View of Approximation of continuous set-valued functions in Fréchet spaces II

-\LVfl IÍr _Ii'I' ùf

I)Ii'IHEOIìII] I)Iì

A'I'I(;A

-

^{rìlÌ\' t il i l )'ANA} L'ÀPPIìOX tNIr\I'ION^{r.\' SI' N t rNIì'iR I QtrE}

L',lli

¿\.tr,Ysfì N Lr ùIIqRIQUIì

ET L,\ TU¡iOtìüt tr)tì L'AIìI'RO

XIilIA'llXOl{

Tomc 20, No'

1-2,

^{'1991, pp.}

25-38

APPII.OXI1\{A'IION Otr CONTINUOUS SIÌT-VAI-UED trUNCTIONS IN trRÉCFIIi'I SPACIiS. IIt

Àl l(:I IìiLtt ^(l¿\NIP t'lI (llari)

Àbstrar'1. Irr this papcl rvt'. appll,somc rcsul[s obLaiuccl in Lhc filsL pal'I to tlte a¡rpt'oxitnatiott 6f sitrglc-r.alued conti¡rrrous ltrnctions dcÏirrcrl on a cornpacl llausclotff loltological ^spacr: ancì

s,ith vaìucs in zr lìrócheI spacc.

Iulrodrrctiull"

'-l.he l.ell-lcno\\¡rì resulbs

on the Korovkin

¿ìpploxima- 1,ion

o[

continlrous leal-valuec1

functions

h¿1,'se playecl

a crucial ^rôle jn the

irrvestig¿ì,tion c¡f set-valuecl continuous

functions (cf. [4],

15]

anil l8l)

and tììe lesnlts obtainecl are

strictl¡.

rela,tecl

to

the cxistence

of a l(orovkirl

set,

of

continuous siugle-valued

functions; in the first parl, of this

paper

[2

] ^r'c

^hat-e

follol'ecl ¿ clifferent

approa,ch

to the

approximation of set- r-aluerl continuous

ftrnctions,

rvhich is indepentleìtt

of Korovlcin

sets of single-vzlluetl

functions; this

aì.lox.s

us to apply the results in

l,he

first

pa,rt

to tho

case of single-valued functions

rvith

values

in a

^{Fréchet spa,ce.}

As a

consequence, 1ys

sþf¿in

^{zì, n¿ìtural} generalizal,ion

of the

rvell-hnorvn

câ,se

of real

valuecl functions.

lYe shall

assurne

the

sarne

notation of

^1;he

first palt ; 1/

clenotcs ^zl, real lìréchel, space,

ll a

ltase

of collvex

olren neighbot'hoods

of 0 in

^JJ

¿t-t't(l(6Go,,t,(Il) is

tlle

cone ^oü

all

Irotr ernpty conr¡ex cornpact sul¡sots

of

-14.

rlloleo \-er', -ve

fix a

courpact

Ilausdorff topological

space

,f

ând tve

slrall

clenote

by _'6 ('Y,I:/) the

spacc

of all

continuous

functions

orr

-Ï

rvitlr

valnes

in

-¡9

anil by ç

^('Y, GÇt'ttu(It)))

the

cone

of aÌlcontinuous

set-

\.alued funotions frorn

X in

G(6

"r,t'(D); ^G(X,

^{1l) and}

G(X,

GG"rzz'(Z)) are

i:oth

e-c¡uippecl

s,ith the topology of the

uniTorm conrrergence.

Irinall-vr ^r\¡e I'ecall the

notation/

< tl

+ ^Ir

^{(1, g}

^eØ(X,

(66ortu(lù)) _ttncl

Ir

^e

tì) to

^inrlicate _.:f(n)

-

fl@)

)- l/ ^for

^cach

^r

^e

^X

^.

;tp¡lroximatiorr

oI

eou[illuorrs vccf,t¡r'-r.alrrotl

fttnctiuns. ln this

See-

tion,

n'e shall appl¡.

the main theoremof the filst

ptr,rt,

l'2, lheoreir.

^2.+) 1;o

obt,ain

sornc

l{orovhin-type

theorerns

for

single--t

aluecl

oontinuolls

furLc t,ions.

\À,'e

shall introduce a

cl¿r,ss

il of linenl'

continuorts opelators on

tlc

spa,cc

çl(X,

1l)

rvhich

câ,n

be

t'egarcìecl as

a

gencralization

of

^rlonotone

t ^\\jor.li performcr[ undcr Urc anspicrcs of l"tre (ì.N.4.ll.4. (C.N.lì.) ancl ]I.U.lì.S"I (60)i) anrl supportc([ b.v I.N.cl.A.tlL

26 Michclc Campiti

i,o opera-

rcolG(x,

appl"t'iiìg

I4irstlr,, r','e nccd t,o consider

the

set

(l .l

) ^,7(X, ßGo.nu(It)):

_{{"/ c}

G(X,

G?,,.t,u(lr)) _| iltere c,:tist grr. . , ,p"

e(t'(X, Il)

suclr, thut

f(t:) :

co(9r,. ^.

.,?o)

(et)

for

^{each n}

e,y|

(fot' _each t ex,c'(p*,....,-ç,)(o) _tlenotes thc

convex

hull oli tht¡

^sc.l;

{qt(r),.' ^{.q^(ø)});} in'iiíe tótt,í*ì"g.'-i-,å"n-" _s,e rist *olnã pro¡re.ties

ori

Ø(X,

(îG.,¿t,(1))).

[,Rllr

sel

.Ø(Xr,6ß,.,,t,(ll))

isttsttl¡c<¡tco,f

,t()r,ilt,, _t,t,(]ll) co.nkú'ryi_rtg

_alu,ad, 'fti¿cli,ons.'

t\[ote

9.o(çt,'.'

', _'9,)

^urxr

_ll : _co

(ür,. . . , þ,,) rtre

in v(,x,

(€G(,tu'U

))

O,'riie n,aíe' ^'"'

r\pirr oxirration of coiltiDuOus set-valui_.d functiolrs I I

Cortsec¡rLenLiy, r,r'e ltat't:

),,

Iír¡:.1

2 27

IXÀ,t'',:I

frqÞ

i-l i...r i=-r

i=.1

Ð),

7 ¹

þ

lt -,u

^{J- ,¿,}

: _E

_À,q,(r)

j:t rùntt

q

1- 'L ^pl|¡@) -

i -,1

()'iV.¡e¡@)

t ^l¡t

¡ü¡(ø))

:

ll

(t.2)

_.[

-f

^g

-: co

(er _{-]_}

ür,..., e, *

^ûr,

(r'3)

_ì,1'

_-.

_{,1r, (),p,,.}

: _{É í:}

À¿p,(?¡

* ^{¡x¿);

i:.1 j I

'Llris .yields !/ e

co (ç, f {r,.

^{. .} ,go

J- ür,.. .,gr

^_j_

þ,r,.,.,

g,, fû,,,)(¿r).

,

.. ^Since y-e90 (pr,...,pu)

(ø) ^-.1-

co (,þr,...,ú,n) (æ)

¿ìnrl ^.il

€ry

^a,r(\

z.lrl;itt'ar'.1', (1.2)

is

truè.

,llinall¡'(1.3) is trir.ial.

ffil

11

n

^>> 1

anrl gtt...t,p,,

e

G(X,,I4),

u'e shall consiclcr.the sct

(1.4) A(pr,, ..,pò - ^[ç . G(X,Ir))

_lJ'or each a e .T' ilm.e emi,sts

i : 7,.

. .,tt, s'uclt, Í,lta,t, c¿(t;)

:

q,(n)1t

i¡Í

all

collti-tìtlous functions orL -X ^{-w,hose graphs}

are

containerl

in ilre

union oJ

tlre

glaphs

of

gr, . . .¡g,,.

tlrli<'s

Llle'alrre

0

olr

Llrs

irrf.l'r,¿rrl- t ,, t

l

^ancr

^{ag'c*i u,itrr} g,

ers.-

*,îrere, lrclongs Lo

A(9r, 9,). ^{L t'} * I'

^u'

l

,..._.,Mot'"ovcr',

il' ?tt...¡g,t

Jriive pai'ü,istr

rlisjoi'û

gr;aphs brrl,

.\- llas

iln

rtìtrnrte

Jrurnt)('r'

oI

corlnecterl crxn¡torrcnts,

l;o harc

ag,aiu ûlra1 t,hc srrL

A (gr,

..

^. _, ^co,, _{) rna.\'} be

'ot, finite

/r'or,

"-rn,rrr1rle, consitler

,f

==-- {0}

, g, _{ ; })

^,

.tLou,t¡r,et'r rve have

the lbllorvilg.

t,r,opositiori.

l)rioposi,r'loN 1.2.

.l,el gr,...¡,g,, þt Þ 1)

^{ba ùt,}

ß(,y, l!))

w,ítlt pa,,ír,nise úisjoint grcr,plts.

-.

.'Ih.e.n,,.the elent,enls ct.f

tl\.gr,.,..,?o)

_ure

all

fltt: .fu;nctiot¿s

in

ß(<{?

]r)

wh

ich

coincitre

uillt

sor¿e'

ç,G'1

^{1 ,.' .'} '.",n)

on

^euter).y

''ionür"tr¿

contþìríerrú

o;[

,l.

(lonsequ,entLy, i,l'

L is

^u, cr¡rrnecled, topoLooi.ul s\tucr;, ^,¿.oe lru,,ua

(f ir) _A(qr,.

_.

_.,rpò=:

{qr, ^{. . .,ent\,}

'¡ _9t -l

_þ,,r. .

., _?, -l

þ,,),

,),'!u).

-

^.l'roo,f.

It

is obvit¡us _LI.nL

.F(X.

i-ulrctions; therr,

iL

suffitre,s

i,,

lsfì"nl l,lris

ii

rviJI I'ollou, tttaL

g(,\,

^(€%.,.,,

,

^.J,ol,

^{./ :.co(g,,}

^{. .}

^.,?,)

^anrl

^rt

^_

rr,nrl,f'jx _{ir e}

)[ ;

^f'or

^crclì i: l;'. ...

e rro(

9,,

^{. ,}

^g,,Xr,) l_

^{tro lrN,,} . . . _{, lJ/,,,(}

co (gr

-l- ü,,. ..,,9, I ü.,...

¡ gt

I ù,,,,..,g,1-

Q,,)(¡r,)- co

(gr,...,g,)

(,1)-,

co

(qrr,...,ú,,)(e,).

Convelsely,let,l/

:

Qt1.a, rvith xL

€

co.(a,,,. . .7,!u)

(*.)antl

?) e oo (ür,. . .

.;;

.r*u)

(:r)

;rthcir, there t,xisí À,,..1,j,o ) 0 ¿¡'cl

.r1t..¡,{,t

> 0 s'ch

1,h¿1,

,)ì

_^, ^=-

i, _I]

_{u., =-}

I anrt r, = Íì

À,q,(¿), ,,

: _i,

_{r¿¡ù¡(c).}

i t j.-t i.t i I

_: aì

ìvIich<:lc _CìanrpiIi

()v,.

. .,),,,,) e R,,, i ^7.,

)0folc¿lcìl i -=

,|,

4 lt Approxirnation _oI c:ontilLroLts s,c[-valucd funcl.ions. II :i t)

cr,rt,il

, if ^.Y

^{ltu,s ttr,}

(ttt ) 1)

cottrtected, contltctttc,n.ts, ilte.tt,

cîìd (/ (?,,.

. .t,9,)) __ _{n,,, .}

ed.l'j l..h cotì1 ilìtìoì.ts Ijrrt¡¿rr, (Jtr)el.¡_r,1 ()l,ri l'ol Iori'irr.g llrï)l)(,1,1.\, ^.

t'tn'l ü','

^{. .} r'!,,, c

A(?t,. ..¡,?,,)

sut,!,

9 e%('Y,

0),

p e r:o(çr,.

..,?,),i, =, ^L(ç) e

co(1,(l/1),...,1, (þ,,,)).

( r. , ) lÍ]t"'ttrttsl

v

,

^th

c itl<¡'tit' opc'r,tcu,

sa,1,i¡fics

r:onrli1,i''

_(l .T

)

þr.[;2, .lll n.e rlcrrrotc b¡, A,,,(rir

) ^I) the

set

E)

sal isl'it,s crlrriil,iolt

(l.l).

';:; ,,!l;,:,;¡','ing

1';r.,¡;er ^1'1;

,g

e,6(X, D),

c¡ c__

co

^(g1

,...r?,,)

^,_

l,(?)e _co

(rl,( o,

),,..,J,(y,)).

i-rr<1, l:'r'(1.10)

l'r,:if Í ?tt''',Q, ^e%'('x, lt)

l¿rt,'L'e

ïtainuise rlisjoiltt

graptrs, [ttetr, .l'rt, ^r,ru:tr

Lj i¿(":)t,,¡)t Lr 1,,(í:,,r,) ^(,,.)i

,e.'/¿/(,\, r..;r¿,) 6r...i.rlc:\//[ \i--'j )'''l

Conrlition _(1.7) uerrer¿llizes

in

abstr¿tct iJpiÌccs 1,he r,ôle

gf

¡r9¡,t,rlric.

oÌrera,iors ii.r sl:r,ccs

oi

'eaì

^r.¿¿lnecl

o.;ntinuorìs-iri"";;;:.",'ì^',:,ì"rj'ir;'r:,;.1;

Ille

ftrllou irrg lì,oposi[íot),

.n e cl.e tl

_c.o ^{r t t 1,t} u c

t

il' u. u,s tt, o r.f

I'

^{I,o 1,t o L o r¡ irn l,}

) ^is

.rt lùtear

o,¡ternlor _,¡i:ì.,nt iC1-t", Et¡

arpn.,rulent

X.

und

X'

o,l'

X

su,ch, Lhat,

_l

^.,,,.

,t

,- U _f

.

_u,ttd

(1.13) qe(ú(X,

_tR),

? )0+ t,(q)>_0on,,yta,nd _L(9) ç0oz-}._.

,4/[orcoaet', i,J'.1, scLtisJ',ies cr,)

or

eqr.Liualent,ly b),

to. ca,

lul;e

(1.14) _{X ,}

.-.1,(t

)

_'(

f0,

^oc)),

_u

==

L (r)

^t ((__ co, 0 l), tohere

1

ilenr.¡t,as Llte oon,strtttt, ft¿nctiot¿ oJ constartt ,ualtte ^.l .

I'rooJ'. a,)

- ^¡¡

^J,ct

jI '

^{i¡lrtl .T}

-t

ÍÌl'(r 1111¡sc6[ ¡ulrscts _ol'

-L

l¡ncl

,[

Ii I

l l I

(l .tì) z\.,,

:.

^I

I

D¿I

,

//i lrtì(l

_Âr

\' 7., ^I

_l't ^,

ir j

qcrrtlil,iott _{(f .7)}

lrir,r, lle

t,cls1,¿ltctl as flolìou,s {J q)

, ^il'

p.,..'..,p,ì,

e,r,() " ^" '

^r"

Lhrr,l, Ior

el,'lt

_?

e,ì1-\-, li¡,

. . . ,),r,) e ^{t),,,, s'tcclt,}

lttal

tr(

(i'1t'1, il' _?,,.

.

.,?,

_e'c(

I.lttr.l,, _.l ot' cttt:lt aue'.I _,

' ^{.,ú,, G}

-l(gr,. .,p,,\

¡ru.t'l¿

e-Y,

t.heru

eristi

('),,,..

, ol

erlnivlt,ltlril¡,

. rú,, ^{e ,,,1 (c¿rr. .} .

¡?,)

^sn.uih

!',

,..r,,

^u'1,.1,, .,r,,,, [''( o)('': 0)

]

^{-=,^,,}

..\)

u

o,,,{'

⁽

¡',, ^+,)

^{r *,rl} }

-

(lrrlrtlitio'(l.ttt)

_{sr¡ r,s}

f Lrat,l.or,r,a<,lr /.eg(,.1-,Ç,C,.,ta(l!))), t'rrr,r,<,xisf,

,ù,, in ^,6(-r', ti)

*u¿rr

¿j;;ì'f -:";; i,i,;

^{.,,,f,,,) arrtl,}

ràll,,Jn",l,

_,r,,

€,,[;

,9,,,r,{tt('"",t ^:,,,,, _H..,,, {" ^{(Ð,i,ü,) tr,.t}}

^"

,i

¡

ii

'r i.c. _{9(.') e} co(gr(.t),. . .,p¿(.r)) tor cach ¿ e _\

:i t) Michele Carnpifi

Tluitirìgfrl'e¿roh

geG(|0,1.|,

[R) ancì

nel.0) Ll,L(ç)(n):

_?(.n0)

-

^{ç( )rl/}

(i

7 Approxirlatiot-r oI c<¡ntinuous set-valuccl ftrtrr:tions II 3l Obsclr.e

that

conclition (1.11ì)

is

ccluivalerrt

to

1,lie follox,ing

{r.1ir) q,þeG(X,

tR),

ç ( _ü *r(ç) _< I(t!)

on

X+,.L(e) _< r(,þ) on.T lìnr¡lnri

1.4.

1. lhe irnplication b) - ^{a) in}

l?loitosition l.B rernains

ii'uc

also

if -I

^is

^not

connected,

but the implication ai+

^{b) ctoes}

^not

^holcl

ìn

^genera,l.

__.__Fg.eTallpl_e,. cousirler -Y

:

_{0,

_1}

_and

_lhe

_operator

_L r.6(X,

_IR)

*

-n(ß(X7 fR) clefined

b¡rputting', for-eacli 9e ^.ó(-I, tR) attùt,e'X.i

(1

.r$) L(p)

(n)

:

_e(0)

-

^{?(r ).}

s:iïì.e

',',i1"

ìiliil ,i,i'$l,1iii',ï"*ì)",,,';ì",,,1..'î:"åìlä'î,1;1

'6(x' ir"îi'jil,ñ.'*,li|,;'^îïll tf,k,i,rÍlJ _¡,[,;

prr . . . _79u e

ß(X,

[R ) and consider. the

Proposition

1.8

a,ncl lìcrnat,l< t.4.1,

(t'7)'1'R).(ß(X,IR)satisfiescondibion

In the

c¿lse

Ð:

^[R ob"qerve

tlial, tbele

exist continuous linea,r, operzù-

tors L zØ(X, R)--, ^Ø@,

fR)

rvhich satisf¡. condition (1.2) antl

are- not trtotrotone;

-

^{G(1u,, D_l,B)}

an

Qt,,b e,ranrple e ^{fR, ra}

is 1ðr)

furnishecl

tlefinedb¡'puttirg,Ioleach p¿q(tä,AL,n) bv the operatol f,:G(la, ltl,

tR)

-,

r.ìncl rr e _l-g,, bl,

pclssible rnt(Il))--+

(/G _ot.zt.,(

:6(X,

^IX

Pnoposrrrox 1.5.

Lct L:(€(X, E) ->G(X, D)

be ta rutn,ti,,uous Lùrcetr tryteratot' sal,isfyittg (l .'i).

!I'lten,, {or ^each,.f e

Ø(X,

G(Gr,;tu(It))) cultl ^,t:

eX,

^th.e sel,

(t'17)

f'1

"'

:-

*nl!,u,{t'(ç) (ù)

i"s

a,-non

cntptrl_coltl)efl golnpctr:(, sul¡set oJ'

It)

a,ntl tll¿ sct-t¡q,ltte¿ .l'g,pcl,,iort.

.f ¡.: -X +.6'(6ont;(It)) d,e.!ùted tni pu,l,tirtg, ,for"ectch

;

"t;

(1.18) _l,,e;) -

^.;¡,,,

is

cot¿tinuolts.

,

M.oreouer, th,e.n.u? !'.,._j

{\X,

GG1tn,(IJ))

-, G(X,

6ßr,tn.,(D)) ù,eji,trctl

bt¡ pttttittçr,

for

^euclt,

f

^e

^Ø(X, (6.€inu(E)),'

(1.r

e) r,.ff)

_{-= 1,.,}

ùs & eo'¡ttinlllott,s tnon,olcnLelly.eiy op,cralor

þon, ^the

stt,Itcottc

Ø(x,

GÇo,tt,(Il)\

9l

_^ß.(Xt

G(r,tt,(7ì)) itt, G(X, ÇC",,,"p¡¡

søti,sfying condìtioin,s (2.2)'ctrid

(2.3) of _I2l.

e

Ø(X,

(6G,,nt'(It))) and

iT e X , ^{Llte set}

[',,"

^is

), the set

L¡,,,

is the ima

,.},,)(*)

^dcfinecl orL

ilrc

1;hereforc

f'¡.,,

rs also conrpact.

_r\or\', considel

the

set-r,alued funcbion ./r, delìrred as

in (l.rg).

l,ct,

*o.e

I

^¿¡,rrtl

I'Se3; try ilre

corrl.irrriLv

of l,(ür),...,/,(ú,j

t,1,,r,"

r,xisls

¿r

neighbolhoorl

À'ol

a'o such l,lral,

l'ol

eãclr æ

c'ñ'arlrl' i':"'i,.

.,tn) t'(,þ¿)(n) eL(lt¡)(no)

+ ^{V, t(+,)(n)}

^e

^t(þt)(n) * ^t.;

thcn, fol

t¡ach ø e

ü

_1cf.

(1.t0), (1.I7)

ancl (1.1E)),

.i't(,r,) L,,.,

oug,,,,lJ,(ù (,r)Ì

,,.,..

.Hu.,, {r(,Íì

^,.,+,)

^t*l}

,

çJ _-

^

^{',', t} )'i(J,(l.1,,)(,,'n) -i- I ')

\, l, ,/.r/¡J cr,/) ¡ I

I'(ç) (t') :

I

_{I e(l)}

df _,

ÍoJ

u'lrerc ^øo

ìs:l

fixecl elerncnt

in the

open int,ervar (a,

ú)

(the

ploof

Lltnt L sai,isfies (1.7)

is

b¿rsecl

on the rnonotonicity of ilie'inîegra,I ãnd ¡re

fact lhi.r,t,

lor

ea,ch g e_Ø([a,_Ö],

R)), ilre valuc t}at

_L(9) tãkes ¿r,t

n el,a,

b'l

deperuìs

onlv on the

values

that g

tahcs

in the

inl,err.al

u.ith enflpoíits ø nnd

ro).

Ilorvever',

condition (l.?)

is

not

satisfiecì

b¡'

ever¡r

continuous lineal

r,rpet'atol

I¡G(X, _{Il) -,} %(X,

D)

; for

c,xampie,'consicier

again the

real

casc ^.l? =-

[R anrl the

operator

.L:,€I0, 1],R)

^'-+

G[0,

1.],

ñ)

rlefinect

by

_fi

$tt ^n,-r,1,¡,¡

-lt _\J

(r.r,..,i,fl¿)el¡i¿

liL -'ì

,J -

f,' i

^,);

C i. I

\i t ,i',,,)e:\,r.

U

À,,r,(rf ,)(u;o)

l) *

^,

rvlrt¡r'e ,t;,,

is

ti.red

in

_10,1

j. l)e¡ote b)'

_{çu ancl 9.}

ilre

consl,ir,u

nonstanl r'¿rlue 0 aucl respeotivel¡' 1

;

^{thcn, fcll eàch ?,} e 10, t

r'(},ps

I G - ^l)qr)) (u):

{)

arid (Í.2) is

uot, satisfiect,' ^'6fl

:,,,,

11,,,o,,{'(É

r"';''')

1'r.,)}n t - ^L1'*uI t" :J'''('t:,,) ! ^}',

i, fun ct;ions of

I

^{¿ì,nd âiel0,}

ll,

Michele Can'rpiti

and sirnilar'l¡,,

l'þù-,f,.(n)lV;

since øo

is arl¡itraly in -I, ^the

set-va1uer1

frrnction fr. ^is

continuous (cf:

l'2,

^(1.6)l).

Nol',

consicler

the

mup _Tr. delincd as

in

(1.1g).

The linearit¡r of Ir

follorvs

from

_[2,

Proposition 1.1]

and

Lenma 1.1; moreover, by

(1.1Tt (1.1.s)

altl

^(!._1_9), i1,

is clear

LhaL

T¡. is monotorre ancl

satisfies (2.2j arrtl (2.3)

of l2l.

. ^thus,

^{rve _have}

^{only to} shorv_that

^,I

r, is continnous. Let

I¡ e t3 _; since 1,

is

coptinuous there

exists [/,

e

!3

such that,

(1.20) çeG(X, E), p@)eU, for

eìach.

,¿eX - ^L(p) (n)elr

for.

e¿clr r

^e

X.

,Norv,1et U e E be such tlnat

t ^c

[/r and consicler,,/,.r7 e

g(X,ÇG

_otru(ú))) satisfying

,f <

tJ

I ^U, !

^{< ,/}

+

^{¿/. T,}

(1.18) ancl (1.19)

there

exists

a

sele

since,/ < f ^-l-

^{U, there exists}

^a

^selec

fol

each I e

X

(inclescl,

it

suffices 1,o

t

t ^+> ç¡(t)

n

(p(ú)

- t/) ^which is

^lo.w

I'roposition 2.5

aucl

Proposition

2.

- ^I'(|t)

^{(ú) e}

I' for

each I e

I/ and

in

tlris

vielcls

y : L(q)

(n) e

I'(þ) (r) *

ancl ø e

.Y _are albitlalv, rve

obtai

$¡a)r, \\'e have

'I¡(g) <

^1',,(,f)

1- V

^a

At this poinl,, s'e make

the

I)r¡rr¡rruroN

).(t. Let L: G(x, rl)

^--+(€(x, rt)) be w contimto¿tslimeu,r ope;r'cttor sat.isJying \t.l_).

fVe

^shal,l

^{suy that u}

^subset

I

o,f

V(X,D) is

øn,

o1'-I{oroultitt, sct

in

%(X,14) if _,

for

^eaclt, eqtri,con,tinuous net

(L,)f., oj

Lineo, ltaru,tort f

rotr 4',X, E)

in, ¡¡s3tl's:t,li1f yin,¡ conditíon, (L.7) ct,n,[I

siclt, tltttt

^tl¿e

nat, (.1', (f _)),'¿, conDeïges to ['(^¡\f6y etcch. ^i el,zue a,l,solt,6,ue that,tlt.e n,et

(L,(Q),1,

conuer(tes to

I'(9) e6(X,ll).

_

Il ! ⁱ⁹ tltg_

eru,tor,

un

f,-Koroulcin, set

itt, G(X,

It))

ui,tl

^he

int,ply

caLlecl u,

K i'n Ç(X, t).

We are no"r'

in

a pcsil,ion

to

state

the

follotying lesul{,.

înr.;oluilr 1.7. r'et :x

l¡e

ø

co,n.n,ectecl compact

Eausttorff

toptoto¡¡ical

sï)(t,c.e. tmtl

.L':6(X, ^H) -'€(:X,11)

^{be et,} con,l,inuous linc¿ar operaior satí,si,¡¡iqg

t:ontlit'íon ^(1.7).

l-J

a

su.bset

I'

o,f

G(X,I/)

satisf'ies ttt,e followirtg ^conclition

(1.21) J'oL

^eqc.h,

9.eG(X,.

^IX).,

uoe;X ancl Y eE,

ttr,a.e

enist

^(u. ^{. .} .

..1 "(,el

^to,í,tlt ptu,irtoise

ûisjoint

g¡raphs ctnil, su,clt, tltut

g e

c0

(^(r,. .

.,

^(ò,

J,(^y)@),, .

.,

I'(y")(no) e

I'(q)@ò *

^V,

then

l'is

^cLn, L-I{o¡'oril¿in set in,

GQY,IÌ).

Proof

.I'eL

(L,),<., be an equicontinuous

net of lineai

olrer,ators frorn

6(X, Iì) in itself

satisf¡'ing conclil,ion (1.?) and such thal, l,he neb

(r,(f))å,.

ApproximaLion of corrtìnuous set-valuetl func:tious II colì\'elgeñ to L(.1)

fol

cach v e l-.

to

_{-/l as}

in

(1.19) ancl, {or ehch

i

st _Yl,. . .f(t¿ e

I with

pairr,r'ise clisjoinú c0

(Yr,...f,")|.

¡

^convorgos

^to

^Z(la)

fol each

tt e

H.

with

pa,irrvise

clisjoint

gr.aphs such

í :i1i,ï : l: ",, ;),,,.,."å:TíJ, i';;

L,(y¡)

(n) e

L(y,)

_@)

I ^{y, L(^iù}

@) ^e

L,(yò

(n)

¡

^¡r.

B¡'(-1.12)and(1.19)welìa,vo, Iolea,crr

æex

aucl ^¿

er,t )

^ø,

r,(tt) (n):,^,.

!j.o,, {å ^{r,r,,r,l r*l}} .

r,.,...,llle

¡, _¿,t,,rt o,)(r)

^-r-

^r,)

-

_(,^,,

,lJ,*o, {å ^úft,)@')l) r ^, : r(tt)

(n)

t ^v

and

sirnilar.ly

!I(k) (n) -

^{I,(tt,) (ml}

I ^y

^;

T(ta).

titJ

ril,;?"*-t" ^corr'er'

^{we observe}

Í/

e?8, there _oxists /¿ e-X _such

that

{e} < h, ^,I(It)

(no)

- ^r({e}) _@l ^_l

V _;

,guo

eA

in

ûhe firsú nr¡rt

o[

the_proof

of _[2,

Tlreorem Z.4l

net (",({E})),.[,

corivergu,,

tä

"ffij'iä ^åi.1,

^{,p e}

^{6(X. Di}

rhar rrre ner (_r,,(p))a.

"oroÀrgä* úo

¿f

oi ^t_ ìtãri

rl'u¡r¡riri r.8.

_\ye

¡roi.t o't

i,riaü

in

(r-.91) ûl¡e r,oclui'errent

on

^(t¡. . .;^{t

[o

have

':ai.rvisc

rìisþint st"pilï"uÀ*òoti*i. ¡or,'ä*ri'pre,

_{.ousicrer the}

('rle'niror

L z6(ln,bl,h¡-"'ã1irï,

_¿,r,

nLiã, i'Ë'n,r,

,i;iääír""clbyp.*ing,

fot' each g e,€(En,

bl,lR)

_and

_ne la,

^bl,

L(p)(r) : _I

cp

ft) ^{tltand let}

J

q

t: _I

e

V(lø,

á_1,

R)l

p(ú)cll

:

6 ì

4

I

3-c. tsogl

34 lv{ichelc (ìampiti _1(l

11 Âpproximation of continuotrs set-valuecl functions

II Ther, for

each g

Fy(¡;a, b),lR),

^#o

^e

^la,,

bl and

^e

) Q, ilrerc

exist

rtt

rz ^e

I

srrch

that

_e(ø)-à

y" úrøí, r"r@i;".i ;;(r,i I q(*ò:

^{^(2(nù.}

Morcover, -/,

0-Korovhin

_ser

satisfies-öòn¿ition'(ï.i)"(ct. in _{%'¡a, b),} R) (ñd¿cà;-;-;öi'^

_Remark

j.4:1, ^J'-'J)) _]:ut f is

nol, a

35

Iìn¡r¡.nn 1.9. ff X is a

comp_act I{ausdorff topological s

\!r> l)

connectecl components, ancl

if we

r,e¿lace condition l,he follorving

(1.221

_for

cacl¡,..16.(+,h)), nseX

and,_ I/_e

tì,

tltetc

crist1r,.

E)

uittt, .jlr,irwise äis.¡oint'graphs" a,nd s,ttalt tiltut

'4 (^ír,. . .ryn)

- |

g e co

(y,.

. .ù,u),

pace

vith

^ito

(1.21)

with

' ' r7,

e$(,X,

L(y)

_@ò

eL (ç)

(no)

-l

^V

_for

eactt, ^¡ eA(^¡r,. . .,^i,u),

tlre,n _,

_E).

that ^to

^accou¡rü

dìsJ; ^th

Pairrvise.^

/¿)),1, _con- . .rï¿ e

ç(X, E) n-ith

pa,ilwise an$ ft,

.:

co(.r,rr...,.f,)Ì.

r.

the

special case rvhere

/,

is

space el

.l ^orfJ. toPcilog['trll"

(i.zs) ; d'itiott'

^:;

þo';,r:ri 'tt

"'¡^(nel.-'¿uí"1:I¡

9(,u) e

co (Tr,.. .,),,) (n) for

^eaclt. n

eX,

^(t@ù,. ^{. .} ;r^(*oj ^e 9@o)

|

^|r,

th,et¿ T'

l/,: [R. Ât ^{rvn lesults in the}

^cast¡

a'¡iX--+canclofinethefurrcll'iclns eX,

^:

a¡(n)

: inf /(ø),

þr(u) ⁼⁼ sup

/(r)

^;

nd "f

.:

c<t (u¡, _þ¡);

(.) _, ^(6,G o,,,,(R)) _1ct.

continuous ljue¿1,

aled

opelaLor ^,1,t,

(cf.

lùcna,t.Ìi

rvlrole

cone (6(X, (6Gonu(R.)).

also shorvs

that

the assocì¿ltóit

ch

continuous set_valuecl function

f

^e

^GlX,

GG.ozu(R)) (intlccd, evcr,,v continuousfunction

f e,6(X.

G(¿,,,,(lR\) can be

nnifbnnl¡,

approxiniaicct'b¡,

tte continu**--.ät.,*ì,iJ,f iiìí,,riäìí_

;;J,î';?,,,1'* ^.,

^(e

) 0),

rvhich sarisf¡r _@.t

- ^,)

^(æ)

⁺

^(p,

{ ^{e)(u) for}

lR)

is a

monotone

¡",$¡líriî::ff¿), (t.24) _T"ff)

_(n)

:

l;L(ar)(n), L(g,)@)1.

Tarcirrg

iuto

_account

trc

above

'er.ar,ri, ^{\vo cau} bricfry

'eüurn to

r'|{irsicror

scl;va.¡ed

"o"i;",.tãüJ tïöuor* in

_o'dcr.

to

eivr

ofiapproxirnation _processes

in

trrð*Jorr"

G(X,

GGoo,..,(R))., somo examples

1.

Consider

p ) I a

_anclarcl

simplex

in pr

^.

l-r,- [,^^

-t- :

i\nt,.

^.

^.,

^{(rr) e}

R,l *, Þ 0

fcn, r_.ac^ ri

:

1,. . ., _7,

^rA f,r- *,l,

tr'or each _z

e

[N:

^.we

recall

,r,,11,._rl]o

"ç¿_th_Bernstein

^operator l,::,,'Y!T;,,TJ

ì,s(x, Rl isir"?iîäb¡:

_scrting,

_ro'e¿rch

e e

,€(x, R)

ancl '(1.25)

I]"(ç) (ru..., n,)

=

,

^nr_

2tr

lt ú,.,, LDetl hLl, ,-lhD<n

It,r!,

-h

^{r) t'}

.,ltrt(n-hr-. r\' " '

^nla

1-

I ^{,,)" h'} "'-t''

,x,)

,

r(',,;,,+).

(1.26) B,,Ifl(rr,.., nr):

_|8*(a¿ì

(e;r,...,ür), _B,(F¡) (tr,...,nr)_];

.

i

i' lr

ir

36 Michele Canpiti

we

finally

obtainr

fol

e¿¡ch

f ^eG(X,

%'Gonu(R))

and

(rr,

...,ar)eX, (1.27) ß"(Í)

^{((at,. .}

.¡

nt)

:

t n!---_==_

slt

^{. .}

^.

^nt:þ

^(L .-

t,,,...fren

hrt....lr,rt.1tt,

-

^1,,

- ... - ^lr,,)! "'' */

\^

h l,_ ...1 hp<t,

I I'(^rr) (øo)

- ^t(yr)

^(¿o)

^I

^<

<

l/,(i,r,r) ^(øo)

_

L(qr)(ao)l

i lL(tzà

^(a,o)

_ r,(qr)(r,)

_I

+

I ilk)

^(ø.,0)

_ _1,(çr)

_(øo))

_I

_<

( _l/,1tt,.¡

_(ro)

- ^f,er,r)

^(øo)l

+

'ilL(yr,")(øo)

_

L(.rr,r)(¿,.)l l_

+ lL (ç) _(nò - ^{r(ç,) (ro) I}

^<

<à+8_þel3(e.

oo""ui|"oåtty

^(1'29)

follows fi'otn (1.28)

_rvit;h

-r,

^equ.at

to the identity lve

can

gile

_arlolrrer

applicaiion.o{,treorern 1.7

ancr

cororra.v

_tr.r0,

rr¡'

consictering

rtre pa.ticuräi, ;;;;-E:_- il; f;. ,ii,rr¡iri"ity, u,"

r,esrr,icr our.

atterrtion to ilre irlentity

_o1réiàio..

coHol'r"rnv l'73' Let 'r

^bc s.conne.at.e. cotttlt*ctIIau,sctor,f! topoLo¡¡icør

'oo"'

,,1,,!,,,t.,17;

yr;ttttset of

ø1.t., _R

j

_sc ür¡yans"äå'í,ìííí¿ïi,,it

.zsl.

lt':o' t.'uuu{ï;""J'!

F;l/ ^tl:::f f!:tf ⁱ :.,',i'-t!,su'ctt'

_ttts't

pr'¡.se,

V(:i,,,Ål:;¡.denoteï ihe't-ei"iuiti"i""oi.p,"'i;,,'å\'# lj

Kotoutti,+t, ser, ín

L,,tt'¡.'¡ ( _{p, < yli} (ç, -

^{pr.¿ o}

⁹⁾ a'tl

_ì,í,(,ro)

-

^yíe,u)

(

^l_

err,. Tt.l,

ear:ìr

,snbset.,/

of

_[1 ,,

...,t1],

cor:sitler.

ilre ïrurcijorr T¿:l ^1R,,

_r_illt corr,_

lrortenls

Tr,,(i : l,

. . .,

n) definta

ì.,.1ì

^(r,r _{1!- ^,'i}

if

i,

t' ./ rrrrì

^.(¡,¿

:_

"¡,,, ill ,í e,L

r,r.".rjì1i:'

^1,

^{-. il:.}

^{. . .,}

^{,;t, 7,}

^is

^u

; ,

_'\,.,; xlltt.:tl ','rl

illtl

_1j'

"

^._

r,.. lt(.lton¡j

_^l¿(,1

. t.], ..

glaplts; r

e co((-¡")¡.1,,...,,,1

f

^an.l

t2

.,

rr)

^the

by

^show-

0b"'¡fip

13 Approximation _{of itruous}

set-vulu ecì funcliolts II

coul

- _Ét *')"-"

^..-hp

Í (':

^lru

t?,

(1.27) easily follo'n's

by clcnotinglvith

An(ür,. ^.

of

(J.27) ancl,

rvitìr the hclp of

(1.25) anct (1.26),

equivalence

_U

eA,,(l) _Gtt.,...,

^fro)

o y eB"(l) bitrary _ll

e 1ì).

. ^2,

Consicle,r

? )

¹ ancl lel, -Y

: [0,1]"

^be

^the

^hypercube

of

[Re.

}r tlris case,¡ve recall that, for

^encln

n efi, thez-th'Èernstein

operator.

I),:G(X,lR)-* ^(6(X,

R.)

is

clefined

by setting, for

each q

e6(X, h)

^nnrl

(nu...,mr)eX,

B,(q)

^{(,rr, . . . ,,} fio)

:

r'i'(L -

frt)",

... *'io(I - ^r)t,a

^g

:f,

hv,..,ho==o

n,

t,,

1L ,ìT

1L It,,

Itt

^ho

As

in

the firsl, exarnplc, also

jn this

^{c¿ìrse ì\¡e} havethal, thecot'respontl- ingassociated secluence

(8,(,f)),.^

convorges

to/foreachfe4(Xrß8oati;(R)).

,{lso in

this

case, the

explicit

expression of Bn(n e [N) can be obt:l,inecl follorving

tbe

same

line of Ilxample

1.11.1.

In tho

^case f?

:

IR, the classical

definition

of

-t-I(orovkin

set involves equicontinuous

nets of monotone linear operators rather than

^line¿u'

operators

sal,isfying

condition (1.7); by virtue of Proponition 1.3

au<l

Iìemark

1.4.1, and

tr-Korovkin

set

in the

sense

of Dofinition 1.6 is

ahva,ys

an J-Korovkin in the

classical sense.

By Ploposition

1.3, rI'heorem 1.7 ancl Corollary 1.10, rve o'btain 1,he

following rosult rvhich is n'ell knou'n in the

case

of

monotolle Iinear continuous operators

( cf. _{[3.] ancl [l-,} 'Iheorom

^3]).

Conolr,Àny

L72. I'et X

lte c,

c

topoloç1fua|

space and,

f,

:G(X

,

) ^be

n

oJ

G(,{,

R \

i,tr, il,sel.f

satislying b)

of

fi

^{u, subset}

I

^satisJ

(1.?8)

for

^caclt, 9

e eX

^ce

.ls€l

suclt,that

Tr ( _{? {.¡2ancl} lL(yr)(nò-L(lr)@ù l(

^r,

thøn

I

^{,ís un,} f'-I(o¡'oukin, set

in

(6(X, R).

Moreouer,

if a

subset

I of 6(X,R)

satisfies Lhe follo'toi,ttg cond,itiott, (1.29)

for

^eu,clt,

^{ge(6'(X, R),}

^ro

eX

an,il e

) ⁰

tl¡ere eni,sl,.(1,

^(zeI'

^sr¿¿'./¿

l,lr,at

yt < ç ^<

^{^¡2 and ^iz@ò}

-

^yr(no)

(

e, th,ett

I ^is

^ct Kot'oal¡i,n set ùt,

ø(-lf,

IR).

.,e Mii:irelc CaurPiti ¹⁴ ì{ A.rr _E.I. Iill\{^TIC¡\

_ RIì\¡t'ì

D,r\NAI_y SE NIJt{litì I eUIì Drl,r.ltiioR IB

t)Iì

t,,r\ppl]\ox IN^.tIoN

l,'ÅN¡tr'I'sIì NruI{EItrQUn rl'l' LryIIIBoItIE _ì)E L,ÅppJtoxlitIÀ.i.

_I(}N

Tome 20, ¡o"

I

_2,

_Igrl.ll,

_pp.

_{llg_zr1}

7

"',''

2

1

co((i'")7.1r,...,,,1

)(ro), ^\\'e llst'u

,tl < anrl

hcncc

_^í,(,r'o)

2

(,1

c, {7,

^{. . ., n,} ) lrclonE;s 1o 1.ho closcrl}

^ball jlt

_IRo

_of

_t:erlter g(r'0) an<l ^r'a-

rlirlr

s. lLrhcn 4,, s¿li,isfics

conditiott (1.2:l). ñ

.L'illnll¡r

^11,p eþ¡rç1'1,1¡ 1.ìra,1

lian¡'exarnples of

su.brrets

['o1'

_'l(,11, ^[R)

sa,tis[r'ine conrlit,jon (1.29)

alc l'cll-linon'n,

^.'ìncl

flom

^{thcrrn \r'e câtt obtain}

rn?ùu)r 1'1¡1'1'p.spoirrìiny,^ c-rrrrnpl<rs

of

suJrscts

I',, of g(X1tl") _{ttefinotl as}

itt (1.30)

ri'hich alc I(ulor-hin

^scl.s

irt G(f

, [R").

]Ioler)r'cr, if

\ve corìsid{'r ^i', srrlrsct

l' ol

^(6(.\-,

R)

s¿ttisf

i.jng

⁽¹ .29) a,lrcl c'otrsisl.ing <-rl

p

{)lcmclll-(,

lllc

t'rrlles¡rrrrrrlini¡

t(olovkin

^{se1 7-,,} i'n (6(.Y, [R") colìsistsexil(]11¡'of'ltuclenri:ttis ;

in

l¡¿ltii<,ullr,r',

i[ ,\

^j¡i

llte

cornpact ]'eftl jnl,er'\¡a.l [0, ^11, u'e catt colrsìtJcrt'iltrr rrri nirrrrrLrr nunrbt-¡r'

p = ^:i

^{¿lrtl obl,ailr tì}

I¡'ortlr-liiit st'f ilt (g(À',lR')

('{)-(lsist.-

iIrg of iirr, elernt:rrts.

A NIÌ\\¡ REIìINTiÌI\{IìNT O]ì .JIìNSiìN,S INIiQUALII'Y

SIi\¡IiIì S. t)tì,.\(ìOtIìP. _nuct NICOLIIT¡\ ìt. TOr--fiSOt.,

(lìriilc I krr.t,ulnne)

P. lì lì li lì ^l:i

"r-

(l ll S '{l¡str¡trlt J. ('c'tnirì r¡r¡rricnrious 1ìris ¡ra¡lc. i¡r corrrcctiori *'c shall ¡roi'I ouI rrirh so¡rc a rrc*'r'efi'c'rort ììr.1il_1,,,orr.,,

il,,irì;ì.;."ii*n

o-[ .].rrse.,s cris<:r,cto gi..",,. irc(Jrìalit\,.

Jn tlte

l'c:cetli; pzll]cìr _{i,3 f}

, ^the filst _nuthor

esta,¡li¡rhccl

ure f'ìì'rving

lt'fitrcntenl,

of

Jensen's ineqr,Lalit¡, _:

(t) _t (;-,Ë!,,,) ^*r") _r';É,,8, ],,t,tt (,+-)<

^{( >)}

_i,;E1,,

^r¡.,;¡),

L llclcrrs, l[., arrd L o ]'cttL't, (',.Q,,Gcontelt'i(lltcorllol'I(oroul:tn sc/.s,.J. i\pplox. 1'ltc'or'¡',

ii5 (1975), rro. il, 161-189.

2. C anr p i Lì, \,I., tl¡>¡ttoxìnuiliott of cottlirtrtoLts sel-ualucd l'ttncliorts in ^I;réchcl spctt:cs, I, l,'r\rralvso nurrréLiquc cl- la théol'ic clc 1'Lrpproxirnatic,r, 20 (1991), 1-2,7i-23,

3. ll¡r'g-rrson, ^{1,, l'1. O., arrrì} lìuslt, ll. l)., I(orool:ítt sels for atl operdot'ott a ^spacc ol ct¡nLittttous ltrtttliorts, ^{l'¿rcilic .1,} ìiaLh., (iÍ ^(1976), uo. 2, 337-?'+-c.

4. Ii ^{c i rrr c} l, I(., ânil Ro t.h , W., :l l(orou]¡t¡t l¡¡pe ap¡:ro,tinutLiott lltcorent _l'or scl-ualLted fuut:li.otts, Ploc. ¡\rner'. I'Iath. Soc., l0Z (1988), 819-823.

5. Ii e irn cI, I(., arÌcl Iì ^{o L} h, \\¡., ^{Ordcrerl concs} attcl approt,itntúiott, pt'cprittl 'Icclruische lfoclrscllr.lc Dalrnstadt, pat'L I, II, III, I\¡, ^1988--89.

6. III i c ìr a c 1, Ij., Cottlittttotts selcclions. I, ;\nrt. i\tIaLJr., (iil, (1956), 2, ³⁶¹ _-382.

?. P l o I1 a , J. ^T),, Appntíntaliort of cotttiiutotts conla.t-corrc-uttlttecl litrtctìorts bg ntortoLortc o¡tcralors, ¡rlc¡rlint l)trivclsitlatlc ^{Jlstadual clc} Carnpiu:ts, Iìrasil, no. 27 (1990),

8. \'i ^t nI c, ìì. r\., ,tTrplolitnrLlion ol cottuc.t scl-ualuerl [Ltttcliotts, J. Appt'ox. :Ì'hcol']',26 (1079), tto. ^.J, 3ll1-316.

1ìcccivcrl 1.l\.1990

I)i _lttr ^r limct tlo tlÍ A,I ttl.cntalica UttiutrsíLìi dclli ShLdi di ]"ari Truut:rstt 200 ltìct J:l.c Daoid, ⁴

't 0 125 B-+IlI (I'l' ;ll,Y )

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as

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l'ttlìou'ine. Ocrtain applications a^r,t¡ also ^q.iverr.

lllr¡¡iou'lr.

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f.:1

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one ltq,s l,.l¡e

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er,

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I

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