x3. FIXED-POINT THEOREMS

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8 Οκτ 2013 (πριν από 3 χρόνια και 8 μήνες)

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x FIXEDOINT THEOREMS
Fixed oin t theorems and xed oin t s f or v ex alued
mappings
W e being b y the w ell kno Banac h con traction principle A mapping
f X Y from a metric space X in a metric space Y d is said
to be a c ontr action if there is a n um ber suc h that inequalit y

d f x x x x holds for ev ery pair of poin ts x x X The
Banac h ed oin t theorem states that ev ery con traction f X X a
complete metric s pace X in to itself has a p oin t x X suc h t f x
x h a p oin t x is called a e d p oint of the mapping f Moreo v er if

x f x and x f x then

d x x d f x x x x

This means that either d x x or In either case w e see that a
con traction f admits a single ed p o in t The standard areas of applications
of this theorem a re existence theorems for in tegral and diren tial equations

F or example the Picard form the solution of the Cauc h y problem y
f x y w ith the initial data y x y

W e are concerned here with m ultiv alued nalogues a of this fact because
certain selection theorems y an essen tial role in their pro ofs F or m ulti
v alued mappings there exists a natural generalization of the of the
ed poin t if x F x then a p oin t x is called a e d p en
m ultiv alued mapping F F or detailed information ab out general e of
the xed oin t theory see the monograph As an example w e s b e
lo w a theorem whic hdaels with relations bet w een ed oin t theorems and
selection t heorems
Theorem Theorem L et C b e a c onvex not c es
sarily close d subset of a Banach sp ac e E let F C C b e a lower
semic ontinuous mapping of C into itself with c onvex close d values If
closur e o f he t set F C is c act in C then F ahsaxe dp oint x Ce

x F x

Pr o of The s tandard con v ex v alued selection theorem is applicable to the
mapping F So let f C C be a contin singlev alued selection of F
Then f x F x Cl f F C g C and w e use classical hauder
ed oin t theorem for the mapping f Hence there exists a p oin t x C

suc htath x f x F x

The tructure s of the pro of of the B anac hcno traction principle for singl
ev alued mapping f X X is as follo ws One tarts s b y n a a rbitrary p oin t
x X and hen t sets x f x for ev ery n I N It is easy to see
n n

that
Sc the can
uous
omp
the
and
ne
tate
cts asp
giv the of oint
notion
pla

of

Suc
hat
of

to
wn
con ets
Fixe d oint the or ems
the sequence f x g is fundamen tal and therefore con v in the
n n I N
plete metric space X to some p oin t x By the con tin uit y of the traction

f w eha v e that
f x f lim x lim f x lim x x
n n n
n n n
Therefore x is a xed p oin tof f

The idea of the pro of in the m ultiv alued case is practically the same
w as st realized b y Nadler in W e b gin e more b y an arbitrary
poin t x X and replace the equalit y x f x y some suitable c hoice
n n
x from the set F x where F is a g iv en m ultiv alued mapping The only
n n
problem is o h w to form ulate the c onditions for m ultiv alued mapping F whic h
w ould guaran tee the desired estimate of distance bet w een x x
n n
F s h an estimate one can obtain that f x g is a C h y s equence in
n n I N
the c omplete metric space X and its limit p in o t ill w be a ed p oin t of the
con traction F
Deition X be a metric space and let D M
the neighb orho o d of the subset M X Supp ose that for closed
subsets A X and B X the follo wing set is onempt n y
f j A D B and B D A g
Then the in of this set is called the Hausdor distance H A B
bet w A and B
It is easy to c hec k that H is a metric on set of all b ounde d
closed subsets of the giv en metric space X a nd it is in fact the standard
Hausdor metric But for our purp oses w e can consider H
un b o unded subsets of X
Deition Am ultiv alued mapping F X X of a m s
X in to itself with closed v alues is said be a c ontr action if some

the inequalit y H F x x x x olds h for v e ery pair of

poin ts x x X
So t he m ultiv alued analogue of the B anac hcno traction principle states
that ev ery con traction F X X of a omplete c metric space X n to
itself admits a ed p in o t x x F x The essen tial dirence bet w een

the m ultiv alued and siglev alued case is that a xed p oin tinm ultiv alued ase c
is not nique u in g eneral F or example recen t esults r of ain S t Ra ymond
sho w that the set Fix F of all ed poin of the traction F y be
nonconnected ev en when all v alues F x x X a re compact and connected
Hence in order to establish some top ological prop erties of the ed oin tste
Fix F one needs to v e some serious restrictions for v alues F x of the
con traction F A simplest example of suc h a restriction giv es the c onvexity
of F x x X Here Con v ex v alued selection theorem pla a crucial
role The follo wing result is due o t Ricceri

ys sets
ha
ma con ts


for to
pace etric
for also

the
een
um

denote Let
auc uc rom
and the

once
It

con
com ergesFixe d oint the or ems and e d oint sets for c onvexalue d mappings
Theorem F or every c action F B B of a Banach
sp ac e B k into itself with c onvex values the e d p oint set F f x
B j x F x g is a r etr act B
Pr o of An ycno traction F is a lo w er semicontin uous nd upp er semicon
tin uous mapping and its ed oin tset Fix F is a nonempt y c s
of B So b yCon v ex v alued selection theorem w e c an d a inglev s alued con

tin uous selection f B B of the lo w semicontin uous s election F

mapping F here w


F x x Fix F

F x

f x g x Fix F
Let F F f Then F is a lo w semicontin mapping with con v ex

closed v alues and for ev x B

dist f x x H F x x H F x f x

x f x k k x f x k x

where B is some contin uous function Theorem A sho ws
that in this situation there exists a selection f of the mapping F suc hthat

k f x f x k x Moreo v er if x Fix F t f x x F x

F f x F x i f is a s election of F o v er closed set Fix F Hence

w e can assume hat t f coincides w ith f o v er ed oin t set Fix F

Let F F f Then F isalo w er semicontin uous mapping with con v ex

closed v alues and for ev x B
dist f x x H F x x H F f x f x


f x f x k x

Moreo v er for x Fix F w e ha v e f x f x x F x

F f x F x Hence there exists a selection f the mapping F

suc htath

k f x f x k x x B and

f x f x f x x for x Fix F

Acno tin uation of this pro cedure yields a sequence of ontin c uous single
v alued mappings f B B h that for ev n I N the mapping f
n n
is iden tical o v er the set Fix F and is a election s F F f with
n n
n n
dist f x x x and k f x f x k x x B
n n n n
The function is lo cally b ounded b ecause of con uit y of B
Hence sequence f f g is lo c al ly Cauc h y i this sequence
n n I N

the
tin the

of
ery suc


of
that


ery

hen



ery
uous er

the of er
ubset losed
of
Fix
ontr Fixe d oint the or ems
has p oin t wise limit f and the con v ergence f f is lo cally uniform
n
Therefore f is a lo cally ence globally contin mapping of the

Banac h pace s B in to itself
Let x f B i x f z lim f z for some z B Then
n
n
dist x F x dist f z f z

f z f z k ist f z z
n n n
n
H F z f z k f z f z k z
n n
f z f z k n
n
Hence x F x i x Fix F Moreo v er b y the construction f x x
n
for all n INadn x Fix F Hence f j d j i f B B is

Fix F F
a r etraction of B on to Fix F
A arametric p v ersion of Theorem w as pro v ed in i a closed
con v ex mapping F X B B w considered suc h that all mappings
F B B F z F x z are con tractions same constan t
x x
Moreo v er the w er semicon uit yof F w replaced b y a quasi
x
lo w er semicon tin uit y ee R esults x f or the d eition
Theorem L et and supp ose the Cartesian pr o duct
X B of a p ar omp act sp ac e X and aBnaachsp ac e B k isap ar ac act
sp ac e or example one c an X b e metrizable or p normal L et
F X B B b e a mapping with close d c onvex values such that

H F x z x z z z k for al l x X z z B dn
The mappings F X B F x F x z ar e quasi lower semic onti
z z
nuous for every z B
Then e exists a c ontinuous singlevalue d mapping f X B B

that f x z F f x z for al l x X z z B
Notice that in Theorems and one can substitute he t Banac h
space B with its closed subset Y B

such ther


ctly erfe let
omp ac
that
eak
as tin lo
the with
as
Fix







uous and
theFixe d oint sets of nonc onvex value d mappings
Fixed oin t sets of noncon v v alued mappings
W e being b y a generalization Theorem to the noncon
v ex v alued mappings More precisely e consider paracon v ex v alued map
pings F or deition of the paracon v exit y see R esults x In the
follo wing selection theorem w as pro v for suc h of m ultiv alued map
pings
Theorem L et and let F X B b e ac onvex
value d lower semic ontinuous mapping fr om a p ac act sp ac e X a
Banach sp ac e B Then
F or every every and every c ontinuous singlevalue d
ele ction f of the mapping F ther e exists a c ontinuous singlevalue d

sele ction f of the mapping F such that

k f x f x k for every x X and


F admits a c ontinuous singlevalue d ction f
W e use a sligh t mo diation of this theorem whic h consists of the re
placemen t of he t constan t in the part b y n a rbitrary a contin
tion X Of course the inequalit yiman ust then b e rewritten
as follo ws
x
k f x f x k ev ery x X


Theorem L and b ec onstants fr om such t hat
Then for every ar ac onvex value d ontr active mapping F B B
of a Banach sp ac e B into itself the e d Fix F F is a r act
of B
Pr o of Ev ery traction F is lo w er semicontin uous nd upp er
contin uous Fix F is a nonempt y losed c subset of B So b y Theorem
b w e an c d a singlev alued con uous selection f of the aracon p


v v alued lo w semicontin uous selection F of the mapping F where


F x x Fix F

F x

f x g x Fix F
Let F F f Then F is also a p aracon v v alued lo w er semicontin uous

mapping nd a for ev ery x B eha v etath

dist f x x H F x x H F x f x

x f x k k x f x k x





ex

er ex
tin
and
semi con
etr of set oint

et
for

func uous
sele

into omp ar
ar an
kind ed
the

of case of
ex Fixe d oint the or ems
where B is some con tin uous function i f is a selection of

F and f j is a selection F j Theorem a sho ws for

Fix F Fix F
an edyx there exists a selection f of the mapping F suc htath

x
k f x f x k ev ery x B


Moreo v er w e can assume that f is an extension f from Fix F o t the whole

space B
W e can alw a ys assume hat t q b ecause of the inequalit y

sues to the con tin uit y of function t at a
t
poin t t Hence
k f x f x k x ev ery x X dn

f x f x x ev ery x Fix F

Let F F f Then F is also an p aracon v ex v alued lo w er semicon

tin uous mapping and for ev x B eha v e that
dist f x x H F x x H F f x f x

f x f x k x

i f is a election s of F

Hence there exists a selection f of the mapping F suc h that k f x


f x k x q x As ab o v e w e can assume

f j d j

Fix F Fix F
Con tin uation of this pro edure c pro duces a s equence o f on c tin uous singl
ev alued mappings f B B suc h that for v e ery n I N the apping m f
n n
is iden tical o v er Fix F and is a selection of F F f ithw
n n
n n
dist f x x x and k f x f x k x
n n n n
The emaining r part of the pro of coincides ith w he t corresp onding ne o of
the p ro of of Theorem
A parametric v ersion of Theorem n spirit of Theorem
can also b e pro v ed

the


that



ery
for
for
the use It

for

that of
Hilb ert sp ec ase
Hilb ert space case
Is the restriction in Theorem essen tial In general the
er to this question is negativ e Namely e pro v e that in a Hilb ert pace s

the inequalit y is suien t f or the existence of con tin

selections So the situation is admissible
in a Hilb ert space The o f is based on a new v ersion of paracon v exit y
namely strong paracon v exit y on relations bet w een paracon v exit y and
strong paracon v exit y in a Hilb ert pace s
Deition Let A nonempt y closed subset P B of
a Banac h space B is said to be ongly ac onvex if for ery op en ball
D B with adius r r and for ev q v D P the follo wing inequalit y
holds dist q D P r
The dirence b et w een paracon v exit y and v y is that in
the latter w e use the inequalit y dist q D P r instead of he t inequalit y
dist q P r Clearly strong paracon v exit y implies the usual
paracon v exit y The onc v erse is false but in the Hilb ert space it is p ossible
to obtain the con v erse implication or f some w eak er degree of paracon v exit y
p

Let for ev ery
for ev ery p ositiv e
Prop osition Each ar ac onvex subset P H a ert
sp ac e H is its str ac onvex subset
Pr o of W e an op en D D c r H with
radius r cen tered t a a p in o t c H and a p oin t q from the c losed on c v ex h ull
of the in tersection D P Only t w o cases re a p ssible o
Case k q c k r
In this case op en ball D q r is a subset of the D
in tersection D q r P is nonempt y due to the paracon v exit y of the
set P Hence
D q r P D q r D P D q r D P
i ist d q D P r
Case r q c k r

Let q be a p oin t in tersection the sphere S c r with the
segmen t c q a nd let b e the t angen th yp erspace to this sphere at the p oin t

q Then the in tersection D is an in cen tered at the poin t

q with the radius r F or ev ery t let q t t q and
let t b e t he h yp erspace parallel to passing through he t p oin t q t W e
put t up fk q z kj z t D g Then is a monotone d ecreasing
p

con tin uous function r k q q k r and

r Therefore there exists a p oin t q q t q uc h hatt t

r F or simplicit yw e call the h yp erspace t the orizon


h yp erspace and w e sa y that p oin ts c q lie elo w Then


the and the
tal




tq
ball en op
of of




The ball the
ball
ar ong
Hilb of
and Then
the
exit paracon strong
con ery
ev ar str
and
pro
uous
answ
ac Fixe d oint the or ems
poin t q lies ab o v e If all poni ts of the in tersection D P are b elo w

then the c on v ex h ull of this in tersection also lies b lo e w Hence the p in o t

q v D P lies b elo w b elongs Con tradiction Therefore

there e xists a p o in tfrmo eth in tersection D P h lies ab o v e

dist q D P t r

So in b th o cases w e o btain that
q D P r
and b y p assing t o the limit when o t w e nd that
dist q D P r
Theorem L et and let F X H b ean ac onvex
value d lower semic ontinuous mapping fr om a p ac act sp ac e X a
Hilb ert sp ac e H Then

F or every every p ositive c ontinuous function
X I R and every c ontinuous singlevalue d ele ction f of the mapping

F ther e exists a c ontinuous singlevalue d ction f the mapping F
such that
x
k f x f x k x X and


F admits a c ontinuous singlevalue d ction f
Pr o of
W xe and put
F x v f F x D f x x g

Then F is a lo w er semicontin uous mapping nonempt y closed con v ex

v alues Hence F admits a selection sa y f The paracon v exit y of the

v alues F x implies t hat
dist f x x x x ev ery x X

Let
F x con v f F x D f x x D f x x g

Then F is a lo w er semicontin uous mapping with closed con v v alues

Moreo v er F x due to Prop osition Hence F admits a conti

n uous selection sa y f aracon v exit y v alues F x implies that

dist f x x x x x X



of The
ex

for
with
con

sele
of sele

into omp ar
ar
tends
dist

Hence whic
to or conHilb ert sp ec ase
Let
F x con v f F x D f x x D f x x g

and so on Hence w e construct a sequence con tin singlev alued map
pings f X H suc h that for ev ery x X k f x f x k x and
n n
dist f x x x where and se
n n n n
quence f g is monotone decreasing and v erges to a nonzer o limit
n n I N
limit



So if w ec ho ose the n um ber suc h that


w e can then d an index N suc h that


N
Therefore the mapping g f is a con tin uous selection of the mapping
N
F and k f x g x k x for ev ery x X

If w e rep eat the pro edure c o v e starting with g then w e d a


election g of F suc hthat

k g x g x k x x X

Con uation of this construction pro duces a tin mapping f
lim g w hic h is the desired selection of F
n
n
F ollo from
One an c rep eat t he pro of of ixed F p o in t t heorem from the previous
section using Theorem instead of Theorem So for a Hilb ert
space H w e btain o the follo wing impro v emen t o f T heorem
Theorem L et and b e c onstants fr om that

Then for every ar ac onvex v alue d c active map
ping F H H of a Hilb sp ac e H into itself the e d p F
of F is a r etr of H
As an example for Fixed p oin t theorem giv es the estimate
for the degree of con tractivit y But ixed F p oin t heorem t giv es
the e stimate for the on tractiv e aracon p v v alued apping m F
whic h g uaran tees the existence of ed p o in ts

ex


act
Fix set oint ert
ontr
such
ws
uous con tin
ab



con
The
uous of

ac Fixe d oint the or ems
An application of selections in the iteimensional
There are sev eral results concerning structure of ed oin t set F
of a con traction F ee s
Here w e men tion an elegan t application of he t selection theory in ite
imensional ase c prop osed b ySina t Ra ymond The riginal o question
w as the follo wing Let F be a on traction the follo wing
implication hold
ix F is a singleton x F x is a singleton x

p
The e xample of a mapping F z z o v er unit circle S in the complex plane
Csho ws that in general the answ er is negativ e
But there t w o cases when the a nsw er is amativ e The rst is the

case when the constan t of the con tractivit yis sles anth The second

one is escrib d ed b y he t follo wing theorem
Theorem L et X b e a close d c onvex subset a Banach
sp ac e and F X X a ontr action with close d c onvex values
Then for e ach x Fix F


diam ix F diam F x


W e r epro duce a p ro of from to the ect that for a iteimensional
Banac h pace s the ab o v e inequalit y c an b e sharp ened as follo ws

diam ix F diam F x


Pr o of Let x F x and let y be a p oin t from F x W e w an t to

d a ed p oin t x Fix F uc h hatt

k x x k k y x k


F or arbitrary w e dee a m ultiv alued mapping F of closed

con v ex space X in to itself b y s etting for x X
F x Cl f F x D y k x x k g

W e conclude from dist y x H F x x k x x k k x x k

that F x Con v ex v alued selection theorem be applied the

mapping F X X i w e an c d a contin uous selection o f F a y f

Let r k y x k and X X Cl D x X is

con v ex and ompact c b ecause of the ite dimensionalit yof balls W e claim

Then

to can


an






of
are


es do When
Fix
caseFixe d oint the or em for e d c omp osablealue dc ontr
that f maps X in to itself In fact for eac h x X w e ha v e x X and

k x x k r Therefore

f x F x X Cl D y k x x k


Hence f x X k y f x k k x x k i

k x f x k k x y k k x x k


y x k r


So w e ha v e pro v that f X X w e can d a ed poin t of f

sa y x F rom x f x F x w e see that x is a ed p oin t of

giv en on traction F Moreo v er
k y x k x x k y x k k y f x k k x x k


So k x x k k y x k


There are exactly t w o p ossibilities

k x x k k y x k for some or



k y x k k x x k k y x k for all


In the p oin t x is desired ed poin t of F In w e x to

b e an accum ulation p oin t of he t sequence

f x j g
n
n
n
Suc hanamcuc ulation p oin t exists due to the c ompactness of the closed balls

Eviden x Fix F nd k x x k k y x k The example F R



I R F x sho ws that the constan t is the b est p ossible

in the inequalit y diam ix F c diam F x

Fixed oin t theorem for decomp osable alued con tractions
There exists another v ersion of Theorem on top ological struc
ture ed p oin t Fix F for noncon v v alued traction F i the
decomp o sabilit y of subsets a Banac h space L as a substitution for con

v exit y F or deition of decomp osabilit ysee R esults xr x b elo w
Let be a measure space with a ite p o sitiv e nonatomic measure
and a Banac h space B k let L be the Banac h space all

lasses B o c hner n tegrable mappings with the norm
Z
k f k k f k d
B


of for

of
con ex set of
the



tly
set the





the
and ed


and

actions Fixe d oint the or ems
Theorem L et M L L b ea c ontinuous mapping of

the artesian C p r o duct of a metric sep ar able sp ac e M a ar able Banach
sp ac e L with nonempty b ounde d close dand de c omp osable alues v L et

b e a c ontr action with r esp e the se c variable i
H m f m g k f g k
L

for some and m M f g L Then ther e exists a c ontinu

ous singlevalue d mapping M L L such that for every m M the

mapping m is a r etr action o f L onto the set Fix of al l e dp oints
m
of the ontr action f m f
m m
In summ ary ed p oin t sets of no tractions decomp osable
v alues are absolute retracts and moreo v er retractions ybe c hosen c onti
n uously ep d ending on the arameter p m M
The pro o f of Theorem is similar to the pro o f Theorems
and with some mo diations Instead of hael con v ex v alued se
lection heorem t one m ust use the election s theorem f or decomp osable v alued
mappings s ee Theorem b elo w
Finally e form ulate the theorem on t he structure of the xed p oin t sets
in whic h s election conditions are assumptions of he t theorem
Theorem L F X X b e a mapping of a Banach
sp ac e X into itself with c onvex values p oint x y
gr aph ther e exists a sele ction f of F such that f x y and which
F x
isac ontr action of X e gr e e c ontr activity in gener al dep ends on x y
Then e d oint F the mapping F is line arly c onne d
Recen Gorniewicz and Marano prop osed some unid approac h to
pro ving of Theorems and They extracted some selection t yp e
prop ert y whic h holds for con v ex v alued and for decomp osable v alued con
tractions as w ell and sho w ed that this prop ert y implies t hat he t ed oin t
set is n a a bsolute retract
Deition Let X b e a metric space a nd F X X be a
lo w er semicontin uous closed v alued apping m f rom X in to itself W esa y t
F has the sele ction p r erty with resp ect to X if for ev ery pair of contin
mappings f X X h X uc htath
G x Cl F f x D f x x x X
f y nonempt y c losed set A X v ery c ontin uous selection g of G j
A
admits a contin uous extension g o v er X suc hthat g is a ontin c uous selection
of G
Theorem L et X b e a c omplete absolute r etr act and F
X X ac ontr action Supp t F has the sele ction p r op r e ct
to X Then the e d p oint set Fix F a r etr act of X

is
esp with erty hat ose
an or and

and
uous op
hat
tly
cte of Fix set the
of
the of any for that such
et

Mic
of
ma
with the

any
ond to ct

sep and
Fixe d oint the or em for e d c omp osablealue dc ontr
In fact one can dee the selection prop ert y ith w resp ect t o a class L of
metric spaces It sues to consider in Deition a pair f Y X
h Y for v e ery Y
Theorem L et X b e a nonempty close d subset of sep ar able
L Then every lower semic ontinuous mapping F X X with

b ounde d de c omp osable values ction pr op erty with r esp e ct to
class of al l sep ar able metric es
It w as sho wn in that t he b oundedness restriction in Theorem
can b e mitted o

ac sp
the sele the has


actions