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

DOI 10.1007/s00186-014-0462-0O R I G I NA L A RT I C L E On topological existence theorems and applications to optimization-related problems Phan Quoc Khanh · Lai Jiu Lin · Vo Si Trong L

DOI 10.1007/s00186-014-0462-0

O R I G I NA L A RT I C L E

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: quasivari-ational 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: pqkhanh@hcmiu.edu.vn

P Q Khanh

Federation University Australia, Ballarat, Victoria, Australia

L J Lin

Department of Mathematics, National Changhua University of Education,

Changhua, Taiwan

e-mail: maljlin@cc.ncue.edu.tw

V S T Long (B)

Department of Mathematics, Cao Thang College of Technology,

Hochiminh City, Vietnam

e-mail: vstronglong@gmail.com

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 fromWu(1959) andHorvath(1991), two directions of deal-ing 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 continu-ous map Very recently, inKhanh 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 prob-lems, (see, e.g.,Ding 2005,2007;Hai et al 2009;Khanh et al 2011;Khanh and Quan

and a definition inChang 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

suffi-cient 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 Section2contains 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 var-ious optimization-related problems are presented Section4 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 andX 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, andR = 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 → 2 Y and F∗ : X → 2 Y,

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), , e n = (0, , 0, 1) of R n+1.

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, , y n } ∈ 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 inPark (2008) under the name “ A-space” (independently and earlier than the GFC-space) These spaces are generalizations of other topological structures as G-convex spacesPark and Kim

(1996), FC-spacesDing(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., Example3.1below

Definition 2.2 (Hai et al 2009;Khanh et al 2011;Khanh and Quan 2010;Khanh

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 , , y n } ∈ Y  and each {y i0, , y i k } ⊂ N, one has

T (ϕ N ( k )) ⊂ ∪ k

j=0F (y i j ), where ϕ N ∈  is corresponding to N and  k is the face

of nformed by{e i0, , e i k}

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 , , a n } ∈ A, there exists N = {y0 , , y n } ∈ Y  such that, for each {i0 , , i k } ⊂ {0, , n}, one has

T (ϕ N ( k )) ⊂k

j=0F (a i j ), where ϕ N ∈  is corresponding to N and  kis the face

of nformed by{e i0, , e i k}

Note that Definition2.3is a natural generalization of Definition2.1ofChang and

of another topological vector space, T is the identity map, and ϕ N (·) = co(·) (the

usual convex hull) Consequently, it also generalizes Definition2.1ofAnsari 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 , , y n } ∈ Y  let ϕ N

be defined byϕ N (e) =n

i=0λ i y i for all e=n

i=0λ i e i ∈  n Clearly,(X, Y, {ϕ N })

is a GFC-space Let F : Y → 2 Z 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 , , a n } ∈ A = Y ,

we take N = {y0 , , y n } = {|a0|, , |a n |} ∈ Y , where | · | denotes absolute value.

It is easy to see that

T (ϕ N ( n )) = [minN, maxN] ⊂ [0, +∞) = F(a i ), ∀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 , , a n } ∈ A, there exists N = {y0 , y1, , y n} ∈

Y  such that, for each {a i0, , a i k } ⊂ N A ∩ H(z), one has ϕ N ( k ) ⊂ T (z),

where  k is the simplex formed by {e i0, , e i k };

(ii) H∗is a g-T∗-KKM mapping.

Proof (i) ⇒ (ii) Suppose to the contrary that N A = {a0 , , a n } ∈ A exists such that, for each N = {y0 , y1, , y n } ∈ Y , there exists {i0 , , i k } ⊂ {0, , n},

T∗(ϕ N ( k )) ⊂

k



j=0

H∗(a i j ). (1)

We choose N given in condition (i) associated with N A By (1), there are x0 ∈

ϕ N ( k ) and z0∈ T∗(x0) such that

z0 /∈

k



j=0

H∗(a i j ) =

k



j=0

Z \ H−(a i j ) = Z \

k



j=0

H−(a i j ).

It follows that{a i0, , a i k } ⊂ N A ∩ H(z0 ) Hence, from (i) one has ϕ N ( k ) ⊂

T (z0) Hence, z0∈ T−(x0), and so z0 /∈ Z \ T−(x0) = T∗(x0), a contradiction.

(ii) ⇒ (i) Suppose there exist z0 ∈ Z and N A = {a0 , , a n } ∈ A such that, for each N = {y0 , , y n } ∈ Y , there exists {a i0, , a i k } ⊂ N A ∩ H(z0 ) such that

Since H∗is a g-T∗-KKM mapping and N is arbitrary, one can take N associated

with N Asuch that

T∗(ϕ N ( k )) ⊂

k



j=0

H∗(a i j ). (3)

Since{a i0, , a i k } ⊂ N A ∩ H(z0 ), one has z0∈k

j=0H−(a i j ), i.e.,

z0 /∈ Z \

k



j=0

H−(a i j ) =

k



j=0

H∗(a i j ). (4)

By (2), there is x0 ∈ ϕ N ( k ) such that x0 /∈ T (z0) This means that

z0∈ T∗(x0) ⊂ T∗(ϕ N ( k )). (5)

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, forα ∈ {α1, α2},

(VR α ) 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 inKhanh and Luc

(2008);Luc(2008) Developments have been obtained in some papers, (e.g.,Balaj

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 non-empty values Assume that there are a nonnon-empty 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 = {¯a0 , , ¯a n } ∈ A such that

Z = n

i=0int H−( ¯a i ) Then, there is a continuous partition of unity {ψ i}n

i=0of Z

associated with the finite open cover {intH−1(¯a i )} n

i=0 From (i) there exists N =

{y0 , , y n } ∈ Y  associated with N A = {¯a0 , , ¯a n} Moreover, due to the GFC-space structure, there isϕ N :  n → X corresponding to N Now, we define the continuous

mapsψ : Z →  n and t : Z → X, respectively, by

ψ(z) =

n



i=0

ψ i (z)e i , t(z) = (ϕ N ◦ ψ)(z).

Suppose to the contrary that t is not a selection of T , i.e., there exists z0 ∈ Z and

t (z0) = ϕ N (ψ(z0)) := x0 /∈ T (z0), or equivalently, z0 ∈ Z \ T−(x0) = T∗(x0).

Furthermore, one has

ψ(z0) =

n



i=0

ψ i (z0)e i = 

j ∈J(z0)

ψ j (z0)e j ∈  J (z0) ,

where J (z0) := { j ∈ {0, , n} : ψ j (z0) = 0} Since H∗is a g-T∗-KKM mapping,

one has

z0∈ T∗(x0) = T∗(ϕ N (ψ(z0))) ⊂ T∗(ϕ N ( J (z0) )) ⊂ 

i ∈J(z0)

H∗(¯a i ).

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

z0 /∈ H−(¯a j0). (6)

On the other hand, by the definition of J (z0) and the partition {ψ i}n

i=0,

z0∈ {z ∈ Z : ψ j0(z) = 0} ⊂ intH−(¯a j0) ⊂ H−(¯a j0),

contradicting (6) Finally, puttingϕ = ϕ Nwe arrive at the conclusion 

Remark 1 Theorem3.1improves Theorem 2.1 ofKhanh et al.(2011) since assump-tion (i) is weaker than the corresponding assumpassump-tion (i) of that result Consequently,

it also improves Theorem 2.1 ofDing(2007), Theorem3.1ofYannelis and Prabhakar

(1983), and Theorem 1 ofYu and Lin(2003) The next example gives a case where The-orem3.1is more convenient than Theorem 2.1 ofDing(2007) in terms of FC-spaces and Theorem 2.1 ofKhanh 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 , , x n } ∈ X and {x i0, , x i k } ⊂ 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 thatN contains also zero), (X, {ϕ N}N ∈X ) is a FC-space We show that

assumption (i) of Theorem 2.1 inDing(2007) is not satisfied For N /∈ N, z ∈ Z, and

{x i0, , x i k } ⊂ 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 inKhanh et al.(2011) is not fulfilled, i.e., there does

not exist any map H : Z → 2 Y such that, for each z ∈ Z, N = {y0 , , y n } ∈ Y  and

{y i0, , y i k } ⊂ 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 = { ¯y0 , , ¯y n } ∈ Y  such that N ∩ H(¯z) = { ¯y i0, , ¯y i k } = ∅ Set N∗ = N ∪ {0.5} Then, ϕ N∗( k ) = {√2} ⊂ T (¯z) = [0, ¯z], a contradiction To apply our Theorem3.1, we take A = N, and H : Z → 2 Adefined

by H (z) = F(z) ∩ N For each z ∈ Z and N A = {a0 , , a n } ∈ A = N, we choose N ≡ N A ∈ Y  to see that, for each {a i0, , a i k } ⊂ N A ∩ H(z) = N A,

ϕ N ( k ) = {0} ⊂ T (z) = [0, z], i.e., (i) of Lemma2.1, which is equivalent to (i)

of Theorem3.1, is fulfilled (ii) of Theorem3.1is satisfied because Z = H−(0) =

intH−(0) By Theorem3.1, 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 →  nare continuous Then,ψ ◦ ϕ :  n →  nis also continuous By virtue of the Tikhonov fixed-point theorem, there exists¯e ∈  nsuch thatψ ◦ ϕ(¯e) = ¯e Setting ¯x = ϕ(¯e), we have

¯x = ϕ(ψ( ¯x)) = t( ¯x) ∈ T ( ¯x).

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 ofBrowder(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 , , a n } ∈ A, there exists N = {y0 , , y n } ∈ Y 

such that, for each {a i0, , a i k } ⊂ 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.

Observe that, for a similar reason as in Remark2, Theorem3.3generalizes Lemma

4 ofFan(1961)

Theorem 3.4 (maximal elements) Let (X, Y, ) be a compact GFC-space and T :

X → 2X 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 → 2X 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

GFC-space, 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 , , a n } ∈ A, there exists N = {y0 , , y n } ∈ Y 

such that, for each {a i0, , a i k } ⊂ 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 (VR αi ), i.e., ∀w ∈ S( ¯x), α i (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⇒ Theorem3.3 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 Theorem3.3is false By (i) of Theorem

3.3and Lemma2.1, (i) of Theorem 3.2 is satisfied Since the two assumptions (ii) are the same, by Theorem 3.2,¯x ∈ T ( ¯x) for some ¯x ∈ X, i.e., ¯x ∈ {w ∈ X : ( ¯x, w) /∈ M},

which contradicts (iii) of Theorem3.3

Theorem3.3⇒ 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 Theorem3.3

holds According to Lemma2.1, for each x ∈ X and N A = {a0 , , a n } ∈ A, there exists N = {y0 , , y n } ∈ Y  such that, for each {a i0, , a i k } ⊂ N A ∩ H(x), one

hasϕ N ( k ) ⊂ T (x) Hence, for each w ∈ ϕ N ( k ), w ∈ T (x), i.e., (x, w) /∈ M It

follows thatϕ N ( k ) ⊂ {w ∈ X : (x, w) /∈ M}, i.e., (i) of Theorem3.3is satisfied Since the two assumptions (ii) are the same, applying Theorem3.3, 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 Theorem3.5, all assumptions of Theorem3.4are fulfilled Therefore, there exists

¯x ∈ X such that T ( ¯x) = ∅, i.e., X \ G−( ¯x) = ∅ Hence, ¯x ∈x ∈X G (x).

Theorem3.5⇒ Theorem3.4 Under the assumptions of Theorem3.4, let G (x) =

X \ T−(x) for x ∈ X Then, assumptions (i) and (ii) of Theorem3.4clearly imply the corresponding assumptions (i) and (ii) of Theorem3.5 From (iii) of Theorem3.4,

one has x ∈ X \ T−(x) = G(x) for all x ∈ X, i.e., (iii) of Theorem3.5is satisfied

By Theorem3.5, 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⇒ Theorem3.4 Suppose that T (x) = ∅ for each x ∈ X Then, (i) of

Theorem3.4implies (i) of Theorem 3.2 Since X =w∈X T−(w), (ii) of Theorem

3.2 is satisfied along with (ii) of Theorem3.4 By Theorem 3.2, T has a fixed point.

This contradicts (iii) of Theorem3.4and we are done

Theorem3.4⇒ Theorem3.6 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 3.6and Lemma 2.1, H∗ is a g-T∗-KKM mapping, i.e., (i) of

Theorem3.4is fulfilled It is not difficult to see that (ii) and (iii) of Theorem3.6imply the corresponding (ii) and (iii) of Theorem3.4 Applying this theorem, we have¯x ∈ X such that T ( ¯x) = ∅ Consequently, ∀w ∈ S( ¯x), α i (z, F( ¯x, w)), R( ¯x, w, z) holds.

Theorem3.6⇒ 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) = {z0} for an arbitrary z0∈ Z,

α i (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 Theorem3.6is fulfilled Clearly,

by (ii) of Theorem 3.2, (ii) of Theorem3.6is fulfilled Since H∗ is a g-T∗-KKM

mapping, by Lemma2.1, (i) of Theorem3.6holds According to this theorem,¯x ∈ X

exists such that, for allw ∈ S( ¯x) = X, α i (z, F( ¯x, w)), R( ¯x, w, z) holds This means

thatw /∈ 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 papersHai et al.(2009);Khanh et al.(2011);Khanh and Quan(2010);

terms of a general KKM mapping (the map T or H∗) with respect to another map.

Example2.1shows that being such a general KKM mapping is properly weaker than being a usual KKM mapping following Definition2.2, 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

quasivaria-tional 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 → 2Z and K : X × X × Z → 2Z For r ∈ {r1 , r2, r3, r4} andα ∈ {α1, α2}, we consider the following quasivariational inclusion problem

(QIP r 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 inHai 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 (QIP r 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 , , a n } ∈ A, there exists N = {y0 , , y n} ∈

Y  such that, for each {a i0, , a i k } ⊂ 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 (QIP r j αi ) has a solution.