View of On some properties of \(K\)-monotone operators

IIA'1'I IL lL\1'I(1,\

-

^{IìltY LtIj D',\NÄL y} SD À-t_'.\tÉÌì _{I e} L U

E'T DD TIIÉOIìIÍ' IJE I-';\PPROXINIATION

L',ANALYSE NIIÌ|IflIìIQUE ET L,\ THÉOÌìIB

DE

L'r\pIltìOXIilLITION- Tomo

16, No

J,

1987,

pp. 69-76

Ol{ ^SOiVIll PROPDIìifItrlS,OF 1I-MONOTONII OIìER,\TO

RS

IìAJ)U I)IìIiCUI)

(Clu j-Napoca)

1. Introduetiou. In an

ear'lier píìper,

f14], ^u'e'have

inllocluced ilre class

oI

If-molrotoIte opolatols, u'hicìr is u'icleil

than

the class of monotone

(in _the

,sonse

ol

r\rintr'-rh'owcler')

operatols. since this

class

of

oper.atofs also iltolucles

a sufficicntl¡' lillge set of

(o)-monotone (rnonotone

in

tho scnse

of

olcler') opclatols,

il,

follon's lJrat

the

invcstigation

of

/f-monotono opelatols

is of

int,elest ab

lcast fot a

unita,r.\¡ ¿tpptoach

to the

theolies of rnouotone ancl (o)-rnolotone operrltols. illhe

fact

hhat this unitar.¡, zr,ppr,oach

is a

natural

one,

follorvs flom lefs.

_{1151, |-16.],}

l'helc

the bcsl,-appiõxima-

tion operatol of the elerncnts of a

Tililber.t

spàce X, bli

elernents

of

a nont'oid closcil convex subsel, O aucl ^n,i1,h r:espgct

to a celtain

^nolrn

orr ,Y, is iuvestigatecl,

In

some conrlitions imposed

to

the

nonn

ancl

to

tìre subset, C, the best-apploxirnation operatol can bc monotone, or' (o)-nono-

tone,

or' (essentially)

onlr.

1{-tnonotone.

ln this papcr', s'c sìrall

extencl some 'n'ell-knorvn l¡asic

results in

monotone operlatoLs

to the

cla,ss of Jf-rnonotone

opelatols.

Pi¡r't,

of thcse

cxl,ensions

rvoto

ah.eacl¡. ,1uon to

t141.

û.

1f-tr{onotone oper.atols"

Let ,{

ancl

Y

l¡e

trvo real linear

spaccs.

By (.,.) *e

clcnote

a bilincar. functional on .X X y. If I{

^c.

X is

^a

cotn)e;Ð cou,o

-(i.e. I{

1-

I{ c Il ancl aK < If ior, ary ø ) ^0), then

^l,he

ltolg,r_cone_I{*

of -Il with

respect,

to the bilinoar

function¿r,l

(.;.) is

defi:.

neil by I{'t:{y.Y; (r,tJ) ) 0 for.all øe,Ií}.

^I_.iet

A:X -+2v ba

^a rrrultir.aluecl

operatol ancl cleuotc ))/ ll(d) the

sei; {ø

e,-l ; ^Än t ^A\.

Tlro operator'

á is

callecl nt'onotone (rvit,h respect

to the bi-liletl

functió-

lal-(.:.))-it ^{fol an¡' n, n'eD(A), tire}

iuequatif,y

(r - ^{ù', !/} - ^{y') >}

^o

Irolds

fol all y

^e

An and,y'eAn'.

The opelator

á ^is

^said.

to

l-te(o)-mono- tona (rnototonc

in

tho sense

of

older') ploviclecl

that

nìrenevet:

n,

a' ^e D(A) ancl

n - ^c'

^{e 11,}

^then

^lJ

^{-* !l'e 1l* for all}

^{y e}

^An

^and.

^q'

^e

^{Án'. fn lef.}

f.14.], 1ìrc operatol

l. ^l'as

callccl l{-ntou,ototta proviclecl

that

for. àtty

u, n'e çDAÐ

such thal,

!! -.rr;'

^{e 11,}

thc

inerlualil,¡'

(r - ^n',

^tJ

- ^y'> ) ⁰

^holds

for all

^{y e}

tLr

ancl

y'

e

An'. Arnonotone

((o)-rnonotonô ot'-,Ií-rnonotonc)

opelator -4 is said to be

ntamintal ntonolona

(ntanintal,

(o)-ntonotone or.

m'animul .kntonotot¿a)

iÎ, l'hencvcr

-B

is

an

opelator'having the

same pr.o-

pelty

_as

zl

anrl

An c Bn lot all r

^e

X, then A : ll.

68 J. E. PEÖARIÓ

REFEIìENCES

I1j' T. Popoviciu' ^{Sur quelques} inégalités entre les fortctionsconueres(prernièrcnote) C. Iì.

Inst. Sc. De Roumauie, 2 ^(1938), 44î1.--454.

[2] - , ^{Sur quelqttes} inégatitës enlre les fottclions cottuexes ( deuxièutc ttole ), ^{Ibid. S ( 1gJ8), ,lir 4} - ^45g.

[3 ] r

' ^{Sur quelEluee} inégalilés enlie I es fotrclions ^{conuettcs (} lroísicnte _{nole )} , tbid ^. 3 (1g 99), 196 -. ^402.

[1]-, ^Les foncliottl conl¿.1]cs, Paris, ^'1g45.

Recaived 10. XIL ¹⁹⁸⁵

Fctcttly of ^C tuil E n¡Jirtccring IJuleuar lleuo[trcíje'lJ

1 1 000 Ilcogrctd, Y u"goslauia

ri _,

70 lì. PlìDCUP

It is

clear

thal,

an opcrator

is

1f-rnonoi,onc

if

ancl onl.v

if it

is

(- Iq-

Dlonotone. AIso, cach ntonobono

opelatol is

lGnrclnotorie fór. aÌr.y coÌl\rc.\

conc

/f andiI I(

U

(-I{) : )l,

l,'[rcrl

the

1f-¡ronotonicii,r- r.ec]¡ccs

to t¡ono-

tonicit¡..

Tt is

also tx,idgnt tha,t c¿tch (o)-nronotone opelzr,t,ol

is

-Ií-lnonot,ouc, l\loleovcr',

iÎ Cc }l is

^¿r, colr¡ex cone

anrl 'f': ^Y

^-+

Y is

a litrear,olte- I'al,ol n'ltich tnaps C

into

1l'l',

i,c.

^?'(

^())r,

lç't', thcu each opcr.a,tol

A: ^X-r2r' rvlrich js

_(11, C)-rnonototri¡,

in the

,sctirjc

thai, l-hcner-cl lor,

:t:,

:r'e eD(A)

one has ))

- ^ü'e

^/1,

^{then y} - ^.!l'e

^{C for.}

^a\l

^y e:1,¿ ancl ^..r¡' e A.7t',

is

/f-rnonol,one

n'ith

t'csptc1,

to the bilincal

^f

unctiorral

( . ,7'(.

)), ^i.e.

^1he

ineclnaliti' (r - r', 'l'(y ^{!l')> > 0}

^holcts

^for

^{cr.et'\. iü, ))' e}

^Ð(,{)

^sntis-

fying

ø

- ^fi'

^e

If

and

l'ol all

u

e iin

anrl _ry' ^e

Aa'.

Also,

if I'

t .Y

-,,Y ^{is a}

^linea,r, opclzl1,or.

I'hich

nra,ps _^11 inbo O'i,, i,c.

L(Ii)

^<-

C*,

rvhcrrc _{__U*}

- [r

^e

-{; (,r, l) ^> 0 for nll

ⁱ⁷ e

C}, thcn

ea,ch

opclat,oÌ

A

:

X + 2y

tt-lticlt

is

(11, O)-rnctnotouo,

is

-11-rnoilo[,onc ritii,]r rcs-

pect, to

1,ltr¡ bi-lino¿lr" TLrncl,iolral

( l,(.), .), i.c. thc

inoc¡rLzr,1it,,1' J,(c:

-- -fr'), y- ^ll'> )0

^lrclltìs

for

cvt¡r,)'

l:, il'e 1)(;t) satisfying ø- a'e](

ancl

fol all

^{y e}

;lr

ancl q' e .,Lt'.

11,

in

ncìclitiotr, ,lJ ancl

Y

Íìr'r. sop¿ìir'¿ìtcr1 iocall¡' conve-\ sll¿ùce¡i,

I{ + ^L

ancl

d ¡

^{Y, then}

^tltclo

^cxisb l:1:3)2,

i.121tn,o non-trivial

continuous linca,r' funol,ionals

/: -l

^--+

[p arttl r/:

-l-

-' [fi ^{such tìrnt}

^,/'(1()

^{c f0,} f ^oo I

^ancl

g(0) c [0, ]-cof.

^{\Àre c¿rn} irnmecliat,el¡' sce i,Lra1, e.zìch

(lfr

O)-nronotoner

opclatoÌ A

^:

_'Y

-+

2r' ^js

-1(-tnonotolrc.

rvith

rc'sltcct

to tire ltilineal

^functio-

nal

clefinecl

b¡' (ø,

^,y)

- ^{lQ) gQ) fot}

^r)

^e

^{.Y ¿rnri u e}

^]i.

3" À

nraxirlrnlitS, I'r,$ull,

olì Jl-rnouot,ole opera[ors.

,Jìhr:ou¡¡hout

this

paper', ,lJ

rvill

be ^¿ rca,I

lincal

normecl str

Ìce, Ii its

ch¡al -f

*

_{a,rril i,ho}

l¡i-linear functional on ,T X ,L't u'ill bo the ltetl'een

,Y

ancl

.Ï*, that is (et,

r'F)

- ^a';(t,)

^I'or,

r

^e

-\

We

sh¿r,ll sa,¡r

¡1tr¡ tlie

oprrlator'

á

^:

lf D(tl) ,- ,\, is

lGltent'ícuxtinx(,ous crt n e D(,4)

if lirn zl(,1 r

lespecl, 1,o

tlre

n'ca,h'i' topolog¡- o[ -lJ'r'r tur, ,uuy.f Ë.¿f

, (-K)

such ^1, ra,t

r

-l-

tll

^e Il'(,1)

for. 0 (

ú

( t. / is iirìid 1,o

Ltt' K-l¿enticonl,inttotts

iT it, is ?-heini-

oolrtiuuous ¿lt ca,ch u e

I)(-4), In

1.hc 1ttr,r.ticn1:lt ^czi,pc

n.lrcn /( ^:= ,\,

l),'e

shä,ll etlrploJ, l,he uStt¿tl

r(hclìlicolLi¡¡ili-" _tet'llr

_i's¡ear1

of '¡I-heu'ico't,i-

nttit¡,t'.

LulrlrÄ 3.L Lat I(

^<-

X

be ct cott,^e:t) colle

milh,

u,on-enr,pty it¿terior cnud

let A:

^D(J,)

+,{* ltc

rt, I{.-h,en¿iconti,u,tr,otts ctpercr,tdy, uhet,à tl1.+¡

ts

u den,se conuer s1[l)sct

of

-Y. IJt a:ne

D(tL),

r:io e

Ili.

^{rt,nd [h:,c} ,inequ,alítr¡

(3.1) (n -.

^{no, ,,1:a}

- r{) ^>

^t)

Itoltls J'or ^cDery

0reD(A)

su,ch, lltu,t

r - ^froe1l

^U

(--I{),

llten,

a;ff:,1.ru.

_. J?ro-oJ.

Iìilsl,,

n'e me.nt,ion th¿ll,

iï

ittt¡

l( *

^{Ø, the,u}

ill,

^,1(

-f (-inl,

I()

:

_{-Y ancl}

int 1f _{u {0} is}

iìr con\ro,\ cone.

Supposc t_!al,

qf I tltru.

'J'hcn, sinco

ir{, li

^{-l- (}

-int, /f) : .l,

^ri'c

irray fincl

n

in X

such iJr¿-¡t

(3.2) t -noeilLt,lf U(-int If)

^:

oN sol\4tt PtìoIrutl-t'l'1ES oll i<-J\4()No'r oNll oI,lJÌlA'I'olìs ì1

ancl

(3.3)

(,r, '--

llo,

^{:l;ìro -'-} ero*)

<

^0.

Sirrcc ^D(.,1) ìs a rlcnsc subsoi,

of ,\l

and rerla,tions (3.2) _and (3.3)

ale

^satis-

jliccl

in

¿r,

l'hole

ncic'hbour'hood o1i

i/,

f\ro rna,y assurlre

tbat in

relations (3.2) a,rrd (3.11) ,L'e

D(.{).

Nex1,,

thc colrvc-rity ot D(A) implies

thal, r¿

:J.,0

_{F f(,r;} -_- _,1:n) e D(-,1

)

^f

rrr'

0

<¿ < 1. ,,\lso, ll1' (3.2) \\'e

have i.ì'¿-,?,0==,

l(,r'-,ro)

^e ir-Lt1i

u (-inl, Il) c 1f u (--K) for'0{ú(l'

Con-

sirrg 1.o Lintit ¿r,s / J 0 arrrl t:-r,kin¡l'itt1o ¿tccotLltt, t,he /f-hcmiool'r.1,inlriby

olì

;1,

n'e obtaiu (r

--

r,,,

;l;ì'0

- ^{l'ot) >} 0, tt'hich contlaclicts lclation

(3.3).

Ilcncc,

;tff

--,{¡ro.

'-llrmonr*r

3.1.

¿/ thr: co¡tue"t: oo¡to

Ii

^<- ,Y

hns

u,r,tu,-ent,1tt,.ry ínlcrí,or, I'lten etrclt Ii-ltctnicon!'it¡ttt¡¿ts I{-non,oton,e operat,or

J

: -,ll

-, .Yf is

nt,ct"oi,nt,ctl

I(- Ln ,t n ul,tt n r'.

I,'t'or-,.f . [*:t, IJ'.

,f.-

^{2 \* be ¿ì} 1L-ltrclnotonc

operalol

such

l,ltat At

^{e ]3,r}

tol all r

^e

-\'.

Ciotr;,lic1cr' (r'"r:bit,r'alilr') íirr)

e

,\:

.- D(A) autl

^{;r¿T e 1lro.}

.,\sBis

/l-tnonototrc,r'oltzr,t-e

(:i;-rnr,,l,l- e;f) ) 0lclr ailre -I

^sr-rch

tJrat

,r-r-Jo e 1( u

(-ll).

Flcnce, l)\'Lelrrln¿ì

3.I, rff.=,Aro.'l'husr ;lro -I]no

fol

evcrt',r' ^L'o e

li,

ri-hich ptovos t,he rnaximalit¡r ¡¡1

4.

fjonoll.ur,rr 3.7.

Lat,l: ,I

--,

.ti'ú¿

u cr,¡ntitntolt,s

litt,etr

^{operu'tor sutis-} Jluing the i,n,equu,tity

(n',

_J,r.r)

':-0 _.for

_ct.t ^'l,aast _{on'a n} e

X.

^T'h,an' tlt'ere euists

c(, col¡L:efi con,c

K --

-Y

tuítlt'

tton-entp.hl

interior,

sttclt' th'ut

A

be nt'a,ni¡ttr.t'1, I{-nt,ott,oton,e.

Proo,f.

If fol

^{n',, e}

X tire

ineclualit,y

(ro,

;1rr,,)

) 0

holtls,

then

as ,{

is

oonl,inuòus,

thero

exists ¿ì oonvex neighboulhoocl

I/ of ro such that (t:,

An)

>0 ^tol

^{cvcl'y Ír e 1r.}

Nol',

ii, is cleal

tltat,4 is

/f-monol,onr:

u'itlt

rrspeci;

to the

coìr\¡e\ r',ont

s'ith

uon-r:inpt¡'

jntciriol: Ii: _{In;

ⁿ

^e l''

and

t

^>-

}ti

ancl rve rn:l-v zr,p¡lly

'f'hcoletn 3.].

Il.cnt,urli,:1.1.

It'

1hc ooltvc'x oono

Jf

s¿r,f.isfies K.u

(--'K) :

,f

,

^thelr

ihe /l-tnoncltorliciLy a,ntl t,he -I(-hcnrico¡rtinrLit,¡' rllc lospectir-cl¡- eqr.rir.alerlt 1o

lhe

urolotrrnicritl- ¿¡ricl

the

ìicLniconbintr-itr'àncl so, a,

l'ell-knol'n

^I'csult'

l-[,

^Ircmtìllì

2l

nlrout

the milxiiìirl

r-ncnol,onicii,t' of irelnicoutinuotls ]lLolÌo-

torrc

ol.lci'atclls

is tlclivetl fi'om

,l.lteot'em _3.1.

4.

troc¿rl h¡nr¡tded¡ltx¡s iulEl

eolltiuuilr-

lrf 1f-ltouof one optla(ot's. ^Àt-L o¡rèr'ator'

Á

:

X -,

^2x*

^is

^s¿i,jt1

^to be \ocatly

bt¡u'txled c¿t n ^e .X.

it

^1,hel'c

r,iists a, neighìroullioocl lr of 'ü ^such tÌrat thc set ^Açlr¡ :

u

{Ay

^;

'y e

D(A)

n ^¡21

is

bolLndetl

in

-l'!'.

illtrucllto¡u

4.1.

I'et 'Y ^be

^{tt, t'ectl} l',u,n,ttclt, sprtce ut¿d'

let I(

c- ,Y be ru coit,ì)c,t) t:on,e u'ítlt, nott-em,pl,y ^í n'terior. ^Tlt,e

n'

cr'tr,¡1 l{.-ttt,ott otr¡tte operat,or

tI

^:

,\.

^--+2x*'is Locr.tll,tl Ltcttutcletl

ut tlte

in'tet'ir,¡r po'i,nts

of

D(A).

['rooj.

AssuLnc

tho

oxisteLLcc

of a ltoint

^{øo e}

^{itrt D(A) itt}

^rvhich

;l

is

not

locall.y l¡ounclcd.

\\¡c

rnav suilpoije tl,:,t'

no:

0 bccause the

/l-mono-

lorricit,¡' is

jnr¡ar:i¿r,nt rrucl.cl tr'¿lnslations.

lhcn thcre

cxist, ¿ì sctltlence

72 II. T'EECUP 4 l-t ON SOMN Ì)IìOPBß'I'IìCS OI¡ K-MONOI'ONIi] OPFJIìATORS

(¡,) -

1)(,,1) arrcl ¿ù sequcncc (.u,f),

r

--+ co &s ^{,t'L -->}

æ.

Sincc

1{ is a

con

can fincl a,

clos i

c int If u _{0}.

:

B(0

;2 ?)

^î,

(4,.1) (n¡ - ^{z, nf) -)} -

^oo

^{as./ +}

^co,

for

a certain subsecluence-

(ri)

ol1n,,¡ (rvrrer,c b.y -B(0; p) rve

have

clcnotecl

the.set {ae

,Y;_l[,ru]l

< g]).

^Assuming

that this js.-t'ttr"

case, u,e shall

tlcrive a

coutracliction._I.¡-c1;

p)0

srich

that for cacr øe

rìIo ilicr.c c*i,st.s

a

constant

c" fot

rdrich the

inequalit¡' (r,, -

^z,

^{n,f) >}

^{c, hoids}

^{for all}

^r¿.

Fol

a,r¡.

rratural nutlbel _I;, tho sct 14ri lre ^{l[r1 (r,,-_ ü, nl)} ) ^_h

for

a1l rr) is olosccl a,uc[ Jlo

:

Û.,r,,.. sirroc

-y

^is cornpletc ir,nd ^ìfois ^closecl,

h is

¿r contl¿r,rÌic,tion. llìhus,

\,c

h¿ì\-e

a z

e

)[o

such that, r.elatiotr (4,[) (ø¡)

of (r,,). Noni

sincc

a,o:.0

€

p_

ì

^0,

^snch

^{i,hat B(0}

;

^{2 p)}

^c

^D(A)

1o belongs

to D(,1). On ilrc oilrcr

füom belorv,

\'e

h.ave arrivecl

to a

c

the local ì:ouncledncss of

:l at ro

as

conor,r,.tnv r..1(,Itil).

ry -r ^is *

reur Lletnuch, s,pu,c(3, trrcn a,*y t,,o,,otQtrc opertttor

A : x

^--+

2x^ is

locally l¡otuttl,ecl

ut

th,e

.interior

_poiruts

"l ^D(.{).

ProoJ. Theolenr 4.1 ca.,n

bc

appliect, n'her.c

Ii :

.y,

Cor¡or,r,¡¡r,y

4.2.

f,et,Y,be nractl lSanuclt, spu"ae ctntllet

Itc_ X

boa cot,t_

1)ù) cone witla non-atnpty inteúor. 'I'lretr,, any ^(o)1 tnonotctne operutor

A:

-y ^_,

-* 2x* is

^tocaUy l¡o¿ntclecl at, tlrc

ittterior

^,poituts

of

D(Ä).

. ^Pl:gul

^{Recall 1,hat} eaoh (o)-rnorrotone opclator,

is -Il-norrotone

aucl

altPl¡, ifheorem

4.1.

g e A(no

l-

¡-r.r.r.).

lllhe

opcral.or'

/ beirìg

(o)-monotone, \ve

have (y, /) ( ( (3/, ø*) < _Ql,

^g>

for ever¡'

u e .R(no) g1'), uÅ'e

-'ln ald

y e

1(.

r\orv, let z bc atry element, ol ,Y. Since ø can be

xritl,en

^iì,s _!/1

- ^y,

^'lvi1,h

^yr,

^{1y, e}

^I(,

n'e get (yr, ,f) - (y* ^g) { Q, r',*) <

Qtt,

{t) - ^Qlr,.f) for all r

^{e B(æo;}

¡rr') ancl n+ ^e

Aø. Using thc

rLniform bouncleclness theorern, t)re ^las1, ine- qurr,lil,ies

itnply that,4(B(øn;

pr)) is l¡ouncletl

itr.I*. llhns, l'e

have plor.ecl

the local

l¡ouncledness

of :l

a,t

ro

as clairnecl.

An

operator _-l1 :

D(A) -, X*, D(tI) < I, is

saicl

to 1:a

detr¿iconti- lln;o"tts

nt

no

if

A!f,,,

-t:løn

^(as

ⁿ - ^Ø)

^weahl¡.

ⁱⁿ ,I*, ^for

a,rry seqtrence

(n,) -

^D(,,1)

^strongly

convergent

to

no

in ,{.

în¡onnu 4.2.

f'et,

X

be u, reJl.enit:e Bu,ttuah space an'd,

let I(

^c:

X

be

a

contten cone tuitl¡ non-entpt'¡1

interior. Let A:D(A) -

^X.'F,

^{D(A) c X,}

be u, IGntonotone r,tpercttor a,nd,'[et no e

int D(A). IJ

the olterntor tL

is

l{-hemi-

continntou,s at ^{no, th,eu,}

it is

euen' denticou,tinttotts at ^no.

Proof

.

^Let

(r,,)

c- inL

D(A)

be such that'

r,,

^{-+ ,f0 ¿ì,s} 't1, ^--+

æ.

Accor-

tling to

Thcore,m

4.1, the

seqrelìce (;lø,,)

is

bounclctl

in X*

ancl,

liy

the

reflerivit¡' of ,I,

passing

it

necessary

to a

subsequencer \\'e m¿ùy assume lTtaL

Aa,,

^{-+ æiÈ} n'eakly

in ,Y* lor

tt, ^--+ oo.

f,et

,r

be any

elemenl,

of Il(A)

for rvhich

fr

-

^{fio e}

^int Il

U

(-int/l). Then for ø la gc

enough, ø

- - ^fr,,eIf

^U

(-Il)

arrcl

by the /(-monotonicity of /, we |ta\e (r -

^r,,,

An - ^An,,)

^>-

^0.

^{Passing 1,o}

^{the limit, u'e}

^obtain

(4.2) (n -

^{no, tLm}

- _"ff) ^>

⁰

for

cl¡cr'-v

n

^e

D(A) sal,isfying n -

^{fro e}

int/f u (-iut Il). Sincc *,0.

e

intD(:l), to any

^?¿e

intll u (-intIl)

there correspontls

ù tu) 0

^sLrch

tlrat

øo

]_'ln,einf D(A) for all

ú

lvith 0<¿ (t,,. lahing r: ^{nol^ttt} in _{by for}

(4.2),

_{tlre every øe} ,Il-hemicontinuity r'e obtain jnt/f

tlnaL _U

(- (u, of inLI().

A(no

A

aT

I Ilut r0,

^tu) \\'e har.c

- rf) since inl,If ^> that ^{0 for} * (-int I(): a,

(tr',

^{0 <}

A

n,

^¿

^< - ^{f,. nf)}

^l)hen,

^{> it}

^O

follos.s that this inequalit¡'

holcls

for all

u,e,T.

Thclcfore, Aro-- nf¡

l,lrat is :l is

clerniconl,inuous nL no.

Conor,r,¿nv

4.3 (tgl). f'et X

be

a

reJlenit¡e l)un,ttclt spüce. Th,en, any tttott,etone henticontittttiu,i oltercttor

A: I)(A)

^-->

X'r, D(A)

^c-.

X, is

ilerai,' aln,ti'ìtttous ^{ou, iu,t} D(A).

Proof. AppI¡'

'Iheort¿rn

4'2, tl'hcrc I( : ^X.

Conor,r,lnv 4.4.

Let X

^bea raJleuiae lìcm,q'clt, sltn,ce rmil let

I(

^{c- Xbe}

ft

^{co,¡tÃeli aone}

^roitl¡

tton-etnpt'y i,n'te,ri'or' ll'hett,,

rnty

(o)-mon,oto'tte ltem,icon,-

linuuts

oper&tol'

A

t

D(tl) *-,Y1, D(A)

^c-

X, is

ilenticottti+¿tt'otts on,

ittt

D(A),

5.

Surjeetivit¡z

6f /f'monolone

opel'ttors.

Let _-T l¡e a leal

^lincar

nor.mcd Spzì,ce. -.\n operator

A: X

^--+

X* is

^{saicl Lo be coerciae}

uitlt'

^respcct to lhe elentent

lt,eX* if

^the,re exists a nurnl¡cr

r>0

^such

that _llrlf 2rim-

plies that,

(u,

Am

-

^/,,)

^{> 0. 'Ihe operator}

^-4

is callecl

coet'cí,oe

it ^(n,

An)lllrll.-+

^oc ars

llrll +

oo.

We shall usc l,he follorving

lxoposition

^(sec [12, ^3.2.8]).

74 R. I'RIICUIJ

IrrJr\Jtf 5.7. Let

X

be u,finitc-tlimens,ir¡tt,nl Ila,naah, s,pa,ce.

I.f A : X _-, -' X* is

ct cclt,linttotts oyteralor ruhiah

is

coet'r'ire u'íIlt, respeet tr¡ tlte c\entent h, e X*", tlt,en, Iltere enists u,t least ottc elentent n: e ,Y suclt, t,ltctt

lt : _Ar.

îrlnonnrtr

5.7. f,d X

^be

u

re,flea,ir:e Butt,aclt, ^s1:tctce cr,t,r,d

let I(

c:

X

be q, _con'uen cone h,o,l)inq non,-ent,1ttt1 in,ierior ui,t,h raspect to tlte uenl; ^toptology

on,

X. IJ A

^:

X -,

^Xoo

^is

^ct, .[(-lten¿'icon,tin,ttct.ts .T{-tnr¡n,olone operu,tor uh,iclt, is coerciae

uitlt,

reslteot I,o h e

X*,

theu, llt,erc eaisl,s

at

least

lne

element n e ^X.

such

tltut

lt,

: At'.

?roof

.

Derrole

by

int,

1l

ancl ru-ittt

I{,

^f.hcr

^jnteriols of ./f with

res- pect

to thc

strong; topology ancl

to the

rvealc topolog;v or-L

,Y,

rcspcctivcly.

Obvionsly, tu-inb

J( c int K antl

since

u int l( 7 Ø,

ri'o Jiavc r,o-int,

/l

^-{-

--l-

(-zu-int I{) :

;Y.

L:et,

A

l¡e coercii.e

rvith

respect,

to

h e

Xt'. Thcn thc

operato'-

ßn : :.Ln -

^/¿

^is

Jl-hernicontinuons, -Il-rnonotouc ancl cocrcive

l'ith

^respccl,

to

0. Thus,

if

one considcrs

the opelatol R

in¡¡'¡carl

of

^:1 1\¡c

mây

âssutne L}l'at lt,

: 0.

'Iherefole, ^sre have

to

_llt'ove

that

l,Ìlcle cxìsts

ân

r,r0 e -lI such

ltlut' ^Aro :

^g.

LeL

lf

be 1,[re

famil¡' of

¿-¡ll finite-tlirnensional linc¿u subripaces -I1 of

X for which

11

n int l{ + ^Ø,

^orclerecl

ì:y

inclusion.

It is

obvious

that

^i1ì

H

e

lf,, then the

convcx conc

1l n

^{1{ has} non-crn1t1,r' irrtc'r'iol

in thc

topo-

logy

inclucccl

on

11.

Fol

each -17 e

//, lct J,,

)te 1,he

injection

rnap

of

-Il

into X

^ancl

let

,/å _l¡c 1,hc chLal projccl,ion

nap

(tìre surjection)

of X*

onto

I1+. We

^set

zlro:,ffiAJr. îhe1,

tfue

opclatr.t' A,r:II

_-> ^F1'F

is ]( n II-.

monotone _ancl

1( _n f/-herlicontinuous anrl, lt¡' llhcolern

^,.L2,

iL is

con- tinuous. Also,

r{r, is

cocrcivc

n'ith

lespcct,

to

0. Accclcling to.Ltrtrma, 5.1,

tlrerc exists an

^,ir1

e,I1

such l,ihat

tL,rrrr: 0.

C.lonseqLreutl¡. (uro,

An:r) : :

^(frn¡

Arrfrr,):0.

Then

since;l is

cocrcivc

rvith

respect l,o

0, it

follorvs

l,hat therc exists rì

constrùnt

úl

inrlepcnrlenl,

of

11 such t'ha,1, llørrll

< t

for

every

II

e

/f.

_

Nou', Iol au¡, IIoe :/f

consiclel t,he

su'biiet lirn,: {rr,r; ^llo

^c-

II}

of

^.11(0

; ^C).

^TlLt'rL

^tlro

l':urrilr' _{{ l'tt,,}

: ^lIu

^e

t(} har llrn

ljnile-jntclsccbiorr

property. Indcerl, it

,24,

Llre lf urtl

rye rlt:notc

by

_J/o

: Éf,

U 11r, then

VHn

c [a, î, l'n". ^Sincc ,l is lei]cxivc, i,hc

bzrll 1]

(0; 0) is

rveahl¡' conlpact,

and

thus

therecxists an

elcnrenl, no e

,T ri'liich

lte,longs.lo thc weal< closuro

o[

cach scb l/¡70 svith .Tf

oe,/(.

lrct r bc an

¿l,tùitlrlly elernent

of

,T such t.ha1; ¿r

--

,?o Gru.ini,

I(

^U

U

(--

^ru

^-inl, 1()

a,ncl

lct IIne //

bo such that, ø e /1n.

Flincc

øn bclongs

1,o the u'ea,li clostrre

of

l/¡¡o,,t,hele _{_e-rists} ¿ù scquoncc (n,,)

- ^{Jr,o such}

^tlial;

ü,,

+

o0 (1or

n --

co) s.oahlJ.

in ,Y. We

rna,¡' ¿ìsslltììe i;ha,t,

ø'--

n,,ey,u.ít-1L

K u

(*-to

-int 1l) c

-Ií

U (_-1l) for

a,11

z. flhen, by

t,he _Ii-

rronotonicity of .4, s'e have that

(5,1) (r -

^nu,

A*) > (n

^{- -} n,,, A,n,,)

llol all ø. On

i,hc

othel

hancl,

b¡' ,, e

[/¡70

ii; follols

f,hat,

thclc

c,rists arL

IL,,e'/f

stLclt

thal, IIo r, |Iu, il.t¿elI,ttttL A¡¡,,nù--=0.llì[cn, n--

t;,,eLIn âncl

x'e rnay

"write

that

_1ø

--

^{iu,,, ,4,fr,,)}

= ⁽ⁿ -

^{3: ,,,} tl¡1,, n;,,)

:0,

Tltus,

by

(5.1)

we finrl that (ø --

^n,,,

,A,t) > 0 for all

n,. l?assing

to t]re limit

^s n-+

^co,

lye obtain thal, (r -

^no,

^An) ) ^{0 Íor}

^c\¡ctJ'

neX

satisfying

ON SO1\4¡] PROPERI'IES OF K-I\ÍONO'IONE OPER.{TORS

ír'

-.

^{rrr,r e ¿u-int}

ll

U

(

^ro -inL

I{). Norv, obsell'tl

^1,li¿t_.

,zl satisfies

all t,heasiurupl,iotrsof Lôlntna3.1,nhi'retllcictol"Lycxcotlt-

-/l':¿¿r-intIf

^U_{0}

h:r,r'ing- rron-crnpty

stlong itrtclior', st:lnils

^{for' .1f.}

Altplying

^{Trelnlrlâ 3.1¡}

tye olttain Arn :0, l'llich

conplet'es

the

ploof.

I1

ryc

take /l - ll, ^tlierr

llhc()r'enì

5,1 itlltlics the follot'ing

x'ell-

lrnorrl

lesrtlt ^:

Conor,l¡nr¿

;j.1 ([2.], tl1.1).#

_'Y ^is

^u

re.fla:rire l)ct'pctclt s'pur:e anil,

A:

-1.

-* ,lt is

^et, cocrci'L:e lteinicot¡titt'tt.i,tts m,ott,ot,ott,e operrttor, then, A.

is

^su,rjec'

tiae.

It ^js

rLat¡r.:r,l

to

aslc

iÎ

ilheor,cur

5.1 is altplicable to

(o)-monotonc 6¡te¡a1ols.

llic

arisri'et'

is

neg^nbivc

lor

infinitc-clilncllsiontùl ^Spa,ccs

,Y,

as

follort's florn:

Rentctrl¡

5.1. Let X bc an

infinite-climensional linear, normccl spa,ce

ancl 1e1,

I( c X

^{lto a} colt¡¡ex colle

h ling

non-en'ipt¡'

interior rvith

^respect

|o tlrc

rvealçtopology on

'Y.

^{IrcL h, e} 'Y*'. flhen, an (o)-monotone operator.4 ^:

,l

_-> -Tå'

l'hicli ^is

cocrcivt¡

tt'ith lespccl to

lt,, clocs

not

exist.

llo ploye this, let us

¿ùssttìllc

t re

cxistence of such anoper,ator" We slrnll

rleriie

a cotrtracliction.

Let rreu-int

^11. flhen, there

exist

fiT,

nl, .'.

.. .,

^fi*;,e

X* artcl

^e

) 0

st"rch

tlat,

Y :{r:eI;l (n - ^r,nrjo)] < ^e, i :1, ^2, ...,tt\ c Ii'

Colr,qidel

the

cont'ex cotre

lf,

^c-

I(, I(.t: {llu;

^:r

elrl

À

> 0}

ancl deuote

1t

S

:- n

lçel e'.Í.

ls is clcilr tirat -Il{' c /lit antl

^).ri,u

-l S c Il, for

cverv

i.>ó]tîLl-,s, i[ øi'e-I(;*, lhctr

(Àøu -l- ^cr, a,'F

> ⁰ for all re S

a,ncl

l>0.

Passing

to limit as

). J

0, tt'e obtain that r, ar') > 0 for all

^{ø e}

S' It

iotlorvÀ

that

(ar, t¡'r)

:0 _{for all}

ø ^e S.I{c'ncc, rScl<el

r* folcvery

.ø* e--Ilf.

Sincc.T is an

infiiite-c-limensiotral

lineal

spâce' onc ha,s

S I {0}'

^\fo\\',

u'c

fix

¿ìrìJ- y e B,

ll I

^0.

^Fol

^)'

)

^{0, rt}

t,hc

(o)-rlonotonicit¡-

of

/,

rvc obl'ai

tlrclc is rf

^e

I{f

such

that

A(),y)

0 s'hich is

il, contr'¿ttlicbion ^l¡ecause t'hc tlitrtension

of

ltcr' (;t(0)-D)

is

irrfi- nitc,

(ì ,.] 75

lìIiFtìRTiN(ìns

f ll ll n r'ìr rr, \i., ^{l)-r.c c ìt p n lt} u, 'I'h., ConDctilr¡ ^rttttL Oplintisrtliott ítt..lJtntaclr ,S2accs, Sij[- llo[[ .t Nr¡r¡r'LcìholI lliLcrlr. l,u]rl. -. lltlitur'¿r r\cacìctttici, JìtrculcçLi, ^1978.

l:21 ll I o \\'d r t', Jr. IÌ., À.ofllirttnr ^alllptit: l¡r¡nndntu-uriluc ptolilctns, l',uì1. .¿\n]cr" ùIath. Soc .

GÐ, rlü2 ^{ß7.1 ( l0rì:ì).}

¡:r] ^lir''oì,1 ¿"., 1,. u.1 Ptoltlit¡tt;; ttott-Lìtrtuircs, Plcssc tìc l'Uliv. clc Àlonlróal, 15, ^1966'

ia TI, ÞRECUP

[a] ^{B r o w cl e} r', F. li,, Nonlineat maximal monotone opet'¿ìtors in Banaqh splce, ùIt,th.

Ann. 173, 89- 113 (1968).

[5] Cristescu, Iì,, 'Iopologit:ctl 1'cclor Spcu,ts, Nor¡rtlholt Intc¡n. I'uL¡I., Lc¡'tlcn- IÌditu¡a Acadcnrici, BücLrrcsti, 1977.

[B] ^{F i} Lt.pat lick, P. ÀI., Suljcctiri[¡, r'csults lol' nonlirrcal nrnppings flotn a Rannch.

spnce to its clual, 11,[olh. tlttn.,20!t, 1'î7- 188 (197:]).

[7] ^{.I a rn c s o} n, [i., Orrlcrccì Lirreat Speccs, Leclure Nof¿s in AIath., l4l, Spri¡gcr-Vcllng, 1970.

[8] I{achulovskii, lì. I., OnrrronotoncopclnLorsnrlrìconlcxluuctionals.(inlìussirur), Uspchi ÀIal. Nari/r, tõ, 213-215 (1960).

[Ð] I{ ^a t o, l'., I)enricontinnity, hcrr.iconLinuii)' arìcì rnonotonicit¡', Dutt. Anter. XIatlt Soc., 70, 548 550 (1964).

[10] I( ^{a t} o, T., I)cl¡icoltinuiLy, hcrnicontirrriLy aucl nlonotonicit]' II, BuIt. A¡ne¡, Ill[at]t Soc., 7il, 886-889 (1967).

[11] ^{À'I i n t} y, G. J., Ou a nronotonicilJ' utcthod fol thc solntiotr o[ nonìinear crlnations in Ratrach spaces, Ploc. Nal . Acatl. Scl. [,I.S.rt., 50, 1034- 1041 (19û3).

[12] PascaIi, l)., Sl:ullan, S., ;\-onlirrerø.Ìtappings of llonolone ?ype, Sijthoff &

Noordhoff Intcrn. Publ. - Ddíttu'a r\catlcmici, l3ucur.csti, 1978.

[13] Peressiui, A. L., Ot'detctl'1'oltologtc;ill/ccloLSpt.ces, lJarper&Rorv, 196?.

[14] ^{P r e c rr} p, R., O generalizate a nolittnii de monolonie 1¡¿ scns¿rl ttti ùlíttty-Bror¿,rle¡, Sem.

itin. ec. Itrnct, aprox. coìtvcx., Cluj-Napoca, o4-64 (1078).

[15] ^{P r c c 11} p, Í\., A![ortoLorticilg propet lics ol ^Lhe bcsL appro:tínttliott operalors, Itinc¡ant Seminar on Iìnnctional Tlquations, Approx. and Convexity, Cluj-Napoca, 223-226 (1986).

[16] ^{P r e c u} p, lì., :I Ii-nronalone besl ap¡trorcímcLtiott operalor p/ri¿l¡ i.s neillLet nto¡rolone and

- (essenliallry) nor (o)-monotone, Alal. ^Nunrcr. 1'hóor,. Approx., ló,2, 753-762 (1986),

[17] Roôlialelllt', lì.'I., Locctlbouncleclnessol¡tottlí¡teu'ntonolo¡teoperalors, trIlchigan t\{nth. J., 16, 397-407 (19q0).

lìc.ccivccl 10.XI.1986

ñt;\.t,lIDl\IA.t.Icrt .._ lìEYL:Fl D'Á.Nr\LYSIj _{NUMÉrìt QUE}

lj'l

DII ,LIIEOIìI]ì

l)E

L,i\ppììOXIÀIATION

r,'aNALySE

NUUÉ_nI0UE

ET r,A Tt{uoRIE r}E L'APPROXIMA'IION

Tonre

16, N" I,

1987,

pp. ZZ*80

PEAIí SIìTS, pROpDTì t¡zicns r\ND IIOUñDAA,IES

]. Iì.,\SÄ (Cluj-Napoca)

., ^{1. rn [1] J. w. B*^ce}

^âncr

^w.

^{R,. zame have} studied the peak sets t,lre pro-per_fac-es a,nrl

the

bo'nclaries of .a rineaii

-;b;;;;; _{H oi C(xi-by}

using_the

rlual

-F1';-the¡.-hzùve _provecl

ilrat a

certain's-irbset

of the

cho- quel, bounda,ry'

of

11

is

also

a liounclaly for

'ral, il.l:t"'Ëi"i,"iuriïi'f,ä'ü3Ìlîåi,l î"îj"TllJu'"ilr"

B. n - ^is

^{also zr,} bòuirclary

to" ¡¡.

*""^"

2. r,ct _-1

bc_a _compact

rra'sclorff

_{space ancl}

let H

^r¡e

a

lineEr ,subsproo

of

O(.\') $'lriclì sc1irur,a{r,s

,l'

:

l,et, rrs

clcrrote

II+ :

_{ltr ë}

_tt

_:

_{lr >}

₀

oalled a, peük sBl

if

l,hetìe cxists a,n /¿

: _{0). r\}

nolt-cIìtpl.v

scl u'llir,ll js

_an

llen er(,l,ize(¿ ?el,lt, ^seL.

I.¡oL us denol,e

lrv

J/

thc farnily of alt

ge-neralized peak _sets and

by

JJl ttr.e

famil.v o{

rull

rni'imal

g.,1'-erí*rir.,r

peak

sets.

Añ apptication

of Zolnts lemma sho.ws tha1, ever,yY

I( eV

contains a,

K,eW,

L,et

P(II) : _{r

e

X: _{{ø} is}

¿ù peak sel,}

I(H):{neX:{n}eV}

B(H) :

_U

_{{K; Ke,T}.}

îhen P(I1) c: I(.II) c ñ(fI).

A lon.empty,

^convex

sribict Itr of II+, is callccl a

^jucc

of

.E¡+

if o t

^ct,

{7, j, geIJ't, (1--

ø)

I ^-t

^ag

^{eZ irnplie,} j, _i1.n.

A non-etnpty

snbset

E of H1 is a face iff (i) n ₊ n ^c:

^ú), øütr

c f) fot all

ø

e

1,,.,., and.

(ii) /e ^D, teZ, ⁰ ( g

_<

Í

^implies

9e ^Ð

^(see l1)1.

A

face Iù

of Ir+ is

saicr

to

be proper

if Ir _{1-H+ ; this is}

_equivarent

to 1t'E.

r-'¡et

's

^denote

^{by /,the family of all}

prope-r faces

of E+

anct

by

-ilf

the family of all

maximar

prorer

fäces

; ^ad

ap^plica,ti;i,

;f

zorn,s remma shorvs

that e'ery

proper faôe

is

containäcl

i" íil^ili,"åt l"opu" t*.".

I

Il iglt

(lal^School ol'Informalics

ecr I'urzÌi ¹ 40a-- ¹ 12

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