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REYI]E D'ANALYSE NUì,ßRIQUE ET DE TTTNONN OE L'APPROXIMATION Tome24, No' 1-2, 1995, pp. 45-52

\

SELECTIONS ASSOCIATED TO THE METRIC PROJECTION

s. coBzA$, c. MUSTÃTA (Cluj-Napoca)

Let

X

be a normed space,

M

a subspace of

X

and

r

an element of

X.

The distance frorn

r

to

Mis

defined by

(l)

d(x,

A[):= inf{ll.r -

yll:

y

e M} .

An

element

y

e

M veriffing

the equality

ll¡- yll= d(x,M)

is called an

element of besÍ approximation for x by elements

inM.

The set of all elements

of

best approximation for x is denoted

by Pr(x)

,i.e.

(z)

P¡a(x):=

{y

e

M:

llx

- yll= d(x,M)}

,

If Pr(x)

+

Ø

(respectively

Pr(x)

is a singleton) for

all

.rc e

X ,thenMis

called a proximinal (respectively a Chebyshevian) subspace

ofX.

The set-valued applicatiotr P¡r:

X -+

ZM is calledthe metric projection of

X

onMand

afiinction p:X

-+

M

stchthat

p(x)

e

Pr(x), forall x

e

X ,is

called

a selecfion for the metric projection Prn . Observe that the existence of a selection

for P* inlplies P*(r)

+

Ø,for all

.r' e

X ,

i.e. the subspace

M is

necessarily proximinal.

The set

(3)

Ker P¡4:=

{x

e

X:0

e

P¡¡(x)},

is called the kernel of the metric projection P", .

In many

situations

for

a

given

subspace

M of X

the problenl

of

best approximation is not considered for the q'hole space

Xbut

rather for a subset

Kof

X. This is the case which u,e consider in tliis paper and to this end we need some definitions and notation.

IfKis

a subset ofthe normed space X and

Pr(r) # Ø

(respectively PuoQ) is a singleton) for all ,r e K, then the subspace

Mis

called K-proximinal (respectively K-Chebyshevian). The restriction of the rnetric proiection Pnnto

K

is denoted by

P*ty

arrrd its kemel by

Ker

Pr4*:

(2)

46 $tefan Co bzaç and C.

(4)

Ker p¡o¡*:=

{x e

K:O

e

p¡ae)}.

Fortwononvoid subsets

u,vof

xdenoteby (J

+v:= {u+v:u

e (J,v

€v}

tlreir algebraic sum.

If

every

x eu +v

can be uniquery written

in

trre fonn x

=u+ v with u euand,v ev, thenu +v

iscared lherrtrectargebraicsuntof the sets

u

and

v

and

is

denoted

by urv. rf K=(Jiv

andthe appricatio' (u,v)

-+

Lt+v, u e (J,

-v

ev,

is a toporogicar Ìrcttteontorprtisnt betN,ee, Lr

xv

(endowed lvith the product toporogy) ãnd K trren K is caneá üte crít.ecÍ. toporogicar sum of tlte sets Uand Z, denotecl

by K =

U @ r/.

F' Deutsch [2]proved

thatif

Misaproxirninal subspace

ofxthen

tlle urehic projection

Pr

admits a continuous and linear selection

if

and only

if

the subspace

Mis

complemented

inxby

a closed subspace

of Kerp,

([2], Theore m2.2).

In [4], one of the autho¡s of th for a closed convex cone K in Xand P¡r,x to admit a continuous, positivel following sufficient coudition fo¡ flre

If

there exist two closed convex cones C

c

Ker

p*

and (J

c M,

such that

P*,,

admits a continuouù-, positively Theorem A).

and prove that

if

p,rrlr admits a contin selection, satisSzing some suplernent decornpositions

K=

C@

U,

with C

co'es (Theorern B). Although the conditions in trreoreurs A,and B are very crose

to

be necessary and sufficient

for

trre existe'ce

of

a

"orrtinuous, positively Iromogeneous anti additive selection

for pr,*,

we weren,t abie

to fi1d

such

;"",'i.Ïïïf;.1;""å]"tio's

rvliich mav occur a¡e illust¡ared

by

some exarnpres By a convex cone in

x

we understand a

no'void

subset K

ofxsuch

that:

a).ri +

x,

e

K, forall

x1,t2 Ç

K,

ar-td

b) À ...r- e

K,

for all -r e

K,

andÀ > 0.

A car.cfull exarnination of the

statem

e proof of Theorern A

in [4]

yields the following more detailed refo

THEOREM t\,. Let M be a

closed

of a ttot.ntecl spaceX ancl

K

a closed conver cone ín X. If o

closerJ

convex

cot.tes C

cKerPr,*

and (J c: M sìuch rhat

K

= C@ (J, tlrcrt the a¡tplicatto,

,;U _; ;:

Metric hojection 47

defnedby

p(x)

= z,

for x

=

|

+

z

e

K, y ec, z e(J,

isa continuous,posifivery homogeneous and additive selection of the ntetric projection poo,* . The subspace

Mis

K-proximinaland C =

p-,(0), U

=

p(K).

The

following

theorem shows that,

in

some cases, the existence

of

a

continuous, positively homogeneous and additive selection for

p*r*

implies the

deconrposability of K in the form

K

= C @ U, with C and [/closed convex cones, THEOREM B. Let

x

be a normed space, M a crosed subspace

ofx

and K a cTosed convex cone ín

x.

suppose that the metric

projection p*r*

admíts a

contínuous, positívely homogeneous and addítíve selection p suclt that:

a) p(K)

is closed and contained in K, and

b) x- p(x) eK, for all x

e K.

Then p-l (0) and

p(K)

are close J convex cones contained

in

Ker

pr,*

and

M

respectively, and

K

=

p](0)@ p(K).

tf p(K)

is a closed subspace

of

K

or M c K

then the conclitions a) and b) ar e auto ntatic al ly fulfi ll ed.

Proof. By the additivity, positive homogeneity

ofp

and the fact rhat

K

is a

convex cone, it follows irunediately thaf p(IÇ is a convex cone contain edin M. By hypothesis a) it is also closed.

By the continuity

ofp

the set

p-r(0) c

Ker poo,* isclosed.

If y

e

pt(0)

and l" > 0

thenp(I

.

!)

= ?".

p(y) = 0,

showing that X.

y .p-1(0).

Similarly,

yy yz

ep-1(0)

and

tlie additivity

of

p imply p(yt

+ yz) =

pbù

+

p(y)

= 0,

showing that

p-t(0)

is a closed convex cone contained

in

Ker

pr,*.

Now

we prove

that K

=

p-'

(0) +

p(K).

rf

x

e K then

by condition

ó),..

y:= x- p(x)

e

K. By

Condition

a), p(x) eK irnplying ;r= y+p(:u)

with y,

p(x)

e

K.

Using the additivify of the functionp and the fact that

p(p(x))

= p(r,)

(irfactp(tn): mforùl

m e

M)we

obtain

p(x)

=

p(y)

+

p(p(x))

=

p(y)

+ p(x),

It follows p(y)

= 0,

i.e. y

e

p-'(0)

and

K c

p-t(g) +

p(K).Since p-'(0)

¿n¿

p(K) are contained in K and K is a convex cone, it follows

that

p'(0)+ p(K) c

Kand

K =

p-1(0)+

p(K).

To show that this is a direct algebraic sunl suppose tlut ari element

x

e

K

adnrits

two representations: x

=

| * p(x)

and x =

!,

+

2,, witlty, !,

e p-l

e)

ancl

z'

e

p(K)

c.

M. It follows p(z')

=

z' and, by the additivify of p, p(x)= p(y')+ p(z')=0+z'=

z',irnplyrng

!,= t- p(x) =yand z,=

p(x),

It remains to sliotv tlrirt the eonmpondence (y,z) -+ y + z, y e p-' (0),

z

e

p(K),

is a homeomorphism between

p-t(0)

x

p(K),

equipped with the product topology,

2

J

(3)

48 and c.

and

K. To this end consider a

sequence

(y,,2,,)

e p_r(O)x

p(K), tt

e N,

converging

to (y,z)

e

p-l(o)x p(K),it

follows

y,

->

yan{l2,,

-+ z, implying

(yn,z,,)

) ! r z,

which proves the continuity of the application (y, z) _+ y + z .

To prove the

continuity of

trre inverse apprication .r

Þ

(_y,

z),

wrrere

x= ))*2, ! ep.(0), z ep(K),

takeagainasequeuce .Én=

!,+zn

e

K, !,,

e

p1(0),z,,ep(K), converging to x=y+ze K, where yep_r(,) ard

z e

p(K)' it

follows

z,

= p(x,,),

n

e

N,

z =

p(.\),

and, by the cortinuify of the

application

p,

2,, = p(x,,) -+ p(¡,,,) =

z, But

then y,, = xtt_ zn

+

x _ z = )), proving that the sequence

((!,,,r,,7),,.*

colìverges

ta (y,z)

rvith respect to tho product topology or p-r1o¡ x

p(K),

This shows that the applicafion

x

r_> (y, z),

x

=

l* z e

¡t-lço)+

p(K),

is continuous too and, conseque'fl¡ the applicatio' (y,

")

r-+ y +

z

is a homeomorphism between p-,

e)

x

p(K)

anrl K.

rf p(K) is a

closed subspace

of K

then

condition a)

holds

a,d, for x

e

K' p(x)

and

-p(x) arcin ¡:(K)c K

so trrat .r

- p(x)e K, srrowi'g

that Condif

ion ó) holds too. If M

c.

K

thett

M

=

p(I[) c p(K) and,

since

p(K) c M' itfolrows

that

p(K)

=

M

isa closed subspace

of'.

Theo¡e'r B is conrpletely proved.

Remarlc' conditions a) andó) are fulfillecl by the selectio'p give' in Theorcm

A', Irdeed, K

=

f

(0)@

p(K) implies

that

p(K) is

a close<l convex colle

containedinK, p(y)=0

Irr the and

following

z = Since

p(x), itfollows every reK

that

canbewritteninthefomr x_ p(x)= x_z

=

y €Kfo¡all x x =!*z eK. witlt

examples, trrere arways exists a continuous, positivery ho'rogeneous

X

= pa (O) @

p(K)

and additive serection is not true in all these

of

the metric projections cases.

bu!

the eq'arity

Exatnpre 1. Take

x

= R2 witrrtheEucritreannonnand

M{(xr,O):x,

eR}.

Then P¡a((x1,x2))={(-",,0)}, foraD(x1,x2) uRr, i.e.Misachebyshevian.subspace of

X

and the only selection

of

tlie metric projectiori

is

p((e.,,

,r)) =(,r:,,0),

for

(*,, rr)

e R2

.

Let

fi,= {(r,,

xr)

e

Rz: x, > o, r:, à o}.

Mehic hojection 49

a) Take

K

= {(xr,

xr)'.xr: xr,xr2oi, In

this case

KerPr,*

=

{(0,0)}

so

that tlre only closed convex cone contained in Ker

Pr,*

is C = {(0,0)} , The subspace

M contains two nontrivial closed cones U* :

{(xr,O):x, >

0}

and

g-

={(x1,0):xr

I 0} pG)=(J*andK+C@(J* =(J*.

b) Let

,< =

{(",,

xr)

e

R2:x, > .x,,:c, >

0}. In this

case

Ker

Pr,*

= {(0, xr):x, >

0}

and the only nontrivial closed convex cone contained

in

Ker

Pr,*is

C = Ker

Pr,*.

Again

p(K)= tI*

but

K

+ C@

p(K)

=

rt.

c) Let K

=

{(r.,,xr):x,

<

x,,x, 2 0}. In this case

Ker Pu,* = {(0,0)}

implying

C = {(0,0)}. We have

p(K)

= U*

c. KlMbutK

+ C@

p(K)

=

p.

d) K

=

{(xr,xr):xr)

0, xz > O}. tn tlús case

KerPr,"

=

{(0, xr):xr>

0}.

C = p-1(0) = Ker

prtx, p(K)

= (J* and

K

= p-1(0)@

p(K).

e) K={(tr,rr)eÀ2:;rr>0}. In this

case

p(K)= M cK, KerP*,*

=

:{(o,rr)t*r> 0}

and

K

= C @

p(K),

where C

=

Ker

P*,*.

Remorla. kr Example 1. a) none ofthe Condition a) and ó) from Theorem A is vuifi ed.

In Exarnple Lb) condition ó) is fulfilled

but p(K)øK,

while

in

Example

l.c), p(K) c Kbut

x

- p(x)

e

K

only ¡o¡ ¡' = (0,0).

In Exarnple 1.d) Conditions a) and ó) are both verified

but p(K)

is not a subspace of K.

In Exanrple

lre) p(K)

= l¡4.

Tlre following example shows that

p(K)

may be a closed subspace of

K

with

p(K)

+ M.

Example2.Let

X

= l?3withtheEuclideannoûn,

M ={(rr,xr,O):xr,x,

e R}

and K :

{(0,

x2,4):x,

e

R,t,

>

0},

Then

p(K)

= {(0,

x,,O):xre.R}'. ltt

and Ker

Pr,*=

{(O,xr,O):x,

> 0}, The equality K

= C @

p(K) holds with

C = Ker Prt*.

4 5

(4)

50 $tefan andC. Mustãta

6

Exampre

3' Let x

=

cfa,ó]

be the Banach space

of a'

continuous rear_

valued functions on the interval [a,

b]

withthe sup_norm.

The set

u:= {f eC[a,tfl:f

(a) =

f(b)

= o]

is aclosedsubspace

of

C[a,b],

Y2={f ec[a,bl:f (a)=.f (b)>o],

is a closed convex cone

in

C[a,

bl

and

M

c. K.

First show that the subspace M

is

K-proxrminar. For

f

e

K,

the fi.rnction

g

defined

by g(x):= f

(x) _ -f (o),

x

e

fa,

bl, is an element of best

apr._*";;:

forfinK.

rndeed, wehave

llf _sll=f@) andllf _hl!>lftÒ_nço¡l,forall h

e M, It follows that

d(f

,

M)

=

f

(a) and

g

e puW(.f).

The kernel of the restricted metric projection is Ker P¡n,* =

{-f eK:

0 e

parc(f)}

=

=

{-f

e

K: - f(a)

<

-f(x)

<

f(o), foiall x e[a,b]].

It

follows

p(-f)

e

Pr¡*(f)

and the inequalities

llpí) - I(DII=

ll¡,

- f,ll+l¡ç"¡

_ .f,(o)l

= 2.ll.n _

f,ll

,

for

frf,

e

K,

imply the continuity of the applicationp.

obviouslythatp ispositivelyhomogeneous

and additive onK. since

M c

K, Tlreorem B can be appried to obtain the equarity t<

=

p-lço¡ @

p(K).In

this case

P-tQ) =

{g

e

K:ic> 0,g(*)= c, foÍall x

e

[o,bl]

and

f(x): -f(o)+(-f(x)

-"f

(")) is

the unique decomposition

of -f

e

K in

the

fo'n /

= g +

rt witlt s

e

p-'(0) and

h e

p(K) (s(x)

=

f(a)

and h(x)

= -f(r)_ f.(o) fo¡ all x e[a,b]).

In Examples ld) and e), the

subspace

M is K-chebyshevia'

and

y

= Ker Pulx @

p(K).The

fotowing corolrary shows that this is a general property of K-Chebyshevian subspaces.

coRoLLARy r. Let K be a crosed convex cone ín trte normed space

x

and

M

a

K-chebyshevían subspace

tf x u-;;ere

exist rwo crosed convqc cones

Metric hojection 51

C cKer

Prlx

and U

c M

srch that

K

= C @ (J, thenC

:

Kerp¡a¡* and U

:

p(K) where p:

K

-+

M

is the only selectíon assocíated to the metric projectíon py1r.

Proof.

Since

C cKerP*,* it

remains

to

show

that

Ker

prt cC. Let

x

eKer

Pr,¡

ând

let y

e C,

z eLl

be such that

x -

y +

z,

By Theorem

A'

the

selectionp is given

by p(*)

=

z

and by the additivity

ofp.

0 =

p(x)

=

p(y)

+

p(z)

= 0 + z = z,

inrplying

x

=

/

e

c.

The equality

u

=

p(K)

follows also

fro'r

Theorem A'.

Apartial

converse of Corollary

I

is also true:

coRotLaRy 2, Let M be a closed subspace of the nornted space

x

ancJ K a closed convex cone in

x. If K =

Ker P¡rlx @

M

tlten the subspace

M

is K- Chebyshevian and the only selection assocíated

to

the meîric

projection

is continuous, positívely homogeneous and addítive on K.

Proof. First we prove that

Pr,"(y)

=

{0}

for every

y

e

Kerpr,".

indeed,

y

e Ker

Pr,*

is equivalent

to

0 e Ker

Pr,"(y). If z

eKer poo,*(y) then,

taki'g

into account the fact

thatMis

a subspace

ofXand z

e

M,

we ob.øin

lly - "ll= inf{lly -

mll:m e

M}

=

=

inf{lly - " - m'll:nt'

e

M)

= d(y

- z,M),

slrowing that 0 e

P*l*(y - z)

or,equivalently, y

- z

eKer

pr,*,

But thenyadrnits tworepresentations

y

=

y+0

and

y=(y-z)+2,

tvith

y, y- z eIàrpr,*

a,,d

0, z e

M.

The unicity of this representation implies z

=

0,

Now, writing an

arbitrary element

x

e

K in

the

form x

=

y-t z, with

y

eKer

Pr,*

and z e

M,

we obtain

Pr*Q)

=

P*t*(y *

z) = z +

pr*(y)

= z

*

0 = z,

slrowing thatzis the onlyelement ofbestapproximationfor

xinM,i.e.

thesubspace

Mis

K-Chebyshevian.

we

conclude the paper by an example of a non-chebyshevian subspace

of

R2 for which the decomposition

K

= C @

M

is true.

Exatnple

4.Let X =

R2 with the sup-norm

ll('',"r)ll= -a*{l-r1l,l¿l},

çx1,x2)

e

R2,

7

(5)

52

and

$tefan Cobzaç and C.Mustãp

luí:= {(xr,0):rq R},

K,= {(t,

xr):x,

e4 ,,

> 0}, It is easily seen that

P*¡*((rr,*r)):

{@t,0):xt

-

xz

1

nr 4

¡,

+

x2},

for

(xr,xr)

e K.

Indeed, xt _ xz < rn 1 x1+ x2 is equivalentto l.r, _

nl<;rr,

irnplying

If

(m',0) is an arbitrary element of

Mthen

ll(r,, rr)

-

(nr ,o)!l= rnax{l;r,

- ,t l,

xr} > xr,

slrowing

that d((x,,:r2),M)=

x2and(nt,,) e

pu6(e1,*r)) if ard

only

if

m

eR

verifìes the inequality lrq _

ntl<

x2.

TLe kemel

of

prr," is Ker

pr,*

=

{(rr,

,r)

e R2: | ;, j <

xr, x, >

0},

and

K

= C@

M,

where

t

= {(0,

xr):xr> O} is

a closed convex cone strìctl¡, contained

in

Ker pro,".

REFERENCES I'E'w'clrerreyandD'E'wulbert,

Thebistenceand(JníciÍyofBestApproxinnliozqÀtlath.Sca'..

. Anal 49 (1 9 83), 269 _2g2.

otrs and Lípschifz Continuot.ts 291_314.

et de Theorie

läpproxirn

s, Revue d'Anal¡'seNumérique lìeceiverl 3 Vtri I

994

Acadenia Romôntï InstíÍutul de Calcul ',Ti b e riu p opot,ic í u,,

p.O. Box 6g 3400 Cluj_Napoca

I

Rontânia

REVUE D'ANALYSE ¡,UMÉRIQUE ET DE rrmOnm DE L'APPROXIMATION Tome 24, No'1-2, 1995, pp. 5F57

A THIRD ORDER AVERAGING THEOREM FOR KBM FIELDS

CÁTÃLIN CUCU-DUMITRESCU ANd CRISTINA STOICA (Bucharest)

The averaging theory

is

one

of

the most powerfrrl tools

in

approaching problems governed by differential equations, The goal of this note is to present a theoretical exteusiou of the averaging method (based on important rvorks in this domain: [1-3]), materialized into a tlúrd order averaging theorem for differential systems having fields with the Krylov-Bogolyrbov-Mitropolskij (KBM)properfy, The theoretical results we shall present here were developed as a consequence of the practical necessities following from problems belonging mainly to celestial mechanics (and space dynamics), but not only. Our theorem and its corollary (for the case ofperiodic fields) describe constructive methods for obtaining approximate solutions

for

the considered differential systems;

this

recommends them for nunrerical applications, Their domain of applicabilify is very large, transcending considerably the celestial mechanics,

DernqntoN 1, Let

zbe

a small positive real parameter,

let

Í. e[0,co) be"a tirne-type variable, and let

x

e D

c R"

be an ¡l-dirnensional (spatial-type) vector,

Let

a:[O,co) x D

-) R"

be a

KBM

function of average a0. Then we define the operator

(t)

A^(a)(2,t,*¡,=

"ltolo,(",r) -

o01r¡]a", and denote

(2) llÃ(a)ll":= sup ltr(a)(z,t,x)l,

0<ztcl,xeD

DerlNltloN

2. Let ¿ be the firnction considered

in

Definition 1, consider á:[0,co) x D -+

R',

and suppose that ¿ and å adrnit spatial derivatives (r.e. with respect to the components of.x), Then we define the operator

(3) B(a,b)(t,x):=Ya(t,x).b(t,x)-Yb(t,x).a0@),

8

li

ri

i

i

ìii

Referințe

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