Abstract convexity. Fixed points and applications. Juan

Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
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Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
UNIVERSIDAD DE ALICANTE
FACULTAD DE CIENCIAS ECONOMICASY EMPRESARIALES
ABSTRACTCONVEXITY.
FIXEDPOINTSANDAPPLICAIIONS.
Memoria presentada pon
JUAN VICENTE LLINARES CISCAR
para optar al grado de doctor
en Ciencias Económicas
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
D. JOSEP ENRIC PERIS i FERRANDO, TITULAR
DE
UNIVERSIDADDEL DEPARTAMENTO
DE FUNDAMENTOS
DEL ANALISIS
ECONOMICODE LA UNIVERSIDADDE ALICANTE,
CERTIFICA: Que la presente memoria "ABSTRACT CONVEXITY.
FIXED POINTS AND APPLICATIONS",ha sido realizada
bajo
su
dirección,
Fundamentos
Universidad
Ciencias
del
de
en
el
Anátisis
Alicante,
Matemáticas
Departamento
Económico
por
D. Juan
el
Licenciado
y que constituye
su Tesis para
grado
de
Ciencias
en
la
en
Vicente . Llinares
Císcar,
Doctor
de
de
optar
al
Económicas
y
Einpresariaies (sección Económicas).
Y para que conste, en cumplimiento de la
r legislación vigente,
de Alicante
la
presento ante
referida
la
Universidad
Tesis Doctoral, firmando
el presente certificado en
Alicante, 27 de Junio de 1994
Fdo. JoSEP ENRIC PERIS i FERRANDO
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
En primer
del
Departamento
Universidad
en
mi
prestado
de
de Alicante
trabajo,
en
asÍ
todo
me gustaría
lugar
del
Fundamentos
por
como
agradecer
Análisis
el constante
ei
apoyo,
momento-
interés
moral
Especialmente
de las primeras
Económico
de
la
que han mostrado
y material
quisiera
y a José A. Silva que "sufrieron"
Begoña Subiza
a mis compañeros
la
que me
agradecer
lectura
han
a
de una
versiones de esta tesis.
Tengo una deuda muy especial hacia Josep E. Peris, que
en todo momento me ha apoyado y animado y sin cuya dirección no
podría haber realizado este trabajo.
Su supervisión y orientación
continua han sido vitales en el desarrollo y finalización
de esta
investigación.
No puedo continuar sin hacer mención de la importancia
que
Mari Carmen Sánchez ha
trabajo,
ella
ha
seguido
tenido
paso
a
en
el
desarrollo
de
este
paso
la
evolución
de
esta
investigación a la que ha contribuido con múltiples discusiones y
sugerencias que han sido para mi de gran ayuda. Además, el apoyo
moral
y
eI
entusiamo
que me ha trasmitido
siempre,
han
sido
fundamentales en la realización de la misma.
También quiero
Enrique Martínez-Legaz
agradecer a Carmen Herrero
el interés
y
a Juan
que siempre han mostrado,
así
como las sugerencias que me han hecho en todo momento. Del mismo
modo quiero agradecer a Charles Horvath su hospitalidad en las
distintas estancias que tuve en la Universidad de Perpignan en las
que tuve ocasión de contrastar opiniones que ayudaron a enriquecer
esta memoria.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Alicante Junio 1994.
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
tn hlo.lú Ganmzn
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
INDEX.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
INDEX
CHAPTER1 : CONVEXITYIN ECONOI'íY.
1.1
INTRODUCTiON
CIIAPTER 2 : ABSTRACTCO¡IVEXITY.
2.O
INTRODUCTION
77
2.7
USUAL CONVEXITYIN R"
1,9
2.2
ABSTRACTCONVEXiTYSTRUCTURES
23
2.3
K-CONVEXSTRUCTURE
35
2.4
PARTiCULARCASES: RESTRiCTiON
ON FUNCTIONK
2.5
2.6
4I
RELATIONSHIP BETWEENTI{E DIFFERENT
NOTIONSOF ABSTRACTCONVEXITIES.
ol
LOCAL CONVEXITY.
72
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
CHAPTER3 : FIXED POINT RESULTS.
3.0
INTRODUCTION
76
3.1
PRELIMINARIES
79
3.2
C L A S S I C A LF I XED POINT THEORN,ÍS
85
3.3
F I X E D P O I N TTHEOREMSiN ABSTRACT
CONVEXITIES
3.3.1.
Fixed point
B9
theorems in c-spaces
90
3.3.2. Fixed point theorems in simplicial
convexities
3.3.3.
Fixed point
theorens in K-convex
continuous structures
.
3.3.4. Fixed point theorens in mc-spaces .....
101
113
CHAPTER4 : APPLICATI0N OF FIXED POINT THEORffiS.
4.0
INTRODUCTION
4.T
EXISTENCEOF MAXiMALELEMENTSIN BINARY
RELATIONS iN ABSTRACTCONVEXITIES
4.2
4.3
125
r 27
EXISTENCEOF EQUILIBRIUM IN ABSTRACT
ECONOMiES.
150
EXISTENCEOF NASH'S EQUILIBRII.JM
176
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
REFERENCES
186
SUBJECT INDEX
201
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
1
CHAPTER
CONVEXITY
IN ECONOMICS.
1.I
INTRODUCTION.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
I.
l:
ConveXity
in
Economics
INTRODUCTION.
Modern
which
allow
the
Most
behavior.
are
Theory,...)
based
of
tools
these
Nonempty
Inequalities,
different
creation
of
uses strong
analysis
economic
models
to
(Fixed
Point
the
on
analyze
concept
complex
Theorems,
Concave
Intersections,
mathematical
human
Variationai
Programming,
convexity
of
tools
to
Game
obtain
the
results.
The
of
concept
(in
set
convex
a
linear
topological
space) is one of the most used mathematical notions in economic
analysis due to the fact
very
regular
preferences,
assumption
However,
consider
that
Convexity
behavior.
choice sets or
.and
many
that
is
is very intuitive
it
"natural"
in
"natural"
(especially
authors
production
decision sets, etc.
considered as
this
in
in
character
and implies
sets,
is
individual
a much used
economic models.
experimental
is
a
not
economics)
presented
in
reality.
Next,
we
are
going
to
comment on several
modelizations where convexity can be justified
as
in
the
case
of
the
existence
of
economic
in a natural
equiiibrium
(from
w?Y,
the
consumers' as well as the producers' point of view), game theory
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
l:
in
Convexity
Economics.
decision problems under
situations,
(expected utility
uncertainty
convexitv
is
not a natural assumption but only a technical requirement will
be
theory).
In
these contexts, other
where
situations
mentioned.
t
The classical way of justifying
of
preferences
would
be
to
consider
the convexity assumption
it
as
the
mathematical
expression of a fundamental tendency of ec<.¡nomicchoice, namely,
the propensity to diversify
(successive units
amounts of
smaller
utility
is a
consequence, on the one hand, of the decreasing marginal
natural
utility
consumption. This diversif ication
function;
of
a consumption good yieid increasingly
which provides the
utility)
and, on the other
means of
decreasing marginal
constant,
it
is increasingly
rate
hand it
concavity of
the
can be justified
by
of substitution
(keeping utility
more expensive to replace units
of
a
consump,tion good by units of another).
Convexitv has been one of the most important
conditions
in many results which ensure the existence of maximal elements in
preference
etc. ).
relations
This
contexts
(Fan, 1961; Sonnenschein, 1971; Shafer, -1974,
assumption allows
where
individuals
us to
have
obtain
classical results
preferences
necessarily given by utility functions.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
which
are
in
not
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
l:
Convexity
All
production.
in
the
Economics
above
mentioned can
we assume that
If
also
be
appiied
perfectly
inputs and outputs are
divisible, then convexity on the production set implies that
point
any initial
at
its boundary, it
to
from
takes an increasingly large
amount of input to produce succesEive additional units of outputs.
Therefore,
convexity
production
in
sets
is
a
characteristic
of
economies with not increasing returns to scale. This statement can
be derived from
divisibility
other
of
two
all
hand the
basic requirements: on the
the
inputs
additivity
(that
property
do not
interfere
returns
to scale and due to the fact
convex,
with
used in
each other).
production
any
system
one hand the
production,
and
the
is, production activities
In the context of
that
which
on
decreasing
the production set is
generates
an
efficient
aggregated production can be supported by means of a price system
in which each firm
maximizes benefits. Furthermore, productions
which maximize benefits do not change discontinuously qith
hence it
is possible to apply Brouwer's or Kakutani's fixed
Theorem which
ensure the existence of
equilibrium
prices,
point
(Debreu, 1959;
Arrow and Hahn, 197?; Cornwal, 1984).
If
an económy whose technology set has constant returns
to scale is considered, then the production set is a closed convex
cone. Economies which verify
there
is
f¡'ee
entry
into
this
and
property
exit
from
are
those in
production.
outputs can be doubled (halved) by doubling (halving) inputs.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
which
That
is,
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
l:
Convexity
Economics.
in
which strengthens the convexity assumption
Another fact
is
economies with
case where
the
a
considered (Aumann, 1964, 1966). In this
continuum of
case, by
agents
is
assuming that
agents do not necessarily have convex preferences it
is obtained
that the aggregated excess demand correspondence is convex valued.
So,
order
in
to
point
fixed
prove
to
results
(Brouwer, t9l2;
existence of
the
equilibrium,
classical
be
applied
can
correspondences
Kakutani, 1941). These kinds of
results
are based
on the application of Lyapunov's Theorem (see Aumann, 1965,1966).
expected
The
aversion,
principle
results
powerful
a
(and therefore
operation
the
form
reinforcement
to the
lotteries
the
preference
expectation
is
von
of
the
of
to
equivalent
mathematically
function.
and
of
risk
diversification
convexity).' In these contexts
takes
to
Moreover,
equivalent
Thus,
to
the
the
the
preferences
the hypothesis that
of
form
that
condition
a
it
statistical
linear
in
hypothesis
is
be
the
aversion
rist
the
concavity
assumption
convexitiy
implied by the model.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
the
spaces) where a convex
has been defined and in which
function
probabilities.
Neumann
theory
the
are considered. Mathematically,
over
utility
in
theorv
are presented in spaces (mixture
combination
the
provided,
has
Morgenstern
utilitv
condition
is
of
the
directly
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
l:
Convexity
In
in
Economics.
context
the
of
convex sets. That
is due to
mixture
space
the
strategy
probability
it
Game Theory
the fact
(which
distributions
is
that
the
usual
to
decisions are
by
obtained
over
is
pure
consider
made on a
considering
strategy
all
space)
of
which
is a convex set.
In
same way,
the
ensure
the
existence
of
games.
In
particular,
a
convexity is
Nash's
compact
usual condition
equilibríum
convex
in
to
noncooperative
strategy
spaces
and
quasiconcave continuous payoff functions will be required.
Although
been justified
criticized
points
of
intrinsic
which it
this
as
in the
a
models mentioned above, convexity
"natural"
requirement,
many
authors
assumption fnom both the experimental
view.
Some of
requirement
of
them
the
argue that
model, but
would be desirable to
convexity
rather
eliminate. Farell
context of competitive markets says,
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
?
has
have
and formal
is
not
technical
an
one
(1959) in the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
"...I
Economics.
in
1: ConvexitY
argue,
shall
relevant
functtons
However
we
shaLl
assurnptüons of
markets
the
to
essential
are
often
see
that
convexíty
world'
real
the
in
not
the
by
are
opttmaltty
and that the assertions
the
convex.
tradítíonal
no
means
compettttve
of
to the contrary
are based on an elementary fallacy."
Next, some of the criticisms of convexity are presented'
Starr
preferences
indivisibility
(1969)
since this
and
anticomplementary
criticizes
convexity
class
a
of
the
requirement
postulates away
relations
(those in which there
which
are two
of
all
one
convex
forms
caII
might
different
of
goods
and the simultaneous use of both of them yields less satisfaction
to the consumer than would the use of one or the other. Examples
could be pep pills
wine;
estetic
versus sleeping pills;
beach holidays
satisfaction
versus
mountain
white
,wine versus red
holidays;.-.Evaluations
tastes also tends not to be convex, as Bacon
(in Arrow and Hanh, 1977) saYs,
"There is no excellent
strange in
of
beauty whích has nothtng
proportton"
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
1: Convexity
In fact,
in
Economics.
problem could be avoided whenever these
this
goods could be kept for
are
"time
Theory,
dated",
as
a
a future
as
in
fact
consumption. However, if
happens in
consequence preferences
goods
Generai Equilibrium
will
not
be
convex
(Starr, 1969).
Although no convexities of
consumption
important
sets
can
because of
be
criticizes
mitigated
the
the fact
by
aggregation,
consequences they imply.
considered by Aumann of
Starr
preference relations
economies with
that
In
or
of
they
are
the
case
a continuum of
agents,
the weight of an individual in the
economy wiII never literally be zero,
"...Though
ít
seems
índLvidual
as
5
economy,
I find
as O of
x
reasonable
1O-e of
ít díf ftcult
to
the
treat
Uníted
to conceíve
an
States
of
htm
it."
Alternatively,
Starr
considers
large
economies where
preference agents are not necessarily convex, and in this context
he obtains the existence of an approximated equilibrium. In order
to do this, he applits Shapley-Folkman's Theorem which allows the
effect
of
a
fall
in
the
degree of
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
non
ccnvexity
degree by
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
Convexity
l:
aggregating
a
Hahn, 1977) to
the equilibrium)
Economics,
number
large
of
considered
be
(that
equilibria
in
is,
a
agents
(Starr,
1969;
the
existence
of
and
configuration
(Starr,
a
Arrow
approximated
distance
negligible
and
from
1969) to be obtained.
On the other
hand, there are some situations
in which
the production set is not necessarily convex. For instance, in the
cases where there are indivisible goods, set up costs, increasing
returns
by
to scale or externalities
Arrow
and
Hanh is
clear
in the model. The example used
this
in
sense when
the
indivisible goods is considered,
"...There
are
productton
ínstruments
whích
if
acttvttíes
for
divístble.
If
the
it
cannot therefore
ín
goods
whtch
partícuLary
(spades, mineral
produced
are
CertaínLy,
goods,
many
they
mtlLs)
units.
indívistble
are
índívísible,
are
used cannot
is possible
the
be
to use a spade ít
be deduced that there extsts a
process where haLf a spade can be used. A more
compLtcated example
storage.
ts that of
The uti-Ltty of
proportional
to
recípíents
one such recLpíent
the surface
area (that
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
for
üs
üs, tf
case of
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
Convexity
l:
in
Economics.
the thickness
perhaps
of
the walls
necessary
to
is constant,
íncrease
volume to resíst
the pressure
This
formally
case
can
of
be
tt is
¿t wíth
the
the contents.
consídered
as
indívLsible, gíven that the geometríc shape of
rectpíent
the
and
recipLents
of
different
stzes,
must be consídered
as dífferent
goods
where
each
produced
whole
one
is
only
ín
quantíties"
A different
and a single
output
case would be an economy with a single input
can only be used in a f ixed
which
amount
( e . g . a i r p o r t s , . ). .
An important
situation in whirch convexity assumptions do
not appear in a natural
returns
to
scale.
constant returns
distribution
of
to
In
way is that
the
particular
scale but
these
fixed
of economies with
between the
produced yields increasing returns
to
could
to
cause
increasing
returns
of
uniform
different
units
scale. Another fact
which
scale
is
organizational advantages in the internal structure
Adam Simth's idea
economies with
set up costs, the
initial
costs
case
increasing
that
of
of
the
production.
of labour productivity being determined through
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
10
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
the
Convexity
l:
in
specialization
returns
to
Economics.
and the
division
of
labour,
allows
be obtained on a scale significantly
individual
labourer for
Generally
in
classical
model
these
a world
situations
as
higher than the
where labour is the only input.
it
is
possible to
not
production
the
increasing
sets
are
apply
not
the
convex
(Mas-Colell, 1987).
If
economies
with
increasing
to
returns
scale
are
analyzed, the non convexity of the production
set is observed. A
technical
is
consequence of
this
non convexity
the
which
Kakutani's)
supply
defined.
ensure the existence of
comespondence might
Another
there
general
existence of
equilibrium.
are
no
equilibrium
with
incompatibility
to
results
Moreover, imperfect
economic equilibrium
inefficient
is
not
due to
be convex valued or
the
so
either
not
consequence is
cómpetition,
it
point results (such as Brouwer's
possible to apply classical fixed
or
that
or
perfect
ensure
competition
individual
even
the
leads to
firms
ending
up large.
In
action
of
economies with
an agent affects
feasible set of
market),
externalities,
either
another and this
classical
equilibrium
the
(that
objective
is,
when
the
or
the
function
action is not regulated by the
results
thev have non convexities.
1l
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
cannot
be-' applied
since
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
Convexity
l:
in
Therefore
preferences
are not
Economics.
out.
of
convex
when there
appropiate
'19'72) pointed
Starret
assumptions
the
These
non convexities
As we have mentioned, the
in
research
it
behavior is
under
expected
is
shown that
not valid
risk,
utility
since
this
experimental
the
In
recent
descriotive
describing a kind of
in
these
cases,
(197,
that
relative
is
based are
individuals
to
put
outcomes which
behavior
is
different
lotteries,
etc.
individual
preferences over
in
the
In particular,
not
more
are
to
convexity
is
experimental
economic
choice problems
preferences
individual
paradox (1952)
out by Kahneman and Tversky
Machina (1982) who illustrate
or
expected utility
show
carried
inherent
model of
for
work
as
on the sets.
systematically violate the axioms. Consider Allais'
or
convex
effects,
are
hypothesis of
models.
and
are external
both the possible action sets and preferences
inherent
sets
principles on which
that
satisfied.
These experiments
emphasis on
certain
results
merely probable, and that
presence of
negative
or
agent
positive
experimental evidence suggests that
lotteries
are
the probabilities.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
2
typically
not
linear
in
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
l:
in
Convexity
Economics.
, the non existence of
In Game Theory
is
an
problem
important
(in
preferences
well-behaved
pure
payoff
strategies),
functions
response
are
of
correspondence
the
to
of
Examples
models.
oligopoly
duopoly
can produce at no cost and where demands arise
models, where firms
from
in
Nash's equilibria
no Nash equilibrium
in which
produced.
are
easily
not
quasiconcave
(which
firm
one
of
action
the
best
response
prof it-maximizing
the
is
firms)
examples
these
the
and
gives
other
In
exists
not
convex-valued
(Vives, l99O).
In
there
literature,
which
some results
are
try
to
ensure the existence of Nash's equilibria under weaker conditions
over payoff
relaxing
functions
convexity conditions. In this
line,
we have to mention McClendon's work (1986), who relaxes convexity
by
considering
(1989),
who
the
relaxes
computational
mentioned
contractibiiity
point
are
of
those
of
Other
Baye,
convexity
from
results
which
have to
Zhou
(1993),
Tian
and
characterize the " existence of
Nash's equilibria
and non-quasiconcave payoffs;
Vives's work
the
existence of
semicontinuous
Nash's equilibria
payoffs
which
Kostreva's work
of
assumption
view.
or
conditions,
with
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
who
(1990) which analyzes
certain
and upper
monotonicity
properties, directly related to strategic complementarities, etc.
IJ
be
discontinuous
considering Iattices
verify
the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten l:
Convexity
in
Economics.
From the comments we have made above, the possibility of
eliminating or generalizing convexity in these models is of great
interest whenever this fact
is compatible with
results which solve
the mathematical problem which appears in economic modelization.
Technically, the problem can be stated in the following
terms:
obtaining
the
existence
of
point
fixed
results
using convexity conditions (since many problems are
applying
appropriate
analyzed this fact
Some works
of
point
results).
Recent
reduced to
works
have
in the context of pure and applied mathematics.
along this
Van de Vel (1993),
generalizations
fixed
without
line
are
...among
the notion of
those of
others
who
Horvath (1987, 199t),
have
introduced
a convex set and have extended
some fixed. point results. The present work
reasearch.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
IA
l-
follows
this
line of
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Volver al índice/Tornar a l'índex
Chapter
1: Convexity
The
in
Economics.
thesis
organized
is
chapters which are dedicated to
properties
usual
of
follows:
there
are
three
steps in the research.
different
convexity is analyzed and the most
In Chapter 2 the notion of
impontant
as
convexity
have been summed up.
Moreover, the notion of abstract convexity is introduced and some
of
the
presented
results
Subsequently,
different
literature
in
notions
of
put
are
abstract
forward.
convexities
are
introduced and the relationship between these new notions and the
ones previously mentioned is given. In Chapter 3 the existence of
f ixed point results relaxing
generalizations
Kakutani's,
of
the convexity is analyzed, and some
classical
Browder'S,... )
are
theorems
(such
as
obtained.
Finally,
in
different
applications
of
existence
of
elements,
abstract
maximal
economies and
fixed
the
point
the
results
non-cooperative games.
i5
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
of
Chapter
4
shown:
the
of
equilibria
in
Nash's
equilibria
in
existence
existence
are
Brouwer's,
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
CHAPTER
2
ABSTRACT
CONVEXITY
2.0. INTRODUCTION.
2.1. USUAL CONVEXITY IN RN.
2.2. ABSTRACT CONVEXITY STRUCTURES.
2.3. K-CONVEX STRUCTURE.
2.4. PARTICULAR CASES: RESTRICTIONS ON
FUNCTION K2.5. RELATIONSHIP BETWEEN THE DIFFERENT
NOTIONS OF ABSTRACT CONVEXITTES.
2.6. LOCAL CONVEXITY.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.O.
2.
Abstract
Convexity.
INTRODUCTION.
As mentioned in the previous chapter, the notion of the
convex set is a basic mathematical tool used in many economic
problems. Many generalizations
made
this
from
c-spaces
concept:
(Horvath,
of very different
sets,
star-shaped
1987,
i99l),
1990,
(Bielawski, 1987) or convexitity
natures have been
contractible
sets,
convexities
simplicial
induced by an order are some of
these generalizations.
general,
In
generalizations.
On the
concrete problems,
optimization
the
can
consider
two
one hand, those which
G.g.
the
existence of
kinds
of
motivated
by
different
are
continuous selections,
problems,... ). On the other hand, those stated from
point
axiomaiic
convexity
we
of
view,
where
is based on the properties
the
notion
of a family
of
abstract
of sets (similar
to the properties of the convex sets in Rn).
In
will
particular,
the
notion
of
abstract
convexity
which
be introduced is in the line of the former one and based on
the idea of substitute the segment which joins- any pair of points
(or the convex hull
of
a finite
set of points) for
plays their role.
t1
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
a set which
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
This
usual
Convex¡ty.
chapter
convexity
and
is organized
some properties
as follows:
related
in S e c t i o n 2 . 1 t h e
to
it
are
presented.
In Section 2.2 some extensions of the notion of convexity used in
Iiterature
are
structure,
which will
analyzed. In
Section 2.3
be called K-convex structure,
In Section 2.4 some restrictions
when additional
analyzed.
structures
In
generalized convexity
a
conditions are
Section 2.5
introduced in
the
the
of
this
structure
imposed on
relatioship
the
is introduced.
which appear
function
between the
previous sections is
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
are
different
presented and
finally, Section 2.6 analyzes the case of local convexity.
t8
K
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 2.
Abstract
Convexity.
2.1. CONVEX SETS. PROPERTIES.
In this Section the notion of the usual convex set in Rt
and some of its properties
(see Rockafellar, 1972; Van Tiel, 1984)
are analyzed.
Definition
2.1.
Let X be a linear topological space over R. A subset AcX is a
convex set iff for any pair of points the segment which joins them
is contained in the subset, that is
{ (l-t)x + ty: t e [O,l] ] c A.
Vx,yeX
This notion can be defined equivalently
in the following
wáY'
Definition
2.2.
Let X be a linear topological space over R. A subset AcX is a
convex
x,x-,..,x
l'z'nl'z'n
set
iff
e A
for
any
finite
family
anil non negative real
of
It
=litisverifiedthat
i =l
ltx
i =l
19
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
in
A
numbers t.,t^,..,1 e IO,i]
nn
suchthat
elements
ii
eA.
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Definition
Let
Convexity.
2.3.
X be a linear
combtnation
of
the
can be represented
points
Xl,x2,..,xn € X
is
a
point
of
convex
X
which
in the form
I a,*, where
i =1
space over R. A (finite)
topological
I t,=l
and
t . eI O , 1 ] V i = I , 2 , . . , n .
i =1
A consequence of
the
definition
of
convex set is
the
following,
Theorem 2.1.
The intersection of an arbitrary
collection of convex sets is
convex.
As an immediate consequence of this result,
who associates for
an operator
any subset of X the smallest convex subset in
which it is contained, can be defined.
Definition
2.4.
Let A be a subset of a linear topological space over R. The
convex huLL of A is the intersection of all the convex subsets of
X containing A.
C(A) = fl { B: B is convex such that A c B }
20
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexity.
properties
Some
verified
by
this
operator
are
the
f ollowing,
l. A c c(A).
2.AcB
+
C(A)sC(B).
3 . C ( C ( A ) )= C ( A ) .
4. C(o) = a.
The following
the
convex hull
of
result
illustrates
another way
in which
a set can be obtained by means of
convex
combinations of finite families of the elements of the set.
Theorem 2.2.
Let
A
be a
subset of
a
linear
topological
space over
IR,
A = {a,: iel}, then
C(A) = j
(l
I .r",r J finite set, Jcl,
i
I €r
The foilowing
by the closure and interior
result
I .r=1,
i€J
shows that
t.e[O,l]
convexity
is
|
)
inherited
of a set.
Theorem 2-3.
Let AcX be convex. Then the interior
of A, int (A), and the
closure of it, A , are convex sets. Furthermore
C(A) is an open set.
2L
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
if
A is open then
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstnact
Convexity.
Now the well known Carathéodory's
Theorem 2.4.
If
result
is presented.
[see Van TteL, 1984J
A c Rn, then for each xeC(A), there exist n+l points of A
such that x is a convex combination of these ooints.
Definition
2.5.
Under Theorem 2.4. conditions,
is defined
the Carathéodory
as n+1, since each element
a convex combination
of
number
of
R"
C(A) can be expressed
as
of no more than n+l elements of A.
By Theorem 2.3. C(A) is open whenever A is open, however
it
is not true that C(A) is closed if
A is closed. But in R', C(A)
is compact whenever A is compact.
Theorem
2.5. [Rockafellar,
1972; Van TLeL, 1984]
If A c Rt is a compact set, then C(A) is a compact set.
22
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.2.
2.
Abstract
Convexity.
ABSTRACT CONVEXITY.
The notion of abstract convexity has been analysed and
used by
among which
many authors,
Womble (1971),
we
can mention Kay
Jamison (1974),
and
Wieczorek (1983, 1992),
Soltan (1984), Bielawski (1987), Horvath (1991),etc.
2.6. [Kay and WombLe, 1971]
Definition
0' of
A family
structure
subsets of
X, with
for
a set X is termed
a convexLty
(X, g) being called a convexíty
the pair
space, whenever the following two conditions hold
I. a and X belong to 0.
2. € is closed under
¡ A. e 0
i€r
Then elements of
of
X.
following
condition holds
3.{x}e0
The
definition
of
abstract
intersections:
for each subfamily {A,},.,
. g.
g are called E-convex (or simply convex)
Moreover
subsets
arbitrarv
0
is
called
a
T -convexítv
1-
iff
the
raise
the
foreachxeX.
convexity
notion
an operator sÍmilar to that
in topology.
23
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
allows
us to
of the closure operator
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2-
Abstract
Definition
Let
defined
generated
2.7.
[Kay and LVombLe, 1971; Van de VeL, 1993]
X
be a
and
let
by
Convexity.
set
A
which
in
be
a
a convexity
an
abstract
subset
of
structure
X,
€,
g
convexity
then
which
the
we
has
hull
will
been
operator
call
0-hull
(or convex hull) is defined by the relation
C , r ( A )= n
{ B e 0 : A s B },
VAcX
This operator enjoys certain properties identical to those of
usual convexity, such as the foilowing one
Proposition 2.1.
C$(A) is the smallest 0-convex set which contains set A.
The convex hull allows us to define an operator between
the family of subsets of X
p: P(X) -------+ ?(X)
in tile following way
p(A) = Cg(A)
and which verifies the following conditions
l.VAeP(X),
Acp(A).
2. VA,B e ?(X), if A c B
3. VA e ?(X),
then
p(A) c p(ts).
p(p(A)) = p(A).
A map p: ?(X) -------+P(X) which satisfies these conditions
is
called convex
huLL on X. Note that
from
a convex hull
abstract convexity structure can be defined in the following way:
24
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
an
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexity.
The
póLytope
segment
convex
and the
hull
hull
of
of
a
is
convexity such that
all
topologtcal
structure.
a topological
finite
set
of
points
set is called
is
called
d
an tntervaL
(Van de Vel, 1993).
equipped with
set
convex
a
a two-point
between these points
If
p(A) =.\
(+
Ae0
both
a
topology
and
a
polytopes are closed, then X is called a
is clear that
It
convex structure,
as well
usual convexity
as the abstract
is
convexity
defined from the family of closed subsets of a topological space.
Furthermore
it
is
possible
of
the
0, has a Carathéodory number c iff
c is
to
extend
the
notion
Carathéodory number to the context of abstract convexity.
Definition
2-8.
[Kay and WombLe, 1971]
A convexity structure
the smallest positive
integer for
which it
is true
that
the 0-hull
of any set A c X is the union of the 0-hulls of those subsets of A
whose cardinality'is
not greater than c. That is
: c A, lBl= c }
c g ( A )= u { c u ( B ) B
where lB I denotes the cardinality oi set B.
25
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstract
Convexity.
A generalization
Definition
2.9.
o f this concept is as follows.
IKay and WombLe, 1971;Van de VeL, 19931
A convexity structure
0' is domaín f inite
iff
the following
condition is satisfied
C g ( A ) = U { C U ( B ) :B c A ,
Obviously an abstract
lBl < +o }
convexity defined on X such that
its carathéodory number is c is a domain finite convexity. However
the converso is not
(Kay and Womble, lg7l).
true
The following
result shows that under certain conditions, a subset A is a convex
set in a domain finite
for
convexity
if
and only if
the convex hull
any pair of points of A belongs to A. To state this result
need the notion of join
we
hull commutative which is defined in the
following way.
Definition
2.1O.
A convexity
huLL commutattve
structure
iff
it
defined on a set X is called a joín
is satisfied
that for
any convex set S in
X and for any xeX it is verified that
=
cuix,s)
rJ
!u [t.l "
"!f,
26
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Theorem
If
Convexity.
Abstract
2 . 6 . [ K a y a n d l V o m b l e , 1 9 7 1 . ;V a n d e V e L , L 9 9 3 )
g is a convexity
commutative
structure
and domain finite,
AcX is 0-convex
In the
context
defined
Cn({x,l})
abstract
authors who consider a different
asking for
is join
hull
then
iff
of
on X which
s a
convexity, there
definition
additional conditions for
Vx,yeA
of
the family
abstract
of
are
some
convexitv
subsets which
defines the convexity. Next, some of the most important ones are
presented.
Definition
2.11. [Wieczorek, 1992]
A family
of subsets g of
closed convexíty iff
a topological space X is called a
it is verified that
l. 0 is a convexity structure on X.
2. VAe0,
Note that
generalize
the
A is closed.
of abstract
this definition
notion
of
usual
convexity
(in
convexity
does not
topological
vector
spaces), therefore it is an alternative definition.
A different
by
notion of abstract
Van de Vel (1982, 1983, 1993)
and
convexity is the one used
Kindler and Trost (1939)
which requires the union of cor,vex sets to verify some conditions.
z7
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Abstract
Chapter 2.
Convexity.
2.12. [Van de VeL, 1982]
Definition
A family of subsets 0 of a topological space X is called an
it is verified that
altgned abstract convexíty iff
l. € is a convexity structure on X.
2. 0 has to be closed under unions of
Many
other
authors
a
consider
convexity by means of assigning for
any finite
cnalns.
I
generalization
family
of
of points a
subset of X which substitutes the convex hull of these points. In
this
case the
works
Bielawski (1987) must
contractible
Horvath (1987, 1991), Curtis (1985) and
of
notion
of
be used to present these results,
is
be mentioned. First
set, which will
of
all
the
given-
2.13. '[Gray, 19751
Definition
if
A topological space X is contractible
H: X x [O,l] ----------+X
in X and a continuous function'
Vx e
there is a point *o
such that
X it is verified
1. H(x,l) = x*
2. H(x,O) = x
So, a contractible
set X can be deformed continuously at
a point of X.
'
A family
of subsets { A }
i
i€I
is
called
ordered by inclusion.
28
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
a
chain
iff
it
is
totallY
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
The
Convexity.
notion
Horvath (1987). The
with
any f inite
of
c-spacez
idea of
subset of
this
X,
was
notion
introduced
consists of
a contractible
set
by
associating
(which can be
interpreted as a "polytope").
Formally
Definition
2.14.
the notion of c-space is as follows,
[Horvath, 1987, 1991]
Let X be a topological space, a c-structure
on X is given by
a mapping
F:<X>---+X
(where <X> is the family of nonempty subsets of X) such that:
l. V A e <X>, F(A)
is nonempty and contractible.
Z- V A, B e <X>, A c B
implies F(A) g F(B).
then, (X, F) is called c-space"
Observe
that
this
definition
includes
the
usual convexity in topological vector spaces as a particular
'
notion
of
case.
Irritiully this concept was called- H-space by Horvath (1987) and
was
used in
this
way
Bardaro and Ceppitelli (1988),
by
(1990,
Tarafdar
199I, I99Z), etc. However Horvath later called it
c-space.
29
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstract
Definition
2.15.
Convexity.
1987, 1991]
IHorvath,
Let (X,F) be a c-space, a subset D c X is called an F-set
iff
it is satisfied
F(A) c D
VA e <D>,
The
following
result
shows that
a
family
of
F-sets
defines an abstract convexitv on X.
Proposition 2.2.
Let
X
defined by
be
a
topological
means of
space in
a function
F.
which
Then the
a
c-structure
family-
{A }
ii€I
is
of
F-sets. i.e.
O, . X
such that
A. is an F-set
is an abstract convexitv on X.
On the
abstract
o.lher h a nd,
convexity
structure
functions. In particular
a continuous function
Bielawski
from
a
he associates with
(1987)3 introduces
family
of
any finite
an
continuous
subset of X
defined on the simplexa whose dimension is
the cardinality of the considered finite set.
'Curtis
?
(1985) works
define it formally.
aThe
with
n-d¿m enstonal si.mplex
this
kind of
structure
i
does not
Ar. g Rt*r is defined as follows,
A = { x€Rt*r; x = xt.e. , t.¿ o, xt.=l }
n
l¡
r
t
where e
but
are the canonical vectors of R.
30
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Definition
Let
convexíty
A,bstnact Convexity.
2.16.
1987]
[Bíelawskt,
space. It
X be a topological
each ne$l, and for
if for
is said to have a
símplictaL
each
(x.x....x ) e X' = X x .?l x X
l-Z'n
there exists a continuous map
-----+X
Q[xr,xr,...*.,],Arr_,
such that it verifies
Q[x](t) = x.
t. VxeX
V n = 2' l ,' Z n lv- (2 x' n n, x- l - , . . . x ) e X t ,
2.
if
t
=O
v(t-,t-,...t ) e a
then
i
é l x r , x r , . . . x r r ] ( t l , t z , . . , . a r r=) o [ x _ , ] ( t _ . )
where x
denotes that
iil'z-n
Note that
x
is omitted in (x ,x ,...x ).
usual convexity can be viewed as a particular
case of this structure
considering the funetion é as follows,
o l x r , x r , . . . x r r l ( t r , t r , . . . , t . r )= t r * , . * t r * , + . . . + t r r x '
Furthermore
sets
which
are
this
stable
convexity structure
under
the
functions
....
which it is defined.
JI
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
is
related
Q[xr,xr,...*r,]
to
the
from
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Definition
Let
Abstract
2.17.
Convexity,
IBteLawskL, 1987]
X be a topological
defíned. A subset A of
space where a simplicial
X i s c a l l e d símplíctal
convex
convexity
iff
is
Vne0.,l and
Vur,^2,..,.r, . A it is verified that
ó [ a r , a r , . . , . a , r l ( ue) A
It
stable
easy to .show
is
under
arbitrary
that
intersections,
VueA
simplicial
therefore
convex
sets
they
define
are
an
abstract convex structure.
Proposition 2.3.
The
family
convex structure
of
can
method that
structures
associate for
mention
is
points
sets
defines
by
is used in literature
means of
interval
any pair of points a subset of X.
Prenowitz
introduce the notion of
defined from
convex
an
abstract
to
generate
on X. Furthermore it is a T,-convexity.
Another
convexity
simplicial
and
Jantosciak's
operators
which
In this line we
results
(r9iq
whó
convexity by means of a union operation
the axiomatic point of view. Thus, for
X,Y € X a nonempty subset of X, x.y,
of the assumptions required are the following,
32
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
any pair
of
is associated..some
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexity.
A.l Existence law of join:
4.2 Commutative law:
Va,b € X,
A.3 Associative laws:
Va,b,c e X,
A.4 Idempotent law:
The
of
idea
a.b * a.
Va,b e X,
a'b
(a.b)-c = a-(b.c).
Va e X,
this
a'a
operation
is
(a,b e X) for
to
joining
up a pair
general
a and b do not belong to the set a.b.
points
of
= b'a.
= a.
substitute
the
set a.b,
the
segment
although
in
The convexity defined by the union operation is given by
means of the following relation,
(+
Ae0
Furthermore,
is
a
particular
case
Va,beA
as in
of
a'beA
the previous cases, usual convexity
this
structure
(defining
the
union
operation as the segment which joins up pairs of points).
Another
joining
particular
case consists of
associatirig
a path
up pairs of points, which is not necessarily the segment.
It is the notion of equiconnected space.
u (..b) ." =
U
Q.c)
z€a'b
33
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Definition
A
Abstract
Convexity.
2.18. [Dugundji, 1965; Htmmelberg, 19651
metric
topological
space X
is
equtconnected. iff
r: X x X x [O,l] --------+Xsuch that
exists a continuous function
'r(a,b,l) = b,
z(a,b,O)= a,
there
r(a,a,t) = a
for any t e [O,l] and for any a,beX.
Let it be observed that for
we can associate the following
verifying z-___(O)=x,
r
xyxy-xy'
any pair of points x,y e X
continuous function
r*u,IO,1] ---+X,
(l)=y. Then ¡_--_represents a path from x to
y, so we can define the abstract convexity in the following terms,
Definition
A
2.19.
nonempty
subset
of
an
equiconnected space
A c X
is
equi-convex iff the path joining points of A is contained in A.
It
is
clear
that
this
family
also def ines an abstract
T -convexity.
In
general,
AR
spaces
equiconnected spaces (Dugundji,
metric
spaces with
finite
(absolute
retracts)6
1965). Moreover, in
dimensionality,
contexts
are
.
of
equiconnected spaces
coincide with AR ones.
6Aset
X is an absolute retract
(AR) if ,it
is a metric space and
for any other metric space Y and any closed subset AcX, it is
v e r i f i e d that any continuous function f:A ----+X can be continuouslv
extended to Y.
J+
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexity.
2.3. K-CONVEX STRUCTURE.
In this Section a different
convexity
presented.
is
mentioned in
structure
pairs
of
is
previous
the
which
functíon
and
idea of
particular
a
the
ones
case. This
considering functions
joining
That is, the segments used in usual convexity
are substituted for
function
includes some of
Section as
based on the
points.
case
This
way of defining an abstract
an alternative path previously fixed on X. The
defines
X
will
this
path
to
be said
will
be called
have a
the
K-convex
Kl-"on ex
structure.
Formally, we give the following definition,
Definition
2.2O.
A K-convex
on the set X is given by means of a
structure
flunction
-_+X
K:XxXxlo,ll
Futhermore (X, K) will be called a K-convex space.
So, for
K(x,y,[0,1]) =
U
any
pair
of
points
{K(x,y,t): te[o,l]]
is
x,y€X
a subset given by
associated (in
a
similar
way to the case of the union operation).
''
From this function we can consider a family 0 of subsets
of X which is an abstract
convex sets are
exactlv
convexity on X and where the abstract
the
sets which
function.
35
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
are
stable
under *.his
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexity.
Propositior: 2.4.
Let (X,K) be a K-conyex space,
----)X
K:XxXxlO,ll
then the family of sets 0 such that
(+
A€g
defines an abstract
Vx,yeA
K(x,y,[O,tl)cA
convexity. The elements of
0 will
be called
K-convex sets.
Furthermore
2.I4.),
that
it
is
is, the union of
an
aligned
convexity
an arbitrary
(Definition
collection of
K-convex
sets totally ordered by inclusion is a K-convex set.
From
this
convexity
a
convex
hull
operator
can
be
defined in the usual way,
= n { B: AcB,' B is K-convex }
C-(A)
f r
(
\
is finite
Another important
property
domainT, that
the
coincides with
is,
of this
convex hull
convexity
of
the union of the convex hulls of
is that
an arbitrary
all of
it
set
its finite
subsets:
c,.(A) = U { c.-(s): B c A, lBl < +o }
KK
'
Proposition [Van de Vet, 1993]
Let X be a set where a convexity_ structure stable under unions of
chains has been defined, then this convexity is finite domain.
36
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexity.
examples show different
The following
K-convex
structure
can be defined
in a natural
situations
where
a
way.
Example 2.1.: Usual Convexity.
If
particular
is
X
a
vector
case of
the
space, then
K-convex
the
usual
structure.
To
convexity
show
is
this
a
we
consider the following function,
K: X x X x [O,l]-----; X
K(x,y,t)=11-1)¡+ty
In this
case, function
K associates for
any pair
of
points
the segment which joins them.
Note that
in this
case the K-convex set5 coincide with
the family of convex sets of X.
3'7
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
Example
2.
Abstract
2.2 :
Log. convexity
Another
.
Convexity.
possible
way
in Rl*.
of
defining
the
path
joining
^n
points
in R-- is by means of the logarithmic image of the segment which
joins them. In this case function K is given as follows,
K :Rl* x Rl* x [O,l] -----+ Rl*
if
x = (xr, ..., *r) ; y - (yr, ..., yr,),
=
K(x,y,t)
which defines a path joining
In this
[*l-'
y;,
x,! € Rl*
, -l-' rlJ
x and y.
case. a subset A c R"
++
is K-convex iff
I o g ( A ) = { ( l o g x r , . . , l o g x , r )(: x r , . . , x , r )e A }
is a convex set.
38
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstract
Convexity.
Example 2-3- [Castagnolt and Mazzolent, 1987J
Let A be a convex set and X a set such that there exists a
biyection from X into A,
h: X ----+ A
particuiar
In this
case it
is possible to define a K-convex
structure by means of function h as follows,
K: X x X x [O,l]-)
X
- t^
( l
\
(l-t)h(x) + th(y)
K(x,y,t) = h
|
l./
In this context a subset B is K-convex iff
h(B) is a convex
subset of A.
Until
all
lo*,
of the examples presented correspond to
convex sets or situations where a bivection fi'om these sets into
convex sets can be stated. The following example (used by Horvath
(1991) in the context
(and therefore
structure
of
c-spaces) shows a non contractible
set
not homeomorph to a convex set) where a K-convex
can be defined.
Example 2.4.
Let X c R' be the following set,
X =
!x
e Rz : O ( a = llxll = b,
a,b e R )
Considering the complex representation,
*-Px
ic(v
ic[x
'Y=Pre
39
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 2.
Abstract
Convexity
a function K can be defined as follows
K: X x X x [O,lJ----+ X
K ( x ,-y , t ) = | . t t - a l o * t p
' Y )l " i ( ( t - t ) c r x + t a v )
x
l.
It
is important
to
note that
in this
example K-convex
sets
cannot be contractible sets.
the
In
imposed on
properties:
the
set
function
generated
convexity
following
for
K
by
in
section,
order
function
instance, that
K(x,y,[O,1J); that
additional
to
K
ensure
are
.So,
presented which verify
ones of usual convexitv.
40
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
the
abstract
"desirable"
points x,y
belongs to
K(x,y,[O,lJ) varies
way whenever the ends do; etc.
structures
of
that
are
certain
satisfies
any pajr
conditions
different
in
a
continuous
abstract
properties
similar
convexity
to
the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
2.4.
Abstract
Convexity.
PARTICULAR CASES: RESTRICTIONS TO FUNCTION K.
If
and it
some continuity conditions are imposed on function K
is required that
structures
Vx,y , x,y € K(x,y,[O,1]), then restricted
can be defined where the meaning of
function
K
is
completely clear.
Now some particular
cases of
K-convex structures
are
presented.
2.4.1. K-convex continuous structure.
2.21.
Definition
X is a topological
If
space, a K-convex
conttnuous
structure
is defined by a continuous function
K:XxXx[O,1]
----+X
such that
K(x,y,l) = y
K(x,y,O) = x
it
.
structures
obvious
structure
continuous
K(x,x,t)
is
= x
can
Vte[O,l].
generalize
that
in
be
any
equiconnected
defined,
moreover
spaces with
Therefore
equiconnected'spaces.
4i
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
set
a
K-convex
verifying
K-convex
that
continuous
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstract
From
Convexity.
function
K,
a
family
of
continuous
paths
joining
pairs of points of X can be defined,
K
xy
K
Furthermore
it
xy
: [O,l] ---+ X
(t) = K(x,y,t)
is verified
that
we consider points
if
which are close together (x' close to x, y' close to y), then the
path which joins x and y and the path which joins x' and y' are
also close together.
Obviously in
any
convex
structure can be defined, although it
in any set. The next proposition
set
a
K-convex
continuous
is not possible to define it
states the conditions which have
to be required by X so that a K-convex continuous structure can be
defined on X.
Proposition 2.5.
Let X be a subset of a topological space, then it is possible
to
define
a
K-convex
continuous
structure
on
X
iff
X
is
a
contractible set.
Proof.
Let K be the function which defines the K-convex continuous
structure.
For
any
fix
a € X, .the
following
considered
H:XxlO,
ll
42
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
----+X
function
can
be
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstract
Convexity,
H(x,t) = K(x,a,t)
It
is
continuous
furthermore
function
is
since
continuous
verifies that
TL
H(x, l)
H(x,O)=x,
so X is a contractible set.
X is
Conversely, if
continuous function
H
set then there
a contractible
which
satisfies
the
exists
a
previous assumptions,
and from which it is possible to define the following function K,
te[o, 0.5]
( ulx,zt)
K(x,Y,t) = '{
telo.S, I l
I nly,z-zt)
which defines a K-convex continuous structure on X.
It
is
structure
continuous
K-convex
to
the
function
equivalent,'
subsets coincide with
fact
that
under arbitrary
abstract
are
the
intersections,
convexity.
K
is
family
Hence
given by
it
and therefore
abstract
the
some of
the
not
it
sets
that
mean
subsets. That
contractible
the
having a K-convex
does
contnactible
of
althoush
that
condition of
condition and the
contractibility
note
to
important
is
not
is
due
stable
does not define an
convexity
contractible
defined
by
subsets of
X
(since it is true that any K-convex set is contractible)Now some examples of sets where a K-convex continuous
structure can be defined are shown
43
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexity.
Example 2.5.: Star-shaped setIf X is a linear space, a subset of X is called a star-shaped
set iff
3x eX
oo
tx + (l-t)x^ € X
such that
VxeX , Vte[O, l]
In this case function K is given as follows,
-----rX
K:XxXxlo,ll
I rr-rrr* + ztxo
K(x,y,t) = .{
I
| 2-2ixo
'
Note
structure
that
on
V t e [O, 1].
function
X
since
However
equiconnected structure
The next
K
it
a
teIO, O'5]
+ (zt-l)Y
does
does
similar
not
not
telo'5, 1l
define
verify
function
an
that
which
equiconnected
= x
K(x,x,t)
defines
an
on X can be considered.
example shows a star-shaped
an equiconnected one.
44
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
set
which
is not
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstract
Convexity.
Example 2-6.
Let A c R2 be the following set,
l] :n€tr',| ]v[0,
A=U{(x,x/n),X€[O,
l]x{O}
A is a star-shaped set which is not equiconnected as it
is
not locally equiconnected8.
Other kinds of
K-convex
These
continuous structure
sets
situations,
sets where it
are
interesting
such as in
the
is
possible to
is the case of
as
they
appear
define a
comprehensive sets.
in
analysis of- production
many
economic
(free
disposal
assumption), bargaining problems, etc.
t
Th"or"-
[Dugundjt,
1965]
Equiconnectéd spaces are
precisely the
equiconnected ones.
45
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
contractible
locally
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Example
Abstract
Convexity.
2-7.
Let X c R'be
a set which satisfies that
if y=x
VxeX
From the definition
then
of the set it is easy to show that
Vx,-veX
This
K-convex
fact
y€X
XA1l=[min(x.,y.)l eX
allows
us to define
continuous structure
a function
K which
provides
the
to the set X.
I O-rrl* + Zt(x¡y)
K ( x , Y , t=
) l
(xny) + (2t-1)y
I Q-zt)
t
si
t € [O,O.s]
si
t e [O.5,1]
observe that this function is continuous since the minimum is
a continuous function in
Rt.
46
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
If
structure,
X
Convexity.
is
a topological
(from
then an operator
convex
Iine
as
the
with
it can be defined.
but
hull,
In order to
space with
K) which
function
which
a K-convex
general
in
continuous
is in the same
does not
coincide
operator, consider the function
define this
z: ?(X) ---+ ?(X)
such that:
z(A) = K(AxAxlO,ll)
since K(x,Y,O) = x and K(x,y,l) = !,
t h e n A c K ( A x A x I O , 1 ] ) ,s o i t
is clear that
Asz(A)cz(z(A))9...
then if
we call
Al = A , A2 = z(Al),
we obtain
... , At = z(At-l),
...
that
Al . A2 c -.. < At-l c At < ...
and it
hull
is
is
the
operator
closely
possible
prove
to
C.-(A) = V Ai,
Ki
that
since the
\
smallest K-convex set which contains A.
the following
related
to
of
definition
the
stability
is
number
Carathéodory
K-convex
From
this
stated which is
associated
with
abstract convexity.
Definition
If
2.22.
neD,,l,then a K-convex structure on a topological space X
is n-stable if it is verified that
vA(x
-
z(At)=At.
41
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstnact
Convexity.
If K defines an n-stable
K-convex
structure
on X. then
C*(A) = A"
so any point
a finite
of the K-convex
hull of a set A can be obtained from
number of elements of A.
Proposition 2.6.
Let (X,K) be a set with
an n-stable K-convex structure, then
it has a Carathéodory number c such that c s zt-r.
Proof.
By the n-stability we have,
if x e C..(A) = A' = K(A"-l x A'-r x [o,1])
K
then there exist
"l-t,
*l-t
€ A'-1, t"-relo,tl
such that
x = K( *l-t, *l-t, t"-1 )
But
since
At-l = K(A"-z x At-2 x [o,l]),
reasoning
same way it is obtained that there exist
*i-t, *f,-',*2-',*i-t € A'-2,
such that
*l-t
\
ti-', ti-'e [o,l]
= K( x'-2, *f,-z, t"-z )
*l-t = K( xl-2, *"-2, tl-z )
therefore
x = K[xt*;-',*i-', ti-'), K(x'-2,*i-', t)-',, ."-'
and hence
x e C*( *l-',
Reasoning in a recursive
*f,-', *n-2, *n-' ).
way it
4B
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
is obtained
that
]
in
the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstnact
Convexity.
x
^,1
I
I
I
€ LKIx1,*2,*3,.. .,X^r_
¿
.
I
J.
where the number of elements is never more than 2n-1.
To obtain a result equivalent to Carathéodory's we need
to introduce the following conditions,
l. Idempotent:
K(a,a,t)=¿.
2. Simetry:
K ( a , b , [ O , t ] )= K ( b , a , [ O , l ] ) .
Vte[O,l].
3. Associativity:
K ( K ( a , bI,o , 1 ] ) , cI,o , l ] ) = K ( K ( a , cI,o , l ] ) , b ,I o , l ] )
Proposition 2.7.
Let X be a topological space with a Kl"onu""
structure which
satisfies
conditions 1, Z and 3. Then the K-convex structure
is
n-stable
iff
to
the
Carathéodory
number is
Iess than
or
2n-1.
To prove this result we need some lemmas.
Lemma 2.1.
Under the assumptions of Proposition 2.7. we have that
C..({x,Y}=
) K(x,Y,IO,l])
K-
49
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
equal
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexitv.
Proof.
It is enough to show that
then apply
that
it
K(x,y,[0,1]) is a K-convex set and
is the smallest K-convex set which contains
the subset {*,y}.
So, if
a, b e K(x,y,[O,l]) we have to prove that
K ( a , b , I O , 1 ]c)
K(x,y,tO,il).
We know that there exist s,t e [O,t] such that
a = K(x,y,s);
If we denotez = K(a,b,r) with
z = K(a,b,r) =
b = K(x,y,t)
r e [O,t], then
K(K(x,y,s),K(x,y,t),r)
and applying conditions I,2,3
we
obtain
that
fr'
e [O,l]
such
that
z = K(x,y,r') e K(x,y,[O,l])
The previous lemma states
that
under conditions I, Z
,and 3, the path which joins any pair of points is a. K-convex set.
The following
lemma presents a useful property of the K-convex
structure to obtain the K-convex hull of any set.
Lemma 2-2Under
the
assumptions of
proposition
2.J.,
the
convexity
generated by the K-convex structure
is join hull commutative, that
is,
subset
if
C(Au{xi)
A
is
a
K-convex
=UC(a.x)
KK
a€A
50
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
and
x e X
then
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstnact
Convexity.
Proof.
Let A be a K-convex subset of X. Since we know that
A v {x} c
U c*(a,x)
U c*(a,x) c cK(A v {x})
and
a€A
if
a€A
were a K-convex set, it
U C--(a,x)
K
would verify
the lemma
a€A
since
would
it
be
the
smallest
K-convex
set
which
contains
A u {x}.
Next it is shown that the set
. U c -(a,x)
a€A
K
U C*(a,x), we have to
is a K-convex set. Consider u,v €
prove
a€A
that the path which joins u and v is contained in this set. Since
u and v are in the union,
3ar, a, e A:
u e C*(ar, x) = K(ar, x, [O,1])
v e C*(ar, x) = K(ar, x, [O,1])
ar, a, e [O,1] such that
so there exist
u = K(ar, x, tr)
v = K(ar, x, tr)
Therefore
Vr e [O,l] we have:
K ( u ,v , r ) = K [ * t . r , * , a , , . K ( a r ,x , t r ) , . J =
and reasoningin the same way as in Lemma2.I. we obtain
t)
= rlr(ar,
where
".
a r , t ) , * , r ' , J = K ( a . , X , r ' ) e C * ( a . , x ,.
= K!.r, a2, t) € A
a9
€ A. C * ( a , x )
since A is a K-convex set.
51
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
Abstract
2.
Finally
2.L. which
from
lemma is a generalization
the following
allows
function
Convexity.
us to
obtain
K in a recursive
the
K-convex
hull
of
of Lemma
a finite
set
wav.
Lemma 2.3.
Under
the
assumptions
of
Proposition
2.7.
it
is
verified
that
C * ( { x r , .. , x r r } ) = K ( . . K ( K ( x , " , , x nI_oI,,l ] ) , x r r _ rI,o ,I I ) . . ) , x r ,I o , l l )
Vt. e IO,t], Vi=l,2,...,n.
Proof.
We prove the result by induction on n. If n=2 then we are in
the case of Lemma 2. l. Assume that the result is true whenever the
cardinality of the set is at the most n.
Consider the set
B = {xr,xr,..,Xrr,Xr.,*r).It
can be expressed
in the following way.
.
B=AU{*rr*r}
where
A={xr,xr,..,*r,).
Then
c(A)cC(B)
KK
{x
}cC(B)
n+1
K
*o."tu".
BcC(A)U{x
}cC(B)
Kn+lK
therefore
C*(B) = CK(CK(A)U {x,.,*r})
and applying Lemma 2.2.
52
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstract
Convexity.
c ( B ) = c K ( c K ( AU
) { x , . , . , }=) U
K
K
K
c * ( i a , x ' , * r } )=
N+I
A€C
(A)
and by the inductive hypothesis
=
. r.Yo, l I
K(K("K(K(xr'*
)'xr'tr)
r'Xrr'trr)'X.,-r't.'-t).''
C K ( B )= K ( K ( . .K ( K ( X . , * r , XIrO
, ,,I I ) , x , . , _trO, ,l ] ) . . . ) , x r ,[ 0 ,l ] )
Proof of ProposítLon 2.7.
Proposition 2.6. proves one of
the implications.
Conversely; let c = }n-r be the Carathéodory number of
the
K-convex structure. Then for anv AGX it is verified that
c (A) = U t c frl'
KK
fFl = c, F c A ).
Let us assume that the structure
exists x € A'*l - A".
is not c-stable, then there
A"*l c CK(A), x e C*(A) and by
But
the
definition of the Carathéodory number we obtain
i
C*(A) =U{C*(F):lFl
=c,FcA}
therefore x e C__(F)for some F such that
lFl
K
Lemma
2.3.
we
obtain
Similar
that
x
€ Ac,
conditions to
introduced by many other
which
those
= c, F c A
and by
is a contradiction.
of
Proposition
2.1 .
were
authorg (see Wieczorek, 1992) in other
contexts of abstract convexitY.
53
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Under
Convexity,
stability
which generalizes a well
conditions,
known result
the
following
from
proposition
the usual convexity
(Theorem 2.5.) is verified.
Proposition 2.8.
Let
X
be
a
topological
continuous structure-
If
space with
an
n-stable
K-convex
A c X is a compact subset, then C*(A) is
also compact.
Proof.
Since K is an n-stable K-convex continuous structure, we know
that
C*(A) = An
but
Ar = A,
A2 = z(A1),
...,
A' = z(At-l)
zÁ) -- K(A x A x [O,l])
\
So, it is clear that z(A) is a compact subset due to the fact that
A and [O,l] are compact and K continuous. Applying this argument
repeatedly it
is obtained that
An is compact and therefore
CK(A) is also a compact subset.
54
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
that
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexity,
2.4.2. mc-spaces.
If
the
the continuitv
K-convex
continuous
generalization of this
any finite
their
family
of
composition
structure
way
to
points, a family
a
is
the
is
we
relaxed,
obtain
a
concept. Now the idea is to associate for
case of
of functions requiring
function.
that
This
composition
the family
of finite
points in a
c-spaces or
simplicial
convexities.
continuous
generates a set associated with
similar
K which defines
condition in function
However in contrast with these cases, no monotone condition on the
associated sets is now required.
Definition
2.23.
A topological space X is rnc-space íf for
of
subset
X,
A c X,
there
exists
a
any nonempty finite
family
of
elements
br,
i=O,..,lAl-1 (not necessarily different) and a family of functions
P f : X x [ 0 , 1 ] - - - +x
i= o,1,..., lAl-1
such that
t. pl (x,o)= x,
ef tx, l) = b.
VxeX.
2. The following function
G : [O, l]" ---+ X
given by
= r:I rl,h:
GA(to,tr,...,t.,_r)
[u",rJ,.._,],."_,J,...],tJ
is a continuous function.
55
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 2.
Abstract
Note
f ield
Convexity.
that
the
possibilities,
of
different.
For
mc-space
since
subset,
AcX,
following
the
way:
since
instance,
it
is
of
mc-space
can
appear
notion
it
every
possible
family
of
nonempty
to
def ine
functions
ranges
over
context
in
topological
for
p1,
any
a
wide
completely
space
nonempty
X
is
finite
Vi=O,...,1A1-1, in
the
aeX (X;¿o) **
if we fix
"on=rd"r
Pl(x,t) = x
VxeX, VteIO,l)
I
pA(x,l) = a
VxeX
i
It
is
clear
G (t ,...t
AOn-l
that
from
these
functions
it
is
obtained
that
) i s a c o n t i n u o u sfunction.
In the previous case, functions Pi are defined independently
¡
of
the f inite
subset A
which is
considered. However, in
cases,functions PA can be directly reiated with
other
elements of A, as
in the case of convex sets:
P1(x,tl = (t-t)x + ta.
VxeX, ütelo,tl
tl
where A = {a-,a,...,á,
0'
l'
,
lAl-1
Moreover, note
} and b
that
= a
i
if
i
X
Vi.
has a
K-convex
continuous
structure and we consider functions P.(x,t) = K(x,a.,t), then they
define an mc-structure on X. Therefore mc-spaces are extensions of
K-convex continuous structures
56
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexity.
Example 2.4. shows an mc-space where it is not possible
to
define
a
K-convex
continuous structure
since
it
is
not
a
of
X
contractible set (it has a "hole").
In
this
structure
(A = {a^, ... ,a }),
nonempty
each element
for
0'ni
any
for
there exists a function
a_ e A
finite
and
subset
for
each x € X,
ef{x), [O, l]----+ X , satisfying that
p l (x)(o )= x
pl(x)(t) = u..
and
ll
If
joins
-ii'ii
x
represents
P:
is
continuous, then it
and b-. Furthermore, if
b.
a continuous path which
represents a
is
joins
equal to
x
path
which
a., PA(x,[O,1])
and a.. These paths
depend, in a sense, on the points which are considered, as well as
the finite
subsets of A which contain them. So. in contrast to the
K-convex
structure,
the
of
nature
these
paths
can
be
very
different.
Therefore, function Go can be interpreted as follows:
the
path
point
which
pA -(U , I
n-l
n
joins
.) = 'pn - 1 .,
-
n-l
b.
with
represents
a
point
of
the
= pn_z i=
br,_r, al_r(Or,_r,^
n-z)
point of the path which joins p
with b ^, etc.
5'7
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
a
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
So
Convexity.
we
if
want
to
ensure
that
the
functions PA is continuous, we need to ask for
i
functions
P A ( z ) : [0, 1] ->
the path joining
b.
i+l
X
in
any point
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
of
the continuity of
"2" which belongs to
and p
' . _, with i = O,...,D-Z.
i+2'
5B
composition
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convex¡ty.
t, = l, then any t,
Finaily note that if
affect
does not
the
if t. = O, then b. will
Moreover
G
function
Aii
such that
(since PA(x,l) = b
j>i,
VxeX).
not appear in this path.
¡I
From
an
mc-space structure
it
is
always
possible to
define an abstract convexity given by the family of sets which are
stable under function G^. To define this convexity we need some
previous concepts.
2.24.
Definition
Let X be an mc-space and Z a subset of X.
that
A^Z + a,
function G
A
AnZ = {.0,"r,..,arr},
we define the
VA e <X>, such
restriction
of
to Z as follows:
Gol=, [o, i]" --+
ca
¡, ( t )
X
( a n , l ) , t , , _ ,) . . . .) , t o )
= t:(...PA_r{PA
where PA are the functions associated with the elements a-eA which
l¡
belong to Z.
From
extension
this
of K-convex
notion
we
define
sets) in the following
59
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
mc-sets
way.
(which
are
an
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Definition
Abstnact
Convexity.
2.25.
A subset Z of an mc-space X is an mc-set iff
G, ([o,l]") sZ
Atz
VAe<X>, A¡Z*a
The
intersections,
family
so it
of
mc-sets
is
defines an abstract
been done previously,
in
it is verified
this
case we
stable
under
convexity on X. As has
can
define
operator in the usual way.
C_"(A) = ¡ {B I AcB,
60
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
arbitrary
B is an mc-set}
an
mc-huli
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.5.
2.
Abstract
Convexity.
RELATION BETWEEN THE DIFFERENT ABSTRACT CONVEXITIES.
In
kinds
of
this
Section the
convexity
between the
relationship
in
introduced
previous
the
analyzed. Some of them are easy to prove, for
different
Sections, is
example that
equiconnected space has a K-convex continuous structure
the
K-convex continuous structure
c-space. Those which
are
not
is
a
particular
or
case of
an
that
the
proved
immediately obtained are
throughout this Section.
The following result shows that a c-space is a K-convex
sDace in which the F-sets are K-convex sets.
Proposition 2.9.
Let (X, F) be a c-space, then there exists a function K such
(X, K)
is
a
K-convex
Since (X,F)
is
a
that
F-sets
Furthermore
space.
are
K-conr¡ex sets.
Proof.
c-space, then
Vx e X
it
is
possible to
choose x* e F({xi), because of F({x}) + o.
Through the monotone of F we obtain that
Vx,y e X, F({x}) c F({x,y})
hence
X*,y* e F({x,y}).
set, there
contained in
and
Applying that
F({y}) c F({x, y})
F({x,y})
exists a continuous path joining
F({x, y}).
6l
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
is
a
contractible
x¡t and y*
which
is
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstract
Convexity.
Therefore we can define function K as follows
K: X x X x [o,l] ---)
where
K(x,y,[O,ll)
X
is the path which joins x* and y*. So it is
satisfied
Vx,y e X
K(x,y,[o,il)c F({x,y})
In order to show that F-sets are K-convex sets, it is enough
to verify that the path which joins any pair of points of an F-set
is included in the set. But it
is immediate from the definition
of
an F-set and due to the way in which K is defined. Therefore it is
verified that
K(x,y,[0,1]) < F({x, y}) .
Proposition 2.1O.
Let
(X,K) be a space with
a K-convex continuous structure,
then it is possible to define a simplicial convexity in X.
\
Proof.
Consider the family of functions O[ar,ar,..a,rl as follows,
Q[ar,ar,..a,.,1(t,,..,1r,_)=K(..K(arr,ar.,_r,t.,_r),?n_2,.
. ). . ),ar,tr)
,
The simplicial
convexity generated by 0 coincides with
one which is obtained from K.
62
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
On
simplicial
Convexity.
the
other
convexity
is
hand,
a
it
is
K-convex
important
structure
as
to
note
the
that
following
result shows.
Proposition 2.11.
Let X be a space with a simplicial convexity generated by O,
then there exists a function K which defines a K-convex structure
on X. Moreover, simplicial convex sets are K-convex sets.
Proof From the simplicial
convexity, we have that
Vx,y e X there
exists a continuous function
Q[x,yl: A, ------>X
Considering the following function
K(x,y,t) = O[x,y](t,1-t)
K: X x X x [O,1]-----+X
it defines a K-convex structure on X.
Moreover, if
A is a simplicial convex set then it
is obtained
that
Vn e 0,.|,V(ar,ar,...arr)e A, Vu € Arr_r O[ar,ar,..,arrl(u)e A
and in particular
V x, ye
K ( x , y , [ o , l l ) = o [ x , y ] ( 4 1 )g A
A
Therefore simplicial convex sets are K-convex sets.
Tl"
next
proposition
contain, as a particular
shows
that
mc-space
structures
case, c-spaces (in the sense that
are also mc-sets).
63
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
F-sets
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 2.
Abstract
Convexity.
Proposition^ 2.I2.
Let
(X,F) be a c-space, then X
is an mc-space. Moreover
F-sets are mc-sets.
Proof.
Since
(X,F)
is
a
c-space,
then
for
any
finite
subset
A={a^,a-,...,a } of X, we assign the contractible set F(A). Thus a
O'l'n
singular face structuree can be defined as follows,
F':<N> -_-_)X
F'(J)=F((a.:ieJ))
where N = {O,1,...,n}. So we can apply one of
Horvath's
results
(1991) to ensure the existence of a continuous function
g:4,, ->X
such that for any J c N
e(Ar) c F'(J) = F(ia. : i e J))
. Next, we are going to prove that X is an mc-space by means
of
function g. If
of R^*r, functions
we denote by {e.r i=0,..,n}
Pf:Xxto,ll ->X
I
the cannonical base
are defined in the following
way:
'
o
Definition
IHorv^ath, 1991]
Let X be a topological space, an N dimensional singular face
structure on X is a map F:<N> --+X such that:
l. VJe<N> F(J) is nonempty and contractible.
2. VJ,J'ecN> if JcJ' then F(J)cF(J').
Theorem [Horvath, 1991]
Let X be a topological space and F:<N>-->X a singular face
structure on X. Then there exists a continuous function f:An ---+X
such that
VJe<N>, f(A")cF(J).
64
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstract
Convexity.
PA(x,l)=g(e,",)=b,.,
PA-r{PA{*'t)'tr,-r)= s[t,.,-r"r,-, * (t-t"-r)e,r]
=
tl-,
[tl-'[eo{*'t)'t,,-,J't"-J
=e
[t"-r"r,-r+(1-t,,-r)
[tr,-r".,-r*(t-t"-r,"")
J
in general
.-J't]=
"lI tl-,["][."'tJ,.,,-,J,,
"-,J,
n
(
tr
-elt.".
l nn+
.
(Functions pi
j=k*l
will
now so that
until
it
,j-,
,)
) .",[ilrr-tlll
L j
t
i=k
be defined in those values not considered
would be an mc-space, that
is PA(x,O) = x,
and pA(x,l¡ = g(e.)).
Finally,
= t:I tl-,[4["",tJ,t,,.,J,t
GA(to,tr,'.',t,,-r)
"-rJ,'],tJ
-
f F
l
) o,t,l
t.?"J
= cl
where
&.
ir
furthermore
are
it
functions
is
which
verified
that
vary
continuously
I o,=1.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
So,
the
with
t.
and
composition
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstract
¡^
, rD
Go: [0, ] i ----+X
means of
Convexity.
a
is
the
continuous
composition
of
function
continuous
since
it
functions,
is
def ined
by
defines
an
so it
mc-space structure.
We only
need to
show
that
F-sets
are
mc-sets.
If
Z
is
an
F-set, then V A' e <Z>, it is satisfied that F(A') < Z.
A e (X),
Consider
A'= AnZ,
then
mc-structure,
as
a
such
AnZ * a,
that
result
of
way
the
and
used
let
to
us
denote
define
the
we have
c A l z ( [ o , l ] - ) c s ( a r ) s F ( A n Z )= F ( A ' ) 9 Z
whereJ=ii:a.eAnZ).
Therefore F-sets are mc-sets.
The
following
example
a c-space, in the sense that
shows
mc-sets
an mc-space
which
do not coincide with
is
not
F-sets.
Example 2.8.
@
Consider the following
X = U [2n,2n+1] (ne[,]).
subset of R:
n=O
.
Then it is possible to prove that X is an mc-space whose mc-sets
are
not
functions:
F-sets.
In
order
to
do
it,
we
def ine
VA = {ar,...,.r,} € <X>
( -¡.
I P'-'(x,O)= x
1¡
I tlf*,t) = max(A) = a*
66
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Vte(O,ll
the
following
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 2.
Abstnact
Convexity.
S o i t i s c l e a r that
G
is a continuous function
AA
G A ( t o , t l , . . . , t n) = P " ( . . . P " ( a, l ) , t
On
n
n- I
since
)...t ) = a*
Vt.eIO,l]
I
therefore X is an mc-space.
In this context we can ensure that the following subsets are
mc-sets:
Z
Thus, Z
that
w
AnZ
=[w,+o)nX
VweX
w
is an mc-set since for
*@
ww
every finite
weknowthat
subset A of X such
a*eZ,therefore
I
-\
c- . 1 II fr vot ¡ rr t * | = a * e Z
nlz*t
w
)
However it
is not
possible to define a c-structure
on X in
which Z* were F-sets (VweX)- It is due to if F:<X> ---)X
defines
a c-structure
VAe<X>
on X, then it has to be verified that F(A)
has to be a contractible set, and therefore to be included in some
interval
[Zn,Zn+Ll. Moreover,
by
the
AsB then F(A)€F(B)) this
interval
Ae<X>), since
case they
in
other
connected components and they
monotonicity
condition
has to be the same for
would
would
not
be
in
two
(if
everv
different
be contractible
sets.
Therefore it is clear now that Z* is not an F-set whenever w)Zn+l
(for every Ae<Z > it is verified F(A) is not included in Z ).
61
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Next,
Proposition
Convexity.
it
2.I2.,
mc-structure,
proved
is
that
a
in
simplicial
in which simplicial
a
similar
convexity
way
as
induces
an
convex sets are mc-sets.
Proposition 2.13.
Let X be a space with a simplicial convexity generated by O.
Then in X there exists an mc-structure where the simplicial convex
sets are mc-sets.
Proof.
Since
X
has
nonempty finite
a
simplicial
subset A
c
convexity
X,
g(O),
then
for
A = {ao,ar,...,"r,), there
any
is
a
continuous function
é[a ,a,...,a ]:A -------+X
such that
1.
é[x](1)= x.
2.
Q I a o , a r , . . . , a n ] ( t 0 , t l , . . , o , . . . , aorI)a=_ . ] ( t _ . )
i)
then it
is
possible to
define an mc-structure
by means of
f u n c t i o n é [ a ^ , a - , . . . , a]
Otn,
pA:Xxlo,1l->X
in the following way
p A ( x , t ) = o I a o , a r , . . . , a , . , 1 ( e=, . ., r) ,
'p A t p A l x l ) , -t r , - 1) '
.,-1" r'^""
= ó- ' [- a
o ',-,t a
' . 1, ." .' -.n, 'u[ "lnf-tl " n - e
l
68
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
')
+ ( l - t" n - t l' -. n J
"
the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexitv.
tl-.
Qlao'a,'"''t.,1
't"-r]=
[t:-,[eo{*'t)''"-,J
[',,-r".,-r*('
-tn-z)
[',,-r""-r*('-'"-r,""J]
in general
tl['tl,[tl["",tJ,'"-,J,'
"-,],"'J,'J=
é[ao,a,,-..,."t
.rf,.,"r[,|[_,t
.,,J
[,n"u
]
f inally
'l[ 4-,[rl[.",tJ,."-,J,."-,J,
=
J,.J
rL
o l ao^. r, a
l, u
. ..",l.\.,l.4
t t)J
where
o,
are
functions
which
)
" . " .I
vary
continuously
with
t,
and
furthermore. it is verified that f c.=1.
So, the composition
Go:[o,l]"-----+X
= t;[. t:-,[t:[.",tJ,."_,J,...J,.J
=
cA(t.,tr,...,t,,-r)
(n\
é[ao,ar,...,u"rl
l"r"rl
\
is
a
continuous function
since it
l=o
is
t
defined by
means of
composition of continuous functions, so X is an mc-space.
69
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
(Functions
Convexity.
P1 will
i
until
now so that
it
be defined
in those
values not
considered
would be an mc-space, that is Pl(x,O) = x,
and P"(x,l) = Ola ,a.....a l(e )).
i0'l-ni
We only need to show that
therfore
if
Z
is
simplicial
convex
set
convex sets are mc-sets;
then
V A' < <Z>,
it
is
é[a_,a ,...,a ](A ) g Z.
verified that
Let
a
simplicial
Olmm
A e (X),
with
AnZ + a,
and
let
us
denote
A'= AnZ = {a. ,...,a. i then as a result of the way used to define
II
0m
the mc-structure, we have
G ^ r _ ( [ o , 1 ] *=) o l a . , . . . , d . l ( ¡ ) s Z
Al'
ro
t-
m
whereJ={i:a.eA¡Z]1.
I
Therefore simplicial convex sets are mc-sets.
'to
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexity.
Summing up we have proved the following relations:
USUAL CONVEXITY
EQU ICONNECTED
CONTINUOUS K-CONVEX STRUCTURE
I
ü
SIMPLICIAL CONVEXITY
C-SPACE
K-CONVEX
11
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
STRUCTURE
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.6.
2.
Abstract
Convexity
LOCAL CONVEXITY.
Finally,
that
the
notion
of
local convexity
(which
assumes
each point has a neighborhood base of convex sets) and its
extension to the context of
abstract convexities are analyzed. It
is important to consider this case due to the many applications of
convexity
that
require
convexity
will
u9
used
propert.ies, so
local
throughout
generalize some results from
the
the
local
following
usual convexity. In this
abstract
chapters
to
Section we
introduce the notions of local convexity in some of the particular
contexts of abstract convexities mentioned in the Iast sections.
Definition
2.26.
[Van de VeL, 1993]
A set X is LocaLLy convex if
each point has a neighborhood
base of abstract convex sets.
In
particular,
in
the 'case
of
usual
convexity,
this
concept can be expressed in the following wayr
Definition
2.27.
A linear topological space X is local Ly convex if the point O
has a neighborhood base of usual convex sets.
A simple
convexity
example
of
localiy convex space in the usual
is the following:
72
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
2.
Abstract
Convexity.
Example 2.9.
A normed space X is locally convex due to it
is possible to
define a base of neighbourhoods at O as follows,
B(O,e)={x:llxll
When we want
apart
from
asking for
se}
to extend the notion of local convexity,
the balls to be abstract
convex sets, we
need to require the balls of abstract convex sets
B(E,e)= { x e Xl d(x,E) < e }
to be abstract convex sets.
Definition
2.28.
A metric
[Horvath, 199L]
c-space (X,d) is a
LocaLLy
"-"p"..to
if
the open
balis of points of X as well as the balls of F-sets, are F-sets.
In this
line, but in the context of K-convex spaces and
mc-spaces, we give the following definitions:
toThi,
concept
introduced
corresponds
by Horvath
to
the
notion
(1991).
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
of
Lc-metrtc
space
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Volver al índice/Tornar a l'índex
Chapter
2.
Definition
Abstract
Convexity.
2.29-
A metric space (X,d) with a K-convex continuous structure
is
a LocaLLy K-convex space iff Ve>O it is verified that
d(x,E)<r}
{xeXl
is a K-convex set whenever E is a K-convex one.
Definition
2.30.
A metric
mc-space is
a
LocaLLy mc-space
verified that
{xeX:d(x,E)<e}
is an mc-set whenever E is an mc-set.
14
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
iff
Ve>O it
is
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
CHAPTER
3
FIXEDPOINTRESULTS.
3.0. INTRODUCTION.
3.I. PRELIMINARIES.
3.2. CLASSICAL FIXED POINT THEOREMS.
\ 3.3. FIXED POINT THEOREMS IN ABSTRACT
CONVEXITIES.
3.3.1. Fixed point theorems in c-spaces.
3.3.2. Fixed point theorems in simplicial
convexities.
3.3.3. Fixed point theorems in K-convex
continuous structures.
3-3-4. Fixed point theorems in mc-spaces.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
point
Fixed
results.
o . INTRODUCTION.
point
Fixed
we can ensure that
into
itself ,
basic
to
or
f ixed
a
or
tool
problems
of
in
the existence of fixed
That
conditions
from
is,
can
a topological
space
exists
These
the
This
be
which
point
a
depending on whether
prove
economics.
under
there
respectively.
used to
point for
the
defined
x e f(x),
problems
these
f,
point.
correspondence
mathematical
several
most
a
analyzes
a function
x = f(x)
such that
function
has
Theory
is
to
ref ormulated
specific
results
existence
due
f
as
is a
are
a
solutions
of
the
x
fact
that
problems
of
functions.
For instance, a classical problem in economic theory is
the
existence
economies,...)
existence
(f :E -+E).
of
of
which
the
(in
equilibrium
in
macroeconomics,
some cases is
solution
of
solved by
\such
equations
Leontief
proving
as
f(d
the
= O
But these problems can be formulated as a fixed point
problem by considering the following
function:
p(z) = z - f(z).
In
the same way, we can mention the problem of the existence of a
solution
to
variational
inequalities
which
also
are
problems
probiems
complementarity
basic
(general
( z >O; z
Í (z' - d-fQ' ) = O ],...,
in
the
solving of
equilibrium,
76
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
among
equilibrium
distributive
f(z) = O),
others,
existence
problems,...)
and
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
which
the
point
Fixed
to
points
case
of
of
{ x e Dom(f):
the
by
In particular,
can be obtained
from
the
g(x) = sup{O,x-f(x)};
in
considering
where
tr(y)
the
following
is
the
} (see Borden, 1985; Harker
Vz e Dom(f)
set
and
1992).
A different
point
inequalities,
xy = zy
problems.
function:
A(x,y) = tt(y)xf(x)
Pang, 199O; Villar,
fixed
problems
following
variational
point
as fixed
complementarity
correspondence:
of
results.
can be reformulated
solution
fixed
the
3.
kind of problem which can be solved by means
theory,
is the
case of
inequality
systems stated as
a nonempty finite intersection problem t fr f. * a ) by making use
'
i=l
of Knaster-Kuratowski-Mazurkiewicz's
result (KKM). In this case it
is also possible to solve the problem by proving the existence of
a
fixed
point
of
a
specific
(see
function
Border, 1985;
Villar, L992).
On the
existence of
other
maximal
hand,
is
it
important
to
note that
the
elements, thc existence of Nash equilibrium
in non-cooperative games, etc. are all problems which can also be
solved by means of fixed point theory.
Therefore,
of
it
is
very
interesting
to
analyze extensions
f ixed poina- results by relaxing the conditions usually imposed
on them in order to cover more situatians than those covered bv
77
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
the
Fixed
known
used
problems
where
classical
is that
the
Horvath's
and
results.
results.
point
f ixed
in
point
of
mapping
results
is
defined.
obtains
the
the
most
order
in
the convexity
(1987,
work
One of
to
important
solve
some
of both the mapping
this
In
line,
we
generalizes
1991) who
existence
of
conditions
and the set
have
f ixed
continuous
economic
to
mention
point
results
selection
to
correspondences.
This chapter is devoted to analyzing fixed
and the existence of
correspondences by
convexities
the
is
convexities
organized
definitions which will
In
Section 3.2
analysis
fixed
presented
and
we analyze this
abstract
problem in some of
Section
in
using
previous chapter.
in the
follows:
point
classical' fixed
new
convexity
approximations to
3.1
the
This
basic
be used in this development are introduced.
in
Finally
theorems to the
and
the
introduced
as
presented.
are
point
relaxing
instead. In fact,
abstract
chapter
continuous selections or
point results
results
theorems used in
Section
context of
in
the
presented in chapter 2 are proved.
18
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
3.3,
abstract
new
economic
extensions
of
convexities are
abstract
convexities
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
I.
3.
Fixed
point
results.
PRELIMINARIES.
In this Section several concepts and well known results
which
be
will
used
throughout
the
work
are
presented
(see
Dugundji and Granas, 1982 or Istratescu l98l).
Definition
3.1.
Let X be a topological space, a family of open subsets of X
such that X = U W. is called an open covertng of X.
{W.}.-I
lel
I
i€I
Definition
3.2.
Let {W,},., be a covering of X. Then if
J is contained in I
and {W.}.., is again a covering, it is called a subcoverLng.
Definition
3.3.
Let
and
{*,},.,
is a ref inement
of
\
be two coverings of X. {tr}r.,
{*r}r.,
{*r}r.,
if
for
every iel
there
exists
jeJ
such that W c w-
Definition
3.4.
A covering {W.).-- oi'a
I lcr
finite
if
for
every xeX there exists a neighborhood V* of x such
that W ¡ Y + a
ix
topological space X is called locally
onlv for a finite number of indexes.
79
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
point
Fixed
Definition
results.
3.5.
A Hausdorff
topological
open covering has a locally
Definition
Let
compact
space X is called. paracompact
finite
if
each
open refinement.
3.6.
X be a Hausdorff
if
for
topological space, X
each open covering
of
x
there
is
said to
exists
be
a finite
subcovering.
First of all we present some previous resuits which are
needed to prove fixed point theorems.
Definition
3.7.
Let {*rirl,
x-
A
{Wiii:l
finite
be a finite
partitíon
of
open covering of
untty
subordinate
is a set of continuous functions
p.:X ___+R
such that
.
1. VxeX
ú.(x)=O
n
2.
VxeX
3.
VxÉW.
Iú(x)=l
t
tI,
II
ú.(x) =O
80
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
a topological space
to
this
covering
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Fixed
Theorem 3.1.
point
results.
IIstratescu,
1981]
Let X be a compact Hausdorff topological space and {W.}.], a
finite
open covering of X. Then there exists a partition
of unity
subordinate to this covering.
Next,
the
notion
correspondence and
of
continuity
for correspondences are formally given.
Definition
3.8-
Let X, Y be topological spaces, and ?(X), ?(Y) the family
all the subsets of X and Y. A, correspondence
from
of
X into Y is a
function from X into P(Y)
f:X ----+ P(Y)
and it will be denoted by
f:X _+_+
Definition
Let
y
3.9.
X,
Y
be
spaces
topological
and
f:X ->+
a
Y
correspondence. It is said that
r-l(y)={xeX:y-ef(x)}
are the ínverse ímages of l.
A correspondence has open tnverse ímages whenever f-l{y)
an open set for everY YeY.
81
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
is
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3,
Fixed
point
results.
The graph of a correspondence f:X ___>_> y is given by
the foilowing subset of X x y:
Gr(f) = { (x,y) e Xxyl y e f(x) }
Definition
Let
3-lO.
X and Y be two
topological
correspondence such that for
to
be
upper
all x e X , l(x)*a.
semícontínuous
neighborhood v of f(xo)
spaces and I-:X ___++ y
at
xo€X
, there exists
if
for
Then it
any
a
is said
arbitrary
a neighborhood u
of *o
such that f(x) < V for all xeU.
The correspondence I'
is
said to
upper
be
semicontínuous
(u.s.c.) on X if it is upper semicontinuous at e a c h p o i n t x e X .
Definition
Let
3.11.
X and Y be two
topological
correspondence such that for
to
be
Lower
neighborhood v
all * I x
semícontinuous
of f(xo)
spaces and I':X -+-+
at
, l(x)+o-
xo€x
, there exists
if
for
Then it
any
y
a
is said
arbitrary
a neighborhood u
of *o
such that Vx e U it is verified that f(x) n y * a.
The
correspondence f
is
said to
be Lower
semicontLnuous
(l.s.c.) on X if it is lower semicontinuous at each point x e X.
82
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Definition
Fixed
point
nesults.
3.12.
Let X and Y be two topological spaces and let f:X --+->
a correspondence with f(x);¿a for all x e X. Then f
conttnuous at x € X if
Y be
is said to be
it is upper and Iower semicontinuous. The
correspondence I- is called continuous on X if
it
is continuous at
eachpointx€X.
One of the problems which will
be analyzed throughout
the work is the existence of fixed points. This concept, which is
stated
in
context
the
of
functions
(x*: f(x*) = x*),
has
been
extended to the context of correspondences in the following way:
Definition
3.13.
LetXbea
topological
space
and
correspondence. Then x* e X is called a fíxed
I-:X -----+---+ Y a
potnt
of
f
if
it is
verified that
x* e f(x*).
Two
different
ways
of
def ining
a function
associated
consider whether a function
whose
images are contained in correspondence images, or a function
whose
with
graph
a corresoondence is to
is
as
close as is
wanted t-9 the
Formally we give the following definitions:
83
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
correspondence graph.
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
Definition
point
results.
3.14.
f :X ----->--) Y be a correspondence.
Let
function
Fixed
f:
X---------; Y
such that
f(x)
Definition
Let
for
A selection
from
I- is a
everv x € X
e I-(x)
3.15.
X,Y
be
metric
correspondence. A f amily
spaces and
{f,},.,
indexed by a nonempty filtering
Ve>O
Hereafter,
3jel:
of
let
f:
functions
X----------++
Y
a
between X and y
set I is an approxímation
Vi=¡
be
for f if
Gr(f.) c B_(Gr(f))
Hausdorff
topological
considered.
B4
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
spaces
will
be
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 3.
2.
point
Fixed
results.
CLASSICAL FIXED POINT THEOREMS.
Perhaps the
most important
result
the
in
f ixed
point
theory is the famous Theorem of Brouwer Q9l2) which says that the
closed unit
ball R'
has the fixed
ball and f:Bt ->Bt
denotes this
point property,
that
is, if
is a continuous function,
B'
then
there exists a point x*€Bt such that f(x*)=;¡*.
One of
the unit
its
possible extensions consists of
substituting
any other compact and convex subset of
sphere for
a
finite dimensional euclidean sDace.
Theorem 3.2.
If
[Brouwer,
1912]
C is a comDact convex set in a f inite
dimensional
soace
and f :C -----+ C is any continuous mapping, then there exists a point
x*eC such that f(x*)=xx.
This
result
was
extended to
the
context
of
nonfinite
dimensional spaces (locally convex) by Schauder-Thychonoff.
Theorem 3.3.
Let
C
topological
[schauder-Thychonoff,
be
a
see Dugundjí and Granas,7982J
compact convex subset of
space. Then any continuous function
t lxecl Dornt.
B5
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
a
locally
convex
f :C ------+ C has a
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Fixed
point
results.
Brouwer's result
point results
these
is the main tool used to extend f ixed
to correspondences because the way used to obtain
extensions
consists
of
considering
a
continuous
approximation to or selection of the correspondence in which the
existence of
fixed
points
is ensured by Brouwer's result.
Since
these fixed points are also fixed points to the correspondence the
problem is solved.
The first
extension of this theorem to correspondences
considered is that of von Neumann (1937).
Theorem 3.4.
[Von Neumann, 7937; Kakutani, 1941]
Let X and Y be two nonempty compact convex sets each in a
finite
dimensional euclidean space. Let
E and F
be
two
closed
subsets of X x Y. Suppose that for each y e y
(x,y)eE}
E(y)={xeXl
is nonempty, closed and convex and also fon each x e X
(x,y)eF)
F(x)={yeYl
is nonempty, closed and convex.
Inthiscase
Kakutani
EnF*a.
(1941) obtains
an
extension
of
Brouwer's
Theorem by considering compact convex valued correspondences in a
compact convex set in the Euclidean space.
86
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
point
Fixed
Theorem 3.5.
results.
IKakutani, 1941]
Let E be an Euclidean space and C a nonempty bounded closed
convex
set
in
X.
I':C ------++C
Let
be
an
upper
semicontinuous
corresDondenceand closed convex valued. Then f has a fixed point.
Another
result
along the
same line
is
that
of
Fan
(1961), which was obtained independently by Browder (1961), where
the
of
existence
a
fixed
point
is
proved
using
conditions
different to those of Kakutani.
Theorem 3.6.
[Fan, 196L;Browder, 7967]
Let E be a topological vector space and C
a nonempty compact
convex subset of E. Let f:C -------) C be a nonempty convex valued
correspondence with open images. Then I has a fixed point in C.
As we mentioned above, the way to prove the existence of
fixed
points
to
correspondences
continuous selection or approximation
applied.
or
Therefore
selections- of
the
existence of
consists
of
constructing
to which Brouwer's
result
continuous approximations
correspondences has been of vital
the theorv of fixed point.
87
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
a
is
to
importance in
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3. Fixed
point
results.
The following
ensure the
and,
results,
by yannelis
existence
of
a continuous
this,
the
existence
from
and Prabhakar (1983),
selection
of
of
a correspondence
points
fixed
to
this
correspondence is obtained.
Theorem 3.7.
Let
x
[YanneLis and prabhakar, 19g3]
be a paracompact space and y
space. suppose that
f: X -------+-+ y
a linear
topological
is a nonempt-v convex valued
correspondence which has open inverse images. Then there exists a
continuous function
f :X -----+Y such that for all xeX
f(x) e f(x).
Corollary
Let
topological
3.1. [YannelLs and prabhakar, 1983]
X
be
a
compact convex nonempty subset of
space E and
suppose that
f :X---+--+X
a
linear
is a nonempty
\convex valued correspondence which has open inverse irnages.
Then f
has a fixed point.
These
results
will
be
extended to
abstract convexities in the following section.
B8
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
the
context
of
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3, Fixed
point
results.
FIXED POINT THEOREMS IN ABSTRACT CONVEXITIES.
In
the
context
of
fixed
point
theory
in
abstract
convexities we have to mention Horvath's work (1987, 1991), lvho
obtains
selection and fixed
Bielawski (1987), who
c-spaces;
context
selection
point
of
simplicial
results
in
convexities. Other
results
proves
to
correspondences in
similar
results
in
convexities
and Curtis
(1985) who
a
similar
that
context
works
which
are
to
also
of
the
obtains
simplicial
interestins
in
this
framework are those of Keimel and Wieczorek (1988) and Wieczorek
(1992) who use closed convexitv-
In this
the abstract
Section the
convexities
existence
of
fixed
points
in some of
mentioned in Chapter 2 is analyzed.
89
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Fixed
point
results.
3.1. Fixed point Theonems in c-spaces.
As we mentioned in
c-space
was
convexity
to
introduced
and to
obtain
conrespondences
following
lower
Horvath (1987, l99l)
by
some results
equivalent
proves
result
semicontinuous
the previous chapten, the
the
to
of
those
existence
of
selection
of
result
is
usual
generalize
and
Dresented in
the
convexity.
extends
the
of
point
and fixed
a continuous
correspondences
Theorem (1956)rl. This
to
notion
The
selection
to
Michael's
context
of
locally c-spaces.
Theorem 3.8.
Let
"-rp.""t'
vxeX f(x)
X
IHorvath,
be a
1991]
paracompact space, (y,F)
and I.:X
a
complete
locally
Y a lower semicontinuous map such that
is a nonempty closed F-set. Then there is a continuous
selection for I-.
The next result
covers the extension of Browder,s and
Fan's Theorems (Theonem 3.6.) to c-sDaces.
rr
Theorem [Michael,
1956]
Let X be a perfectly normal topological space, then every
l.s.c. correspondence from X in itself with closed convex and
nonempty images admits cont!.nuousselection.
t'
S"" Definition 2.28.
90
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Fixed
point
results.
Theorem 3.9.13
(X,F)
Let
I-: X -->--+
be
a
compact c-space
and
a
correspondence
the
existence
X such that
1. VxeX, f(x) is a nonempty F-set.
z. x = U {int r-l(y): yex}.
Then f has a fixed point.
The
following
continuous approximation
whose images are
result
to
F-sets.
states
of
a
upper semicontinuous correspondences
Therefore
it
is
used io
prove
the
extension of Kakutani's Theorem (Theorem 3.5.) to c-s-.aces.
Theorem 3.1O.
Let (X,F) be a compact locally c-space and a correspondence
I':X ------+>X such that
1. f is upper semicontinuous.
2. Vk e X
f(x)
is a nonemtpty compact F-set.
Then Ve > O there exists a continuous function f.:X -->X such.
that
Gr(f )
a
c
B(Gr(f), e).
Furthermore,- f has a fixed point.
^'This
13_.
result
is not stated exactly as Horvath (1991) did;
in fact derived from Horvath's result.
9l
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
it
is
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Fixed
point
nesults.
To prove this result we need the follolvins
lemma.
Lemma 3.1.
Let X be a metric compact topological space and f: X ----_¡+ X
an upper semicontinuous correspondence
Then vc > o there exist
x o , . . . , x n € X a n d p o s i t i v e r e a l n u m b e r s6 0 , . . . , ó ns u c h t h a t
{ r t*.,u.r oi:
is
an
open
partition
of
covering
of
X.
i=o, ,"}
Moreover,
thereexistsafinite
unrty tú,i,=o subondinate to
forallx€X
this
covering
such
that
3j e {O,...,n} such that
r' i/ . ( x ) > O
i m^p l i e s
f(x)s
i-
B ( fk . ) , e / 2 )
J
Proof.
Fix
every
e > O. Since f
x e X
z e B(x,6(x)),
there
is upper semicontinuous
6(x) > O
exists
such
we have that
that
for
for
an-v
\
it is verified
r(z) c e ( r(x),|)
(t)
a
6(x) < 9
Moreover, we can take
The family
space X. Let
a finite
partition
of
2
balls
ig(*,
a(*) /a))
*ex,
covers the compact
rÍr
{ B ( x , ó , z c )' i}=" o b e a f i n i t e s u b c o v eI rn'g a n o' 1r 'W
,),=o
i'
i
of unity
I- ' i r y ' . ( x )= t ,
subordinate
r y ' . ( x )> O ,
to this subcover
r / . ( x ) > O+ x € B & . , 6 . / q )
¡
r
92
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
r
l
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
point
Fixed
nesults.
J ( x ) = { i : r' iú ( x ) > O } . i t i s s a : i s f i e d t h a r
Let
x e B(x., 6.,2+)
g. =
If we take
máx
jt
vi e J(x)
n'e have Vi e J(x),
{6.: ieJ(x)},
x e B(x.,ó.,/+), thus
x.e B(x., 6./z),
IJIJJ
hence
B(x., 6./+) c B(x.. ó.),
Therefore by (l), for any i e J(x)
r ( x¡.t ¿) ( B ( r t x . l . * )
I
Proof of Theorem 3.10.
a. Existence of continuous approxímatícn.
By Lemma 3.1, there exists a covering {B(x.,6./+)}n_o and a
partition
finite
function
of
unity
V :X ------+ A
en
subordinate to
iúr),lo
it.A
continuous
can be defined b,.'
ilrr(x)=(ry'o(x),..
. ., r/n(x ) )
Moreover, for any i=0,1,..,n we cari choose y. e f(x.).
Hence, since (X,F) is a c-space and applying one of Horvath's
results
(1991), we
have that
for
an]- y0,yr,...,!'n € X
there
exists a continuous function defined on the n-dimensional simplexla
toA =Co{"^,...,€
nO'n
{eo,.. . , e,.,}
is
}
is
the
the
standar simplex of
cannonical
base
A -CO{e: ieJ}.
93
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
of
dimension N,
RN*t
and
f or
where
JgN
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
point
Fixed
results.
s:'
o
-aa ,
-----J \
such that if J g N={O,...,n}
P( A)e
F ( i-vi : i e J ) )
J
Since
V x
€ X,
' A nV- ^
C (Jx( x) ) ' €
A,
in particular
ü (x) e A
therefore
g(9^(x)) € g( 4...) s F(i_v.:ie
¿
J(XJ
r
Moreover since I'(x.) is an F-set
J(x)))
and X a locally c-space then
J
B(l'(x.), ¿/z) is an F-set, and appiying Lemma 3.1.
J
y. e I'(x.) c B(f(x.), e.zz) ¡,vhenever i e J(x)
ll
so it is verified
that
S(Vr(x)) e F({y. : i e J(x)}) c
Therefore if we denote
f,
B( f(x.), e.zz)
= go \lr, , rve have
fr(x) e B(f(x.), t/z)
so there exists y' e I'(x.) such that
J
'¿(r (x), y')
< t/z
a-
and by the proof of Lemma 3.1. x e B(x., 6.)
JJJ
with ó.< e/2 ,so
d((x, f^(x)), (x., y')) - d(x, x.) + d( f^(x) ,.v') < e
ajje
and we can conclude
(x, f
where
f^
is
a-a
a
continuous
a
(x)) e B(Gr(i-), e).
function
continuous
94
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
because
C
and
V-
are
also
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
Fixed
b. Extstence
Let
of
point
nesults.
points.
fíxed
{c.}, e.> O
V i e 0 , 1a d e c r e a s i n g s e q u e n c e o f r e a l
It
numbers
which converges to
O. Let
approximations to f
which are obtained b.r' reasoning as above for
every
{f. )
be ihe sequence of
continuous
c.. Then
( x , f . ( x ) ) e B ( G r í i - ) ,e . )
lt
If we take the following
V
it
is
continuous from
function
o og:' - A
---------+
A
¡nn
compact convex subset into
a
itself,
so
applying Brouwer's Theorem we can ensure the existence of a fixed
point 2..
I
32.
€A:
[iú- "91 2.)=2.
INI¡I
e ( ú¡ t.l( e ( 2 . )=) g Q . )
and if we denote yx = g(2.) we have that
II
g ( Ü . ( x T )=) x *
¡ll
Therefore
f(
iii
x* ) =
x*
that is,' i l f- has a fixed point x*. Hence
(xl , f. (xx)) e
Itlr
and from this sequence {xl}
B{Gr(f), e.)
and by considering that X is compact,
I
we
know
that
there
exists
a subsequence which
pornt x.
95
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
converges to
some
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Fixed
point
results,
Therefore
d((;, ;), Gr(t-)) = Lim d f.,*- , f. (x*)),Gr(r)l =
(iii)
Since Gr(f) is a closed set
(i, ;)
e
that is
xef(x)
96
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Grf
o
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter- 3.
Fixed
3.2. Fixed
point
point
nesults.
Theorems
in simplicial
Bieiawski (1987) introduces
proves
In
several
particular,
semicontinuous
generalizations
he
of
obtains
the
simpliciai
classical
a
correspondences.
convexities.
fixed
selection
This
resuit,
convexity
ooint
result
in
the
and
results
to
case
Iower
of
open
inverse image correspondences is as follows:
Theorem 3.11. IBtelawski,
1987]
Let 0(0) be a simplicial convexity on a topological space X.
Let
Y
be
a
paracompact
space
correspondence such that f(x)
and f-l(y)
and
let
I-: Y ------+> X
be
a
is a nonempt-v simplicial convex set
is an open set for each x € X. Then I' has a continuous
selection.
Originally,
since
he
convex
does
values
function
not
this
require
(which
prove
comespondences with
structure
is
obtained is in fact,
To
was
result
the
wrongly
stated
correspondence
needed
to
ensure
to
that
by
Bielawski
have
simplicial
the
continuous
a selection).
the
existence
of
a
fixed
point
to
open invense images, Bielawski considers a
less general
than
simplicial
91
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
convexity
which
he
calls
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
f initely
shows
locai
that
simplicial
¡'esults.
the
same
convexity.
will
.^^-'15
(Bielawski,
convexity
correspondence
will
point
Fixed
result
can
Firstly,
a
l9S7)-be
The
stated
continuous
be obtained and, a f t e r l v a r d s ,
in
next
the
theorem
context
of
selection
of
Brouwer's
Theorem
the
be applied to this selection.
Theorem 3.12.
Let C(O) be a simplicial convexity on a compact topologicai
space X and let I': X -----++X be a correspondence with open inverse
images and nonempty simplicial convex values.
Then f has a continuous selection and a fixed ooint.
Proof.
Since f-l(y)
is an open set for
each y€X and f(x)
+ a for
each xeX, then { f-t(y) }.rex is an open covering of X, which is a
compact set. So there exists a finite subcovering
-l
{ r-'(y.)}."^
tu
D"fi.,ition
1987]
[Bíelawiki,
A convexity
€' on a topological
space X is called f i.nttely
Local
if
there
exists
convexity
C(O) such that
a simplicial
0 c 0(é) and for each f inite subset {*,,*r,..,x,,}cX
there exists a
perfectly
property
normal
set
for compact maps such that
ó- .[-x- 1- , x- -_2 , .-., , x ] ( A ) c C -[ x , x
n--n_1.
having
C[xr,xr,..,xn1
-
L, 2,..'^nr
98
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
1 ' \'
the
r c . ^t -l ' ^ 2-'
f ixed
..,x ]
n
point
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
and
a
Fixed
point
results.
finite
continuous
ñ2.f iti^ñ
of
unitv
subordinate
to
this
subcovering,
)1
f r ¿i . ( x =
{ ',i/ . } . " ^ , r' i/ . ( x ) = 0 ,
i=0
We define the following
V : \-------+ A
function
r y ' . ( x ) > o +x € r r ( y . )
r
r
ú:
V ( x ) = ( r / ( x ) , r / l( x ) , . . . , r i ' ( x ) )
nOln
if we take J(x)={ i:
r y ' . ( x )> O } , t h e n w ' e h a v e
I
y ef(x)
(t)
VieJ(x)
I
On
structure,
the
lve
other
hand,
can
define
since
a
X
has
continuous
a
simplicial
function
convexity
f:X ---+X
as
follows:
f(x) = 0[yo
. . , y . l ( r y ' o ( x ) ., . . , ú " ( x ) )
And applying that f(x) is a simplicial convex set anci by (l)
for every
i e J(x),
y. e l(x)
we have that f is a continuous selection of f, that is
f(x) e I'(x)
This function f is a composition of the following functions:
V : X€
<D: A,", -+
4.,,
X
So if lve take g = ü o é, it is a continuous function defined
from
a compact convex set (An) into
itself.
Therefore, we can
apply Brouwer's Theorem and conclude that g has a fixed point.
f x
€A
onoo
g(x)=x
:
99
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Fixed
point
results.
Hence
0( v(o(x)))
=é(x)
oo
and if we denote x* = O( x ) we have
o
f(x*)=x*
that is, f
has a f ixed point which is also a f ixed point to the
correspondence f.
100
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
point
Fixed
results.
Fixed point Theorems in K-qonvex continuous structures.
3.3.
Although the results obtained in the context of K-convex
continuous
structures
are
immediate
consequences of
the
ones
obtained in the context of mc-spaces (which u'ill be presented in
the next
proving
section), we will
constructive
is
selection
as
constructed.
paths
well
In
which
as
present them noiv since the way of
and it
the
is
to
approximat.ion to
both cases, the
vary
interesting
correspondences is
basic idea is
continuously
and
that
show how the
the
the
existence of
functions
are
constructed by means of the composition of these paths.
Theorem 3.13.
Let
X
be
a
continuous structure
compact topoiogical
and let f:
valued correspondence with
space with
a
K-convex
X -------+> X be a nonempty K-convex
inverse open images. Then there exists
a continuous selection of I' and I has a fixed point.
Proof.
As
in
Yo,! ,,Y,,.. ,! n
Theorem. 3.12.,
and
subordinate to if-r(y.¡¡."0
y.- e f(x)
a
we
f inite
can
ensure
Partition
that
or
such that
Vi e J(x) = { i: rl.(x) > 0 )
tol
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
there
'
untry
-
exist
¡ 'ñ
\w.¡.
ii=O
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
point
Fixed
results.
Moreover, by considering that
ensure
f (x),
that
the
assuming
for
path
the
f(x)
y. e f(x)
every
joins
which
existence
of
is a K-convex set,
, i e J(x)
them
will
these
arcs,
and
be
an-v point
contained
the
we can
p
of
f (x).
So,
of
the
def ine
the
in
construct,ion
selection by composing them is presented now'.
a. Construction
From
the
of the continuous
<el eafi
partition
finite
an
f
unity,
of
we
can
following family of functions,
to
_j
if
I
t.(x)=
,¿.t"1
Il r
if
ln
If
ú .t ( x ) * O
Ir/.(x)
I
l..J
ú.(x) = O
J=r
we take hr,_l= yn , then both hn_, and yn._, b"long to I-(x)
lwhenever
ú,.,(x)>0,
K(hr,_r,yr,_r,[O,l])joining
úr,_r(x)>O
therefore
),
these points (
which we
the
call
path
gr,_ )
r
is
contained in f(x) since it is a K-convex set.
If we compute gn_r in t,r_r(x), we will have
h.,-z = g,.r-l(tn-l(x))= K( hn-r' -vn-r' 1,.,-r(x))
and by construction h.,-z e I'(x). By reasoning in this wáy but with
the path which joins h.,_z .nd y.,_, (which we will
call
computing it in t,.,_r(x), we obtain,
h.,-3 = 9,.,-, (t,.,-r(*)) = K( hn-z' !n-z'
102
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
tr'--tx))
En_z) and
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 3.
Fixed
point
nesults.
In the same way we repeat this
element
Finally
h^ rvhich will
o
this
and we will
be
linked
continuous function
to
will
reasoning untii
'y0
b -v
\,ve obtain the
means of
be computed in
a
path
? o.'
a
to(x) = ry'o(x)
obtain the selection imase in x- that is
so( r/o(x))= K( ho, yo, to(x)) = f(x)
f(x)=K[ -
{-[*,r,,,r,.,_,,..,_,(*)),],.,_2,."_r,*,J,-u^_.,t"_.{x)}...]
\
By
the
way
of
defining
f(x) e I'(x),
Vx e X, since the
f(x).
that
Note
if
f
it
is
"relevant"
ú.(x) = O for
any
proved immediately
paths are
i,
then
that
contained in
we
have that
t.(x) = O, so
K( h., y., t.(x)) = K( h., y., O) = h.
and y. will
not appear in the construction of function f.
Hence to
I
construct
selection f we only need points y. which have ry',+¿.
lo3
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Fixed
point
results,
b . Conttnuíty of selectíon f.
Selection f can be rewritten
d>
the following composition
( ü(x) ))
f(x)=f(
where
R: [o,t]" --------+\
( r
f
\
K ( to^ , . . , 1 ) = K l . . x l f l v , v . t l .-vr , - z ' t
n-l
n-r.¡
L
iI¡ : X----)
t
,]
1.-n-n-l
ú(x) = (ry'o(x),ry',(x),...,ún(*))
A:
n
f:A-----+Rn:
n
T.(z)=
I
z =O
if
{i
I
J
j=i
9(V(x ) ) = ( t ( x ) , t ( x ) . . . . . t
ol
f(x)=I(tgt
((
L
l. [n
n-l
I
\ I=,
f
i-O
z *O
if
(x))
n-l
))
ú ( x ) ) ) = K ( t ( x ) . t ( x ) , . . . , 1 . , _ r ( x=
or
= K l . . K l K l y , y . , t ( x ))L v , t ( * )\ \ ' l
l , . .l , I ^ , t ^ ( x ) l
n-l' n-l
n-2' n-2'
In order to prove the continuity
x, firstly
be
of f =
we are going to prove that
continuous function.
would
)))
)''
If
inmediately
this
is
obtained
_.
true
(since
continuous functions : R."g and ü).
lo4
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
K ( f
( ú)) at any point
K " g , A ----+ X
n
then
f
the
is
a
¡antinrri+"
uvrrL¡r¡urLJ
is a
^{.
9l
composition
f'
I
of
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
point
Fixed
results.
To analyze the continuity
it is important
. J
[
of function
at anv point zeAn
if z >O then
to note that
n
z
T.(z)=
i
tn
fz
,:,
is
a
continuous function,
j
is
since it
quotient of
a
continuous
function whose denominator is not null.
In
other
denominator
case, I.Q)
is
zero,
could not
that
is
be
when
2,.
continuous (when
zero
are
for
its
all
k = i, ... ,n-l).
In the f irst
in not problem: since I( .9
case, the continuity
is a composition of continuous functions and therefore continuous.
In the second case, we define
j=max{i:2.>o}
then
z j * l ' n= O , . . . , 2 = O
hence .
O , .. . ,
T. .tz)-
T (z) = o
and
T.Q) = |
J+INJ
Tlz)=
'i
because
j
z+
+...+z
z
jj*lnj
Futhermore
because their
f
(a=0,...,j)
are
=l
z
continuous functions
at
z
a
denominators are non nuls, Q.>O and z.> O V k*i,
njk
therefore I tu , O, Vk=O,...,j)
K=a
ro5
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
Fixed
By the
point'
way
results.
in
rvhich function
K
has been def ined. it
is
verified that
K ( 7 0 ( z ) ,. . , J . Q ) , . . , 9 , . , , _ , () z= )
K { f o { z ) , . . ,1 , O , . . , O ) =
'1
-* .1r. _
. *.[(*_f .t (" , t " _ ), , o^)_, tr ,) o ) , . . tr),,J, , .). . J , r o , r o r z t l
and since K(a,b,l) = b,
V a e X, then the part of the function
*f..*|'*|.r-,r
= -yj
y.,il
- n -,,rr ' n -,{z)'l
'i
^,r -(zll,
r ) ',y,
l. [-n
L
'
)
)
"-2' "-z'
and it is independentof the values of 7,.,-,(z), ln_re),
..., f
r*re)
that is,
* [ . . * [ * [ r n , r . , - , , ^ , , - , ) , y n - r , t r n - r )r, r , r ) = y
j
V t r r , _ t.,. . , ^ r * , e I o , t l
Sot
To
simplify, we
call
T=(fo(z),..,I) and
t h u s r { Í o { = ) , . . , 1 , ^ r * r , . . , } r , _=r ) r ( T , l )
ro6
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
^ =,^r*,,...,),.,_,),
VreIo,l]t (m=n-j-t).
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Fixed
point
results.
In order to show that
function
Kof
is continuous we are
going to prove that
V z € A , r ,V W e N ( K " g ( z ) ) , f V ' e
By applying that X.g(=)=f (t,f )
K"f(V') c !V
N(z) :
V¡,elO,tlm and that
K is a
continuous function. we have that
v w € N (R (T ,I)),' T A T3AV 1
"
Mo¡'eover, since the
V- e N( ( T.) ,) ) : r f i' l ., v, ) s w
family
of
tr e [O,l]m is a covering of [O,l]t,
neighborhoods
when
which is a compact subset, we
recovering .,shich will
known that there exists a finite
V¡,
( l)
be denoted
as follows
¡^
-rITl
{V^. : i=1,...,p}
lo,ll"'=,
Hence, if
we
take
ul
t
,
Vi=I,. .,p,
and
we
)¡
Vr.= ^ {Vi': Vi=I,..,p},
then V, is a neighborhood of T. But by considering that
T = ( f o ( z ) , . . , f r _ r ( z t)),
we can rewrite
vtri = vtrtt ...t vlt
TTOTjTkk
hence
V
-V
x...xV
TTOTJ
where
vtrt
<
N(7 (z)).
where
V r u =^ , u l | , i = 1 , . . , pk) = o , . . . , i .
l07
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
consider
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3,
Fixed
H e n c e-,
defined
as
Moreover,
V_
Tk
a
is
results.
a
finite
these
there exists
point
neighborhood
of
i- k - 2- ')
intersection
of
neishborhoods
functions
!,.
are
neighborhoods U. of
k
continuous
since
at
z
it
has
of
been
!
(d.
k
Vk=O,.,.,j,
so,
zsuch thar
Tk(Uk) . Vrn
Finally, on the one hand, if we denote
V'=n{Uu:k=o,...,j}
then V' is a neighborhood of z, and it is verified
VweV', (90(w),..,9.(w)) .
On the
is verified
other
hand,
for
that
Vro x...x Vr. = Vr c Vlt
the
rest
of
indexes
Vi=I,..,p
(k=j+l,...,¡r¡
it
that
( 9 . , ( w ) , . . . , T _ . ( w ) ) e I o , l ] ' = U { V - . .: i = 1 , . . . , p }
j+l
n-l
Ai
so there exists an index i
o
such that
\
( 9 r * r ( * ) , . . . , f , . , - r ( w ) ). u ^ , o , i o e { 1 , . . . , p }
Thus we can ensure that
( 7 ^ ( w ) , . . , f. ( w ) , 7 . . ( w ) , . . , 7 - ( w ) ) e V - x V . - . V l t o
u
n-r
T
Aio
,
J
J+r
and since we had obtained, (1), that rcfV| ^ VI) s W
can conclude that for any w € V' it is verified that
t < ( g ^ t * ) , . . , 7 . ( w ) , 7-.( w ) , . . , 9 . ( w ) ) < W
j
j+t
u
n-l
108
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
t
ulto
V).elo,tlm, we
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
Fixed
point
results
c. Ftxed point exístence.
Selection f can be written as the following composition
f(x) = X f g ( ú(x) )) = é ( v(x) )
where é=K-"f
ú ( x ) = ( r / ^ ( x ) , r(/x ) , . . . , ú ( x ) )
V : X-----+ A :
nOIn
Consider
0
are
compact
continuous,
set
g = ü " é: An --------) A,.,. Since V and
now the function
into
it
is
itself,
a
continuous
so Brouwer's
function
Theorem
from
can
a
convex
be applied
and
we have
3x
Therefore,
eA:
g(x)=x
onoo
=
O( g(xo).)
Q(xo) and, thus
f(O((xo)) =
O(xo),
xx = é(x ) we have obtained that
so if we call
o
f(x*) = x*,
that
is,
f
has
a f ixed
point
which
is
also
a f ixed
point
to
the
conrespondence.
Note that
selection f
in the last theorem, the way of defining the
could have been done in the
paracompact space.
lo9
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
case of
considerins a
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
The
next
approximation
nonempty
to
the
3.13.
results.
result
an
K-convex
constructing
Theorem
point
Fixed
shows
upper
values.
approximation
stated in locally
the
K-convex
existence
semicontinuous
compact
although
the
In
context
is
continuous
a
correspondence
this
similar
is
of
case,
the
way
of
the
one
used
in
to
generai
less
with
since
it
is
continuous structures.
Theorem 3.14.
Let X be a compact locally K-convex space , and let
f: X-_>
x
be an upper semicontinuous correspondence with nonempty K-convex
compact values. Then
l.
V e > O, f f
.
2.
a
: X-+
c
Gr(fr)
X
continuous such that
B(Gr(i-), e)
I has a fixed point.
Proof.
By applying Lemma 3.1., we have that
there
exists
a
finite
partition
of
for
unity
every fixed
subordinate
{B(x., ó,,/+)} and
3j e J(x):
Vi € J(x),
x e B(x., é../z)
r(x.)c B ( r(x.) ,|)
If
we take y, e f(x.)
y. e f(x.)
for
c B( f(x.),
any i = O,...,n, then
e/z)
uo
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
whenever
i e J(x)
e>0,
to
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Fixed
point
results.
a. Construct¿on of the approxtmatíon f .
Since X is a K-convex continuous srrucrure and I-(x) is a
K-convex set, we have that Vc > 0, B(f(x).e) is also a K-convex
set.
Then for
any pair
joins
K(a,b,[0,1])
them
points a,b
of
and
is
in
B(f(x), e)
contained
in
that
the
path
ball.
By
neasonÍng in the same way as in Theorem 3.I3. we can define the
following function
.
r
((
L
t
\
) I
f ^ ( x ) = K l . . K l K l K-(ny-,-yn -. 1, t' n -. l( x' )' ')n, -y2^' ,nt- z^ ( x ) 1 , _ ^v , t ^ ( x' ) 1 . . . 1
e
[
J'-n-3'n-3
I
It is easy to verify that f"(x) e B(f(x-),e/z), Vx e X, since
all of the paths are contained in B(f (x.),e,/z). So,
3y' e f(x.) tal que d(fr(x), y') < e/z
thus
d((x, f.(x)), (x., y')) = d(x, x.) + d( fr(x) , -y')< e
and therefore
The
(x, f"(x)) e B(Gr(f), s).
continuity
of
the
approximation
is
proved
as
in
Theorem 3.13.
b. Fixed
Let
poínt
{e.},
exLstence.
a.>
O
Vi
e
D',1,be a
decreasing
sequence of
real
II
numbers which
converges to
O. Let
{f. }
be
the
corresponding
sequence of approximations to f which are obtained by reasoning as
above for every e.. Then
Iil
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Fixed
point
results.
( x , f . ( x ) ) e B ( G r ( f) , e . )
tl
From the last
theorem we knorv that
f,
has a fixed
point
f.(x*) = x*
Since
(x* , f (x*)) e B(Gr (l-),e )
iiii
from
this
set, there
sequence {xi},
and
by
considering
that
x
is
a compact
exists a subsequence which converges to some point i.
Hence,
d ( ( i , ; ) , G n ( f ) ) = L i m d f . , " : , f . ( x * ) 1 ,c r ( r ) l = o
ti¡i)
Since Gr(l') is a closed set, then
(i, il
e
that is
xef(x)
t12
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Gri'
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
Fixed
3.4. Fixed
point
The
point
results.
theorems
in mc-spaces.
following
points and selection of
presents the
result
correspondences with
existence
of
f ixed
open inverse images
in the context of mc-spaces.
Theorem 3.15.
Let X be a compact topological mc-space and I-: X ------>> X a
correspondence with
values. Then f
open inverse
images and
nonempry mc-ser
has a continuous selection and a flixed point.
Proof.
In the same way as in Theorem 3.12. it is possible to ensure
that
there exists
a finite
partition
of
unity subordinate to
this
covering
{ r - r (- iy.)i =} O."Let J(x) be the following set
J(x)={i:r/.(x)>O}
so
If
we take
if
i e J(x)
y. e I-(x).
then
A = {yo,yr,...,Irr}; since X is an mc-space, there
éxist functions
PA:XxlO,
ll----+X
I
such that
Pf(x,o) =x
and
PA(x,l)=b.
¡ll
lt3
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 3.
Fixed
point
t n such a way that
nesults-
Go:[O,ll" -----+ X defined
a5
.\
GA(to,
... , t.,_r,=.:I eo_r[e^
, r ) . tn - l j l . t
,fnorv
n
t>
.,.J
"_,J,.
a continuous function.
Constructton of
the seLection.
From the partition of unity, we can define
if
ry'.(x)= O
I
t(x
O,1,...,n-l
i
if
ty'.(x) ;t O
I
so, functi
as follows
= G (t (x), ...
t
Aon-l
(x)) =
(t|I
I
l\ ne ^ ( yn ^ , t )n,-tr - , ( lx, )tn_- .z( x )1 . . . . ,ot ^ ( |x )
)
)
) = O, and ú,r_r(x) > 0 ,
Note
I
then
t,.,_r(x) = l,
therefore
tl _ r(b",."_r ( x)=) r l_,( 0.,1)= bn_,
that is, b,, is not in the path defined by G¡. 81' applying the same
reasoning repeatedly, if
ú',(xo) = ú,r-r(xo) = o
n4
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
and
ú,.,-2(*o)> o
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
Fixed
point
results.
t,.,_r("o) -
then we have that
p^ ^(pA (b ,t
n-2
n-l'
Therefore
to
t.,_z(xo) = l, so
O y
( x- )- ") , t- n - Z(' -x-l") = p
- 'r (b
- n '.-i' ) = b
-
n' n-l-
n-Z'
construct
the
approximation
f
points b. such that y. e I'(x). Ivforeover since f(x)
IT
the
mc-convex hull
of
y.ef(x)) and since f(x)
the points yi
such that
n-2
rve only
need
is contained in
ieJ(x) (that
is,
is an mc-set, w-e obtain that i-(x) contains
that mc-convex hull. In particular we have
f(x) e'f(x)
The wa-v of proving the continuity of the selection is similar
to that of the K-convex case (Theorem 3.13.).
b. Ftxed point existence.
In the same way as in the last theorems, function f
can be
written as the following composition
9:X -----+A,r,
f:An ---+ [o, t]n
Go,[O, l]n --+X
\
Let g be the following function
8=Ú"Go"7
It
is
a continuous function
convex set into itself.
and is defined from
a comDact
By applying Brouwer's Theorem we have that
there exists a point xo such that
-so( xo ) = x
115
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
Fixed
3.
po¡nt
So, if we call
results.
x* = [Go o T](g(xo)) =
f(xi)
Therefore
f
has a fixed
(7(x ))
AO
we have
= x*
point
w'hich is a l s o a f i x e d
point
to
the correspondence.
As an immediate consequenceof this theorem, some of the
results previously presented can now be obtained. That is the case
of Fan's and Browden's results (Theorem 3.6.) or Horvath's result
(Theorem 3.9. )
in the
context
of
c-spaces. rv'foreover, and since
every K-convex structure defines an mc-structure. Theorem 3.13. is
also obtained
as a consequence of
it
as '"vell as yannelis
and
Prabhakar's results (1983).
The
Kakutani's
proves
the
next
Theorem to
existence
result
corresponds
the
of
context
a
of
to
the
extension
mc-spaces. That
continuous approximation
is,
to
semicontinuous correspondences and the existence of a fixed
In the same line as last approximation theorems, it
the context of locally mc-spaces.
116
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
of
it
upper
point.
is stated in
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
Theorem
point
results.
3.16.
Let
upper
Fixed
X
be
a
compact
semicontinuous
values.
locally
mc-space
correspondence
with
and
I-: X
----> X
nonempt-v compact
an
mc-set
Then
l.
V e > O, 3f
e
Gr(f
2.
: X-+
,)
X
c
continuous such that
B ( G r ( f) , e )
f has a fixed point.
Proof.
a. Construction of the approxtmation.
From Lemma 3.1. for every e>O there exists a finite partition
of
unity
subordinate
i,/.i.n^
¡ l=u
to
the
covering
{B(x.,6..2+)}n^,
I
I
l=u
verifying that if
>O},
J(x)={i:ú.(x)
then
f(x.) ce
ijz
( r(x.),i)
with
where j is in such a way that... U, =
ieJ(x)
máx {ó.: ieJ(x)}, 6.<e/2).
If we choose y. e f(x-), for any i=O,..,n, and we consider
tl
A = { yo, yr, ... , y.,}
we have
y. e i'(x.)c B( f(x.), e./z)
i e J(x)
llJ
and we can define f - in the following
a
fr(x)
way
= Go(to(x), t,(x), ... , tn_,(x))
r r ?I
LL
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 3.
rvhere G
Fixed
is
mc-structure
of unity
point
th e
nesults.
continuous function
def ined
b_v means
on X, and t. are functions d e f i n e d f r o m
of
the
the partition
V i=O,1 , . . . , n - i
T
loif
I
--l
(x)=
t
ú.(x) =
I
u'txr
I
r-
ry'.(x)+ 0
;f
ln
It - i I ú
I
(x)
J=r
So,
if and only if r/.(x) - O. Furthemore if
t.(x) = O
I
ú.(x) > o, then
I
y. e f(x.)c B( f(x.), s,/z)
t¡J
and since f (x.)
is
an mc-set,
then B(l'(x.), x / z )
)
is
a l s o an
mc-set because of its being in a locally mc-space. Hence
f
a
(x) = G (t (x),t (x),...,t
Therefore,
there
from
d(x,x.)
the
< ó. < e/2,
proof
J
y' e f ( x . )
)
exists
dtr
and
( x ) ) e B ( l - ( x. ) , x / z )
AOln-l
of
^.,^l>uLtt
that
< t/z
,(x),I')
Lemma
3.1.,
it
is
verif ied
that
then
JJ
d((x, f (x)), (x, y')) = d(x , *,) * d( fr(x) , y') < e
ei'that is
( x , f r ( x ) ) e B ( G r ( f ) , e ) , and
f (x)
is
an aporoximation
to the correspondence.
The continuity of function
can be proved
as Theorem 3.13.
118
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
in a s i m i l a r
way
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 3.
b. Fixed
point
Fixed
potnt
results.
exístence.
point
The fixed
existence can be proved in the
Theorem 3.1-1. by substituting
function
K fo.
function
same way
G
as
.A
I
Some of
stated
results
now as immediate
case of
and
the
Kakutani's
last
results
(Theorem 3.5.),
obtained
in
the
as weil
result
the
well
is the
as fixed
context
(Theorem 3.lO) and K-convex continuous structures
Next,
sect.ions can be
consequences of Theorem 3. 16. That
Theorem
approximation
present.ed in the
of
point
c-spaces
(Theorem 3.13.).
known Knaster-Kuratowski-Ifazurkiewicz
(KKM) (which is equivalent in the usual convexity to
the
existence of fixed point) is presented.
Theorem 3.17. [Knaster-Kuratowskt-Mazurkiewícz,
see Border, 1985]
^m,_m
Let X c R-", if we associate a ciosed set F(x)cR"' for
xeX which verifies the following conditions,
] . F o r a n' yl n f i n i t e s e t { x , . . . , X } c X w e h a v e
t\
c l { x - , . . . ,nx}) l c U F ( x . )
i
I
2. Flx)
Then
is compact for
j=r
some xeX.
¡ F(x) is nonempty and compact.
x€X
It9
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
every
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
F¡xed
poinr
results.
We prove
this result
in the context
of rnc_spaces
Itherefore it
wirl arso be
true in all
of the structures
particular cases
which are
of them)' To
prove the result
we need the
of generalized
notion
KKM correspondence
which is defined
as follows.
Definition
3.16.
Let X be an
rnc_space and
B a subset of
X. A correspondence
f: B------_____+>
X is a genera¿í.zed
KKIvI correspond.ence
verified that
if
it
is
for every fjnite
subset { * , , * r , . . , * n } .
B
tn"({*r'*r,",xn})
where c-.
n
c Ur(xl
i =¡
i. the mc-convex
hull.
i
Theo¡em 3.18.
LetXbea
compact mc_space
and B c X. If
l-: B _______>
genera.lized
X is a
KKM correspondence
wit)r nonempty
compact values,
then
fi{f(a):aeB}*a.
Proof.
By contradiction,
assume that
X=X\
nt(.)=
a€B
U (x r rra)l
a€B
So, X has aa
open covering
and since it
is a co¡npact
can obtain
set we
a finite subcovs¡i¡g.
So, there exists
a fjnjte famill
{ar'ar,",a,.r} such
that
120
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
point
Fixed
results.
X = U (X \ f(a.))
I
i=l
Since {X\f(a.)}.".
i
there
exists
a
i=l
is a finite subcovering of X, we know that
partition
f inite
of
unity
subordinate
{,/ . }.".
I
to
l=r
it.
Thus,
(x) > O if and only if x e X\l-(a. ).
V¡.
ll
is a KKM correspondence, in particular
But since f
({a.}) s f(a.)
C
Vi=I,..,n.
mcll
So, if
we take A = {ar,ar,..,a.,}, since X
is an mc-space,
there exists a continuous function
G : [O,1]" -----------+
X
A
From the way of defining G^ and the partition of unity,
G (t (x),...,t
(x)) e C
AOn-lmci
(l)
( a: ieJ(x) )
are defined in the same way as in Theorem 3.13.
where functions t
I
On
the
other
hand,
since
f
is
a
generalized
correspondence, we have
C
mc
(a:ieJ(x))s
U
t
'r€J(x)
I'(a)
i
Moreover, the following composition
f
'
=G
o9o!ú
A
üÍG
X#
A.-r
----------+
[O,tl"
r2l
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
--j-+
X
KKM
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
3.
Fixed
I
nesults.
(since Goog is continuoustb as well
is continuous
consider
point
= G
A
o T,
continuous one and it
itself.
Therefore,
¡1, o .q: A
function
is defined from
n-l
as ú).
-------------) A
n-l
So if
is also
vve
a
a compacr convex set into
we can apply Brourver's Theonem and conclude
that there exists a fixed ooint
Y€Ar,_rrVlg(f')l
=I
and if we call x* - g(y) we obtain
f(x*) = x*
By (l) we have
xx=G^[f(V(x*)]
eC
A
(a.:ieJ(x*)) c
mc
U
i
f(a.)
i-
i€J(x*)
Furthermore, if ieJ(x*) then x* e X\l-(a.), hence
xr É
,.?,**rx\f(a')
= X\,.Y,**) f(ai)
which is a contradiction.
The last result can be extended by considering weaker
continuity
considered
conditions,
by
in
Tarafdar
particular
(1991, lggZ\.
conditions
The
similar
to
generaiization
those
is
as
follows,
tuBy
reasoning as in the proof of Theorem 3.i3. but replacing K
f o r G . , i t w o u l d b e o b t a i n e dt h a t G . 9
i s a c o n t i n u o u sf u n c t i t _ ¡ n .
AA
r22
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Volver al índice/Tornar a l'índex
Chapter 3.
Theorem
point
Fixed
results.
3.19.
X a generalized
Let X be a compact mc-space and i-: X ---------+->
the set \ \ I-(x) contains an
KKM correspondencesuch that Vx e X
open s u b s e t O w h i c h v e r i f i e s U O
x
x€X
Then
=f,.
x
n I' ( x) * a.
x€X
Proof By contradiction, if we assume that the result fails we have
X=X\¡f(x)=
U(xrr(x))
x€X
x€X
On the other hand, since X =
and X is a compact set,
U O*
x€X
exists
there
subordinate
a
finite
subcovering
partition
finite
a
of
unity
to it,
f,=
ü
i=o
o
{ , /I . l}=:u^
*i
such that if r/.(x) > O then x e O
c X \ f(x.).
IX,'
If
and
we take A = {xo,xr,...,xn}, since X is an mc-space we
can ensure that there exists a continuous function
Ga, [O,t]" -->
that
such
a
parallel
way
X
reasoning to
of
the
one
obtain the existence of
Theorem 3.15 can be applied to
used
in
a fixed
point. From this point, and by reasoning now as in the last part
of
the
n
llx)
previous
Theorem,
we
obtain
= a.
x€X
r23
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
a
contradiction
with
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
CHAPTER
4
APPLICATIONS
OFFIXEDPOINTTHEOREMS.
4.O. INTRODUCTION.
4.1. EXISTENCE OF MAXIMAL ELEMENTS IN BINARY
RELATIONS IN ABSTRACT CONVEXITIES\
4.2. EXISTENCE OF EQUILIBRIUM IN ABSTRACT
ECONOMIES.
4.3. EXISTENCE OF NASH'S EQUILIBRIUM.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
4.O.
Applications.
INTRODUCTION.
In this chapter, applications to different
economic analysis
of
selection,
approximation
topics in the
and
point
fixed
results obtained in the last chapter are presented.
Firstly,
some generalizations
of
in the area of the existence of maximal
the
classical
results
elements are presented.
the theorems in this
These generalizations extend many of
area
including those based on convexity conditions such as Fan (1961),
Sonnenschein (1971), Borglin
which
Keiding
acyclic
consider
$976),
binary
and
Yannelis
(1993), etc.
(1985), Tian
Prabhakan (1983), Border
results
and
as
relations
well
as
such
as
Bergstrom (1975), Walker (1977), etc.
for
In Section 2, the existence of equilibrium
economies with
non-convex
general
assumptions is
and non-compact
(K-convex sets, mc-sets...)
agents without
infinite
proved:
in
contexts
dimensional strategy
where a countable infinite
convex preferences
abstract
with
spaces
number of
is considered. Thus our result
generalizes many of the theorems . on the existence of equilibrium
in abstract economies, including those of Arrow and Debreu (1954),
Shafer and Sonnenschein(1975), Border (1985), Tulcea (1988), etc.
r25
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
Finally,
non-cooperative
obtained.
considered,
result
In
in
games
this
with
case,
a
(1951) and the
Section
in
the
context
infinite
non-f inite
theorems
3,
Nash
of
abstract
dimensional
quantity
of
of
agents,
126
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
in
convexities
strategy
Martinez-Legaz
are generalized.
equilibria
spaces
thus,
and Marchi
is
are
Nash's
(1991)
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4,
4.1.
Applications.
EXISTENCE OF MAXIMAL ELEMENTS IN BINARY RELATIONS
ON SPACES WITH AN ABSTRACT CONVEXITY
The notion of preference relation
a
fundamental
theory.
concept
Then, when a
particular
economics, in
in
consumer is
or utility
with
faced
the
choosing a bundle of products, in the end, he will
function
in
is
consumer
problem of
look for
the
bundle which maximizes his oreference relation from those which he
can afford.
The problem of
looking for
sufficient
conditions
which
ensure the existence of maximal elements of a binary relation
been studied
Walker, 1977;
or
Continuity
by
several
(Fan, 1961; Sonnenschein, 1971;
authors
Yannelis y Pnabhakar, 1983;
convexity
has
conditions
of
the
Tian, 1993;
set
of
more
etc.).
(less)
prefered elements
U-r(x) - {yeXl xpy}
U(x) = (yeXl ypx)
or
transitivity
required.
conditions
Some of
unrealistic,
of
the
preference
relation
them, have been criticized
especialli
the
transitivity
Starr,1969).
r27
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
are
usually
as being strongly
condition
(Luce,
i956;
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
purpose
The
(especially
problem
the
of
transitivity
existence
independent
condition
ways,
and
conditions
of
on
the
the
one
other
the
elements
hand,
hand
sets U(x).
If
condition
transitivity
acyclic
binary
considered
that
binary
subset
a
has
a
relations.
relation
maximal
is
In
is
In
results
maximal
elements as in Walker's or Bersgtrom's
Theorem 4.1.
give
[Bergstrom,
us
sufficient
different
the
the
and
transitivity
convexity
relaxed,
it
could
be
it
must
be
this
several
that
two
carried
considering
sense,
acyclic
element.
condition
has
relaxing
by
on the upper contour
the
transitivity
indiference)
maximal
the
considered
finite
of
of
on
avoiding
if
this
and
only
area,
conditions
if
any
there
are
to
obtain
results,
7975; Walker, 1g77]
Let X be a topological space, and let p be a binary relation
on X, such that it verifies:
-t
^(x)
l.
U
2.
P is an acyclic binary relation.
are open sets V x e X.
Then every compact subset of X has a p-maximal
r28
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
element.
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
In the second approach the results are mainiy based on
convexity conditions on the
set
and on the
upper contour
sets
(u(x)). To obtain the existence of maximal elements in this case,
most of
results apply classical results
Browder, ... )
or
of f ixed points (Brouwer,
nonempty intersection
results
(KKM), therefore
convexity conditions are required on the mapping and on the set
where it
is defined. In this context, an element x*
element for
a
binary
relation
P
if
the
following
is a maximal
condition
is
verif ied
U(x*) = a
In the approach based on convexity
conditions,
a first
result is Fan's theorem (196I)
Theorem 4.2. [Fan, 7967;see Border, 1985]
Let X be a compact convex subset of R" and let p be a binary
relation defined on X, such that:
1. Gr(U) = {(x,y) | yPx} is an open set.
2. VxeX,
x É U(x)
and
U(x) is a conve.x set.
Then the set of maximal elements, {x*: U(x*)=o}, is nonempty
and compact.
r29
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
There
result
as
are
some situations
lexicographic
which
(Gr(u)
order
presents
a
conditions
weaker than those of Fan.
Theorem
Let
relation
first
4.3.
extension
[Sonnenschetn,
X be a compact
of
is
Fan's
are not
not
covered
open).
result,
by this
Sonnenschein
by
considering
1971]
convex
subset of R' and let
p be a binarv
defined on X, such that:
I. Vx e X
x e C ( { x o , . . . , x 0 } )V, x . e U ( x ) ,V i = l , . . , p , p s o + I .
y e U-l{x),
2. If
then there exists some x, € X such that
y e int U-l(x').
Then the set of maximal elements, {x*: U(x*)=a}, is nonempty
and compact.
In
spaces
this
and
a
line,
but
similar
considering
continuity
infinite
condition,
dimensional
yannelis
and
Prabhakar (1983) prove the existence of maximal elements.
Theorem
Let
space
4.4.
X
and
[YanneLís
be
a
let
and
compact
U: X--->>
Prabhakar,
convex
X
7983]
sübset
be
a
of
a
linear
topological
correspondence
which
verifies:
i. VxeX
2. VxeX
x É C({x^,...,x_}), - x.eU(x), pe[rl,isp, pe0.l.
0pi
U
-t
^(x)
is an open set in X.
Then the set of maximal elernents, {x*: U(x*)=a}, is nonempty.
130
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
Next, an existence result of maximal elements in binary
relations,
which constitutes the
union point
of
the
two
focuses
commented previously is presented. So, this result generalizes the
previous one which considers acyclic binary relations, as well ás,
those which consider usual convexitv conditions.
This
general
Sonnenschein's but
by
result
is
stated
in
a
similar
way
to
considering mc-spaces and mc-sets rather
than usual convex sets.
The method used to prove this result is based on a fixed
point result obtained in Chapter 3. In this way, it is remarked on
point technique includes these two different
as the fixed
analyzing the proble-r?
The
ways of
of the existence of maximal elements.
continuity
condition
which
considered is
is
that
rused by Sonnenschein, but it could be argued in an analogous way,
by considering Tarafdar's
condition
'J9gZ)18(which
in this context
is equivalent) and obtains the same conclusion.
L'7
In this line, Tian (1993) presents a result which tonsiders the
case of acyclic relations,
but only in the context of convex sets
is also a consequence
and topological
vector spaces. This result
of that which will be presented as follows.
18
vxeX,
u-l(x)
conditionthat
vO
x€X
contains
an
oDen subset O
x
-X.
x
131
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
which fulfills
the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 4.
Applications.
Theorem 4.5.
Let x
be a compact mc-set and let p be a binary relation
defined on X, such that:
1. VxeX
and
verified
V AcX,
finite,
AnU(x) + o
it
is
x É Golut*,([O,l]-)
y e U-l{x),
2. If
A
then there exists some x. e X such
thatyeintU-l(x,).
Then the set of maximal elements, {x*: U(x*)=z}, is nonempty
and compact.
Proof.
Suppose u(x) + o, for
each x e X, then for
each x there is
y e U(x) and so x e U*l(y).
Thus,
{ u-r(y) : y e X } covers X. By 2. \{ intu-l(y), yex }
is an open cover of X.
Since X is a compact set, then there exists a finite
subcover
{ int U-l(y.) : i=o, ...,n }
and a partition
of the unity subordinate to this finite
{ r' i/ . ) : ^
r/.(x) >o
i=0
Let
A = {y^,
-O'
consider this
¡
, y_i
n
xeintu-l(y.)
Hence X is an mc-space
continuous function
f:X -+X
t32
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
subcover.
as follows
then we can
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
f(x) = G (t-(x), ...,
AOn-l
((\\
P l . . p ^ l P . ( P (-yn,' l ) ,' t
oI
n-2 n-l n
1
where the functions t,
(x)) =
t
. ( x ")') , t ^ ( x- ')) 1' , . .'.-,ot '^ (- 'x) ) l
n-l'
"-2'
are defined as in the fixed
(Theorem 3.13.), and the proof
of the continuity
point results
is the same as
the continuity of the selection in Theorem 3.13, in the same way
there exists a fixed point of f.
Let
x* = f(x*)
and consider the set
J(x*) ={i:ry'.(x*)>O}
it is verified that
xx e int U-l(y.),
i e J(x*)
hence
y.eU(x*),
'
it
is
ieJ(x*)
Moreover, as a result of the way used to define functions t.,
verified
function
if
that
ú.(x*)=O, then
PA(2,t.(x*)) = pl(z,o) = z,
t.(x*)=O,
(that
iii
is,
in
which
identity
case,
function),
so, the composition
f(x*) = GA(to(x*),... , t,r_r(x*))=
(('t\
P ^ 1 . . P _ l e _ , ( P _ ( r ^ , 1 ) , t _ , ( x * ) )_, (t _
x*)1,...,t^(x*)l
Ol
n-2[
n-r
only the P, functions
n-n
n-l
n-'¿
)
u
)
which correspond to those index i e J(x*)
will appear, hence
f ( x * ) = G o ( t o ( x * ) , . . . , ! , . , _ r ( x * )=
) G¡(qo,...,e_)
133
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
where
the
Applications.
coordinates
q. are those non null
rj
of
t (x*).
Then
x* = f(x*) . Golu,**,([o,l]^)
which contradicts l.
Therefore, the set
{x: u(x) = o} is nonempty, that is, there
exist maximal elements. Furthermore, this is a closed set because
its complement is open. This is proved as follows
if w e {x: U(x) =
ñ}
U(w) + u,
then
therefore, there exists
y e X: y e U(w), That is
w e U-t(y)
and by 2. there exists y' € X such that
w e intu-t{y') . u-l{y'),
thus if
z e intU-l(y') then y'etJ(z), that is, tJ(z)*a thus
intu-r(y') c X\{x: U(x)=o}
Consequently it
\
set then compact.
{x: U(x) = al
is obtained that
is a closed
I
Let
hipothesis f.
us
notice
that
condition
f.
in
Theorem 4.5.
is
in Theorem 4.3 when the functions po which define
function G. are defined as segments joining pairs of points, that
A
is,
pA(x,t) = (l-t)x
+ ta.
where
,
A={ao,...,a,.,} in
which
is verified that
pA-(pA(x,l),t
n-r
n
n_I
)=t
n_l
134
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
a
+(l-t
n_i
n_l
)a
n
case
it
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
=
rl_,
,r"_J
[rl_,[eor*,rr,."_,J
tr'-2".,-z*(I -t.'-r)
[t"-rtrr-r*t'-t"-r
)t^J
in general
G A ( t o , . . . , a r r )=
't
't
't'l
(_ I
)
Po
't"-,J
't"-rJ
'"',J''"] =
l"P"-,[t" ['"'t,J
_l-_
j-l
n
1
n
'
i=o
rl
(n
to'o*L',',lll
"-"1¡- L',*,
f
j=t
The next
defined
on
a
i=o
Lemma shows how. from
topological
space, it
mc-structure
on X such that
irreflexivity
condition
(1.)
the
in
is
an
acyclic relation
possible to
upper contour
Theorem 4.5.,
define
sets verify
then
an
the
Theorem 4.1.
(Walker, i977) will be a particular case of Theorem 4.5.
Lemma- 4.1.
Let
relation
X
be
defined
a
topological
on X.
space, and P
Then there
exists
an
acyclic
binary
an mc-structure
on X
such that
VxeX
and
V AcX, A finite,
verified
AnU(x) * a it is
x e Golur*,([O,l]-)
135
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
Proof.
As P is an acyclic binary relation,
every
finite
subset
element, that
is,
then it
A = {xo, xt,...,xn} c X,
there
is
an
element in
is verif ied that
has
A,
a
*o
maximal
such that,
U(xo) nA=4.
It is possible to define the following mc-structure
pA:Xx[O,l]--------+X
i=O,l,...,n
pA(x,o) = x
PA(x,t)= xo
where x^
0
is one of
te(o,ll
the maximal elements of
the set A. Then,
composition Go will be as
GotIO,l]" --_-+X
Go(,0,
. . .,t,,_,
) = t:
as'
PA(x,.,,l)= xo
ff
[
.eo_,
1),1,,_r),."_rJ,.
.,.0J
[no_rtpA(x,,,
in the end composition Go will
be .a constant
function equai to xo,
G o ( a 0 , . . . , t r . , _=r )x o
t o , . . . , t r r _ le I o , 1 ]
Let us see that this mc-structure verifies
VxeX
and
V AcX, A finite, AnU(x) + ñ it is
verified
x é Golu(*)([O,1]*)
136
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
If
4.
Applications.
it
is
supossed that
it
is not
nonempty subset A of X such that
true,
then
there
is
a finite
* o and which verifies
Anu(x)
x e Golu(*)([O,l]-)
therefore,
maximal
by
the
construction
element on A, that
AnU(x) ;¡ o-
because
then condition
It
of
function
Go
x
is, AnU(x) = o which
Hence,
if
P
is
an
has
to
be
a
is a contradiction
acyclic
binary
relation,
l. in Theorem 4.5. is verified.
is important
unif ied treatment
to
to remark that
analyse the
Theorem 4.5. yields an
existence problem of
elements
in
preference
relations
when
relations
or
convexity
conditions
are
either
maximal
acyclic
considered.
binary
Thus,
this
result allows most of the results obtained until now by means of
these
two
immediate
different
consequences of
(Theorem 4.5. ),
and Tianle
results
ways
Fan
to
this
theorem
(Theor:m 4.2.),
(1993) among others
which
generalized
be
are
the
and
extended.
results
of
As
Walker
Sonnenschein (Theorem 4.3.)
obtained. Furthermore,
the
are presented below and which correspond to
the
lgTheorem
4.5. can be rewritten as a characterization in a similar
way te that of Tian's result (1993) in the usual convexity, since
it is immediate that if there exists a maximal element then the
set is an mc-space (by defining Go(ao,..,trr)= x*, where xx is the
maximal).
1?'7
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
existence
of
structures
and simplicial
maximal
elements
in
convexities
c-spaces,
K-convex
continuous
are also obtained.
Corollary 4-1.
Let x be a compact c-space and let p be a binary relation
defined on X, such that:
t.VxeX
2. If
xeC.(U(x)).
y e U-l(x),
then there exists some x. e X such
thatyeintU-l(x,).
Then the set of maximal elements, {xx: U(x*)=o}, is nonempty
and compact.
Coroilary
Let
4.2.
X
be
a
cont.inuous structure
compact
topological
space with
and let P be a binary relation
a
K-convex
defined on X,
such that:
t.VxeX
2. If
xeC*(U(x)).
y e U-l(x),
then there exists some x. e X such
thatyeintU-l(x.).
Then the set of maximal elements, {x*: U(x*)=o}, is nonempty
and compact.
138
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
Corollary 4.3.
Let
X
be
a
compact topological
space with
a
simplicial
convexity. Let P be a binary relation defined on X, such that:
l.VxeX
2. If
xeCO(U(x)).
y e U-l{x),
then there exists some x' e X such
thatyeintU-l(x').
Then the set of maximal elements, {x*: U(x*)=o}, is nonempty
and compact.
The following
and
shows
a
example
(Corollarv
is
situation
simple
Sonnenschein's
result
covered
4.2. )
example
(I97i)
by
is based on the
of
nonconvex
cannot
some
of
be
the
euclidean
preferences
applied.
previous
X = { (x,y)eR2: ll(x,y)ll = b, y=o }.
where ll . ll denotes the euclidean norm in R2.
The binarv relation P is defined as follows:
( x . x ) P ('vr .' 'v ) < + l l ( x . x ) l l > l l (' vt '.-v ) l l .
r' z
z
t' 2
2
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
which
this
generalizations
\
139
in
However
Example 4.1.
Let
distance
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
App.lications.
Notice
that
result because
However
by
Corollary
continuous
this
example
VxeX
this
4.2.)
rings
not
covered
since
in
in
covered
it
the
upper
case) K-convex
U-l(x)
is
an
(in
def ine
sets.
a
K-convex
(u(x))
sets
This
particular
structure
are
is
way
= [(l-t)p
y e U-'(x)
open set
,
X
* tp ]"tt(t-t)ctx
v
the complex representation
Furthermore, if
to
contour
K: X x X x [O,ll------+
considering
Sonnenschein's
by Theorem 4.5.
possible
is
which
this
defined in the following
K(x,y,t)
by
x e C(U(x)) = X
case is
structure,
(semicircular
is
of the points in R2.
then x e U(y)
VxeX,
Corollary 4.2.
140
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
+ tc(vl
then
it
is
€
llxll > llyll, so
possible to
apply
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications
In fact,
this
relation
is
continuous and acyclic. Then it
is
possible to apply walker's result (Theorem 4.1.) which witl ensure
the existence of maximal elements. However, this relation
modif ied
by
considering
another
which
is
not
acyclic,
can be
hence
Walker's Theorem cannot be applied.
Example
Let
4.2.
X be the following subset of R2,
X = { ( x , y ) e R z : l l ( x , y ) t l = 1, Y=O )
Let us consider the following subsets of X.
A = { ( x , y ) e R z : l l ( x , y ) l l = 1 , xcO, y>O ).
B = ( ( x , y ) e R z : l l( x , y ) l l = l , x=O, y>O ).
The preference relation (P) is defined on X as follows:
This
VbeB,VxeX\B
bPx
VaeA,xeXrelsu{y*}
aPx
xx =( -I/2,O), y*=(-l,O)
x*Py*
Vx,y < X\ AvBu{x*,y*}
xPy
(+
llxll >llyll
x e {x*,y*), Vz e X\{x*,y*}
xPz
€+
llxll >llzll
x e {x*,y*), Yz e X\{x*,y*}
zP
(+
llzll > llxll
is
in
fact
a
non
acvclic
relation
cycle
y*PG3/4,O)Px*Py*
141
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
x
because
there
is
a
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
However,
Coroliary
this
preference
relation
4.2. as is shown immediately
verifies
every
condition
in
after.
From the structure defined in example 4.1., X has a K-convex
continuous
structure.
we
going
are
verify
to
the
continuity
condition of the preference relation, that is,
y e U-l{x)
+
3x': y e int U-l(x')
in this case the problematic point is y*
y* e U-l(x*)
¿3x': y* e int U-l(x')?
but in this case it is verified
y*eU-'(z)=X\B
VzeB
moreover, since X\B is an open set, then
y* e int u-l(z)
Finally,
as
for
the
if z e B
irreflexivity
condition
we
must
prove
that
x e C*(U(x))
In this
other
case
definition
VxeX.
case, the problematic point
this
condition
is
verified
is X*, because, in any
obviouslv
of the preference relation P, but x*
from
the
also verifies. this
condition because
U(x*) = { xeX: ttxtl > ilx*il } t {y*}
is a K-convex set due to the fact
t42
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
that y* is an extreme point,
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
that
U(x*)
4.
Applications.
is, yx does not
(except
if
y*
belong to
is one of
we pick up the point
K-convexity
K-convex
will
not
y* from
be
joining
any path
the
extremes
of
and
the
the
of
set
points
in
then
if
path),
llxll > llx*ll
the set { xeX:
broken
pairs
U(x*)
} then its
will
be
a
set, so it is verified
x*eU(x*) =cx(u(x*))
AIso
it
U(y*) = B u {x*}
is
verified
and
the
that
K-convex hull
y* É CK(U(y*)),
of
this
set
because
does not
contain the point y*, just as is shown in the following graph
Then all the conditions of Corollarv 4.2. are verified
can be concluded that there exist maximal elements.
143
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
and it
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
In
results
the
which
context
give
of
sufficient
usual
convexity,
conditions to
there
are
ensure the
existence
of maximal elements weakening the compacity of the set x.
this
kind of
some
with
results a boundary condition must be introduced. In
due to Border (19s5), the compacity condition is
the next result
replaced by the o-compacityzo.
Theorem 4.6.
Let
[Border, 1985]
XcRt be convex and o-compact and let p be a binary
relation on X satisfying:
l.VxeX
xeC(U(x)).
2. U-L({ is open (in X) for all x in X.
3. Let DcX be compact and satisfy
VxeX\D
there exists
zeD with z e U(x).
Then the set of maximal elements, {x*: U(>¡*)=6}, is nonempty
i
and a compact
sübset of D.
The following result is an extension of Theorem 4.6. in
the context of an n-stable K-convex continuous structure.
20
A set Cc{R', is called a-compact
compact
subsets of C satisfying:
if
C =
144
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
there
,.,H*
is a sequence
C,.,.
{C } of
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
Theonem 4.7.
Let
continuous
X
be
a
o-compact
structure
topological
n-stable
and let
space
with
a
P be a binary
K-convex
relation
on X
satisfying:
I . VxeX
2.
If
x e C*(U(x)).
y € U-l(x),
then there exists some x' € X such
thaty€intU-l(x').
3. Let DcX be compact and satisfy
VxeX\D
3zeD:z€U(x)
Then the set of maximal elements, {x*: U(x*)=o}, is nonempty
and a compact subset of D.
Proof.
Since X is o-compact,
subsets of X satisfying
there
a
is
sequence
{X )
n
of
compact
- X.
u X
Set
/\
Tn= c l ü x u D
l
j=t
K[
j
)
then, by applying proposition 2.8., {Trr} is an increasing sequence
of compact K-convex sets each containing D with X = u Trr; moreover
by Corollary
P-maximal
condition 3.
4.2. it
element
follows
Xrr, that
implies that
x
1- and 2. .t}i.at each T,., has a
from
n
is,
e D.
U(x.,)¡Tr, = o.
Since D is
Since
D c T.
compact, we
.can
extract a convergent subsequence {x } --+ x*.
Suppose that xx is not a maximal element, that is, U(x*) + a .
Let
z e U(x*),
by 2- there
is
a z'
(x*
145
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
e
int
U-t(z'))
and
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
neighborhood W of x* contained in U-t(z'). For a large
x
€ W
rI¡NN
and
maximality
D,
and
P-maximal
thus
2.
3. implies
implies
that
that
z' e U(x )nT ,
contradicting
the
= a.
of xn. Thus U(x*)
Hypothesis
to
z' e T_,
enough n,
the
any p-maximal
element must
p-maximal
is
set
closed.
belong
Thus
the
set is a compact subset of D.
Theorem 4-5. can be extended by
general family
of
considering a
correspondences than the one utilized
more
in
that
Theorem. The idea is to see that if for each x it is verified that
A(x) c B(x)
B(x)
=
a
introduced
by
Borglin
then
the
existence of
introduce
the
implies
VxeX
that
A(x)
=
a.
This
and Keiding (1926) to
maximal
notion
of
elements. In
KF
and
order
KF
technique
extend
to
results
do this,
locally
was
on
they
majorized
correspondences as follows:
Definition
4.1. [Borgltn y Keídíng, 1qf6]
A co*espondence
P: X----+> X defined o., .
"onu"x
a topological vector space, is said to be KF-majorízed
another correspondence O: X------++ X
1. VxeX
subset X
if there l:>^
such that verifies
p(x)c0(x)."
2. Gr ó
is an open set.
3. VxeX
x e e(x)
146
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
o1
and O(x) is convex.
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
r\pplications.
Any correspondence which satisfies
conditions 2.
and 3.
called a KF correspondence. Moreover, a correspondence f
on
X
is
LocaLLy KF-majorízed
neighborhood
lV* of
x
at
x,
;¿ o
f(x)
and a KF-correspondence
if
iñ.
x
defined
there
X---+>
is
is a
X such
that
VzeW
f(z)<aQ)
x
This notion will
be extended
convexity, in particular to the context
Definition
to the context of
abstract
of mc-spaces.
4.2.
Let X be an mc-space. A correspondence I-: X -----)
KF* correspondence
X is called
if it is satisfied:
l. VyeX
l-r(y) is open.
2. Vxe\X
f(x) is an mc-set.
3. VxeX
x e f(x).
A correspondence
P: X -;+
there is a correspondence KF*,
X is called
KF*-majorized
f: X ---+> X, such that VxeX it
if
:l >^
verified that P(x) c f(x).
It
can be seen that
a KFx corresoondence satisfies the
conditions of Theorem 4.5., thus an immediate consequencewould be
the following result.
t4'7
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Corollary
Applications.
4.4.
LetXbeacomDact
KF*-correspondence.
mc-space
and
Then the set of maximal
f:
X---+>
elements {x*:
X
a
I-(x*)=z}
is nonempty and compact.
is possible to work with the local
In some situations it
version of KF* corresDondencewhich is defined as follows:
Definition
4.3.
Let X be an mc-space. A correspondence l:
X --->> X is called
Vx e X such that i'(x) + o, there exists
LocaLLy KF*-majortzed
if
an open neighborhood V
of x and a KF* correspondence é : X ------+>X
such that
I'(z) s 0 (z)
VzeY
The following
correspondence
has
to
preference
relations).
Proposition
the
result
some
equivalent
x
existence
shows
point
with
of
maximal
a locally
empty
KF*-majorized
(which
is
contexts
of
images
elements
in
4.1.
Let X be a compact topologicál
locally
that
KF*-majorized
mc-space.
correspondence,
nonempty.
148
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
then
If
set
f : X ->
{x*:
X is a
i'(x*)=g}
is
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
The way of
usual convexity,
but
proving this
result
by substituting
mc-spaces and KF correspondences for
is similar
convex sets for
to
result in usual convexity can be found in Schenkel (1993).
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
of
mc-sets or
KF* ones (a proof
t49
that
of this
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.2.
4.
Applications.
EXISTENCE OF EQUILIBRIUM IN ABSTRACT ECONOMIES.
In the
elements
in
presented.
in
the
choice
the
feasible
set
problem
(or
of
decisions
of
the Droblem of
only
one agent
section the problem
In this
which
Section
last
the
of
other
equilibrium)
over
the
of
is
affect
the
raised.
Then
appears among the
maximal
factible
of an economy with
anyone.
agents
existence
set
n-agents
preferences
a
is
or
compatibility
decision
maximizers
of
individual agents.
In this
context
some of the most notable results
the existence of equilibrium in abstract
In
classical
the
abstract economy I
of
model
about
econiomies are analyzed.
(1954),
Arrow-Debreu
an
= (X., 4., u.): - ir defined as follows:
N = { 1 , 2 , . . , n }i s t h e set of agents.
is the set of choices under agent i's control;
X
I
X=IIX
&
ii
: X---+ X
is
i
feasibility
the
or^
correspondence of agent i.
u:
i
X-+
R
i s t h e utility
150
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
function
of agent i.
constraint
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
For an abstract
vector
economy 6,
an equíLíbríum
is a strategy
Vi = 1,2,...,n,
xxeX such that
1. x* e d-(x*).
tl
u . ( x-*¡ . , x l ) = u . ( x-*t . , z )
2.
The
I
!
result
following
Debreu (1954), and
r
shows
Vz.e&.(x*).2r
¡
r
t
proved
was
by
conditions
sufficient
to
Arrow
ensure
and
the
existence of equilibrium of an abstract economy.
Theorem 4.8.
Let
[Arrow and Debreu,1954J
6 = (X., d., u.)l
r
t
I
r=I
be an abstract economy such that for
each i, it is verified i=l,..,n:
1. X c Rp is a nonempty compact and convex set.
i
Z. u.(x.,x.)
-t
r
3. 4
i
t
on x. for each x - l..
is quasiconcave2?
I
is a continuous and closed graph correspondence.
4. V x e X, l-(x) is a nonempty and convex set.
Then there is an equiiibrium of 8.
An extension of this
result
do have not in general transitive
21
x - i denotes the vector
is obtained when the agents
or complete preference relation,
in which i's coordinate
22L
is said quasíconcave
,e l function f:X---+R
=
=
a} are convbx sets.
sets X q.
{xeX I f(x)
151
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
is omitted.
if
for
any aeR the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
so
in
4.
Applications.
this
functions.
case, preferences
are
not
representable by
In this line, Gale and Mas-colell's work
be mentioned and Shafer
utility
(1975) has to
and Sonnenschein's work (19?5), which
analyzes the problem of the existence of equilibrium in abstract
economies I
= (X., 1,, a,)l=,
in
which
utility
functions
are
replaced by best response corresponclences.
P.: X-
X,
defined by
) . x}
P.(x) = {z.eX.l (x_.,2.p
where p. is the preference
relation
of agent i.
In this context, the equilibrium notion is introduced in
the following way:
x* is an equíLíbrLurn of the economy I
if it is satisfied:
l. x* e l. (x*).
2. d.(x*) n P.(x*) = o
This
kind
of
formulation
which cannot be represented by utility
Vi=1,2,..n.
ailows
preference
relations
functions to be considered.
By using this point of view, Shafer and Sonnenschein(1975) prove
the following equilibrium resültt
r52
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
Theorem 4.9.
and Sonnenschein, 1g7S].
[shafer
= (X., 1,, a,)i=,
Let I
be an abstract economy such that for
each i:
l. X.S Rp is a nonempty compact and convex.
2. 4,
is a continuous
correspondence
such that
VxeX, d.(x)
I
is a nonempty convex and closed set.
3. P. is an open graph correspondence.
i
4. VxeX
e C(p (x)).
x
t¡
Then there is an equilibrium.
There are many results
existence
of
considering
equilibrium
conditions
in
in literature
abstract
different
to
the
which
results
correspondences
consider different
*i
have
to
of
either
Shafer
by
and
on the one hancl, among
continuity
mention
Border (1985, p.p.93) and rarafdar
analyze the
economies,
those
Sonnenschein or by extending that result.
which
conditions
the
in
works
the
by
(L992) who also consider choice
sets as c-spaces. On the other hand, among those which generalize,
Yannelis and Prabhakar's work (1983) can be mentioned in which an
infinite
number of goods and a countable infinite
number of agents
ane considered; or Yannelis' work (1987) who considers a measure
space
of
agents
work (1988)
who
and
infinitely
replaces
many
the
commodities or
continrrity
constraint correspondences by its semicontinuity.
153
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
condition
Tulcea's
of
the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
In general, one of the conditions which is maintained in
most
the
of
results
on
the
existence
of
equilibrium,
sets
as
as
the
is
the
convexity
of
the
values of
the
constraint
or
best response correspondences. This is due to
the
fact
the proof
of
of
points.
that
existence
choice
fixed
well
these results is based principally
So,
if
it
is
on the
necessary to
apply
Brouwer's or Kakutani's results, then the convexity condition will
be required. In the rest of the section these results are extended
to the context of mc-spaces.
The
existence
problem
of
equilibrium
in
abstract
economies is reduced to the existence problem of equilibria
unipersonal
economy
Keiding, 1976).
equilibrium
in
(following
Therefore,
an
abstract
the
first
method
the
economy with
of
Borglin
and
result
of
existLnce
only
in a
one
agent
is
presented and then this case is generalized to infinite agents.
Some previous lemmas which are required to prove this result
are now shown.
154
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 4.
Applications.
Lemma 4.2.
Let X be a compact mc-space and let
l. l:X-----+> X
be a correspondence
1.1. VxeX
l-l(x)
1.2. VxeX
l(x)
such that:
is an open set.
is a nonempty mc-set.
2. P:X----+>X is a locally KF* majorized correspondence.
Then there is x* e X such that
x* e TIxTl-
d(x*) n P(x*) = o
and
Proof.
The correspondence
@: X---+>
X is defined as follows:
I d(x)
d(x)=1"t*lnr(x)
t
If
is
ó(x)
locally
if
x é J{x)-
if
xe¿rx]-
KF*-majorized,
then
by
applying
Proposition 4.1. we have
fxx e X
As &(x*) + a
and
such that
then it
0&*)
-- o.
is obtained P(x*) n l(x*)
= Q(xx) = o
x* e l--(x*I
It
locally
only
remains
KF*-majorized.
to
In
prove
that
order
to
the
correspondence
do this,
let . xeX
ó
is
such that
Q&) * s.
Casel:
In
xel(x).
this
case it
is
possible to
prove that
open neighborhood of x, V_- such that Vy e V*
155
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
there
y 4 A(1.
exists
an
Then the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 4.
following
Applications.
correspondence is defined:
( e,$)
y€v*
[.,
yÉV*
ó* g) = I
So,if
yeV*
yeÁTvl-wehavethaty4óU),
as
which
implies that S* majorize locally to @.
CaseZ:
x e l--(xl
In this case as correspondence P is locally KF*-majorized,
if
P(x) + a then there is an open neighborhood of x, V*, and a KF*
correspondence Q*:X -------+>X
such that
Vy e V*, p(y) c e*(V).
Let's consider the following correspondence
vA f\rr¡ r) =- J
(e(y)
if
y*@
letyl.'¿tyl
if
yefil)-
tx
Therefore @ is a corespondence
which majorizes to @. To see
'
|x
that
Q* is KF*, first
inverse images, that is,
zeq-r$)
of all
it
will
be proved that
it
has open
O -ttyl is an open set (VyeX).
(eQ)
( + v e d * ( z )= . 1
e(z)nd(z)
Ix
ze$-(¿l
=.a{ff
If z * ÁTzl, then there is a neighborhood V" of z such that:
VweV,
wéllw-
Therefore, as d has open inverse images and as z e ú-r(y),
then there is a neighborhood Vl of, z such that
lsÁ
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
-I(y)
v' c J
W = \,I n V'
By taking
we have that
zzzz
w€l-r(y)
and
Vw e W
wed-(wJ
then @.-(w)= 4(w), hence it will be had that
x
W_ . O_t(y)
')r
z
On the other hand. if
z e l7l
so,if
ye0(z)=
Q(z) nd(z)
xx
. x.
and as it
óx xb) = Q (z) n l(z)
then
then
z€Q-t(y)nl-r(y)
is an open set,
it
can be concluded that
there
is a
neighborhood
wr.(e*nd)-l(y)
then y e l(w),
henceif weW
y € Q*(w) n l(w),
zx
wed[w)-
d(w)=l(w)
w e J(wJ-
d x x(w) = Q (w)nd(w)
^(y),
-1
.
therefore W c @
zxx
so @ has open inverse images.
We can conclude that @* has mc-set values since it is defined
from
0x
the intersection of mc-set valued correspondences. Moreover,
does not have a fixed
point, since if
it
had a fÍxed
point
there would exist an element z such that z e Q*(z), which would
imply that
ze&(z)
with
ze7(71
which is impossible, or
157
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
ze&Q)^Q*(z)
but
correspondence Q*(z) is KFx, which is also a contradiction.
Finally, correspondence Q* has open inverse images, so it
can be
concluded that it majorizes locally to d.
Note that the previous lemma, apart from
providing the
existence of quasiequilibrium23, covers two results which are of a
different
nature:
the
existence of
maximal
elements and
the
existence of fixed Doints.
That is, if
we consider l(x)
= X for
every x € X, then
the hypothesis of Lemma 4.1. is verified and we obtain
3x*eX:P(x*)=z
Therefore,
this
generalizes
result
and
covers
the
existence of maximal elements in binary relations (Theorem 4.5.).
On the other hand, if we consider the correspondence
P: X ---+> X with empty images, then the hypotheses of the lemma
are verified too and it is obtained
23
Let
8=(X,,d,,Pr)r.,
be an abstract
quasiequilíbrtum iff x* e lTxrj
iirr
economy, then x*eX
and d. (x*)nP. (x*) = z.
1s8
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
is
a
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
That
Applications.
is,
the
3xxeX
:
existence
of
x*€d(x*)
fixed
points
in
the
clousune
correspondence.
Lema 43.24
Let
locally
X be a subset of
mc-space.
corespondence
correspondence
'a
f : X --+>
If
with
a paracompact
mc-set
Y
values,
is
topological
space and y a
an
semicontinuous
then
upper
Ve>O
there
exists
a
H : X--------+> Y such that
l. H_ has an open graph.
a
2. H
a
has mc-set
3. Gr(f)
Theorem
images.
c Gr(H_) < B(Gr(f),e).
e
4.1O.
If X is a compact locally
l. d:X ------+; X
is
mc-space,
a correspondence
and it is verified
with
a closed
that
graph
and i
such that d(x) are nonempty mc-sets.
2. P:X ------> X is a locally KF*-majorized
correspondence.
3. The set {x e X / P(x) n 4(x) = o } is closed in X.
Then there exists x* e i
x* e d(x*)
such that
and
l(x*)nP(x*)=¿
24
that of the usual
The way to prove this result is similar to
convex case (Schenkel, 1993) by substituting usual convexity for
mc*spaces and convex sets for mc-sets.
1.59
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
Proof.
From Lemma 4.3.. we have
V e > O 3 H, such that Gr(l) c Gr(H") c B(Gr(l), e)
where H, is an open graph correspondence whose values are mc-sets.
If we consider (x, Hr, P) and by applying Lemma 4.1. we can ensure
that there exists an element x
x
eHIx-T
eacee'-a'
a
such that
and
H(x)np(x)=a
Let {e,.,} be a sequence which .converges to O and reasoning as
above we obtain another sequence {-
such that
}
t a.J.'eo'{
vneD.,r,
- [ - r " Jllj =a
fr["^l.,pf"
ll .fu-|'*-]^pf*
LIr,rJ
le,.,JJ Le,,[e,.,J
and since it is in a comDact set due to
x.
an
e {xeXl
d(x)np(x)=a}
then there exists a convergent subsequence to a point X*, which
will be an element of the set since it is a closed set.
x*€
{xeX/&.k)np(x)=a)
We have to prove now that x* is a fixed point of A.
Vn e 0,,1we have
r\
lx_
, x^ | e Gr(H- ) c
cn'
I tr,
"r, J
B(Gr(l), a )
n
and since A is a compact set then
r\
l x ^t^ , x -tn)| - )
I
(x*, x*) e Gr(l)
160
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
Next a result which ensures the existence of equilibrium
in
an abstract
economy with
an countable number of
agents is
presented.
Theorem 4.11.
Let
I
= (X-,1.,P,),a, be an abstract economy such that
1. X is a compact
locally mc-space, X =
'
i
ii X,.
i€I
2- 4. ís a closed graph correspondence süch that
d.(x) is a nonempty mc-set VxeX.
3. P. is locally KF*-majorized.
4. The set {xeXl d.(x) n P.(x) = o} is closed in X.
Then,
there
exists
an
equilibrium
for
the
abstract
economy.
Proof.
X ---+-)
Consider the correspondence l:
I e l(x)
r
that is
l ( x- ) =
(+
y. e l.(x)
X as follows:
VieI
II
l(x).
n
ll
i'
i€I
Moreover,
for
each
iel
we
define
the
following
correspondences:
a)
yeP*(x)
Px:X----->X:
(+
¡III
that is, PT(x) = X x..x P.(x) x..x X.
¡t
b).- P: X--->
X
in the following way:
IOI
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
y.eP.(x)
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 4.
Applications.
(
I
p(x) =l
n P*(x)
r
ielr*l
I
I\
if I(x) + s
o
in any other case
where I(x) = {iell P.(x) n 4.(x) * s).
First
of all,
will
it
be proved that under theorem conditions
it is verified that
l. P is locally KF*-majorized.
2. 4
is
a closed graph correspondence with
nonempty
mc-set values.
l. Consider xeX such that P(x) * o, then there exists ioel(x)
such that
P. (x) n 4. (x) + a,
'o
to
Since the set {xeXl l.(x)
n P.(x) ;¿ o} is an open set, there
r¡\
exists a neighborhood V of x such that
P. (z) n 4. (z) + a
VzeY
that is i
o
to
to
e I(z). so
p(z) =
VzeY
Moreover,
since
p,
pl(z) c p*. e)
n
'o
i€I(z)
is
a
locally
KF*-majorized
o
correspondence, Pl
is locally
KF*-majorized
and therefore
there
o
exists
G:
X
a
X---+-)
neighborhood
X
w
of
x
such that
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
and
a
KF*
correspondence
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
VyeW
P* (y) c G*(I)
o
Hence, G__is a KF* correspondence which majorizes p in the
x
neighborhood
WnV
of x.
2. Immediate.
Finally,
we
show that
the
set
ixeX I
n P(x) = s)
l(x)
is
closed.
VieI we define the following correspondence:
Q.: X-->
X.
(
e.{xl n d.(x)
e.(x) = .{
I
t
d.(x)
if iel(x).
in any other case.
It is clear that
I
| t t e ¡. ( x )
p(x) n41*¡= jier
I
I s
CorrespondencesQ., X+
P(x) n 4(x) = o
Therefore
(+
Xi
P(x) = z,
ir
P(x) * o
in any other case
have nonempty values, thus
that is I(x) = u.
we have
{xeXI P(x) n l(x) = a) = {xeKl I(x) = a} =
rl
=
nJxeXl P.(x) n &.(v)=6 |
I
r
)
ier(
Hence {xeX I P(x) n &(x) = a) is closed because of
the
intersection
of
the
closed
sets.
163
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
So
(X,d,P)
its
being
verifies
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
Theorem 4.10.
hypothesis
and
we
obtain
that
there
exists
an
element x*eX such that:
x*el(x*)
y
P(x*) n&(x*)=a.
y
P(x*)nd(x*)
I(x*) = o and finally
so
x*el(x*)
Shafer and Sonnenschein's(1975),
p.p.93)
and
Tulcea's
(1988)
results
=¿.
(19SS,
Border,s
can
be
obtained
as
consequences of the last theorem since the conditions they impose
on the conrespondences imply the conditions of rheorem 4.11. First
of
all
we
present
rul.cea's
result
which
is
an
immediate
consequence.
Corollary
[Tulcea, j.988]
4.5.
Let X be a locally
I
'
convex topological linear space and
= (X.,4.,P-)._- be an abstract economy such that VieI:
i' i- i i€I
1. X. is a compact and convex subset of X.
2. ú..(x) is closed and convex VxeX.
3. d
i'
is upper semicontinuous.
4. P is lowen semicontinuous and KF*-majorized.
i
5. The set G. = {xeXl d (x) n p (x) = o} is closed.
Then there exists an equilibriüL ror the abstract economy g.
164
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
Next, several lemmas in which it
that
the hypotheses of
is explicitly
presented
Shafer and Sonnenschein's Theorem (1925)
and Border's (1985) in the context of usual convexity imply those
of Theorem 4.11.
Lemma 4.4.
Let X be a compact and convex set, the following conditions
1. d.: X ---->
X
is
a
cohtinuous correspondence with
nonempty convex and closed values.
2. P.: X ----->
3.VxeX
X is an open graph correspondence.
xeC(P.(x)).
imply
P:
i
X ---------)-)X is a locally
KF-majorized
correspondence.
Proof.
Since
images.
P-
i"i
has
Next,
it
an
is
open
shown
graph,
that
then
P. is
C(P.)
locally
has
open
inverse
KF-majorized
by
c(P.).
Consider -x € ( c ( P ) ) - 1 ( y ) , t h a t
exist
i x , . . . , x ) c P(x)
ln
x. e P(x)
then
x
open neighborhoods
such that
is,
y e C(P(x)); then
y e C(xr, ... ,*.)
.
there
and since
< P-l(x-) which are open sets. So, there exist
of x V. e B(x) such that V. c P-r(x.) for
i =1,.--rn-
165
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
each
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
If
4.
Applications.
we
V = nV.,
take
x' € p-l(x.)
Vx'eV
then
x. e p(x')
i=I,2,..n. Therefore C(x., ... ,x ) c C(p(x')), hence
y e C(p(x'))
Thus,
V
c
(CP)- I (y),
locally majorized
(+
x' € C(p-l(y)).
t
so (Cp)- (y)
is
an open set
and p.
is
by C(P.)
Lemma 4.5.
Let X be a compact and convex set, the following conditions
1. d-: X ----+> X
is
i
a
continuous
correspondence
with
nonempty convex and closed values.
2. P.: X ----+)
i
3.VxeX
X is an open graph correspondence.
xeC(p.(x)).
imply
A = {x e X / P.(x) n l.(x) = a ) is closed in X.
Proof.
Let x e A,
a) z e ú..&)
l-i
then 3z e p.(x) n dr(x), hence
which implies x e &-rb).
b) z e P.(x) , the pair
(x, z) e Gr(p.) and since it
open set there exists a neighborhood v
V
x V
xzi
z<4.(A,
rzzi
by
of (x, z) such that
x v
c Gr(P).
Ontheotherhand,
and
xz
is' an
applying
the
lower
and zey
semicontinuity
(since it is continuous) we have
t66
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
soV
of
nl-ix);¿o.
correspondence
a,
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
fU
eB(x)
xzix
If wetakeW=U
a) x'eV
xx
^V
V
eB(x),
then{x'}xV
xzi
then V
and b) we
nd.(x')*a
wehave
n l(x')
Vx'eU
Vx'€W:
cGr(P.) ,soV
xztz¡
b) x' e U
From a)
suchthat
cP.(x').
* s.
obtain that
P.(x') n d.(x') + o
l¡
,
therefore
W c X\4, and A is a closed set.
As a consequenceof Lemma 4.4 and Lemma 4.5.., it is obtained
that
Shafer and Sonnenschein's result
(1975) (Theorema 4.9. ) is a
consequenceof Theor-ema4.11.
Lemma
4.6,
Let X be a compact and convex set, the following conditions:
1. 4: X ------+) X is an upper qemicontinuous correspondence
with nonepty compact and convex values which satisfies:
1.1.Vx e X,
l(x)
=int-"G].
has open graph.
1.2. x---+int d(x)
Z. P: X--------+) X is an open graph correspondence.
3.VxeX
xeC(P(x)).
imply,
a. P is a KF-majorized correspondence.
b. The set A = {xeX I l(x)
n P(x) * z} is open.
10/
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
Proof.
From
hypotheses 1.,
Z.
and 3.,
inmediately since c(P) majorizes the
conclusion a.
is
obtained
correspondence. So we only
have to prove that b. is also verified.
(+
xx € A
l(x*)
n P(x*) ¡¿ s
zeP(x*)
€
{+
jz e d(x*) n p(x*)
(x*,2)e Gr(P)
Since it is assumed that Gr(p) is open, we have
fV
xü
e B(x*), jV
z
e B ( z )' l V _ x V c G r ( p )
x*
z
in particular
V**t{zicGr(P)
hence zeP(x')
Vx'eV
x*
On the other hand, z e &.(x*) = lnTZTxT)-.
distinguish two different
From now we can
cases:
Case 1: z e int l(x*).
In this
case, since the correspondence x ->int
l(x)
has an
open graph, by reasoning as above *e huí",
3V', € B(x*)l z e int d(x')
xl
Vx, € V,
x*
If we take V = V** n V**, we obtain that V c A, so A is an
open set.
168
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Case 2:
Applications.
z e 4(x*) \
intJ(x*).
In this case, we denote M(x) = int d(x), hence l(x)
Since z € MT;TI- \M(x*),
we know that for every U= e B(z) we have
U= n M(x*) * a, in particular
since w e M(x*)
= TTxl.
for U= = Y=. Thus, if w e V" n M(x*)
and M has open graph,
there
exist
W e B(w),
V'- € B(x*) such that V'- x W c Gr(M), so V'- x {w} c Gr(M), thus
xr
xr
xt
w € M(x')
Vx' € V'
xf
Since w e V
we have that
V _ x {w} c Gr(P),
zx*
w e p(x')
If we take
so
Vx'e V**
V = V** n Vi* we have:
ifxeV,then
(
I xeV
Jx*
I
It t
therefore
xe V'*
so
weP(x)
so
weM(x)=intd(x)cl(x)
w e P(x) n l(x),
that is
P(x) n &(x) + a
VxeV.
So, it has been proved that in this case V c A and therefore
that A is an open set.
As a
Lemma 4.6.
consequence of
Border's result (1985) is obtained.
r69
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
and Theorem 4.11.,
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
Corollany 4.6.
IBorder, 1985]
Let 6 = (X., J,, t,)l=, be an abstract economy such that
1. X-c Rp is nonempty, compact and convex.
i
2.
&
i
is
an upper semicontinuous correspondence with
compact convex values such that
Z.t. V x € X, .d.(x)= *a
qt¡
The correspondence intd.: X -->
2.2.
intJ.(x) = int[,
r
[r
t"ll
X.
given by
has an open graph.
)
3. P is an open
graph correspondencein
--- X
-- x
--i'
" X.
i
4.VxeX,
x.eC(p.(x)).
Then, there exists an equilibrium for the abstract economv g.
The next
abstract
result
shows the existence of
equilibrium
economies when the compacity condition is relaxed.
in
In the
same way as in the result of the existence of maximal elements, we
need to
impose an
additional
boundary
condition.
Moreover
the
result is presented in the context of n-stable K-convex continuous
structures.
170
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten
4.
Applications.
Theorem 4.12.
Let f = (X.,/.,P.). -_ be an abstract economy such that VieI:
i
i' i i€I
L X,
is
a
nonempty space with
an
n-stable
locally
K-convex continuous structure.
2.4
i'l
is an upper semicontinuous correspondence: l.(x)
is nonempty, K-convex and compact VxeX.
3. P, is locally KF*-majorized.
4. The
set
{xeXl J.(x) n P-(x) = o}
is
closed
in
II
X, Viel.
Z. = C,.[D. v l.(D)]
5. 3D c X. compact such that if
ir-iK-rr
then it is verified that
4&)nZ
i
i
*a
VxeX
V x - e X \ D . , x -. i e X--i
i
i
-i
xZ
3-yi . e & - | x ) n Z .
i
i
i
i
:
y- i . e P . ( x )
i
Then, there exists an element x* e X such that VieI
x*
e l(x*)
d(x*) n P(x*) = o
and
iiii
Proof.
Consider the following compact set
Since lby applying
structure,
,8,
D,.
semicontinuous, d.(D) is a compact set and
is upper
--
i
¡
the
n-stability
we have that
compact K-convex
If
D =
we
of
condition
the set
K-convex
continuous
Z- = C_-lD.v d.(D)l
is also a
the
¡Kir
set.
take
Z =
,8,
Z,
and
correspondence:
T7T
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
consider
the
folüwing
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
H :Z ---------->--+
Z:
i
H (x) = 4 (x) n Z
-i
i
i-
i'
This correspondence H, has nonempty compact and K-convex
values since
it
given
is
by
the
intersection
of
two
compact
K-convex sets. Furthermore, it has a closed graph, so it would be
an upper semicontinuous correspondence.
From the
last
correspondence, we can define the following
one:
L
: Z---------->+ Z
ii
( e.&)
irx
'
L.(x) ={
ll
d.(x) n Z.
I
L. has a closed graph,
I
i
€ D
i
in any o t h e r c a s e
since if
(x,y)eGr(L.)
yéL.(x)
+
then the following situations can occur:
a)x.eD.
a.tl
b)x
a)
If
have that
ii
x. e D.,
i
i-
eD
then L.(x) = d.(x),
i-
y e 4.(x), that
i
and since y e L.(x)
I
is, the pair (x, y) é Gr(/.).
i
has
a
closed
graph
due
to
the
fact
that
it
semicontinuous, there exists a neighborhood W x V of
that
WxVnGr(&.)=s
I
I-72
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
we
Since &
-i
is
(x,y)
upper
such
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
Therefore, it is verified
V(x',y') e W x V
y' e 4.(x') > L.(x')
hence
W x V n Gr(L) = a
i
b)
If
x e D.,
y e L.(x) = &.(x) n Z
yéd.(x)
the
case of
and
y e l.(x)
yéZi
or
,
by
reasoning as
in
a)
obtained that
3W x V neighborhood of (x, y): W x V n Gr(&.) = o
and thus,
y' 4 &.(x') > d.(x') n Z. = L.(x')
V ( x ' , - v ' )e W x V
TI¡I
hence
WxVnGr(L.)=o
In the case of y é Zi, then since Z. is a closed set,
:lV
neighborhood of
y: V n Zr, =
"
and so, for any neighborhood W of x we have
WxVnZxZr="
as a result of
V(x',y') eWxV
if
it implies that
¡ii'
In
L.(x) = 4.(x) n 2.,
then
as
y'ÉZi.wehave
therefore
WxVnGr(L-)=s
173
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
(x',Y')eZxZ.
it
is
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
That is, correspondence L.
Now the following
8 ' = ( 2 . , L . , P, ).
' l |z
has a closed graph.
subproblem can be considered:
It
verifies
all
of
the
conditions
ll
of
Theorem 4.11. and we only need to prove that
A' = {xl L.(x) n P.' (x) = z}
,
,,=
is ciosed for every i e I.
From
hypothesis 5. we
solutions to the subproblem 6'
have
ih2t
the
A'
set of
possible
is a subset of D . lvloreover, since
L . ( x ) c 2 . , t h e n L '. ( x ) n P . | ( x ) c Z . , thus
i
iilz
¡
L . ( x )n a , 1 " , * ,
L.(x) n P. (x)
but
'¡ ¡
( a . & ) n z . n P . ( x )= [.d.(x) n P . ( x ) l n Z .
J',
plvl-
L.(x) n
¡
I
I
I l.(x) n P. (x)
Since the possibie solutions are those elements x such that
x.e D. and by the definition of Z. , &.(x) c Z. then
lt
ld.(x) n P.(x)l n Z, =.4.(x) n P.(x)
that is
A' = {xl
By 4.
it
ccrnditions from
3x* e Z:
L.(x) n P.l (x) = o) = {xl l.(x) n P.(x) = z}
I
rl
is
verified
i
that
A'
is
a closed
Theorem 4.11. are verified,
x*. € L.(x*)
i
¡
and
la^
r ra
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
i
set,
so all
of
the
therefore
L(x*)
i
n P | (x*) = a
ii-
(l)
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4,
Applications.
Moreover, x* € D, since in other case, that
3iel:
is
if
xrFéD
x* e X.\D.
ttl
and by applying hypothesis 5 we would have that
d.(x*) n Z.
3y.e
-i
i
i
y. e P.(x*)
such that
I
r
but it is a contradiction with (l), L.(x*) n P. | (x*) = o since
t
tlt
L.(x*) n P.l (x*) = d.(x*) n P.(x*) n 2..
¡
il_
i
i
i
Therefore
x*. € l.(x*)
rtt¡
l.(x*) n P.(x*) - a
and
r75
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter- 4.
4.3.
Applications.
EXISTENCE OF NASH EQUILIBRIUM.
In
chapter
this
3 are
applied
equilibrium
in
inequalities
when
a line
Section,
in
fixed
order
to
non-cooperative
similar
to
abstract
that
of
point
obtain
games
convexities
Marchi
theorems
the
and
are
analyzed
existence
of
in
Nash,s
solutions
to
minimax
considered.
To
do this,
and Martinez-Legaz,s
(199r)
work,
is considered.
A game is
partial
control
and
a
situation
general
in
in
which
conflicting
several players
preferences
have
regarding
the outcome. The set of possible actions under player i's
control
is denoted by x..
Elements of X, are called strategies and X, is
the
Let
strategy
set.
N = {1,2,...n} denote the
set
of
players,
and X =,E*X, is the set of strategy vectors. Each strategy vector
determines an outcome given by a function g (g:X ---+lR") which will
be called the payoff function.
The idea of Nash equilibrium
to choose a feasible
own
payoff
strategy
with
in a noncooperative
point
x*,
for
which
respect
to
his
.own strategy
choices of the other players.
116
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
each player
game is
maximizes
choice,
given
his
the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4,
Definition
Applications
4.4.
Let 9 = (N, X, g) be a game, an element x* € X is called a
Nash equilíbrium
for that game if
VieN
it is satisfied:
g.(x*.,x*) = g.(x*.,y.)
Vy
€X
II
The
following
gives
result
sufficient
conditions to
ensure
the existence of Nash's equilibrium.
Theorem 4.13. INash, 195L]
I = (N, X, g)
Let
be a game of
complete information
that
satisfies:
1. X, . R- is compact and convex Vi e N.
,. Zi, X -*-+
R is continuous
3. g-(x
is quasiconcave Vx
-i
- i-,')
'
i
Vx e X
and
e X- i
and
---
Vi e N.
Vi e N.
Then I has at least one equilibrium point (of Nash).
This result
has been generalized in many ways. On the
one hand, some authors
strategy
payoff
have relaxed
sets and continuity
functions
(Kostreva,
Zhou, 1993). On the
inf inite
dimensional
players
(
Marchi
other
strategy
or
the convexity
quasiconcavity
1989; Vives,
conditions
i99O;
hand, . some authors
Baye,
,1991). In
r't'7
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
in
Tian
in
the
y
have considered
spaces and an inf inite
and Martínez-Legaz
condition
our
qr'r".rtity
of
case, this
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
section
Applications.
runs
condition.
parallel
In
order
to
to
function to the context
'work,
this
do
of
this,
relaxing
the
mc-spaces
notion
the
of
generalized
is
convexity
quasiconcave
in
a natural
way.
Definition
4.5.
Let X be an mc-space and
function
f
is
mc-quasiconcave
X- ={xeX:f(x)
=a}
a
f: X ------+R a real function.
The
iff
sets
for
every
a e R the
aremc-sets.
Nash's generalization is as follows:
Theorem 4.14.
Let
I
mc-spaces;
be
any
for
set
each
of
indices, let
iel
X, be
X- i = ITX
let
j*i
O: X- i --------> X
i
compact mc-set
be a continuous
i
images and
function such that g.(v,. )
let
--+
g.: X
R
is mc-quasiconcave for
¡
g.(x*) =
I
which will
with
nonempty
a
continuous
each v e X
'i
proving
g.(x*.,y).
Max
^
y€(r. (xr.)
I
Before
be
let
x* € X such that
x x e @ . ( x- l * . ) ,
¡
and
j
correspondence
I
Then there is
compact locally
I
this
generalization,
be applied in his proof
are introduced.
178
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
-l
-¡
several
results
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
Lemma
4.
Applications.
4.7.
[Marcht
Let
y Martinez-Legaz,
X be a topological
space and
f.:
X --+
the
space, I any
y,
index set,
Y. an u.s.c. correspondence
and compact values for
Then
1.99L]
a compact
with
nonempty
each iel.
y,
f:X -----+-+
correspondence
where
Y=ITY.,
I
f(x) = iI f.(x)
defined by
is also u.s.c.
t
Lemma 4.8.
Let
I
be
any
set
of
indices
and
for
each
iel
Iet
X.
be
a
I
compact
locally
correspondence
Q.: X -+
mc-space,
with
nonempty
X-
compact mc-sets
continuous
a
X = II X., and
where
t
f- : X x X. ----+ R
a
ii
continuous
function
such
f.(x,.)
that
is
t
mc-quasiconcave
Then
the
for
any
x in X.
\ ¿ : X --------+--+X
correspondence
Q(x) = IT O.(x) has a fixed
point,
I
f.(x*, x*.) =
r
I
x*eQ(
x* ),
defined
by
such that
f.( x*, y ).
Max
y€Q- (x*)
r
Proof.
It
X=II
i s possible
X
to
prove
is also a compact
that
the
locally
product
topological
mc-space.
Moreover,
for
I
ieI let f :
i
X -----+-+
X
I'.(x)={zeA.k)/
be the correspondence
f.(x, z)=
defined
Max
y€Q. (x)
r'79
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
by
f,(x, y) )
I
space
each
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
Since A.(x)
is
a
continuous
correspondence
and f.(x,
z)
nonempty
compact
l-¡
continuous
values
function,
and
is
then
u.s.c.
Berge 1963, p.p.116 ),
we
by
have
that
Berge's
and since
f,
has
(Maximum
result
f.(x,.)
is
is
a
Theorem,
mc-quasíconcave, then
I'.(x) is mc-set valued, due to if we consider
c( =
Max
f.( x, y ))
yeQ.(x)
.
then
I'.(x) = Q.(x) n { z / f.(x,z) > a
By applying Lemma 4.7., the correspondence
given by f(x)=
iI f.(x),
f : M-----+-> M
also has these properties. Therefore,
by
Theorem 3.16, f has a fixed point xx € M, indeed
x* e f(x*)
Clearly x* satisfies
f.(x*,xl
rll-
) =Max
f.(x*,y)
y€lJ. (x*)
¡
Proof of Theorem 4.14By
applying
Lemma
Q.: X ------)--+ X., defined by
i
functions f.
i
4.8.
to
CI.(>i)= O.(x- l.)
r
defined by
180
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
l
the
correspondences
and considering the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapten 4.
Applications.
f . ( x , y ) = g . ( x- l . , y ) - g . ( x ) ;
¡
I
we obtain the existence of
x*
€ X, such that
f.(x*,x*)=Max
I
I
x* e O.(x*.)
f.(x*,y)
I
Vie
and
I
Y€@-{x*¡t
I
the last condition yields the desired quality for
the g. , then we
obtain the expression of the existence of Nash's equilibrium.
g.(x*) =
t
The
functions
in
Max
y€o. (**)
mc-quasiconcavity
the
previous
g . ( x-*t . , y )
t
condition
theorem
could
assuming a weaker hipothesis such as that of
imposed
have
on
been
f.(x,. )
weakened,
I'.(x) being mc-sets.
I
The next example shows a situation where Nash's Theorem
cannot be applied because of
the
non convexity
of
the
strategy
sets. However our result covers this situation.
Example.4.3.
Consider
a
game
with
two
players,
each
one
following strateg! space:
X.= { x eRz : O < a = llxll - b }
I
181
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
i=1,2
with
the
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter 4.
Applications.
x X -)
-si:r 2 X
g.(x,y)=llxll
It
is
obvious
particular
it
is
same function
that
possible
X,
are
to
define
R
+llyll
compact
a
and
mc-sets
structure
by
because
considering
in
the
as in example 4.1
K : X x X x [ O , l ]- +
K(x,y,t) = [(l-t)p
xv
X
+
* tp ].tt(t-t)Gx tctvl
P.(x,t) = K(a.,x,t)
The functions
are
defined to
construct
the mc-structure.
Sets X-
a"i
functions
will
existence
of
are circular
be
rings, so they are mc-setszs, then g.'s
continuous
Nash's
and
mc-quasiconcave.
equilibrium
is
Thus,
ensured
by
strategy
sets
the
applying
Theorem 4.14.
Notice
contractible,
that
in
this
and the payoff
case the
functions
are
not
are not quasiconcave, then
it is not possible to apply the classical results.
'uI.,
this case abstract convexity
given by circular rings only.
182
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
which
is
considered is
that
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Chapter
4.
Applications.
The previous Theorem has been proved by applying the
extension of
similar
Kakutani's
point
fixed
(Theorem 3.16). In
result
a
way, as a consequence of the extension of Browder-Fan's
Theorem (Theorem 3.15.) the existence of
inequality
in the context
of
a solution to
minimax
mc-spaces, is obtained, generalizing
the analogous result of Marchi and Martinez-Legaz (1991).
Theorem
4.15.
For
each
let
Q :X ---------+-> X
and
i=1,...,n, let
X,
be
be a nonempty
a
continuous
compact
f. : X x X. ------;R
ii¡
is
mc-set
values,
mc-space
correspondence
tl
nonempty
compact
where
X =
,E,
Xr,
with
and
is a semicontinuous function26 such that f.(x,.)
mc-quasiconcave
for
any
x
in
X,
and
f.(.,y.)
is
continuous.
Then
I
inf I max
xeQ(x) li=r,..,n
where the correspondence
(
f.(x,y) - f.(x,x.)
{ max
II
lyee. l*) t
'
)]=
I
Q : X---++
X is defined as
Q(x) = II f2.(x).
¡
26
Let X be a topological space. A function f:X --+R is called
upper semíconttnuous if for each creR the sets {xeX I f(x) > a} are
closed in X.
183
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Volver al índice/Tornar a l'índex
Chapter 4.
Applications.
Proof.
Given e>O,
let
f.r(x) = { ze
Q.(x) :
f . ( x ,-y ) - f . ( x , z ) < e )
i
max
t
i
ueQ.(x)
fr:X -----> X
and denote by
the product correspondence
fr(x) =
The mc-quasiconcavity of
that
f"(x)
are
mc-sets.
semicontinuity of
f,
n
r
t, (*J =r!,
--1,
t
,r(*l
Moreover,
and the continuity
On the
other
f (.,y),
max
of
hand, from
it
i
y€O. (.)
--1,
f.r(x)
,I,
follows
e, imply
the
that
upper
the sets
are open sets in X.
I-"(x) * a
V x e X, then from Theorem 3.15, the
correspondence f^(x) has a fixed point
x
^
A
a
which
- - - - - - satisfies
---n
x
e
e Q(x^)
and
€:
x_ e I'_(x) =
e
e'--'
tT
ilf
I- (x)
ia
then for i = l, ..., n
f.(x^,x-^) >
r e ia
Max
yee(x)
f ( x . v ) - ei-e'''
ie
,='-i1,"{
r'(x''Í)- r'(x''x") ' '
}
T:á ,-,
I
since
a
can
be
made
arbitrarily
result.
184
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
small,
we
obtain
the
desired
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
REFERENCES.
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
ALLAIS, M.
(1952). "Fondements d'une
comportant
I'ecole
Nattonal
un risque
Américaine".
de
La
et critique
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positive
des postulats
Colloques
Internationaux
Recherche
Sctenttftque
des
choix
et axiomas
de
du
Centre
40,
(1953)
p.p. 257-332.
ARROW, K. & G. DEBREU (1954). "Existence of
equilibrium
for
a
competitive economy". Econometrica 22, p.p. 265-290.
ARROW, K. & F.H. HAHN (1911).
"Análisis
General
competitivo".
Fondo Cultura Económica.
AUBIN' J.P- & A. CELLINA (1984).
"Differential
Inclusions".
Springer-Verlag.
AUMANN, R.J. (1964). "Market,
with
a
Continuum of
Traders,,.
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of
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
SUBJECT
INDEX
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
SUBJECT INDEX
abst r act e con omy
150
cor r espondence
- quasiequilibrium
158
- KF*
- equilibrium
151,152
majorized
r41
-
148
theorem
- Arrow
& Debreu
151
-
Bord er
r70
-
Shaf er
& S o n n e n s c h e i nr 5 3
- Tulcea
r64
141
locally
- KKlvf
119
generaI i zed
- selection
84
-
82
I ower semi continuous
- upper
Carathéodo ry
( R")
t20
semicontinuous
82
22,25
equ i I ibrium
- Nash
convexi ty
117
- abstract
23
- al i gned
28
- closed
27
hul I
-
Iocal
72
- abstract
z4
-
log.-
38
- usual
zo
31
- g-
98
- join
23
- K- convex
- simplicial
f inite
ly
local
-T
- topologi
cal
existe nc e
25
convex
z4
- conmutativity
*"-
20
corr e spondenc e
81
- approximat i on
84
- cont inuo us
83
- graph
az
-KF
r47
- - majorized
t46
-
loca I ly
141
IT '
- mc -quasi co.ncave
178
- upper
183
semicontinuous
idempotent
33
int e rval
25
open
sub
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
Jb
fun c t ion
- c o vering
201
26
60
,
convex combination
177
79
79
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
partition
of unity
80
structure
-
polytope
25
r28
- max imal of
r29
-
qn2
a
29
P
- K-convex
relation
- acyc I ic
a
35
cont inuous
A1
TI
-
A1
at
n-stable
- mc-
55
singular
face
64
set
- compact
80
- comprhensive
46
- contractible
28
- equ i convex
34
-F-
30
- K-convex
36
- mc-
59,60
- star
shaped
- simplicial
convex
simp I ex
c-
32
30
29
I ocal
-
equ i connected
73
J+
-H-
29
-
35
K-convex
local
- mclocal
-
74
55
74
202
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994
existence
-
of maximal
Bergstrom
128
Border
r29,r44
Fan
r29
Sonnenschein
130
Wal ker
rzB
Yannelis
45
space
-
The o r ems
fixed
& Prabhakar
130
point
Bro uwer
85
Browder
a]
Fan
87
Kakutani
87
Schauder
von
- KKM
Thychonoff
Newmann
85
86
119
Abstract convexity. Fixed points and applications. Juan Vicente Llinares Ciscar
Volver al índice/Tornar a l'índex
zo3
Tesis doctoral de la Universidad de Alicante. Tesi doctoral de la Universitat d'Alacant. 1994