Existence theorems in nonlinear analysis and applications to optimization related models

VIETNAM NATIONAL UNIVERSITY - HCMC
UNIVERSITY OF SCIENCE

Nguyen Hong Quan

EXISTENCE THEOREMS IN NONLINEAR
ANALYSIS AND APPLICATIONS TO
OPTIMIZATION-RELATED MODELS

PhD THESIS IN MATHEMATICS

Ho Chi Minh City - 2013


VIETNAM NATIONAL UNIVERSITY - HCMC
UNIVERSITY OF SCIENCE

Nguyen Hong Quan

EXISTENCE THEOREMS IN NONLINEAR
ANALYSIS AND APPLICATIONS TO
OPTIMIZATION-RELATED MODELS
Major: Mathematical Optimization
Codes: 62 46 20 01

Referee 1: Assoc.Prof. Dr. Nguyen Dinh
Referee 2: Dr. Huynh Quang Vu
Referee 3: Assoc.Prof. Dr. Lam Quoc Anh
Independent Referee 1: Prof. D.Sc. Nguyen Dong Yen
Independent Referee 2: Assoc.Prof. Dr. Mai Duc Thanh

SCIENTIFIC SUPERVISOR
Professor Phan Quoc Khanh

Ho Chi Minh City - 2013


Confirmation
I confirm that all the results of this thesis come from my work under the supervision of Professor Phan Quoc Khanh. They have never been included in other
papers in the literature. Moreover, the paper (Q5) is a joint work with Dr. Nguyen
Xuan Hai and we have discussed in detail, and based on the contributions of each
coauthor we have agreed that its whole content is used in this thesis, while the
whole joint paper (Q1) was included in the thesis of Nguyen Xuan Hai in 2007.
October 2013

Nguyen Hong Quan


Contents

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

IV

1

Existence theorems in nonlinear analysis and applications . . . . . . . .
1.1 Notions and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Existence theorems in GFC-spaces . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3.1 Variational inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1.3.2 Minimax theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1
1
6
23
23
31

2

Nonlinear existence theorems for mappings on product
GFC-spaces and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Existence theorems on product GFC-spaces . . . . . . . . . . . . . . . . . . .
2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37
37
42

Topological characterizations of existence in nonlinear analysis
and optimization-related problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Topological existence theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 The existence of solutions to optimization-related problems . . . . .
3.2.1 Variational relation problems . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Invariant-point theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Equilibrium problems of the Stampacchia and Minty types
3.2.4 Minimax theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.5 Nash equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

52
52
70
70
74
77
80
88

3

4

Generic stability and essential components of generalized KKM
points and applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1 Notions and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.2 Generic essentialness of T -KKM mappings . . . . . . . . . . . . . . . . . . . 94
4.3 Essential components of sets of T -KKM points . . . . . . . . . . . . . . . . 98
4.4 Applications to maximal elements and variational inclusions . . . . . 99
4.4.1 Essential components of T0 -maximal elements . . . . . . . . . . . 100
4.4.2 Essential components of solutions to variational inclusions . 100


Contents

III

4.4.3 Essential components of particular cases of variational
inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

List of the author’s papers related to the thesis . . . . . . . . . . . . . . . . . . . . . . 105
List of the author’s conference reports related to the thesis . . . . . . . . . . . 106
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Glossary of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111


Preface

The existence theorems, including theorems about various points like fixed
points, coincidence points, intersection points, maximal elements, and other results as KKM theorems, minimax theorems, etc., constitute one of the most important parts of mathematics. They are crucial tools in the solution existence study
of wide-ranging problems of optimization and applied mathematics.
The existence theorems have a long history of development passing more than
a century with the following major milestones: the Brouwer fixed point theorem
(1912, [14]), Classical KKM principle (1929, [57]), Kakutani fixed point theorem
(1941, [50]), KKM-Fan principle (1961, [33]). From 80s of the 20th century, to
meet demands of practical situations, many classes of problems in optimization
have appeared. One of the first and most important issues of such a class is to
know if solutions exist or not. This requires more new effective mathematical
tools. Hence, existence theorems have been intensively developed to response
that requirement. Especially, in recent years the theory of existence theorems has
obtained many significant achievements.
Like other mathematical theories, the existence theorems have been built from
simple basic results by generalization and abstraction methods. In early forms of
these fundamental results, the convexity played a central role in formulating results. Therefore, most of the later results mainly focus on improving assumptions
on convexity or replacing them by purely topological assumptions. According
to our observations, for the last three decades the existence theorems have been
developed in three ways. First, some researchers renovated classical notions of
convexity based on linear structures. For instance, the KKM-Fan mapping (Fan
[33]) was extended into the S-KKM mapping (e.g., Chang and Zhang [15], Chang
and Yen [16], Chang, Huang, Jeng and Kuo [17]). In terms of these notions,

new existence results were achieved. Second, many authors replaced the classical convexity by abstract convexity notions, not using linear structures, and they
extended earlier existing notions and results to these structures. In the framework of this research direction, types of spaces with generalized convexity structures were proposed and studied. Started with Lassonde ([60]) where a convex
space was created, the following spaces (in the chronological order) were in-


Preface

V

troduced: S-contractible spaces (Horvath [45]), H-spaces (Horvath [46-47]), Gconvex spaces (Park and Kim [79-85]), and FC-spaces (Ding [24-32]). These
spaces have been used in studying existence theorems and nonlinear problems. In
the third approach, a number of authors (e.g., Wu [93], Tuy [91-92], Geraghty and
Lin [38], Kindler and Trost [54], Kindler [55-56], Konig [59], Tarafdar and Yuan
[90]) proved existence results which did not require any convexity structures,
where convexity conditions were replaced by connectedness conditions which
are purely topological conditions.
One of the main purposes of this thesis is to develop further the theory of
existence theorems, focusing on the last two approaches. Based on analyzing the
earlier notions and results, we introduce several new structures and use them to
formulate new definitions and establish new or more general results. By providing
illustrative examples, we show the existence of our structures and their usefulness
in many applications. Moreover, our notions and results in this thesis improve or
include as special cases a number of known notions and results.
It is worthwhile noticing that the equivalence of mathematical theorems is
meaningful in applications because it allows us to approach to a problem from
different angles. Therefore, researchers pay much attention on proving equivalence relations between existence results. For examples, in Ha [40], an extension
of Kakutani fixed-point theorem was proved to be equivalent to a section theorem and a minimax result, an equivalence between the KKM-Fan theorem and
a Browder-type fixed-point theorem was shown in Tarafdar [89]. Later on many
researchers discovered similar equivalence relations for other kinds of results like
coincidence theorems, matching theorems, intersection theorems, maximal-point

theorems, section theorems and some geometric results. One of our attempts in
this thesis is to show the equivalence between many of our existence theorems.
On the other hand, any mathematical result needs be applicable for certain situations. Typical applications of existence theorems are in optimization problems.
Therefore, using our existence theorems to establish solution existence results
for optimization-related models is also a purpose of this thesis. The harvested
results include many new solution existence theorems for various problems as
minimax problems, equilibrium problems, generalized inclusions, variational relation problems, or practical problems as traffic networks, Nash equilibria, abstract economies, etc.
One of the important topics in nonlinear optimization, which have been attracting many mathematicians recently, is properties of solution sets and solution maps. The properties of solution sets such as closedness, connectedness,
convexity, etc, were studied in many papers (e.g., Fort [37], Jones and Gowda
[49], Khanh and Luc [53], Papageorgiou and Shahzad [78], Rapcsak [86], Zhong,
Huang and Wong [105]). The properties of solution maps, like semicontinuities,
continuity, differentiability, etc., which are commonly called the stability properties, have also been intensively investigated during recent years (e.g., Anh and
Khanh [1-7], Khanh and Luc [53], Xiang, Liu and Zhou [94], Yang and Yu [96],


Preface

VI

Yu and Xiang [98], Yu, Yang and Xiang [99], Zhou, Xiang and Yang [101]).
Based on relationships between sets of particular points of set-valued maps and
solution sets of optimization problems, we propose solution map notions for these
points, consider their stability and apply the obtained results to optimization problems. Stability issues considered in this thesis are included in the generic stability
study.
The thesis consists of four chapters and contains the results of 10 papers (from
the list of 14 related papers of the author):
Chapter 1: “Existence theorems in nonlinear analysis and applications” is
based on the papers (Q2), (Q3), (Q4), (Q6), (Q7);
Chapter 2: “Nonlinear existence theorems for mappings on product GFCspaces and applications” is based on the paper (Q5);
Chapter 3: “Topological characterizations of existence in analysis and optimization related problems” is based on the papers (Q10), (Q11), (Q12);

Chapter 4: “Generic stability and essential components of generalized KKM
points and applications” is based on the paper (Q8).

Acknowledgments
I express my deep gratitude to Professor Phan Quoc Khanh, my supervisor, for
a continuous guidance, encouragement and valuable suggestions. I would like to
thank very much the University of Science of Hochiminh City for providing me
all conditions and facilities for my work. I am also indebted to the Vietnam Institute for Advanced Study in Mathematics (VIASM) and its members. During my
stay there as a visiting young researcher, they facilitated me with both a financial
support and a perfect research environment for the completion of a part of this
thesis.
Ho Chi Minh City, October 2013

Nguyen Hong Quan


1
Existence theorems in nonlinear analysis and
applications

In this chapter, we propose a definition of GFC-spaces to encompass G-convex
spaces, FC-spaces and many earlier existing spaces with generalized convexity
structures. Existence theorems are then established with underlying structures
being GFC-spaces under relaxed assumptions. These results contain, as properly
particular cases, a number of counterparts which were recently developed in the
literature. As applications, using these results we prove the existence of solutions
to a general variational inclusion problem, which contains most of the existing
results of this type, and develop in detail general types of minimax theorems.
Examples are given to explain advantages of our results.

1.1 Notions and definitions
We recall notions used in the whole thesis. Let Y be a nonempty set, Y stands
for the set of all finite subsets of Y . For n ∈ N, the set of all natural numbers, ∆n
stands for the n-simplex with the vertices being the unit vectors e0 = (1, 0, ..., 0),
e1 = (0, 1, ..., 0), ..., en = (0, 0, ..., 1) of a basis of Rn+1 . For N = {y0 , y1 ..., yn } ∈
Y and M = {yi0 , yi1 , ..., yik } ⊂ N, let ∆|N| ≡ ∆n , and ∆M ≡ ∆k be the face of
∆|N| corresponding to M, i.e., ∆M = co{ei0 , ei1 ..., eik }. If A, B ⊂ X, X being a
topological space, then A (or clA), AB (or clB A), intA, intB A and Ac signify the
closure, closure in B, interior, interior in B and complement X \ A, respectively
(shortly, resp), of A. Let X, Y be nonempty sets and F : X ⇒ Y be a set-valued
map. For x ∈ X and y ∈ Y , the sets F(x), F −1 (y) = {x ∈ X | y ∈ F(x)} and F ∗ (y) =
X \ F −1 (y) are called an image, a fiber (or inverse image) and a cofiber, resp. The
map F −1 (F ∗ ) is called the inverse map (dual map, resp) of F. The graph of F
is GphF := {(x, y) ∈ X × Y | y ∈ F(x)}. Now let X and Y be topological spaces,
F : X ⇒ Y , and f : X → R. F is called closed (open, resp) if its graph is closed
(open, resp). F is said to be upper semicontinuous (usc, for short) (resp, lower
semicontinuous (lsc)) if for each open (resp, closed) subset U of Y , the set {x ∈
X | F(x) ⊂ U} is open (resp, closed). F is said to be continuous if it is both usc
and lsc. f is said to be usc (resp, lsc), if, for all α ∈ R, the set {x ∈ X | f (x) ≥ α}
(resp, {x ∈ X | f (x) ≤ α}) is closed.


1.1 Notions and definitions

2

The following concepts are taken from [22, 25]. A subset A of a topological
space X is called compactly open (compactly closed, resp) if, for each nonempty
compact subset K of X, A ∩ K is open (closed, resp) in K. The compact interior
and compact closure of A are defined by, resp,

cintA = {B ⊂ X : B ⊂ A and B is compactly open in X},
cclA = {B ⊂ X : B ⊃ A and B is compactly closed in X}.
F : Y ⇒ X is called transfer open-valued (transfer closed-valued, resp) if ∀y ∈ Y ,
∀x ∈ F(y) (∀x ∈
/ F(y), resp), ∃y ∈ Y such that x ∈ int(F(y ) (x ∈
/ cl(F(y ), resp).
F is termed transfer compactly open-valued (transfer compactly closed-valued)
if ∀y ∈ Y , ∀K ⊂ X: nonempty and compact, ∀x ∈ F(y) ∩ K (∀x ∈
/ F(y) ∩ K),
∃y ∈ Y such that x ∈ cintF(y ) (x ∈
/ cclF(y ), resp). Of course intA ⊂ cintA (clA ⊃
cclA), and transfer open-valuedness (transfer closed-valuedness) implies transfer
compact open-valuedness (transfer compact closed-valuedness, resp). Moreover,
a set-valued mapping has open values (closed values) then it is transfer openvalued (transfer closed-valued). We will use these notions in order to compare
our results directly with many known existing ones.
Lemma 1.1.1 (e.g., [22]) Let Y be a set, X be a topological space and F : Y ⇒ X.
The following statements are equivalent
(i) F is transfer compactly closed-valued (transfer compactly open-valued, resp);
(ii) for each compact subset K ⊂ X,

y∈Y

y∈Y

(K ∩ intK F(y)), resp .

(K ∩ cintF(y)) =

(K ∩ F(y)) =
y∈Y

y∈Y

(K ∩ clK F(y))

(K ∩ cclF(y)) =

(K ∩ F(y)) =
y∈Y

y∈Y

We propose the following definition of a GFC-space to unify a number of earlier existing notions of spaces with generalized convexity structures, but without
linear structures. This notion is proposed based on observing that although the
abstract convexity structures associated with the earlier existing spaces such as
convex spaces ([60]), H-spaces ([46-47]), G-convex spaces ([79-85]), FC-spaces
([24-32]) are different, all of them use the image of a simplex through a continuous map.
Definition 1.1.1 Let X be a topological space, Y be a nonempty set and Φ be a
family of continuous mappings ϕ : ∆n → X, n ∈ N. Then a triple (X,Y, Φ) is said
to be a generalized finitely continuous topological space (GFC-space in short) if
for each finite subset N ∈ Y , there is ϕN : ∆|N| → X of the family Φ (we also
use (X,Y, {ϕN }) to denote (X,Y, Φ)).


1.1 Notions and definitions

3

The above mentioned existing spaces are the examples for GFC-space. In particular, a convex subset A of a topological vector space is a GFC-space, where
X = Y = A and each N = {a0 , a1 , ..., an } ∈ A , there is ϕN : ∆|N| → A which is

defined by ϕN (e) = ∑ni=0 λi ai for all e = ∑ni=0 λi ei ∈ ∆|N| . GFC-spaces are properly
more general than known existing spaces. Therefore it is reasonable and valuable
to study existence theorems and nonlinear problems in GFC-spaces without linear
structure.
The following example shows that GFC-spaces are properly more general
than G-convex spaces. Recall that, a G-convex space is [79-85] a triple (X,Y,ϒ ),
where X and Y are as Definition 1.1.1 and ϒ : Y ⇒ X is such that, for each
N ∈ Y , there exists a continuous map ϕN : ∆|N| → ϒ (N) such that, for each
M ∈ N , ϕN (∆M ) ⊂ ϒ (M). A G-convex space (X,Y,ϒ ) is called trivial iff, for
all N ∈ Y , ϒ (N) = X. Of course, any above-mentioned space can be made into
a trivial G-convex space, but a trivial G-convex space has no use.
Example 1.1.1 Let Y =]0, +∞[, X = {(x, x) | x ∈]0, +∞[} ⊂ R2 and, for each
N ∈ Y , ϕN (e) = (α(N), α(N)) ∈ X for all e ∈ ∆|N| , where α(N) = min N. max N.
Then, (X,Y, {ϕN }) is a GFC-space. Suppose there exists ϒ : Y ⇒ X such that
(X,Y,ϒ ) is a nontrivial G-convex space, and (X,Y, {ϕN }) can be made into
(X,Y,ϒ ). Then, since (X,Y,ϒ ) is a G-convex space, we have, for all N ∈ Y
and y ∈ Y , ϕN∪{y} (∆|N| ) = (α(N ∪ {y}), α(N ∪ {y})) ∈ ϒ (N). It follows that, for
all N ∈ Y ,
Λ := (α(N ∪ {y}), α(N ∪ {y})) | y ∈ Y ⊂ ϒ (N).

(1.1.1)

On the other hand, for all N ∈ Y and x ∈]0, +∞[, taking y = maxx N if x ≤ α(N)
and y = minx N if x > α(N), we have (x, x) = (α(N ∪ {y}), α(N ∪ {y})) ∈ Λ . Thus,
X ⊂ Λ . This and (1.1.1) imply that, for all N ∈ Y , ϒ (N) = X, i.e., (X,Y,ϒ ) is a
trivial G-convex space.
Next, we define several concepts in a GFC-space.
Definition 1.1.2 Let (X,Y, Φ) be a GFC-space, D,C ⊂ Y and S : Y ⇒ X be given.
(i) D is called an S-subset of Y (S-subset of Y wrt C) if for all N ∈ Y and for all
M ⊂ N ∩ D (for all M ⊂ N ∩C, resp), ϕN (∆M ) ⊂ S(D).

(ii)If in addition to (i), S−1 (ϕN (∆M )) ⊂ D, then D is called an SGFC -subset of
Y . The GFC-hull wrt S of C is defined by GFCS (C) = {D ⊂ Y | D is SGFC subset of Y containing C}.
Clearly, if D is an SGFC -subset of Y , D must be an S-subset of Y . Roughly
speaking, if D is an S-subset of Y then (S(D), D, Φ) is a ”pre” GFC-space, and if
D is an SGFC -subset of Y then (S(D), D, Φ) is a ”full” GFC-space. When X = Y ,
i.e, (X,Y, Φ) = (X, Φ) is an FC-space, and S = I is the identity map, then being
an SGFC -subset or an S-subset of X coincides with being an FC-subspace of X
([30]). Moreover, the notion of a GFC-hull wrt S of a set extends the notions of


1.1 Notions and definitions

4

an FC-hull of a set in a FC-space ([25]) and a G-convex hull of a set in a Gconvex space ([64]). The extension of concepts of subspaces and convex hulls as
above also shows the rationality of the GFC-spaces. The following Lemma 1.1.2
extends Lemma 2.1 of [95] from FC-spaces to GFC-spaces.
Lemma 1.1.2 Let (X,Y, Φ) be a GFC-space, C a nonempty subset of Y and
S : Y ⇒ X. Then,
(a) GFCS (C) is an SGFC -subset of Y ;
(b) GFCS (C) is the smallest SGFC -subset of Y containing C;
(c) GFCS (C) = N∈ C GFCS (N).
Proof. (a) and (b) are obvious.
(c) For every N ∈ C , it is clear that GFCS (N) ⊂ GFCS (C). Hence,
C⊂

GFCS (N) ⊂ GFCS (C).
N∈ C

Therefore, it suffices to show that the union in these inclusions, which is now

denoted by B, is an SGFC -subset of Y . Assume that N0 ∈ Y and M0 ⊂ N0 ∩ B. By
the definition of B, there is {N1 , ..., Nl } ∈ C , such that M0 ⊂ li=1 GFCS (Ni ) ⊂
B. Since li=1 Ni ∈ C , one has GFCS ( li=1 Ni ) ⊂ B. As, ∀ j = 1, ..., l, GFCS (N j )
⊂ GFCS ( li=1 Ni ), one has further M0 ⊂ GFCS ( li=1 Ni ). Since each GFC - hull
wrt S is SGFC -subset, by Definition 1.1.2, S−1 (ϕN0 (∆M0 )) ⊂ GFCS
Hence, again by this definition, B is an SGFC -subset of Y .

l
i=1 Ni

⊂ B.

The notion of a GFC-space helps us also to extend the notion of generalized
KKM maps which plays an essential role in the theory of existence.
Definition 1.1.3 Let (X,Y, Φ) be a GFC-space, Z be a topological space, F :
Y ⇒ Z and T : X ⇒ Z be set-valued mappings. F is said to be a generalized KKM
mapping with respect to (shortly, w.r.t) T (T -KKM mapping in short) if, for each
N ∈ Y and each M ⊂ N, one has T (ϕN (∆M )) ⊂ y∈M F(y).
The definition of T -KKM mappings was introduced for X being a convex subset of a topological vector space in [16] and extended to FC-spaces in [24]. Definition 1.1.3 includes these definitions as particular cases. It encompasses also many
other kinds of generalized KKM mappings. We mention here some of them. Let
(X, {ϕN }) be an FC-space, Y be a nonempty set and s : Y → X be a mapping. We
define a GFC-space (X,Y, {ϕN }) by setting ϕN = ϕs(N) for each N ∈ Y . Then, a
generalized s-KKM mapping wrt T introduced in [25] becomes a T -KKM mapping by Definition 1.1.3. A multivalued mapping F : Y ⇒ X, being an R-KKM
mapping as defined in [20], is a special case of T -KKM mappings on GFC-space
when X = Z and T is the identity map. The definition of generalized KKM mappings wrt to T in [61] is as well a particular case of Definition 1.1.3.


1.1 Notions and definitions

5

Definition 1.1.4 Let (X,Y, Φ) be a GFC-space and Z be a topological space.
A multivalued mapping T : X ⇒ Z is called better admissible if T is usc and
compact-valued such that for each N ∈ Y and each continuous mapping ψ :
T (ϕN (∆|N| )) → ∆|N| , the composition ψ ◦ T |ϕN (∆|N| ) ◦ ϕN : ∆|N| ⇒ ∆|N| has a fixed
point. The class of all such better admissible mapping from X to Z is denoted by
B(X,Y, Z).
The class of better admissible mappings generalizes classes of admissible
mappings proposed in [24, 82, 85].
Definition 1.1.5 Let (X,Y, Φ) be a GFC-space. We say that a set-valued mapping T : X ⇒ Z has the generalized KKM property if, for each T-KKM mapping
F : Y ⇒ Z, the family F(y) : y ∈ Y has the finite intersection property, i.e. all finite intersections of sets of this family are nonempty. By KKM(X,Y,Z) we denote
the class of all mappings T : X ⇒ Z which enjoy the generalized KKM property.
Lemma 1.1.3 Let (X,Y, Φ) be a GFC-space, Z be a topological space, S :
Y ⇒ X, D be an S-subset of Y , and T ∈ KKM(X,Y, Z). Then, T |S(D) ∈ KKM
S(D), D, T (S(D)) .
Proof. Assume that R : D ⇒ T (S(D)) is a T |S(D) -KKM mapping. Then, for each
N ∈ D ⊂ Y and each M ⊂ N,
T (ϕN (∆M )) = T |S(D) (ϕN (∆M )) ⊂

y∈M R(y).

Define a set-valued mapping F : Y ⇒ Z by
F(y) =

R(y) if y ∈ D,
Z
if otherwise.

Clearly F is a T-KKM mapping. Since T ∈ KKM(X,Y, Z), the family F(y) : y ∈
Y has the finite intersection property. It follows that the family R(y) : y ∈ D

has this property too. Thus, the lemma is proved.
Lemma 1.1.4 Let (X,Y, Φ) be an GFC-space and S : Y ⇒ X. Assume that Ω :
Z ⇒ Y has nonempty values such that Ω −1 is transfer compactly open-valued.
Let GFCS ◦ Ω : Z ⇒ Y be defined by (GFCS ◦ Ω )(z) = GFCS (Ω (z)) (GFCS ◦ Ω
is called a GFC-hull mapping wrt S of Ω ). Then (GFCS ◦ Ω )−1 is also transfer
compactly open-valued.
Proof. Let K ⊂ Z be any nonempty compact subset, y ∈ Y and z ∈ (GFCS ◦
Ω )−1 (y) ∩ K. Then y ∈ GFCS (Ω (z)). By Lemma 1.1.2(c), there is N ∈ Ω (z)
such that y ∈ GFCS (N). We have z ∈ Ω −1 (y ) ∩ K for all y ∈ N. Since Ω −1 is
transfer compactly open-valued, there exists N ∈ Y with |N| = |N| such that
z ∈ intK Ω −1 (y)
ˆ ∩ K ⊂ Ω −1 (y)
ˆ ∩ K for all yˆ ∈ N. Then, N ∈ Ω (z) and hence


1.2 Existence theorems in GFC-spaces

6

N ⊂ GFCS (Ω (z)). Fixing yˆ0 ∈ N and choosing yˆ = yˆ0 we have yˆ ∈ GFCS (Ω (z)) =
(GFCS ◦ Ω )(z), i.e., z ∈ (GFCS ◦ Ω )−1 (y).
ˆ Moreover, z ∈ intK Ω −1 (y)
ˆ ∩K
−1
−1
⊂ intK (GFCS ◦ Ω ) (y)
ˆ ∩ K . Thus, (GFCS ◦ Ω ) is transfer compactly openvalued.

1.2 Existence theorems in GFC-spaces
Using two elementary topological tools: the finite intersection property of compact sets and the existence, for a finite covering of a compact set, of a partition of

unity associated with this covering, we first prove the following three versions of
results.
Theorem 1.2.1 (Intersection) Let (X,Y, {ϕN }) be a GFC-space, Z be a topological space. Let T ∈ B(X,Y, Z) and F : Y ⇒ Z satisfy the following conditions
for each y ∈ Y , F(y) is compactly closed;
F is T-KKM;
either of the following three conditions holds
there are N0 ∈ Y and a compact subset K of Z with y∈N0 F(y) ⊂ K;
there is S : Y ⇒ X such that for each N ∈ Y , there exists an S-subset LN of Y,
containing N, so that S(LN ) is a compact subset and, for some nonempty and
compact subset K of Z, T (S(LN )) ∩ y∈LN F(y) ⊂ K;
(c) there are S : Y ⇒ X and a nonempty subset Y0 ⊂ Y such that K := y∈Y0 F(y)
is compact and that, for each N ∈ Y , there exists an S-subset LN of Y containing Y0 ∪ N so that S(LN ) is compact.

(i)
(ii)
(iii)
(a)
(b)

Then, K ∩ T (X) ∩

y∈Y

F(y) = 0.
/

Theorem 1.2.2 (Coincidence points) Let (X,Y, {ϕN }) be a GFC-space and Z
be a topological space. Let S : Y ⇒ X, T : X ⇒ Z and F : Z ⇒ Y be multivalued
mappings with T ∈ B(X,Y, Z). Assume that
(i) for each x ∈ X and each z ∈ T (x), F(z) is an S-subset of Y;

(ii) for each y ∈ Y, F −1 (y) contains a compactly open Oy (some Oy may be empty)
of Z such that K := y∈Y Oy is nonempty and compact;
(iii) either of the following three conditions holds:
(a) there is N0 ∈ Y such that y∈N0 Ocy ⊂ K;
(b) for each N ∈ Y , there is an S-subset LN of Y, containing N such that S(LN )
is compact and T (S(LN )) ∩ y∈LN Ocy ⊂ K;
(c) K = Z; there is a nonempty subset Y0 of Y such that y∈Y0 Ocy is compact or
empty; and for each N ∈ Y there is an S-subset LN of Y containing Y0 ∪ N
so that S(LN ) is compact.
Then, a point (x,
¯ y,
¯ z¯) ∈ X ×Y × Z exists such that x¯ ∈ S(y),
¯ y¯ ∈ F(¯z) and z¯ ∈ T (x).
¯


1.2 Existence theorems in GFC-spaces

7

Theorem 1.2.3 (Maximal-elements) Let (X,Y, {ϕN }) be a GFC-space, Z be a
topological space and K ⊂ Z be nonempty and compact. Let F : Z ⇒ Y and
T : X ⇒ Z be multivalued mapping such that T ∈ B(X,Y, Z) and the following
assumptions are satisfied
(i) for each y ∈ Y , F −1 (y) includes a compactly open subset Oy (some Oy may
be empty) of Z such that y∈Y (Oy ∩ K) = y∈Y (F −1 (y) ∩ K);
/
(ii) for any N ∈ Y and M ⊂ N, T (ϕN (∆M )) ∩ y∈M Oy = 0;
(iii) either of the following conditions hold:
(a) there is N0 ∈ Y such that Z\K ⊂ y∈N0 Oy ;

(b) there is a multivalued map S : Y ⇒ X such that, for each N ∈ Y , there is an
S-subset LN of Y containing N so that S(LN ) is compact and T (S(LN ))\K ⊂
y∈LN Oy .
Then, a point z¯ ∈ K exists such that F(¯z) = 0.
/
Theorem 1.2.4 (Maximal-elements) Let (X,Y, {ϕN }), Z, F and T be defined as in
Theorem 1.2.3 such that (ii) is satisfied and (i) and (iii) are replaced respectively
by
(i’) for each y ∈ Y, F −1 (y) contains a compactly open subset Oy , which may be
empty, of Z such that, y∈Y (Oy ∩ Z0 ) = y∈Y (F −1 (y) ∩ Z0 ) for each nonempty
compact subset Z0 of Z;
(iii’) there are a multivalued map S : Y ⇒ X and a nonempty subset Y0 of Y such
that K := y∈Y0 Ocy is compact and that, for each N ∈ Y , there exists an
S-subset LN of Y containing Y0 ∪ N so that S(LN ) is compact.
Then, an element z¯ ∈ Z exists with F(¯z) = 0.
/
Remark 1.2.1 Our Theorem 1.2.1 contains Theorem 3 of [83] and Theorems 1-3
of [22] as special cases for the case of G-convex spaces. Theorem 1.2.2 generalizes Theorem 1 of [80], Theorem 1.2.3 extends Theorem 2.2 of [24], and Theorem
1.2.4 includes Theorem 2.1 of [24] as a special case for the FC-space setting.
Proof.

To prove the above theorems, we first prove the following lemma.

Lemma 1.2.1 Let (X,Y, {ϕN }) be a GFC-space and Z be a topological space.
Let F : Y ⇒ Z and T : X ⇒ Z be multivalued mappings. Assume that
(i) for each y ∈ Y , F(y) is compactly closed;
(ii) T ∈ B(X,Y, Z) and F is T-KKM.
Then, for each N ∈ Y , T (ϕN (∆|N| )) ∩

/

y∈N F(y) = 0.

Proof of Lemma 1.2.1. Suppose to the contrary that N = {y0 , y1 , ..., yn } ∈ Y
exists such that


1.2 Existence theorems in GFC-spaces

8

(Z\F(yi )) ∩ T (ϕN (∆|N| )) ,

T (ϕN (∆|N| )) =
yi ∈N

n

i.e. the family (Z\F(yi )) ∩ T (ϕN (∆|N| )) i=0 is an open covering of the compact
set T (ϕN (∆|N| )). Let {ψi }ni=0 be a continuous partition of unity associated with
this covering and ψ : T (ϕN (∆|N| )) → ∆|N| be defined by ψ(z) = ∑ni=0 ψi (z)ei .
Then ψ is continuous. Since T is better admissible, there is a fixed point of ψ ◦
T |ϕN (∆|N| ) ◦ϕN , i.e. there is z0 ∈ T (ϕN (∆|N| )) such that z0 ∈ T (ϕN (ψ(z0 ))). Setting
J(z0 ) = j ∈ {0, 1, ..., n} : ψ j (z0 ) = 0 and M(z0 ) = {y j ∈ N | j ∈ J(z0 )}, We have
ψ(z0 ) =

∑

ψ j (z0 )e j ∈ ∆M(z0 ) .

j∈J(z0 )

As F is T -KKM, we also have
z0 ∈ T (ϕN (ψ(z0 ))) ⊂ T (ϕN (∆M(z0 ) )) ⊂

F(y j ).
j∈J(z0 )

Hence, there exists j ∈ J(z0 ), z0 ∈ F(y j ).
On the other hand, by the definitions of J(z0 ) and of the partition {ψi }ni=0 ,
z0 ∈ z ∈ T (ϕN (∆|N| )) : ψ j (z) = 0 ⊂ (Z\F(y j )) ∩ T (ϕN (∆|N| )) ⊂ Z\F(y j ),
a contradiction.
Proof of Theorem 1.2.1. Case of (a). For y ∈ Y , set U(y) = T (X)∩( y ∈N0 F(y ))∩
F(y). By (i) and (a), U(y) y∈Y is a family of sets which are closed in K. For each
N ∈ Y , setting M = N ∪ N0 , by Lemma 1.2.1 we have
F(y) ⊂ T (X) ∩

0/ = T (ϕM (∆|M| )) ∩
y∈M

F(y) =
y∈M

U(y).
y∈N

Since K is compact, this implies that
U(y) ⊂ K ∩ T (X) ∩

0/ =
y∈Y

F(y).
y∈Y

Case of (b). By (i), K ∩ T (X) ∩ F(y)
K (K is given in (b)). Suppose that
0/ = K ∩ T (X) ∩

y∈Y

is a family of sets which are closed in

K ∩ T (X) ∩ F(y) .

F(y) =
y∈Y

y∈Y

Then there exists N ∈ Y such that
K ∩ T (X) ∩ F(y) = K ∩ T (X) ∩

0/ =
y∈N

F(y),
y∈N


1.2 Existence theorems in GFC-spaces

9

i.e.,
T (X) ∩

F(y) ⊂ Z\K.
y∈N

In view of the assumption (b), T (S(LN )) ∩

y∈LN

F(y) ⊂ K. On the other hand,

F(y) ⊂ T (X) ∩

T (S(LN )) ∩

F(y) ⊂ Z\K.
y∈N

y∈LN

Thus, T (S(LN )) ∩ y∈LN F(y) = 0.
/
As LN is an S-subset of Y , by virtue of Lemma 1.2.1 we have, for each M ∈
LN ,
0/ = T (ϕM (∆|M| )) ∩
F(y) ⊂ T (S(LN )) ∩

F(y).
y∈M

y∈M

By the compactness of T (S(LN )) (as T is usc, compact-valued and S(LN ) is compact), this implies that T (S(LN )) ∩ y∈LN F(y) = 0,
/ a contradiction.
Case of (c). If K is compact, then K ∩ T (X) ∩ F(y) y∈Y is a family of closed
subsets of K. Suppose that K ∩ T (X) ∩ y∈Y F(y) = 0.
/ Then there is N ∈ Y
such that
F(y).
0/ =
K ∩ T (X) ∩ F(y) = T (X) ∩
y∈Y0 ∪N

y∈N

Therefore,
T (S(LN )) ∩

F(y) ⊂ T (X) ∩

F(y) = 0.
/
y∈Y0 ∪N

y∈LN

Now we can argue similarly as for the case (b) to get a contradiction.

Proof of Theorem 1.2.2. Define a new multivalued mapping G : Y ⇒ Z by setting, ∀y ∈ Y, G(y) = Ocy , which is compactly closed by (ii), i.e. assumption (i) of
Theorem 1.2.1 for G in the place of F is fulfilled. It is clear that (a), (b), and (c)
imply the corresponding assumptions of Theorem 1.2.1 for G. By the definition
of K in (ii),
K ∩ T (X) ∩
G(y) ⊂ K ∩
Ocy = 0,
/
y∈Y

y∈Y

which means that the conclusion of Theorem 1.2.1 for G in the place of F does
not hold. Therefore assumption (ii) of this theorem must be violated, i.e. G is not
T -KKM. This means that there are N ∈ Y and M ⊂ N such that
T (ϕN (∆M )) ⊂

Ocy .

G(y) =
y∈M

y∈M

This in turn is equivalent to the existence of x¯ ∈ ϕN (∆M ) and z¯ ∈ T (x)
¯ such that
−1
z¯ ∈ Oy , for all y ∈ M. Since Oy ⊂ F (y) by (ii), y ∈ F(¯z), which is an S-subset

1.2 Existence theorems in GFC-spaces

10

of Y . Hence x¯ ∈ ϕN (∆M ) ⊂ S(F(¯z)), which means that there is y¯ ∈ F(¯z) such that
x¯ ∈ S(y).
¯
Proof of Theorem 1.2.3. We check the assumptions of Theorem 1.2.1 in order
to apply it for, instead of F, a new multivalued mapping G : Y ⇒ Z defined by
G(y) = Ocy . Assumption (i) is clearly fulfilled. For (ii), with arbitrary N ∈ Y and
M ⊂ N, we have, by (ii) of Theorem 1.2.3,
Ocy =

T (ϕN (∆M )) ⊂
y∈M

G(y),
y∈M

i.e. G is T -KKM as required. By (a) of this theorem, we obtain (a) since
Ocy ⊂ Z\(Z\K) = K.

G(y) =
y∈N0

y∈N0

From (b) of this theorem it follows that
T (S(LN )) ∩

G(y) = T (S(LN )) ∩ Z\
y∈LN

Oy
y∈LN

⊂ T (S(LN )) ∩ Z\(T (S(LN ))\K)
⊂ K,
i.e. (b) is satisfied. Now that all the assumptions of Theorem 1.2.1 have been
checked, we obtain from this theorem
0/ = K ∩ T (X) ∩

G(y)
y∈Y

= K ∩ T (X) ∩ Z\

Oy
y∈Y

⊂ K ∩ T (X) ∩ Z\

(Oy ∩ K)
y∈Y

(F −1 (y) ∩ K) .

= K ∩ T (X) ∩ Z\
y∈Y

Therefore an element z¯ ∈ K exists such that z¯ ∈
/ F −1 (y) ∩ K for every y ∈ Y , i.e.
F(¯z) = 0.
/
Proof of Theorem 1.2.4. It is not hard to see that all the assumptions (i), (ii) and
(c) of Theorem 1.2.1 are fulfilled with G : Y ⇒ Z defined by G(y) = Ocy in the
place of F. By this theorem we have,
0/ = K ∩ T (X) ∩

G(y)
y∈Y


1.2 Existence theorems in GFC-spaces

⊂ K ∩ T (X) ∩ Z\

11

(Oy ∩ K)
y∈Y

(F −1 (y) ∩ K) .

= K ∩ T (X) ∩ Z\
y∈Y

Consequently, there is z¯ ∈ K such that z¯ ∈
/ F −1 (y) for each y ∈ Y , which means
that F(¯z) = 0.

/
The above theorems are very close to each other, but they are not completely
equivalent. The coincidence point and maximal-element theorems are deduced
from intersection theorems. To underline a possible equivalence, we establish a
maximal element theorem and show the equivalence between the three theorems
as follows.
Theorem 1.2.5 (Maximal elements) Let (X,Y, {ϕN }) be a GFC-space, Z a topological space, T ∈ KKM(X,Y, Z) and S : Y ⇒ X a multivalued mapping such that
Y is an S-subset of itself. Assume that Ω : Z ⇒ Y satisfies the following conditions
(i)
(ii)
(iii)
(iv)

Ω −1 is transfer compactly open-valued;
for each (y, z) ∈ Y × T (S(y)), y ∈
/ GFCS (Ω (z));
for each compact subset D ⊂ X, T (D) is compact;
there is a nonempty compact subset K of Z such that for each N ∈ Y , there
is an S-subset LN of Y containing N such that S(LN ) is compact and for each
z ∈ T (S(LN )) \ K, there exists y ∈ LN such that z ∈ cint(GFCS ◦ Ω )−1 (y).

Then, there is zˆ ∈ Z such that Ω (ˆz) = 0.
/
Theorem 1.2.6 (Coincidence points) Let (X,Y, {ϕN }) be a GFC-space, Z a topological space, T ∈ KKM(X,Y, Z) and S : Y ⇒ X a multivalued mapping such
that Y is an S-subset of itself. For multivalued mappings H and Ω from Z into Y ,
impose the following conditions
(i)
(ii)
(iii)
(iv)

H has nonempty-values and H −1 is transfer compactly open-valued;
GFCS (H(z)) ⊂ Ω (z) for all z ∈ Z;
for each compact subset D ⊂ X, T (D) is compact;
there is a nonempty compact subset K of Z such that for each N ∈ Y , there
is an S-subset LN of Y containing N such that S(LN ) is compact and for each
z ∈ T (S(LN ))\K, there exists y ∈ LN with z ∈ cint(GFCS ◦ H)−1 (y).

Then, a point (x,
ˆ y,
ˆ zˆ) ∈ X ×Y ×Z exists such that xˆ ∈ S(y),
ˆ zˆ ∈ T (x)
ˆ and yˆ ∈ Ω (ˆz).
Theorem 1.2.7 (Nonempty intersection) Let (X,Y, {ϕN }) be a GFC-space, Z a
topological space, T ∈ KKM(X,Y, Z) and S : Y ⇒ X a multivalued mapping such
that Y is an S-subset of itself. Assume P, Q : Y ⇒ Z satisfy the following conditions
(i) P is transfer compactly closed-valued;


1.2 Existence theorems in GFC-spaces

(ii)
(iii)
(iv)
(v)

12

for all y ∈ Y , T (S(y)) ⊂ Q(y);
for all N ∈ Y , Q(GFCS (N)) ⊂ P(N);

for each compact subset D ⊂ X, T (D) is compact;
there is a nonempty compact subset K of Z such that, for each N ∈ Y , there is
an S-subset LN of Y containing N such that S(LN ) is compact and T (S(LN )) ∩
y∈LN cclP(y) ⊂ K.

Then,

y∈Y

P(y) = 0.
/

Remark 1.2.2 Theorem 1.2.5 includes the following known theorems: Theorem
3.1 of [95] for FC-spaces, Theorem 3.1 of [23], Theorem 6 of [62], Theorem 3
of [63] and Theorem 3.1 of [18] for G-convex spaces. Theorem 1.2.6 generalizes
theorem 3.1 of [40], Theorems 2.5-2.6 of [65], Theorem 3.5 of [67] for convex
spaces. If X = Y and S = I, Theorem 1.2.6 implies Theorem 3.1 of [36]. Theorem 1.2.7 generalizes Theorem 3.3 of [66], Theorems 3.2-3.3 of [67] for convex
spaces. If X = Y and S = I, Theorem 1.2.7 implies Theorems 3.2-3.3 of [36].
The following example illustrates Theorem 1.2.5, showing in particular that
its assumptions are relaxed and not hard to be checked.
Example 1.2.1 Let X = [−2, 2], Y = [−2, 0[ and, for all N = {y0 , y1 , ..., yn } ∈ Y ,
ϕN (e) = ∑ni=0 λi yi , where e = ∑ni=0 λi ei ∈ ∆|N| , ei being the ith unit vector of Rn+1 .
Then, (X,Y, {ϕN }) is a GFC-space. Let Z = [0, 2] and S : Y ⇒ X, T : X ⇒ Z and
Ω : Z ⇒ Y be defined by S(y) = {y}, T (x) = [0, |x|] and Ω (z) = ]−z, 0[. Then, it is
not hard to see that T ∈ KKM(X,Y, Z) and, for each z ∈ Z, Ω (z) is an SGFC -subset
of Y , i.e. GFCS (Ω (z)) = Ω (z). It is also clear that, for all (y, z) ∈ Y × T (S(y)) =
Y × [0, |y|], y ∈
/ ] − z, 0[ = GFCS (Ω (z)). One has Ω −1 (−2) = 0/ and Ω −1 (y) =
{z ∈ Z| − z < y < 0} = ] − y, 2] for y ∈ ] − 2, 0[. Then, Ω −1 (y) is open in Z. Hence,
Ω −1 is transfer compactly open-valued. Thus (i) and (ii) of Theorem 1.2.5 are

satisfied. (iii) is obviously fulfilled. (iv) is checked with K = Z and LN = Y for
each N ∈ Y . Theorem 1.2.5 implies the existence of z ∈ Z such that Ω (z) = 0.
/
One easily sees directly that z = 0 is a maximal-element of Ω .
Proof. We first prove Theorem 1.2.5.
Proof of Theorem 1.2.5. Define F : Y ⇒ Z by F(y) = Z \ (GFCS ◦ Ω )−1 (y) for
y ∈ Y . By Lemma 1.1.2 and the transfer compact open-valuedness of Ω −1 , F
is transfer compactly closed-valued. We prove that F is T -KKM. Take arbitrary
N ∈ Y , M ⊂ N and suppose that T (ϕN (∆M )) ⊂ y ∈M F(y ) is not satisfied.
Then, there are x ∈ ϕN (∆M ) and z ∈ T (x) such that z ∈
/ F(y ) for every y ∈ M, i.e.,
−1
z ∈ (GFCS ◦ Ω ) (y ) for every y ∈ M. Hence y ∈ GFCS (Ω (z)) for every y ∈ M.
Since GFCS (Ω (z)) is an S-subset of Y , ϕN (∆M ) ⊂ S GFCS (Ω (z)) . Therefore,
y ∈ GFCS (Ω (z)) exists such that x ∈ S(y) and hence z ∈ T (S(y)). Thus, there
is (y, z) ∈ Y × T (S(y)) such that y ∈ GFCS (Ω (z)), which contradicts assumption
(ii).


1.2 Existence theorems in GFC-spaces

13

Now suppose ad absurdum that, for all z ∈ Z, Ω (z) = 0.
/ Then, for all z ∈ K,
(GFCS ◦ Ω )(z) = 0.
/ By the transfer compact open-valuedness of (GFCS ◦ Ω )−1
and the compactness of K, there is N ∈ Y such that
K=

y∈Y

(GFCS ◦ Ω )−1 (y) ∩ K

=

y∈Y

cint(GFCS ◦ Ω )−1 (y) ∩ K

=

y∈N

cint(GFCS ◦ Ω )−1 (y) ∩ K

⊂

y∈LN

cint(GFCS ◦ Ω )−1 (y)

= y∈LN Z \ cclF(y)
= Z \ y∈LN cclF(y),
where LN is given in assumption (iv). Since, for each z ∈ T (S(LN ))\K, there exists
y ∈ LN with z ∈ cint(GFCS ◦ Ω )−1 (y), we have
cint(GFCS ◦ Ω )−1 (y)

T (S(LN )) \ K ⊂
y∈LN

= Z\

cclF(y).
y∈LN

Therefore,
T (S(LN )) = K ∩ T (S(LN )) ∪ T (S(LN )) \ K ⊂ Z \

y∈LN

cclF(y). (1.2.2)

Consider GFC-space (S(LN ), LN , ϕN ) and mappings Tˆ = T |S(LN ) , Fˆ : LN ⇒ Z
ˆ
by F(y)
= cclF(y) ∩ T (S(LN )). As F is T -KKM, by Lemma 1.1.3 we see that
T |S(LN ) ∈ KKM S(LN ), LN , T (S(LN )) and Fˆ is T |S(LN ) -KKM. Hence the family
ˆ
{clF(y)
: y ∈ LN } = {cclF(y) ∩ T (S(LN )) : y ∈ LN } has the finite intersection
property. Since this is a family of closed subsets of compact set T (S(LN )), one
has
ˆ
cclF(y) ∩ T (S(LN )) =
F(y)
= 0.
/
y∈LN

y∈LN

It follows that there is z¯ ∈ T (S(LN )) such that z¯ ∈
dicts (1.2.2).

y∈LN

cclF(y), which contra-

Theorem 1.2.5 implies Theorem 1.2.6. Suppose the conclusion of Theorem 1.2.6
is false. Then, for all (y, z) ∈ Y × T (S(y)), y ∈
/ Ω (z). By (ii), y ∈
/ GFCS (H(z)).
Thus, the assumptions of Theorem 1.2.5 are satisfied for H. Hence, we have zˆ ∈ K
such that H(ˆz) = 0,
/ contradicting (i).
Theorem 1.2.6 implies Theorem 1.2.7. Define H, Ω : Z ⇒ Y by H(z) = Y \ P−1 (z)
and Ω (z) = Y \ Q−1 (z). We verify the assumptions of Theorem 1.2.6. For y ∈ Y ,


1.2 Existence theorems in GFC-spaces

14

H −1 (y) = {z ∈ Z : y ∈ Y \ P−1 (z)} = {z ∈ Z : z ∈
/ P(y)} = Z \ P(y). By (i), H −1
is transfer compactly open-valued as required in (i) of Theorem 1.2.6.
To check (ii) of Theorem 1.2.6 suppose to the contrary the existence of
z ∈ Z and y ∈ GFCS (H(z)) such that y ∈
/ Ω (z) = Y \ Q−1 (z), i.e. z ∈ Q(y).

By Lemma 1.1.2(c), there exists N ∈ H(z) with y ∈ GFCS (N). Then, by
(iii), z ∈ Q(y) ⊂ Q(GFCS (N)) ⊂ P(N). Hence, y¯ ∈ N ⊂ H(z) exists such that
z ∈ P(y)
¯ = Z \ H −1 (y),
¯ i.e. y¯ ∈
/ H(z), a contradiction. (iii) of Theorem 1.2.6 is
assumption (iv).
To see (iv) of Theorem 1.2.6, for each N ∈ Y , by (v) one has an S-subset LN
of Y containing N such that S(LN ) is compact and T (S(LN ))∩ y∈LN cclP(y) ⊂ K.
This means that for each z ∈ T (S(LN ))\K, there is y ∈ LN such that z ∈ Z \cclP(y)
= cint(Z \ P(y)) = cintH −1 (y) ⊂ cint(GFCS ◦ H)−1 (y) as required.
Now, by (ii), for all (x, y, z) ∈ X × Y × Z with x ∈ S(y) and z ∈ T (x), one has
z ∈ Q(y) = Z \ Ω −1 (y), i.e. y ∈
/ Ω (z). Then, the conclusion of Theorem 1.2.6
is not true for H and Ω . Thus, assumption (i) of Theorem 1.2.6 must be violated. Consequently, there exists zˆ ∈ Z such that 0/ = H(ˆz) = Y \ P−1 (ˆz). Hence
P−1 (ˆz) = Y , i.e., zˆ ∈ y∈Y P(y).
Theorem 1.2.7 implies Theorem 1.2.5. Define P, Q : Y ⇒ Z by P(y) = Z \ Ω −1 (y)
and Q(y) = Z \(GFCS ◦Ω )−1 (y). We check the assumptions of Theorem 1.2.7 for
P and Q. By assumptions (i), (ii) of Theorem 1.2.5, assumption (i), (iv) of Theorem 1.2.7 are clearly fulfilled. For N ∈ Y , (GFCS ◦Ω )−1 (GFCS (N)) ⊃ Ω −1 (N).
Hence Q(GFCS (N)) = Z \ (GFCS ◦ Ω )−1 (GFCS (N)) ⊂ Z \ Ω −1 (N) = P(N), i.e.,
assumption (iii) of Theorem 1.2.7 is satisfied. For (ii) of Theorem 1.2.7, for all y ∈
Y and z ∈ T (S(y)), from (ii) of Theorem 1.2.5 we see that z ∈
/ (GFCS ◦ Ω )−1 (y).
−1
Then, z ∈ Z \ (GFCS ◦ Ω ) (y) = Q(y). This means that T (S(y)) ⊂ Q(y). To see
(v) of Theorem 1.2.7, by (iv) of Theorem 1.2.5 we have a nonempty compact
subset K ⊂ Z and for each N ∈ Y , one has an S-subset LN of Y containing
/ K,
N such that S(LN ) is compact. Take any T (S(LN )) ∩ y∈LN cclP(y). If z ∈
then z ∈ T (S(LN )) \ K and by (iv) of Theorem 1.2.5, there is y ∈ LN such that

z ∈ cint(GFCS ◦ Ω )−1 (y), i.e., z ∈
/ Z \ cint(GFCS ◦ Ω )−1 (y) ⊂ Z \ cintΩ −1 (y)
= cclP(y) which is a contraction. Thus, by Theorem 1.2.7, y∈Y P(y) = 0.
/ Hence,
−1
there is zˆ ∈ Z such that zˆ ∈
/ Ω (y) for every y ∈ Y , i.e. Ω (ˆz) = 0.
/
The following consequence is an extension of Theorem 1 in [11].
Corollary 1.2.1 Let (X,Y, {ϕN }) be a GFC-space, Z be a nonempty set, S : Y ⇒
X, Ω : X ⇒ Y , F : X ⇒ Z and G : Y ⇒ Z. Assume that
(i) G(Y ) = Z and F −1 is onto and transfer compactly open-valued;
(ii) for all x ∈ X, Ω (x) is an SGFC -subset of Y ;
(iii) for each x ∈ X, y ∈ Y : F(x) ∩ G(y) = 0/ ⊂ Ω (x);


1.2 Existence theorems in GFC-spaces

15

(iv) there is a nonempty compact subset A of X such that for each N ∈ Y , there is
an S-subset LN of Y containing N such that S(LN ) is compact and S(LN ))\A ⊂
/ .
y∈LN cint x ∈ X : F(x) ∩ G(y) = 0
Then, a point (x,
ˆ y)
ˆ ∈ X ×Y exists such that xˆ ∈ S(y)
ˆ and yˆ ∈ Ω (x).
ˆ
Proof. Define H : X ⇒ Y by H(x) = {y ∈ Y : F(x) ∩ G(y) = 0}.

/ We check the
assumptions of Theorem 1.2.6 for S, T = I and Ω (with Z = X). Let K ⊂ X be
any nonempty compact subset, y ∈ Y and x ∈ H −1 (y) ∩ K. Then, there is z ∈
F(x) ∩ G(y) such that x ∈ F −1 (z) ∩ K. By (i), there exists z ∈ Z such that
x ∈ intK (F −1 (z ) ∩ K)
⊂ y∈Y z∈G(y) intK (F −1 (z) ∩ K)
⊂

y∈Y

intK

⊂

y∈Y

intK

−1
z∈G(y) (F (z) ∩ K)
(H −1 (y) ∩ K).

It follows that there is y ∈ Y such that x ∈ intK (H −1 (y ) ∩ K), i.e. H −1 is transfer
compactly open-valued. By (ii) and (iii), GFCS (H(x)) ⊂ Ω (x), i.e. assumption
(ii) of Theorem 1.2.6 is fulfilled. By (iv) we have (iv) of Theorem 1.2.6. Thus,
applying this theorem we are done.
From the Theorems 1.2.5-1.2.7 we derive equivalent forms of a general alternative theorem. Alternative theorems of this type have been rather rarely
studied but they are also important. For H : X ⇒ Y we define H c : X ⇒ Y by
H c (x) = Y \ H(x) for x ∈ X. The mentioned forms of a general alternative theorem are stated as follows.
Theorem 1.2.8 Let (X,Y, {ϕN }) be a GFC-space, Z be a nonempty set, S : Y ⇒ X,

F : X ⇒ Z and G : Y ⇒ Z. Assume that
(i)
(ii)
(iii)
(iv)

F −1 is transfer compactly open-valued;
for all x ∈ X, y ∈ Y : F(x) ⊂ Gc (y) is an SGFC -subset of Y ;
for each (y, x) ∈ grS, F(x) ⊂ Gc (y);
there is a nonempty compact subset A of X such that for each N ∈ Y , there is
an S-subset LN of Y containing N such that S(LN ) is compact and S(LN ))\A ⊂
c
y∈LN cint x ∈ X : F(x) ⊂ G (y) .

Then, at least one of the following assertions holds:
(a) there exists x¯ ∈ X such that F(x)
¯ = 0;
/
−1
(b) there exists z¯ ∈ Z such that G (¯z) = 0.
/
Theorem 1.2.9 Let (X,Y, {ϕN }) be a GFC-space, Z be a nonempty set, S : Y ⇒ X,
P : X ⇒ Z and Q : Y ⇒ Z, impose that
(i) P−1 is transfer compactly closed-valued;
(ii) for all x ∈ X, y ∈ Y : Qc (y) ⊂ P(x) is an SGFC -subset of Y ;


1.2 Existence theorems in GFC-spaces

16

(iii) for each (x, y) ∈ grS, Qc (y) ⊂ P(x);
(iv) there is a nonempty compact subset A of X such that for each N ∈ Y , there is
an S-subset LN of Y containing N such that S(LN ) is compact and S(LN ))\A ⊂
c
y∈LN cint x ∈ X : Q (y) ⊂ P(x) .
Then, at least one of the following assertions holds:
(a) z∈Z P−1 (z) = 0/
/
(b) y∈Y Q(y) = 0.
Theorem 1.2.10 Let (X,Y, {ϕN }) be a GFC-space, Z be a nonempty set, S : Y ⇒
X, F : X ⇒ Z and Q : Y ⇒ Z, assume that
(i)
(ii)
(iii)
(iv)

F −1 is transfer compactly open-valued;
for all x ∈ X, y ∈ Y : F(x) ⊂ Q(y) is an SGFC -subset of Y ;
for each (x, y) ∈ grS, F(x) ⊂ Q(y);
there is a nonempty compact subset A of X such that for each N ∈ Y , there is
an S-subset LN of Y containing N such that S(LN ) is compact and S(LN ))\A ⊂
y∈LN cint x ∈ X : F(x) ⊂ Q(y) .

Then, at least one of the following assertions holds:
(a) there exists x ∈ X such that F(x) = 0;
/
(b) y∈Y Q(y) = 0.
/
Theorem 1.2.11 Let (X,Y, {ϕN }) be a GFC-space, Z be a nonempty set, S : Y ⇒

X, P : X ⇒ Z and G : Y ⇒ Z, impose that
(i)
(ii)
(iii)
(iv)

P−1 is transfer compactly closed-valued;
for all x ∈ X, y ∈ Y : G(y) ⊂ P(x) is an SGFC -subset of Y ;
for each (x, y) ∈ grS, G(y) ⊂ P(x);
there is a nonempty compact subset A of X such that for each N ∈ Y , there is
an S-subset LN of Y containing N such that S(LN ) is compact and S(LN ))\A ⊂
y∈LN cint x ∈ X : G(y) ⊂ P(x) .

Then, at least one of the following assertions holds:
(a) z∈Z P−1 (z) = 0;
/
(b) there exists z¯ ∈ Z such that G−1 (¯z) = 0.
/
Remark 1.2.3 If X = Y and S = I, Theorem 1.2.8 implies Theorem 3 of [11]
and different to Theorem 7 of [10].
Proof. Theorem 1.2.5 implies Theorem 1.2.8. Suppose to the contrary that, for
all (x, z) ∈ X × Z, F(x) = 0/ and G−1 (z) = 0.
/ Then G(Y ) = Z. Define Ω : X ⇒ Y by
c
Ω (x) = y ∈ Y : F(x) ⊂ G (y) . We check the assumptions of Theorem 1.2.5 for
Z = X and T = I. By (iv) assumptions (iii) and (iv) of Theorem 1.2.5 are obvious.
(ii) and (iii) imply assumption (ii) of Theorem 1.2.5. Arguing for Ω similarly as
for H in the proof of Corollary 1.2.1 we see that Ω −1 is transfer compactly openvalued as required in assumption (i) of Theorem 1.2.5. Applying this theorem

1.2 Existence theorems in GFC-spaces

17

we have x¯ ∈ X such that Ω (x)
¯ = 0.
/ Then, F(x)
¯ ∩ G(y) = 0/ for all y ∈ Y . Hence,
F(x)
¯ ∩ Z = 0,
/ i.e. F(x)
¯ = 0,
/ a contradiction.
Theorem 1.2.8 implies Theorem 1.2.9. Setting F = Pc , G = Qc and applying Theorem 1.2.8 to F and G we are done.
Theorem 1.2.9 implies Theorem 1.2.10. Make use of Theorem 1.2.9 for the given
Q and P = F c .
Theorem 1.2.10 implies Theorem 1.2.11. Employ Theorem 1.2.10 for F = Pc and
Q = Gc .
Theorem 1.2.11 implies Theorem 1.2.8. Set P = F c and use Theorem 1.2.11.
Next, we establish KKM type theorems, which encompass a large number of
known KKM type theorems.
Theorem 1.2.12 Let (X,Y, Φ) be a GFC-space, Z be a topological space, S :
Y ⇒ X and F : Y ⇒ Z be set-valued mappings and T ∈ KKM(X,Y, Z) such that
Y is an S-subset of itself. Let the following conditions hold
(i) T (S(Y )) is a compact subset (of Z);
(ii) F is T-KKM and transfer compactly closed-valued.
Then, T (S(Y )) ∩

y∈Y

F(y) = 0.
/

Proof. We define a new set-valued mapping F : Y ⇒ T (S(Y )) by, for y ∈ Y ,
F(y) = T (S(Y ))∩cclF(y). Then F has clearly closed values in T (S(Y )). We claim
that F is T -KKM. Indeed, for N ∈ Y and M ⊂ N, as F is T -KKM one has
T (ϕN (∆M )) ⊂ y∈M F(y). Since Y is an S-subset of Y, T (ϕN (∆M )) ⊂ T (S(Y )).
Therefore
T (ϕN (∆M )) ⊂ T (S(Y )) ∩

T (S(Y )) ∩ F(y) ⊂

F(y) =
y∈M

y∈M

F(y).
y∈M

As T ∈ KKM (X,Y, Z), the family {F(y) : y ∈ Y } = {F(y) : y ∈ Y } has the finite intersection property. Since this is a family of closed subsets of compact set
T (S(Y )), by Lemma 1.1.1 one has
T (S(Y )) ∩

T (S(Y )) ∩ F(y)

F(y) =
y∈Y

y∈Y

T (S(Y )) ∩ cclF(y)

=
y∈Y

=

F(y)
y∈Y

= 0.
/