View of Selections associated to the metric projection

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*:

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

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

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

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