai0 , ..., aik } ⊂ N A ∩ H (x), ϕ N ( k ) ⊂ {w ∈ X :
α3−i (z, T (x)), z, x − w < 0};
(ii) w∈X {x ∈ X : α3−i (z, T (x)), z, x − w < 0} ⊂ a∈A int H − (a).
Then, the traffic problem has a solution.
Proof Apply Theorem 3.7 with r = r1 , X = Rm , Z = (Rm )∗ (the dual of X ), Z = R,
F(x, w) = T (x) for all w ∈ X , G(x, w) ≡ (−∞, 0) and K (x, w, z) = z, w − x .
E. Saddle points. Let B, D be topological spaces and f a real function on B × D.
We consider the saddle-point problem
¯ d)
¯ ∈ B × D such that, for all (b, d) ∈ B × D,
(SPP) find (b,
¯
¯
f (b, d) ≤ f (b, d).
Theorem 3.10 For problem (SPP), assume that there are Y and such that (B ×
D, Y, ) is a compact GFC-space, a nonempty set A, and H : B × D → 2 A such
that the following conditions hold
(i) for each (b, d) ∈ B × D and N A = {a0 , ..., an } ∈ A , there exists N =
{y0 , ..., yn } ∈ Y such that, for each {ai0 , ..., aik } ⊂ N A ∩ H (b, d), ϕ N ( k ) ⊂

{(b , d ) ∈ B × D : f (b, d ) − f (b , d) < 0};
(ii) (b ,d )∈B×D {(b, d) ∈ B × D : f (b, d ) − f (b , d) < 0} ⊂ a∈A int H − (a).
Then, (SSP) has a solution.
Proof We simply apply Theorem 3.7 with X = B × D, Z = R, r = r1 , α = α1 ,
S(b, d) ≡ B × D, G(b, d) ≡ (−∞, 0) and K ((b, d), (b , d )) = f (b, d ) − f (b , d).
F. Constrained minimization problems. Let X be a topological space, f : X → R
and g : X × X → R. Consider the following constrained minimization problem
(MP) find x¯ ∈ X such that f (w) ≥ f (x)
¯ for all w satisfying g(x,
¯ w) ≤ 0.
Theorem 3.11 For problem (MP), assume that there are Y, such that (X, Y, ) is
a compact GFC-space, a nonempty set A, and H : X → 2 A such that the following
conditions hold
(i) for each x ∈ X and N A = {a0 , ..., an } ∈ A , there exists N = {y0 , ..., yn } ∈ Y
such that, for each {ai0 , ..., aik } ⊂ N A ∩ H (x), one has ϕ N ( k ) ⊂ {w ∈ X :
g(x, w) > 0, f (w) − f (x) < 0};
(ii) w∈X {x ∈ X : g(x, w) > 0, f (w) − f (x) < 0} ⊂ a∈A int H − (a).
Then, (MP) has a solution.
Proof To apply Theorem 3.7, we set Z = R, r = r1 , S(x) = {w ∈ X : g(x, w) ≤ 0},
G(x) ≡ (−∞, 0), and K (x, w) = f (w) − f (x).

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4 Existence theorems on product GFC-spaces and applications
4.1 Existence theorems on product GFC-spaces
Theorem 4.1 (collective fixed points) Let I be an index set, {(X i , Yi , {ϕ Ni })}i∈I be a
family of GFC-spaces, Ti : X → 2 X i for i ∈ I and X = i∈I X i be a compact space.

Assume that there exist a nonempty set Ai and Hi : X → 2 Ai , for i ∈ I , such that the
following conditions hold
(i) for each i ∈ I , Ti has nonempty values and Hi∗ is a g-Ti∗ -KKM mapping;
(ii) X = a i ∈Ai intHi− (a i ).
Then, there exists x¯ ∈ X such that x¯i ∈ Ti (x)
¯ for all i ∈ I .
Proof For each i ∈ I , by Theorems 3.1, Ti has a continuous selection ti = ϕi ◦ ψi ,
where m i ∈ N and ϕi : m i → X i , and ψi : X → m i are continuous maps. Let
→ m i be the canonical projection of onto m i . We
= i∈I m i and pi :
define two mappings : → X and : X → by, for e ∈ and x ∈ X ,
(e) =

i∈I ϕi ( pi (e)),

(x) =

i∈I ψi (x).

Then, and are continuous and so is ◦ : → . By virtue of the Tikhonov
fixed-point theorem, there exists e¯ ∈ such that ( ◦ )(e)
¯ = e.
¯ Setting x¯ = (e),
¯
we have
x¯ =

( (x))
¯ =

(

¯
i∈I ψi ( x))

=

i∈I ϕi ( pi (

¯
i∈I ψi ( x)))

=

i∈I (ϕi

◦ ψi )(x).
¯

It follows that x¯i = (ϕi ◦ ψi )(x)
¯ = ti (x)
¯ ∈ Ti (x)
¯ for all i ∈ I .
Note that Theorem 4.1 sharpens Theorem 3.1 of Khanh et al. (2011), and hence
also improves Theorem 2.2 of Ding (2007), since assumption (i) here is weaker than
the corresponding ones in those results, as illustrated by the following.
Example 4.1 Let I = {1, 2}, X 1 = X 2 = R, Y1 = Q and Y2 = N. For each N1 ∈ Y1
and N2 ∈ Y2 , let continuous functions ϕ N1 : n 1 → X 1 and ϕ N2 : n 2 → X 2 be
defined by, for e1 ∈ n 1 and e2 ∈ n 2 ,
ϕ N1 (e1 ) =

0,
if N1 ∈ N ,
√
2, otherwise

and
ϕ N2 (e2 ) = 1.
Then, (X 1 , Y1 , {ϕ N1 }) and (X 2 , Y2 , {ϕ N2 }) are GFC-spaces. Let T1 : X := X 1 × X 2 →
2 X 1 and T2 : X → 2 X 2 be given by T1 (x) = {0} and T2 (x) = {1}. Assumption (i)
of Theorem 3.1 in Khanh et al. (2011) is not fulfilled, since there does not exist
any map H1 : X → 2Y1 such that for each x ∈ X , N1 = {y01 , .., yn11 } ∈ Y1

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On topological existence theorems and applications...

and {yi10 , ..., yi1k } ⊂ N1 ∩ H1 (x), one has ϕ N1 (
1

k1 )

⊂ T1 (x). Indeed, suppose to the

exists. We take x ∗

∈ X and N 1 = { y¯01 , ..., y¯n11 } ∈ Y1 such that
contrary that such a H1
√

N 1 ∩ H1 (x ∗ ) = { y¯i10 , ..., y¯i1k } = ∅. For N1∗ = N 1 ∪ {0.5}, one has ϕ N1∗ ( k1 ) = 2 ⊂
1
T1 (x ∗ ) = {0}, a contradiction. To apply our Theorem 4.1, we take A1 = A2 = N,
H1 : X → 2 A1 and H2 : X → 2 A2 defined by H1 (x) = H2 (x) = N. For each x ∈ X
and N A1 = {a01 , ..., an11 } ∈ A1 = N , we choose N1 ≡ N A1 ∈ Y1 to see that, for
each {ai10 , ..., ai1k } ⊂ N A1 ∩ H1 (x) = N A1 , ϕ N1 ( k1 ) = {0} ⊂ T1 (x) = {0}, i.e., (i)
1
of Lemma 2.1, which is equivalent to (i) of Theorem 4.1, is fulfilled. It is not difficult
to see that the other assumptions of Theorem 4.1 are satisfied. On the other hand, we
easily check directly that x¯ = (0, 1) is a (collective) fixed point, i.e., 0 ∈ T1 ((0, 1))
and 1 ∈ T2 ((0, 1)).
Now, we pass to systems of coincidence points.
Theorem 4.2 (systems of coincidence points) Let I and J be index sets, {(X i , Yi ,
{ϕ Ni })}i∈I and {(X j , Y j , {ϕ N })} j∈J be families of GFC-spaces, T j : X → 2 X j for
j

j ∈ J , Fi : X → 2 X i for i ∈ I , X := i∈I X i be a compact space and X :=
Aj
j∈J X j . Assume that there exist nonempty sets Ai , A j and maps H j : X → 2 ,
G i : X → 2 Ai , for i ∈ I , j ∈ J , such that the following conditions hold
(i) for each j ∈ J , T j has nonempty values and H j∗ is a g-T j∗ -KKM mapping;
(ii) for each i ∈ I , Fi has nonempty values and G i∗ is a g-Fi∗ -KKM mapping;
(iii) X = a j ∈A intH j− (a j ) and X = a i ∈Ai intG i− (a i ).
j

¯ for all
Then, there exists (x,
¯ x¯ ) ∈ X × X such that x¯i ∈ Fi (x¯ ) and x¯ j ∈ T j (x)
(i, j) ∈ I × J .
Proof For each j ∈ J , by (i), (iii) and Theorem 3.1, T j has a continuous selection

t j : X → X j and hence we obtain a continuous map t : X → X defined by
t (x) =

j∈J t j (x).

For each i ∈ I , define two new set-valued maps Pi : X → 2 Ai and Q i : X → 2 X i by
Pi (x) = G i t (x) andQ i (x) = Fi t (x) .
We see from (ii) that, for each i ∈ I and NiA = {a0i , ..., ani i } ⊂ Ai , there exists
Ni = {y0i , ..., yni i } ⊂ Yi such that, for each {l0 , ..., lki } ⊂ {0, ..., n i }, one has the
following string of equivalent statements
Fi∗ (ϕ Ni (

ki
ki )) ⊂

G i∗ (alih ) ⇔ X \ Fi− (ϕ Ni (

ki
ki )) ⊂

h=0

X \ G i− (alih )

h=0

⇔ t − X \ Fi− (ϕ Ni (

ki
ki )) ⊂

t − X \ G i− (alih )

h=0

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P. Q. Khanh et al.
ki

⇔ X \ t − Fi− (ϕ Ni (

ki )) ⊂

X \ t − G i− (alih )

h=0

⇔ Q i∗ (ϕ Ni (

ki
ki ))

Pi∗ (alih ),

⊂
h=0

i.e., Pi∗ is a g-Q i∗ -KKM mapping. While from (iii), we have

intG i− (a i ).

X =
a i ∈Ai

It follows that
intG i− (a i ) =

X = t − (X ) = t −
a i ∈Ai

a i ∈Ai

intt − G i− (a i ) =

=
a i ∈Ai

t − intG i− (a i )

int Pi− (a i ).
a i ∈Ai

Now that the assumptions of Theorem 4.1 hold for Pi and Q i , it yields a x¯ ∈ X such
that, for all i ∈ I ,
¯ = Fi (t (x)).
¯
x¯i ∈ Q i (x)
¯ one sees that x¯i ∈ Fi (x¯ ) and x¯ j ∈ T j (x)
¯ for all (i, j) ∈ I × J .

Setting x¯ = t (x)
4.2 Applications
A. Systems of variational relations. Let I be an index set, {X i }i∈I a family of sets, and
X = i∈I X i . For i ∈ I , let Si : X ⇒ X i and Ri (x, xi ) be a relation linking x ∈ X
and xi ∈ X i . We consider the following system of variational relations
(SVR)

¯
find x¯ = (x¯i )i∈I ∈ X such that, for all i ∈ I, x¯i ∈ Si (x)
¯ x¯i ) holds.
and Ri (x,

For problem (SVR), we set E i = {x = (xi )i∈I ∈ X : xi ∈ Si (x)}.
Theorem 4.3 For problem (SVR), assume that X := i∈I X i is a compact space and
there are {Yi }i∈I and {ϕ Ni }i∈I such that {(X i , Yi , {ϕ Ni })}i∈I is a family of GFC-spaces,
a nonempty set Ai and Hi : X → 2 Ai for i ∈ I such that the following conditions
hold
(i) for each i ∈ I , x ∈ X , and N Ai = {a0i , ..., ani i } ∈ Ai , there exists Ni =
{y0i , ..., yni i } ∈ Yi such that, for each {a ij0 , ..., a ijk } ⊂ N Ai ∩ Hi (x),
i

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On topological existence theorems and applications...

ϕ Ni (
(ii) X =

a i ∈Ai

ki )

⊂

{xi ∈ X i : Ri (x, xi ) holds}, if x ∈ E i ,
if x ∈
/ Ei ;
Si (x),

int Hi− (a i ).

Then, (SVR) admits a solution.
Proof For i ∈ I , we define Ti : X → 2 X i by
Ti (x) =

{xi ∈ X i : Ri (x, xi ) holds}, if x ∈ E i ,
if x ∈
/ Ei .
Si (x),

By (i) of Theorem 4.2, it is not hard to see that (i) of Lemma 2.1, which is equivalent to
(i) of Theorem 4.1, is satisfied. Clearly, (ii) of Theorem 4.2 is just (ii) of Theorem 4.1.
According to Theorem 4.1, there exists x¯ = (x¯i )i∈I ∈ X such that x¯i ∈ Ti (x)
¯ for all
¯ a contradiction.
i ∈ I . Suppose that x¯ ∈
/ E i . Then, by the definition of Ti , x¯i ∈ Si (x),
¯ x¯i ) holds for all i ∈ I .
Hence, x¯ ∈ E i , i.e., x¯i ∈ Si (x), and Ri (x,

B. An abstract economy. Let I be a finite or infinite set of agents. For each i ∈ I ,
let X i be a nonempty set. An abstract economy, see Yannelis and Prabhakar (1983),
is a family of triples ϒ = (X i , Si , Pi )i∈I , where Si : X = i∈I X i → 2 X i and
Pi : X = i∈I X i → 2 X i are correspondences. A solution of ϒ is a point x¯ ∈ X
¯ and Si (x)
¯ ∩ Pi (x)
¯ = ∅ for each i ∈ I . For a point x ∈ X , xiˆ
satisfying x¯i ∈ Si (x)
stands for its projection onto X iˆ = j =i X j .
For abstract economy ϒ, we set
E i = {x = (xi )i∈I ∈ X : xi ∈ Si (x)}.
Theorem 4.4 For abstract economy ϒ, assume that X := i∈I X i is a compact
topological space and there are {Yi }i∈I and {ϕ Ni }i∈I such that {(X i , Yi , {ϕ Ni })}i∈I is
a family of GFC-spaces, a nonempty set Ai , and Hi : X → 2 Ai for i ∈ I such that
the following conditions hold
(i) for each x ∈ X and N Ai = {a0i , ..., ani i } ∈ Ai , there exists Ni = {y0i , ..., yni i } ∈
Yi such that, for each {a ij0 , ..., a ijk } ⊂ N Ai ∩ Hi (x),
i

ϕ Ni (
(ii) X =

a i ∈Ai

ki )

⊂

{xi ∈ X i : Si (x) ∩ Pi (xiˆ , xi ) = ∅}, if x ∈ E i ,
Si (x),

if x ∈
/ Ei ;

int Hi− (a i ).

Then, ϒ has a solution.
Proof Apply Theorem 4.2 with the relation Ri defined by, for i ∈ I ,
ˆ

Ri (x, xi )holds ⇔ Si (x) ∩ Pi (x i , xi ) = ∅.

123


P. Q. Khanh et al.
Acknowledgements This work was supported by the Vietnam National University Hochiminh City
(VNU-HCM) under the grant number B2013-28-01. A part of the work was done during a stay of the
first author as a visiting Professor at the Department of Mathematics, National Changhua University of
Education, Taiwan, whose hospitality is gratefully acknowledged.

References
Ansari QH, Lin YC, Yao JC (2000) General KKM theorems with applications to minimax and variational
inequalities. J Optim Theory Appl 104:41–57
Balaj M, Lin LJ (2010) Equivalent forms of a generalized KKM theorem and their applications. Nonlinear
Anal 73:673–682
Balaj M, Lin LJ (2011) Generalized variational relation problems with applications. J Optim Theory Appl
148:1–13
Browder FE (1968) The fixed point theory of multivalued mappings in topological vector spaces. Math Ann
177:238–301
Chang SS, Zhang Y (1991) Generalized KKM theorems and variational inequalities. J Math Anal Appl

159:208–223
Ding XP (2005) Maximal element theorems in product FC-spaces and generalized games. J Math Anal
Appl 305:29–42
Ding XP (2007) Collectively fixed points and a system of coincidence theorems in product FC-spaces and
applications. Nonlinear Anal 66:2604–2617
Fan K (1961) A generalization of Tichonoff’s fixed point theorem. Math Ann 142:305–310
Hai NX, Khanh PQ, Quan NH (2009) On the existence of solutions to quasivariational inclusion problems.
J Global Optim 45:565–581
Hai NX, Khanh PQ, Quan NH (2009) Some existence theorems in nonlinear analysis for mappings on
GFC-spaces and applications. Nonlinear Anal 71:6170–6181
Horvath C (1991) Contractiblity and generalized convexity. J Math Anal Appl 156:341–357
Khanh PQ, Long VST (2013) Invariant-point theorems and existence of solutions to optimization-related
problems. J Global Optim. doi:10.1007/s10898-013-0065-y
Khanh PQ, Long VST, Quan NH (2011) Continuous selections, collectively fixed points and weakly KnasterKuratowski-Mazurkiewicz mappings in Optimization. J Optim Theory Appl 151:552–572
Khanh PQ, Luc DT (2008) Stability of solutions in parametric variational relation problems. Set Valued
Anal 16:1015–1035
Khanh PQ, Luu LM (2004) The existence of solutions to vector quasivaria-tional inequalities and quasicomplementarity problems with applications to traffic network equilibria. J Optim Theory Appl 123:533–548
Khanh PQ, Quan NH (2010) Intersection theorems, coincidence theorems and maximal-element theorems
in GFC-spaces. Optimization 59:1–10
Khanh PQ, Quan NH (2013) A unified study of topological existence theorems and applications, submitted
for publication
Khanh PQ, Quan NH, Yao JC (2009) Generalized KKM type theorems in GFC-spaces and applications.
Nonlinear Anal 71:1227–1234
Lin LJ (2012) Variational relation problems and equivalent forms of generalized Fan-Browder fixed point
theorem with applications to Stampacchia equilibrium problems. J Global Optim 53:215–229
Luc DT (2008) An abstract problem in variational analysis. J Optim Theory Appl 138:65–76
Luc DT, Sarabi E, Soubeyran A (2010) Existence of solutions in variational relation problems without
convexity. J Math Anal Appl 364:544–555
Park S (2008) Comments on the KKM theory on φ A -space. Pan Am Math J 18(2):61–71
Park S, Kim H (1996) Coincidence theorems for admissible multifunctions on generalized convex spaces.

J Math Anal Appl 197:173–187
Wu WT (1959) A remark on the fundamental theorem in the theory of games. Sci Rec New Ser 3:229–233
Yannelis NC, Prabhakar ND (1983) Existence of maximal elements and equilibria in linear topological
spaces. J Math Econ 12:233–245
Yu ZT, Lin LJ (2003) Continuous selections and fixed point theorems. Nonlinear Anal 52:445–455

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