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 ofX
andr
an element ofX.
The distance frornr
toMis
defined by(l)
d(x,A[):= inf{ll.r -
yll:y
e M} .An
elementy
eM veriffing
the equalityll¡- yll= d(x,M)
is called anelement of besÍ approximation for x by elements
inM.
The set of all elementsof
best approximation for x is denoted
by Pr(x)
,i.e.(z)
P¡a(x):={y
eM:
llx- yll= d(x,M)}
,If Pr(x)
+Ø
(respectivelyPr(x)
is a singleton) forall
.rc eX ,thenMis
called a proximinal (respectively a Chebyshevian) subspace
ofX.
The set-valued applicatiotr P¡r:
X -+
ZM is calledthe metric projection ofX
onMand
afiinction p:X
-+M
stchthatp(x)
ePr(x), forall x
eX ,is
calleda selecfion for the metric projection Prn . Observe that the existence of a selection
for P* inlplies P*(r)
+Ø,for all
.r' eX ,
i.e. the subspaceM is
necessarily proximinal.The set
(3)
Ker P¡4:={x
eX:0
eP¡¡(x)},
is called the kernel of the metric projection P", .
In many
situationsfor
agiven
subspaceM of X
the problenlof
best approximation is not considered for the q'hole spaceXbut
rather for a subsetKof
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 andPr(r) # Ø
(respectively PuoQ) is a singleton) for all ,r e K, then the subspaceMis
called K-proximinal (respectively K-Chebyshevian). The restriction of the rnetric proiection PnntoK
is denoted byP*ty
arrrd its kemel byKer
Pr4*:46 $tefan Co bzaç and C.
(4)
Ker p¡o¡*:={x e
K:O
ep¡ae)}.
Fortwononvoid subsets
u,vof
xdenoteby (J+v:= {u+v:u
e (J,v€v}
tlreir algebraic sum.
If
everyx eu +v
can be uniquery writtenin
trre fonn x=u+ v with u euand,v ev, thenu +v
iscared lherrtrectargebraicsuntof the setsu
andv
andis
denotedby urv. rf K=(Jiv
andthe appricatio' (u,v)-+
Lt+v, u e (J,-v
ev,
is a toporogicar Ìrcttteontorprtisnt betN,ee, Lrxv
(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 subspaceofxthen
tlle urehic projectionPr
admits a continuous and linear selectionif
and onlyif
the subspaceMis
complementedinxby
a closed subspaceof 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 Cc
Kerp*
and (Jc M,
such thatP*,,
admits a continuouù-, positively Theorem A).and prove that
if
p,rrlr admits a contin selection, satisSzing some suplernent decornpositionsK=
[email protected]U,
with Cco'es (Theorern B). Although the conditions in trreoreurs A,and B are very crose
to
be necessary and sufficientfor
trre existe'ceof
a"orrtinuous, positively Iromogeneous anti additive selection
for pr,*,
we weren,t abieto fi1d
such;"",'i.Ïïïf;.1;""å]"tio's
rvliich mav occur a¡e illust¡aredby
some exarnpres By a convex cone inx
we understand ano'void
subset Kofxsuch
that:a).ri +
x,
eK, forall
x1,t2 ÇK,
ar-tdb) À ...r- e
K,
for all -r eK,
andÀ > 0.A car.cfull exarnination of the
statem
e proof of Theorern Ain [4]
yields the following more detailed refo
THEOREM t\,. Let M be a
closed
of a ttot.ntecl spaceX anclK
a closed conver cone ín X. If o
closerJconvex
cot.tes CcKerPr,*
and (J c: M sìuch rhatK
= [email protected] (J, tlrcrt the a¡tplicatto,,;U _; ;:
Metric hojection 47
defnedby
p(x)
= z,for x
=|
+z
eK, y ec, z e(J,
isa continuous,posifivery homogeneous and additive selection of the ntetric projection poo,* . The subspaceMis
K-proximinaland C =p-,(0), U
=p(K).
The
following
theorem shows that,in
some cases, the existenceof
acontinuous, positively homogeneous and additive selection for
p*r*
implies thedeconrposability of K in the form
K
= C @ U, with C and [/closed convex cones, THEOREM B. Letx
be a normed space, M a crosed subspaceofx
and K a cTosed convex cone ínx.
suppose that the metricprojection p*r*
admíts acontínuous, positívely homogeneous and addítíve selection p suclt that:
a) p(K)
is closed and contained in K, andb) x- p(x) eK, for all x
e K.Then p-l (0) and
p(K)
are close J convex cones containedin
Kerpr,*
andM
respectively, andK
=p](0)@ p(K).
tf p(K)
is a closed subspaceof
Kor 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 rhatK
is aconvex 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 setp-r(0) c
Ker poo,* isclosed.If y
ept(0)
and l" > 0
thenp(I
.!)
= ?".p(y) = 0,
showing that X.y .p-1(0).
Similarly,yy yz
ep-1(0)
andtlie additivity
ofp imply p(yt
+ yz) =pbù
+p(y)
= 0,showing that
p-t(0)
is a closed convex cone containedin
Kerpr,*.
Now
we provethat K
=p-'
(0) +p(K).
rfx
e K thenby condition
ó),..y:= x- p(x)
eK. By
Conditiona), p(x) eK irnplying ;r= y+p(:u)
with y,p(x)
eK.
Using the additivify of the functionp and the fact thatp(p(x))
= p(r,)(irfactp(tn): mforùl
m eM)we
obtainp(x)
=p(y)
+p(p(x))
=p(y)
+ p(x),It follows p(y)
= 0,i.e. y
ep-'(0)
andK c
p-t(g) +p(K).Since p-'(0)
¿n¿p(K) are contained in K and K is a convex cone, it follows
thatp'(0)+ p(K) c
KandK =
p-1(0)+p(K).
To show that this is a direct algebraic sunl suppose tlut ari element
x
eK
adnritstwo representations: x
=| * p(x)
and x =!,
+2,, witlty, !,
e p-le)
anclz'
ep(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
ep(K),
is a homeomorphism betweenp-t(0)
xp(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)xp(K), tt
e N,converging
to (y,z)
ep-l(o)x p(K),it
followsy,
->
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 .rÞ
(_y,z),
wrrerex= ))*2, ! ep.(0), z ep(K),
takeagainasequeuce .Én=!,+zn
eK, !,,
ep1(0),z,,ep(K), converging to x=y+ze K, where yep_r(,) ard
z e
p(K)' it
followsz,
= p(x,,),n
e
N,
z =p(.\),
and, by the cortinuify of theapplication
p,
2,, = p(x,,) -+ p(¡,,,) =z, But
then y,, = xtt_ zn+
x _ z = )), proving that the sequence((!,,,r,,7),,.*
colìvergesta (y,z)
rvith respect to tho product topology or p-r1o¡ xp(K),
This shows that the applicafionx
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)
xp(K)
anrl K.rf p(K) is a
closed subspaceof K
thencondition a)
holdsa,d, for x
eK' p(x)
and-p(x) arcin ¡:(K)c K
so trrat .r- p(x)e K, srrowi'g
that Condifion ó) holds too. If M
c.K
thettM
=p(I[) c p(K) and,
sincep(K) c M' itfolrows
thatp(K)
=M
isa closed subspaceof'.
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
thatp(K) is
a close<l convex collecontainedinK, p(y)=0
Irr the andfollowing
z = Sincep(x), itfollows every reK
thatcanbewritteninthefomr 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 theseof
the metric projections cases.bu!
the eq'arityExatnpre 1. Take
x
= R2 witrrtheEucritreannonnandM{(xr,O):x,
eR}.Then P¡a((x1,x2))={(-",,0)}, foraD(x1,x2) uRr, i.e.Misachebyshevian.subspace of
X
and the only selectionof
tlie metric projectioriis
p((e.,,,r)) =(,r:,,0),
for(*,, rr)
e R2.
Letfi,= {(r,,
xr)e
Rz: x, > o, r:, à o}.Mehic hojection 49
a) Take
K
= {(xr,xr)'.xr: xr,xr2oi, In
this caseKerPr,*
={(0,0)}
sothat tlre only closed convex cone contained in Ker
Pr,*
is C = {(0,0)} , The subspaceM contains two nontrivial closed cones U* :
{(xr,O):x, >0}
andg-
={(x1,0):xrI 0} pG)=(J*[email protected](J* =(J*.
b) Let
,< ={(",,
xr)e
R2:x, > .x,,:c, >0}. In this
caseKer
Pr,*
= {(0, xr):x, >0}
and the only nontrivial closed convex cone containedin
KerPr,*is
C = KerPr,*.
Againp(K)= tI*
butK
+ [email protected]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 havep(K)
= U*c. KlMbutK
+ [email protected]p(K)
=p.
d) K
={(xr,xr):xr)
0, xz > O}. tn tlús caseKerPr,"
={(0, xr):xr>
0}.C = p-1(0) = Ker
prtx, p(K)
= (J* andK
= p-1(0)@p(K).
e) K={(tr,rr)eÀ2:;rr>0}. In this
casep(K)= M cK, KerP*,*
=:{(o,rr)t*r> 0}
andK
= C @p(K),
where C=
KerP*,*.
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,
whilein
Examplel.c), p(K) c Kbut
x- p(x)
eK
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 ofK
withp(K)
+ M.Example2.Let
X
= l?3withtheEuclideannoûn,M ={(rr,xr,O):xr,x,
e R}and K :
{(0,x2,4):x,
eR,t,
>0},
Thenp(K)
= {(0,x,,O):xre.R}'. ltt
and KerPr,*=
{(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 spaceof 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
andM
c. K.First show that the subspace M
is
K-proxrminar. Forf
eK,
the fi.rnctiong
definedby g(x):= f
(x) _ -f (o),x
efa,
bl, is an element of bestapr._*";;:
forfinK.
rndeed, wehavellf [email protected]) andllf _hl!>lftÒ_nço¡l,forall h
e M, It follows thatd(f
,M)
=f
(a) andg
e puW(.f).The kernel of the restricted metric projection is Ker P¡n,* =
{-f eK:
0 eparc(f)}
=
=
{-f
eK: - f(a)
<-f(x)
<f(o), foiall x e[a,b]].
It
followsp(-f)
ePr¡*(f)
and the inequalitiesllpí) - I(DII=
ll¡,- f,ll+l¡ç"¡
_ .f,(o)l= 2.ll.n _
f,ll
,for
frf,
eK,
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 caseP-tQ) =
{g
eK:ic> 0,g(*)= c, foÍall x
e[o,bl]
andf(x): -f(o)+(-f(x)
-"f
(")) is
the unique decompositionof -f
eK in
thefo'n /
= g +rt witlt s
ep-'(0) and
h ep(K) (s(x)
=f(a)
and h(x)= -f(r)_ f.(o) fo¡ all x e[a,b]).
In Examples ld) and e), the
subspaceM is K-chebyshevia'
andy
= 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
andM
a
K-chebyshevían subspacetf x u-;;ere
exist rwo crosed convqc conesMetric hojection 51
C cKer
Prlx
and Uc M
srch thatK
= C @ (J, thenC:
Kerp¡a¡* and U:
p(K) where p:K
-+M
is the only selectíon assocíated to the metric projectíon py1r.Proof.
SinceC cKerP*,* it
remainsto
showthat
Kerprt cC. Let
x
eKerPr,¡
ândlet y
e C,z eLl
be such thatx -
y +z,
By TheoremA'
theselectionp is given
by p(*)
=z
and by the additivityofp.
0 =
p(x)
=p(y)
+p(z)
= 0 + z = z,inrplying
x
=/
ec.
The equalityu
=p(K)
follows alsofro'r
Theorem A'.Apartial
converse of CorollaryI
is also true:coRotLaRy 2, Let M be a closed subspace of the nornted space
x
ancJ K a closed convex cone inx. If K =
Ker P¡rlx @M
tlten the subspaceM
is K- Chebyshevian and the only selection assocíatedto
the meîricprojection
is continuous, positívely homogeneous and addítive on K.Proof. First we prove that
Pr,"(y)
={0}
for everyy
eKerpr,".
indeed,y
e KerPr,*
is equivalentto
0 e KerPr,"(y). If z
eKer poo,*(y) then,taki'g
into account the factthatMis
a subspaceofXand z
eM,
we ob.øinlly - "ll= inf{lly -
mll:m eM}
==
inf{lly - " - m'll:nt'
eM)
= d(y- z,M),
slrowing that 0 e
P*l*(y - z)
or,equivalently, y- z
eKerpr,*,
But thenyadrnits tworepresentationsy
=y+0
andy=(y-z)+2,
tvithy, y- z eIàrpr,*
a,,d0, z e
M.
The unicity of this representation implies z=
0,Now, writing an
arbitrary elementx
eK in
theform x
=y-t z, with
y
eKerPr,*
and z eM,
we obtainPr*Q)
=P*t*(y *
z) = z +pr*(y)
= z*
0 = z,slrowing thatzis the onlyelement ofbestapproximationfor
xinM,i.e.
thesubspaceMis
K-Chebyshevian.we
conclude the paper by an example of a non-chebyshevian subspaceof
R2 for which the decomposition
K
= C @M
is true.Exatnple
4.Let X =
R2 with the sup-normll('',"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 thatP*¡*((rr,*r)):
{@t,0):xt
-
xz1
nr 4¡,
+x2},
for(xr,xr)
e K.Indeed, xt _ xz < rn 1 x1+ x2 is equivalentto l.r, _
nl<;rr,
irnplyingIf
(m',0) is an arbitrary element ofMthen
ll(r,, rr)
-
(nr ,o)!l= rnax{l;r,- ,t l,
xr} > xr,slrowing
that d((x,,:r2),M)=
x2and(nt,,) epu6(e1,*r)) if ard
onlyif
meR
verifìes the inequality lrq _ntl<
x2.TLe kemel
of
prr," is Kerpr,*
={(rr,
,r)
e R2: | ;, j <xr, x, >
0},and
K
= [email protected]M,
wheret
= {(0,xr):xr> O} is
a closed convex cone strìctl¡, containedin
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 I994
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
oneof
the most powerfrrl toolsin
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 solutionsfor
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 letx
e Dc R"
be an ¡l-dirnensional (spatial-type) vector,Let
a:[O,co) x D-) R"
be aKBM
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 consideredin
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)[email protected]),
8
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ri
i
i
ìii