Một số định lý tồn tại trong giải tích phi tuyến và áp dụng vào các bài toán liên quan đến tối ưu

Main tools used to study the existence of solutions of a problemare existence theorems for important points in nonlinear analysis such as fixed points,coincidence points, maximal points,

VIETNAM NATIONAL UNIVERSITY - HO CHI MINH CITY

UNIVERSITY OF SCIENCE

VO SI TRONG LONG

SEVERAL EXISTENCE THEOREMS

IN NONLINEAR ANALYSIS AND APPLICATIONS TO

OPTIMIZATION-RELATED PROBLEMS

PhD THESIS IN MATHEMATICS

Ho Chi Minh City - 2015

SEVERAL EXISTENCE THEOREMS

IN NONLINEAR ANALYSIS AND APPLICATIONS TO

OPTIMIZATION-RELATED PROBLEMS

Major: Optimization Theory

Code: 62 46 20 01

Reviewer 1: Assoc Prof Dr Lam Quoc Anh

Reviewer 2: Dr Nguyen Ba Thi

Reviewer 3: Assoc Prof Dr Nguyen Ngoc Hai

Anonymous Reviewer 1: Assoc Prof Dr Lam Quoc Anh

Anonymous Reviewer 2: Dr Ha Binh Minh

SCIENTIFIC SUPERVISOR: Prof DSc PHAN QUOC KHANH

Ho Chi Minh City - 2015

To the memory of my mother.

To my beloved children, Delta and Lambda.

I hereby declare that this dissertation, done under the supervision of Professor PhanQuoc Khanh, is entirely the result of my own work, and this study and its findingshave never been published by any other researchers

I also obtained the consent of Nguyen Hong Quan, co-author of the joint paper [43]referred to in Chapter 2, to let me include in this thesis some of the results of the saidjoint paper, which were not included in his thesis defended two years ago

Ho Chi Minh City, november, 2015

The author

Vo Si Trong Long

Second, I am thankful to the University of Science, Vietnam National University

Ho Chi Minh City, for providing me with favorable working conditions and facilitiesduring all the time of my PhD program Many thanks also go to all the members ofthe Optimization Group of Southern Vietnam, especially Dr Nguyen Hong Quan Hiscomments and advice have been crucial for my research

Finally, I could have not done this work without my family’s support I would like tothank my father, my wife, and all the family members for their love and encouragement

Ho Chi Minh City, november, 2015

The author

Vo Si Trong Long

1.1 Basic definitions and properties 1

1.2 Abstract convexity structures and generalized KKM mappings 2

1.3 Problem settings 9

1.3.1 Variational relations 9

1.3.2 Quasivariational inclusions 10

1.3.3 Stampacchia-type vector quasiequilibria 10

1.3.4 Nash equilibria for non-cooperative games 11

1.3.5 Traffic networks 11

1.3.6 Constrained minimization and maximization 12

1.3.7 Saddle points 13

2 Several existence theorems in nonlinear analysis related to generalized KKM mappings and applications 14 2.1 Existence theorems and applications to optimization-related problems 15 2.1.1 Existence theorems 15

2.1.2 Optimization-related problems 22

2.2 Existence theorems on product GFC-spaces and applications 28

2.2.1 Existence theorems on product GFC-spaces 28

2.2.2 Applications 31

2.3 Conclusions 34

3 Fixed points and existence of solutions of optimization-related prob-lems 35 3.1 Locally GFC-spaces 36

3.2 Fixed points and coincidence points 39

3.3 Existence of solutions of optimization-related problems 46

3.4 Conclusions 54

4 The weak finite intersection property and characterizations of the solution existence in optimization 55 4.1 The weak finite intersection property and its characterizations 56

4.2 Characterizations of the solution existence in optimization problems 64

4.3 Conclusions 72

5 Invariant-point theorems in metric spaces and applications to optimization-related problems 73 5.1 Problem settings 74

5.2 Invariant-point theorems in metric spaces 76

5.3 Existence of solutions of variational relation and inclusion problems 80

5.4 Applications 85

5.5 Conclusions 99

List of the author’s conference reports related to the thesis 102

List of Symbols and Notations

2X the family of all nonempty subsets of a set X

hY i the family of nonempty finite subsets of a set Y

R = R ∪ {−∞, +∞} the set of the extended real numbers

∆M the face of ∆n corresponding to a finite set M

(X, Y, Φ) or (X, Y, {ϕN}) a finite continuous topological space (GFC-space)

(X, Y, U , Φ) a locally-GFC-space

KKM (X, Y, Z) the class of the mappings enjoying the weak KKM property

(·, ·] and [·, ·) the half-open intervals

{xα} the net or sequence with elements xα

N (x) the class of all the neighborhoods of an element x

f : X → Y a single-valued mapping from X to Y

F : X → 2Y a set-valued mapping from X to Y

F−: Y → 2X the inverse mapping of F : X → 2Y

F∗ : Y → 2X the dual mapping of F : X → 2Y

limsupx 0 →xf (x0) infV ∈N (x)supx 0 ∈Vf (x0) for f : X → R

liminfx0 →xf (x0) supV ∈N (x)infx0 ∈Vf (x0) for f : X → R

Optimization studies contains various topics However, the existence of solutions ofoptimization problems and optimization-related problems such as variational inequal-ities, equilibrium problems, minimax problems, etc, always take a central place in theoptimization theory Main tools used to study the existence of solutions of a problemare existence theorems for important points in nonlinear analysis such as fixed points,coincidence points, maximal points, intersection points, etc

One of the most celebrated existence theorems in nonlinear analysis is the classicalKKM theorem (KKM being an abbreviation of Knaster-Kuratowski-Mazurkiewicz; thetheorem is known also as the Three Polish lemma) Knaster-Kuratowski-Mazurkiewicz[56] in finite-dimensional spaces, which is equivalent to the well-known Brouwer fixed-point theorem and the Sperner lemma There have been many extensions and appli-cations of this KKM theorem in optimization theory First, it was extended to generaltopological vector spaces by Fan [25] Later, we will mention the subsequent maindirection of developing this result Various generalized linear/convex structures havebeen proposed and corresponding types of KKM mappings have been defined togetherwith these spaces Lassonde [58] introduced convex spaces and obtained many funda-mental results A well known generalized convex structure was developed by Horvath[34] by replacing a convex hull with a continuous map from a simplex to a contractibleset Park-Kim [71] proposed the notation of a G-convex space Ding [18] introduced theconcept of a FC-space and then Khanh-Quan [48] generalized and unified the previousspaces into a notion called a GFC-space

The Fan-Glicksberg fixed-point theorem in Fan [24] and Glicksberg [29], a ization to locally convex spaces of the well-known Kakutani fixed-point theorem, and

general-many classical fixed-point results, have been also developed by various researchers Amajor attention has been paid to generalizing it to abstract convex spaces The firstresult of this direction appeared in Himmelberg [35] Later, Tarafdar [73] dealt withthe case of a locally H-convex space So far, it has been considered for other generalspaces such as locally G-convex uniform spaces and locally FC-spaces, etc Moreover,

it has been also extended in another way by replacing the convex-valuedness of a mapwith more general conditions (see for example Jeng-Hsu-Huang [37]; Khanh-Long [41]and Kuo-Jeng-Huang [57])

Another kind of existence theorems, which has also been attracting increasing tention, is the invariant-point theorem in Dancs-Hegedus-Medvegyev [17], which isequivalent to the Ekeland variational principle Ekeland [22] Several efforts have beenmade to establish sufficient conditions for the existence of solutions of optimization-related problems by using this result (see Khanh-Lin-Long [39]; Lin-Chuang [60] andLin-Chuang-Wang [62])

at-On the other hand, many contributions have been made in recent decades to duce general setting problems related to optimization and then deriving as consequencescorresponding results for particular cases For these problems, the above existence the-orems have been employed as crucial mathematical tools to achieve existence results.Among such general models, variational relation problems and variational inclusionproblems have been attracting increasing interests of researchers Variational relationproblems were investigated first in Luc [63] for existence and in Khanh-Luc [44] for sta-bility and have been developed in Balaj-Lin [8, 9]; Khanh-Long [40] and Khanh-Long-Quan [43] It is worth stressing that this model contains many important problems

intro-in optimization such as constraintro-ined mintro-inimization, variational intro-inequalities, equilibriumproblems, minimax problems, Nash equilibria, etc The variational inclusion problemwas proposed in Hai-Khanh-Quan [31] and has also been used, e.g., in Khanh-Long-Quan [43] and Khanh-Quan [49] This problem also includes special cases, as many asthe variational relation model

The major focus of this thesis is to develop several existence theorems in ear analysis in order to apply them to considering the existence of solutions of various

nonlin-optimization-related problems as mentioned above It includes five chapters We begin,

in Chapter 1, with several basic concepts, definitions and preliminary facts for our lateruse In chapter 2, we consider some continuous selection results, and use them to estab-lish purely topological-based sufficient conditions for the existence of important points

in nonlinear analysis and prove the equivalence of these conditions Then, we applythem to investigating the solution existence of optimization-related problems stated

in Chapter 1 The results in Chapter 2 were published in Khanh-Long-Quan [43] andKhanh-Lin-Long [39] Chapter 3 is devoted to an extension of the Fan-Glicksberg fixed-point theorem and applications to problems concerning optimization The contents ofthis chapter are based on the paper Khanh-Long [41] In Chapter 4, we propose a gen-eral notion, called FI-map (finite-intersection map) and use it to establish necessaryand sufficient conditions for the solution existence of various optimization-related prob-lems such as variational relation problems, minimax problems, saddle-point problems,etc, in pure topological settings This chapter includes the results of the submittedpaper Khanh-Long [42] Finally, Chapter 5 develops several invariant-point theorems

in metric spaces also as tools for establishing conditions for the existence of solutions tomany of the above-mentioned problems However, applications of results on invariantpoints are based on techniques quite different from those on other encountered points.Chapter 5 also contains discussions about the equivalence of these conditions Manyexamples and comparison remarks are provided to illustrate advantages of the obtainedresults Chapter 5 is written by using results of the paper Khanh-Long [40] which hasalready been published

Chapter 1

Background and Preliminaries

1.1 Basic definitions and properties

In this thesis, we use standard notations For a set X, by 2X and hXi we denote thefamily of all the subsets and the family of all the nonempty finite subsets, respectively(resp), of X If X is a topological space and A ⊂ X, then intA and clA signify theinterior and closure, resp, of A Let X and Y be nonempty sets For F : Y → 2X,

we define F− : X → 2Y and F∗ : X → 2Y, resp, by F−(x) = {y ∈ Y : x ∈ F (y)}and F∗(x) = Y \ F−(x) F− and F∗ are called the inverse and dual, resp, map of F grF := {(y, x) ∈ Y × X : x ∈ F (y)} denotes the graph of F N, Q, and R stand forthe set of the natural numbers, the rational numbers, and the real numbers, resp, and

R = R ∪ {−∞, +∞} For n ∈ N, ∆n signifies the standard n-simplex, whose verticesare the points e0 = (1, 0, , 0), ,en= (0, , 0, 1) of Rn+1

Definition 1.1.1 Let X be a topological space A function f : X → R is said to be:(i) upper semicontinuous (u.s.c.) at x ∈ X if, limsupx0 →xf (x0) ≤ f (x);

(ii) lower semicontinuous (l.s.c.) at x ∈ X if, liminfx0 →xf (x0) ≥ f (x)

The function f is called upper semicontinuous (lower semicontinuous) if it is uppersemicontinuous (lower semicontinuous, resp) at every point of its domain

Proposition 1.1.1 (classical) A function f : X → R is upper semicontinuous (lowersemicontinuous, resp) if and only if, for every α ∈ R, {x ∈ X : f (x) < λ} ({x ∈ X :

f (x) ≤ λ}) is an open (a closed resp) set

Definition 1.1.2 (classical) A set-valued mapping F : Y → 2X between two ical spaces is said to be:

topolog-(i) closed if its graph is closed;

(ii) compact if F (Y ) is contained in a compact subset of X;

(iii) upper semicontinuous at y0 ∈ Y if, for each open set U ⊃ F (y0), there is aneighborhood N of y0 such that F (N ) ⊂ U ;

(iv) lower semicontinuous at y0 if, for each open subset U ⊂ X with F (y0) ∩ U 6= ∅,there is a neighborhood N of y0 such that F (y) ∩ U 6= ∅ for all y ∈ N

Proposition 1.1.2 [(6)] Let Y and X be topological spaces, and F : Y → 2X Then,(i) F is l.s.c at y0 iff, for any net {yα} ⊂ Y with yα → y0 and any x ∈ F (y0), thereexists xα ∈ F (yα) such that xα → x

(ii) If F is compact-valued (i.e., F (y) is a compact set for each y ∈ Y ), then F isu.s.c at y0 iff, for any net {yα} ⊂ Y with yα → y0 and any xα ∈ F (yα), thereexist x0 ∈ F (y0) and a subnet {xλ} of {xα} such that xλ → x0

1.2 Abstract convexity structures and generalized

KKM mappings

Definition 1.2.1 ([48, 49, 52]) A triple (X, Y, Φ) is said to be a generalized finitelycontinuous topological space (GFC-space) if X is a topological space, Y a nonemptyset, and

Φ := {ϕN : ∆n→ X : ϕN is a continuous function, for each N = {y0, , yn∈ hY i}.Later, we also write (X, Y, {ϕN}) instead of (X, Y, Φ)

Definition 1.2.2 ([41]) Let (X, Y, Φ) be a GFC-space, D ⊂ X, and C ⊂ Y Then,(i) D is called GFC-convex with respect to (wrt) C if, for each N := {y0, , yn} ∈

hY i and M := {yi0, , yik} ⊂ N ∩ C, ϕN(∆M) ⊂ D, where ∆M is the face of thesimplex ∆|N |−1, corresponding to M (i.e., ∆M is the simplex formed by ei0, , eik,and hence, sometimes we write ∆k instead of ∆M if no confusion can occur) If

Y = X and C = D, then D is said to be GFC-convex simply

(ii) If, in addition, Y ⊂ X, D is said to be weak GFC-convex if, for all N ∈ hY ∩ Di,

ϕN(∆|N |−1) ⊂ D

Remark 1.2.1 (i) Note that a GFC-space is equipped only with topological-basedstructures, without linear or convex structures The same notion was introduced

in [70] under the name “ΦA-space”

(ii) If Y = X, then (X, Y, Φ) is written simply as (X, Φ) and becomes a FC-spacedefined in [18-21] Let D and C be subsets of (X, Φ) D is called a FC-subspace

of X relative to C if, for each N := {x0, , xn} ∈ hXi and M := {xi 0, , xik} ⊂

N ∩ C, ϕN(∆M) ⊂ D If C = D then, D is termed simply a FC-subspace.Note that, when Y = X, being a GFC-convex (wrt C) coincides with being aFC-subspace (relative to C) However, the definition of weak GFC-convexity is

a proper extension of that of a FC-subspace as seen in Example 1.2.1 below.(iii) Another special case of a GFC-space is a G-convex space defined as follows.Definition 1.2.3 ([69]) Let X be a topological space, Y a nonempty set, and Γ :

hY i → 2X be such that, for each N ∈ hY i, there exists a continuous map ϕN :

(X, Y, Γ) is called a G-convex space

For the special case where Y ⊂ X, a subset D of X is said to be Γ-convex if, foreach N ∈ hY i, N ⊂ D implies Γ(N ) ⊂ D

Note that the notions of a FC-space and G-convex space are incomparable andgeneral enough to include many spaces which are met in the literature, but both areparticular cases of a GFC-space

In comparison with a FC-space, a GFC-space is more convenient and easier to bebuilt suitably for a particular purpose, since Y can be chosen different from X, andhence a family Φ can be easily chosen, see Example 2.1 of [43] Observe that to define

a G-convex space, we need a set-valued map Γ : hY i → 2X and a family of continuousmaps ϕN : ∆n → Γ(N ) such that ϕN(∆k) ⊂ Γ(Nk) for each Nk ⊂ N (This condition

is clearly satisfied if Γ(N ) = X for each N ∈ hY i, and we call this case trivial.) For

a GFC-space, any family Φ of continuous maps, having a map corresponding to each

N ∈ hY i, without additional conditions, can be used

Moreover, the notion of weak GFC-convexity includes properly both the definitions

of a FC-subspace (a GFC-convexity with Y = X and C = D) and Γ-convexity, asshown by the following two examples

Example 1.2.1 Let X = R, D0 = {0}, D1 = {1}, and D2 = {0, 1} We showthat there is no FC-space (X, Φ) such that D0, D1 and D2 are FC-subspaces Indeed,suppose to the contrary that they are FC-subspaces in a FC-space (X, Φ) Then, bydefinition, one must have

To find a GFC-space (X, Y, Φ) so that D0, D1 and D2 are weak GFC-convex sets,

we simply take Y = X = R and define a family {ϕN}N ∈hY i by ϕN(e) = minN for

e ∈ ∆|N |−1

Example 1.2.2 Let X = Y = R and D = [0, 1] ⊂ X We build a family Φ to have aGFC-space such that D is a weak GFC-convex set by defining, for each N ∈ hY i with

|N | = n+1, ϕN(e) := minN +maxN2 for all e := Pn

i=0λiei ∈ ∆n Clearly, ϕN is continuousand hence (X, Y, {ϕN}) is a GFC-space It is easy to see that D is weak GFC-convex.However, D is not GFC-convex Indeed, take N = {−1, 0} Then, N ∩ D = {−1}.However, ϕN(∆0) = −12 6⊂ D (∆0 is the face of ∆|N |−1 corresponding to {0})

On the other hand, suppose (X, Y, {ϕN}) can be made to become a G-convex space(X, Y, Γ) Then, for all N ∈ hY i and y ∈ Y , ϕN ∪{y}(∆n) ⊂ Γ(N )

For each x ∈ X, if x < 12(minN + maxN ), choose y = 2x − maxN Then,

Definition 1.2.4 ([32, 48, 49, 52]) Let (X,Y ,Φ) be a GFC-space and Z be a logical space Let F : Y → 2Z, T : X → 2Z be set-valued mappings F is called aKKM mapping wrt T (T -KKM mapping) if, for each N = {y0, , yn} ∈ hY i and each{yi0, , yik} ⊆ N , one has T (ϕN(∆k)) ⊆ Sk

topo-j=0F (yij), where ϕN ∈ Φ is corresponding

to N and ∆k is the face of ∆n formed by {ei0, , eik}

Definition 1.2.5 ([39]) Let (X, Y, Φ) be a GFC-space, Z a topological space, A anonempty set, F : A → 2Z, and T : X → 2Z F is said to be a generalized KKMmapping wrt T (g-T -KKM mapping) if, for each NA= {a0, , an} ∈ hAi, there exists

N = {y0, , yn} ∈ hY i such that, for each {i0, , ik} ⊂ {0, , n}, one has T (ϕN(∆k)) ⊂

Sk

j=0F (aij), where ϕN ∈ Φ is corresponding to N and ∆k is the face of ∆n formed by{ei0, , eik}

Note that, if A ≡ Y , then Definition 1.2.5 implies Definition 1.2.4 While Definition1.2.5 is a natural generalization of Definition 2.1 of [16], where X = Y = Z is atopological vector space, A a convex subset of another topological vector space, Tthe identity map, and ϕN(·) = co(·) (the usual convex hull) Consequently, it alsogeneralizes Definition 2.1 of [4] 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 1.2.3 Let X = Z = R and Y = Q For each N = {y0, , yn} ∈ hY i, let ϕN

be defined by ϕN(e) = Pn

i=0λiyi for all e = Pn

i=0λiei ∈ ∆n Clearly, (X, Y, {ϕN}) is aGFC-space Let F : Y → 2Z be given by F (y) ≡ [0, +∞) and T be the identity map.Let N = {−1} Then,

T (ϕN(∆0)) = {−1} 6⊂ F (−1) = [0, +∞)

Hence, F is not a T -KKM mapping

Now, for each NA = {a0, , an} ∈ hAi = hY i, we take N = {y0, , yn} ={|a0|, , |an|} ∈ hY i, where | · | denotes the absolute value It is easy to see that

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

This means that F is a g-T -KKM

Lemma 1.2.1 ([39]) Let (X, Y, {ϕN}) be a GFC-space, Z a topological space, A anonempty set, F : Z → 2A, and T : Z → 2X Then, the following statements areequivalent

(i) for each z ∈ Z and NA = {a0, , an} ∈ hAi, there exists N = {y0, , yn} ∈ hY isuch that, for each {ai0, , aik} ⊂ NA∩ F (z), one has ϕN(∆k) ⊂ T (z), where ∆k

is the simplex formed by {ei0, , eik};

We choose N given in condition (i) associated with NA By (4), there are x0 ∈ ϕN(∆k)and z0 ∈ T∗(x0) such that

(i) Let (X, Y, Φ) be a GFC-space, Z a topological space, F : Y → 2Z, and T : X →

2Z F is said to be a weak KKM mapping wrt T (weak T -KKM mapping) if, foreach N := {y0, y1, , yn} ∈ hY i,

(ii) We say that a mapping T : X → 2Z has the weak KKM property if, for eachweak T -KKM mapping F : Y → 2Z, the family {clF (y) : y ∈ Y } has the finiteintersection property, i.e., all the finite intersections of sets of this family arenonempty By KKM(X, Y, Z), we denote the class of the mappings T : X → 2Zwhich enjoy the weak KKM property If Z = X, we simply write KKM(X, Y ).Definition 1.2.6 relaxes Definition 1.2.4, since the inclusion in (i) is required onlyfor ∆n, not for all the faces of this simplex Of course, a T -KKM mapping is a weak

T -KKM mapping, but the converse is not true in general as seen in Example 1.2.4.Moreover, for the special case where X = Y is a convex subset of a vector topologicalspace and, for each N = {y0, , yn} ∈ hY i = hXi, the function ϕN : ∆n → X is given

by ϕN(e) =Pn

i=0λiyi for all e =Pn

i=0λiei ∈ ∆n, Definitions 1.2.6 and 1.2.4 collapse to

a generalized KKM mapping in [15, p 225], which is an extension of the classical KKMmapping concept Moreover, in Section 3.2, imposing the weak T -KKM property, weobtain useful sufficient conditions for the existence of fixed points, fixed points of acomposition, and coincidence points of set-valued mappings Moreover, in Section 3.3,

we show that, applying these conditions, we can establish criteria for the existence ofsolutions of most optimization-related problems

Example 1.2.4 Let X = Y = Z = R, F : Y → 2Z be defined by F (y) = {y}, and

T : X → Z be the identity map First, assume there exists a F which is a T -KKMmapping in the sense of Definition 1.2.4 in a GFC-space (X, Y, Φ) Take N∗ = {0, 1}.Then, by definition, one must have

T (ϕN ∗(∆0)) = {ϕN ∗(∆0)} ⊂ {0} = F (0), (9)where ∆0 is the face of ∆|N ∗ |−1 corresponding to {0},

T (ϕN ∗(∆00)) = {ϕN ∗(∆0)} ⊂ {1} = F (1), (10)where ∆00 is the face of ∆|N ∗ |−1 corresponding to {1}, and

T (ϕN∗(∆|N∗ |−1)) = {ϕN∗(∆|N∗ |−1)} ⊂ {0, 1} = F (0) ∪ F (1) (11)

However, there hardly exists such a continuous function ϕN∗ : ∆|N∗ |−1 → X for theconstruction of a GFC-space (X, Y, Φ) such that ϕN∗ satisfies (9), (10), and (11) to get

a T -KKM mapping Of course,

T (co(N∗)) = [0, 1] 6⊂ {0, 1} = F (0) ∪ F (1),where co(·) is the usual convex hull operator Therefore, F is not a generalized KKMmapping wrt T in the sense given in [16, p 225] either

Now, let (X, Y, Φ) be the GFC-space given in Example 1.2.1 Then, for each N ={y0, , yn} ∈ hY i,

For any sets U, V , a point x and a, b ∈ R, we adopt the notations:

r1(U, V ) means U ∩ V = ∅; r2(U, V ) means U ⊂ V ;

r3(U, V ) means U ∩ V 6= ∅; r4(U, V ) means U * V ;

Let X, W , and Z be nonempty sets, S1 : X → 2X, S2 : X → 2W, K : X × W → 2Z

be set-valued mappings with the nonempty values, and R(x, w, z) be a relation linking

x ∈ X, w ∈ W and z ∈ Z Our variational relation problem is, for α ∈ {α1, α2},(VRα) find ¯x ∈ S1(¯x) such that, for all w ∈ S2(¯x), α(z, K(¯x, w)),

R(¯x, w, z) holds

Originally, the first variational relation problem, investigated in [44, 63], is for

α = α1 The reason for our change is: varying α causes the model to include moreparticular problems and hence it is worth for the notations to become more complicated.Definition 1.3.1 ([63]) The relation R is called closed if the set {(x, w, z) ∈ X × W ×

Z : R(x, w, z) holds} is closed

Let X, W, Z, S1, S2 and K be the same as for problem (VRα) Let eZ be a nonemptyset, H : X × W × K(X, W ) → 2Z e and G : X × H(X, W ) → 2Z e For each r ∈{r1, r2, r3, r4} and α ∈ {α1, α2}, we consider the following quasivariational inclusionproblem

(QIPrα) find ¯x ∈ S1(¯x) such that, for all w ∈ S2(¯x), α(z, K(¯x, w)),

r(H(¯x, w, z), G(¯x, z))

This formulation was proposed first in [31] and has been used, e.g., in [43, 49] Itlooks complicated However, the used notations cause it to include many more specialcases and help to unify proof techniques for the existence of solutions in many cases

Let X, W, Z, S1, S2, K and H be the same as for problem (QIPrα) Let eZ be atopological vector space, and C : X → 2Z e a set-valued mapping with the nonempty-closed-convex-cone values For each r ∈ {r1, r2, r3, r4} and α ∈ {α1, α2}, we considerthe following Stampacchia-type vector quasiequilibrium problem

(VEPrα) find ¯x ∈ S1(¯x) such that, for all w ∈ S2(¯x), α(z, K(¯x, w)),

r(H(¯x, w, z), (−C(¯x) \ {0}))

Note that, if S1(x) ≡ X, r ∈ {r1, r4} and α ∈ {α1, α2}, (VEPrα) collapses to thecorresponding four problems investigated in [59]

1.3.4 Nash equilibria for non-cooperative games

Let I := {1, , n} be a set of players, X1, , Xn be nonempty sets, and Gi : Xi →

2X i, where Xi := Πj6=iXj Each ith player has a nonempty strategy set Gi(Xi) and apay-off function fi : X := Πi∈IXi → R The 2n-tuple

Γ := (G1(X1), G2(X2), , Gn(Xn), f1, f2, , fn)

is called a n-person non-cooperative game x(−i) denotes the projection of x ∈ X on

Xi A point ¯x := (¯x1, ¯x2, , ¯xn) ∈ X is called a generalized Nash equilibrium point ofthe game Γ if, for all i ∈ I, ¯xi ∈ Gi(¯x(−i)) and wi ∈ Gi(¯x(−i)),

fi(¯x) ≥ fi(¯x(−i), wi)

We define a bifunction ΦΓ : X × X → R by ΦΓ(x, w) := Pn

i=1(fi(x) − fi(x(−i), wi))(the Nikaido-Isoda bifunction) Then, ¯x is a generalized Nash equilibrium point of Γ

if and only if ¯x is a solution of the quasi-equilibrium problem

(QEP) finding ¯x ∈ X such that, for all i ∈ I, ¯xi ∈ Gi(¯x(−i)) and w ∈ Πni=1Gi(¯x(−i)),

ΦΓ(¯x, w) ≥ 0

Let a network consist of nodes and links (or arcs) Let Q := (Q1, , Ql) be the set

of pairs called origin-destination (O/D) pairs Each of them consists of an origin nodeand a destination one Assume that Pj, j = 1, , l, is the set of paths connecting thepair Qj and that Pj includes rj ≥ 1 paths Let m := r1+ + rl, and x := (x1, , xm)denote a path (vector) flow Assume that the constraint of the capacity of the paths

denote the travel vector demand by g := (g1, , gl) Denote the Kronecker numbers by

equilib-(QVIα) find ¯x ∈ S(¯x) such that, for all w ∈ S(¯x), α(z, L(¯x)),

hz, w − ¯xi ≥ 0

Let X be a nonempty set, f : X → R, S : X → 2X and ρ ∈ {ρ1, ρ2} Theconstrained minimization and maximization problems we consider in this thesis arestated as follows:

(MPρ) find ¯x ∈ X such that ¯x ∈ S(¯x) and ρ(f (w) ≥ f (¯x)) for all w ∈ S(x)

1.3.7 Saddle points

Let B, D be nonempty sets, f a real-valued function on B × D and S : B × D →

2B×D The generalized saddle-point problems we consider are stated as follows:(SPP) find (¯b, ¯d) ∈ B × D such that (¯b, ¯d) ∈ S(¯b, ¯d) and, for all (b, d) ∈ S(¯b, ¯d),

f (¯b, d) ≤ f (b, ¯d)

Chapter 2

Several existence theorems in

nonlinear analysis related to

generalized KKM mappings and

applications

The study of the existence of solutions takes a central place in the theory forany class of problems and plays also a vital role in applications The study of theexistence of solutions of a problem is based on existence results for important points

in nonlinear analysis such as fixed points, maximal points, intersection points, etc.During a long period in the past, it was believed that such existence results neededboth topological and linear/convex structures However, originated from Wu [81] andHorvarth [34], two directions of dealing with pure topological-based existence theoremshave been developed The first approach is based on replacing convexity assumptionswith connectedness conditions, and the second one on replacing a convex hull with animage of a simplex through a continuous map Very recently, in [50, 51], a combination

of the two ways was discussed This chapter follows the idea of the second approach.Recently, this idea was intensively developed in combination with the KKM theory toobtain pure topological-based existence theorems and applications in the study of theexistence of solutions of optimization-related problems, e.g., in [20, 21, 30, 31, 43, 48,

49, 52, 55].

On the other hand, continuous-selection theorems also play an important role in

nonlinear analysis, optimization and applied mathematics The first result was

ob-tained by Michael [66] Since then, many contributions have been made to develop this

result in general settings: C-spaces [35], G-convex spaces [68], and C∞-spaces [83]

Inspired by the above results, in this chapter we use Definition 1.2.5 to establish

some continuous-selection theorems and derive equivalent topological sufficient

con-ditions for the existence of many important points in nonlinear analysis Then, we

apply these conditions to various optimization-related problems Our results improve

or generalize a number of recent ones in the literature

2.1 Existence theorems and applications to

optimization-related problems

In this subsection, the existence of important objects in applied analysis in pure

topological settings of GFC-spaces are established Let us begin with the existence

of continuous selections of set-valued maps For a set-valued map T : Z → 2X on

topological spaces Z and X, recall that a (single-valued) continuous map t : Z → X is

called a continuous selection of T if t(z) ∈ T (z) for all z ∈ Z In this chapter, all the

topological spaces will be assumed Hausdorff

Theorem 2.1.1 Let Z be a normal (topological) space, (X, Y, {ϕN}) a GFC-space, and

T : Z → 2X Assume that there exists F : Z → 2Y such that the following conditions

hold

(i) for each z ∈ Z, T (z) is GFC-convex wrt F (z);

(ii) Z =Sm

i=0intF−1(¯yi) for some M = {¯y0, , ¯ym} ⊆ Y

Then, T has a continuous selection t of the form t = ϕ ◦ ψ for some continuous maps

ϕ : ∆m → X and ψ : Z → ∆m

Proof By (ii) and the normality of Z, there exists a continuous partition of unity{ψi}m

i=0 of Z associated with the finite open cover {intF−1(¯ai)}mi=0 Then, for each z ∈

Z, let J (z) be a subset {¯yi0, , ¯yik} of {¯y0, , ¯ym} such that ψil(z) 6= 0 for l = 0, , k.Since ψil(z) 6= 0, z ∈ intF−1(¯yil) ⊆ F−1(¯yil), i.e., ¯yil ∈ F (z), ∀l ∈ {0, , k} Hence,

t(z) = (ϕM ◦ ψ)(z) ∈ ϕM(∆J (z)) ⊆ T (z),where the last inclusion is true by (i) Finally, putting ϕ = ϕM, we arrive at the

Theorem 2.1.2 Let Z be a compact topological space, (X, Y, Φ) be a GFC-space, and

T : Z → 2X have the nonempty values Assume that there are a nonempty set A and

F : Z → 2A such that the following conditions hold

(i) F∗ is a g-T∗-KKM mapping;

(ii) Z =S

a∈AintF−(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 NA= {¯a0, , ¯an} ∈ hAi such that Z =

Sn

i=0intF−( ¯ai) Then, there is a continuous partition of unity {ψi}n

i=0 of Z associatedwith the finite open cover {intF−1(¯ai)}n

i=0 From (i) there exists N = {y0, , yn} ∈ hY iassociated with NA= {¯a0, , ¯an} Moreover, due to the GFC-space structure, there is

ϕN : ∆n → X corresponding to N Now, we define continuous maps ψ : Z → ∆n and

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

ϕN(ψ(z0)) := x0 ∈ T (z/ 0), or equivalently, z0 ∈ Z \ T−(x0) = T∗(x0) Furthermore, onehas

F∗ is a g-T∗-KKM mapping, one has

Example 2.1.1 Let Z = [0, 1], X = [−2, 2], and T : Z → 2X be defined by T (z) ≡[−1, 1] For each N := {x0, , xn} ∈ hRi, ϕN : ∆n → X be defined by, for e ∈ ∆n,

Then, (X, {ϕN}N ∈hXi) is a FC-space However, we claim that it is impossible to find acontinuous selection of G by using Theorem 2.1 of [20] Indeed, suppose to the contrarythat there exists a mapping F : Z → 2X such that assumption (i) of this theorem (foreach z ∈ Z, T (z) is a FC-subspace on X wrt F (z)) is fulfilled That is, for each z ∈ Z,

N = {x0, , xn} ∈ hXi and {yi0, , yik} ⊂ N ∩ F (z), one has ϕN(∆k) ⊂ T (z) Wetake ¯z ∈ Z and M = {¯x0, , ¯xm} ∈ hXi such that ∅ 6= {¯xi0, , ¯xil} ⊂ M ∩ F (¯z) Set

M∗ = M ∪√

2 Then ϕM∗(∆l) = 2 6⊂ [−1, 1] = T (¯z), a contradiction

Now, we take Y = Q and a set-valued mapping F : Z → 2Y defined by F (z) = Q.Then, for each z ∈ Z, N := {y0, , yn} ∈ hY i = hQi, and {yi 0, , yik} ⊂ N ∩F (z) = N ,one has ϕN(∆k) = {sinPk

j=0yij} ⊂ [−1, 1] = G(z), i.e., (i) of Lemma 1.2.1, which isequivalent to (i) of Theorem 2.1.2, is fulfilled Assumption (ii) of Theorem 2.1.2 isclearly satisfied By this theorem, T has a continuous selection

Now, apply Theorem 2.1.2 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 2.1.3 (fixed points) Let (X, Y, Φ) be a compact GFC-space and T : X → 2X.Assume that there are a nonempty set A and F : X → 2A such that the followingconditions hold

(i) T has the nonempty values and F∗ is a g-T∗-KKM mapping;

(ii) X =S

a∈AintF−(a)

Then, T has a fixed point ¯x ∈ X, i.e., ¯x ∈ T (¯x)

Proof According to Theorem 2.1.2, T has a continuous selection t = ϕ ◦ ψ, where

ϕ : ∆n → X and ψ : X → ∆n are continuous Then, ψ ◦ ϕ : ∆n → ∆n is alsocontinuous By virtue of the Tikhonov fixed-point theorem, there exists ¯e ∈ ∆n suchthat ψ ◦ ϕ(¯e) = ¯e Setting ¯x = ϕ(¯e), we have

¯

x = ϕ(ψ(¯x)) = t(¯x) ∈ T (¯x)

Remark 2.1.2 Theorem 2.1.3 sharpens Corollary 3.1 (ii1) of [43] since assumption (i)

is weaker than the corresponding assumption (i) of that result Applied to the specialcase where X = Y = A is a nonempty compact convex subset of a topological vectorspace, T = H, and ϕN(·) = co(·), Theorem 2.1.3 generalizes Theorem 1 of [12]

Theorem 2.1.4 (sectional points) Let (X, Y, Φ) be a compact GFC-space and M asubset of X × X Assume that there are a nonempty set A and F : X → 2A such thatthe following conditions hold

(i) for each x ∈ X and NA = {a0, , an} ∈ hAi, there exists N = {y0, , yn} ∈ hY isuch that, for each {ai0, , aik} ⊂ NA∩ F (x), ϕN(∆k) ⊂ {w ∈ X : (x, w) /∈ M };

(ii) X =S

a∈AintF−(a);

(iii) (x, x) ∈ M for all x ∈ X

Then, there exists ¯x ∈ X such that {¯x} × X ⊂ M

Note that, for a similar reason as in Remark 2.1.2, Theorem 2.1.4 generalizes Lemma

4 of [25]

Theorem 2.1.5 (maximal elements) Let (X, Y, Φ) be a compact GFC-space and T :

X → 2X Assume that there are a nonempty set A and F : X → 2A such that thefollowing conditions hold

(i) F∗ is a g-T∗-KKM mapping;

(ii) S

w∈XT−(w) ⊂S

a∈AintF−(a);

(iii) x /∈ T (x) for all x ∈ X

Then, T has a maximal point ¯x ∈ X, i.e., T (¯x) = ∅

Theorem 2.1.6 (intersection points) Let (X, Y, Φ) be a compact GFC-space and G :

X → 2X Assume that there are a nonempty set A and F : X → 2A such that thefollowing conditions hold

a∈AintF−(a);

(iii) x ∈ G(x) for all x ∈ X

Then, T

x∈XG(x) 6= ∅

Theorem 2.1.7 (solutions of variational relations) Let (X, Y, Φ) be a compact space, Z a nonempty set, S : X → 2X, K : X × X → 2Z, R(x, w, z) be a relationlinking x ∈ X, w ∈ X, z ∈ Z, and i ∈ {1, 2} Assume that there are a nonempty set

GFC-A and F : X → 2A such that the following conditions hold

(i) for each x ∈ X and NA = {a0, , an} ∈ hAi, there exists N = {y0, , yn} ∈ hY isuch that, for each {ai0, , aik} ⊂ NA∩ F (x), one has ϕN(∆k) ⊂ {w ∈ S(x) :

α3−i(z, K(x, w)), R(x, w, z) does not hold};

(ii) S

w∈X{x ∈ S−(w) : α3−i(z, K(x, w)), R(x, w, z) does not hold} ⊂S

a∈AintF−(a);(iii) x /∈ {w ∈ S(x) : α3−i(z, K(x, w)), R(x, w, z) does not hold} for all x ∈ X

Then, there exists a ¯x ∈ X such that ∀w ∈ S(¯x), αi(z, K(¯x, w)), R(¯x, w, z) holds.Now, the equivalence of the above five theorems will be proved following the diagram

Theorem 2.1.3 ⇒ Theorem 2.1.5 ⇒ Theorem 2.1.7 ⇒ Theorem 2.1.3

Theorem 2.1.4 Theorem 2.1.6

Theorem 2.1.3 ⇒ Theorem 2.1.4 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 the proof is complete.Suppose T (x) 6= ∅ for all x ∈ X and the conclusion of Theorem 2.1.4 is false By(i) of Theorem 2.1.4 and Lemma 1.2.1, (i) of Theorem 2.1.3 is satisfied Since thetwo assumptions (ii) are the same, by Theorem 2.1.3, ¯x ∈ T (¯x) for some ¯x ∈ X, i.e.,

¯

x ∈ {w ∈ X : (¯x, w) /∈ M }, which contradicts (iii) of Theorem 2.1.4

Theorem 2.1.4 ⇒ Theorem 2.1.3 Assume that all the assumptions of Theorem2.1.3 are fulfilled and set M = {(x, w) ∈ X × X : w /∈ T (x)} Suppose to the contrarythat x /∈ T (x) for all x ∈ X Then, (x, x) ∈ M for all x ∈ X, i.e., (iii) of Theorem 2.1.4holds According to Lemma 1.2.1, for each x ∈ X and NA = {a0, , an} ∈ hAi, there

exists N = {y0, , yn} ∈ hY i such that, for each {ai0, , aik} ⊂ NA∩ F (x), one has

ϕN(∆k) ⊂ T (x) Hence, for each w ∈ ϕN(∆k), w ∈ T (x), i.e., (x, w) /∈ M It followsthat ϕN(∆k) ⊂ {w ∈ X : (x, w) /∈ M }, i.e., (i) of Theorem 2.1.4 is satisfied Sincethe two assumptions (ii) are the same, applying Theorem 2.1.4, one obtains ¯x ∈ Xsuch that {¯x} × X ⊂ M It implies that w /∈ T (¯x) for all w ∈ X, contradicting theassumption that T has the nonempty values

Theorem 2.1.5 ⇒ Theorem 2.1.6 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 2.1.6, all the assumptions of Theorem 2.1.5 are fulfilled Therefore, thereexists ¯x ∈ X such that T (¯x) = ∅, i.e., X \ G−(¯x) = ∅ Hence, ¯x ∈T

x∈XG(x)

Theorem 2.1.6 ⇒ Theorem 2.1.5 Under the assumptions of Theorem 2.1.5, letG(x) = X \ T−(x) for x ∈ X Then, assumptions (i) and (ii) of Theorem 2.1.5 clearlyimply the corresponding (i) and (ii) of Theorem 2.1.6 From (iii) of Theorem 2.1.5,one has x ∈ X \ T−(x) = G(x) for all x ∈ X, i.e., (iii) of Theorem 2.1.6 is satisfied

By Theorem 2.1.6, there exists ¯x ∈ T

a fixed point This contradicts (iii) of Theorem 2.1.5

Theorem 2.1.5 ⇒ Theorem 2.1.7 Let T : X → 2X be defined by

T (x) = {w ∈ S(x) : α3−i(z, K(x, w)), R(x, w, z) does not hold}

By (i) of Theorem 2.1.7 and Lemma 1.2.1, F∗ is a g-T∗-KKM mapping, i.e., (i) ofTheorem 2.1.5 is fulfilled It is not difficult to see that (ii) and (iii) of Theorem 2.1.7imply the corresponding (ii) and (iii) of Theorem 2.1.5 Applying this theorem, wehave ¯x ∈ X such that T (¯x) = ∅ Consequently, ∀w ∈ S(¯x), αi(z, K(¯x, w)), R(¯x, w, z)

Theorem 2.1.7 ⇒ Theorem 2.1.3 Let the assumptions of Theorem 2.1.3 be satisfied

We define two mappings S : X → 2X, K : X × X → 2Z and a relation R by, for

x, w ∈ X,

S(x) ≡ X, K(x, w) = z0 for an arbitrary z0 ∈ Z,

αi(z, K(x, w)), R(x, w, z) holds ⇔ w /∈ T (x)

Then, one has, for all x ∈ X,

{w ∈ S(x) : α3−i(z, K(x, w)), R(x, w, z) does not hold} = T (x)

Suppose, for all x ∈ X, x /∈ T (x) Then, (iii) of Theorem 2.1.7 is fulfilled Clearly,

by (ii) of Theorem 2.1.3, (ii) of Theorem 2.1.7 is satisfied Since F∗ is a g-T∗-KKMmapping, by Lemma 1.2.1, (i) of Theorem 2.1.7 holds According to this theorem,

¯

x ∈ X exists such that, for all w ∈ S(¯x) = X, αi(z, K(¯x, w)), R(¯x, w, z) holds Thismeans that w /∈ T (¯x) for all w ∈ X, contradicting assumption (i) of Theorem 2.1.3that T has nonempty values

Remark 2.1.3 The existence of the above-mentioned points has been obtained in anumber of contributions, to various extents of generality and relaxation of assumptions,see, e.g., recent papers [32, 43, 51] Our assumption (i) here is weaker than the existingcorresponding conditions, and furthermore, it is of a very simple form and directly interms of KKM-type mappings

Now, we apply topological existence theorems to establish sufficient conditions ofthe solution existence of many typical optimization-related problems Note that, in thenext chapters, we still consider other existence theorems and also apply them as tools

to investigate such conditions

A Quasivariational inclusions Now we consider the quasivariational inclusionproblem mentioned in Chapter 1 with S = S1 = S2

Theorem 2.1.8 For problem (QIPrjαi), j = 1, , 4 and i = 1, 2, assume that thereare Y and Φ such that (X, Y, Φ) is a compact GFC-space Assume, further that thereare a nonempty set A and F : X → 2A such that the following conditions hold

(i) for each x ∈ X and NA = {a0, , an} ∈ hAi, there exists N = {y0, , yn} ∈

hY i such that, for each {ai0, , aik} ⊂ NA ∩ F (x), ϕN(∆k) ⊂ {w ∈ S(x) :

Then, problem (QIPrjαi) has a solution

Proof Employ Theorem 2.1.7 with the relation R defined by

R(x, w, z) holds if and only if rj(H(x, w, z), G(x, z))

Under the assumptions of Theorem 2.1.8, the assumptions of Theorem 2.1.7 are ily seen to be satisfied Hence, ¯x ∈ X exists such that, ∀w ∈ S(¯x), αi(z, K(¯x, w)),R(¯x, w, z) holds Consequently, ¯x is a solution of (QIPriαj) in this case 2

eas-B Stampacchia-type vector quasiequilibrium problems Since problem (QIPrjαi)includes most optimization-related problems, sufficient conditions for the existence oftheir solutions can be derived directly from Theorem 2.1.8 Here, we mention onlysome important problems as examples, also for the sake of comparison with severalrecent existing results First, we discuss a relatively general model of Stampacchia-type vector equilibrium problems, and then apply the obtained result to other ones.Theorem 2.1.8 becomes a sufficient condition for the existence of solutions of problem(VEPrα) in Chapter 1 with S = S1 = S2 and the simple replacement of G(x, z) with

−C(x) \ {0} Note that, here we do not need eZ to be equipped with a topology, and

C be closed-valued, as assumed in [59] The following example provides a case whereTheorem 2.1.8 for problem (VEPrα) is applicable while a recent existing result is not

Problem (VEPr4α) in this case is: find ¯x ∈ X such that, ∀w ∈ S(¯x),

To check assumption (i) of Theorem 2.1.8, consider x ∈ X and NA= {a0, , an} ∈ hAi

If x 6= 1, we take N = {a0 + 2, , an+ 2} ∈ hY i to see that, for each {ai0, , aik} ⊂

Suppose 1 ∈ S−( ¯w) for some ¯w ∈ X such that ¯w2(12+ 1) ∈ (0, +∞) By the definition

of S, ¯w = 0, it implies that ¯w2(12 + 1) = 0 /∈ (0, +∞), a contradiction This meansthat assumption (ii) of Theorem 2.1.8 is satisfied For each x ∈ X, it is easy to seethat

x /∈





{w ∈ S(x) : w2(x2+ 1) ∈ (0, +∞)} = {1, 1 − x}, if x 6= 1{w ∈ S(1) : w2(12+ 1) ∈ (0, +∞)} = ∅, if x = 1

Thus, (iii) of Theorem 2.1.8 is checked We can also verify directly that ¯x = 1 is asolution However, Theorem 4.1 of [59] does not work, since, for each x ∈ X, there is

no y ∈ X such that w2(x2+ 1) /∈ [0, +∞) for all w ∈ S(y), i.e., assumption (ii) of thattheorem is not satisfied

C Nash equilibria for non-cooperative games Pass now to the Nash equilibriumproblem introduced in Chapter 1 with G(x(−i)) ≡ Xi for all x(−i) ∈ Xi and i ∈{1, , n}

Theorem 2.1.9 For the game Γ, assume that there are Y and Φ such that (X, Y, Φ) is

a compact GFC-space Assume further that there are a nonempty set A and F : X → 2A

such that the following conditions hold

(i) for each x ∈ X and NA = {a0, , an} ∈ hAi, there exists N = {y0, , yn} ∈ hY isuch that, for each {ai 0, , aik} ⊂ NA ∩ F (x), one has ϕN(∆k) ⊂ {w ∈ X :Φ(x, w) < 0};

(ii) S

w∈X{x ∈ X : Φ(x, w) < 0} ⊂S

a∈AintF−(a)

Then, Γ has a Nash equilibrium point

Proof We simply apply Theorem 2.1.8 with r = r1, S(x) ≡ X, eZ = R, G(x) ≡(−∞, 0), and H(x, w) = ΦΓ(x, w) (problem (NEP) does not include Z, α, and F ) Inthis case, we note that (iii) of Theorem 2.1.8 is always satisfied 2Example 2.1.3 Consider a 2-person non-cooperative game Γ with X1 = X2 = [0, 1] ∪[2, 3], f1(x1, x2) = 2x1 − 3x2, and f2(x1, x2) = x1 + 2x2 This game is equivalent

to the equilibrium problem: find ¯x = (¯x1, ¯x2) ∈ X := X1 × X2 such that, for all

(X, Y, {ϕN}) is a compact GFC-space We choose A = Y and F : X → 2A defined by

ϕN(∆k) = {(3, 3)} ⊂ {w ∈ X : Φ(x, w) < 0} = {w ∈ X : x1+ x2− w1− w2 < 0}

If x = (3, 3), then NA ∩ F (3, 3) = ∅ Hence, assumption (i) of Theorem 2.1.9 isfulfilled Clearly, intF−(a) = int(X \ {(3, 3)}) = X \ {(3, 3)} for all a ∈ A, and soS

a∈AintF−(a)=X \ {(3, 3)} Condition (ii) of Theorem 2.1.9 is satisfied because

{y ∈ X : Φ(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 For the traffic networks problems stated in Chapter 1, we havethe following

Theorem 2.1.10 For the traffic network problem with α ∈ {α1, α2}, assume that thereare Y and Φ such that (X, Y, Φ) is a compact GFC-space Assume further that thereare a nonempty set A and F : X → 2A such that the following conditions hold

(i) for each x ∈ X and NA = {a0, , an} ∈ hAi, there exists N = {y0, , yn} ∈ hY isuch that, for each {ai0, , aik} ⊂ NA∩F (x), ϕN(∆k) ⊂ {w ∈ X : α3−i(z, L(x)), hz, x−

Then, the traffic problem has a solution.

Proof Apply Theorem 2.1.8 with r = r1, X = Rm, Z = (Rm)∗ (the dual of X), eZ = R,K(x, w) = L(x) for all w ∈ X, G(x, w) ≡ (−∞, 0), and H(x, w, z) = hz, w − xi 2

E Saddle points Next, we get the following solution existence result for the point problem stated in Chapter 1

saddle-Theorem 2.1.11 For problem (SPP), assume that there are Y and Φ such that (B ×

D, Y, Φ) is a compact GFC-space, a nonempty set A, and F : B × D → 2A such thatthe following conditions hold

(i) for each (b, d) ∈ B×D and NA= {a0, , an} ∈ hAi, there exists N = {y0, , yn} ∈

hY i such that, for each {ai0, , aik} ⊂ NA∩ F (b, d), ϕN(∆k) ⊂ {(b0, d0) ∈ B × D :

f (b, d0) − f (b0, d) < 0};

(ii) S

(b 0 ,d 0 )∈B×D{(b, d) ∈ B × D : f (b, d0) − f (b0, d) < 0} ⊂S

a∈AintF−(a)

Then, (SSP) has a solution

Proof Employ Theorem 2.1.8 with X = B×D, eZ = R, r = r1, α = α1, S(b, d) ≡ B×D,G(b, d) ≡ (−∞, 0), and H((b, d), (b0, d0)) = f (b, d0) − f (b0, d) 2

F Constrained minimization Let X be the same as for problem (MPρ), stated

in Chapter 1, and g : X × X → R We study the problem (MPρ) with ρ = ρ1 andS(x) := {w ∈ X | g(x, w) ≤ 0}

Theorem 2.1.12 For problem (MP), assume that there are Y and Φ such that (X, Y, Φ)

is a compact GFC-space, a nonempty set A, and F : X → 2A such that the followingconditions hold

(i) for each x ∈ X and NA = {a0, , an} ∈ hAi, there exists N = {y0, , yn} ∈ hY isuch that, for each {ai0, , aik} ⊂ NA ∩ F (x), one has ϕN(∆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∈AintF−(a).

Then, (MP) has a solution

Proof To apply Theorem 2.1.8, we set eZ = R, r = r1, S(x) = {w ∈ X : g(x, w) ≤ 0},

2.2 Existence theorems on product GFC-spaces and

applications

Theorem 2.2.1 Let I be an index set, {(Xi, Yi, {ϕN i})}i∈I be a family of GFC-spaces,

Ti : X → 2X i for i ∈ I, and X = Πi∈IXi be a normal space Assume that there exists

a Fi : X → 2Yi, for i ∈ I, such that the following conditions hold

(i) for each x ∈ X, each Ni := {yi0, , yini} ⊆ Yi and each {yji0, , yij

ki} ⊆ Ni∩ Fi(x),one has ϕNi(∆ki) ⊆ Ti(x) for all i ∈ I;

(ii) for all i ∈ I, there exists a finite subset {¯yi0, , ¯yimi} of Yi such that either of thefollowing two conditions holds

(ii1) X =Sm i

j=0intFi−1(¯yji) for all i ∈ I;

(ii2) for each compact subset K of X, one has K ⊆S

y i ∈YiintFi−1(yi), and either the set

Then, there exists ¯x ∈ X such that ¯xi ∈ Ti(¯x) for all i ∈ I

Proof For each i ∈ I, by Theorems 2.1 and 2.2 of [43], Ti has a continuous selection ti,and there are continuous maps ϕi : ∆mi → Xi and ψi : X → ∆mi such that ti = ϕi◦ ψi