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-tdb) 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

acontinuous, 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 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 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 theapplication

### 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'arityExatnpre _{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 caseP-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,,d0, 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