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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,

a subspace of

and

an element of

### X.

The distance frorn

to

defined by

d(x,

yll:

e M} .

element

e

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

,i.e.

## (z)

P¡a(x):=

e

llx

,

+

(respectively

### Pr(x)

is a singleton) for

.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

onMand

-+

stchthat

e

e

### X ,is

called

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

+

.r' e

### X ,

i.e. the subspace

### M is

necessarily proximinal.

The set

Ker P¡4:=

e

e

### P¡¡(x)},

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

situations

a

subspace

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

is denoted by

### P*ty

arrrd its kemel by

### Ker

Pr4*:

(2)

46 \$tefan Co bzaç and C.

Ker p¡o¡*:=

{x e

e

### p¡ae)}.

Fortwononvoid subsets

xdenoteby (J

e (J,v

### €v}

tlreir algebraic sum.

every

### x eu +v

can be uniquery written

trre fonn x

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

iscared lherrtrectargebraicsuntof the sets

and

and

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 proved

### thatif

Misaproxirninal subspace

### ofxthen

tlle urehic projection

### Pr

admits a continuous and linear selection

and only

the subspace

complemented

### inxby

a closed subspace

### of Kerp,

(, Theore m2.2).

In , 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

Ker

and (J

such that

### P*,,

admits a continuouù-, positively Theorem A).

and prove that

### if

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

C@

### U,

with C

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

### to

be necessary and sufficient

trre existe'ce

### of

a

"orrtinuous, positively Iromogeneous anti additive selection

we weren,t abie

such

### ;"",'i.Ïïïf;.1;""å]"tio's

rvliich mav occur a¡e illust¡ared

### by

some exarnpres By a convex cone in

we understand a

subset K

that:

a).ri +

e

x1,t2 Ç

ar-td

b) À ...r- e

for all -r e

### K,

andÀ > 0.

A car.cfull exarnination of the

### statem

e proof of Theorern A

in 

yields the following more detailed refo

THEOREM t\,. Let M be a

### closed

of a ttot.ntecl spaceX ancl

closerJ

cot.tes C

### cKerPr,*

and (J c: M sìuch rhat

### K

= C@ (J, tlrcrt the a¡tplicatto,

### ,;U _; ;:

Metric hojection 47

defnedby

= 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 =

=

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*

contínuous, positívely homogeneous and addítíve selection p suclt that:

### a) p(K)

is closed and contained in K, and

e K.

Then p-l (0) and

### p(K)

are close J convex cones contained

Ker

and

### M

respectively, and

=

### tf p(K)

is a closed subspace

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

the set

### p-r(0) c

Ker poo,* isclosed.

e

and l" > 0

.

= ?".

showing that X.

Similarly,

yy yz

and

of

+ yz) =

+

= 0,

showing that

### p-t(0)

is a closed convex cone contained

Ker

we prove

=

(0) +

rf

e K then

ó),..

e

Condition

with y,

e

### K.

Using the additivify of the functionp and the fact that

= p(r,)

m e

obtain

=

+

=

+ p(x),

= 0,

e

and

p-t(g) +

¿n¿

that

Kand

p-1(0)+

### p(K).

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

e

=

and x =

+

e p-l

ancl

e

c.

=

z',irnplyrng

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

p(x),

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

e

### p(K),

is a homeomorphism between

x

### p(K),

equipped with the product topology,

2

J

(3)

48 and c.

and

sequence

e p_r(O)x

e N,

converging

e

follows

yan{l2,,

(yn,z,,)

### ) ! r z,

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

To prove the

### continuity of

trre inverse apprication .r

(_y,

wrrere

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

takeagainasequeuce .Én=

e

e

z e

follows

= p(x,,),

e

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

colìverges

### ta (y,z)

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

### p(K),

This shows that the applicafion

r_> (y, z),

=

¡t-lço)+

### p(K),

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

r-+ y +

### z

is a homeomorphism between p-,

x

anrl K.

closed subspace

then

holds

e

and

so trrat .r

that Condif

c.

thett

=

since

that

=

### 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

=

(0)@

that

### p(K) is

a close<l convex colle

Irr the and

z = Since

that

=

### y €Kfo¡all x x =!*z eK. witlt

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

= pa (O) @

### p(K)

and additive serection is not true in all these

### of

the metric projections cases.

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

p((e.,,

for

e R2

Let

xr)

### e

Rz: x, > o, r:, à o}.

Mehic hojection 49

a) Take

= {(xr,

this case

=

### {(0,0)}

so

that tlre only closed convex cone contained in Ker

### Pr,*

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

{(xr,O):x, >

and

={(x1,0):xr

,< =

xr)

### e

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

case

Ker

= {(0, xr):x, >

### 0}

and the only nontrivial closed convex cone contained

Ker

C = Ker

Again

but

+ C@

=

=

<

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

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

### implying

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

= U*

+ C@

=

=

### {(xr,xr):xr)

0, xz > O}. tn tlús case

=

0}.

C = p-1(0) = Ker

= (J* and

= p-1(0)@

case

=

and

= C @

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

while

Example

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

with

+ M.

Example2.Let

### X

= l?3withtheEuclideannoûn,

e R}

{(0,

e

>

Then

= {(0,

and Ker

{(O,xr,O):x,

= C @

### p(K) holds with

C = Ker Prt*.

4 5

(4)

50 \$tefan andC. Mustãta

6

Exampre

=

### cfa,ó]

be the Banach space

### of a'

continuous rear_

valued functions on the interval [a,

### b]

withthe sup_norm.

The set

(a) =

### f(b)

= o]

is aclosedsubspace

C[a,b],

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

is a closed convex cone

C[a,

and

### M

c. K.

First show that the subspace M

### is

K-proxrminar. For

e

the fi.rnction

defined

(x) _ -f (o),

e

### fa,

bl, is an element of best

rndeed, wehave

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

e M, It follows that

,

=

(a) and

### g

e puW(.f).

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

0 e

=

=

e

<

<

follows

e

### Pr¡*(f)

and the inequalities

ll¡,

_ .f,(o)l

= 2.ll.n _

,

for

e

### K,

imply the continuity of the applicationp.

obviouslythatp ispositivelyhomogeneous

### M c

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

p-lço¡ @

this case

P-tQ) =

e

e

and

-"f

### (")) is

the unique decomposition

e

the

= g +

e

h e

=

and h(x)

subspace

and

= 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

and

### a

K-chebyshevían subspace

### tf x u-;;ere

exist rwo crosed convqc cones

Metric hojection 51

C cKer

and U

srch that

= C @ (J, thenC

Kerp¡a¡* and U

p(K) where p:

-+

### M

is the only selectíon assocíated to the metric projectíon py1r.

Since

remains

show

Ker

eKer

ând

e C,

be such that

y +

By Theorem

### A'

the

selectionp is given

=

0 =

=

+

= 0 + z = z,

inrplying

=

e

The equality

=

follows also

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

Ker P¡rlx @

### M

tlten the subspace

### M

is K- Chebyshevian and the only selection assocíated

the meîric

### projection

is continuous, positívely homogeneous and addítive on K.

Proof. First we prove that

=

for every

e

indeed,

e Ker

is equivalent

0 e Ker

### Pr,"(y). If z

eKer poo,*(y) then,

### taki'g

into account the fact

a subspace

e

we ob.øin

mll:m e

=

=

e

= d(y

### - z,M),

slrowing that 0 e

### P*l*(y - z)

or,equivalently, y

eKer

=

and

tvith

a,,d

0, z e

### M.

The unicity of this representation implies z

0,

### Now, writing an

arbitrary element

e

the

=

eKer

and z e

we obtain

=

z) = z +

= z

### *

0 = z,

slrowing thatzis the onlyelement ofbestapproximationfor

thesubspace

K-Chebyshevian.

### we

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

### of

R2 for which the decomposition

= C @

is true.

Exatnple

### 4.Let X =

R2 with the sup-norm

çx1,x2)

### e

R2,

7

(5)

52

and

\$tefan Cobzaç and C.Mustãp

luí:= {(xr,0):rq R},

xr):x,

### e4 ,,

> 0}, It is easily seen that

{@t,0):xt

xz

nr 4

+

for

### (xr,xr)

e K.

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

irnplying

### If

(m',0) is an arbitrary element of

ll(r,, rr)

### -

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

xr} > xr,

slrowing

x2and(nt,,) e

only

m

### eR

verifìes the inequality lrq _

x2.

TLe kemel

prr," is Ker

=

{(rr,

e R2: | ;, j <

0},

and

= C@

where

= {(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

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

e D

### c R"

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

a:[O,co) x D

be a

### KBM

function of average a0. Then we define the operator

## (t)

A^(a)(2,t,*¡,=

### "ltolo,(",r) -

o01r¡]a", and denote

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

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