SEMIGROUPS OF OPERATORS,
APPROXIMA TION AND SA TURA TION
IN BAN A CH SPA CES
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CA. Timmermans
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STELLINGEN
behorende bij het proefschrift
SEMIGROUPS OF OPERATORS,
APPROXIMATION AND SATURATION
IN BANACB SPACES
door
C A . Timmermans
De openbare verdediging van hel proefschrift en de
stellingen vindt plaats op
donderdag 17 december 1987 om 16.00 uwin de aula van de Technische Universiteit
Mekelveg
Delft,
te Delft
U bent hierbij van harte welkom. No afloop van de promotie
is er een
receptie.
Theorem 1
■
I
I
Let X = C[0.1]. equipped with the supretnum norm. Suppose *>'■ [0.1 ]x [R —> P. be
continuous. Assume there exists a continuous function •en on (0.1) such that
0
0 < * 0 (x) < -p(x.f)
for x € (0.1). f € P..
Let A: D(A) C X — • X be defined by
= {u € C[0.1] O C 2 (0.1):
D(A)
(Au)(x) = *(x.u(x)) u"(x)
lim f(x.u(x)) u"(x) = 0. j = 0.1)
for u € D(A) and x € (0.1).
Then the operator A is densely defined and m-dissipative in X.
References:
1. J.A. Goldstein. C-Y.Lin: Singular nonlinear parabolic boundary value
problems in one space dimension. Journ. Diff. Equations 68. pp. 429-443.
(19S7).
2. J.A. Goldstein. C-Y.lin: Highly degenerate parabolic boundary value
problems, October 19S7, preprint.
3. C-Y.Lin: Degenerate nonlinear parabolic boundary value problems. Thesis,
Tulane Univ.. july 19S7.
Theorem 2
The sufficient condition
lim inf
sup
| (n (B f-f) - g) (x) | = 0.
n
n-«° x€[0.1]
where B
is the n-th Bernstein operator and f.g € C[0,1] for the assertion
f" € C(0.1) and x(l-x) f(x)/2 = g(x). 0 < x < 1. in [Mi], Th 3.2, is also a
necessary condition, provided that g(0) = g(l) = 0 .
Reference: [Mi]: Micchelli, C.A. The saturation class and iterates of the
Bernstein polynomials. Journ. of Approx. Th. S. pp 1-1S. (1973).
Theorem 3
Let the sequence (K ), n = 1,2,... of Kantorovic operators be defined by
V
c
Co.iD — c[o.i]
n
(<k-H)/(n+l)
(K f)(x) = (n+l) 2 p
(x)
f(t)dt. n = 1.2....
n
J
k=0 n , K
k/(n+l)
where P n k (x) = [ £ J x (l-x) n " .
Let the space C; [0.1] (nt i 1) be equipped with the norm II«II defined by
llfll :=
m
2
k=0
II D f II
.
f € d"[0.1].
k
where D f denotes the k-th derivative and II»II the supremum norm. Then for
f € C^O.l]
II KRf - f llm = o(n _ 1 )
i)
. n —»»
if and only if f is a polynomial of degree < 1.
i i ) II K J - f llm = 0 ( n - 1 )
. n -♦ »
m
i f and only i f u:= D f s a t i s f i e s u € C ^ O . l ) . u - € AC.
(0.1) n L^O.l),
#u" € L " ° ( 0 . 1 ) . where *(x) = x ( l - x )
Theorem 4
With the definitions and notations of chapter 1 of the thesis, the following
holds:
fa
€ L (x 0 > r_). then r„ is not natural and moreover
)
r 2 is regular
<=* W € L J (x 0 > r 2 ) A (aW)' 1 €
i)
r 2 is exit
<=» W € L
ii)
r 2 is extrance c* W C L ^ X Q . ^ ) A (oW)" 1 €
1
l\xQ.T2)
^ . ^ ) A (aW)" 1 « L ^ X Q , ^ )
L^XQ.^)
and
fa
-V>
\
C L (x_. r 2 ). then r_ is not regular and moreover
)
r2 is natural
=* W € L 1 ^ . ^ ) V (oW)"1 € L ^ x ^ )
i)
W € L (x Q .r 2 )
*
ii)
(aw)
€ L (x 0 .r 2 ) *
r 2 is exit or natural
r 2 is entrance or natural
Theorem 5
Let J = (r
r_). - a> < r
< x. < r, < • ; j is the two points
compact ification of J: a.0 € C(J). a > 0 on J.
W(x) = exp | - J^ Pa-1dt j . x € (lyi^).
Assume p. .p„ € ( - = • . j r ) . X = C(J) equipped with the supremum norm, let
A: D(A) C X —. A be defined by
D(A) = {u € C(J) n C ^ J ) ! lim
Au
u(x)sin p $ ♦ (W"V)(x)cos p. = 0. i = 1.2}.
= ou" ♦ 0u' for all u € D(A)
and suppose that the boundary points r. are regular in the sense of Feller,
i.e.
W € l\j).
(aW)" 1 € L J (J).
Let u. and u- be solutions of
u - Au = 0,
satisfying
lim U j (x) = 0. lim
(W -1
V ) ( x ) = (-1).1-1. 1 = 1
< -» r.
x -» r,
Finally, let G be the open
subset of
(- | . | ) x
(- | . | ) in the
xy-plane bounded by the curves x = - =• , y = =■ .
(u 2 (rj) tan x + (W„-1 u 2 )(rj)) tan y + tan x = 0.
x i 0.
y < 0. and
(Uj(r 2 ) tan y ♦ (W„-1 uj)(r 2 )) tan x - tan y = 0,
x>0.
y i 0-
Then A is densely defined and m-dissipative in X if and only if (p.,p„) € C.
Theorem 6
Let X. A. G. u.. u_. W, p and p„, be defined as in the preceding theorem.
Lei
Let
H = {(x.y) €
(-|.| )x
(
. | , | ) | - | <
x
< - arctan (W 1
_1
- (W^ujH^))/
arc tan (({u2(rj)tan x - ( w ' ^ M r j ) )
u^Mr,).
u,(r2)) ^ y <
TL
Then we have G C H. and H\C * <}>. Moreover, if (p., p„) € H\G. then
i)
A is densely defined.
11)
I-A: D(A) — » X is a bijection. (I-A)" 1 is positive, but (1-A)" 1 is not
ill)
A is not m-dissipative.
a contraction on X and thus
Stelling 7
In R
(n \ 2) is B een bol met straal a en R een overal even dikke bolschil
met uitwendige straal b en inwendige straal c.
n> punten worden op willekeurige wijze op het inwendige van B gestrooid.
Indien p € W. zodat
/% n
n*
(a + b )
n
p < "'t0 -c )
dan is de bolschil altijd zodrjiig neer te leggen dat de bolschil
p + 1
punten bevat.
Stelling 8
De regels van het spel Mastermind (Invicta P.Ltd) in de oorspronkelijke vorm
met 6 kleuren en 4 plaatsen worden bekend verondersteld.
Een strategie voor Mastermind is een algoritme om door middel van het stellen
van vragen en het verkrijgen van de betreffende antwoorden volgens de regels
van het spel een Mastermind code te breken.
Zij A de verzameling der strategieën om een willekeurige Mastermind code te
breken, en zij B
T
de verwachting van het aantal vragen bij een strategie
€ A.
Dan geldt
JS ^ f S S h 4 - 3 8 8
Litt. R.W. Irving, Towards an optimum Mastermind Strategy. J. Recr. Math. Vol
11(2). pp. S1-S7 (19S7).
Stelling 9
Bij een tennistoernooi voor gemengde t u t t i - f r u t t i dubbels met 16 manlijke en
16 vrouwelijke deelnemers kan een wedstrijdrooster voor acht - en niet meer
dan acht - volledige ronden van acht wedstrijden worden opgesteld op een
zodanige wijze dat elke deelnemer elke andere deelnemer in de wedstrijden
hoogstens éénmaal ontmoet a l s dubbelpartner of a l s tegenspeler.
Een v o l l e d i g wedstrijdrooster i s af te lezen uit het onderstaande schema,
waarbij de vrouwelijke deelnemers genummerd zijn van 1 t/m 16 en de man­
nelijke
van 17 t/m 32.
(14e Hompestomper tennistoernooi, okt. '67. Heerenveen)
1
2
1 X
2 b
3
4 q
5 li
6 t
7 y
8 n
9 e
10
11 h
12
13
14
15
16
17 a
16 c
19 d
20 J
21 g
22 P
23 r
24 1
25 s
26 V
27 f
26 m
29 o
30 V
31 1
32 z
B
X
q
3
4
5
6
7 6
9 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 25 26 27 26 29 30 31 32
0
K T
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b X
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K T
T K
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1 z
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f M o v 1 z A c D J
H f v o z > c A J D
s « 1 z o V D J A c
V
f S z
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1 z s » r M C P r I
z 1 ■
SN f P C 1 r
V
0
r M s « r 1 C P
V
M
f
« S 1 r P C
0
h
y n e
q k
t k n y
e
h
q
X b y n k t h
e
b X n y I k
h
e
y n X b . q
n y b X q
k t
q X b
t
k q
b X
X b
h
q
b X q
h
e
q X b
b X
e
q
h
k 1 y n
e
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h I k n y
h
e
y n k i
1i
e n y 1 k
J C P r
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D
1
r
P
C
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V
0
i
z
7
«
1
z
S
z
0
V
1
V
0
V
1
f M
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f H S *
M f * S
C r r 1
P c 1 r
r 1 C P
1 r P C
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c A J D
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Opmerking: In dit achem hebben "kleine letters" dezelfde bcickrni* a l s
hoofdletter». De "kleine l e t l e r " - i n f o r i « t i e kan afpclrid worden
uit de hoofdletter-informtie.
«edsirijd rooster
Ie ronde: dubbel
2e ronde.' dubbel
3e ronde: dubbel
4e ronde: dubbel
?* ronde: dubbel
fic ronde: dubbel
7e ronde: dubbel
6c ronde: dubbel
X
X
X
X
X
X
X
X
A speelt
D speelt
C speelt
J speelt
M speelt
P speelt
S speelt
V speelt
tegen dubbel B ♦ C
tegen dubbel E * F
tegen dubbel H * I
tegen dubbel K ♦ L
tegen dubbel N ♦ O
tegen dubbel 0 ♦ R
t c g c dubbel T < V
tegen dubbel ï * Z
Men nene in elke r i j voor de diverse letters de correspondcicndc kolomnummers.
Voorbeeld 6e ronde'dubbel I ♦ 22 - dubbel 4 •• 23
dubbel 2 * 21 - dubbel 3 ♦ 24
dubbel 5 •» 18 - dubbel 6 •• 19
enz. .
Stelline 10
De aantrekkelijkheid van het voetbal als kijkspel kan verhoogd worden door de
afmetingen van het doel te vergroten.
Stelling 11
Zoals de leerlingen in het voortgezet onderwijs een schoolrapport ontvangen
opgesteld door hun leraren, dienen omgekeerd de leraren een rapport te
ontvangen opgesteld door de klassen waaraan ze lesgeven.
Stelling 12
Leraren bij wie het ontvangen van slechte rapporten - als bedoeld in de
vorige stelling - zich voordoet als een chronisch verschijnsel, dienen
verplicht te worden een bijscholingscursus te volgen.
Stelling 13
In de discussie over de mogelijke oorzaken van de zure regen wordt ten
onrechte meestal niet genoemd: de invloed van electromagnetische straling
opgewekt door radio- en radarzenders. Wetenschappelijk onderzoek naar deze
invloed dient ter hand genomen te worden.
Stelling 14
De demokratisearring fan bestjoer fan de Fryske wetterskippen is earst
foldwaande motivearre as ek de demokratisearring fan it behear fan it Fryske
oerflaktewetter syn beslach krijt en yn hannen lein wurdt fan in wetter- en
suveringskip. wylst de provinsje him beheine moat ta tafersjoch.
u i <l*7
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(\<J
SEMIGROUPS
APPROXIMATION
OF
OPERATORS,
AND SATURATION
CA.
IN BAN ACH SPACES
Timmermans
Delft University Press
TR diss
1598
2
SEMIGROUPS OF
APPROXIMATION
OPERATORS,
AND SATURATION
IN BAN ACH SPACES
Aan
Hanny
Ellen
Henkjaap
Aan mijn moeder
3
SEMIGROUPS OF OPERATORS,
A P P R O X I M A T I O N A N D SATURATION IN BANACH SPACES
PROEFSCHRIFT
ter verkrijging van de graad van
doctor aan de Technische Hogeschool Delft,
op gezag van de rector magnificus,
Prof.dr. J.M. Dirken,
in het openbaar te verdedigen
ten overstaan van een commissie door
het College van Dekanen daartoe aangewezen
op donderdag 17 december 1987
te 16.00 uur
door
CORNELIS ALBERTUS TIMMERMANS,
geboren te Dordrecht,
wiskundig ingenieur
Delft University Press
4
Dit proefschrift is goedgekeurd door de promotor
Prof.dr. Ph.P.J.E. CLÉMENT
Samenstelling van de commissie:
Prof.dr. Ph.P.J.E. Clément (promotor)
Prof.dr. O. Diekmann
Prof.dr. A.W. Grootendorst
Prof.dr.ir. R. Martini
Prof.dr. CL. Scheffer
Prof.dr.ir. F. Schurer
Dr.ir. C.J. van Duijn
5
CONTENTS
INTRODUCTION
CHAPTER 1 - SOLUTIONS OF GENERAL SECOND ORDER
DIFFERENTIAL EQUATIONS IN VARIOUS
SPACES
1.1. Solutions of u - aD u - 0Du = f in C(J) and L°°(J).
Classification of boundary points
1.1.1. Preliminaries
1.1.2. Solutions of the homogeneous equation
1.1.3. Classification of the boundary points
1.1.4. Solutions of u - Q D 2 U - 0Du = f in C(J)
1.1.5. Solutions of z - aD 2 z - /3Dz = k in L°°(J)
1.2. Solutions of v - D(aDv -/9v) = g in L ] (J)
1.2.1. Solutions of v - D(aDv - 0v) = 0
1.2.2. Solutions of v - D(D(av) - 0v) = g in L ^ J )
1.3. Solutions of w - (D(aDw) - jJDw) = h in NBV(J)
1.3.1. Solutions of w - (D(aDw) - 0Dw) = 0
1.3.2. Solutions in NBV(J) of w - (D(aDw) - j5Dw) = h
13
13
13
14
30
40
48
50
50
56
60
62
66
CHAPTER 2 - SEMIGROUPS GENERATED BY DIFFERENTIAL
OPERATORS SATISFYING VENTCEL'S BOUNDARY
CONDITIONS AND THEIR DUALS
67
2.1. Semigroups in Banach spaces
2.2. Two propositions j)n adjoint operators
2.3. Semigroups in C(J)
2.4. Dual semigroups in 3R x NBV(J) x ]R
2.5. Restricted dual semigroups in 3R x L ^ J ) x }R
2.6. "Bidual" semigroups in K x L°°(J) x ]R
67
72
77
88
102
106
6
CONTENTS (continued)
2.7. On CQ-semigroups in a space of bounded continuous
functions in the case of entrance boundary points
2.8. Dual semigroups in NBV(J), NAC(J) and L ] (J)
2.9. "Bidual" semigroups in L°°(J)
CHAPTER 3 - SATURATION PROBLEMS FOR BERNSTEIN
OPERATORS IN C m [0,l]
3.1. Introduction
3.2. A unified approach to pointwise and uniform
saturation for Bernstein polynomials
3.3. Saturation and Favard classes
3.4. Application: Uniform saturation class for Bernstein
operators on C[0,1 ]
3.5. Uniform saturation classes for Bernstein operators in
C m [0,l] norms (m > 1)
116
126
130
135
135
136
153
159
163
REFERENCES
186
SAMENVATTING
189
ACKNOWLEDGEMENT
191
7
INTRODUCTION
This thesis deals with aspects of the theory of semigroups of operators in
Banach spaces as well as aspects of the theory of approximation by means of
linear operators. In fact we investigate operators in Banach spaces which
generate semigroups of operators and we apply the obtained results to satura­
tion problems in approximation theory. Saturation is an interesting phenome­
non in approximation theory. This concept was introduced by Favard in 1947,
[Fa].
Definition (cf. [BB], p. 87)
Let (L ) be a sequence of linear operators in a Banach space X strongly
convergent to the identity operator in X. Then the sequence is (uniformly)
saturated if there exists a sequence of positive numbers (<f> ), which tends to
infinity if n tends to infinity, and a class s(L ) closed in X such that the
following holds
(i)
lim
n—K»
6u ||f - L n f|| = 0
if and only if f e s(L ),
(ii)
(1)
there exists at least one f e X\s(L ) for which
'I W
- f )ll = 0 ( l ) , n - o o .
The class of functions f, satisfying (1) is called the saturation class of the ap­
proximation process (L ), and 0(#~ ) is the saturation order.
Thus the saturation problem concerns the determination of the optimal order
O(0
) of approximation and the (non-trivial) class of elements which can be
approximated with this optimal order. Saturation problems in approximation
theory are investigated by many authors by different methods. In particular
we mention the theorems of Lorentz and Schumaker ([LS], Th. 4.3) and Becker
and Nessel ([BN], Satz 3.6).
Many approximation processes (L ) in a Banach space X strongly convergent to
the identity operator in X, satisfy the so-called Voronowskaya property
(2)
lim
n—►<»
<j>
(L f - f) = Af,
nx n
f e D(A) C X,
where A : D(A) c X —
► X is the infinitesimal generator of a strongly
continuous semigroup T . in X. An important class of approximation processes
for which (2) holds is the class of (^„-semigroups T . on X. In that case
L := T(l/n) and <j>
= n. It is known that a semigroup, considered as an ap­
proximation process, is saturated ([BB], p. 88). The saturation class of T . is
sometimes called the Favard class of T . . It is known that if the condition (2) is
satisfied, the saturation class of the approximation process'(L ) is the same as
the Favard class of T . , denoted by Fav (T ) (see e.g. [BN], Satz 3.6 and ([B],
3.1). A nice example of an approximation process which is saturated is the
sequence (B ) of Bernstein operators, defined by
B : C[0,1] -+ C[0,1] (equipped with the sup-norm),
(3) '
(B f)(x) = V
II
f(k/n) p
k-Q
(x),
11,K
with p
n,k
(x) = ( P ) x (1 - x) n " .
This sequence has the Voronowskaya property
(4)
lim ||n(B n f - f) - Af|| = 0,
f G D(A),
n—►<»
where A is defined by
D(A)= {f G C [ 0 , l ] | f e C 2 ( 0 , l ) ,
lim x(l - x)f"(x)= lim x(l - x)f"(x) = o\,
x—0
x—1
J
(5)
. Af(x) = x(l - x)P(x)/2.
In fact, Voronowskaya proved that
9
lim n(B n f(x) - f(x)) = x(l - x)f'(x)/2
n—*oo
for functions f which are twice differentiable in x, [Vo]. In 2.3 it will be
proved that A, defined by (5), is densely defined and m-dissipative, while in
3.3 it will be proved that
(6)
D(A) = {f | lim n(B f - f) exists}.
n
*• n—»oo
i
Berens and Lorentz proved (4), [BL]. The description of D(A) in [BN] and
[Fb], namely D(A) = {f G C[0,1] | <£f' G C[0,1]}, where <f>(\) - x(l - x)/2, has
to be replaced by D(A) as given in (5).
Applying Theorem 3.3.1 to the sequence of Bernstein operators we obtain that
the uniform saturation class of the Bernstein operators is Fav ( T . ) .
The remaining problem is to describe Fav (T ). Several methods are known.
Butzer (cf. [BB], p. 92) characterized the Favard class for certain restriction
of a dual semigroup T® as the domain of the infinitesimal generator of the
non-restricted dual semigroup T . Berens [Be] gave a characterization by
means of the concept relative completion of a Banach subspace in a Banach
space. For practical purpose the following characterization is useful (cf. [CH],
3.36):
(7)
Fav (T ) = D(A®*) n X,
where A© is the adjoint of A®, and A® is certain restriction of the adjoint of
A. We compute D(A® ) in section 3.4. It is known that for a function
f G C [0,1] the m-th derivative of B f converges to the m-th derivative of f
in the uniform norm topology. In section 3.5 we compute the uniform satura­
tion class of the Bernstein operators in C [0,1], equipped with its usual norm,
by using the same method as in section 3.4. Partial results concerning the asso­
ciated Voronowskaya formula for Bernstein operators in C [0,1] can be found
in [Fb].
10
As mentioned above, for the uniform saturation class of an approximation
process satisfying the Voronowskaya property it is sufficient to have a charac­
terization of D(A®*), where A is a densely defined m-dissipative operator. In
section 3.4 as well as in section 3.5 the considered operator A is a second order
differential operator. It appears in section 3.4 that
' D ( A ) = { f e C [ 0 , l ] | f SC 2 (0,1),
lim x(l - x ) f ( x ) = lim x(l - x)f"(x) = o ) ,
x—0
x—1
>
{ (Af)(x)=x(l - x)f'(x)/2.
In chapter 2 we investigate the general case
' D(A)= { f e C [ r i , r 2 ] | f e C 2 ( r r r 2 ) ,
lim (aD 2 f + /3Df)(x) = 0, i = 1,2),
x-q
J
(8)
Af =
QD
f + £Df.
We give necessary and sufficient conditions on a and fi for A to be the infi­
nitesimal generator of a semigroup in C [ r . , r - ] , equipped with the supremum
norm. Moreover, for m-dissipative operators of this type we will give an ex­
plicit characterization of D(A® ).
It is interesting to note that A defined by (8) is an infinitesimal generator if
and only if both boundary points are not entrance boundary points in the
terminology of Feller.
We shall use the Feller classification of boundary points in regular, exit, en­
trance and natural boundary points. In section 3.4, where a(x) = x(l - x)/2
and f) = 0, both boundary points are exit boundary points. In section 3.5
however, it appears that the considered operator A is such that both boundary
points are entrance boundary points.
In Chapter 1 we give a self-contained explanation of Feller's theory and the
classification of boundary points in a more modern setting. The results obtained
11
here are slightly more general, since we do not assume any differentiability on
a, as is done in Feller's paper [Fe, 1]. The classification rests on properties of
two 'minimal' positive solutions of the homogeneous equation
u - Au = 0.
We also study solutions of second order differential equations in other spaces.
More explicitly, we investigate solutions of the other following problems
' u-
QD U
- £Du = 0
u e C(J) n C^J), Du e AC Joc (J).
' v - D(D(av) - 0v) = 0
1
1
v e C(J) n L (J), av G C (J), D(av) - 0v e AC(J).
w - D(aDw) - 0Dw = 0
1
w € C (J) n NBV(J), aDw e AC loc (J), D(aDw) - 0Dw G NBV(J).
Here NBV(J) is the space of functions of bounded variation which are nor­
malized by
(i)
f(xf)) = 0, x_ G J is a fixed point,
(ii)
f(x) = (f(x+) + f(x-))/2
for all x G J.
By means of two special positive solutions of the homogeneous problem we
construct a Green operator to solve the corresponding inhomogeneous problem.
A number of results in Chapter 1 are conveniently arranged in state diagrams
in section 1.4.
13
CHAPTER 1 - SOLUTIONS OF GENERAL SECOND ORDER
DIFFERENTIAL EQUATIONS IN VARIOUS SPACES
OO
2
1.1. Solutions of u - a D u - ffDu = f in C(J) and L (J).
Classification of boundary points
1.1.1. Preliminaries
Let J be a non-empty open interval of 1R (not necessarily bounded), J the two
points compactification of J and dJ := J\J. C(J) denotes the set of real-valued
continuous functions on J, C (J) (k = 1,2,
) is the set of functions in C(J)
which are k-times continuously differentiable, X := C(J) is the Banach space
of real-valued continuous functions on J equipped with the supremum norm.
r. (resp. r«) denotes the left (resp. right) boundary point of J, and xft denotes
an arbitrary fixed point in J. Thus -oo < r. < x„ < r_ < oo, and for each
f G C(J) the limits
lim
f(x) and lim
f(x) exist. Since limits at the
x—»rj+
x—>r2~
boundary points are always one-sided we shortly denote these limits by
lim
x—+rj
f(x), i = 1,2. C c (J) denotes the set of functions in C(J) with compact
support inside J and CQ(J) the set of functions f in C(J) with lim f(x) = 0.
1
x—»3J
L (J) is the space of equivalent classes of Lebesgue measurable functions on J
for which
J
|f|dx < oo. This space, equipped with the usual L -norm,
J
llfllj:- J
Ifldx,
is a Banach space. L ( r i , x n ) and L (* 0 ,r_) have a similar meaning. For sake
of convenience we sometimes denote L (x n ,r.) for L (r. , x A
Let a and fi be real-valued continuous functions on J with a(x) > 0 for all
x 6 J. a and /J may be unbounded on J, and a(x) may tend to zero if x tends to
one of the boundary points. We consider the differential expression
(1.1.1)
Au = a D 2 u + /3Du
14
2
2
for u e C (J). Here Du and D u denote the first, respectively the second
derivative of u. In order to investigate the solutions of the ordinary differ­
ential equation
(1.1.2)
u-AAu = f
we introduce the real-valued functions W, Q and R on J as follows:
.x
W(x):=exp{- f
(/3a 1 )(s)ds},
(1.1.3)
1
Jx
x e J,
J
o
Y
Q(x):=(aW) _ 1 (x) f
(1.1.4)
W(s)ds,
x e J,
(aW) _I (s)ds,
x e J.
J x
0
Y
(1.1.5)
R(x):=W(x)f
Jx
o
Note that W(x ) = 1, W(x) > 0 for x e J; Q(x) > 0 and R(x) > 0 for x > x •
Q(x) < 0 and R(x) < 0 for x < xQ.
1.1.2. Solutions of the homogeneous equation
We start with an investigation of the homogeneous equation
u - (aD 2 u +/3Du) = 0
(1.1.6)
with u e C (J).
Proposition 1.1.1
Let a and p be real-valued continuous functions on J with a(x) > 0 for all
x e J. Then there exists a unique positive increasing solution u. of (1.1.6)
and a unique decreasing solution u- of (1.1.6) satisfying u.(x„) = 1 (i = 1,2),
such that u. (resp. u?) is minimal on (r ,x~) (resp. (xn,rj,
i.e. if u. (resp.
u-) is any positive increasing (resp. decreasing) solution of (1.1.6)
ing u.(x„)
xefX(),r2)).
= 1, then u.(x) < uJx),
x e (r x „ / (resp. uJx)
satisfy­
<
uJx),
15
For the proof see Lemma 4 of section 2.3.
D
The importance of Proposition 1.1.1 is that u. and u» are uniquely determined
independent monotone positive solutions of (1.1.6). Thus each solution of (1.1.6)
is a linear combination of u. and u».
Remark. Since for all positive values of A the functions Aa and \p satisfy the
same conditions as a and 6, Proposition 1.1.1 remains valid if (1.1.6) will be
changed into
(1.1.6a)
u - A(aD2u + /?Du) = 0,
A > 0.
As a consequence all assertions in this section remain valid if (1.1.6) will be
changed into (1.1.6a). Then of course u. and u . depend on A. Finally it is
remarked that the function W defined by (1.1.3) is independent of A.
For many purposes it is convenient to rewrite (1.1.6) in an other form:
(1.1.7)
u - Q W ( W " 1 U 1 ) ' = 0.
or
(1.1.8)
(aW) _ 1 u - ( W " 1 ! ! ' ) ' = 0.
The next relation follows from (1.1.8) and is very useful:
(1.1.9)
u , (x) = W(x).{u'(x 0 )+ |
((aW)" 1 u)(s)ds}.
Note that W(xQ) = 1.
The next lemma is also a direct consequence of (1.1.8).
Lemma 1.1.2
For each positive decreasing solution u of (1.1.6)
creasing function with
Urn (W
x—*.r?
u')(x)<0.
W
u ' is a negative in­
16
In order to make a useful classification of the boundary points r and r- we
will investigate the behaviour of the "minimal" solutions u. and u_. We define
M.1 :=
M 1, :=
m.1 :=
lim u.(x) = sup u.(x)
x-»r2 " *
XGJ " ]
lim u-(x) = sup
x-»rj " z
uJx)
x £ j ~z
lim u.(x)
x-»rj _ 1
= inf u.(x),
xeJ - 1
i = 1,2.
The following lemmata will be proved for the boundary point r«. It is clear
that the proofs for the boundary point r. are similar.
Lemma 1.1.3
M < oo ^=>. R e L (x0,r ) ,
M < oo ^=» R e L (r
xQ).
Proof
Necessity. From (1.1.9) we obtain for all x e J
x
(1.1.10)
Uj(x) = J
t
W(t){|
((aWf ^jXsJds + U j ' U ^ J d t
+
Ul(x0).
Since u. is positive increasing we have for x > x_
X
0<Uj(x)[
Jx
X
R(t)dt< [
o
Jx
t
((aW)" 1 u 1 )(s)ds<u (x) < M
W(t) [
o
Jx
o
Thus, if M. < oo, then R e L (x n ,r«).
Sufficiency. Since u . is increasing we have from (1.1.9) for x e (x n ,r-)
Y
0 < Uj'(x)< Uj'(x0)W(x) + Uj(x)W(x)
[
Jx0
or
(aW)" l (s)ds
17
(*)
O < u|(x) < C.W(x) + R(x).Uj(x),
with C = UJ'UQ) > 0.
Following [Fe,l], p. 483 we consider the differential equation
y ' = CW + Ry,
yec'fJ),
(**)
y(xQ) = 1
where C > 0, R G L (x»,r-) and thus also W G L ( x 0 , r A
It is standard that this equation has a unique solution y with
y(x) = v(x) exp
with
{ƒ
.x
x
R(.)d.}
o
x
v(x) = C |f
t
W(t).exp/- f
W(t).exp
R(s)ds]- d t + 1.
C
0
Since
v '(x) = CW(x).expl - f
R(s)ds]- > 0
JxQ
and
Y
y
«(X) = {v '(x) + v(x)R(x)} exp-T f
R(t)dt]- > 0
for x G (x f t ,r-), it follows that v and y are increasing for x G (x 0 ,r„).
Moreover, since R G L (* 0 ,r_) and W G L (x 0 ,r~) we see that v and y are
bounded. Thus
N := lim y(x) =
sup
y(x) < oo.
x r
^ 2
xG(x 0 ,r 2 )
We define the function z on [x n ,r_) by
z := Uj - y.
18
Then
z'(x)<0,
I
z(x
x e [x Q ,r 2 ),
0 ) = - 1 ( V " y ( x 0 ) = 0'
Thus z(x) < 0 for x e [x 0 ,r 2 ), or
Uj(x) < y(x)
and thus also
M. < N < oo.
Since u_ is positive decreasing, u- is also bounded on (x_,r-), so we have
Lemma 1.1.4
Each solution of (1.1.6) on J is bounded on (xn,r.)
•<==► R e L
(x~,r.),
i = 1,2.
a
As a consequence of Lemma 1.1.3.we have
Lemma 1.1.5
There exists a positive monotone solution u of (1.1.6) with
j
R $ L (xn,r.), i = 1,2 .
Or
lim u(x) = oo
x^r;
D
Lemma 1.1.6
For all a,b e TR the boundary value problem (1.1.6) with u(xn)
1
lim u(x) = b, has a unique solution <=*■ R = L (xn,r.), i = 1,2 .
nT'
= a,
19
The next lemma concerns the functions R and Q.
Lemma 1.1.7
If R G L (xQ,r ) and Q G L (xQ,r ) then
(i)
lim (W u% )(x) exists,
x-*r2
for the
he "minimal"
"minimal" decreasing
decreasir solution u. (see Proposition 1.1.1) we
have
for each solution u of (1.1.6)
(ii)
(1.1.11)
(Hi)
lim uJx)
x-*r2 '^
= 0,
lim (W'1 u\)(x) = L < 0,
x-*r2
'^
there exists a positive decreasing solution u, of (1.1.6)
(1.1.12)
lim uJx)
x-*r2 J
= 1,
lim (W'^lXx)
~~+r2
satisfying
= 0.
x
Proof
Let R G L ^ X Q , ^ ) and Q G L 1 ^ , ^ ) .
ad (i). Let u be a solution of (1.1.6). From Lemma 1.1.4 we know that u is
bounded and since Q G L (x Q ,r 2 ) we also have
-
(Q\V)~
G L (x 0 ,r 2 ). From
1
(1.1.8) it follows that (W u ) ' is integrable, hence
ad (ii). That
lim
X r
-1
1
^ 2
(aW)
G L (x Q ,r 2 ) w e
lim (W~ u')(x) exists.
x-r2
u_(x) = 0 is a consequence of Lemma 1.1.6. Since
have from
(1.1.9)
(W _1 u 2 ')(x) = u 2 '(x 0 ) + ƒ
(1.1.13)
I
lim
2
((aW) _1 u 2)(s)ds
2
((aW) - 1 u.(s)ds.
u ' is negative increasing so lim (W - uJ)(x)
< 0, and since
~*
x—»r2
"l
1
JI 2 ((aW)'
((aW) u.
u09)(s)ds = 0 it follows that
The function W
X—tri
r
2
J Y
-^
20
(1.1.14)
lim (W _ 1 u')(x)
L :=
X—*T2
= u 2 ' ( x 0 ) + | 2 ((aW _ 1 u 2 )(s)ds<0.
""0
Suppose L = 0. Then we have from (1.1.13) and (1.1.14) for all x e (x Q ,r 2 )
0 < - ((u 2 ) _1 u^Xx) = (u 2 )" 1 (x).W(x) j "
2
(aW)" 1 (s).u 2 (s)ds
< W(x) f 2 (aW)_1(s)ds.
Jx
Integration of this inequality gives
0 < - log u 2 (x) < f
W(x) f 2 (aW) _1 (s)ds < oo.
J xQ
Jx
This implies that
lim u„(x) is positive, which is a contradiction. Thus L < 0.
X-T2
l
ad (Hi). Let u denote the unique increasing (bounded) solution of (1.1.6)
satisfying u(x.) = 0, u ' ( x n ) = 1- Then it follows from (1.1.9) that W
u ' is a
positive bounded increasing solution and thus
k:=
lim (W" 1 u')(x)> 0.
x-r2
We define the function ü by ü := u« - Lk" u. It follows that
m :=
lim ü(x) = -Lk
x—*X2
. lim u(x) > 0,
x—»r2 ~
lim (W" 1 ü')(x)= lim (W _ 1 u')(x)
c-»rT
x-+ro
'*■
x->r 2
- LLk
k " 1l. lim (W" 1 u , )(x) = 0,
x^r2
and
(W" 1 G l ) l (x) = (W" 1 u')'(x) - L k _ 1 ( W " 1 u , ) ' ( x ) > 0
21
for x G (x„,r-). So W
G' is a negative increasing function and thus ü is de­
creasing. We see that the function u , := m ü is a positive decreasing solution
of (1.1.6) satisfying (1.1.12).
D
We continue with two lemmata concerning the function Q.
Lemma 1.1.8
There exists a positive decreasing solution u of (1.1.6) with
1
Q G
Ll(xQ,r2).
Urn u(x) > 0
x~*r2
Proof
Necessity. Let u be a positive decreasing solution of (1.1.6) with
lim u(x) =
2
L > 0. Since u *(x) < 0 for all x G J it follows from (1.1.9) that for x £ (x 0 ,r.)
x-+r
Y
0 < f
((aW) _1 u)(s)ds< - u ' ( x 0 ) .
Moreover
x
,
„x
0 < L f (aW) _1 (s)ds< f
((aW)_1u)(s)ds
" JxQ
Jx0
for all x G (x 0 ,r_), so
(1.1.15)
(aW)"1
eLl(x0,r2).
From Lemma 1.1.2 we know that lim (W~ u ')(x) exists, and that
x r
. j
- 2
k :=
lim (W u')(x) < 0. It follows from (1.1.8) and (1.1.11) that for all
x-+r 2
x G (x 0 ,r 2 )
(1.1.16)
- ( W _ 1 u ' ) ( x ) = -k + f ^ V ((aW) _1 u)(s)ds.
Jx
22
So
-(W V x x ^ L . f 2 (aW) _1 (s)ds> 0
" Jx
or
2
-u'(x) >L.W(x) f
(aW) 1 (s)ds>0.
0
Since u ' e L (x 0 ,r-) we have
2
f
u(x Q ) - L > L
W(t) f
2
(aW) _1 (s)dsdt
v
0
and with Fubini's theorem we get
r
u(x ) > L . [
2
.ss
(aW)"'(s) [ W(t)dtds
J XQ
=L . I
2
J xxr
0
J XQ
Q(s)ds,
so Q E L (x Q ,r 2 ).
Sufficiency.
Assume Q G L (x«,r») and let u be a positive decreasing solution
of (1.1.6) with
lim u(x) = L. Then L > 0. As above
X—*T2
~
lim (W _ 1 u'Xx) =
X-*T2
k < 0, and also (1.1.16) holds. Then we obtain for all x e (x„,r_)
(1.1.17)
- u ' ( x ) = - k W ( x ) + W(x) f
2
(aW)_1(s)u(s)ds
thus
-u'(x) < -kW(x) + u(x)W(x) f
2
(aW) 1(s)ds.
Jx
If k = 0, then we get
0 < -u'(x)/u(x) < W(x) f
2
(aW) _1 (s)ds
23
and integration over (x_,r«) and Fubini's theorem shows that
0 < -lim log (u(x)/u(xj)
< f 2 W(t) [ 2 (aW) _1 (s)dsdt
u
x—r 2
J XQ
Jt
r
s
= f 2 (aW)"'(s) f W(t)dtds
J x0
J xQ
=[
2
Jx0
Thus, if k = 0, then
Q(s)ds.
lim u(x) = L > 0.
x^r2
If k < 0 we obtain from (1.1.17) for all positive decreasing solutions of (1.1.6)
-u'(x) > -k.W(x)> 0
for all x G (x_,r«), and since u ' e L (x„,r„) we see
(1.1.18)
WELVXQ,^).
Since Q e L ( x 0 , r . ) it is clear that
(1.1.19)
(aW)" 1
GLl(xQ,T2),
and from (1.1.5) it follows that R e L (x 0 ,r_). In this case the assertion will
follow from Lemma 1.1.7(iii).
D
The following lemma is a corollary of Lemma 1.1.8.
Lemma 1.1.9
For each positive decreasing solution u of (1.1.6)
1
Q 0
LJ(x0,r2).
D
lim u(x) = 0 holds <=>■
x-^ri
24
Together with Lemma 1.1.7(iii) the next lemma shows the importance of the
condition R £ L (x n ,r„), while Q G L ( x - . r . ) , for the value of
(W
lim
u')(x) if u, is a positive decreasing solution of (1.1.6).
Lemma 1.1.10
If R £ L (x„,r ) and Q G L (xn,r.)
of (1.1.6)
then each positive decreasing solution
satisfies
lim u(x)>0,
x->r2 J
lim (W'^'
x-*r2
6
)(x) = 0.
Proof
Assume Q G L (x_,r.A From Lemma 1.1.8 we know there exists a positive
decreasing solution u_. of (1.1.6) with L~ :=
lim u„(x) > 0. Let u, := L„ u„.
x—^2
Then u, is a positive decreasing solution of (1.1.6) with lim u.(x) = 1. As in
•*
x—►r?
the proof of Lemma 1.1.8 we have k =
lim
(W
_1
x—»r2
■*
^
u')(x) < 0, and also
■*
(1.1.16) holds. From (1.1.16) we obtain for all x G (x n ,r„)
(1.1.20)
u^(x) = k.W(x) - W(x) f
Assume moreover R £
(aW)" 1 G Ll(x0,r2)
2
(aW) _1 (s).u 3 (s)ds.
L (x„,r_). Since Q G L (x n ,r~) we then have
and W £ L 1 ^ , ^ ) . From (1.1.16) we obtain
u^(x)<k.W(x)<0,
xG(X(),r2).
Since u^ G L (x Q ,r 2 ) it follows that k = 0. So from (1.1.20) we get
(W_1up(x)=- J
and thus
D
lim (W _ 1 u')(x)
= 0.
3
x—>ro
2
(aW)"1(s).u3(s)ds
25
The next lemma gives the relation between the function W and strictly
monotone solutions of (1.1.6).
Lemma 1.1.11
Let u j be a positive increasing and u ? be a positive decreasing solution of
(1.1.6), then
(1.1.21)
u u
) 2~
U U
2 1
= K W
0
Witk
K
0
= ( U U
~ \ 2 ~ U2U1^X0^
>
°'
Proof
Since u. en u- satisfy (1.1.6) we have
(au"j +)8uJ)u 2 = (ou"2 + / 8 u p u r
So
U U
1 2 - U2U1
U
U
U
U
5 2" 2 1
A
~a
or
("j"2-u2ul>'_
u!u- - u ' u .
0
a'
Integration over (x»,x) leads to (1.1.21).
D
Lemma 1.1.12
If u is a positive increasing solution of (1.1.6), then
Urn (W~1ux)(x)
x-*r2
< oo «=► Q(EL}(xn,r
°
z
).
Proof
Necessity. By (1.1.8) W~ u ' is a positive increasing function. Assume
M:=
lim (W _ 1 u')(x) < oo.
x^r2
26
Then there exists an x G ( x n' r 2^
s u c h t h a t f o r a11 x e
^ x l' r 2^
(W"1u')(x)>^.
Thus for x > x. we have by integration
u(x)>u(x0) + ^
f
W(t)dt,
X
0
or
((aW) _1 u)(x) > u(x 0 ).(aW) _1 (x) + ^ .Q(x),
and thus
(W _ 1 u')(x) - (W" 1 u , )(x_)= f
0
((aW) _1 u)(t)dt
= f
Jx
(W'^'J'Wdt
JxQ
(by (1.1.8))
0
X
>u(x0).J
(aW)_1(t)dt + ^ J
X
Q(t)dt.
By taking the limit for x —
► r . we obtain
f 2 (aW) _1 (t)dt + ^z f 2 Q(t)dt.
M - u ' ( x nu ) > u(xA
u
Jx0
Jx0
Thus Q G L (x Q ,r-).
Sufficiency.
If Q e L (x~,r») there exists a positive decreasing function u .
with
lim u.(x) = 1. Since W~ u ' is a positive increasing function there
x-+r 2 *
exists an N, N G (0,oo], such that
N :=
So
lim ( W ^ u ' X x ) .
x^r2
27
lim (W V x x J . u 3 (x) = N.
x-r2
From Lemma 1.1.11 we have for all x e (x n ,r»)
(W" 1 u')(x).u 3 (x)<K 0 <cx>.
Thus N < K» < oo. This completes the proof of the lemma.
D
Lemma 1.1.13
Let u.M-f and
(aWf1
(1.1.22)
£
Kn
1,
1.1.11. If R E L (x-,r
be as in Lemma
) and
L](x0,r2),then
lim (W'KlXx)
x—*r2
^
= -K
/
u
lim
x—*r2
u(x)<0.
l
Proof
Let R e Ll(x0,r2)
and (aW)" 1 £ L 1 ^ , ^ ) . Then clearly Q jÊ L 1 (x ( ) ,r 2 ).
Moreover u.1 is a bounded positive increasing function, thus M :=
lim u.(x)
x—>r2
exists and is positive. The function W u i is negative, and since (W u i ) ' =
-1
-1
-1
(QW) U . > 0, W u i is increasing. Thus
lim (W ul)(x) exists and is
1
l
x-+r2
non-positive, say
(1.1.23)
lim ( W - 1 u ' ) ( x ) = L < 0.
x—»r2
On the other hand, since Q £ L (x„,r_) it follows from Lemma 1.1.12 that
lim (W _ 1 u!)(x) = oo.
X—*T2
Furthermore, as Q £ L (x»,r») we have from Lemma 1.1.9 we see
28
lim u 9 (x) = 0.
x—»r2
For the product of W~ u ! and u- we have by Lemma 1.1.11 and (1.1.23)
(1.1.24)
lim (W" 1 u!u-)(x) = K n + LM,
X—*T2
thus
lim (W
x—»r2
u!u-)(x) exists and since u ! > 0 this limit is non-negative,
say
(1.1.25)
lim ( W _ 1 u ! u , ) ( x ) > 0.
X—»T2
'
Let e > 0. By (1.1.23) there exists a number x^ e (x„,r-) such that for all
x,t e (x»,r.) with x- < t < x < r»
(1.1.26)
0 < (Wu p(x) - .(Wu p ( t ) < c/(2M).
Let T e (x-,r_). Then for all x e (T,r-) we have
0 < (W _ 1 u;u 2 )(x) = u 2 (x) {J*
(W _1 uJ)«(s)ds + u | ( x 0 ) ] .
(aW)" 1 (s)u 1 (s)ds + u j ( x 0 ) }
= u 2 (x) { f
0
< u 2 (x) { u j ( r 2 - ) |
(oW)" \%)ds + u J(X())]>
T
(aW) _1 (s)ds + u j ( x 0 ) }
= u 2 (x) { u j ( r 2 - ) [
X
0
+ u 2 (x)Uj(r 2 -) ƒ
(aW) _1 (s)ds
T
< u 2 (x) { u , ( r 2 - ) J"
(aW)" ^sjds + u " ( X Q ) }
Y
+ Uj(r2-)J
(aW) _1 (s)u 2 (s)ds
29
.T
= u 2 (x) { u j ( r 2 - ) |
(aW)"1(s)ds + u ' ( x 0 ) }
O
Uj(r 2 -)
x
^ ( W ^ u ^ d s
(aW) _1 (s)ds +
= u 2 (x) { u ! ( r 2 - ) J
u\(xQ)}
+ Uj(r 2 -) { ( W ^ u ^ x ) - ( W _ 1 u p ( T ) }
T
(1.1.27)
< u 2 (x) { u j ( r 2 - ) |
(aW)_1(s)ds + u ^ ) } + e/2.
Since
lim u-(x) = O there exists a number S e (T,r_),
such that for all
z
x-+ro 2
xe(S,r2)
T
(1.1.28)
0 < u 2 (x) < { u j ( r 2 - ) j*
(QW)"1(s)ds + u' 1 (x () ))-" 1 £/2.
From (1.1.27) and (1.1.28) we obtain for all x e (S,rJ
0 <(W_1uJu2)(x)<e.
Thus
lim (W _ 1 u!u-)(x)
= Ku + L M = 0.
l l
x—»r2
(1.1.29)
Combining (1.1.23), (1.1.24) and (1.1.29) we obtain
lim (W _ 1 u')(x) = L = - K M " 1 < 0.
l
u
x—r2
This completes the proof of the lemma.
D
Remark. This lemma improves the result of Feller ([Fe,l], Th. 11) which
stated L < 0.
30
1.1.3. Classification of the boundary points
After the investigations on the solutions of the homogeneous equation (1.1.6) it
is possible to give a very useful classification of the boundary points r and r .
of the interval J. We give the classification for the right boundary point r_.
The classification for the left-boundary point is similar.
The classification given here is Feller's classification [Fe,l]. We mention that
Feller assumed that a is positive and continuously differentiate on J, whereas
we only assume that a is positive and continuous on J. Thus here we drop the
condition on the differentiability of a.
Firstly we give the classification of Feller [Fe,l].
Definition 1
Let a and P be as in section 1.1.1. The boundary point r . is called
(aW)" 1 G
Regular
if
W e \}{xQ,x2),
Exit
if
(aW)' 1 É L ^ X Q , ^ ) , R G L ^ X Q , ^ ) ,
Entrance
if
W £ L (x Q ,r 2 ), Q G L (x Q ,r 2 ),
Natural
in all other cases.
L\XQ,T2),
It is also possible to state the criteria only using the functions Q and R. An
equivalent definition is
Definition 2
Let a and p be as in section 1.1.1. The boundary point r_ is
Regular
if
Q GL1^,^), R G
Exit
if
Q £ L^XQ,^), R G
Entrance
if
Q G L (x Q ,r 2 ), R £ L ( x 0 ) r 2 ) ,
Natural
if
Q £ LVQ,^), R £ L1^,^).
L1(XQ,T2),
L\X0,T2),
From now on we will use Definition 2. From this definition and the lemmata
31
of section 1.1.1, it follows that the criteria for the boundary points can be
given by means of monotone solutions of the homogeneous equation (1.1.6). We
will summarize and prove - as far as not proved before - these results in the
following Lemmata 1.1.18 - 1.1.21.
State diagram for the boundary point r?
state-1
WGL^XQ,^)
(aWr^L^XQ,^)
RSL^XQ,^)
Q€L 1 (x ( ) ,r 2 )
r . Regular
yes
yes
yes
yes
r . Exit
yes
no
yes
no
r . Entrance
no
yes
no
yes
r» Natural (i)
(ii)
(iii)
yes
no
no
no
yes
no
no
no
no
no
no
no
Remark.
The equivalence of the Definitions 1 and 2 follows from the
implications:
(i)
Q G L ^ X Q , ^ ) =► (aW)" 1 G
(ii)
RGL1(X0,T2)=>
(iii)
W 6 L \ x 0 , r 2 ) A (aW)" 1 e L 1 ^ , ^ ) => Q,R G L 1 ^ . ^ ) .
Simple
h\x0,T2)
W GL^XQ,^)
examples
1. J = (-1,1), XQ = 0, a(x) = 1, 0(x) = 0. The points -1 and 1 are regular
boundary points. The functions u
and u 2 with u (x) = sinh(l - x), u.(x)
= sinh(l + x) form a fundamental set of solutions of u - u" = 0,
2
u G C (-1,1), satisfying
32
lim u . ( x ) = lim u_(x)
= 0.
x-+l l
x—»-l z
2. J = (-7T/2,IT/2), x Q = 0, Q(X) = 1, £(x) = 2 tan x. The points TT/2 and -ir/2
are ex/7 boundary points. The functions u. and u- defined by
u.(x) = cos x + (TT/2 + x) sin x
u^(x) = cos x - (7T/2 - x) sin x
form a fundamental set of solutions of
u - u" - 2 tan( . ) u ' = 0, u e
C2(-n/2,ir/2),
satisfying
lim
X^-TT/2
u.(x)=
l
lim
u,,(x) = 0.
2
X-+7T/2
3. J = (-ir/2,jr/2), xQ = 0, a(x) = 1, /?(x) = -2 tan x. The points -JT/2, TT/2 are
entrance boundary points. The functions u . and u . defined by
u,(x) = (TT/2 - x)/cos x
U 4 ( X ) = (7T/2 + X)/C0S X
form a fundamental set of solutions of
u - u" + 2 tan( . )u' = 0,
satisfying
lim
X^-TT/2
u .(x) = lim
4
X-7r/2
u,(x) = 1.
3
33
4. J = (-00,00), x„ = O, Q(X) = 1, y9(x) = 0. The "points" -00 and 00 are natural
boundary points. The functions u. and u» defined by
/ \ = ex
Uj(x)
u 2 (x) = e
form a fundamental set of solutions of
u - u" = 0, u £ C(-oo,oo)
satisfying
lim u.(x) = lim u»(x) = 0.
x—»-oo *
x—*oo ^
Lemma 1.1.14
If r. is a regular boundary point, then there exists a positive decreasing
solution u~ of (1.1.6)
satisfying
lim u(x)
x—r 2 z
= 0,
lim (W'1 u\)(x)
z
x^r2
and a positive decreasing solution « , of (1.1.6)
lim uJx)
x-*r2 J
= l,
positive and decreasing
satisfying
lim (W~1 u\)(x)
i
x->r2
Each solution of (1.1.6) is bounded on (xn,r?),
= -1
= 0.
but a solution u of (1.1.6) is
if and only if there are non-negative numbers p?
and p-., not both zero, such that u = p?u7 + p,u,. There also exists a positive
increasing solution u.1 of (1.1.6) for which
lim W
x-+r2
u\(x)
< 00.
J
Proof
The existence of u_, resp. u- follows from Proposition 1.1.1, and the Lemmata
1.1.7 and 1.1.8. Assume u = P2U-) + P^ u ? *s positive decreasing with p» > 0,
34
p
> O, not both zero. Then (W _ 1 u'Xx) = p 2 ( W _ 1 u p ( x ) + p 3 (W _ 1 u^)(x) < 0
for x G (x ft ,r_), so u is decreasing. Conversely, if u is a decreasing solution of
(1.1.6) then there are p - , p , such that u =
P2UT
+ P i u i - **y taking the limits
for x -* r . we see
lim u(x) = p . . lim . u,(x) = p . > 0
x— r
x—>T2
*2
and
lim (W" l u')(x) = p
x-+r2
lim (W _ 1 u»)'(x) = p > 0.
x-+r2
If u is decreasing then p . , p , are not both zero.
Since each (positive increasing) solution u of (1.1.6) is a linear combination of
u z? and u-,
lim (W
u)(x) is finite.
■* x—»r?
D
Lemma 1.1.15
If r is an exit boundary point, then there exist a positive decreasing solution
u- of (1.1.6)
satisfying
lim uJx)'0,
x-+r2 l
lim (W'1 u\)(x)
z
x-*r2
and a bounded positive increasing solution u. with
1
= -1,
lim (W
u \)(x) = oo.
x^r2
l
Proof
The existence of u_ follows from Proposition 1.1.1 and Lemma 1.1.13. Let u.
be a positive increasing solution of (1.1.6). By Lemma 1.1.3
_1
From Lemma 1.1.12 it follows that lim (W u')(x) = oo.
x—»ro
u. is bounded.
l
D
Lemma 1.1.16
If r? is an entrance boundary point, then there exists a positive decreasing
solution u, of (1.1.6)
satisfying
35
lim (W~1 u\)( x) = O,
J
x-^r2
lim u(x)=l,
x-+r2 5
and an unbounded positive increasing solution u. of (1.1.6) with
lim
x^r2
(W~1 1u))(x)<oo.
Proof
The existence of u , follows from the Lemmata 1.1.8 and 1.1.10. The existence
of a positive increasing solution u. of (1.1.6), such that
lim u.(x) = oo folx^r2 ]
_j
lows from Lemma 1.1.5. From Lemma 1.1.12 it follows that lim (W u!)(x)
x—*X2
< oo.
D
Lemma 1.1.17
If r7 is a natural boundary point, then there exists a positive decreasing
solution Uy of (1.1.6)
satisfying
lim u(x)
x—>r2 z
= 0,
lim (W~}u\)(x)
l
x-*r2
= 0,
and an unbounded positive increasing solution u. of (1.1.6) with
lim (W~1u\)(x)
x—r2
i
= oo.
Proof
The existence of a positive decreasing solution u . of (1.1.6) with lim u.(x)= 0
follows from Proposition 1.1.1 and Lemma 1.1.9. Let
lim (W
r
x—►ro
2
u')(x) = L,
z
as;
then L < 0. The existence of an unbounded positive increasing
solution u. of
(1.1.6) follows from Proposition 1.1.1 and Lemma 1.1.4. Assume L < 0, then
lim (-W" 1 u»)(x).u.(x)
= oo,
z
x-+rT
1
36
in contradiction with the boundedness of W
u i u . , which follows from
(1.1.21). Thus L = 0.That
lim (W~ u !)(x) = oo follows from Lemma 1.1.12.
x—^2
D
State diagram for monotone solutions of u - aZ) u - f)Du = 0, u e C (J)
I
II
III
IV
r- Regular
yes
yes
yes
no
r_ Exit
yes
no
yes
yes
r- Entrance
no
yes
no
no
r_ Natural
no
no
yes
yes
I
- There exists a positive increasing solution u. with lim u.(x) < oo.
1
x-»r2 l
II - There exists a positive decreasing solution u , with lim u(x) = 1.
■*
x-+r2
HI - There exists a positive decreasing solution u_ with lim u(x) = 0.
^
x—>r2
IV - For each positive decreasing solution u lim u(x) = 0 holds.
x-r2
Remark. The type of the boundary point r- is completely determined by the
columns I and II. The columns III and IV give extra information.
37
-1
State diagram for lim
x^r2
(W
2
2
u ' )fx), where u - aD u - 0Du = 0, u € C (J)
I
II
III
IV
r„ Regular
yes
yes
yes
no
r . Exit
no
yes
no
no
r» Entrance
yes
no
yes
yes
r„ Natural
no
no
yes
yes
III
lim (W u ')(x) is
x-r2
There exists a positive decreasing solution u-with lim (W
^
x—>r*>
There exists a positive decreasing solution u , with lim (W
IV
For each positive decreasing solution u
I
II
For each positive increasing solution u
■*
lim (W
x^r2
X—>f)
finite.
ul)(x)<0.
^
ul)(x)
= 0.
J
u ')(x) = 0 holds.
Remark. The type of the boundary point r- is completely determined by the
columns I and II. The columns III and IV give extra information.
Now we will show that the classification of the boundary points is intrinsic in
the following sense. If <j> is a twice differentiable diffeomorphism from an
open interval J not necessarily bounded) with the two points compactification
J^, onto J, such that the boundary points jr. and £- of J are carried into the
boundary points r
respectively r_, then the boundary points £. and r.
(i = 1,2) are of the same type. We will state this in the next lemma.
38
Lemma 1.1.18
Let the continuous bisection <f>: J —
► J satisfy the following
conditions:
2
<t> e C (J), '<j> *(t) > 0 for all t e J, <f>(tQ) = xQ, <j>d.) = r. (i - 1,2) and let <
be normalized by <j> %(tn) = 1. Let a and 0 as in Proposition 1.1.1, u := u o <f>,
let the differential
expression Au be given by (1.1.1) and assume that <t>
carries Au into the differential
(1.1.30)
expression
Au := aD2u + fSDu.
Let the function W on J be defined by (1.1.3) and let W on J be defined by
(1.1.31)
(ior1)(s)ds].
W(t) = exp{-\
Then the boundary points £. and r. are of the same type, (i = 1,2).
Note that a and £ are defined by (1.1.30).
Proof
Let x = <j>(t). Then u'(t) = u '(\).<f> '(t), u"(t) - u"(x).(0 '(t)) 2 + u '(x)^"(t). Sub­
stitution of u'(x) and u"(x) in (1.1.1) and a comparison with (1.1.30) shows that
a(t) = (aoMt).'(*'(t))~ 2
g(t) = (/J o <j>)(t).(<t> '(t))" 1 - (a o Mt).f(t).(tf '(t))" 3 .
Now
f
(£a _1 )(s)ds = f
Jt
Jt
0
tfa^Xfls^'feJds
0
(sW(s)) - 1 ds
" It,
*
0
v
*(t)
f•YK
'
JxQ
-1
39
So by (1.1.31) we have
W(t) = (W o <t>)(t)-<t> *(0
and it follows that
W e L
1
^ ) <=* W e
L\X0,T2).
Moreover,
(«W)" 1 (t) = ((aW)" 1 o 0(t).* "(t),
hence
(aW)" 1 G h\t0,T2)
(aW)" 1 G
^
Ll(xQ,r2).
Let the functions Qand R. be defined similar to (1.1.4) and (1.1.5):
9(t):=(aW)_1(t) f
W(s)ds,
°
(aW)_1(s)ds.
R(t):= W(t) f
0
Then it is easily verified that
Q(t) = (Q o 0(t).tf '(t),
R(t) « (R o <f>)(t).<f> '(t).
Hence
Q G L!(t0,r2) * ^ Q e L ^ X Q , ^ ) ,
R. G L ! ( t 0 , r 2 ) ^ * R 6
Ll(x0,r2).
With Definition 2 it is easily seen that £ - and r- are boundary points of the
same type. This proves the lemma.
D
40
1.1.4. Solutions of u - aD2u - QDu = f in C(J)
A well-known technique to obtain solutions of an inhomogeneous equation is
the construction of solutions by means of Green functions, cf. [Y,l].
In this section we will define a special Green function V: JxJ -+1R, called a
regular Green function in Feller's terminology, in order to obtain a special
solution of the inhomogeneous equation
(1.1.32)
u - (aD 2 u + 0Du) = f,
f e C(J),
u e C(J) n C 2 (J),
denoted by u f .
Then the general solution of (1.1.32) in C(J) is given by
u = u f + c . u . + c^u 2
where u. and u» are two independent solutions of the homogeneous equation
(1.1.6) and where c.,c_ GlR. We will see that if r. (resp. r-) is an entrance or
a natural boundary point c_ (resp. c.) is equal to zero. If r . (resp. r_) is a
regular or an exit boundary point an extra condition will be necessary for
uniqueness of the solution of (1.1.32).
In particular we are interested in the case we have so called Ventcel's
boundary conditions on D(A):
(1.1.33)
lim ( Q D 2 U + £Du)(x) = 0,
x-> r i
i = 1,2.
It will appear that in the case of a natural boundary point this condition is
automatically satisfied.
Let u. and u . be the unique special solutions of (1.1.6) as given in Proposition
1.1.1. Thus u.(x_) = 1, u. is an increasing solution which is minimal on
(r ,x„) and u„ is a decreasing solution which is minimal on ( x „ , r A
From the Lemmata 1.1.7, 1.1.9 and (1.1.10) we know that
lim u.(x) = 0 if
x-*r: _ 1
41
and only if R e L (x„,r.) or Q £ L (x-,r.), i.e. in the case that r. is a
regular, an exit or a natural boundary point, and thus
lim u.(x) > 0 if
and only if R £ L (x„,r.) and Q e L (x„,r.), i.e. r. is an entrance boundary
point.
We define the Green function T: JxJ —>3R of the equation (1.1.6) as follows:
T(x,s) = ( K 0 Q W ) " 1 ( S ) U 1 ( S ) U 2 ( X )
forrj<s<
x < r2
(1.1.34)
= ( K . Q W ) " (s)u-(s)u .(X)
for r. < x < s < r-.
Here K~ is the positive constant defined (as in Lemma 1.1.11) by
K Q = u j(x Q ).u 2 (x 0 ) - U ^ X Q X U ^ X Q ) = U J U Q ) - u 2 '(x 0 ). For each f e C(J) we
define the function u f by
(1.1.35)
(uf )(x)
)(x)== || ^2 r(x,s)f(s)ds.
1
Then the mapping f - » u f induces an operator K on C(J). We have
Proposition 1.1.19
Let for each f G C(J )
Kf be defined by Kf = uf Then
(i)
K is a positive, linear contraction operator from C(J) into C(J),
(ii)
Kf is a particular solution of (1.1.32).
Proof
(i)
Let f G C(J), then f is bounded, and
|u,(x)| < ||f||
2
f
Jr
r(x,s)ds
l
((K 0 aW)" 1 u 1 )(s)ds
= llfll {u 2 (x) J
1
+ Uj(x) J
2
(K0aW)'1u2)(s)ds}
42
(W" 1 up'(s)ds
^ j ' l l f H {u2(x)J
1
2
+ Uj(x) ƒ
Since W
u ! is positive increasing and W
(W_1up'(s)ds}.
uJ is negative increasing both in­
tegrals exist. It follows that for each x € ( r . , r . ) u f (x) is well-defined.
We have for all x e (r. ,r-)
((K Q aW)" J u j f)(s)ds
u f (x) - u 2 (x) I
+ Uj(x) J
2
((K ( ) aW)" 1 u 2 f)(s)ds.
2
((K 0 aW)" 1 u 2 f)(s)ds
-Xhen-u^-is-twice-continuously-dif-ferentiable-with
Y
((K 0 aW) _ 1 U l f)(s)ds
u£(x) = u ^ x ) J
H{W fx
and
Y
u"{(x) = u 2 (x) I
((K 0 aW)" l u j f)(s)ds
u"(x) J
2
((K n aW)" 1 u 0 f)(s)ds
' 0 " " ' -2 1
((uJu2-u^u1XK0aW)"1f)(x).
From Lemma 1.1.11 we have
( u j u 2 - u^u 1 )(x) = K 0 .W(x).
By substitution it is easily verified that for all x e (r ,r 0 )
43
u f (x) - au'j.(x) - pu '(x) = f(x).
It is clear that K is linear and since T is a positive function the operator K is
positive.
Let f be a non-negative function in C(J), and let ((a ,b )) be a sequence of
intervals such that
r. <
1
<a
. <a <
n+1
n
< a. < b . <
1 1
<b <b . <
n
n+1
< r„.
2
Moreover, let (f ) be a sequence of continuous functions such that
f n (x) = f(x)
x e [a n ,b n ],
0 < f v(x) < f(x)
v
n '
'
x e v(a , , a ) u ( b , b
.),
n+1 n
n n+1"
f v(x) = 0
n '
x € v(r.,a , ) u v(b , , r . ) .
1 n+1'
n+1' 2'
Then for each x e (r ,r_) the sequence (f (x)) is non-decreasing and
lim f (x) = f(x).
n—»oo "
Let n e l . Since K is positive uf is a non-negative solution of (1.1.32) with
'n
f = f . On the subinterval (r. ,a ,) we have
n
. 1' n+1'
u rf ( x ) - ( a u "l )(x)-(/Ju!.l )(x) = 0,
n
n
n
and for x e (r.
,a .) we have by the definition of u r
v
1 n+1
'
f
u f (x) = Uj(x) . |
2
((KoaW) _1 u 2 f n Xs)ds > 0
n+1
and similarly for x e (b
.,r_)
u f (x) = u 2 (x) ■ ƒ
n+1
((K 0 aW)" 1 u 2 f n )(s)ds > 0.
44
Thus u f
is positive increasing on (r.,a
.) and positive decreasing on
(b
v
,,r_). Hence u,. has a positive maximum at a point x e (a , ,b .).
n+1 2'
fn
m
n+1 n+1
Then u ' v (x ); = 0 and u" v(x ) < 0. So
fn m
fn m'
u , v (x ) = f (x
) + (au" )(x )
fn m'
nv nv v fn' nv
< f \(x / ) < ii||f||.
ii
-
n
m
Thus certainly
(1.1.36)
l|uf||<||f||
'n
for all n e l .
Because of~the posiïiVity of-KT~the sequence (u f (x))_is~increasing for each
n
x G J, and since
[
2
J rj
r(x,s).(f .(s) -
f(s)) ds
n
converges to zero, for each x G (r.,r_) the sequence (u r (x)) converges from
l
Z
tn
below to u f (x). Thus also
(1.1.37)
||u f || < || Uf ||.
Applying Lebesgue's Monotone Convergence Theorem on the integral repre­
sentation (1.1.35) for u f and using (1.1.36) we obtain
'n
(1.1.38)
||Kf|| = ||uJ| = sup u f (x) = sup lim u f (x) < sup sup \xf (x)
1
XGJ l
XGJ n-+oo l n
XGJ nQN ^n
< sup sup ||uIf || = sup ||ur f || < sup ||f|| = ||f||
XGJ n&N n
nelT
n
nGJN
for all non-negative functions in C(J). Because of the positivity of K, (1.1.38)
holds for all functions in C(J).
45
The boundedness of u f follows from (1.1.38), but this does not guarantee that
u f1 e C(J). So we have to prove that
lim u f (x) exists for i = 1,2. We will
x—>r; '
prove the existence of the limit for the boundary point r-.
Let f G C(J). Then by (1.1.35)
(1.1.39)
u f (x) = a f (x) Uj(x) + b f (x) u 2 (x)
with
a f (x)= f
2
( ( K a W ) " ' u f)(s)ds
and
bf(x)=J
((K 0 aW) _ 1
Ul f)(s)ds.
We have to investigate the various cases. We formulate the results in two
lemmata.
Lemma 1.1.20
If r. is a regular or an exit boundary point, then
(1.1.40)
lim (Kf)(x)
x-+r2
= lim u/x)
-x-r2 I
= 0.
Proof
If r„ is a regular or an exit boundary point, then lim u.(x) = 0, lim u.(x)
z
x-*i"2 ~l
is finite and
x—>*2
lim a f (x) = 0. Moreover, if r»l is regular, then also lim bf f (x)
x-+r2 *
x-»r2
is finite. In view of (1.1.39) we then have lim u f (x) = 0.
x-+r2 *
If r_ is an exit boundary point, then clearly lim a f (x) u.(x) = 0, so it rez
x-»r2 '
mains to show that
(1.1.41)
lim b f (x) u-(x) = 0.
x-+r-j
r
~l
46
we have
|b f (x)|.u 2 (x) < K Q 1 ||f||.u 2 (x) ƒ
((aW)" 1 u 1 )(s)ds
= KQ1||f||.u2(x)|
(W^ujVWds
= KQ 1 ||fl|.u 2 (x){(W" 1 u 1 ')(x)-
lim ( W _ 1 u ; X t ) }
sK^llfll.u^xMW^ujXx).
From (1.1.29) and the proof of Lemma 1.1.13 we have
lim u (x).(W" 1 u')(x) = 0.
x-+r2
Thus also (1.1.41) holds. The proof for r. is similar.
Lemma 1.1.21
If r 7 is a natural boundary point, then
(1.1.42)
lim (Kf)(x)
x—*r2
= lim u/x)
x-*r2 '
= lim f(x).
x-^rj
Proof
Let r» be a natural boundary point and let e > 0. Since f e C(J) there exists,a
point b e (x Q ,r-) such that for all x e (b,r«)
|f(x) -
lim f(x)| < e.
x-r2
Then we have for x G (b,r_)
(1.1.43)
u f (x) = f(x) f
Jr
with g(x) = |
2
Thus
*
2
r(x,s)ds + g(x)
l
r(x,s).(f(s) - f(x))ds.
47
(1.1.44)
2
|g(x)| < e f
Jr
r(x,s)ds,
xÊ(b,r,).
2'
l
By the definition of V (1.1.34) we have
Y
r
f
2
r(x,s)ds = u 2 (x) ƒ
(K 0 aW)" 1 (s)u 1 (s)ds
2
+ Uj(x) J
(K 0 aW) _ 1 (s)u 2 (s)ds
= U2(X).KQ1.|
(W'^V^ds
2
1
+
u1(x).K-0 .^
(W" 1 uj)'(s)ds
(by 1.1.8)
2
= u^.K^.^W^Uj'Xx) - (W^'Xr,*)}
+ u ^ x J - K J ^ . / o - (W _ 1 u^)(x)\ (by Lemma 1.1.17)
= 1 - KQ 1 .u 2 (x).(W" 1 u 1 l )(r 1 +)
(by Lemma 1.1.11).
Since
lim (W u.')(x)
is finite (compare Lemma 1.1.2) and lim u_(x)
=0
]
x-+ri
x—^2 l
if r , is a natural boundary point, it follows that
(1.1.45)
-P
lim
I
2
T(x,s)ds= 1
x-+r 2 J r j
»r-> J r .
if r_ is a natural boundary point. Finally it follows from (1.1.43) - (1.1.45)
that
lim u.(x) - f(x) f
i-*2
f
Jr
and with (1.1.45) we obtain (1.1.42).
D
2
l
T(x,s)ds = 0
48
Returning to the proof of Proposition 1.1.19, in all occurring cases Kf e C(J).
This completes the proof of the Proposition.
□
As a corollary of Lemma 1.1.21 we have
Corollary 1.1.22. If r? is a natural boundary point, then for each solution u in
C(J) of u - Au = f, f e C(J), we have
Urn (Au)(x) = 0.
x—r2
D
Corollary 1.1.23
i
sup
xeJ
rr?
\Jr
T(x,s)ds < 1.
Proof
If eJs) = 1 for all s e J, then
(1.1.46)
||
2
r(x,s)ds| = | J
2
r(x,s).e0(s)ds
= l(Ke 0 )(x)| < ||Ke0H < ||e0H = 1
which proves the corollary.
D
1.1.5. Solutions of z - aD z - 0Dz = k in L
(J)
oo
Since (C(J), ||.||) is a closed subspace of (L (J), ||.|| ) the results of section 1.1
are valid in the space L (J). Here ||.||
|z|| 0 0 = ess sup |z(x)
XGJ
denotes the norm defined by
49
Let us consider the equation
(1.1.47)
z - aD 2 z - /?Dz = k
in L (J). For each k £ L (J) we define the function z. by
(1.1.48)
(x) - f '2
z R (x)= | 2 r(x,s)k(s)ds,
Jrj
where T is the regular Green function defined by (1.1.34). Then the mapping
k - t z , induces a linear operator K on L (J). As an extension of Proposition
1.1.19 we have
Proposition 1.1.24
Let for each k G L (J) Kk be defined by Kk = z , . Then
(i)
K is a positive, linear contraction operator from L (J) into L
(ii)
Kk is a particular solution of (1.1.47).
(J).
Proof
The proof is essentially the same as the proof of Proposition 1.1.19, so we can
refer to that proof.
oo
Since k e L (J) and T is continuous, Tk is locally integrable. Moreover, for
each x e J we have by (1.1.46)
(1.1.49)
•J;
|z k (x)| < HkH^. | r2 r(x,s)ds < HkH^ < oo.
'l
z.(x) is well-defined for each x e ( r . , r - ) . It follows from (1.1.48) that
z, G C (J), z ' G AC. (J) and that z. is a particular solution of (1.1.47). That
K is a contraction follows from (1.1.49).
D
50
1.2. Solutions of v - D(aDv - ftv) = g in L (J)
In this section we will investigate solutions of the equation
(1.2.1)
v - D(aDv - M = g
in L (J), where J is an open interval ( r . , r . ) (not necessarily bounded) and a
and P are real-valued continuous functions on J with a(x) > 0 for all x e J
just as in section 1.1. The differential expression
(1.2.2)
Bv = D ( a D v - £ v )
which appears in (1.2.1) is the formal adjoint of the differential expression Au
given in (1.1.1). So we may expect that there is a relation between the solutions
of the homogeneous equation (1.1.6) in C(J) and
(1.2.3)
in L .
00
v - D(aDv - £Dv) = 0
(J). Here L.
for which
fb
(J) denotes the set of all Lebesgue measurable functions f
°°
|f(x)|dx < <x> for each compact subset [a,b] of J.
Ja
It will be shown that indeed there is a close relation between the solutions of
(1.1.6) and (1.2.3). In the various lemmata and propositions we will exploit this
relation. We will start with an investigation of the solutions of the homo­
geneous equation (1.2.3).
1.2.1. Solutions of v - D(aDv - 0v) = 0
Let V be the set of all solutions of the homogeneous equation (1.2.3)
v - D(D(av) - /Sv) = 0,
51
satisfying
(a)v 6 L ^ J ) ,
(b)ov e AC loc (J),
(1.2.4)
(c)(av)' - fiv e AC loc (J).
Note that v e V also satisfies v e C(J), av e c ' ( J ) , (av) 1 - /9v e C (J), and
aWv e C 2 (J).
Let U be the set of all solutions of the homogeneous equation (1.1.6)
u - (aD 2 u + /9Du) = 0
with u e C (J).
Then we have
Lemma 1.2.1
The mapping $.■ U -* V, defined by
(1.2.5)
<t>(u) =
(aW)~1u
is a linear bijection.
Proof
1
$(U) c V. Assume u 6 U and set v := (aW)
_1
2
u. Clearly v satisfies (a).
1
Since av = W uu w
with u e C (J) and W" e C (J) we have av e C (J), thus
(b) holds. Moreover,
( a v ) ' - pw = W - 1 ( u ' + /3a ' u ) - j5.(oW)"'u
= W"1u,)
i
i
which implies that (av) ' - j J v e C (J), thus (c) holds.
52
It is standard that U and V are two dimensional linear spaces. (Here we use the
assumption a > 0). Finally, $ is linear and injective (since (aW)
> 0), hence
surjective.
a
Lemma 1.2.2
Let v E V be positive and such that aWv is non-increasing,
Urn (W }(aWv) ')(x)
x-*r2
and is non-positive and v £ L
then
exists
(xn,r~).
Proof
Assume v e V such that aWv is non-increasing. Since (1.2.3) can be rewritten
as
(1.2.6)
v - (W'^aWv)')'= 0
and since QWV is a twice continuously differentiable function it follows that
Y
W" l (aWv) '(x) = f
(1.2.7)
Jx0
Thus
W
lim W
x-»r2
v(t)dt + (aWv) ' ( x A
0
(QWV) ' is a non-positive non-decreasing continuous function, so
(QWV)'(X)
exists and is non-positive. Further, from (1.2.5) we
1
-1
then obtain v e L (x n ,r ). Similarly, if v e V, then x—>r
lim (W (aWv)')(x)
exists and is non-negative, and v e L ( r . , x n )
In the Lemmata 1.1.14 - 1.1.17 we gave for the different types of b o u n d a r y
points special solutions u . , u« and u , of the homogeneous equation (1.1.6).
F r o m t h e preceding Lemma 1.2.1 it follows that the functions v . = $ ( u . )
1
1
1
(i = 1,2,3) are special solutions of (1.2.3) in L. (J). In the next Lemmata
1.2.3 - 1.2.6 we will summarize some properties of these solutions.
53
Lemma 1.2.3
Let r . be a regular boundary point, let u- and u , be as in Lemma 1.1.14, and
let V: = $(u.), i = 2,3.
Then v and v, are positive continuous solutions of (1.2.3),
v. G L (xn,r7),
2
U i
av72GC}(J)
aWv7 is decreasing, Urn (aWvy)(x)
i
x—*r2
and
Urn ((av)' z - pv )(x) = -1;
x->r2
v, e L (xn,rj,
aWv, is decreasing, Urn (aWv,)(x)
i
5
u 2
3
x—>r2
av i G Cl(J)
= 0,
and
= 1,
Urn ((av J ) • - /3vJ(x)
= 0.
J
x-yrj
For each solution v of (1.2.3) we have v e C(J), v G L (x„,r 2), aWv is
bounded on (x„,r .), av G C (J) and Urn ((av) ' - fiv)(x) exists. There also
u
^
x—*r2
exists a positive solution v. for which aWv. is increasing.
For each positive solution v of (1.2.3) we have Urn ((av)' - 0v)(x) < 0 if
x->r2
and only if
V = P V
22
+ P V
33
with p2> 0, p.. > 0, not both zero.
Lemma 1.2.4
Let r? be an exit, boundary point, let u. and u ? be as in Lemma 1.1.15 and let
v.*9(u.),
i = 1,2.
Then v . and v2 are positive continuous solutions of (1.2.3),
v fc L (x~,r ) , aWv. is increasing and bounded,
av 1 EC (J) and
v. G L (x',r),
Urn ((av 1 ) • - /?v1.)(x) = oo;
x-*r2
aWv j is decreasing,
Urn (aWv 2)(x) = 0,
54
av.eC^J)
1
and
Urn ((av.)J • - 0v 1 )(x) = -1.
x->r2
Only solutions of (1.2.3)
which are scalar
multiples
of v? belong to
Ll(xQ,r2).
D
Lemma 1.2.5
Let r- be an entrance boundary point, let u. and u , be as in Lemma 1.1.16,
and let v. := $(u.), i = 1,3.
Then v. and v, are positive continuous solutions of (1.2.3),
v. G L (x„,r .), aWv. is increasing,
Urn (aWv.) = oo,
x *r2
av 1 G C (J) and Urn ((av) ' - 0v)(x) < oo;
x-+r2
v, e £ (x0,r2),
aWv ? is decreasing,
av J G C!(J) and
Urn (aWvJfx)
x *r2
= 1,
Urn ((av) ' - 0v)(x) = 0.
x-*r2
For each solution v of (1.2.3) we have
v e L (xnu,r z ) and
Urn ((av) ' - 0v)(x)
x-+r2
exists.
a
Lemma 1.2.6
Let r2 be a natural boundary point, let u. and u- be as in Lemma 1.1.17', and
let v. = *(u.), i = 1,2.
i
i
Then v. and v, are positive continuous solutions of (1.2.3),
v * fi L (XQJ.),
aWv J is increasing,
Urn (aWv J(x) = oo,
~*r2
av 1 G C (J) and Urn ((av 1 ) ' - PvJ(x)
= oo;
1
x-*r2
x
55
v7 6 L (xn,r 7), aWv- is decreasing,
av G C](J)
z
and
Urn (aWv7)(x)
= O,
Urn ((av ) ' - fiv )(x) = 0.
x—*r2
Only solutions of (1.2.3)
which are scalar
multiples
of v . belong to
J
L (x0,r2).
a
The proofs of these lemmata rest on the proofs of Lemmata 1.1.14 - 1.1.17.
The only thing we have to prove yet is v £ L (x ft ,r-) in the case of an exit
or a natural boundary point.
This is a consequence of
Lemma 1.2.7
Let u be a positive increasing solution of (1.1.6) and let v = $(u), then
v G L (xQ,r2)
<s=s* Q G L
(xQ,r2).
Proof
By (1.2.5) we have
X
f
Jx
X
|v(t)|dt= f
o
Jx
ItfaW^uXOIdt
o
Y
= f
(W _ 1 u')'(t)dt
v
J x„
0
(W"1u')(x)-u'(xo).
Then the assertion follows from Lemma 1.1.12.
D
We define the functions v and v- by
(1.2.8)
y. = (aW) _ 1 u.,
i - 1,2,
(by (1.1.8))
56
where u and u_ are the special solutions of (1.1.6) as given in Proposition 1.1.1.
It follows that if r. and r» are not entrance boundary points, then
(1.2.9)
lim (aWv.)(x) = 0,
x-»r;
'i
i = 1,2.
The general solution of (1.2.6) is given by
(1.2.10)
v = c
l^l+c2-2'
C C
e R
1' 2
' '
From the Lemmata 1.2.3-1.2.6 the following lemma is easily obtained
Lemma 1.2.8
Let the general solution of (1.2.6) be given by (1.2.12).
If r■ is an exit or a natural boundary point, then v G L (xn,r J implies c . = 0.
If r. is a regular or an entrance boundary point, then for uniqueness of the
solution of (1.2.6) an extra condition on v is needed.
a
1.2.2. Solutions of v - D(D(av) - 0v) = g in
L](J)
As in section 1.1 we will construct solutions of the inhomogeneous equation
(1.2.11)
v - D(D(ov) - M = g,
geL^J),
veL^J)
by means of special Green functions. It will appear that it is possible to use the
Green function T, defined in (1.1.34), to obtain solutions of (1.2.11).
Let u. and u_ be the unique "minimal" solutions of (1.1.6) as given in
Proposition 1.1.1 and let the Green function T be defined by (1.1.34).
For each g e L (J) we define the function v by
(1.2.12)
v
g(X):=I
2 r(s,x)g(s)ds
-
57
The mapping g - » v
induces an operator L on L (J).
As a counterpart of Proposition 1.1.19 we have
Proposition 1.2.9
Let for each g G L (J) Lg be defined by Lg = v . Then
8
(i)
(ii)
]
j
L is a positive linear contraction operator from L (J) into L
Lg is a particular solution of (1.2.11).
(J).
Proof
Let g e L (J) and x e J. Then
(1.2.13)
2
v g (x) = ( K ^ W r ^ U j U ) j
u2(s)g(s)ds +
Y
+ (K Q aW)~ l (x).u2(x) j "
u j (s)g(s)ds,
where K» is defined by (1.1.21).
Since
j
2
u2(s)g(s)ds < u 2 (x)J
2
|g(s)|ds < u 2 (x) . Hgllj
and
(s)g(s)ds < U j ( x ) |
Jr
l
1s(s)|ds < Uj(x) . Usllj
1
the integrals in (1.2.13) are convergent. Thus the function v
:= x -♦ v (x),
x 6 J is well-defined and the function aWv is locally absolutely continuous.
For the derivative of aWv we have
g
(1.2.14)
2
J
(aWv
vg )'(x)
)'(X) = K -; VU lV' ( x^) f f'2
?
2
(s)g(s)ds
x
K^u^x)]"
Thus (aWv ) ' is also locally absolutely continuous.
u1(s)g(s)ds
58
We obtain
(W" 1 (aWv g )')'(x)= K Q V W ' ^ P ' W ƒ
2
u2(s)g(s)ds
+ KQ1(W'1U^XX)
J
u1(s)g(s)ds
+ K Q 1 W " 1 ( X ) ( - U 1 ' U 2 + U^U 1 )(X) g(x)
= K;1(aW"1u1)(x)|
^0
2
a.e.
u 2 (s)g(s)ds
KQ 1 (aW" 1 u 2 )(x) [ X u1(s)g(s)ds - g(x) a.e.
= v (x) - g(x)
a.e.
It follows that
(1.2.15)
v
- ( W " ' ( a W v ) ' ) ' = g,
thus v is a particular solution of (1.2.11) in L. (J).
Note that aWv
g
ec'(J).
That L is linear is clear, and that L is positive follows from the fact that T is
positive, so for all g e L (J) |v | < v.
and in order to prove that L is a
contraction, it is sufficient to prove the inequality
IIVgHj <||g||j
for functions g in L (J) which are positive a.e. Since C (J), the set of functions
1 c
in C(J) with compact support in J, is dense in L (J) it is even sufficient to
restrict ourselves to positive functions in C (J). Let g G C (J) be positive. Then
c
—
c
there is a compact interval [a,b] C J such that g(x) = 0 for x e J\[a,b]. Because
of the positivity of L the function v is non-negative and satisfies the equation
59
v(x) - ((av)' - pv) '(x) = 0
for
x e J\[a,b].
As in the proof of Proposition 1.1.19 there are positive constants c. and c„
such that
v (x) = CjV^x),
xG(rra)
= c 2 y 2 (x),
x G (b,r 2 ),
where the functions v and v . are defined by (1.2.8).
Regarding Lemma 1.1.2 we see that
and moreover
lim
(1.2.16)
(W
lim (W"^(aWv )')(x) exists (i = 1,2)
x-n
8
^ Q W V ) ' ) ( X ) > 0,
x—»r j
8
lim (W"'(aWv )')(x) < 0.
L x-r2
8
By integration of (1.2.15) with g := g we obtain
I 2 v (s)ds= f 2 g(s)ds+ lim (W _ 1 (aWvJ'Xx)
J r.
§
J r, "
x-r2
lim (W _1 (aWv ) ' ) ( X ) .
x—»ri
8
In view of (1.2.16) we have
l|v g ||j = j
vg(s)ds < J
g(s)ds = ||g|i
This implies that L is a contraction in L (J), and this completes the proof of
Proposition 1.2.9.
D
60
1.3. Solutions of w - (D(aDw) - ftDw) = h in NBV(J)
Let BV(J) denote the set of all functions h which are of bounded variation
over J, i.e. for which the total variation over J, Var-(h) is finite (cf. [Tay],
Ch. 9). It is well-known that functions of bounded variation are continuous
except at most on a countable set. Moreover, for each x e J the limits h(x+)
and h(x-) exist and thus at points of discontinuity the jumps are finite. It is
useful to identify two functions h. and h- if for each x 6 J
(1.3.1)
hj(x+) - hj(x-) = h 2 (x+) - h 2 (x-).
This means that h
and h . have the same points of continuity, respectively
points of discontinuity. Let J, be the subset of J of points of continuity of h.
n
Then it follows that two functions h. and h_, which differ from a constant
and for which the jumps h.(x+) - h.(x-) for x G JV»h- (i = 1,2) are equal, will
be identified. Clearly Var T (h.) = V a r J h . ) and Var (h ) = 0 if and only if h.
is constant. In each class of functions which are identified with the function
f G BV(J), there is exactly one function h for which
(i)
h(x) = (h(x+) + h(x-))/2
(ii)
h(x 0 ) = 0.
for all x G J,
(1.3.2)
Such a function is called normalized.
Let NBV(J) denote the set of all functions h of bounded variation over J
which are normalized by (1.3.2). NBV(J) is equipped with the total variation
norm IUIy a r ( J ) , defined by
' | h | l Var(J) : =
also denoted
T
Var (h
J >'
|dh|, see e.g. [Tay], Ch. 9.
61
An important closed subspace of NBV(J) is the space NAC(J) of all functions
in NBV(J) which are absolutely continuous on J. If h e NAC(J), then it is
well-known that h is differentiable almost everywhere, and that
Var(J)= Ijl d h ' = Jjlh'Wtfx-llhV
k(s)ds
and conversely, if k e L (J), then the function h, defined by h(x) :=
Jx
i
o
belongs to NAC(J). Thus NAC(J) and L (J) are isometric isomorphic spaces.
In this section solutions w>G NBV(J) of the equation
(1.3.3)
w - (D(aDw) - /3Dw) = h,
satisfying
a) w e C (J)
h G NBV(J)
b) aw • G AC loc (J)
will be investigated. Since it will be not necessary that D(QDW) - /?Dw is
normalized, (1.3.3) is an equation in BV(J).
We see that h and D(aDw) - £Dw are continuous in the same points of J, and
in the points of discontinuity we have
(W _ 1 (aWw')')(x+) - (W" 1 (aWw') , )(x-) = h(x+)- h(x-).
Thus the pointwise interpretation of equation (1.3.3) is
(1.3.4)
w(x) - (W" 1 (aWw'))'(x) = h(x) + c
where
c = (QWW ') ')(x ) = ((QWW ') '(x 0 +) + (aWw •) \x
The differential expression
Cw = D(aDw) - £Dw
-))/2.
62
is closely related to the differential expression Bv in (1.2.2), so we may expect
that there is a close relation between the solutions of (1.1.6), (1.2.3) and
w - (D(aDw) - 0Dw) = 0
in BV. (J). Here (N)BV. (J), respectively
(N)AC1QC(J)
denotes the set of
functions h with h G (N)BV([a,b]), respectively (N)AC([a,b]) for each compact
subset [a,b] of J. In the next subsection this relation will be investigated.
1.3.1. Solutions of w - (D(aDw) - /3Dw) = 0
Let W be the set of all solutions in NBV. (J) of the homogeneous equation
(1.3.5)
w(x) - ( W " 1 ( Q W W ' ) ' ) ( X ) = 0,
satisfying
a) w e AC. (J),
b) aw • G AC loc (J).
1
1
2
Note that w also satisfies w G C (J), o w ' e C ( J ) and aWw' e C (J).
Let V be defined as in section 1.2.1. Then we have
Lemma 1.3.1
The mapping *.- W -* V, defined by V(w) = w' is a bijection.
Proof
(i)
V(W) c V. Let w G W and v = w ' . Then v satisfies (1.2.3) and the
conditions (a), (b) and (c) of (1.2.4).
(ii)
* is infective. If w ' = 0 then w = 0 since w is normalized.
(iii)
* is surfective. If v G V, then it is easily verified that w defined by
x
w(x) = f
v(t)dt
v
0
belongs to W.
D
63
As a corollary we have
Corollary 1.3.2
The mapping Ü := $
<P: W —* U is a bijection. Moreover,
nw - aWw',
w eW.
The mapping $ and the set U are defined in section 1.2.1).
D
Lemma 1.3.3
Let r? be a regular boundary point, let Uy and w, be as in Lemma 1.1.14 and
let w. = n " 7 u . - Urn (n'}u.)(x),
'
' x-*r2
'
Then n>
BV
loJJ>--
i = 2,3.
and vv> are continuously differentiable
solutions of (1.3.5)
"
w2 < 0, w' > 0, w' G L (xn,r J, aWw' is decreasing,
Urn (aWw 2\)(x) = 0,
x-^r2
aw*1 £ C](J)
and
Urn ((aw z'J « - f3w\)(x)
= -1;
z
x-*r2
w, < 0, w' > 0, w' G L (xn,r
) , aWw' is decreasing,
Urn (aWw\)(x)
x—r 2
*
aw• G CJ(J) and Urn ((aw\)'
J
x—*r2
•>
- pw\)(x)
•>
= 1,
= 0.
Each solution w of (1.3.5) belongs to AC(xn,r J while
w EL C (J), w*G L (xn,r.),
aw% G C (J) and
aWw' is bounded on
Urn ((aw ' ) • - 0w \)(x)
x->r2
3
5
(xn,r2)
exists.
in
64
There also exists a solution w., for which w. > 0, w' > 0 and aWw' is in­
creasing.
For each increasing solution w of (1.3.5) we have Urn ((aw *) ' - fiw *)(x) < 0
x->r2
if and only if
w = p2w2 + p3w3
with p ?> 0, p , > 0 not both zero.
Lemma 1.3.4
Let r? be an exit boundary point, let u. and u? be as in Lemma 1.1.15 and let
w.1 := n" 7 M. - Urn (ü'^.Xx),
l
' x-*ri
i - 1,2.
Then w. and w- are continuously differentiable
BV
loc(J>
solutions of (1.3.5)
in
:
w. > 0, w' > 0, w' £ L (xn,r.),
aw\1 eC
(J) and
Urn ((aw 1' J • - 0w l\)(x) = oo;
x->r2
w2 <0,w2>0,w'2GL
aw' 1 E CJ(J) and
aWw' is bounded,
(xQ,r ) ,
Urn aWwUx) = 0,
x—*r2
Urn ((aw z\) • - pw z\)(x) = -1.
x-*r2 .
Only solutions of (1.3.5)
which are scalar
multiples
of vv? belong to
BV(xQ,r2).
Lemma 1.3.5
Let r2 be an entrance boundary point, let u. and « , be as in Lemma 1.1.16
and let w := if
1
Then w.
BV
loc^
u - Urn
v
x-*r i
((l~!u
)(x), w .= fT 7 u
l
i
i
and w, are continuously differentiable
- Urn
(n"7u Jfxj.
x—>r2
i
solutions of (1.3.5)
in
65
w. > O, w\ > O, w\' G L (xn,rj,
l
l
1
aw' 1 e C1 (J) and
j
Urn ((aw l\) • - /5w'\)(x)
< oo;
l
5
aw J' e C',1 (J) and
= oo,
x—*r2
x-r2
]
w , < 0, w • > 0, w' e L (xn,r
5
3
Urn (aWw\)(x)
UZ
UZ
),
Urn (aWw \)(x) = 1
x—►/•?
Urn ((aw J\) • - pw \)(x) = 0.
x—^2
•?
For each solution w of (1.3.5) we have
w G BV(xn,r7)
u
Lemma
Let r
l
and
Urn ((aw*)* - fiw')(x) exists.
x-^>r2
1.3.6
be a natural boundary point, let u. and u., be as in Lemma 1.1.17, and
let w. := n~]u.
Then w.
- Urn (Cl~}u.)(x),
x—>r
i = 1,2.
and w , are continuously differentiable
solutions of (1.3.5) in
BV
loc^:
w. > 0, w\ > 0, w\ £ L (xn,rj,
l
l
l
u z
aw\1 G C (J) and
Only
and
= oo,
Urn ((aw J\) • - Pw1\)(x) = oo;
x-r2
H> < 0, w* > 0, w ' G L1(xn,r
aw' G C!(J)
z
Urn (aWw\)(x)
1
x—*r2
) , Urn (aWw\)(x)
= 0,
Urn ((aw z'J ' - pw z\)(x) = 0.
x-*r2
solutions of (1.3.5)
which are scalar
multiples
of vv? belong to
BV(x0,r2).
D
The proofs of the Lemmata 1.3.3-1.3.6 rest on the proofs of the Lemmata
1.1.14-1.1.17, 1.1.3-1.2.7 and 1.3.2 will therefore be deleted here.
66
1.3.2. Solutions in NBV(J) of w - (DfaDw) - 0Dw) = h
In the sections 1.1 and 1.2 we constructed solutions of inhomogeneous equations
in C(J) and L (J) by means of special Green functions. In this section we will
also use the Green function T, defined in (1.1.34). To obtain a particular
solution w, in NBV(J) of the equation
(1.3.7)
w - (D(aDw) - 0Dw) = h,
h e NBV(J)
the Green function r , defined in (1.1.34) can also be used.
Note that (1.3.7) has the pointwise interpretation (1.3.4).
For each h e NBV(J) we define the function w, by
«=o?
w h (x) = |
|
2
r(s,t)dh(s)dt.
The mapping h —► w, induces an operator M on NBV(J). As a counterpart of
the Propositions 1.1.19 and 1.2.9 it can be proved (cf. [Fe,l], Th. 13.3)
Proposition 1.3.7
Let for each h E NBV(J)
Mh be defined by Mh = w,. Then
(i)
M is a linear contraction operator from NBV(J) into NBV(J),
(ii)
Mh is a particular solution of (1.3.7),
(Hi)
if h is non-decreasing, then Mh is
D
non-decreasing.
67
CHAPTER 2 - SEMIGROUPS GENERATED BY DIFFERENTIAL
OPERATORS SATISFYING VENTCEL'S BOUNDARY
CONDITIONS AND THEIR DUALS
2.1. Semigroups in Banach spaces
Let X be a real Banach space with norm ||.||
and let L(X) be the Banach
algebra of all bounded linear operators from X into itself.
Let T be an operator-valued function, mapping the non-negative real axis
into the Banach algebra L(X) and satisfying the following two conditions:
(i)
T(s + t) = T(s)T(t),
(")
T(0) = I,
s>0, t>0;
(2.1.1)
where I denotes the identity operator. Then the family (T(t) | t > 0}, shortly
denoted by T, is called a semigroup of bounded linear operators in L(X) or a
semigroup on X. The semigroup T is said to be of class C„ if it satisfies in
addition the condition
(2.1.2)
(Hi)
lim ||T(t)f - f||Y = 0
tj.0
x
for each f e X.
A semigroup of class C„ in L(X) is strongly continuous at each point of the
positive real axis. Therefore, a semigroup of class C~ is also called a strongly
continuous semigroup. For each strongly continuous semigroup in L(X) there
exist positive constants M and w, not depending on t or f e X, such that for
each f e X, and each t > 0
68
(2.1.3)
l|T(t)f|lx < MeWt||fHx.
A strongly continuous semigroup in L(X), satisfying (2.1.3) with w = 0 and
M = 1 is called a contraction semigroup in L(X).
The set of all strongly continuous contraction semigroups in L(X) will be
denoted by CSG(X).
Let T e CSG(X). The infinitesimal generator of the semigroup T is an opera­
tor A in X with domain D(A) c X defined by
(2.1.4)
D(A) = ( f e X | lim t _1 (T(t)f - f) exists)
J
^
tj.0
Af
= lim t" J (T(t)f - f)
t|0
for all f e D(A).
The first part of the Generation Theorem of Hille-Yosida ([BB], (1.3.2);
[CH], Ch. 2) tells that A satisfies the following properties:
(i)
(2.1.5)
for each f G D(A)
l|f|l x < llf - AAf|l x
(ii) R(I - A A) = X
for every A > 0,
for every A > 0,
(iii) D(A) is dense in X.
Here R(I - AA) denotes the range of I - AA.
A linear operator A: D(A) c X -♦ X, satisfying (2.1.5)(i) is called dissipative.
A dissipative operator A satisfying (2.1.5)(H) is called m-dissipative.
We
denote the set of all densely defined m-dissipative operators in X by D(X).
Remark. A linear operator A is called (m-)accretive if -A is (m-)dissipative.
It follows from (2.1.5)(i) and (ii) that A is a closed operator. Therefore the
domain of A, D(A), equipped with the graph norm l l l l n / A V i.e.
69
(2.1.6)
l|u|l D ( A ) = ||u|| x + ||Au|| x ,
ueD(A),
is a Banach subspace of X.
The second part of the Generation Theorem of Hille-Yosida states that the
mapping G: CSG(X) — D(X) which associates with T e CSG(X) its in­
finitesimal generator A := G(T), is a bijection.
Let A e D(X), then it follows from (2.1.6)(i) and (ii) that for each A > 0,
I - AA is a bijection from D(A) onto X, and that R,(A) := (I - AA)
is a
bounded operator from X onto D(A) C X satisfying
I|R A (A)||<1,
(A>0).
The operator R.(A) is sometimes called the resolvent of A for A > 0.
Example
Let BUC(IR) be the Banach space of all bounded uniformly continuous
real-valued functions on TR, equipped with the supremum-norm. For each
f £ BUC(IR) and each x GTR, the semigroup T of Gauss-Weierstrass is defined
by
(T(t)f)(x) = (27rt)" 1 / 2 f
exp / - j t _ 1 (x - s) 2 Vf(s)ds
if t > 0,
(T(0)f)(x) = f(x).
It appears ([Y], Ch. IX, 5; ex. 2) that T is a strongly continuous semigroup and
that the domain of its infinitesimal generator is the subspace of functions f in
BUCCR) for which the second derivative f" exists and belongs to BUC(IR), and
that Af = - | f", f e D(A).
Next we will recall some facts on dual semigroups on the dual space X . For
these results we refer to [BB], Ch. 1.4 and [CH], Ch. 3.4.
70
Let X be the space of all bounded linear functionals u on X. X is a Banach
space with the norm
|u*|| =
sup
|<u*,u>|.
||u|l x <l,u€X
Here <.,.> denotes the pairing X x X -+3R.
Let L be a linear operator with domain D(L) dense in X and range R(L) in X.
Then the Jadjoint (or dual) operator L
of L with domain D(L ) C X
is
defined by
' D(L*) = / u* G X* | (3v* e X*)(Vu G D(L)) (<v*,u> = <u*,Lu>)}
(2.1.7)
and
L*u* = v*
for all u* G D(L*).
Let T be a C„-semigroup of linear contraction operators on X. Then the dual
semigroup T* := {T*(t) | t > 0} of T, where T*(t) is the adjoint of T(t) in the
dual space X , is also a semigroup of linear contraction operators. T is not
necessarily a semigroup of class C„. If X is a non-reflexive Banach space we only know that T is weak -continuous. Such a semigroup is called of class C n in
X . The dual operator A of the infinitesimal generator A of the semigroup T
is equal to the weak infinitesimal generator of the dual semigroup T of T, i.e.
u* G X* belongs to D(A*) if and only if for all u G X, lim < t" l (T*(t)u* - u*),u>
tj.0
_.
exists, and u* G D(A*). A*u* is defined as the weak* limit of t~ (T*(t)u* - u*).
We use the notation X» for the closed subspace of X on which T is strongly
continuous. The subspace X~ is invariant under T (t), t > 0 and the restriction
of T to this subspace is denoted by T». Then T_ is again a strongly continuous
semigroup on X„. Its infinitesimal generator is denoted by Afl.
We have the following
71
Proposition 2.1.1
(i)
D(A*Q) C D(A*) C X*Q, .
(ii)
X' is equal to the strong closure of
D(A*)
(Hi)
and D(A*Q),
A* is the part of A* in X* i.e. it is the largest restriction of A with
both domain and range in X~,
(iv)
if X is reflexive, then X' = X and A() = A .
In this chapter we will use the notation X® for a specific representation of the
space X-, and T® for the corresponding representation of T . in X®. (Thus Xand X® will be isometrically isomorphic.)
Let T e CSG(X), then T® 6 CSG(X®). We can imbed X into X®*, the dual
space of X®, by means of the natural imbedding i defined by
(i(u))(u®) := <u®,u>
for all u e X and all u® e X®.
It is known that i is an isometry not necessarily surjective, and thus i(X)
is closed in
In what follows, we identify X with i(X).
For all u e X, u® £ X® and t > 0 we have
<u®,T(t)u> = <T(t)*u®,u> = <T(t)®u®,u> = <u®,T(t)®*,u>
where T(t)® denotes the adjoint of T(t)®.
Thus T(t) C T®*(t) := T(t)®* for all t > 0. We denote by X® 0 the closed
subspace of X®* on which the semigroup T® is strongly continuous. The
restriction of this semigroup to X®® is denoted by T®®. Its infinitesimal
generator is denoted by A®®. From Proposition 2.1.1 we immediately obtain
72
Proposition 2.1.2
(i) Jif00 is equal to the strong closure of the domain D(A® ) of A® in X®*,
(ii) / 4 0 Q is the part of AQ* in X 0 0 .
Note that X C X© 0 , T C T 0 0 and A c A 0 0 .
For application in approximation theory we define the Favard class of a
strongly continuous contraction semigroup on X.
Definition:
Let T e CSG, then the Favard class Fav(T) of T is defined as:
Fav(T) := / u e X: SUD ||t _1 (T(t)u - u ) | | x < 00V
Two characterizations of the Favard class of T are given by
Proposition 2.1.3 ffCHJ, Ch. 3.4)
Let T e CSG(X), then
(i)
Fav(T) = X n D(AQ*),
(ii)
Fav(T) = \u e X] there is a constant M > 0 and a sequence (u ) in
D(A) such that (a)
lim
\\u - u\\Y = 0
(b) \\Au^\x <M,
n = 7,2,...}.
2.2. Two propositions on adjoint operators
In this section we establish two propositions on adjoint operators. If L:
D(L) C X —
► X is a linear operator with domain dense in X, then the adjoint
L* D ( L * ) c X * -f X* is defined by (2.1.7). If <.,.> denotes the pairing
X x X -*1R, then in many cases the characterization of the elements of D(L )
gives difficulties. This characterization must be obtained in one way or
another from the definition of L , especially from the equation
73
<v ,u> = <u ,Lu>,
(see (2.1.7)).
Since in our case the pairing is defined by means of integrals, the difficulties
arise from the non-integral terms after integration by parts. These difficulties
can be highly reduced if one starts with functions in a suitable dense subset of
X and X . Afterwards the obtained results are extended to spaces X and X by
means of continuous extension of the considered operators.
The next two propositions will serve to that purpose.
Proposition 2.2.1
Let A: D(A) C X —» X be a linear m-dissipative operator with domain D(A)
dense in X, and let B: D(B) c X —» X be a closed linear operator.
Let there exist a dense linear subspace M of X and a dense linear subspace
N of X* such that
(i)
N c R(I - B)
(ii)
<v,Au> = <Bv,u>
for all u G D(A) such that u - Au G M
and all v e D(B) such that v - Bv G N.
Then B = A*.
Proof
I - B is infective. Let v e D(B) such that v - Bv = 0. Since A is dissipative,
I - A is injective. For f e M set u f = (I - A)" f, then for all f G M we have
<v,f> = <v, u f - Au~>
= <v,u f > - <v,Au f >
= <v,u~> - <Bv,u f >
= <v - Bv, u f >
= 0.
Since M = X it follows that v = 0. Thus I - B is injective.
74
I - B: D(B) -* X* is surjective
and \\(I - B)~ \\ < 1. Let g e X*, then we
have to prove that there exists a v e D(B) such that v - Bv = g.
Since N is dense in X , there exists a sequence (g ) in N such that
Hm ||g - gu || = 0.
n—»oo
*
Since N c R(I - B) there exists a sequence (v ) in D(B) such that v
For all f e M we have
<v n ,f> = <v n , u f - Au f >
■ <V n " B v n ' V
= <g n , ( I - A) _1 f>.
Then for all f G M = X we also have
!
<v ,f> = <g
ö , v(I - A) f>.
n
n
'
So for all n £ l
llvJL =
sup
l<v_,f>|
n
||f||<l,feX
sup
|<g
l|f||<i,fex
<
sup
| | ( I - Af^lLHg II
l|f||<l,fex
<
sup
||f||<l,fex
||f||.||gJL
n
(because of the m-dissipativity of A), thus
l|v N
(I - A)" f>|
n * -
ll8 !l
n *'
n G:N
-
- Bv = g .
75
In the same way we find
l|v
n " VmN* * l l 8 n " 8 n A ,
m.n eW.
The sequence (g ) is a Cauchy sequence. From the last inequality it follows
that (v ) c D(B) is also a Cauchy sequence in X .
Let v := lim vn in X . Since Bv = v - g , lim Bv exists and is equal to
n-»oo
n
n
n „^QO n
v - g. Since B is closed we have v G D(B) and Bv = v - g. Thus v - Bv = g.
This proves the surjectivity of I - B.
Moreover, for all g e X we have
||(I - B) _ 1 g|| =||v|| = lim ||v
< lim | | gu j | = ||g|| .
n—+00
*
*
Thus ||(I - B ) _ , | | < 1.
B = A*. For f e M and g G N set u f = (I - A) _ 1 f and v = (I - B) _ 1 g. For all
f E M and all g e N we have
< ( I - B ) _ 1 g , f> = <v g ,f>
= <v g , u f - Au f >
= <V ,\lr>
~ <V
,AUr>
g' f
g
f
= <v ,uc> - <Bv ,uc>
g' f
g f
= <v - Bv , u r >
g
g f
= <g, ( I - A) _1 f>.
sion for all f G X and all g e X
Since (I - A)
and (I - B)
-1
are contractions we obtain by continuous exten­
<(I- B) g, f> = <g, ( I - A ) _ 1 f > .
76
By using ([K], Theorem 5.30) we obtain from the last relation for all f e X
and all g 6 X*
< ( I - B ) _ 1 g , f> = <((I- A) - 1 )*g, f>
= <((I- A)*) _1 g, f>.
So for all g S X we have
( I - B ) _ 1 g = ((I- A ) V g .
So
( I - B ) " 1 = ( I - A*) - 1
or
I - B = I - A*.
Thus B = A . This completes the proof of the proposition.
D
Remark. If B is not closed, but B is closable and the conditions (i) and (ii) are
satisfied, then the conclusion of the theorem reads B = A , where B denotes
the closure of B.
If in the preceding proposition N = X, then we have from condition (i)
R(I - B) = X, and condition (ii) holds for all u G D(A) such that u - Au G M
and all v G D(B). So one has more information about B from the conditions (i)
and (ii) than in the case that N is a proper subspace of X. It appears that the
condition on the closedness of B can be dropped. In fact we have
Proposition 2.2.2
Let A: D(A) C X —» X be a linear m-dissipative
operator with domain D(A)
dense in X, and let C: D(C) c X —
► X be a linear operator, such that
(i)
R(I - C) = X\
77
Let there exists a dense linear subspace M of X, such that
(ii)
<v,Au> = <Cv,u>
for all u G D(A) with u - Au G M
and all v G D(B).
Then C = A*.
Proof
As in the proof of the preceding proposition it follows from the m-dissipativity
of A and condition (ii) that I - C is injective. The surjectivity of I - C is
given by condition (i). Thus, if g G X , there exists a unique v G D(C) such
that v - Cv = g. Then we have (as in the proof of the preceding proposition)
for all f G X and all g G X*
<v f> = <g, (I - A)" f>
or
< ( I - C ) " 1 g , f> = <g, ( I - A) _ 1 f>
= < ( I - A*) _ 1 g, f>.
So (I - C)~ ] = (I - A*)" l and it follows that C = A*.
D
2.3. Semigroups in C(J)
As in chapter 1, J := (r,,r_) denotes an open interval o f H (not necessarily
bounded), J the two points compactification of J, dJ := J\J and X := C(J),
equipped with the supremum-norm. Let x„ denote an arbitrary but fixed
point in J. Thus -oo < r. < x» < r» < oo.
Let a and P be real-valued continuous functions on J with a(x) > 0 for all
x G J. Let D(A) be given by
78
D(A) := / u e C(J)|u e C 2 (J),
{
lim (aD 2 u + £Du)(x) .
x-*d3
x-*dJ
o\
J
and let
(2.3.1)
Au := aD 2 u + £Du
for all u e D(A).
Then (2.3.1) defines an operator
A: D(A) c C(J) -v C(J).
In 1975 Martini [Ma,2] gave sufficient conditions on a and /? for A to be the
generator of a C--contraction semigroup on C[J], for a bounded interval J. In
this section we give sufficient
and necessary conditions for A to be the
generator of a C„-contraction semigroup on C(J), where J is not necessarily
bounded.
MATHEMATICS
Proceedings A 89 (4), December 15, 1986
On Co-semigroups generated by differential operators satisfying
Ventcel's boundary conditions
by Ph. Clément and C A . Timmermans
Department of Mathematics,
University of Technology, Delft, the Netherlands
Communicated by Prof. A.C. Zaanen at the meeting of April 28, 1986
1. INTRODUCTION
In [4], [5], [6], Martini and Boer investigated contraction semigroups
(7V)),>o of class C 0 on the Banach space C[a, b] (equipped with the supremum
norm) generated by an operator A of the form
(1.1)
Au = aD2u + PDu,
where a and /? are continuous real-valued functions on [a,b], with a(x)>0 for
a<x<b. The domain of A, denoted by D(A), is given by
(1.2)
D(A): = {ueC[a,b]\ueC2(a,b),
lim Au(x)= lim Au(x) = 0}.
x-*a
x-<b
The boundary conditions lim^._a Au(x) = \imx-.b Au(x) = 0 are usually called
Ventcel's boundary conditions [9], [10] and arise in a natural way in approxi­
mation theory [6].
In [5, Th. 1 p. 17-18] sufficient conditions on a and /? are given for A to be
the generator of a Co-contraction semigroup on C[a,b], namely it is assumed
that a and /? satisfy:
a) a,peC2(a,b)nC[a,b],
b) a(x)>0 for xe(,a,b), a(a) = P(b) = 0,
c) a" 1 is not integrable over neighbourhoods of a and b,
d) a/? _ i is bounded on (a, b).
The goal of this note is twofold: firstly we consider not necessarily bounded
open intervals J and secondly we give necessary and sufficient conditions for
79
A to be the generator of a C0-contraction semigroup on C(J), where J denotes
the two points compactification of J. In section 4 it is shown that our results
drastically extend those of [5], even when a vanishes at a and b. The proofs are
partly based on arguments of Feller [3] concerning semigroups generated by
second order differential operators. Since we are concerned with special
boundary conditions and since the lecture of [3] is not completely straight­
forward, we shall give a self-contained proof here. In Proposition 1 of section
2, we observe that under very general assumptions on a and /?, A defined by
(1.1) and (1.2) is a densely defined closed dissipative [7] operator on C(J) with
positive resolvent. In Theorem 2 of section 2 we give necessary and sufficient
conditions for A to satisfy
(1.3)
R{I-A) = C(J),
where R(I-A) denotes the range of I-A. It is known [7, p. 14] that the
condition (1.3) for closed densely defined dissipative operators A in a Banach
space (X, || I) is necessary and sufficient for A to be the generator of a Cocontraction semigroup on X.
2. MAIN RESULTS
Let J denote a non-empty open interval of IR (not necessarily bounded), 7 the
two points compactification of J and dJ: =J\J. C(J) is the Banach space of
real-valued continuous functions on J equipped with the supremum norm,
denoted by | • |.
PROPOSITION 1. Let a and /? be real-valued continuous functions on J with
a>0 on J. Let
D(A): = {ueC(J)\ueC2(J),
lim (aD2u + 0Du)(x) = O}
x~dJ
and let
Au = aD2u + (]Du
for u e D(A). Then A : D(A)-*C{3), and the following conditions are satisfied:
(i) D(A) is dense in C(J),
(ii) A is closed,
(iii) if we have u - kAu>Q for some A > 0 and some u eD(A), then either u is
strictly positive on J or u is identically zero,
(iv) A is dissipative.
In order to state the conditions on a and P in Theorem 2, we define the
function W as follows:
W(x):=exp f- j.-£ (t)dtl.
I
*o a
)
Moreover, L (resp. R) denotes the left (resp. right) boundary point of J and x0
denotes a point in J.
80
THEOREM 2. Let A be defined as in proposition 1. Then the range of I-A is
C(J), i.e. A is the generator of a C0-semigroup on C(J), if and only if a and
p satisfy
(HL)
WeL\L, x0) or j W(x) J a"' W~ \t)dt dx=oo or both
L
L
R
R
x0
x
and
(HR)
WeL\x0,R)or
J W(x) ] a~lW-\t)dt
dx=oo or both.
REMARK. The reader familiar with Feller's terminology will recognize that
(HL) (resp. (HR)) is satisfied if and only if L (resp. R) is not an entrance
boundary point. Note that in the original paper of Feller a misprint occurs in
the definition of entrance boundary point [3, p. 516], but it is clear from the
context that it is meant that R is an entrance boundary point if
R
R
X0
X
W(SL\x0,R) and j W(x) J a-lW~l(t)dt
or, equivalently W$Ll(x0,R) and QeLl(x0,R),
Q(x): =a~llV-\x)
dx<oo,
where
j W(t)dt.
x0
Observe that QeL\xQ,R)
implies a" 1 W~x
eL\x0,R).
3. PROOFS
PROOF OF PROPOSITION 1
(i) D(A) is dense in C(J). Let us denote by C*(7) (resp. Ci(7)) the subset of
functions of C{J) (resp. of C(J)C\C\j)) which are constant outside of a
bounded closed subinterval of J. Clearly C*(7) is dense in C(J) and a simple
regularization procedure shows that C\(J) is dense in C(J). Finally we note that
Cl(J)cD(A).
(ii) A is closed. Let (un)CD(A), u and v in C(J) be such that lim,,^
| U „ - H | = 0 and lim,,-^ ||^4u„ —1>|| = 0. We have to show that usD{A) and
Au = v. Let us denote A un by v„. For every a, b e [R such that [a, b]CJ there is
a constant c > 0 such that a(x)>c for all xe [a,b]. Therefore the restriction on
[a, b] of aD2u + pDu is a regular Sturm-Liouville operator [2]. Since u„(a)
(resp. u„(b)) converges to u(a) (resp. u(b)), it follows from the classical
theory [2] that u'„ and «Ó' a r e Cauchy sequences in C[a,b], and therefore that
ueC2[a,b], and au" + Pu' = u on [a,b]. Since a and b are arbitrary, ueC2(J)
and au" + Pu' = v on J. Since t»eC(J), \\mx^aj un(x) = 0 and u„ converges
uniformly to v, we see that lim,<._ay («" + Pu')(x) exists and is equal to zero.
Thus ueD(A) and Au = v.
(iii) IfX>0 and ueD(A) then u-XAu>0 implies w>0. It follows from the
boundary condition that limx_ay u(x)>0. If there is any eJ such that u(y)<0,
81
then w possesses a negative minimum at some point zsJ. But then u'(z) = 0 and
u"(z)>0, thus u(z) - Aau"(z) - Xfiu'(z) <0, a contradiction. Note that from the
strong maximum principle [8, p. 6, Th. 3] it follows that either u(x)>0 for every
xeJ or u is identically zero.
(iv) A is dissipative. It follows from (iii) that for X>0, I-XA :D(A)~*
-*R(I-XA) is injective. Indeed, if ueD(A) and u-XAu = 0, then «>0 and
- u > 0 , hence w = 0. Let us define Jx = (I-XA)'1.
Then 7^ is a positive
operator (w>0=>JA«>0)) and JX 1 = 1, where 1 is the constant function
equal to 1. (Note that \eD(A)). Then it follows that ||y A «|< | « | for every
ueR(I-XA). This completes the proof of Proposition 1.
The proof of Theorem 2 is based on the following lemma.
3. Let A be defined as in Proposition 1. Then I-A is surjective in
C(J) if and only if there exist two functions uL and uR, such that
(i) uL and uR are in C2(J) and satisfy
LEMMA
(3.1)
u-aD2u-0Du
=O
on (L,R),
(ii) uL is positive, increasing on (L,R) and lim^._i uL(x) = 0,
(iii) uR is positive, decreasing on (L,R) and lim^^^ uR(x) = 0.
PROOF OF LEMMA 3
Necessity. We assume R(I-A) = C(J). Let a,be IR be such that L<a<b<R.
Let ƒ e C(J) be positive on (a, b) and zero outside of (a, b). Let w be the unique
function in D(A) satisfying a - aD2ü - fiDü =ƒ. From Proposition 1, (iii), it
follows that ü is positive on J. Moreover, since \\mx^dJ f(x) = Q, we have
limx_ay u(x) = 0. ü satisfies a - aD2Q - 0Du - 0 outside of (a,b). For x>b,
u'(x)i=0. Otherwise there is an x > b such that ü'{x) = 0 and u"(x)>0, hence w
has a local minimum of a. Since w>0 and limA._R Q(x) = 0, there must be a
local maximum at some y>x, hence u'(y) = 0 and Q"(y) = (a -! ü)(y)>0, a
contradiction. Thus « is positive and decreasing on (b,R) with lim^-.^ «(x) = 0.
Next define uR to be the unique solution of (3.1) on (L,R) satisfying
uR(b) = a(b) and u'R(b) = a'(b). Note that u'R(b)<0 and u'R does not vanish on
(L, Z?), because otherwise «R would have a local minimum >, which (by a
similar argument as above) is impossible. Hence u'R<0 on (L,R). It follows
that uR satisfies the conditions of the lemma. Similarly we prove the existence
of an increasing function uL satisfying the conditions of the lemma.
Sufficiency. Since A is closed and dissipative, R(I-A) is closed. Since C*(7)
is dense in C(J), it is sufficient to prove that R(I-A)DC+(J). Let ƒ be in
C*(7). Thus there are a, b,c,deU such that L<a<b<R,f(x) = c for x< a and
f(x) = d for x>b. Note that u'L(a)>0 and u'R(b)<0. For y.JelR we define:
uy = c\ + yuL on (L,a], with l(x)= 1,
«,5 =rfl+<5Mflon [Ö,-/?).
82
Then uyeC2(L,a]r\C[L,a],
uy satisfies uy-auy-Pu'y=f
on (L,a] and lim x _ L
(au" + Pu')(x) = 0. Similarly for uó on [b,R). Let us denote by Uj the unique
solution in C2[a,b] of u- au" -fiu'=f satisfying Uf(a) = uf(b) = 0. For/i, veIR,
we set
on [a,b]. If we can find y,ö,n,ve(R, such that u defined by uy on (L,a), u^v
on [a,Z?] and u6 on [&,/?) is CX(L, R), then it follows from the differential
equation that « e C2(J), u belongs to D(A) and u- Au = ƒ. The continuity of u
and u' is insured at a and 6 if and only if the following system possesses a
solution:
c
c
A
E
-A
0
-K
B
-B
F
0
V
C
0
G
-G
d
Y
D
-M
H
0
-H
[s J
where/I = Mi.(a)>0, B = u'L(a)>0, C = uL(b)>0, D = u'L(b)>0,
E=UR(a)>0,
F=u'R(a)<0,
G = uR(b)>0, H=u'R(b)<0,
K=u'f(a) and M=u'f{b), and
where Uj (a) = Uj (b) = 0. The determinant of the system is equal to
(AF-BE)(GD-CH)<0.
Thus JU, v, y and S are uniquely determined. This
completes the proof of Lemma 3.
In the next lemmata we shall find necessary and sufficient conditions for the
existence of uL and uR as in Lemma 3. Since the conditions for uL are similar
to those for uR we shall restrict ourselves to the case of uR. Since uR is positive
on (L, R) we can assume without loss of generality that uR(x0) = 1 for some
x0e(L,R).
From now on x0 denotes an arbitrary fixed element of (L,R).
4. Let a and /? be as in Proposition 1. Then there exists a unique
minimal positive decreasing solution a of (3.1) satisfying Q(x0) = 1, i.e., if u is
any positive decreasing solution of (3.1) satisfying u(x0)= 1, then Q<u*
LEMMA
)F.
PROOF
ForcoeIR, let us denote by uw the unique solution of (3.1) satisfying
H ( X O ) = 1 , U'(X0) = CD. Set
B: ={we[R|there is a £e(x0,R)
such that
uw(£)<0}.
For <x> = 0 the solution uw satisfies u^(x0) = 0, u"(x0) = (u(x0)/a(x0))>0. Thus «
is increasing in a right neighbourhood of x0, and since no solution has a
positive maximum it follows that uw is strictly increasing for x>x0. Thus
0$B. Moreover, B is not empty. Indeed, let £e(x0,R). Since a > 0 on [x0,(^],
there is a unique « satisfying (3.1) and ü(x0) = \, u(£)- - 1 . B is open. For
coeB, let £e(x 0 ,fl) be such that «a)(<J)<0. Then there is an e > 0 such that
"»/(£)<0 f ° r |/7-cu|<£- Moreover, if u>xeB and if ( y 2 ^ ^ i , then U>2BB.
Finally, if a>eB, then u'w<0. First note that M^(JC 0 )<0. Let JP denote the first
zero of u'w, if it exists. If uui(x)>0, then by (3.1) uw has a minimum at Jf and
since uw cannot have a positive maximum and uw vanishes somewhere, this is
* ) on [ x ,R)
o
83
not possible. If uw(x) = 0, then uw has to be identically zero, which is not the
case. If uu(x)<0, then x would be by (3.1) a negative maximum of uw which
is also impossible. Thus u'w<0 on (x0,R). Set cö = sup B and ü = ua. Then a is
the desired function. Indeed, «>0 on (x0,R) since co$B, B being open.
Furthermore, ü is non-increasing since a is the supremum of decreasing
functions. ü>0 follows from the maximum principle [8, Th. 3 p. 6]. Finally if
u>0 is a decreasing solution of (3.1) on [x0,r] satisfying M(JC0)= 1 and « < S ,
then w: =a-u satisfies (3.1) on [x0,R), w>0 and either w = 0 or w'(x0)>0. If
w'(x0)>0, then u'(x0)«ü and u'(x0)eB, a contradiction. Thus w = S, and w is
minimal. This completes the proof of Lemma 4.
In [3], Lemma 4 is proved under the additional assumption ore
eCl(J). We shall define
REMARK.
y = lim ü(x)=
x—R
inf
ü(x).
xe(xo,R)
Then it is clear that condition (iii) of Lemma 3 is satisfied if and only if y = 0.
It will be easier to find necessary and sufficient conditions for y to be positive.
This is done in the next lemmata.
LEMMA
5. Ify>0,
then W<SLl(x0,R).
We shall denote by a the solution of Lemma 4. First we define u
to be the unique solution of u-au"-fiu' = 0 on [x0,R) satisfying w(x0) = 0 and
u'(x0) = 1. Note that v and v' are positive on (x0,R). Let
PROOF.
M=
sup v(x).
Then, if y>0, M=oo. Otherwise, M: =ü-yM~]v
would be decreasing,
satisfying lim_f_J? u(x) = 0, and «<«, contradicting the minimality of Q. Next
observe that W=o'ü-ü'v. Indeed (v'ü-ü'v)(x0)= 1 and an easy computation
shows that
(v'a-a'vy _
v'ü-ü'v
(note that u'a-a'v>0).
p
a'
Hence
(v'ü-ü'u)(x) = exp j - j (^-\t)dtl
= W(x).
Since -ü'v>0 we have v'ü< W, and since u'>0, u(x0) = 0 and y<«, we get
yu{x)< \ W(t)dt.
Since y>0 and M=<x, we obtain
LEMMA
.84
6. If a'lW'xeL](x0,R),
W$Ll(x0,R).
then
(i) 7i: = ü'{x0) + J* üa~' W~ ldt<0,
(ii) yZ-nW<
-ü'<üZ-r\V,
where ü denotes the solution defined in Lemma 3 and
a~lW-\t)dt.
Z(x): = W{x) J
PROOF.
From the differential equation satisfied by Q we get
(3.2)
ü\x) = W(x){ü'(x0) + J f l a " ' r
Since u'{x)<0,
{
dt}.
W(x)>0, n = l\mx-.R u\x)W~\x)<0).
ü\x) = nW(x)-{
Next rewrite (3.2) as
üa-lW-1dt}W(x).
\
X
Since y<u, we get
yZ-nW^-Q'.
On the other hand, since « is positive decreasing and Z is positive, we have
- Q \x) < - n W(x) + a(x)Z(x).
LEMMA 7.
Ify>0,thenZeL\x0,R).
PROOF. From (3.2) we get, for every
y \ a'^-'dts
xe(x0,R),
j öa-'wV^-ö'W.
Since y>0, it follows that a~]Wl eLl{x0,R).
yZ< -a'. But y > 0 and -fi'eL'fXo.^) imply
LEMMA
8.
IfZeLl(x0,R)
and W<SL\x0,R)
By using Lemma 6, we get
ZeL\x0,R).
then y>0.
If ZeLl(x0,R),
it is easily seen that a~liV~l eLl(x0,R).
From
Lemma 6 it follows that -nW< -a', with 7r<0. Since Q'eL\xQ,R) and
W$Lx(x0,R),
it follows that 7r = 0. By using Lemma 6 again, we obtain
-u'<üZ,
and since w>0,
PROOF.
0 < - — <ZeL'(x0,/?).
a
It follows that
|log ü(i?)| = |log y - l o g w(*0)| = j
<oo.
Hence y > 0 .
;85
It follows from Lemmata 5, 7 and 8 that y = 0 is satisfied if and
only if WeL\x0, R) or Z$Lx(x0,R).
This completes the proof of Theorem 2.
COROLLARY.
4. COMPARISON WITH MARTINI'S RESULTS
In this last section we show how the conditions a)-d) given in the introduction
can be improved. Let a,PeC[a,b] be such that a > 0 on (a,b). We look for
sufficient conditions which imply that b is not an entrance boundary point. If
P = P + - / ? " , where p + (resp. /?") denotes the positive (resp. negative) part of
P, it is easily seen that WeLl((a +b)/2,b) if a""1/?" eLl((a + b)/2,b) and thus
b is not an entrance'bouhdary point. In particular, if a(b)>0, or a(b) = 0 and
a~,eLl((a + b)/2,b), then a~xP~ eLl((a + b)/2,b). These cases are not
contained in Th. 1 of [5].
Moreover, if a(b) = 0, a~^L\{a
+ b)/2,b) and P(b)>0, then a'xP' e
eL\(a + b)/2,b). This case is also not included in Th. 1 of [5].
Next we show that, if a " ' $Ll((a +b)/2,b) and er*/8~ eL°°((a + b)/2,b),
then b is no entrance boundary point. Observe that condition HR of Theorem
2 is satisfied for a and P if and only if it is satisfied for la and Xp with A > 0 .
Thus, without loss of generality we can assume a~' $L'((a + ö)/2,b) and
a~^P' < 1 on ((a + 6)/2,6). By using Proposition 1 and Lemmata 3 and 4 it is
sufficient to show that y = 0. Let w denote the function defined in Lemma 4.
We have
ü = aü" + Pü' on {a, b),
and since w'<0 on (a,b),
a<au" + P'(-a')
on (a,b).
Since a > 0 and a "•/?"< 1 on ((a + 6/2), b), we obtain
a
_i _2
-„-
u<uu+a
-t
/
,x
(a + b , .
' « ( - « ) on ( — 7 - , 6 ).
Hence, using Young's inequality,
f
a
u dt<
ö + 6 \ _( a + b
-a
(o + 6)/2
(a + fc)/2
+|
a"'fi2rff + !
J
(o + W/2
J
(a +
a+b
,b
(Q'fdt for xe
fc)/2
V
2
Thus
a~iü2dt<2
j
fl + 6
'a + b\ .
f a+b ,
• Ml —
] for x e (
,b
,( a + b
'a + b\
(a + b , ,
«I -^r~ . xe[ ——-,6 ,
(o + W/2
If y > 0 , we obtain
y2
J
a~ldt<2
(D + W / 2
contradicting the fact that a ' $ L '((A + £)/2, è). Thus y = 0
86
We have shown that the conditions a~l0~ e L\(a + b)/2,b) and a~-fi~ e
e Lx((a + b)/2, b) are sufficient for b not to be an entrance boundary point. If
a = 0, b= 1, a(x)- 1 - x and /}{b) < 0, both conditions are not satisfied. An easy
computation shows that if fi(b)> - 1, then b is not an entrance boundary point.
Thus, if a~x $Ll((a+ b)/2,b), 0(b) may even be negative.
ACKNOWLEDGEMENT
We would like to thank C. Scheffer for bringing reference [3] to our atten­
tion, and A.C. Zaanen for his critical reading and his valuable suggestions.
REFERENCES
1. Arendt, W., P.R. Chernoff and T. Kato - A generalization of dissipativity and positive
semigroups, J. Operator Theory 8, 167-180 (1982).
2. Coddington, E.A. and N. Levinson - Theory of Ordinary Differential Equations, Mc Graw
Hill, N.Y., 1955.
3. Feller, W. - The parabolic differential equations and the associated semi-groups of transfor­
mations, Ann. of Math. 55, 468-519 (1952).
4. Martini, R. and W.L. Boer - On the construction of semi-groups of operators, Indag. Math.,
36, 392-405 (1974).
5. Martini, R. - Differential operators degenerating at the boundary as infinitesimal generators
of semi-groups. Thesis, TH Delft (1975).
6. Martini, R. - A relation between semi-groups and sequences of approximation operators,
Indag. Math., 35, 456-465 (1973).
7. Pazy, A. - Semi-groups of Linear Operators and Applications to Partial Differential
Equations, Appl. Math. Sc. Vo. 44, Springer-Verlag (1983).
8. Protter, M. and H. Weinberger - Maximum Principles in Partial Differential Equations,
Prentice-Hall, Englewood Cliffs, New-Jersey, (1967).
9. Taira, K. - Semi-groups and Boundary Value Problems JI, Proc. Japan Acad. 58, Ser. A,
277-280 (1982).
[10] Ventcel', A.D. - On boundary conditions for multidimensional diffusion processes, Th. of
Prob. Appl., 4, 164-177 (1959).
87
88
2.4. Dual semigroups in 3R x NBV(J) x 3R
It is known that the dual space C(J) of C(J) is isometrically isomorphic to the
space NBV(J) the space of all normalized real functions w of bounded
variation on J. The normalization chosen here is — just as in section 1.3 —
given by
(i)
W(XQ) = 0
(2.4.1)
(ii) w(x) = (w(x+) + w(x-)}/2
for each x e J, but other choices for (i) and (ii) are possible, (cf. [Tay], Ch. 9).
NBV(J) is a closed subspace of BV(J), the space of all real functions of
bounded variation on J.
The pairing C(J) x C(J) -+ 3R is given by
<w,f>
',f> ==
f._ fdw.
J
J
NBV(J) is a Banach space under the total-variation norm, i.e.
||w||
- := Var(w) = |
Var(J)
d|w|.
(J}
Considered as representations for elements from the dual space C(J) , w and
w 2 , satisfying (2.4.1)(ii) are equal in (N)BV(J) if and only if for all f e C(J)
<w ,f> = <w»,f>
or
h
_ fd( W ] - w 2 ) = 0,
thus if and only if w
- w- = c, where c is a real constant. If w e
and w . e NBV(J) then c = 0, but in general c e R
NBV(J)
89
Because of the boundary conditions on D(A) it is handier to isolate possible
jumps in the boundary points r. and r_. So we identify the space NBV(J) with
the space X = TR x NBV(J) xJR with the same normalization (2.4.1). The
corresponding pairing X x X -» 1 is
(2.4.2)
<(w. ,w,w-),u>* := (w(r. +) - w, ).u(r +)
1''
"i'-^i
+ (w 2 - w(r 2 -)).u(r 2 -) +
J ( r ib, rr2 )
udw,
and for the norm of X we have
||(w 1 ,w,w 2 )|| !(i = |w(rj+) - Wj| + |w 2 - w(r 2 -)| + 1
d|w|.
*
*
*
*
Let A : D(A ) c X —
► X denote the adjoint of A, defined in section 2.3,
then we have the following representation theorem for A :
Theorem 2.4.1
Let X = C(J), let a and 8 be continuous real-valued functions defined on J
satisfying
(2.4.3)
WGL^Xpr.)
or
Q if: L1 (xQ,r.)
(i - 1,2)
where W and Q are defined by (1.1.3) and (1.1.4) and let A: D(A) C X -► X
be defined as in Proposition 1 of section 2.3. Then the adjoint operator A of
A with domain D(A ) is given by:
D(A*) = {(w]tw,w2)
e X*\we
NAC(J), aw 1 e ACl
Caw'; 1 - Pw' - ((aw*)'
(2.4.4)
Urn (oWw *)(x) = 0
.
x—>rj
(J),
- pw*)(xQ) G NBV(J),
if r.l is a regular boundary
point (i = 1,2)}.
A*(wrw,w
) = (0, (aw') ' - Bw\ 0)
for all (w w.wj e D( A*).
90
Note that from (2.4.3) it follows that r. is either a regular, or an exit or a
natural boundary point.
Proof
We start with the definition of an operator B with domain
D(B) c X * = l x NBV(J) x TR and range in X as follows:
' D(B) = {(Wj,w,w2) e X* | w G NAC(J), aw • E A C ^ J ) ,
(aw • ) ' - /3w • - ((aw • ) ' - /?w ' X x J G NBV(J),
and
(2.4.5)
lim (aWw ')(x) = 0
x-q
point, ( i = 1,2)}.
B ( w r w , w 2 ) = (0, (aw 1 ) 1 - j8w', 0)
if r. is a regular boundary
»
for all (Wj.w.Wj) G D(B).
Thus D(B) is equal to the right-hand side of the first equality of (2.4.4). We
will show that the conditions of Proposition 2.2.1 are satisfied. Then it follows
that B = A and Theorem 2.4.1 will be proved.
We divide the proof in 5 steps.
Step 1
A is a linear m-dissipative operator with domain dense in X. The linearity of
A is clear. In [CT, 1], Proposition 1 it is proved that D(A) is dense in C(J) and
that A is dissipative. Further, since r. and r- are not entrance boundary
points it follows from [CT,1], Theorem 2 that R(I - A) = C(J). So I - A is
surjective, and then it follows from [CH], Proposition 3.7 that A is m-dissi­
pative.
Step 2
B is a closed linear operator. The linearity of B is clear. Let ((w
w ,w -))
be a sequence in D(B), converging in X to (w w,w.) and assume that the
sequence ( B ( w n l ' w n > w n 2 ) ) converges in X* to ( y r y , y 2 ) .
91
The assertion is proved if we have shown that (w w,w_) G D(B) and
•B(w r w,w 2 ) = ( y r y , y 2 ) .
Since
lim B(wu l wu wll/ ) = lim (0,(aw •) • - fiw • 0) = (y. ,y,y.) in X*
n—>oo
n—xx>
it follows that y. = y- = 0 and
(2.4.6)
lim Var ((aw ' ) • - £w • - y) = 0.
n
n
n^oo (j)
Note that (aw
' ) ' - /?w ' = W ' ^va W w ' ) 1 and y(xA) = 0.
v
n'
^ n
n'
0
Let us denote c n := ((aw * ) ' - /?w ' )(x_).
nO
n
n/v 0
Let x G J. R e g a r d i n g the pointwise interpretation of an equation in (N)BV(J),
we then have
| ( ( o w ^ ) ' - /Jw^Xx) - c n Q - y(x)|
= |(W"1(aWw^)'Kx)-cn0-y(x)|
= |(W _ 1 (aWw^)' - y)(x) - (W - 1 (aWwJ)' - y)(x 0 )|
(2.4.7)
< Var ( W ' ^ a W w ' ) ' - y),
(J)
"
and moreover
|(aWw^)'(x)-W(x).(y(x) + c n 0 )|
= W(x)|(W" 1 (aWw^) , )(x) - c n ( ) - y(x)|
(2.4.8)
< W(x).Var((W" l (aWw •) • - y).
(J)
"
From (2.4.7) we see that the sequence W (aWw ' ) ' - c „ converges pointwise
to y on J. Thus for each x G J we have
lim
W" \ a W w n•) '(x) - cn . = y(x)
n—>oo
^
and
(2.4.9)
lim ( ( a W w ' ) ' ( x ) - W(x).c n ) = (Wy)(x).
n
n—+oo
nU
92
Further, since
w.1 = lim
n—oo
lim
(w
w ,w _) = (w w,w_) in X we have
wni. (i = 1,2). Since wn (x n ) = w(x n ) = 0 and
°
°
lim Var (w n - w) = 0
n - o o (j)
the sequence of functions (w ) converges pointwise to w on J.
Let a,b e
TR, such that r
< a < x~ < b < r„. Then there exists a positive
constant c such that Q(X) > c and W(x) > c for all x e [a,b]. Moreover, if e > 0,
then we obtain from (2.4.6) and (2.4.7)
sup
|(aWw 'J '(x) - W(x).(y(x) + c nQ )|
xe[a,b]
< sup |W(x)| . Var (W V a W w ' J ' - y )
xG[a,b]
[a,b]
"
< €
for sufficiently large values of n, so
lim
n—KX>
(aWw ' ) '(x) - W(x)c n - (Wy)(xj
'nO
holds uniformly on [a,b].
Then by integration we obtain
lim
(aWw')'(s) - c AW(s))ds =
(Wy)(s)ds
n
nu
n-»oo J x_
J xn
or
A X
lim (aWw'Xx) - ( a W w ' X x J - c _
n-»oo
"
n U
nu J x
A X
W(s)ds =
(Wy)(s)ds,
J
x
or
nïoo {wn(x)-(aW)"1(x)-(QWwn)(xO)"CnO(aW)1(x)
|
W s
0
= (aW)"'(x) f
(Wy)(s)ds
JxQ
( >ds}
93
uniformly in x on [a,b].
Again by integration we get, since w (x_) = 0 for all n G UN,
lim
(aW) -1 (s)ds
( w n ( x ) - (aWw^)(x 0 ) f
X
"Cn0
t
(aW)_1(t)f
f
X
t
(aW)_1(t)f
W(s)dsdt)-= f
(Wy)(s)dsdt.
Thus for all x G [a,b]
lim { ( a W wn' X xu J f
(aW) _1 (s)ds
n—«x> *J x~
w(x) -
X
C
n0
t
X
(aW) _ 1 (t) f
f
w
0
0
t
(aW) _ 1 (t) f
W(s)dsdt)-= f
(Wy)(s)dsdt.
0
So w G C [a,b], and by differentiation and substitution of x»
(2.4.10)
(aWw'XxJ"
lim (aWw • )(x )
n
u
n—«x>
and
(2.4.11)
(aWw'Xx) - (aWw')(xn)=
lim c n . f
W(s)ds
n-+oo n U J x»
u
+ f
Jx
(Wy)(s)ds.
o
By differentiation and dividing by W we get
(W"'(aWw ') ')(x) =
lim c n + y(x),
n-+oo n u
Thus
(2.4.12)
and
c 0 = (W" 1 (aWw')')(x ( ) )=
lim
cn()
x G [a,b].
94
(2.4.13)
(W '(aWw ') ')(x) - c n = y(x),
'0
x G [a,b].
Since a and b are arbitrary, and y G NBV(J), we have aWw • G AC. (J),
w G C J (J) n NBV(J) and W ' ^ a W w 1 ) ' - cQ G NBV(J). Hence (Wj,w,w 2 ) G
D(B) and B(Wj,w,w 2 ) = (0,y,0).
Finally we have to verify that
lim (aWw')(x) = 0 if r. is a regular
1
x^rj
boundary point. Assume r . is a regular boundary point. Since a and b are
arbitrarily chosen we have from (2.4.11) and (2.4.12) for each x G J
(2.4.14)
(aWw »)(x) = (aWw ')(x ) + f
Jx
Since (w
(2.4.15)
W(s).(y(s) + cQ)ds.
0
,w ,w _) is a sequence in D(B) we have
lim (aWw')(x)
= 0.
n
x-^r 2
Since r» is a regular boundary point we have W G L (x„,r 2 ) and by the
boundedness of y we have W(y + c„) e L (* 0 ,r-) and then by (2.4.15)
2
(aWw^)(x 0 )+ f
X
=- f
W(s).(y(s) + c n() )ds =
0
2
( ( a W w p 1 - W(y + cnQ))(s)ds + lim (aWw^)(x)
J X„
= - [
Jx
X—*T2
2
((aWw^)' - W(y + c n0 ))(s)ds.
0
Then from (2.4.8) we obtain
2
|(aWw^Xx 0 ) + f
x
2
J x
(Ws).(y(s) + c n0 )ds|
0
| ( ( a W w ^ ) ' - W(y + c n0 ))(s)|ds
95
< {var ( W _ 1 ( a W w ' ) ' - y)} . f
L
n
J
J x
(J)
2
J:o
<e. I
2
W(s)ds
W(s)ds
Jx
for sufficiently large values of n. Thus
(2.4.16)
lim (aWw')(x
)+ f 2 W(s).(y(s) + cn ft)ds = 0.
n
u
n—»oo
Jx.
"
Then finally from (2.4.14), (2.4.12), (2.4.10) and (2.4.16) we have
lim (QWW ')(X) = (QWW'XXQ) + f
x-r2
=
lim
2
W(s).(y(s) + c Q )ds
*" J :
2
(aWw n )(x )+ I
X
W(s)(y(s) + c n 0 )ds
0
= 0.
If r. is a regular boundary point, the proof is similar.
Step 3
Next we define the linear subspace NBV^J) in NBV(J) by
NBV*(J) := {w e NBV(J) | there exist numbers a,b e IR, such that
r
< a < x- < b < r-,
w(x) = w(a)
for x e (r. ,a) and
w(x) = w(b)
for x G (b,r 2 )}.
We will prove
N := TR x NBVJJ)
x 1R is dense in X*. Let (Wj,w,w 2 ) G 3R x NBV(J) x JR
and let e > 0. Then we have to prove that there exists an element (w. ,w,w.) G IT,
such that
96
||(Wj,w,w 2 ) - (Wj,w,w 2 || t < e.
Take w. = w. (i = 1,2) then we have only to prove the existence of a function
w e NBV*(J) such that
(2.4.17)
Var (w - w) < e.
(J)
Let (a ) be a sequence converging to r. and (b ) be a sequence converging to
r.. We define the functions w , n e M in NBV+(J) by
w n (x) = w(x)
,
X £ (a n ,b n ),
w n (x) = w(a n )
,
Xe(rran],
w n (x) = w(b n ) ,
Xe[bn,r2).
Then we have
Var (w - w ) = Var
(w - w ) + Var
(w - w )
n
n
n
(J)
((ri,a n ])
([b n ,r 2 ))
(2.4.18)
=
Var
(w - w(a )) + Var
(w - w(b ))
n
((ri,a n ])
"
([b n ,r 2 ))
=
Var
(w) + Var
(w).
((ri,a n ])
([b n ,r 2 ))
Since w is of bounded variation
lim
n_t0
°
Var (w) = lim
(w) = 0.
(ri.an)
([b n ,r 2 ))
Thus, if we take w = w
(2.4.18) that (2.4.17) holds.
with n sufficiently large then it follows from
97
Step 4
N c R(I - B). Let (z z,z.) G TR x NBV*(J) x R S o there are real numbers a,
b, c and d such that r. < a < x_ < b < r-, z(x) = c for x < a and z(x) = d for
x > b. Note that z(x.) = 0.
Let us consider the equation
(I - B)(w r w,w 2 ) = (Zj,z,z 2 )
for (w w,w 2 ) G D(B).
By writing out we obtain
(Wj,w-W
1
(aWw 1 )', w 2 ) = (Zj,z,z 2 )
w. — z , ,
w_ — z .
so
and
(2.4.19)
w(x) - (W~ , (aWw') , )(x) + cQ = z(x), cQ = (aWw')'(x Q ).
Since r. and r_ are not entrance boundary points there exist two functions w
and w . satisfying the following conditions (see Lemmata 1.3.4-1.3.7)
1
2
w. e C (J), aWw.' G C (J), w. is a solution in BV. (J) of the homogeneous
equation
,-1
w - W
(aWw')' = 0
w- 1 is positive increasing and w z. is negative increasing and lim (aWw.')(x)
= 0,
_1
~
x—»rj
i = 1,2. Note that
W ] (a)
> 0, w ^ a ) > 0, Wj(b) > 0, w ^ b ) > 0,
w 2 (a) < 0, w2'(a) > 0, w 2 (b) < 0, w ^ b ) > 0.
We return to the equation (2.4.19).
98
Since z(x) = c for x 6 (r ,a], the restriction of the solution of (2.4.19) to (r ,a]
has to be of the form
(2.4.20)
w = (c - c Q )e 0 + ivr{ + /iWj,
where e„ is the constant function with e„(x) = 1, and 7,5 are constants.
Since (w.,w,w_) has to belong to D(B) we have the following conditions on
W|
' ,a]
%l,s
(2.4.21)
(i)
w'GL^rj.XQ)
(ii)
lim (aWw')(x) = 0
x^H
if r. is a regular boundary point.
!
In the Lemmata 1.3.3, 1.3.4 and 1.3.6 we gave some properties which will be
used here.
If r
is an exit or a natural boundary point, then w ' e L ( r . , x „ ) and
w ' £ L (r.,x-)thus n = 0 in (2.4.20). If r. is a regular boundary point we
have from (2.4.20) (2.4.21 (ii)) and Lemma 1.3.3
lim (aWw ')(x) = n . lim aWw'(x)
= 0,
x—»ri
x—»r]
~z
and since
lim aWw'(x)
> 0 we have n = 0.
x—>r\
~l
Thus if r. is either a regular, or an exit or a natural boundary point, then
H = 0 and
l^ar^-'o^v^iWr
The restriction of the solution wl .
ieM
-
. can be investigated in a similar way.
This motivates the next definitions. For p,q e I w e define the functions
w p := (c - c 0 )e 0 + pwj
W
q
:= ( d
" C0)e0
+ qv
^2
on ( r r a ] ,
on [b r
' 2)-
99
Then w e C (r. ,a] and w satisfies (2.4.19) on (r. ,a]. Similarly for w .
Let us denote by w the particular solution of (2.4.19) as given in Proposition
1.3.7.
For r,s e 3R we define the function w
: [a,b] —
► 3R by
r,s
w
r SW
=
™ZW
+ r
^ j ( x ) + sw 2 (x),
x e [a,b].
It is the aim to determine p, q, r and s such that the function w, defined by
w on (r.,a), by w
tiate.
Then
it
on [a,b] and by w on (b,r.) is continuously differen­
follows
from
the
differential
equation (2.4.19) that
aWw' G ACj (J), a w ' e AC. (J), (z w,z.) e D(B) and w satisfies (2.4.19).
w is continuously differentiable on J if and only if
lim w v(x) = w v(a)
p '
r,s '
x—»ahm w *(x) = lim w ' v(x)
p '
r,s '
x—»ax—»a+
hm w (x) = w v(b)
.
q
r,s '
x—»b+
lim w ' ( x ) =
lim w ' (x).
By writing out these equations we obtain the following system:
c - c„ + pA = rA + sE + K
pB = rB + sF + L
d - cQ + qG = rC + sG + M
qH = rD + sH + N
where
100
A = Wj(a),
E = w 2 (a),
K = w z (a),
B = w«(a),
F = wj(a),
L = wz'(a),
C = Wj(b),
G = w 2 (b),
M = w z (b),
D = w^b),
H = w2Hb),
N = wz'(b),
or
' A
E
-A
0 '
r
B
F
-B
0
s
- L
C
G
0
-G
P
d - cQ - M
D
H
0
-H
q
- N
' c - cQ - K '
The determinant of the system is equal to (AF - BE)(GD - CH) < 0. So p, q, r
and s are uniquely determined, and the proof of step 4 is completed.
Step 5
Let M := C„,(J) be the subset of functions of C(J) which are constant outside of
a bounded closed subinterval of J. It is clear that CM) is dense in C(J). In this
step we will prove:
For all u e D(A), such that u - Au G M and all (w .,w,wj
(z .,z-z(x„),z2)
(2.4.22)
e iV where (z .,z,z.)
:= (w,,w,w-)
6 D(B) such that
- B(w .,w,w?), we have
<(w .,w,w2),Au>% = <B(w .,w,w ■)),u>%-
Assume u e D(A) and (w w,w„) e D(B) satisfy the above-mentioned condi­
tions.
Let (Zj,z,z 2 ):= (I - B)(w 1 ,w ) w 2 ).Then(Zj,z-z(x Q ),z 2 )G N =HRx NBV^J)xIR,
z. = w
z« = w 2 , z(x) = w(x) - (W~ (aWw')')(x). Thus there are constants
a,b e 3R, such that
101
r < a < x- < b < r •
z(x) = z(a) and w(x) = z(a) - c„ + p.w
for all x e (r.,a],
cQ = (aWw V ( x 0 ) , P j G3R;
z(x) = z(b) and w(x) = z(b) - c» + P ? w_ for all x e [b,r»), p^E IR.
Then we have from Lemmata 1.3.4, 1.3.5 and 1.3.7
lim (QWW ')(x) - p.i lim (aWw.')(x)
= 0.
1
x—»rj
x—>rj
From the proof of Lemma 3 of section 2.2 it follows that (W
u ')(x) =
7(W
ul)(x) near the boundary point r«. Since W uJ is a negative in-1
-1
creasing function
lim (W u ' )(x) exists, and similarly lim (W u ')(x)
x—>r2
x-»ri
exists as a finite number.
Then
<(I - B)(w 1 ,w,w 2 ),u> +
= <(Zj,z,z 2 ),u> +
= (z(rj+) - z 1 )u(r 1 +) + (z 2 - z(r 2 -)).u(r 2 -) +
udz
= (z( r ] +) - ZjMrjH-) + (z 2 - z(r 2 -)).u(r 2 -) + J
udw
- f ud(W"1(aWw')')
= (z(rj+) - ZjMrj+J + (z 2 - z(r 2 -)).u(r 2 -) + I udw
- ( ( W ' V w w V . u X ^ - ) + ((W" 1 (aWw , ) , .u)(r 1 +)
+
(W
u ')d(aWw ')
(by integration by parts)
= (w(rj+) - WjXuCrjH-) + (w 2 - w(r 2 ~).u(r 2 -)
+ f udw + lim
((W'^'MaWw'JXx)
JJ
x—^2~
102
lim ((W^u'MaWw'JXx) - f aW(W ' u ' ) ' d w
—>rj+
J
(since z. = w., z = w - W
(aWw ' ) ' and
integration by parts)
(w(rj+) - w 1 ).u(r ] +) + (w 2 - w(r 2 ~)).u(r 2 -) +
(since lim (W
(u - Au)dw
u')(x) is finite and
lim (aWw')(x) = 0, (i = 1,2)
x-+r;
= <(w 1 ,w,w 2 ),(I -A)u>„,.
Then (2.4.22) follows immediately.
At this stage all conditions of Theorem 2.2.1 are satisfied and it follows that
B = A . This completes the proof of Theorem 2.4.1.
2.5. Restricted dual semigroups in 3R x L (J) x TR
From section 2.1 we know that A is the weak infinitesimal generator of the
dual semigroup T of T.
For the closed subspace X„ of X on which T is strongly continuous we have
by Proposition 2.2.1
X* = D(A*).
More explicitly we have
Lemma 2.5.1
X*Q = OR x NAC(J) x 1R.
103
Proof
Since NAC(J) is a closed subspace of NBV(J), IR x NAC(J) x JR is a closed
subspace of IR x NBV(J) x TR. Let (w w,w ) G 3R x NAC(J) x IR, and let
(. > 0, then we have to prove that there exists an element (w w,w ) in D(A )
such that
(2.5.1)
ll(w r w,w 2 ) - ( w r w , w 2 ) | | + < e.
Since C (J) is dense in L (J), there exists a z. G C (J) such that
J
| w ' - Z j | d x < e/2.
Here C (J) denotes the set of continuous functions with compact support inside
J.
Set S := supp z. and m := min (aW)(x), then m > 0. Now aWz
XGS
2
G C (J), so
I
C
there exists a z . e C (J) with supp z_ <r S such that
|aWz
- z J d x < m.e/2.
Now we define the function w on J by
(aW) _1 (t).z 2 (t)dt.
w(x)=[
0
Then w is constant outside the support of z-, w G NAC(J) n C (J),
oWw' G C 2 (J), and ( w r ^ a W w » ) ' G C J (J) c BV(J). If w = w
we see (w.,w,w_) G D(A ). Moreover,
||(wj,w,w 2 ) - (Wj,w,w 2 )|| #
= Var(w - w)
(J)
w =w
104
=
|w ' - w '|dx
<
|w ' - z.|dx +
<
|w • - z.|dx + m
|z. - w '|dx
|aWz. - zJdx
< e.
G
Now we are able to give the infinitesimal generator A„ of the C„-contraction
semigroup T~ on X . , being the largest restriction of T which is strongly
continuous.
Lemma 2.5.2
X* = JR x NAC(J) x ]R
D A
( *0) = {(Wj,w,w2) e X*0\we
(aw')'
(J),
- Pw' G AC(J),
Urn aWw %(x) = 0
x—>rj
A*0(wrw,w2)
NAC(J), a>v' G AC,
= (0, (aw*)'
if r. is a regular boundary point (i = 1,2)}.
'
- pw', 0) for all (wyw,w2)
G D(A*Q).
D
Set X® := H x L (J) x ]R, equipped with the norm ||.||
ll(v 1 ,v,v 2 )|| 0 = |v 1 | + |v 2 | + J
defined by
|v|dx
for all (v v,v„). With this norm X® is a Banach space. (The sign O is called
"sun".)
We define the mapping
f Ï: X * - + X ®
(2.5.2)
I(Wj,w,w 2 ) = (w(rj+) - Wj, w ' , w 2 - w(r 2 -)).
105
Since
l|I(w 1 ,w,w 2 )|| 0 = |w(r 1 +) - Wj| + |w 2 - w(r 2 -)| + J
i w i| d x
= ll(w r w,w 2 || # ,
and since I(w
w,w«) = 0 implies (w w,w.) = (0,0,0), I is an isometric
isomorphism. So as Banach spaces, X„ and X® may be identified. Moreover,
I" (Vj,v,v 2 ) = (y(rj+) - Vj, y, y(r 2 ~) + v 2 ) where
(2.5.3)
y(x)= f
Jx
v(t)dt.
o
In X® we define the operator A®: D(A°) c X® — X® by
D(A®) = f(D(A*))
(2.5.4)
A® = I o A* o I 1
Then it is easily verified that
' D(A®) = {( V ] ,v,v 2 ) G X® | av G AC lQc (J), (av)' - /?v G AC(J),
lim (aWv)(x) = 0
x-vr;
boundary point),
(2.5.5) .
if r. is a regular
A®( V l ,v,v 2 ) = (((av)'- /3v)(r1+),((<*v)»- M »,-((av) •- /?v)(r 2 -))
for all ( v p v , v 2 ) G D(A®).
The pairing <.,.>~: X® x X -> 1R, induced by I, is given by
7-1.
<(Vj,v,v 2 ),f> 0 = <I
(Vj,v,v 2 ),f> +
= Vj^rj+J + v 2 f(r 2 -) + j
vfdx.
106
By the definition of A® it follows that A® is the infinitesimal generator of a
C„-contraction semigroup T® := (T®(t); t > 0 } i n E x L (J) x JR.
2.6. "Bidual" semigroups in 3R x L°°(J) x 3R
The dual space X®* := ( 1 x L (J) x TR)* of 1R x L (J) x 1 is isometrically
isomorphic to the space ]R x L°°(J) x 1R. The pairing <.,.>fl*: X® x X® —
► IR
is given by
<(z ] ,z,z 2 ),(v 1 ,v,v 2 )> 0 + = ZJVJ + z 2 v 2 + I zvdx.
]R x L°°(J) x TR is a Banach space under the norm
||(z 1 ,z,z 2 )|| 0 + = maxflZjUz^, ess sup |z(x)|}.
xeJ
For the adjoint operator A®*: D(A®*) c X®* -» X®* of A® we will prove the
following representation theorem:
Theorem 2.6.1
Let A®: D(AQ) c X® — X® be defined by (2.5.5). Then the adjoint A®* of
A® is given by
D(A®*) = {(zvz,z2)
e X®* | z e C^J),
n r / i t r " ■*
aW(W
z•6
I t •
z')'
AChc(J),
*- / v \ f r \
e L°°(J), and
Urn z(x) exists and is equal to z.
(2.6.1)
if r. is a regular or an exit bound­
ary point, i = 1,2},
AQ*(zyz,z2)
= (0,aW(W
1
z ') \0), for all (zyz,z2)
e D(A®*).
107
Note that from the definition of A, and thus from the definition of A® it
follows that r. (i = 1,2) is a regular, an exit or a natural boundary point.
Proof
The main line of the proof of this theorem is the same as in the proof of
Theorem 2.4.1. We start with an operator C: D(C) c X®* -► X®* and we will
prove in a number of steps that there is satisfied the conditions of Proposition
2.2.2. Then it follows that C = A®* and Theorem 2.6.1 is proved. So we start
with the definition of the operator C with domain D(C) C X°* = 3R x L°°(J)
x JR and range in X® as follows:
D(C) := {(z r Z ,z 2 ) e X®* | z G C J (J), z ' e A C ^ J ) ,
aW(W
1
z ' ) ' GL°°(J), and
lim z(x) exists and is equal to z.1 if r.l
x—>q
(2.6.2)
is a regular or an exit boundary point,
i = 1,2}.
C ( z r z , z 2 ) = (0, aW(W
z 1 ) ' , 0), for all (Zj,z,z 2 ) e D(C).
Thus D(C) is equal to the right-hand side of the first equality of (2.6.1).
Note that if z • e AC. (J), then
a W ( W " I z ' ) 1 = az" + fiz\
In 3 steps we will verify the conditions of Proposition 2.2.2.
Step 1
A® is a linear m-dissipative
operator with domain dense in X®. The linear­
ity is clear. Since A® is the infinitesimal generator of the C„-contraction
semigroup T® it follows from the first part of the Generation Theorem of
Hille-Yosida (section 2.1) that A® is m-dissipative with domain dense in X®.
108
Step 2
C is a linear operator and R(I - C) = X®*. Let (kj,k,k 2 ) e X ° * =
3R x L°°(J) x 3R. We consider the equation
(2.6.3)
( I - C)(z r z,z 2 ) = ( k r k , k 2 )
for (z. ,z,z.) G D(C). By writing out we obtain
( z 1 , z - a W ( W " 1 z ' ) ' , z 2 ) = (k 1 ,k,k 2 )
so
Z
l=kl'
Z
2
= k
2'
and
z - a W ( W _ 1 z ' ) ' = k.
(2.6.4)
Let the function z, be defined by (1.1.48), then we know from Proposition
1
1.1.24 (and its proof) that z, e C (J), z ' e AC. (J) and that z, is a
particular solution of (2.6.4). Thus the general solution of (2.6.4) is given by
Z=Z
where u
k
+ C
l^l+C2y2'
C
l'C2enR'
and u- are the special solutions of (1.1.6) as given in Proposition
1.1.1.
If r 1 (resp. r„)
is a natural boundary point, then
l
lim u.(x) = oo), thus C- = 0 (resp. c, = 0).
X—*T2
l
l
lim u«(x)
= oo (resp.
l
x—>x\
l
If r is a regular or an exit boundary point, then
lim z(x) = lim z.(x)
+ cl. . lim _u.(x)
+ c«.
lim u„(x)
l
x-+rj
x—>rj K
x—*rj 1
x—»rj ~z
= c-1 • lim u.(x)
x—»r] " z
109
and we have
c 2 = z r (u 2 (r 1 +))"
=k 1 (u 2 (r 1 +))
.
If r» is a regular or an exit boundary point, then we have
C =Z
l
2-(ul(r2-))"1=k2(^l(r2-))"1-
It follows that if r. and r» are not entrance boundary points the equation
(2.6.3) has a unique solution in D(C), which completes the proof of step 2.
Step 3
E x C (J) x JR is a dense linear subspace of ÜR x L (J) x JR and for all
(y v,v J such that (I - AQ(v.,v,v
) E IR x C (J) x 3R, and all (z ,z,z J e
D(C)
<(zrz,z2).
A®(vrv,v2)>Q%
= <C(zrz,z2,
(vrv,v2)>Q,.
That IR x C c (J) x IR is dense in 3R x L (J) x JR follows from the density of
I
C (J) in L (J). We denote by V the subset of tripels (v ,v,v.) in D(A 0 ) such
C
J
that (I - A®)(v v,v.) G
( v r v , v 2 ) G V.
Z
IR x C (J) x ]R Assume (z.,z,z 2 ) G D(C) and
Then we have by integration by parts
<C(z 1 ,z,z 2 ),(v 1 ,v,v 2 )> 0 „ t - <(z 1 ,z,z 2 ),A°(v 1 ,v,v 2 )> =
=
<(0,aW(W~lz')',0),{vrv,v2)>Qt
-<(z ] ,z,z 2 ),((W" 1 (aWv) , )(r 1 + ),(W" 1 (aWv)')',-(W" 1 (aWv)')(r 2 -))> 0 < t
110
z.(W _ 1 (aWv)')'dx
= f aWCW^z'J'.vdx- f
- z 1 .(W" 1 (oWv)')(r 1 +) + z ^ W ' ^ a W v ) ' ) ^ - )
= [(aWvXxMW^z'Xx)] J 2
- f
W _ 1 (aWv)'dz
- [zCx^W'^aWvJ'Xx)]^ + f
W'^aWvJ'dz
- z ] .(W" 1 (aWv)'Xr 1 +) + z ^ W ' ^ a W v ) ' X ^ - )
=
lim (aWvXx).(W _1 z'Xx) - lim (aWvXx).(W~ ! z')(x)
x->r 2
x—rj
lim (z(x) - z.).(W" 1 (aWv)'(x))
-
X-T
Z
2
+ lim (z(x) - Z . ) . ( W " 1 ( Q W V ) ' ( X ) ,
l
x-ri
(2.6.5)
provided that these limits exist. In the next lemmata we will prove that these
limits exist and are equal to zero.
Lemma 2.6.2
(a)
If r. is a natural boundary point, and (v .,v,v?) E V, then
(b)
x—*rj
If r. is a regular or an exit boundary point, and (v .,v,vj
lim
(W~1(OLW\)
%
)(X) = 0,
i = 1,2.
G D(A®),
then
lim (W~1(aWv)%)(x)
exists.
x—>r
Proof
It is sufficient to prove the lemma for the boundary point r„.
(a)
Assume r- is a natural boundary point, and let (v. ,v,v ) e V.
Set g := v - (W~ (aWv)')', then g e C (J). So there exists a b G J, such that
for all x G (b,r.), g(x) = 0.
Ill
On (b,r_) v satisfies the homogeneous equation v - (W
(aWv)')' = 0. Let u.
-1
and u . be as in Proposition 1.1.1. From Lemma 1.2.6 we have v := (aW) u.
£ L (x~,r_), and it follows from the fact that v e L (x 0 ,r-) and Lemma 1.2.2
that for x G (b,r-)
v(x) = c Q v 2 (x) := c 0 ((aW)" 1 u 2 )(x), cQ G TR.
Then (aWv)(x) = c,.u 2 (x), x G (b,r«), so
lim (W _ 1 (aWv)')(x) = cun lim (W - 1 uI)(x)
x—^2
x—>T2
=0
by Lemma 1.1.17.
(b)
Next we assume that r^ is a regular or an exit boundary point, and let
(v v,v_) G D(A®). Then the assertion follows from the fact that W
(aWv)'
belongs to AC(J).
D
Corollary of Lemma 2.6.2
(a)
If r. 'is a natural boundary point, (v .,v,v-) G V and (z .,z,z .) G D(C),
then
(2.6.6)
(b)
lim (W ^aWv)
x—>rj
%
)(x).(z(x)
- z.) = 0, (i = 1,2).
l
If r. is a regular or an exit boundary point, (v .,v,v?) G D(A®) and
(zrz,z2)
G D(C), then (2.6.6) holds.
Lemma 2.6.3
(a)
(z
If r. is an exit or a natural boundary point, (v.,v,v.)
z,z ) G D(C), then
(2.6.7)
lim (aWv)(x).(W~1z,)(x)
x-*rj
= 0,
(i = 1,2).
G V, and
112
(b)
If r. is a regular boundary point, (v ,.v,v?y) G D(A®) and (z ,,z,z ) e
D(C), then (2.6.7) holds.
Proof
(a)
It suffices to prove the lemma for the boundary point r-. Assume r- is
an exit or a natural boundary point, let (v.,v,v_) G V and let (z.,z,z„) e
D(C). From the Lemmata 1.2.2, 1.2.4 and 1.2.6 we know that (aW)~ u g
1
1
1
L (x ft ,r„) and (aW) u- e L (x,.,r»). Then, as in the proof of Lemma 2.6.2(a)
we have in a sufficiently small left neighbourhood (b,r-) c (x 0 ,r„) of r .
V(X) = C 0 ( ( Q W ) " 1 U 2 ) ( X ) ,
cQeK.
Thus
lim (a\Vv)(x) = lim c_u»(x) = 0.
x-+r2
x—*T2
Further, since aW(W
z ' ) ' G L°°(J), there exists a number K > 0, such that
on J
KW^z'Xx)! < K(aW) - 1 (x)
(a.e.)
Then we have for x e ( x n , r . )
|(W" 1 z')(x)| = | z , ( x 0 ) + f
(QW)" 1 (t)(aW(W" 1 z')'Xt)dt|
(aW)" 1 (t).|(QW(W" 1 z') , )(t)|dt
< |z'(x 0 )| + J"
0
.x
|z'(x 0 )| + K . [
x
So for x G (b,r_) we have
0
(aW)_1(t)dt.
113
(2.6.8)
|(aWv)(x).(W" l z 'Xx)! <
,
< | c 0 |.|z (x 0 )|.u 2 (x) + |c 0 |.K.u 2 (x) |
(aW)" 2 (t)dt.
If x tends to r_, the first term in the right-hand side of (2.6.8) tends to zero,
and it remains to prove that the second term in the right-hand side of (2.6.8)
tends to zero.
Since u is decreasing we have for x„ < y < x
(aW) _1 (t)dt < u 2 (x).M y + |
u 2 (x) ƒ
((aW)" 1 u 2 Xt)dt
fy
(with M =
y
-1
((aW)
)(t)dt)
Jx0
f'2
-1
<u 2 (x).M y + J
((aW) '^XOdt.
(26.9)
Let e > 0. Because of Lemmata 1.2.4 and 1.2.6 we have v . = (aW)
u« e
L ( x 0 , r . ) , so there exists a number y„ such that for all y with y_ < y < r
r
(2.6.10)
2
-1
0 < J| v ((aW)
y
u-)(t)dt < e/2.
~z
Let y. G ( y 0 , r . ) . By Lemmata 1.1.15 and 1.1.17 we have lim u.(x) = 0, so
x
~* r 2
there exists a number x. with y. < x. < r ? such that for all x G (x. ,r_)
(2.6.11)
0 < u.(x).M
-2
<e/2.
yj
Then from (2.6.9)-(2.6.11) it follows that for all x G ( y . , ^ ) we have
r.
x
u 2 (x) j
_1
(aW) (t)dt < u 2 (x).M y + J
< e/2 + e/2 = e.
This completes the proof of part (a).
((aW) _1 u 2 Xt)dt
114
(b)
If r_ is regular, and (z z,z-) e D(C), then
(W" 1 z')(x) = z , ( x 0 ) + [
Jx
(aW)~ W a W f W ^ z ' V X O d t ,
o
which is bounded since (aW)" 1 e L ^ x ^ r . ) and a W ( W _ 1 z ' ) ' e L°°(J).
Moreover, for all (v.,v,v_) e D(A®), by assumption
lim
(aWv)(x) = 0.
1
Then it follows that (2.6.7) holds.
D
We return to (2.6.5). It follows from the lemmata just proved that the occur­
ring limits are equal to zero. Hence
<C(z 1 ,z,z 2 ),(v ] ,v,v 2 )> 0<1 - <(z 1 ,z,z 2 ),A®(v ] ,v ; y 2 )> = 0,
proving the assertion in Step 3.
In the Steps 1-3 we proved that the conditions of Proposition 2.2.2 are
satisfied. Then from this proposition it follows that C = A® . This completes
the proof of Theorem 2.6.1.
□
As in section 2.5 we know that A® is the weak infinitesimal generator of the
dual semigroup T® of T®. For the closed subspace X®® of X® , on which
T® is strongly continuous we have
(2.6.12)
X®® = D(A®).
Explicitly we have
Lemma 2.6.4
If r j and r. are regular or exit boundary points, then
(2.6.13)
*® Q = {(z
K
z,zJelRxC(J)xlR\
l
*
lim z(x) = z., i = 1,2\.
I x-*rj
'
J
115
Proof
Assume r. and r„ are regular or exit boundary points. Let Z be the set in the
oo
right-hand side of (2.6.13). Then Z is a closed subspace of E x L (J) x 1R.
Let (z
z,z_) G Z .
i
z
2
Then there exists a function z G C + (J) such that
~
hm
V
2
if
r
z. = z., ï = 1,2. Set
—
1
1
*^ j
y :- z - z, then y G C„(J). Let e > 0. Since C (J) is dense in C J J ) in the sense
2
of the supremum norm, there exists a n y e C (J) such that ||y - y|| < e. Now
it is easily verified that (z., y + z, z») G D(A® ) and
||(z 1 ,z,z 2 ) - ( z r y + z, z 2 )|| 0 + < e.
Hence D(A®*) is dense in Z and thus Z = X®®.
D
Let r ,r» be regular or exit boundary points. We define the mapping
r
I : X 0 0 - + X = C(J)
(2.6.14)
I ( z r z , z 2 ) = z.
Then
(Zj,z,z 2 || 0 O = IKZj ,z,z 2 )|| OJ|t
= max { I Z j U z ^ J I z H ^ }
= z
and
since z = 0 G C(J) implies (z. ,z,z_) = (0,0,0) I is an isometric isomorphism. So
the spaces X®® and X may be identified. If this is the case, the space X is
0-reflexive with respect to the operator A. Note that then also A®® = A.
116
2.7. On CL-semigroups in a space of bounded continuous functions in the case
of entrance boundary points
CA. Timmermans
Delft University of technology
Reports of the faculty of Mathematics and Informatics, no. 87-24 (1987)
Submitted for publication
1. INTRODUCTION
In [1], Clément and the author investigated contraction semigroups (T(t)),
t > 0, of class C„ on the Banach space C(I), where I is the two points
compactification of an open interval ( r . , r - ) , -oo < r
< r . < oo, (equipped
with the supremum norm) generated by an operator A„ of the form
AQu = Q D 2 U +
(1.1)
£Du.
Here a and /? are continuous real valued functions on I, with a(x) > 0 for
x e I. The domain of A-, denoted by D(A n ), is given by
(1.2) D(A u ) := ( u S C(I) I u e C 2 (I), lim Au u(x) = 0, lim Aun u(x) = o } .
*•
x—»rj
x—»r2
^
The boundary conditions in (1.1) are usually called Ventcel's boundary condi­
tions, [7].
In order to state the conditions on a and /9 in the next theorems, we
define the functions W and Q as follows
(/?.<*" J)(t)dt},
W(x):=exp{- f
L
Jx
J
o
Y
_1
Q(x):=(aW) (x) f
J x.
W(t)dt,
117
v
(aW)_1(t)dt.
R(x):= W(x) f
Jx
o
Here x~ denotes an arbitrary fixed point in I.
In [1] we proved the following theorem, which gives necessary and sufficient
conditions for A n to be the generator of a C„-contraction semigroup on C(I).
Theorem 1
Let a and p be real-valued continuous functions on I, and let A„ be given by
(1.1) and (1.2), then An is the generator of a C^-semigroup on C(I) if and
only if a and fi satisfy
(Hj)
W G l}(rrx0)
or
Q$ L^r^)
or both,
(H2)
WEL^Xpr^
or
Q $ L1\xQ,r2)
or both.
The reader familiar with the terminology of Feller [3], will recognize that
(H.) (i = 1,2) is satisfied if and only if r. is not an entrance boundary point,
thus if r. is a regular, an exit or a natural boundary point.
Let
(1.3)
D(A) := {u G C(I) | u € C 2 (I), Q D 2 U + £Du G C(ï)},
and let A: D(A) — C(I) be defined by
(1.4)
Au = aD 2 u + 0Du.
In this paper we will give necessary and sufficient conditions for A to be the
generator of a C„-semigroup on C(I).
118
2. MAIN THEOREMS
Let I denote a non-empty open interval of IR (not necessarily bounded), I the
two points compactification of I, and 31 := I\I. C(I) is the Banach space of
real-valued continuous functions on I equipped with the supremum norm,
denoted by ||.||.
The main theorems read:
Theorem 2
Let a and p be real-valued continuous functions on I with a > 0 on I. Let
D(A) := (u G C(I) | u E C2(I), aD2u + 0Du G C(~I)},
(2.1)
and let
2
Au = a.D U + fiDu
for u G D(A). Then A: D(A) —* C(I), and the following conditions are satis­
fied:
(i)
D(A) is dense in C(I)
(ii)
A is closed
(Hi)
I - A is surjective in C(I).
Theorem 3
Let A be defined as in Theorem 2, then A is the generator of a C„-semigroup
on C(I) if and only if the following condition
(K)
is satisfied.
We also have:
f ° R(x)dx = oo and
Jry
[
2
)x0
R(x)dx = <x>
119
Theorem 4
Let a and fi be real-valued continuous functions on I with a > 0 on I. Let
:= \u e C(I) | u G C2(I), aD2u + fiDu e C(I),
D(A})
Urn ((aD2u) + f3Du)(x) = Jo\,
X—»ry
and let
A .u = aD u + (3Du
for u e D(A .). Then A .: D(A .) —* Cfl) is the generator of a C^-semigroup
on C(I) if and only if the following conditions are satisfied:
(i)
W&L^TJ.XQ)
or Q$
(ii)
\ 2 R(x)dx = <x>.
JxQ
L*(rrxQ)
orboth,
In the next section we will prove Theorems 2 and 3. The proof of Theorem 4
is similar.
3. PROOFS
The proof of Theorem 2 is partly based on the following lemmata.
Lemma 5
Let a and fi be as in Theorem 2, and let X > 0. Moreover, let A . (resp. A?) be
the set of positive increasing (resp. decreasing) solutions on I of
(3.1)
satisfying
u - X(aD2u + pDu) - 0
ufx„) = 1. Then there exists a unique u. G A. (resp. a unique
Uy e A2) which is minimal on (r xnJ (resp. [x„,r )), i.e. if u e A . (resp.
u e A.) then u. < u on (r .,xnJ (resp. u2< u on
fxn,rj).
120
The proof for u» can be found in [1], the proof for u. is similar.
We define
M. = sup u.(x),
l
J:
iv
xGl
l = 1,2.
Then we have
Lemma 6
1,
> ,
„ _ ,/.
M < oo (resp. M < <x>) if and only if R £ L (xQ,r ) (resp. R e L
(r^XQ)).
Proof
We only prove the lemma for M..
Necessity. Since u. satisfies the equation
(3.2)
( W " 1 ! ! ' ) ' =(aW) _ 1 Uj
we have for all x G (r. ,r»)
x
(3.3)
Uj(x) = |
t
(aW)_1(s).u(s)dx + u j(x 0 )}dt + U ^ X Q ) .
W(t) { ƒ
Since u. is positive increasing we then have for x > xfi
X
Uj(x 0 ).f
Jx
X
R(t)dt< f
0
J x
t
(QW)" 1 (s).u(s)ds<u 1 (x)<M ] .
W(t) f
0
X
0
So, if M. < oo, then R G L ( x - , r A
Sufficiency. Since u. is increasing we have for x G (x„,r»)
0 < Uj'(x) < Uj'(x0).W(x) + Uj(x).W(x) J x
= Ul '(x 0 ).W(x) + Ul (x).R(x).
(aW) _1 (s)ds
121
Here R e L (x,.,r«), and thus also W e L (x..,r.A
It is standard that the differential equation
y • - Ry = C.W,
y e C1^,^),
C e l
y(x Q )= 1
has a unique bounded solution on [x~,r_). Since y 1 > 0 and y(* 0 ) = 1 w e have
y is increasing. Moreover we have
0 < u.(x)
< y(x) < lim y(x) < oo.
-l
x—r2
Thus, if R £ L (x n ,r-), then M. < oo.
D
Let V be the Wronskian u!u» - u l u . , then clearly V > 0. We define the
Green function G: Ixl -► JR for (3.1) by
G(x,s) = U 1 (X).(QV)" (S).U 2 (S),
= u 2 W - ( a V ) " (s).Uj(s),
X
< S,
x > s,
and for each g e C(I) the function u : I —
► 1R by
o
(3.4)
u g (x) = J
2
G(x,s)g(s)ds.
Then it follows by verification that u
G D(A), and that u
g
equation
(3.5)
u - AAu = g.
(See also [3], Th. 13.1).
satisfies the
g
122
Lemma 7
The mapping / - A/4: D(A) —
► R(I - A) is injective if and only if condition
(K) holds. (I is the identity operator).
Proof
If condition (K) is satisfied it follows from Lemma 6 that M. =M_ = oo. Then
the only bounded solution of (3.1) is u = 0. Thus I - AA is injective.
Conversely, if I - AA is injective, then u
is the unique solution of (3.5).
o
However, if M. < oo, i = 1 or 2, then for each constant C also u + Cu. should
i
g
i
be an element of D(A) by the fact that AAu. = u. e C(I). This is a contradic­
tion, thus M. = oo, and, with Lemma 6, condition (K) holds.
D
Proof of Theorem 2
(i)
D(A) is dense in C(I). In [1, prop. 1] it is proved that DQ(A) (see (1.2))
is dense in C(I). Since D n (A) <r D(A) for D(A) the same holds.
(ii)
A is closed. Let (u ) c D(A), u and v in C(I) be such that
lim ||u - u|| = 0 and lim ||Au n - v\\ = 0. We have to show that
n—»oo n
n—»oo
u G D(A) and Au = v. Let us denote Au by i/ . For every a,b e IR
such that [a,b] C I there is a constant c > 0 such that Q(X) > c for all
2
x e [a,b]. Therefore the restriction on [a,b] of aD u + /JDu is a regular
Sturm-Liouville operator [2]. Since u (a) (resp. u (b)) converges to u(a)
(resp. u(b)), it follows from the classical theory [2] that u • and u" are
2
Cauchy sequences in C[a,b], and therefore that u e C [a,b], and
2
au" + £ u ' = v on [a,b]. Since a and b are arbitrary, u E C (I) and
au" + /3u' = v on I. Since v e C(I), we see that
exists. Thus u € D(A) and Au = v.
(iii)
I - A is surjective in C(I). This is a direct consequence of (3.4)-(3.5)
with A= 1.
D
lim (au" + /3u ')(x)
x->dl
123
For the proof of Theorem 3 we need the following lemmata.
Lemma 8
If X > 0 and u G D(A), then u - XAu > 0 implies u > 0 if and only if condi­
tion (K) is satisfied.
Proof
If R £ L (r ,x_) and R £ L (x-,r_), then I - AA is injective and since G > 0
it follows from (3.4) that g := u - AAu > 0 implies u = u > 0.
1
1
^
On the other hand, if R e L ( r . , x . ) o r R e L (x„,r_)then M. < oo or M_ < oo.
Say M. < oo. Let ü e D(A) be such that u > 0 and u - Au > 0. Let
-I
C > (AM.)
lim u(x) and let u = u - Cu Then u - AAu > 0, however not
1
x—*r*>
"
1
-
-
u > 0.
D
Lemma 9
A is dissipative if and only if condition (K) holds.
Proof
From Lemma 7 we know that condition (K) holds if and only if I - AA is in­
jective. Assume I - A A is injective. We define J. = (I - A A)
. Then J. is a
positive linear operator (u > 0 =>■ J. u > 0, Lemma 8) on R(I - AA), and J. e„ =
e„ e D(A), where e„ is the constant function e„(x) = 1. Then it follows that
IUAgll < llgll for all g 6 R(I - AA), or
(3.6)
||u|| < ||u - Au||
for all u e D(A),
thus A is dissipative.
Conversely, assume A is dissipative, then it follows from (3.6) that u - AAu =
0 implies u = 0, thus I - AA is injective.
D
124
Proof of Theorem 3
It is known [4, Th. 3.1] that a necessary and sufficient condition for a closed
operator A, with dense domain and for which R(I - A) = C(I) to generate a
strongly continuous semigroup of contraction operators, is that A is dissipative.
Then Theorem 3 is a direct consequence of Theorem 2(i),(ii),(iii) and Lemma
9.
D
Examples
22
1. Let A be defined as in Theorem 2 with I = TR, x» = 0, a(x) = (1 + x )
and )9(x) = 0. Then W(x) = 1, (aW) _1 (x) = (1 + x 2 ) " 2 and R(x) = -^ arctan x +
1
2
x
1+x 2 '
Thus condition (K) is satisfied and A is the infinitesimal generator of a
C„-semigroup on C([-oo,oo)].
2. Let A be defined as in Theorem 2 with I = ^- -z-, -z) , x„ = 0, a(x) = 1,
, (QW)~ (X) = cos 2 x, R(x) = -= tan x
£(x) = -2 tan x. Then W(x) = (cos x)
+
z— . Thus condition (K) is satisfied and A is the infinitesimal
2cos2x
generator of a CL-semigroup on C ( [- -r, -z] J .
Remark. This example can be obtained from example 1 by means of the
C
TT
TT ~\
diffeomorphism <j>: (-00,00) —» [- — — J with <f>(x) = arctan x.
Remark. Let a and /3 be such that condition (K) is satisfied. Then if moreover
condition (H.), i = 0 or 2 is satisfied, then one can prove that
lim Au(x) = 0.
^ri
In this case the boundary point r. is called natural (see Feller [3]). In the case
that both boundary points are natural Theorems 1 and 3 are both applicable.
1
x
Example
I = 1R, xQ = 0, a(x) = 1, 0(x) = 0. Then W(x) = 1, (aW)~ l (x) = 1, Q(x) = R(x) = x.
D(AQ) = {u £ C([-oo,oo]) I u e C2(-oo,oo), u" e C([-oo,oo])},
D(A.)
= {u e C([-oo,oo]) I u G C2(-oo,oo)
1
lim u"(x) = 0}.
x—»ioo
Then D(A„) = D(A.), and A« = A. is the infinitesimal generator of a
CQ-semigroup in C([-oo,oo]), the so-called Gauss-Weierstrass semigroup.
125
ACKNOWLEDGEMENT
The author would like to thank Ph. Clément for his valuable comments.
REFERENCES
1. Clément, Ph. and CA. Timmermans, On C~-semigroups generated by
differential operators satisfying Ventcel's boundary conditions.
2. Coddington, E.A. and N. Levinson, Theory of ordinary
differential
equations, McGraw-Hill, N.Y. (1955).
3. Feller,
W., The parabolic differential
equations and the associated
semigroups of transformations, Ann. of Math. 55, 468-519 (1952).
4. Lumer, G. and R.S. Phillips, Dissipative operators in a Banach space,
Pacific. J. Math. 11, 679-698 (1961).
5. Martini, R., A relation between semigroups and sequences of approxima­
tion operators, Indag. Math. 35, 456-465 (1973).
6. Protter,
M.
differential
and
H.
Weinberger,
Maximum
principles
in
partial
equations, Prentice-Hall, Englewood Cliffs, N.J. (1967).
7. Ventcel's, A.D., On boundary conditions for multidimensional diffusion
processes, Th. of Prob. Appl. 4, 164-177 (1959).
126
2.8. Dual semigroups in NBV(J), NAC(J) and L V J )
In this section we will derive a characterization of the adjoint operator A :
NBV(J) — NBV(J) of the operator A: C(J) — C(J) defined by 2.6, (1.3)-(1.4)
in an analogous way as in section 2.4.
We will prove the following theorem.
Theorem 2.8.1
Let X = C(J),
let a and 0 are continuous real functions defined on J,
satisfying
(aW)'1
G L^x^r.)
and
R £ L^x^r.),
(i = 1,2),
where W and Q are defined by (1.1.3) and (1.1.4), and let A: D(A) c X -» X
be defined as in Theorem 2 of section 2.6.
Then the adjoint operator A
of A with domain D(A) C X
= NBV(J)
is
given by
D(A*) = {w £ X* \ w e NAC(J), aw' G AC.
%
(aw ) • - Pw • - ((aw')'
(2.8.1)
- 0w )(xn)J G NBV(J)
Urn (aWw*)(x) = 0, i = 1,2},
A w = (aw') ' - /3w', for w G D(A ) .
Note that r . and r 7 are entrance boundary points.
(J),
%
0
127
Proof
We define an operator B with domain D(B) c X and range in X as follows
D(B) = {w £ X | w G NAC(J), aw G A C j ^ J ) ,
(aw • ) ' - /3w • - ((aw ') • - jSw ')(x 0 ) G NBV(J),
lim (aWw')(x) = 0, i = 1,2),
x-»r:
(2.8.2)
Bw = (aw ' ) ' - Bw',
for all w G D(B).
Thus D(B) is equal to the right-hand side of the first equality of (2.8.1). We
follow the line of the proof of Theorem 2.4.1.
Step 1
A is a linear m-dissipative
operator with domain dense in X. This follows
from Theorem 2 and Lemma 9 in section 2.7.
Step 2
B is a closed linear operator. Let (w ) be a sequence in D(B), converging in
-*
*
X to w, and assume the sequence (Bw ) converges in X to y. Then we have
to prove w G D(B) and Bw = y. The proof is similar to that of Step 2 in
Theorem 2.4.1.
Step 3
NBV*(J) is dense in X* and NBV*(J) C R(I - B).
Step 4
For all u G D(A) such that u - Au G C*(J) and all w G D(B) such that
w - Bw + Bw(xn) G NBV*(J) we have
<w,Au>* = <Bw,u>.
128
Let u G D(A) and w G D(B) such there is satisfied the assumptions. Let z := w - Bw.
Then there are constants a,b G E , such that r. < a < x,. < b < r» and
z(x) = z(a), w(x) = z(a) - (Bw)(xQ) + PJWJ
for all x G (r.,a], p . 6 1R
and
z(x) = z(b), w(x) = z(b) - (Bw)(xQ) + p 3 >y 3
for all x G [b,r 2 ), p 3 G 3R.
Here w and w are the special functions given in Lemma 1.3.6.
From the boundary conditions on D(B) we have
lim (aWw ')(x) = p
x—H2
lim (aWw )(x) = 0.
i
■* x—*r2
From Lemma 1.3.7 we have
lim (aWw')(x) = 1, so p . = 0. Analogously
i
x—>r2
~^
p. = 0. Thus, on (r. ,a] we have
w(x) = z(a) - (Bw)(x 0 ),
on [b,r.) we have
w(x) = z(b) - (Bw)(x 0 ).
As in the proof of Theorem 2.4.1 we have
lim (W
x—»rj
number. Then we have
<Bw,u>+=f
u ')(x) exists as a finite
udCW'^aWw')')
= u(x).W" ] (aWw ') '(x) ]
T2
r +
r1-
2
(W" l u ')d(aWw ')
- f
J
= uW.W'Vww'VW]^ .,. - (W'u'KxMaWw'Xx)]^
129
+ I aW(W" 1 u 1 )' dw
!-°*
(2.8.4)
= <w,Au>^
provided that
1
(2.8.5)
lim u(x).(W
x->r;
(2.8.6)
ii
lim (Wr i u„, )(x).(aWw')(x)
= 0,
x-»r;
( Q W W ' ) ' ) ( X ) = 0,
1
i = 1,2,
i = 1,2.
(2.8.5) follows from the boundedness of u and (2.8.3). (2.8.6) follows from the
boundedness of W
u ' and the boundary conditions on D(B). So we have
<Bw,u>* = <w,Au>*.
Now all conditions of Proposition 2.2.1 are satisfied and it follows that B = A .
This completes the proof of Theorem 2.8.1.
D
Analogous to Lemma 2.5.2 and (2.5.5) it is easy to prove that
X Q = D(A ) = NAC(J)
D(Aj) = {w G X* | aw • G A C 1 Q C ( J ) , ( a w ' ) ' - pw • e AC(J),
(2.8.7)
lim (aWw ')(x) = 0}.
A* w = (aw ' ) ' - 0w ', for all w G D(A*),
and
130
f X® = L 1 (J)
D(A®) = {v € X® | av e AC loc (J), (av) • - /3v e AC(J),
(2.8.8)
lim (aWv)(x) = 0}.
x-+rj
A®v = ((av) • - 0v) ',
for air v e D(A®).
Note that A® is an infinitesimal generator of a Q,-contraction semigroup T®
in L J (J).
2.9. "Bidual" semigroups in L (J)
It is known that X®* = (hl(J)f
= L°°(J). Analogous to Theorem 2.6.1 we will
prove the following theorem on the adjoint operator A® .
Theorem 2.9.1
Let AQ: D(AQ) c X® -» X® be defined by (2.8.8). Then the adjoint A®* of
A® is given by
D(A®*) = {zex®*\ze
(2.9.1)
C!(J),
z• e
aW(W~1zi)%
ACjJJ),
eL°°(J)}
A®*z = aW(W~]z V • = az" + pz '.
Note that from the definition of A, and thus from the definition of A® it
follows that r. and r- are entrance boundary points.
Proof
The proof is similar to the proof of Theorem 2.6.1.
We define the operator C with domain D(C) c X® and range in X® as
follows:
131
f D(C) = {z e X®* | z e c ' d ) , z ' e AC loc (J),
aW(W" 1 z 1 )' G L°°(J)}
(2.9.2)
Cz = aW(W" 1 z»)'.
Thus D(C) is equal to the right-hand side of the first equality in (2.9.1).
Since A® is an infinitesimal generator of a (^.-contraction group in X® this
operator is linear and m-dissipative. C is a linear operator and R(I - C) = X® .
This is easily seen as follows.
Let k e L
and let
z - aW(W
(2.9.3)
1
z ' ) ' - k.
Let the function z, be defined by (1.1.48). Then it follows from the proposi­
tion that z. G D(C). Thus the general solution of (2.9.3) is given by
z = zR + CJUJ + c 3 u 3 ,
CJ,C3GK
where u. and u , are given in Lemma 1.1.16. Since r. and r_ are entrance
boundary points, the functions u. and u . are unbounded on J, thusc. = c . = 0,
and (2.9.3) has a unique solution in D(C).
Finally, we will show that for all v such that v - A®v G C (J), and all z G
D(C):
<Z A V>
' °
0*
=
<Cz v>
- 0*-
Therefore assume z e D(C) and v - Av G C (J). Then we have
<Cz,v> 0 + - < z , A ® v > Q t =
= f
aW(W _ 1 z')'.vdx - f
z.(W ' ( a W v ' J ' d x
132
= [(aWv)(x).(W lz')(x)~]T2
- f
1
= [z(x).(W _ 1 (aWv)»(x)]' 2 ~ + f
(2.9.3)
=
W ^aWvJ'dz
JJ
W'VaWvVdz
lim (oWv)(x).(W" 1 z'Xx)- lim (aWv)(x).(W"'z ')(x)
x-+r2
x—»rj
-
lim z(x).(W
z(x).(W _1^(aWv)
a W v J '')(x)
X x ) *+ 1lim Z(X).(W" 1 (QWV)')(X)
x—»r2
x—>rj
provided that these limits exist.
In the next lemmata we will prove that these limits exist and are equal to zero.
Lemma 2.9.2
If r. is an entrance boundary point and v - /4®v G C (J), then
lim (W~1(aWv)%)(x)
x-*rj
= 0,
i = 1,2.
Proof
If v - (W" (aWv)')' = g G C (J), then there exists a b G J, such that for all
x G (b,r_): g(x) = 0. Let u . and u . be as in Proposition 1.1.1. From Lemma
1.2.5 we have
v1:=(aW)-1u1^
h\xQ,r2).
Then from Lemma 1.2.2 it follows that
v(x) = c Q v 3 (x) := c 0 (aW) _ 1 u 2 (x), cQ G 1R, for x G (b,r 2 ),
and from Lemma 1.1.16 with u . := u- that
lim (W" 1 (aWv)')(x) = cun
x-+r2
lim (W - 1 u')(x)
= 0.
z
x-»T2
~
133
The proof for r. is similar.
D
Lemma 2.9.3
If r. is an entrance boundary point, v - A®v G C (J) and z G D(C), then
(2.9.4)
lim (aWvXx).(W'1
z')(x)
= 0,
i = 1,2.
Proof
Assume v - A®v G C (J) and z G D(C). As in the proof of Lemma 2.9.2 there
exists a number b G ( x n , r . ) such that
v(x) = c 0 (( Q W)" 1 u 3 )(x),
cQGK.
Thus
(2.9.5)
lim (aWv)(x) = c n . lim u,(x) = c
x—>X2
x—»T2
Set g := a W ( W " 1 z ' ) ' , then g G L°°(J)
(W" 1 z , )(x) = z , ( x 0 ) +
Since (a\V)"
[
J x„
0
((aW) _1 g)(t)dt.
G L (x n ,r ) and g is bounded, it follows that
u
z
lim (W_ z')(x)
x—»r?
exists and
(2.9.6)
lim (W" 1 z , )(x) = z'(x f ,)+ f
((aW) _1 g)(t)dt.
x-r2
0
J xQ
Moreover
x
z(x) = ƒ
t
{z '(x 0 ) + I
((aW)" l g)(s)ds} W(t)dt + z(xQ).
Since W £ L (* 0 ,r 2 ) and z is bounded it is necessary that
134
f'
(2.9.7)
lim z ' ( x . ) +
t^r 2
-1
((oW)
°
g)(s)ds = 0
J xQ
Finally, (2.9.4) follows from (2.9.5)-(2.9.7).
The proof for r. is similar.
D
Returning to (2.9.3) it follows from the Lemmata 2.9.2 and 2.9.3 that
<Cz,v>Q* = <Cz,v>0!|t
for all v such that v - A®v G C (J) and all z G D(C). Then the conditions of
C
O*
Proposition 2.2.2 are satisfied and it follows that C = A
. Theorem 2.9.1 is
proved now.
D
0*
As in section 2.6 A
*
is a weak infinitesimal generator of the dual semigroup
T°* of T®. For the closed subspace X®® of X®*, on which T®* is strongly
continuous we have
X®® = D(A®*).
135
CHAPTER 3 - SATURATION PROBLEMS FOR BERNSTEIN
OPERATORS IN C ^ C U ]
3.1. Introduction
In this chapter we shall consider saturation classes for Bernstein operators in
various spaces. In section 3.2 we first consider saturation problems both
uniform and pointwise in C[0,1] for Bernstein operators. We will do this in a
unified way in the sense that the pointwise saturation theorem of Lorentz [Lo]
and the uniform saturation theorems ([BN], [LS]) are obtained as special cases
of two general theorems. The pointwise saturation class is well-known. The
uniform saturation class is investigated, in different ways by different
authors. Concerning the investigations of Becker and Nessel, [BN] and
Felbecker [Fe], there seems to be a mistake in the definition in the domain of
the limit operator
lim n(B
n—KX>
- I) and in the paper of Lorentz and Schumaker
n
[LS] there seems to be a mistake in Remark 2 on Theorem 4.3 and the applica­
tion with Bernstein polynomials. In section 3.2 these claims will be justified,
and an other approach to saturation problems will be given.
In section 3.3 we state and prove a variant of a result of Becker and Nessel
about uniform saturation. This theorem associates the saturation class of an
approximation process with the Favard class of the semigroup generated by the
operator occurring in the so-called Voronowskaya formula. The proof given
here is direct and does not use the Theorem of Trotter-Kato and the corre­
sponding semigroup.
In section 3.4 we use the theorem of section 3.3 in order to obtain the uniform
saturation class of the Bernstein operators for the supremum norm.
Finally, in section 3.5 we give a characterization of the uniform saturation
class for the Bernstein operators with respect to the usual C [0,1]- norm (and
thus for all equivalent norms). By a suitable transformation we can reduce this
problem to problem of uniform saturation of positive contractions in C[0,1].
Again we use the general theorem of section 3.3 and for the characterization
of the Favard class we invoke the results of Chapter 2.
136
3.2. A unified approach to pointwise and uniform saturation for Bernstein
polynomials
by
Ph. Clément and CA. Timmermans
Delft University of Technology
Report 86-45
1. INTRODUCTION AND MAIN RESULTS
In [9] Lorentz proved the following saturation theorem for the Bernstein
operators B n : C[0,1] -► C[0,1], n = 1,2,..., defined by
(Bnf)(x) = j
^ ( " J x k (l - x ) n " k f [±] ,
x e [0,1],
with f G C[0,1].
Theorem 1.1 (pointwise
saturation)
If f is a continuous function on [0,1], and M > 0, then
(1-1)
\(Bnf)(x)
- f(x)\ <
(M/n)a(x)
for all x G C[0,1], n = 1,2,3,..., is equivalent to
ƒ ' G C!(0,1) and
(1.2)
\f'(x)
(1.3)
- f'(y)\
< M\x - >i for 0 < x,y < 1.
Here a(x) = ~ x( 1 - x).
137
On the other hand Becker and Nessel [1] proved
Theorem 1.2 (uniform
saturation)
If f is a continuous function on [0,1 J, then
(1.4)
\(Bnf)(x)
- f(x)\ < M/n
for some M > 0 and all x e [0,1 J, n = 1,2,3
is equivalent to
a(x)\(D2hf)(x)\<M>
(1.5)
for some M' > 0 and for all x,h such that x+h,x-h E (0,1), h > 0.
Here
(D2hf)(x)
(1.6)
= h~2(f(x
+ h) - 2f(x) + f(x - h)),
and a(x) as in Theorem 1.1.
The goal of this paper is to provide a unified approach to these two saturation
problems. In order to do that we first need to compare conditions (1.2) and
(1.5). This is done with the next lemma.
Lemma 1.3
Let "if be a non-negative
continuous concave function on [0,1 J and let f be a
continuous function on [0,1 J.
a. If f
satisfies
r
a(x) \D2hf(x)\ < *(x)
(1.7)
for all x,h such that x-h,x+h e (0,1), h > 0
then the following condition holds
138
fee1 (0,1)
(1.8)
X
\f'(x)
- f*(y)\<
y(t)>OL~1(t)dt,forallx,ymthO<y<x<l.
f
Jv
b. If (1.8) holds, then we have
(1.9)
a(x) \D2hf(x)\ < 29(x)
for all x,h such that x-h,x+h G (0,1), h > 0.
We give a proof of this lemma in the Appendix.
In Theorem 1.4 we will show that pointwise and uniform saturation are special
cases of a general saturation problem.
Theorem 1.4
Let ty be a non-negative
continuous concave function on [0,1]. If f is a
continuous function on [0,1 ] satisfying
(1.10)
\(Bnf)(x)-f(x)\<j;V(x)
for x G [0,1 J, n > 1, then f satisfies
(1.8).
Remark. If * = a we recover one part of Theorem 1.1 and if * = M we get by
using Lemma 1.3 one part of Theorem 1.2 with M ' = 2M.
For the converse we have
Theorem 1.5
Let * (x) = ap(x), x G [0,1], 0<p<l.lffe
C[0,1] n C^O.l) and satis­
fies
X
(1.11)
\f'(x)
- f'(y)\
a1(t)^(t)dt
< \
y
for 0 < y < x < 1, then
139
(1.12)
\(Bnf)(x)
21-p^p(x)/n
- f(x)\ <
for O < x < 1 and n > 1.
Remarks. If p = 1 we recover the second part of Theorem 1.1. If (1.5) is
satisfied with M ' = 1 then it follows from Lemma 1.3 that (1.11) holds with
* = *
= 1. Then from Theorem 1.5, with p = 0, we obtain (1.4) with M = 2.
Hence we also recover part two of Theorem 1.2.
1
n
One may ask whether the constant 2
in Theorem 1.5 could be 1 in order to
have a true converse of Theorem 1.4 with * = * .
P
This is not the case as it is shown in section 5. Concerning the proofs, we
partly follow the analysis of [1] for Theorem 1.5. As in Micchelli [12], we
introduce the semigroup generated by the iterates of Bernstein operators. In
section 5 we mention a mistake occurring in the literature (cf. [1], [6]) con­
cerning the definition of the domain of the infinitesimal generator of this
semigroup.
2. PROOF OF THEOREM 1.5
As observed in [2], we may restrict ourselves to functions in C[0,1], which
vanish at the endpoints, since the Bernstein operators preserve affine func­
tions. Moreover, since (B f)(0) = f(0) and (B f)(l) = f(l) for all f G C[0,1] it
is sufficient to prove (1.11) for x e (0,1). Let p e [0,1] and
let
f e C [ 0 , l ] n C ! (0,1) satisfy (1.11) and f(0) = f(l) = 0.
Then
|f'(x) - f'(y)| < [
J
a P _ 1 (s)ds, for 0 < y < x < 1.
y
It follows that f' is locally Lipschitz continuous on (0,1), therefore f"(x) exists
a.e. on (0,1) and |f'(x)| < a p - 1 ( x ) a.e. on (0,1).
Let the kernel k e C([0,l]x[0,l]) be defined by
140
t ( l - x),
0 < t < x < 1,
x ( l - t),
0 < x < t < 1.
k(x,t) = .
If f is as above and 0 < x < 1, we have
k(x,t) |P(t)| < 2
almost everywhere in t on (0,1), and thus k(x,»)f"(«) is integrable on (0,1).
Then it is easily verified that
(2.1)
f(x) = - f
k(x,t).f"i(t)dt
Jo
and
(2.2)
(B n f)(x) = - J
(B n k(.,t))(x>P(t)dt, for 0 < x < 1,
and n = 1,2,3,.
Hence we obtain
(2.3)
f(x) - (Bnf)(x) = J
(k(x,t) - (B n k(.,t))(x)) a P _ 1 (t)dt,
for 0 < x < 1.
Note that k(x,t) - (B k(«,t))(x) is non-negative since k(»,t) is concave for all t,
[4]. From (2.1) and (2.2) with f = a and the fact that a - B a = —, we have
n
n '
(2-4)
J
{k(x,t)-B n k(.,t)}(x)dt = ^ - , x e ( 0 , l ) .
Since for each x e (0,1)
J
na
(x) f
{k(x,t)- Bnk(.,t)(x)}-dt = 1
it follows from Jensen's inequality for integrals with 1-p e (0,1) that
141
na
-1 (x) j
1
{k(x,t) - B n k(.,t)(x)}(a- 1 (t) 1 _ P dt
O
< {na _ 1 (x) |
(k(x,t) - Bnk(.,tXx)}a" ^ O d t } 1 " "
= ( n a - ^ x W ^ ^ K X x ) - K(x)} 1 _ P ,
(2.5)
where K e C[0,1] is defined by
K(x) = 2x log x + 2(1 - x) log (1 - x), 0 < x < 1, K(0) = K(l) = 0,
by observing that K" = a
Thus, from (2.3) and (2.5) we get
(2.6)
|f(x) - (B n f)(x)| < [ ^ - )
P
.{(B n K)(x) - K(x)) 1 _ P .
In order to estimate the right-hand side of (2.6) we need the following.
Lemma 2.1
For all x e f 0,1] and all n G U
(2.7)
n((BnK)(x)
- K(x)) < 2.
Proof
Since (B K)(x) - K(x) = (B K)(l - x) - K(l - x) it is sufficient to restrict
ourselves to x e [0,1/2]. For x = 0 the left-hand side of (2.7) vanishes, so (2.7)
is trivially satisfied.
For x e (0,1/2] we define the function
Fx:[0,l]-K
by
2
F x (t) = ^ " ^
- {{K(t) - K(x) - (t - x)-K '(x)}.
142
Then
F x (0) = J^J - \(-K(x)
Fx(l) =
I
+ xK '(x)} = j
^ + log (1 - x) > 0
^+logx>0
F (x) = 0.
xv '
Further,
■^-«-Wi-*».
thus
F '(x) = 0, Hm Fx'(t) = oo and lim Fx '(t) = -oo.
x
x->lt_o+
2
F"(t)
xv ' = x(l - x)
1
t(l - t) '
so F"(x) > 0, and thus F has a minimum in x. Observe that F" has two zeroes
x
x
x
x. and x_, with 0 < x. < x < x« < 1. It follows that F is concave on [0,x.],
strictly convex on ( x . , x . ) and concave on [x„,l]. Thus F possesses only one
local minimum on (0,1), which is taken in x and has the value zero.
It follows that for each x e (0,1/2], F is a non-negative function on [0,1],
and since B
is a positive.operator, that B F is a non-negative function.
Especially,
(B n F x )(x) = £ - | (B n K(x) - K(x)) > 0.
This proves the lemma.
D
Returning to the proof of Theorem 1.5, from (2.6) and (2.7), (1.12) immedi­
ately follows.
D
143
3. T H E SEMIGROUP G E N E R A T E D BY THE ITERATES O F BERNSTEIN
OPERATORS
2
It is w e l l - k n o w n that for u G C[0,1] n C (0,1)
lim n(I - B n )u(x) = -a(x)u"(x), x e (0,1).
n—»oo
This is the so-called Voronowskaya formula, see ([8], p. 22).
Moreover it is known ([2], p. 703) that for f,g G C[0,1] the following holds
lim
n—KX)
||n(B il f - f) - g|| = 0,
2
where || || denotes the supremum norm, if and only if f e C (0,1),
X(1
2 X) r'(x) = 8to, x
G
«M>.
and
lim g(x) = lim g(x) = 0.
x-»0
x-*l
(For the sake of completeness we give a proof of the if-part in Step 3 of the
proof of Theorem 3.1). Therefore it is natural to define the operator
A: D ( A ) C C [ 0 , 1 ] ^ C [ 0 , 1 ] , by
D(A):= -fu G C[0,1]| u e C 2 ( 0 , l ) a n d lim a(x)u"(x)
^
x—0
= lim a(x)u"(x) = o }
x—>1
J
Au(x) = -a(x)u"(x), x G (0,1), for u G D(A).
It is known ([11],[3]) that -A generates a positive contraction C„-semigroup
on C[0,1] (equipped with the supremum norm), which we shall denote by T(t),
t > 0. Then we have
144
Theorem 3.1
Let f be in C[0,1], then
\\B[nt]f
Urn
max
^°°
tefOJJ
n
n
- T(t)f\\ = 0
for every T > 0, where {T(t), t > 0} denotes the semigroup generated by -A
and [nt] denotes the largest integer not larger than nt.
Proof
We divide the proof in 4 steps.
Step 7. Let f e C 2 [0,1]. Then
|n(I - B n )f(x)| < a(x)||P||,
x G (0,1)
by Theorem 1.1. Thus
|n(I - B n )f(x) - Af(x)| < 2a(x)||ni,
x e (0,1).
2
1
So for f e C [0,1] and for each e > 0 there exists a 5 with 0 < 8 < y , such
that for x e [0,5] u [1-5, 1] and for all n > 1,
|n(I - B n )f(x) - Af(x)| < e.
Moreover, it follows from ([9], Th. 2) that for n sufficiently large
|n(I - B )f(x) - Af(x)| < €, for x [5, 1-6].
Then we have for all f e C 2 [0,1], lim
||n(I - B )f - Af|| = 0.
n
n—too
Step 2. Next we show that C [0,1] is dense in D(A), with respect to the graph
norm of A:
2
Hu||A := ||u|| + ||Au||, for u e D(A).
145
2
Given u e D ( A ) , if suffices to construct a sequence (u ) c C [0,1], such that
(3.1)
lim ||u - u|| = 0
m—>oo m
and
Following [5], we define the functions u
fu(1
^)
+ u
W' x e &
u(l-l)
*-
(m = 3,...) on [0,1] by
.(l).Cx-l)+l»-(ii).(x.ii)2.xe[0l1
U
u (x) =
nr '
lim ||Au - Au|| = 0.
m
m—too
+ U
J
' m)
'(l-i-].(x-l+-^)
m-*
v
mJ
+
mJ
+
|u"(l-^).(x-l
+
-^)
2
,XG[l--L,l].
An easy computation shows that (u ) satisifes (3.1).
Step 3. Next we prove that
lim ||n(I - B )f - Af|| = 0
n—*oo
for all f e D(A). It follows from Theorem 1.5 that ||n(I - B )f|| < 2 ||Af||
every f e D(A) and thus
||n(I - B n )f|| < 2 ||f|l A
for every f e D(A).
We see that the operators, n(I - B ) are uniformly bounded on (D(A), ||«|| A ,
n
A
while lim n(I - Bn )f = Af for f belonging to the dense subset C [0,1] of
n—«x>
(D(A), ||.|I A ).
It follows that
146
lim ||A n f - Af|| = O
n—>oo
for all f e D(A).
Step 4. Now we define the family of contractions {V(t); t > 0} in C[0,1] by
means of
V(0) = I,
V(t) = B [ 1 / t ] , f o r t > 0 .
Then Step 3 yields
lim i|t _1 (I - V(t)f - Af|| = 0
U0
for all f e D(A).
Then from Chernoff's product formula (see [7], Th. 8.4) we have for each
feC[0,l]
lim ( v ( n^ )
n—«x>
n
f = T(t)f
uniformly for t in compact subsets of ]R , where (T(t); t > 0} is the semigroup
by -A. It follows that for each f e C[0,1]
lim
m—»oo
||B™m
l /il
f - T(t)f|| = 0.
Let T > 0, 0 < t < T and set [■"] = n, then
[nt] < m < [nt + t] < [nt] + [t] < [nt] + [T].
Since lim Bnk f = f, k = 1,2,...,[T], it follows that lim ||Bn[ n t ] f - T(t)f|| = 0
n—»oo
n—*oo
for all f e C[0,1]. This proves the theorem.
147
4. PROOF OF THEOREM 1.4
As in Micchelli [12] the proof of Theorem 1.4 will be done in two steps.
First we show that if f and \P are as in Theorem 1.4 then
(4.1)
|f - T(t)f| (x) < ttf(x), x e (0,1) and t > 0.
Then we show that (4.1) implies (1.3). Thus we obtain a generalization of the
results of [12], where * = M > 0.
For k e l
we have
k l
k
Bf-f=
~
V
to
I
B (B f - f), w i t h f e C [ 0 , l ] .
nn
t
Since B are positive operators, we obtain by (1.10)
|B„f-f|<
E |B(Bf-OI<^
E B;*.
£=0
£=0
Since * is positive and concave,
k-1
([4]), hence
B
n
k-7
<B
n
< ... < B * < »,
n ~ '
k
|B
f - f| _< - * .
1
n
' n
By choosing k(n) such that lim
n—»oo
(4.2)
|T(t)f - f| < t*,
^ ' = t, it follows from Theorem 3.1 that
n
t > 0.
For s > 0, we define
f :
S fdl
s -7j„ ™ -
148
It is well-known that f e D(A). Moreover
(4.3)
Af - -f(I - T(s))f, and lim ||f - f|| = 0.
s
s
s
sjO
From (4.2) and (4.3), we have
|Afs(x)| < *(x),
xG(0,l).
Thus
(4.4)
a _ 1 *(t)dt,
a(x).|P(x)| < *(x) on (0,1), and |f'(x) - f'(y)| < f
S
S
S
J y
holds for 0 < y < x < 1.
Since f converges uniformly to f, it follows from (4.4) that (f') is bounded
S
ï*
and uniformly equicontinuous on [e, 1 - e], for each e e (0,1/2).
Thus by Arzela-Ascoli's theorem, we obtain
|f'(x) - f'(y)l < [ a - 1 tt(t)dt, for 0 < y < x < 1.
Jy
This completes the proof of Theorem 1.4.
5. REMARKS
1. An easy calculation shows that
(5.1)
(BjK) ( y ) - K ( y ) = log 4 = 1,386294
It follows that the smallest value of the constant M, such that
(5.2)
0 < n(B K(x) - K(x)) < M
holds for each n and each x e [0,1], is strictly larger than 1. From Lemma
2.1 and (5.1) we see that for this smallest value Mft
149
log 4 < M < 2.
This result improves the result of Berens and Lorentz ([2], Lemma 3) who
proved (5.2) with M = 7. Although we suspect that M„ = log 4 we did not
succeed in giving a proof of this fact until now.
1 r\
2. We moreover see from (5.1) that the constant 2
in Theorem 1.5 cannot
be replaced by 1 as it is claimed in [10], p. 422. Thus Theorem 1.4 has not
a true converse.
3. It is worthwhile to note that it is claimed in [1], p. 39 and [6] that if
f 6 C[0,1] and x(l - x)f"(x) is bounded on [0,1], then lim ||A f - Af||
n—*oo n
°°
= 0. However since n(B f(0) - f(0)) = n(B f(l) - f(l)) = 0, this cannot be
true if
lim x(l - x)f'(x) # 0 or lim x(l - x)f"(x) * 0. The function
x-»0
x->l
f = K (from Lemma 2.1) yields a simple counterexample.
6. APPENDIX
Here we will give a proof of Lemma 1.3.
Part (a). Suppose f e C[0,1] satisfies (1.7). Then
(6.1)
|(D2f)(x)| < ^ f 4 , h>0
1
n
/v
"
a(x) '
for all x,h such that x-h,x+h e (0,1). As in [1] we define
f h (x) := - y
f
_h
2
h
_h
rl
f h f(x + s + t)dsdt
(h > 0),
then f^(x) = (D^f)(x). Thus, with y < x,
X
(6 2)
|fj|(x) - f^(y)| < J
X
|f^(t)|dt = ƒ |(Dj;f)(t)|dt < ƒ
X
*(t)a_1(t)dt.
150
Let [a,b] c (0,1), a < y < x < b. Since *a~
is bounded on [a,b], the set
F = {f' | h > 0, 0 < a - h, b + h < 1} is uniformly equicontinuous on [a,b].
Moreover, f = converges uniformly to f on [a,b] if h tends to zero.
From the theorem of Arzela-Ascoli it follows that there is a subsequence in F
converging uniformly on [a,b] to a g G C[a,b], and since f, (x) - f,(a) =
f'(t)dt, we obtain from the limit of this expression that f G C (a,b), and
fj(x) converges to f '(x) for each x G (a,b). Taking the limit in (6.2) we obtain
(1.8) for x,y G (a,b). Since [a,b] is arbitrarily chosen inside (0,1), (1.8) holds
for x,y G (0,1) with 0 < y < x < 1.
Part b. Suppose f G C^O.l) and (1.8) holds. Then for each [a,b] c (0,1),
f G C [a,b] and f' G AC[a,b]. Hence f • is differentiable a.e. on [a,b] and
|f"(x)| <*(x>a _ 1 (x)
a.e. on [a,b].
Since [a,b] is arbitrary
|f'(x)l <*(x)a _ 1 (x)
a.e. on (0,1)
or
a(x)-|f"(x)| < *(x).
Now, with x-h,x+h G (0,1), h > 0
a(x) |(D^f)(x)| =
= " ^ - |f(x + h) - 2f(x) + f(x - h)|
h
(h - u){f"(x + u) + f"(x - u)} du
151
x(l -JÜLL
- x) f" h ( - u) I
* ( x + U)
h
(h
U;
J0
l(x + u)(l - x - u)
h2
^(x - u)
1 .
+7
\Tt—
\ r du(x - u)(l - x + u)J
With the elementary inequalities, (see [ 1 ]), for O < u < h, x - h > O, x + h < 1,
x
x +u
1 - x
1-x+u
h - u
1 - x - u •(1 - x ) < h ,
h - u
x - u • x < h,
and
*(x + u) + *(x - u) < 2*(x)
we obtain
a(x) |(DjJf)(x)| < -£- f
<
i
i
.h
{tf(x + u) + *(x - u)} du
¥ Ioo 2*(x)du
= 2*(x)
This completes the proof of the lemma.
152
[1] M. Becker and R.J. Nessel, Iteration von Operatoren und Saturation in
lokal konvexen Raümen, Forschungsbericht des Landes Nordhein-Westfalen, Nr. 2470, Westdeutscher Verlag Oplagen, 1975, pp. 27-49.
[2] H. Berens and G.G. Lorentz, Inverse theorems for Bernstein poly­
nomials, Indiana Univ. Math. J., 21 (1972), pp. 693-708.
[3] Ph. Clément and CA. Timmermans, On C n -semigroups generated by
differential operators satisfying VentceFs boundary conditions, Indag.
Math., 89 (1986), pp. 379-387.
[4] P.J. Davis, Interpolation and approximation, Blaisdell Publ. Company
(1963).
[5] C.J. van Duijn, personal communication.
[6] G. Felbecker, Linearkombinationen von iterierten Bernsteinoperatoren,
Manuscripta Mathematica, 29 (1979), pp. 229-248.
[7] J.A. Goldstein, Semigroups of linear operators and applications, Oxford
University Press (1985).
[8] G.G. Lorentz, Bernstein polynomials, Univ. of Toronto Press (1953).
[9] G.G. Lorentz, Inequalities and saturation classes of Bernstein poly­
nomials. In: On Approximation
Theory,
pp. 200-207, Proc. Conf.
Oberwolfach, 1963, Birkhaüser Verlag, Basel (1964).
[10]
G.G. Lorentz and L.L. Schumaker, Saturation of positive operators,
Journ. of Approx. Theory, 5 (1972), pp. 413-424.
[11]
R. Martini, A relation between semigroups and sequences of approxi­
mation operators, Indag. Math., 35 (1973), pp. 456-465.
[12]
C. Micchelli, The saturation class and iterated of the Bernstein poly­
nomials, J. Approx. Theory, 8 (1973), pp. 1-18.
153
3.3. Saturation and Favard classes
Let X be a real Banach space and let (L ) be a sequence of contractions in X ,
strongly convergent to the identity operator I in X. Moreover, let us assume
there exists a positive function <j>: U —♦ UN with lim
n-+oo
defined m-dissipative operator A such that
<£(n) = oo and a densely
lim
|Wn)(L n f - f) - Af|| = 0
n—>oo
for all f e D(A) C X.
Let us define for f e X
A Q f := 0(n)(L n f - f),
n=l,2,....
It is easy to prove (see [CH], Lemma 2.9) that for each n £ {1,2,...} A
is a
bounded m-dissipative operator.
In the next theorem it will be shown that if
lim 0(n)(L n f - f) exists,
n—voo
then f G D(A) and that
sup fln) HflnXL f - f)|| < oo
n>l
if and only if f £ Fav(T^), where T ^ is the semigroup with infinitesimal
generator A. It follows that Fav(T^) is the saturation class of the approxima­
tion process (L ) (see the Introduction). Indeed, if
tf(n)(Lnf - f ) = o(l),
n^oo,
then we have f £ D(A) and Af = 0. Moreover,
154
|Lnf-f|| = O(^n)'])
if and only if f e Fav(T A ). Thus, if A # 0, then N(A) c Fav(T A ), and
N(A) ^ Fav(T A ), where N(A) denotes the null space of A.
Theorem 3.3.1 is a variant of a theorem of Becker and Nessel ([BN], Satz 3.6).
They proved their theorem by means of the Theorem of Trotter ([Tr]).
Here we shall give a direct proof without using the iterates of L and without
using the semigroup generated by A.
We shall use Theorem 3.3.1 in the last sections in order to find the saturation
class of the Bernstein operators in the usual C [0,l]-norm (m = 0,1,2,...).
There we shall use the fact that
(3.3.1)
Fav(T A ) - D(A 0 *) n X
(see Proposition 2.1.3) and the results of Chapter 2 concerning D(A® ).
Theorem 3.3.1
Let X be a Banach space. Let A: X —
► X be linear and m-dissipative
with
dense domain D(A) and let for n = 1,2,... A : X —
► X be a linear, bounded
n
and dissipative operator with D(A ) = X. Then we have: if
(3.3.2)
Urn A x = Ax,
n—Hx n
then
(i)
\x G X \ Urn A n x existsf
^
(ii)
for all x e D(A),
\x&X\
*■
n—►oo
= D(A)
J
sup \\A x\\ < oo} =
J
nëN
{x G X | sup \\A,x\\ < oo} =
*■
where Ah := h'^E*
h>0
}
- I), E^ ; - (I - hA)"1, h > 0.
Fav(TA),
155
Proof
(i)
Assume A and (A ) satisfy the conditions of the theorem and assume
that (3.3.2) holds. Define the operator B: D(B) c X -+ X
' D(B) : = | x e X |
(3.3.3)
Bx
:= lim
n-K»
lim
A nx ,
A x exists}x 6 D(B).
By the dissipativity of A we have for all x e D(B)
|x|| < ||x - AA x||,
for all A > 0.
By taking the limit for n —» oo it follows that
||x|| < ||x - ABx||,
for all A > 0
and all x e D(B), thus B is dissipative.
From (3.3.3) it follows that
A C B.
Since A is m-dissipative, hence maximal dissipative (see [CH], Prop.
3.8) we have A = B.
This completes the proof of (i).
(ii)
(3.3.4)
We recall (see [CH], Prop. 3.18) that
Fav(T A ) = {x G X | sup \\A x|| < oo}.
L
h>0
J
Assume x e X such that there exists a positive number M. for which
(3.3.5)
l|Anx||<M,
n=l,2,.
156
Firstly we will prove that for each n A is m-dissipative. Let n e ]N.
n
Since A
is dissipative we have only to prove that I - hA is surjective for all
h > 0. It is sufficient to prove that R(I - h_A) = X for some h„ > 0 (see e.g.
[CH], Prop. 3.7). Let us take h„ > 0 such that
h0HAnl|<l.
We consider the equation
(3.3.6)
v
x - hAA x = f,
On
f e X.
We define the operator T: X —» X by
Tx := h~A x + f.
0 n
Then for all x,y e X we have
||Tx - Ty|| < h 0 ||A n x - A n y||
< h 0 ||A n ||.||x - y||.
By the Banach Contraction Theorem it follows that there exists a fixed point x
such that Tx„ = x_. So x~ is a solution of (3.3.6). This proves the surjectivity
of I - h A A .
0 n
Secondly, we will prove that x G Fav (T^), or with (3.3.4)
sup \\A, x|| < oo.
h>0
n
l
n
Set A , := h ' W
- I), where E, n := (I
v
v - hA ) , h > 0 and ||E. | | < 1
n,h
h
'
h
n' '
" h "~
by the dissipativity of A . Then
157
A , = (-h
n,h
-1
+h
A
-1
I
= (-h
A
= E,
h
n
-1
A
E,
h
n
)x
n
n
Eh .(I-hAn) + h
-1
J
A
n
EhV
Ax.
n
So
"Vhx|l = l | E h n Vi'
<||E h n ||.||A n x||
< l|A n x||
<Mj
An
by assumption (3.3.5). It follows from (3.3.2) that E,
^
x converges to E, x,
therefore that A , x converges to A.x. Thus \\A,x\\ < M. and then from
(3.3.4) we see x e Fav (T^).
For the converse suppose x e Fav(T^). Then by (3.3.4) there exists a positive
number M- such that for all h > 0
P h x | | < M2.
A
Since E, x = x + hA, x we have
h - h| | E A x - x|| < M 2 h.
(3.3.7)
Let X ^ be the Banach space D(A) equipped with the graph norm 11-11™ A Y
Since (A ), n e 3N,
U , is a seq
sequence of bounded linear operators in X we have for
all x G X A and all n e 3N
HAnx|| < ||A n l|.||x|| < HA n l|.||x|| D(A) .
Thus for each n 6 3N, A is a bounded linear operator from X A into X. Since
for each x e X A
lim A x exists it follows that for each x e X A
n-+oo n
158
sup ||A x|| < oo.
n
nëlN
Then by the uniform boundedness principle there exists a constant M . such
that for all x e X A
HAnx|| < M 3 l|x|| D ( A )
(3.3.8)
= M 3 (||x|| + ||Ax||).
For x e Fav(T^) we obtain
||A n x|| < HA n E^x|| + ||A n x - A n EJ^x||
< M 3 (||E*x|| + ||AE^X||) + ||A n ||.|| x - E ^ x ||
< M 3 (||x|| + |M h x||) + ||A n || M 2 h
<M3(||x||
+
M 2 ) + ||A n ||.M 2 h.
Since the left-hand side is independent of h we have
||A n x|| < M 3 (||x|| + M 2 ),
thus also
sup ||A. x H < ob.
neU
n
This completes the proof of the theorem.
(by (3.3.8))
(by (3.3.7)),
159
3.4. Application: Uniform saturation class for Bernstein operators on CfO, 11
Let (B ) be the sequence of Bernstein operators in C[0,1]. As well-known for
n
k
each n e ]N B is a positive operator and B e« = efl. Here e, (x) = x for all
x e [0,1], k = 0,1,... .In section 2.3 we proved that
lim
n—HX>
n(B n f - f) = Af,
for f € D(A),
where A is defined by
' D(A)= {f e C [ 0 , l ] | f e C 2 (0,1),
lim x(l - x)f"(x) = lim x(l - x)f"(x) = o \ ,
x->0
x—1
J
(Af)(x) = x(l - x)P(x)/2, for f e D(A).
Let for n G 1
A : C[0,1] -» C[0,1] be defined by
A n f = n(B n f - f),
feC[0,l],
where (B ) is the sequence of Bernstein operators on C[0,1]. Since the Bern­
stein operators are contractions in C[0,1], it follows that A is m-dissipative in
C[0,1], (see the discussion preceding Theorem 3.3.1).
The operator A is a special case of the operators investigated in section 2.3,
[CT,1], with a(x) = <f>(x) = x(l - x)/2, ^(x) = 0, J = (0,1).
With the notations of Chapter 1 we find after calculations with x_ = 1/2
W(x)= 1,
Q ( x ) = ( 2 x - l)/(x(l - x ) ) ,
R(x) = 2 log x - 2 log (1 - x).
So
Q £ L ^ O . l ^ ) , R e (0,1/2)
Q £ \}(\/2,\\
R e (1/2,1)
160
and it follows that 0 and 1 are exit boundary points. Then it follows from sec­
tion 2, Theorems 1 and 2 that A is m-dissipative with domain D(A) dense in
C[0,1], and thus A is the generator of a (^-contraction semigroup on C[0,1].
Therefore the conditions of Theorem 3.3.1 are satisfied. It follows that
(i)
(ii)
D ( A ) = { f e C [ 0 , l ] | lim n(Bn f - f) exists}.
^
n—«x>
J
The saturation class of the sequence of Bernstein operators (B ) is
Fav(T^), i.e. the Favard class of the semigroup on C[0,1] with
infinitesimal generator A.
As an application of the theory developed in sections 2.4-2.6 we will deter­
mine the saturation class of the sequence of Bernstein operators by means of
Proposition 2.1.3, which says
Fav(T A ) = X n D(A 0 *).
From Theorem 2.4.1 we obtain
' X * = l x NBV(0,1) x TR, xQ = 1/2, flx) = x(l - x)/2,
D(A*) = { ( w r w , w 2 ) G X* | w G NAC(J), 0 w ' G A C ^ J ) ,
( ^ w ' ) ' - ( ^ w ' ) ' (1/2) G NBV(0,1)|-,
A*( W j,w,w 2 ) = (0, Ww')», 0),
By Lemmata 2.5.1 and 2.5.2 we have
for all (Wj,w,w 2 ) G D(A*).
161
f X*Q = M x NAC(O.l) x ÜR, xQ = 1/2,
D(A*) = { ( w r w , w 2 ) e X*Q | w G NAC(O.l), ^w • G AC1(JC(0,1),
W w ' ) ' G AC(0,1)\
[
A
0 ( w r w , w 2 ) = (0, ( ^ w 1 ) ' , 0),
forall(wrw,w2)GD(A*).
By means of the mapping I : X j - X 0 = ] R x L°°(0,1) x JR defined in (2.5.2)
we obtain by (2.5.5)
f X 0 = n R x L](0,l)xIR,
D(A®) = {(Vj,v,v 2 ) G X® | ^v G AC l Q c (0,l),
(0v)' GAC(0,1)}
L A®( V l ,v,v 2 ) = ((^v)'(0 + ), (*v)", (tfv) '(I-)).
Finally, by Theorem 2.6,1 we obtain
f X®* = 3 R x Loo.
^(0,l)xlR,
D(A®*) = { ( z r z , z 2 ) G X®* | z G C^O.l),
z » G A C l o c ( 0 , l ) , «^ Z »GL W (0,1),
lim z(x) and lim z(x) exist,
x-»0
x-+l
lim z(x) = z.l and lim z(x) = z-2 V
x->0
x—1
J
[ A® (z 1 , Z ,z 2 ) = (0,^z",0).
With the identifieation £ defined a second representation for A® , namely
162
f X©* = L°°[0,1]
D(A 0 *) = {z e L°°[0,l] | z e c\o,A)
n C[0,l],
z'6ACloc(0,lUz"6L
A®*z = </>z",
oo r
[0,1]}
for all z G D(A©*).
Thus with (3.3.1) and Theorem 3.3.1 we have for the saturation class of the
Bernstein operators:
Theorem 3.4.1 (flvf, Th. A)
For f e CfOJJ, (i) and (ii) are equivalent, where
= 0(n-1),
(V
\\f-Bnf\\00
«-co,
(ii)
f e C-(0,1), ƒ ' 6 AChc(0,l),
<t>f" e
L°°(0J).
Remark: If f belongs to the saturation class of the sequence of Bernstein
operators, then f • does not necessarily belong to AC(0,1). The function K,
defined by
K(x) = 2x log x + 2(1 - x) log (1 - x),
K(0) = K(1) = 0
introduced in section 2.1 belongs to the saturation class of the Bernstein
operators. How<
However K '(x) = 2(log x - log(l - x)). Thus K ' e AC.
K ' £ AC(0,1).
(0,1) and
163
r
3.5. Uniform saturation classes for Bernstein operators in C-,m[0,1]
norms
In this section we will introduce a class of positive operators approximating the
identity, constructed from the Bernstein operators. We will investigate a
number of its properties and determine its saturation class with the help of the
theory of the Chapters 1 and 2 and Theorem 3.3.1. This class will be used for
the determination of the uniform saturation class of the Bernstein operators in
C m [0,l].
Let m be a positive integer, then we define the differential operators D
(k = l,2,...,m) by
D 1 = D: C ^ O . l l - ^ Q O . l ] ,
Df = f',
and
Dk:Ck[0,l]-C[0,l],
D k f = D(D k ~ 1 f),
k = 2,...,m.
Moreover we define the operators D~ (k = l,2,...,m) by
D'^QOJl-C^O.l],
(D_1f)(x)= [ f(t)dt,
Jo
and
D"k:C[0,l]^Ck[0,l]
D" k f = D _ 1 ( D " k + 1 f ) ,
Note that for f 6 C[0,1] we have
D m D " m f = f,
k = 2,...,m.
164
and that for f G C m [0,l]
D
D f - f is a polynomial of degree < m.
Next we define the sequence (F
), n = 1,2,..., by
Fm,„=C[0,l]-C[0,l]
(3.5.1)
F
m,n
f = DmB
n
D" m f,
'
where B is the n-th Bernstein operator.
For investigations of this sequence the following lemma is useful:
Lemma 3.5.1 f f Da I, Th. 6.3.3)
Let m and n be positive integers with n > m. If there are constants p and P
such that
P < (Dmf)(x)
<P,
0<x<
1,
then
_ m (n - m)! ,nm„ ,:, . n n
P<n J
j - i - (D B f)(x) < P, 0 < x < 1.
n!
n
' - •
D
In the next proposition we list some properties of the sequence (F
).
Proposition 3.5.2
Let the sequence (F
) (with fixed m > 1) be defined on the Banach space
C[0,1 ], equipped with the supremum norm, by (3.5.1), then
(i)
for each n, F
(ii)
for each
is a positive contraction on C[0,1J,
f&C[0,l]
Urn F
f = f,
„ - o o m,n'
(Ui)
for each n: n(F
1
m,n
- I) is
m-dissipative.
165
Proof
(i)
Assume f > 0 and let u := D" f, then u > 0 and D u > 0. Applying
Lemma 3.5.1 with p = 0 we obtain D m B u > 0 and then
n ~
F
m,n
f = D m B D" m f = D m B u > 0.
n
n
Thus F
is a positive operator.
m,n
_
_1
Moreover F
e„ = D B D~ e„ = D B u with u = v(m!) e . Then
m,n 0
n
0
n m
m
'
m
D u = e n . If we apply: Lemma 3.5.1 with P = 1 we obtain
m m
0
,m (n-m)!(DmB
u
)(x)
^ ,
Since
n
-m
n!
,
.—,
r. < 1
(n - m)
we get
|F
e J | = | | D m B u || < 1.
m,n 0" "
n m" ~
Thus for all f G C[0,1],
l|F
m,nf|1 * "f||-||Fm,ne0" <
^
So F
is a contraction.
m,n
(ii)
Let f G C[0,1] and let u = D _ m f . Then u G C m [ 0 , l ] and it follows from
[Lo], par. 1.8, that
lim D m B n u = D m u
n—*oo
or
lim
n—KX>
DmB n D _ m f = D m D " m f = f
in the uniform norm topology.
166
(iii)
This is a direct consequence of the discussion preceding Theorem 3.3.1.
D
In what follows we denote n(F
operators G
and C
- I) by C
m,n
in C[0,1] by:
. Moreover, we define the
m,n
D(G m ) = { f e C[0,1] | f e C 2 (0,1), 4>(" + m^'f • e C[0,1]},
where ^(x) = x(l - x)/2,
G f :=^f" + m^'f • - / " " V V ) ' , for all f e D(C m ).
D(C
) = D(G ),
v
m
nv
C f := G f
m
For G
m
m " f-D f' where ^)
:= m m
( " ] )/ 2 ' for a11 f e D ( c m )
the following lemma holds:
Lemma 3.5.3
G
is densely defined and
m-dissipative.
Proof
2
Since C [0,1] C D(G ) is dense in C[0,1] it follows that G is densely de­
fined. Next we investigate the boundary points 0 and 1 of the interval [0,1].
With the notations of Chapter 1, and x„ = 1/2 we have
Q(X) = ^(X) = X(1
- x)/2,
£(x) = m(l - 2x)/2,
W(x) = 2 - 2 n V m ( l
-xfm,
(aW)- 1 (x) = 2 1 + 2 m x m - 1 ( l - x ) m - 1 ,
,-1.
'x
Q(x) = (aW) '(x) f __ W(t)dt,
J 1/1/2
R(x) = W(x) f
(aW) ! (t)dt.
J 1/2
For x e (1/2,1) we have for m = 2,3,...
V
0 < Q(x) = 2x
(1 - x)
t (1 - t) dt
J 1/2
Y
«m+1 m - 1 , ,
,m-l f
,. ^ - m , .
<2
x
(1 - x)
(1 - t) dt
J 1/2
= 2m+1(m - i r ' x ^ O
- x)m-l{(l
- x) 1 -" 1 - 2 m - !
<2m+1{l - 2 m _ 1 ( l -x)m_1}
<2m+1.
For x e (1/2,1) and m = 1 wehave
O<QW- 2 J; /2 r ! (i - t ) - 1
J 1/2
(i - o"■'dt
< -4 log ( 1 - X ) .
Thus for m = 1,2,... Q e L (1/2,1). In a similar way we can prove that
QeL^O.l/Z).
Next we investigate the integrability of R on (1/2,1) for m = 1,2,... .
For x e (1/2,1) we have
168
_,, . _ -ni/,
x-m f x . m - 1 . .
^m-lj*
R(x) = 2x (1 - x)
t
(1 - t)
dt
J 1/2
/2
Y
»2-m,.
,-m f
.,
,m-l ,
>2
(1 - x)
(1 - t)
dt
J 1/2
->2-ni - 1 , . \-m, ,.
.m ~-m.
=2
m (1 - x) {-(1 - x) + 2 )
=2
m
{-1 + 2
(1 - x)
}.
Thus R 0 L (1/2,1). In a similar way it can be proved that R g L (0,1/2).
Then it follows that 0 and 1 are entrance boundary points. Therefore, the
theory of section 2.7 is applicable. Lemma 9 of that section shows that G
is
dissipative. The surjectivity of I - G m follows from Theorem 2 of that section.
Then also I - AG (A > 0) is surjective, ([CH], Th. 3.7).
This completes the proof of Lemma 3.5.3.
D
For C an analogous lemma holds.
m
Lemma 3.5.4
C
is densely defined and
m-dissipative.
Proof
Since D(C ) = D(G ) C is densely3 defined. Assume A > 0. From the dissim
m m
pativity of G it follows that
(1 + A ( m )
rtfll
< ||f|| < ||f - A(l + A ( m ) ) _1 G m f||.
Hence
llfll < llf + A ( m ) f - AGmf||
or
llfll < llf - AC f||.
169
This proves the dissipativity of C . For the surjectivity of I - AC
let us
consider
f - A C m f = g,
geC[0,l],
A>0,
or
(1 + A ( m ) )f - AGmf = g,
or
f-Ad + A ^ r ^ j - d + A ( « b - V
From the surjectivity of I - pG
for each n > 0 it follows that there exists a
solution f. This proves the surjectivity of f - AC
for each A > 0.
D
The next proposition is of crucial importance for the application of Theorem
3.3.1.
Proposition 3.5.5
Let D(A) - just as in section 2.3 - be given by
D(A):={feC[0,l]\feC2(0,l),
Urn x( 1 - x)f(x)
x-»0
= Urn x( 1 - x)f'(x)
x— 1
= 0},
and let m> 1. Then
D(Cm) =
C1fO,lJnD(A).
Proof
C^O.l] n D(A) c D(C ) is clear, so we have only to prove D(C m ) c c ' [ 0 , l ]
n D(A).
(i)
D(C ) c C]fO,lJ. Let f s C [ 0 , l ] n C 2 (0,l)and g e C[0,l]such that
<t>f" + m 0 l f ' = g,
where
170
0(x) = x(l - x ) / 2 .
Then
,1 -m, ,m~,. ,
or
(4>mfy
= g4>m-1 6 C [ o , i ] ,
thus
(A')(x)= f
g^ m _ 1 dt + K,
JO
xG(0,l),
K e E
or
(*)
f'(x) = ^ _ m (x) f
g* m - 1 dt + K0~ m (x).
Jo
By I'Hospital's Rule we have
Y
lim <j>~m(x) g0 m " dt
\-*0
J0
= lim ( g ^ X x V C m . - r ' W ^ ' W )
x—0
= lim g(x)/(m(l - 2x))
x—0
= g(0)/m.
Hence the function ip on [0,1) defined by V>(0) = g(0)/m,
X
Jo
belongs to C[0,1). From (*) it follows that
,
f(x) = f [\]
c
- j
1/2
1/2
rt
.
^ m ( t ) j o (g^ m - l Xs)dsdt
f
1/2
K
Jx
tf"m(t)dt.
171
Since <j> £ L (0,1/2) it follows from the boundedness of f near zero that
K = 0. Thus
f'(x)=^" m (x) f g^m_1dt
Jo
implying f'
e C[0,1). In a similar way we can prove f'
e C(0,1]. Thus
f ' e C[0,l]or f e c ' [ 0 , l ] ,
(ii)
D(C ) c D(A). Let f and g be as in part (i). Since f' e C[0,1] and
g e C[0,1], it follows that <j>r = g - m<f>H • e C[0,1]. So lim (<£f")(x) and
x—>0
lim (<f>f")(\) exist. Assume
x—>1
L := lim (tff")(x) > 0.
x-»0
Then there exists a number 6 > 0, such that for all x e (0,5)
x ( l - x)f"(x)/2 > L/2
and thus for x e (0,5)
f"(x)> L(x(l - x ) ) " 1 > L x _ 1 > 0.
Then it follows that
f ( l / 2 ) > f'(x) + L
f1/2
t
-1
dt
= f '(x) - L log 2x.
Since f' is bounded and -log 2x is positive and unbounded for x e (0,1/2), L
has to be zero, a contradiction. Similarly if L < 0 we also obtain a contradic­
tion. Therefore L = 0. Similarly lim (4>P)(x) = 0 and thus D(C ) C D(A).
D
172
In order to apply Theorem 3.3.1 it is sufficient to prove that
lim C
f = Cm f
n-»oo m > n
for all f G D(C
). With D" m f = u we have
v
m
C
f = DmA D " m f = D m A u
m,n
n
n
where
A u = n(B u - u).
v
n
n
'
Concerning the convergence of D A u, Felbecker [Fe] proved that for
uGCm+2[0,l]
lim D m A n u = Dm(<iu")
n—+00
in the uniform norm topology. For our purpose we need a sharper result.
First we will prove the following.
Lemma 3.5.6
For all u e (f1*1
[0,1]
lim n(D Bu-B
D u) = mê%D
u - I ,1 D u
in the uniform norm topology.
Proof
TV» J- 1
Let u e C
[0,1] and n > m. Following the lines of Lorentz, [Lo], par. 1.4
we first compute (D B u)(x), 0 < x < 1, where
n
(3.5.2)
(B u)(x) = E u(k/n).p n . (x),
n
n,K
k=Q
173
, ,
P
fnï k,,
.n-k
n,k(x) = U x ( 1 - X )
•
Let Au(x) denote the first order difference u(x + 1/n) - u(x) of the function u
at the point x. In general, the difference (A uXx) of £-th (£ > 1) order
corresponding to the increment 1/n at x is defined by
(A 1 u)(x) = Au(x),
(A £ u)(x)= A(A £_1 u(x)).
Then, as well-known,
£
(A £ u)(x)= E ( - l / M ( f ) u ( x
i=0
(3.5.3)
+
i/n).
By differentiation of 3.5.2 we obtain
n
(DB u)(x) = E u(k/n)(Dp n )(x)
n
n,K
k=Q
= E u ( k / n ) . ( " ) { k x k " 1 ( l - x ) n _ k - (n - k)x k (l - x ) " ^ " 1 }
K
k=0
n-1
n-1
= E u((k+ l)/n).n.p
( x ) - E u(k/n).n.p
(x)
n_1
n _ I , K
k=0
k=0
n-1
= n. E (Au)(k/n).p n (x).
n_1,K
k=0
By repeated differentiation we obtain in this way
n-m
n-m
(D B u)(x) = , " ,, E
E (A m u)(k/n).p
v
/v
n
n ' (n - m)! i,_o
k=0
m
(3.5.4)
!
(x).
m K
'
From Taylor's Theorem we have
u((k
+
i)/n) = m+1
_E i C ^ ^ u X k / n ) , - ^ - ^
(i) m+1 h(k/n,^ k/n )
174
where h is continuous on
V : = { ( x , t ) e [ 0 , l ] x [ - l , l ] | 0 < x + t < 1}
with £.
. G [0,i/n] and h(k/n,0) = 0. Applying this formula we obtain for
n
(A m u)(k/n):
I,K/
m
(A m u)(k/n)= E ( - D ^ r j u C k + iVn)
i=0
*
m+l
m
=E
E (-ir'Cp E j (t)Vu)(k/n)
:_n
i=0
» j
+
= 0
J-
n
v^ , n m-ifm")
1
riïm+1,,. , t
..
{ l)
J
h kn
£0 ^i (SrTT)rW
< / ^i,k/n>
= mE+1J {nVu)(k/n).E ( - ^ ( " V
j=0
'
i=0
1
-m-1 ^ ( - 1 )
(m+ 1)!
+ ~,
rrr n
(3.5.5)
=
l.J,
h(k/n,qk/n)
m+l ,
E I T n"J(DJu)(k/n)S(m,j)
j=0 J— i ,\m-i (rcn .m+l, ,. , ,
>,
1
-m-1 Tm
EQ(-D
liJi
Mk/n,eik/n)
(m + 1)! n
where S(m,j) are known as the Stirling numbers of the second kind (see [AS],
24.1.4). For these numbers we have
S(m,j) = 0
(3.5.6)
if
j < m - 1,
S(m,m) = 1,
S(m,m+1)=
rt1).
Now from (3.5.4)-(3.5.5) we obtain
175
(D m B n u)(x) =
(n
+
(3.5.7)
_" m) ,
E
{n _ m (D m u)(k/n)
+
(nTTT) ( m + 2 1 ) n " m " , ( D n , + lu)(k/ll) }- P n-m,kW
+ (R
v
m,n
u)(x),
/v
"
where
(R
1
u ) ( x ) = , n ! ,, n^^ttm^l)!)E E <-l)m_1 P ) i
vv
m,n A ' (n - m)!
k=0 i=0
h(k/n
^i,k/n)
P
n-m,k«-
Thus
n-m
m_, u)(x)
.. . = 7 — n!
^ /T(D
^m u)(k/n).p
(D"'B
~ r - n -m E
. (x) +
/v
v
n
' (n - m)!
. ^
'y ' ' n-m,k v '
,
, n-m
.
m
n!
- m - 1 v-. ,_.m+l ... . ,
,.
+ T • "7
ü" n
E (D
u)(k/n).p
. (x)
/v
2 (n - m)!
.^' ' n-m,k '
(3.5.8)
+ (R
u)(x).
m,n /v '
Next, we will investigate each of these terms.
Clearly
(n
_ n! m)! n " m - 1(1 - l/n)(l - 2/n)
(1 - (m - l)/n)
= 1 - m(m - l)/(2n) + o(n~ ), n —
► oo
m
= 1 - (■2) n ~ + o(n" ), n
= 1 +o(l), n — oo.
(3.5.9)
Since for all x e [0,1]
n-m
E P
, (x) = (B
e«(x) s 1
£Q
n-m,k v ' v n-m 0V '
we have
*
176
n-m
11
£ (D*m
u)(k/n).p
n
k=0
m,K
(x) < IIDmu||
and
.
n-m
7 —n!^ n -m ,-.
T /T
(D
. (x) =
^m u)(k/n).p
A
k=0
(n - m)!
*-•- v
' ' K n-m,k v '
r
n-m
TïT\ - 1
= (1 - ( m ) n" 1 + oOT 1 )). £ (D m u)(k/n).p
2
k=0
(x), n - oo
'
n-m
= (1 - ( m ) n _ 1 ) . E (D m u)(k/n).p n
(x) + o(n" J ), n - oo,
z
n-m,K
k=Q
(3.5.10)
uniformly in x G [0,1].
Similarly
,__...
(3511)
m
,
n;
yTiT^jT
.
-m-i
n
_.
= m(2n)"
n-m
.
*-■ /r^m+l .,, . .
k?0
(D
U)(k/n)p
, .
n-m,k
(x)
n-m
£ ( ° m + u)(k/n).p
(x) + o(n" ), n - o o
k=0
n-m,K
uniformly in x G [0,1].
Since u G C
[0,1] the function h is continuous on V, and thus also uni­
formly continuous on V. Then, if e > 0 is arbitrarily chosen, there exists a
number 6(e) > 0 such that for all (x,t) G [0,l]x(-6,5) n V
|h(x,t)| < e.
In particular, if n > mS~ , then for all k G {0,l,...,n-m} and all i G {0,1,...,m}
|h(k/n,e u / n )| < ,
With 3.5.9 it follows that for all x G [0,1]
177
|(R m,n u)(x)| < n _ 1 ( l + o(l))((m + l)!)" 1
n-m m
E E ( ?i ) ™+1 ***n-m,K
m k(x)'
R = 0 i=Q
n —»oo
n-m
= n" ( l + o ( l ) ) . e . Q m . E P n _ m > k W . » - oo,
k=0
whereQm-((m+l)!)-,i: (m).im+1,
m
i=0
= n _ 1 ( l + o(l)).€.Q m (B n _ m e 0 )(x), n - oo
= n" (1 +o(l)).€.Q m , n - K »
uniformly in x e [0,1]. This implies
(3.5.12)
n ( R m y O ( x ) = o(l), n - o o ,
uniformly in x e [0,1].
Returning to (3.5.8) we obtain with (3.5.9)-(3.5.12) for n > m
m
n(D
B u - B
D m u)(x)
=
v
/v
n
n-m
'
= n.
n-m
E {(D m u)(k/n) - (D m u)(k/(n - m))}.p
k=0
n-m
- ( m 3- E o (Dmu)(k/n).pn.m)k(x)
n-m
.
+ (m/2). E ( D m uXk/n).P n
k=0
+ 0(1), n -» oo,
and by the Mean Value Theorem
(x)
'
m
. (x)
'
178
n-m
= n. E ( k / n - k / ( n - m ) ) ( D m + 1 u ) ( x
- (?)
z
n-m
E (D m u)(k/n). P
k=Q
).p
(x)
n-m,K
n-m
.
£ (D m + 1 u)(k/n).p
k=0
+ (m/2)
(x)
(x)
'
+ o(l), n — oo,
with k/n < x,
< k/(n - m)
= -m
n-m
Eo(k/(n-m))(Dm+1u)(xM).pn_mJc(x)
- I?]
z
n-m
E (D m u)(k/n).p
k=Q
(x)
n-m,K
n-m
E (D m + 1 u)(k/n).p n
(x)
k=0
n-m,K
+ o(l), n - K » ,
(3-5.13)
+(m/2)
uniformly in x. By the well-known convergence properties of the Bernstein
operators ([Lo, 1 ]) we finally obtain
lim
n _KX)
n{(DmB^uXx)
- v(B„ _D m U )(x)} =
v
n '
'
n-m
- mx(D m + 1 u)(x) - ( m )(D m u)(x)
+ (m/2)(D m+1 u)(x)
= (m/2)(l - 2x)(D m+1 u)(x) - ( m ) (D m u)(x),
uniformly in x. This concludes Lemma 3.5.6.
D
179
If f G D(A), and u = D~ m f then clearly D m u G D(A). Therefore, the next
lemma is a consequence of Step 3 of the proof of Theorem 3.1 in section 3.2.
Lemma 3.5.7
For all u G Cm+2(0,1),
such that Dmu G D(A) we have
Um n(B
n^oo
n-m
D u - D u) = 6D
u
in the uniform norm topology.
D
After these preparations we are able to prove the following proposition.
Proposition 3.5.8
For all f G D(C
) , m> 1, we have
1
m
Um Cm,nJf = CJ
nf
„-►oo
in the uniform
norm
topology.
Proof
Assume f G D ( C m ) . Set u = D~ m f. By Proposition 3.5.5 we have f e c ' t O J l n
D(A), or u G C m + l [ 0 , l ] and D m u G D(A). Using the Lemmata 3.5.6 and 3.5.7
we see
lim
C
f = lim
n(F
f - f)
= h m n(D B D
n
n—>oo
..
/T^nir,
f - D D
_.m .
= lim n(D B u - D
n
n—»oo
u)
f)
180
= lim n ( D m B n u - B n m D m u ) +
n-^oo
"
+ lim n(n - m) . lim (n - m)(B
D
n-m
n—*oo
n—»oo
,,_jn+l
rmi _ m
,T~wm+2
= m<£'D
u - L 2 J D u + <£D
u
u - D
u)
= <^D2f + m<f> ' D f - ( m ) f
= C f
m
in the uniform norm topology.
D
We arrive to the saturation theorem for the operators F
, which is an
m,n
application of Theorem 3.2.1.
Theorem 3.5.9
Let in the Banach space C[0,1 ], equipped with the supremum norm, for m > 1
the sequence of operators (F
) , strongly convergent to the identity operator
I, be defined by 3.5.1. Then for f G Cf 0,1 ] the conditions
(')
1
\\F
" m,nf ~ A\<
" Mn' for some M > 0, n = 1,2,...
(ii)
f£Fav(T
(iii)
f e C](0,l),
(iv)
f G C](0,1), ƒ ' G ACloc(0,l)
)
ƒ ' e AC[oc(0,l),
<f>f" + m<l>,f> e
L°°(0,l)
n L°°(0,1), <j>f" G L°°(0,1)
are equivalent.
Proof
,
If A
:= C
= n(F
- I) and A := C , then it follows from the Lemmata
n
m,n
m,n
m
3.5.4-3.5.8 that the conditions of Theorem 3.3.1 are satisfied and the equi­
valence of (i) and (ii) follows from the second part of the theorem. Further,
the equivalence of (ii) and (iii) will be proved by applying of the theory of
the sections 2.8-2.9 and by using the fact that
181
Fav(T^ ) = D(C®*)nC[0,l]
m
'm
Since D(C ) = D(G
) and
v
nr
nr
G -C = 0 1 ,
it follows that D(C°*) = D(G 0 *).
From Theorem 2.8.1 we have
X =NBV[0,1], x Q = 1/2,
D(G m ) = {w G X* | w e NAC[0,1], <£w' G AC l o c (0,l),
( 0 w ' ) ' - m ^ ' w 1 - ( ( ^ w ' ) ' - m ^ ' w ' X x ^ e NBV[0,1],
lim ( ^ I " m w ' ) ( x ) =
x-+0
G
mw
= (<
^w')' "
lim ( ^ 1 " m w , ) ( x ) = 0 } ,
x-*l
m<
^ ' w '* f o r all w G D(G* ).
Moreover we obtain from (2.8.7)
X Q = NAC[0,1],
D(Gm)0) = ( w G X * 0 | ^ w . G A C l o c ( 0 , l ) ,
( ^ w ' ) ' - m ^ ' w ' G AC[0,1],
lim ( ^ 1 " m w ' ) ( x ) = lim ( / ~ m w ' ) ( x ) = 0},
x-»0
x-»l
G
m 0W
=
^
w
' ) ' " m^ ' w '
f o r a11 w € D
(G*
0).
182
and from (2.8.8)
X® = L 1 (0,1),
D(G®) = ( v e X ® | ^ v e A C l o c ( 0 , l ) ,
( W - n t f ' v e AC(0,1),
lim ( / " m v ) ( x ) = lim (^ 1 _ m v)(x) = 0},
x-+0
x->l
G®v = ((0v) • - m<j> ' v ) ' , for all v G D(G®).
Finally from Theorem 2.9.1 we obtain
f X®* = L°°(0,1),
D(G®*) = {z G X®* | z 6 C^O.l), z ' G AC, v(0,1),
'loc
m
<f>z" + m<f>'z' G oo.
L (0,1)},
G®*z = <j>z" + m0 ' z ' , for all z G D(G®*).
It follows from (3.3.1) that
Fav(T^ ) = D(G®*) n X
m
'm
= {f G C[0,1] | f G C'tf),!), f • G AC l o c (0,l),
^ r + m^'f» G L (0,1)}.
Next we prove the equivalence of (iii) and (iv). The implication (iv) =>■ (iii) is
trivial.
Let f e C[0,1] n C J (0,1) with f' e AC J o c (0,l) and let g := <j>f" + m ^ ' f ' G
L°°(0,1), where 0(x) = x(l - x)/2. Then
183
A<i>mr)> = g<t,m-1 e
1^(0,1)
and
(* m f ')(x) = f g ^ m _ 1 d t + K, K £ IR, x e (0,1),
J
0
,x
f(x) = ^ m ( x ) f g ^ d t + K ^ x ) .
0
or
Since g s L
(0,1) we have for x e (0,1)
x
x) f g^ m _ 1 dt << ||g||
^ m ( x ) fI At)dt.
llg||^""'(x)
J
J
0
0
In a similar way as in the proof of Proposition 3.5.5 we can show that
x - ^ _ m ( x ) f g<f>m~l dt e AC(0,1 /2),
J
0
TV»
1
and that since <j> £ L (0,1/2) the constant K has to be zero. It follows that
f1 e L°°(0,l/2).
f'eL°°(0,l)
Analogously
we
can
obtain
f • e L°°(l/2,1).
Thus
and then also </>(" e L°°(0,1). This completes the proof of
Theorem 3.5.9.
Remark 1. From (iv) we see the saturation class of the sequence F .,F
-,...,
where m = 1,2,..., is independent of m.
Remark 2. The class of functions Z for which the convergence of F
f to f
j
m,n
is faster than 0(n ) is the so-called "trivial class" of the sequence (F
).
Assume f e Z , then we have
n(F
m,nf " 0 - o ( l ) , n - o o
and since Z C D(A) (by Theorem 3.2.1) we have
lim n(F
f - f) = C f = 0,
n-Kx,
m,n
m
184
and thus f belongs to the null space of C , denoted by N(C ). It follows that
Z c N(C
). Since N(C ) c Z we have
v
m
m
Z = N(C
).
v
m'
Thus f e Z if and only if
(3.5.14)
^ r + m^'f' - ( m ) f = 0.
This equation is a hypergeometric differential equation (cf. [Bie], p. 210).
Two independent solutions are the functions f. and f- defined by f. (x) = x
,
f 2 (x) = (1 - x ) 1 _ m if m > 2, and fj(x) = 1, f 2 (x) = log(x/(l - x)) if m = 1. It
follows that f G Z if and only if f is a linear combination of f. and f». Since
f. and f~ are unbounded if m > 2 and f_ is unbounded if m = 1 we have:
f G Z if and only if f is constant for m = 1 and f is zero if m > 2.
We conclude this section by solving the problem of uniform saturation for
Bernstein operators in C [0,1]. Let f G C m [0,l], m G UN, n > m, such that
sup ||D m (n(B n f - f))|| < oo.
n>m
We can rewrite f as
f =p
. +D
m-1
u,
'
where P m _i is a polynomial of degree less or equal m-1 and u e C[0,1]. It
follows from Lemma 3.5.1 with p = P = 0 that D m B p
, =0
n K m-1
Hence
sup ||D m (n(B D" m u - D - m u ) | | < oo
n>m
or equivalently
185
sup ||C
u|| < oo.
n>m
'
Then u satisfies (iv) of Theorem 3.5.9, and thus D f satisfies (iv).
Conversely, let f G C [0,1] such that D m f satisfies (iv). Then
sup ||C
1
n>m
m,n
D m f|| < oo
"
by Theorem 3.5.9, or equivalently
sup | | D m ( n ( B n f - f))|| < oo.
n>m
Finally, if
Jim
||D m (n(B n f - f))|| = O
n—»oo
then D f belongs to the null space of C . Thus the trivial class Z„ consists of all
f G C m [0,l], such that u := D m f satisfies equation (3.5.14). Thus f G C m [0,l]
belongs to Z~ if and only if Df is constant for m = 1 and D f = 0 for m > 2.
If C [0,1] is normed by
m
11*11:=
k
E IID fll
k=0
we have the following saturation result:
Theorem 3.5.10
For m> 1 and f G C^fOJJ,
d)
\\Bnf-I\\m
=
o(n'1),n^oo,
if and only if ƒ is a linear function,
(ii)
\\Bnf-j\\m
=
0(n-1),n^°o,
if and only if D f satisfies condition (iv) of Theorem 3.5.9.
a
186
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189
SAMENVATTING
In dit proefschrift houden we ons bezig met de theorie van halfgroepen van
operatoren in Banach ruimten alsmede met saturatieproblemen in de approxi­
matietheone.
Saturatie is een interessant verschijnsel in de approximatietheorie. Indien (L )
een rij lineaire operatoren is in een Banach ruimte X, sterk convergent naar
de identiteit-operator in X, dan betreft het saturatiep rob leem de bepaling van
de optimale approximatie-orde en de (niet-triviale) klasse van elementen in X
die met deze optimale orde geapproximeerd kunnen worden.
In hoofdstuk 3 worden saturatieproblemen voor Bernstein operatoren in C[0,1]
en C [0,1], m = 1,2,... beschouwd. De resultaten voor C [0,1] zijn nieuw,
(Theorem 3.5.10). De methode van het gebruik van halfgroepen van operato­
ren om saturatieproblemen te onderzoeken werd eerder toegepast, onder andere
door Becker en Nessel in 1975. Het aantal toepassingen op concrete approximatieprocessen was echter zeer beperkt.
De hier ontwikkelde methode is nieuw en bouwt voort op de door Butzer en
Berens (1967) ontwikkelde theorie van duale halfgroepen. Sterk continue
halfgroepen van operatoren kunnen op injectieve wijze geassocieerd worden
met hun infinitesimale generator. De hier gehanteerde methode maakt slechts
gebruik van de infinitesimale generator en laat de halfgroep zelf buiten be­
schouwing. Daardoor is er een goed hanteerbare theorie voor het oplossen van
saturatieproblemen ontstaan.
In hoofdstuk 3 worden de toepassingen gegeven. De algemene theorie wordt
behandeld in hoofdstuk 2. De Banach ruimte waarin we werken is C(J),
waarbij J een open (niet noodzakelijk begrensd) interval op de reële as is. Het
opzetten van de theorie is mogelijk geworden door de randpunten van J te
onderscheiden in reguliere, exit, entrance en natuurlijke randpunten. Dit is
190
een verdeling die Feller in 1952 maakte bij een onderzoek naar het verband
tussen parabolische differentiaalvergelijkingen en halfgroepen van operatoren.
Mede door deze classificatie is het mogelijk om noodzakelijke
en voldoende
voorwaarden te geven voor a en f) opdat de operator A, gedefinieerd door
' D(A) = {f G C[r j ,r 2 ] | f e C 2 (r
{,r2),
lim (aD 2 f + 0Df)(x) = 0, i = 1,2}
x-+ri
i
Af = aD 2 f + £Df voor f E D(A)
een infinitesimale generator is van een halfgroep in C[r.,r-] (met supremumnorm).
De classificatie van de randpunten en de voorbereidende theorie wordt gege­
ven in hoofdstuk 1.
191
ACKNOWLEDGEMENT
The completion of this thesis and the development to its final form would not
have been possible without the help of others.
In particular I wish to express my sincere gratitude to Professor Philippe
Clément for his valuable and constructive criticisms. He also provided the
needed expertise and encouragement during the preparation of this manuscript.
Further I wish to thank Angelina de Wit for her very efficient typing of the
manuscript. Finally, I wish to thank my wife Hanny and children Ellen and
Henkjaap for their tolerance and understanding throughout the entire devel­
opment of this thesis.
Kr raia
p.5
1.2 and 1.21: ü(av) for aDv
p.50
lines 1,3,7.11.18: D(av) for aDv
p.60 after (1.3.1): and h (x) - h„(x) = c (constant) at all points of
continuity of h
and thus also of h„.
w,w ) = (O.(uw')' - pw' - ((aw1)' - Pw'))(x ).0)
p.SO
(2.4.4): A*(w
p.90
(2-1.5): B(w r w.w 2 )
= (O^aw')' - |3w" - ({«')' - Pw))(xQ). 0)
p.161 last line: ... I defined as in (2.6.14), a second representation
0*
tor A
is obtained, namely ...