Semigroups of operators: Approximation and saturation in banach spaces

APPROXIMA TION AND SA TURA TION

IN BAN A CH SPA CES

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TR diss

1598

CA. Timmermans

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 uw-in de aula van de Technische Universiteit Delft, Mekelveg te Delft

U bent hierbij van harte welkom. No afloop van de promotie is er een receptie.

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

I

0

0 < *0(x) < -p(x.f) for x € (0.1). f € P..

Let A: D(A) C X — • X be defined by

D(A) = {u € C[0.1] O C2(0.1): lim f(x.u(x)) u"(x) = 0. j = 0.1)

(Au)(x) = *(x.u(x)) u"(x) 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-«° x€[0.1] n

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).

Let the sequence (K ), n = 1,2,... of Kantorovic operators be defined by V cCo.iD — c[o.i] n (<k-H)/(n+l) (K f)(x) = (n+l) 2 p (x) f(t)dt. n = 1.2.... n k=0 n , K Jk/(n+l) where Pn 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 := 2 II D f II . f € d"[0.1].

m k=0 k

where D f denotes the k-th derivative and II»II the supremum norm. Then for f € C ^ O . l ]

i) II KRf - f llm = o(n_ 1) . n — » »

if and only if f is a polynomial of degree < 1.

i i ) II K J - f llm = 0 ( n- 1) . n - ♦ »

i f and only i f u:= Dmf 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:

f a € L ( x0 >r _ ) . then r„ is not natural and moreover ) r2 is regular <=* W € LJ( x0 >r2) A (aW)'1 € l\xQ.T2)

i) r2 is exit <=» W € L1^ . ^ ) A (aW)"1 « L ^ X Q , ^ )

ii) r2 is extrance c* W C L ^ X Q . ^ ) A (oW)"1 € L ^ X Q . ^ )

and

-V> \

f a C L (x_. r2) . then r_ is not regular and moreover

) r2 is natural =* W € L1^ . ^ ) V (oW)"1 € L ^ x ^ )

i) W € L (xQ.r2) * r2 is exit or natural

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 € ( l y i ^ ) .

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 u(x)sin p$ ♦ (W"V)(x)cos p. = 0. i = 1.2}.

Au = 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 € LJ(J).

Let u. and u- be solutions of

u - Au = 0,

satisfying

-1 .1-1

lim U j( x ) = 0. lim (W V ) ( x ) = (-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 = =■ .

„-1

(u2(rj) tan x + (W u2)(rj)) tan y + tan x = 0. x i 0. y < 0. and

„-1

(Uj(r2) tan y ♦ (W uj)(r2)) tan x - tan y = 0, x > 0 . y i

Lei Let

Let X. A. G. u.. u_. W, p and p„, be defined as in the preceding theorem.

H = {(x.y) € (-|.|) x ( . | , | ) | - | <x < - arctan (W1 u ^ M r , ) .

arc tan (({u2(rj)tan x - ( w ' ^ M r j ) )_ 1 - ( W ^ u j H ^ ) ) / u,(r2)) ^ y < T L

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

a contraction on X and thus ill) A is not m-dissipative.

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*

p < "'t

0

-

c

)

(a + b )n

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 de verwachting van het aantal vragen bij een strategie

T € 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).

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 - v o l l e d i g e 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 u i t het onderstaande schema, waarbij de vrouwelijke deelnemers genummerd z i j n van 1 t/m 16 en de man­ n e l i j k e van 17 t/m 32.

(14e Hompestomper tennistoernooi, okt. '67. Heerenveen)

1 2 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 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 16 19 2 0 21 22 23 24 25 26 27 26 29 30 31 32 X b q li t y n e h a c d J g P r 1 s V f m o V 1 z B X q t k n y e h c a J d P K 1 r w s m f V o I 1 Q X b y n k 1 h d e c r 1 g P f m s V 1 z 0 V 0 D X n y t k h e J d c a 1 T P g m f * s z 1 V o K T Y T K N V N K N Y T X B . b X 0 q X q t h . 1 g P 1 r 1 a < C 1 <l j < o v < i z s w f 1» b h e ■» > r i 1 g ' P : d > J a i c i 1 J Z E 0 V » f l t i II S r w N V T K 0 B X h * 1 r P C J d c a z t V o m f * 8 E H X b q k t y n • « f m o V 1 z a c d J g P r 1 E H B X q t k n y V • m f V 0 z 1 c a J d P g 1 r H E Q X b y n k t f ■ • « 1 z 0 V d J a c r 1 g P H E Q B X n y t k ■ f V s z 1 V 0 J d c a 1 r P g E H K T Y l« X b q o V i z 1 m f * g P r 1 e c d J É H T K N Y B X <J V o z 1 « a * f P g ) r c a J d H É Y h K T Q X b 1 z o V f ■1 > * r 1 g P d J a c H E N Y T K 0 B X z 1 V 0 m 1 V s 1 r P g J d c a A c D J C P r 1 S * f K o V 1 z X b q k t y n e h c A J 0 p c 1 r I S K f V o z 1 b X q t k n y e h D J A c r 1 C P f H s f 1 z 0 V q X b y n k t h e J D c A I r P C M f « S z 1 V 0 q b X n y t k h e C P r P C 1 r C 1 r P A c D c A J 0 J A J D c o v 1 v o z 1 z o z V s » r ■ S N r M s M f « k y t k n y n k n y I X b . b X q q X q e h 1 b h E e i 1 r P C J D c A z > V o M f « S n y t k q b X h e S « f M o V 1 z A c D J C P r 1 e h X b q k I y n t' S H f V o z 1 c A J D P C 1 r e h b X q 1 k n y r H S 1 1 z 0 V D J A c r 1 C P h e q X b y n k 1 M f « s z 1 V o J D c A I r P C h e q b X n y i k o V 1 z S * f M C P r 1 A c D J e h k t y n X b q V 0 z 1 « S H f r c 1 r c A J D e h l k n V b X q i z 0 V f M S * r 1 C P D J A C h c y n k t Q X b 7 1 V 0 M f * S 1 r P C .1 D c A h t n y t k q b X

Opmerking: In d i t achem hebben "kleine l e t t e r s " 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 h o o f d l e t t e r - i n f o r m t i e .

A speelt tegen dubbel B ♦ C D speelt tegen dubbel E * F C speelt tegen dubbel H * I J speelt tegen dubbel K ♦ L M speelt tegen dubbel N ♦ O P speelt tegen dubbel 0 ♦ R S speelt t c g c dubbel T < V V speelt tegen dubbel ï * Z

Men nene in elke r i j voor de diverse l e t t e r s 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. . « e d s i r i j d Ie r o n d e : 2e ronde.' 3 e r o n d e : 4 e r o n d e : ?* r o n d e : fic ronde: 7e r o n d e : 6c r o n d e : r o o s t e r dubbel dubbel dubbel dubbel dubbel dubbel dubbel dubbel X X X X X X X X

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.

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Uv (\<J

SEMIGROUPS OF OPERATORS,

APPROXIMATION AND SATURATION IN BAN ACH SPACES

CA. Timmermans

Delft University Press

TR diss

1598

SEMIGROUPS OF OPERATORS,

APPROXIMATION AND SATURATION IN BAN ACH SPACES

Aan Hanny Ellen Henkjaap

SEMIGROUPS OF OPERATORS,

A P P R O X I M A T I O N A N D S A T U R A T I O N 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

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

CONTENTS

INTRODUCTION

CHAPTER 1 - SOLUTIONS OF GENERAL SECOND ORDER DIFFERENTIAL EQUATIONS IN VARIOUS

SPACES 13 1.1. Solutions of u - aD u - 0Du = f in C(J) and L°°(J).

Classification of boundary points 13

1.1.1. Preliminaries 13 1.1.2. Solutions of the homogeneous equation 14

1.1.3. Classification of the boundary points 30 1.1.4. Solutions of u - Q D2U - 0Du = f in C(J) 40 1.1.5. Solutions of z - aD2z - /3Dz = k in L°°(J) 48 1.2. Solutions of v - D(aDv -/9v) = g in L](J) 50 1.2.1. Solutions of v - D(aDv - 0v) = 0 50 1.2.2. Solutions of v - D(D(av) - 0v) = g in L ^ J ) 56 1.3. Solutions of w - (D(aDw) - jJDw) = h in NBV(J) 60 1.3.1. Solutions of w - (D(aDw) - 0Dw) = 0 62 1.3.2. Solutions in NBV(J) of w - (D(aDw) - j5Dw) = h 66

CHAPTER 2 - SEMIGROUPS GENERATED BY DIFFERENTIAL OPERATORS SATISFYING VENTCEL'S BOUNDARY

CONDITIONS AND THEIR DUALS 67

2.1. Semigroups in Banach spaces 67 2.2. Two propositions j)n adjoint operators 72

2.3. Semigroups in C(J) 77 2.4. Dual semigroups in 3R x NBV(J) x ]R 88

2.5. Restricted dual semigroups in 3R x L ^ J ) x }R 102 2.6. "Bidual" semigroups in K x L°°(J) x ]R 106

CONTENTS (continued)

2.7. On CQ-semigroups in a space of bounded continuous

functions in the case of entrance boundary points 116 2.8. Dual semigroups in NBV(J), NAC(J) and L](J) 126

2.9. "Bidual" semigroups in L°°(J) 130

CHAPTER 3 - SATURATION PROBLEMS FOR BERNSTEIN

OPERATORS IN Cm[0,l] 135

3.1. Introduction 135 3.2. A unified approach to pointwise and uniform

saturation for Bernstein polynomials 136 3.3. Saturation and Favard classes 153 3.4. Application: Uniform saturation class for Bernstein

operators on C[0,1 ] 159 3.5. Uniform saturation classes for Bernstein operators in

Cm[ 0 , l ] norms (m > 1) 163

REFERENCES 186 SAMENVATTING 189

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 6 ||f - L f|| = 0

n—K» u n

if and only if f e s(L ),

(ii) there exists at least one f e X\s(L ) for which (1) 'I W -f) l l = 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).

(2) lim <j> (L f - f) = Af, f e D(A) C X, n—►<» nx n

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

(3) '

B : C[0,1] -+ C[0,1] (equipped with the sup-norm), (B f)(x) = V f(k/n) p (x),

II k - Q 11,K

with p (x) = ( P ) x (1 - x )n" . n,k

This sequence has the Voronowskaya property

(4) lim ||n(B f - f) - Af|| = 0, f G D(A), n—►<» n

where A is defined by

(5)

D(A)= {f G C [ 0 , l ] | f e C2( 0 , l ) ,

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.

lim n(B f(x) - f(x)) = x(l - x)f'(x)/2 n—*oo n

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—»oo n 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].

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 S C2( 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

(8)

' D(A)= { f e C [r i, r2] | f e C2( rrr2) ,

lim (aD2f + /3Df)(x) = 0, i = 1,2),

x - q J

Af = Q D 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

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 - Q D U - £Du = 0 u e C(J) n C ^ J ) , Du e ACJoc(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 ACl o c(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.

CHAPTER 1 - SOLUTIONS OF GENERAL SECOND ORDER DIFFERENTIAL EQUATIONS IN VARIOUS SPACES

2 OO 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 f(x), i = 1,2. C (J) denotes the set of functions in C(J) with compact

x—+rj c

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 |f|dx < oo. This space, equipped with the usual L -norm, JJ

llfllj:- J Ifldx,

is a Banach space. L (ri , xn) and L (*0,r_) have a similar meaning. For sake of convenience we sometimes denote L ( xn, 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

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 - A A u = f

we introduce the real-valued functions W, Q and R on J as follows: .x (1.1.3) W ( x ) : = e x p { - f (/3a 1)(s)ds}, x e J, 1 J x

o

J Y (1.1.4) Q(x):=(aW)_ 1(x) f W(s)ds, x e J, J x0 Y (1.1.5) R ( x ) : = W ( x ) f (aW)_I(s)ds, x e J. J x

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 (1.1.6) u - (aD2u +/3Du) = 0

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) satisfy­ ing u.(x„) = 1, then u.(x) < uJx), x e (r x „ / (resp. uJx) < uJx), xefX(),r2)).

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 "1U1) ' = 0.

or

(1.1.8) (aW)_ 1u - ( W "1! ! ' ) ' = 0.

The next relation follows from (1.1.8) and is very useful:

(1.1.9) u,( x ) = W(x).{u'(x0)+ | ((aW)"1u)(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) W u ' is a negative in­ creasing function with Urn (W u')(x)<0.

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. := lim u.(x) = sup u.(x) 1 x-»r2 " * XGJ "] M , := lim u-(x) = sup uJx)

1 x-»rj "z x £ j ~z

m. := lim u.(x) = inf u.(x), i = 1,2. 1 x-»rj _ 1 xeJ - 1

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 t

(1.1.10) Uj(x) = J W ( t ) { | ((aWf ^jXsJds + U j ' U ^ J d t + U l( x0) .

Since u. is positive increasing we have for x > x_ X X t

0 < U j ( x ) [ R(t)dt< [ W(t) [ ((aW)"1u1)(s)ds<u (x) < M

J xo J xo J x

o

Thus, if M. < oo, then R e L (xn,r«).

Sufficiency. Since u . is increasing we have from (1.1.9) for x e ( xn, r - )

Y

0 < Uj'(x)< Uj'(x0)W(x) + Uj(x)W(x) [ (aW)"l(s)ds J x0

(*) 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, y e c ' f J ) , y(xQ) = 1

where C > 0, R G L (x»,r-) and thus also W G L ( x0, r A It is standard that this equation has a unique solution y with

.x y(x) = v(x) exp with

{ƒ R(.)d.}

x

o

x t f W(t).exp/- f R(s)ds]-v(x) = C | W(t).exp C0 d t + 1. Since and v '(x) = CW(x).expl - f R(s)ds]- > 0 J xQ Y y «(X) = {v '(x) + v(x)R(x)} exp-T f R(t)dt]- > 0

for x G (xf t,r-), it follows that v and y are increasing for x G (x0,r„). Moreover, since R G L (*0,r_) and W G L (x0,r~) we see that v and y are bounded. Thus

N := lim y(x) = sup y(x) < oo. x^r2 xG(x0,r2)

We define the function z on [xn,r_) by

Then

z ' ( x ) < 0 , x e [xQ,r2),

I z ( x

0

) = - 1(

V " y ( x

0

) = 0'

Thus z(x) < 0 for x e [ x0, r2) , 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 lim u(x) = oo

j x^r; R $ L (xn,r.), i = 1,2 .

Or D

Lemma 1.1.6

For all a,b e TR the boundary value problem (1.1.6) with u(xn) = a, 1

lim u(x) = b, has a unique solution <=*■ R = L (xn,r.), i = 1,2 .

nT'

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) for each solution u of (1.1.6) lim (W u% )(x) exists, x-*r2

he "minimal" decreasir have

(ii) for the "minimal" decreasing solution u. (see Proposition 1.1.1) we

(1.1.11) lim uJx) = 0, lim (W'1 u\)(x) = L < 0,

x-*r2 '^ x-*r2 '^

(Hi) there exists a positive decreasing solution u , of (1.1.6) satisfying

(1.1.12) lim uJx) = 1, lim (W'^lXx) = 0.

x-*r2 J x~~+r2

Proof

Let R G L ^ X Q , ^ ) and Q G L1^ , ^ ) .

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 (xQ,r2) we also have (Q\V)~ G L ( x0, r2) . From (1.1.8) it follows that (W- u1) ' is integrable, hence lim (W~ u')(x) exists.

x - r2

ad (ii). That lim u_(x) = 0 is a consequence of Lemma 1.1.6. Since

-1 1 X^r2 (aW) G L (xQ,r2) w e h a v e f r o m (1.1.9) (1.1.13) (W_1u2')(x) = u2'(x0) + ƒ 2 ((aW)_1u

I

2)(s)ds 2((aW)- 1u.(s)ds.

The function W u ' is negative increasing so lim (W- uJ)(x) < 0, and since

~* x—»r2 "l

J

2

((aW)'

1

u.

lim I ((aW) u0)(s)ds = 0 it follows that r2

9

(1.1.14) L := lim (W_ 1u')(x)

X—*T2

= u2' ( x0)+ | 2 ((aW_ 1u2)(s)ds<0. " " 0

Suppose L = 0. Then we have from (1.1.13) and (1.1.14) for all x e (xQ,r2)

0 < - ((u2)_ 1u^Xx) = (u2)"1(x).W(x) j " 2 (aW)"1(s).u2(s)ds

< W(x) f 2 (aW)_1(s)ds. J x

Integration of this inequality gives

0 < - log u2(x) < f W(x) f 2 (aW)_1(s)ds < oo. J xQ J x

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"1u')(x)> 0. x - r2

We define the function ü by ü := u« - Lk" u. It follows that

m := lim ü(x) = -Lk . lim u(x) > 0,

x—*X2 x—»r2 ~

lim (W"1ü')(x)= lim (W_ 1u')(x) c-»rT x-+ro '*■ x->r2 - L k "1 x ^ r2 Lk l. lim (W"1u,)(x) = 0, and (W"1Gl)l(x) = ( W "1u ' ) ' ( x ) - L k_ 1( W "1u,) ' ( x ) > 0

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 Urn u(x) > 0

1 x~*r2

Q G Ll(xQ,r2). Proof

Necessity. Let u be a positive decreasing solution of (1.1.6) with lim u(x) =

x - + r2

L > 0. Since u *(x) < 0 for all x G J it follows from (1.1.9) that for x £ ( x0, r . )

Y

0 < f ((aW)_1u)(s)ds< - u ' ( x0) .

Moreover

x , „x

0 < L f (aW)_1(s)ds< f ((aW)_1u)(s)ds

" J xQ J x0

for all x G (x0,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

. j x-r2

k := lim (W u')(x) < 0. It follows from (1.1.8) and (1.1.11) that for all x-+r2

x G ( x0, r2)

(1.1.16) - ( W_ 1u ' ) ( x ) = -k + f ^V((aW)_1u)(s)ds.

So or -(W V x x ^ L . f 2 (aW)_1(s)ds> 0 " J x - u ' ( x ) >L.W(x) f 2 (aW) 1(s)ds>0. 0 Since u ' e L (x0,r-) we have u(xQ) - L > L v0 and with Fubini's theorem we get

.s

f

2

W(t) f

2

(aW)

_1

(s)dsdt

r s u(x ) > L . [ 2 (aW)"'(s) [ W(t)dtds J XQ J XQ J xr = L . I 2 Q(s)ds, x0 so Q E L (xQ,r2).

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 lim (W_ 1u'Xx) =

X—*T2 ~ 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. J x

If k = 0, then we get

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

x—r2 u J XQ J t r s

= f

2

(aW)"'(s) f W(t)dtds

J x

0

J x

Q

= [

2

Q(s)ds.

J x0

Thus, if k = 0, then 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) W E L V X Q , ^ ) .

Since Q e L ( x0, r . ) it is clear that

(1.1.19) (aW)"1 GLl(xQ,T2),

and from (1.1.5) it follows that R e L (x0,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) lim u(x) = 0 holds <=>■

1 x-^ri Q 0 LJ(x0,r2).

Together with Lemma 1.1.7(iii) the next lemma shows the importance of the condition R £ L (xn,r„), while Q G L ( x - . r . ) , for the value of lim (W 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.) then each positive decreasing solution of (1.1.6) satisfies

lim u(x)>0, lim (W'^' )(x) = 0.

x->r2 J x-*r2 6

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? ■*

_1 ^

the proof of Lemma 1.1.8 we have k = lim (W u')(x) < 0, and also

x—»r2 ■*

(1.1.16) holds. From (1.1.16) we obtain for all x G (xn,r„)

(1.1.20) u^(x) = k.W(x) - W(x) f 2 (aW)_1(s).u3(s)ds.

Assume moreover R £ L (x„,r_). Since Q G L (xn,r~) we then have (aW)"1 G Ll(x0,r2) and W £ L1^ , ^ ) . From (1.1.16) we obtain

u ^ ( x ) < k . W ( x ) < 0 , x G (X ( ), r2) .

Since u^ G L (xQ,r2) it follows that k = 0. So from (1.1.20) we get

( W_ 1u p ( x ) = - J 2 (aW)"1(s).u3(s)ds

and thus lim ( W_ 1u ' ) ( x ) = 0.

x—>ro 3

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)u2~ U2U1 = K0W Witk K0 = (~U\U2 ~ U2U1^X0^ > °' Proof

Since u. en u - satisfy (1.1.6) we have

(au"j + ) 8 u J ) u2= (ou"2 + / 8 u p ur So or U1U2 - U2U1 A U5U2 "U2U1 ~a ( " j "2-u2ul > ' _ 0 u ! u - - u ' u . 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) < oo «=► Q(EL}(xn,r ) .

x-*r2 °

z

Proof

Necessity. By (1.1.8) W~ u ' is a positive increasing function. Assume

M:= lim (W_ 1u')(x) < oo. x ^ r2

Then there exists an x G (xn'r2^ s u c h t h a t f o r a 1 1 x e ^xl 'r2 ^

( W "1u ' ) ( x ) > ^ .

Thus for x > x. we have by integration

u ( x ) > u ( x0) + ^ f W(t)dt, X0

or

and thus

((aW)_1u)(x) > u(x0).(aW)_1(x) + ^ .Q(x),

(W_ 1u')(x) - (W"1u,)(x_)= f ( W ' ^ ' J ' W d t 0 J xQ

= f ((aW)_1u)(t)dt (by (1.1.8)) J x0

X X

>u(x

0

).J (aW)

_1

(t)dt + ^ J Q(t)dt.

By taking the limit for x —► r . we obtain

M - u ' ( xn) > u ( x A f 2 (aW)_1(t)dt + ^ f 2 Q(t)dt.

u u J x0 z J x0

Thus Q G L (xQ,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-+r2 *

exists an N, N G (0,oo], such that

N := lim ( W ^ u ' X x ) . x ^ r2

lim (W V x x J . u (x) = N.

x - r2 3

From Lemma 1.1.11 we have for all x e (xn,r»)

(W"1u')(x).u3(x)<K0<cx>.

Thus N < K» < oo. This completes the proof of the lemma.

D

Lemma 1.1.13

1,

Let u.M-f and Kn be as in Lemma 1.1.11. If R E L (x-,r ) and (aWf1 £ L](x0,r2),then

(1.1.22) lim (W'KlXx) = -K / lim u(x)<0.

x—*r2 ^ u x—*r2 l

Proof

Let R e Ll(x0,r2) and (aW)"1 £ L1^ , ^ ) . Then clearly Q jÊ L1(x( ),r2). Moreover u. is a bounded positive increasing function, thus M := lim u.(x)

1 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- 1u ' ) ( x ) = L < 0. x—»r2

On the other hand, since Q £ L (x„,r_) it follows from Lemma 1.1.12 that

lim (W_ 1u!)(x) = oo.

X—*T2

lim u9(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"1u!u-)(x) = Kn + LM,

X—*T2

thus lim (W u!u-)(x) exists and since u ! > 0 this limit is non-negative, x—»r2

say

(1.1.25) lim ( W_ 1u ! 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_ 1u;u2)(x) = u2(x) {J* (W_1uJ)«(s)ds + u | ( x0) ] . = u2(x) { f (aW)"1(s)u1(s)ds + u j ( x0) } 0 < u2(x) { u j ( r2- ) | (oW)" \%)ds + u J(X())]> T = u2(x) { u j ( r2- ) [ (aW)_1(s)ds + u j ( x0) } X0 + u2(x)Uj(r2-) ƒ (aW)_1(s)ds T < u2(x) { u , ( r2- ) J" (aW)" ^sjds + u " ( X Q ) } Y + U j ( r2- ) J (aW)_1(s)u2(s)ds

.T = u2(x) { u j ( r2- ) | (aW)"1(s)ds + u ' ( x0) } O x Uj(r2-) ^ ( W ^ u ^ d s = u2(x) { u ! ( r2- ) J (aW)_1(s)ds + u\(xQ)} + Uj(r2-) { ( W ^ u ^ x ) - ( W_ 1u p ( T ) } T (1.1.27) < u2(x) { u j ( r2- ) | (aW)_1(s)ds + u ^ ) } + e/2.

Since lim u-(x) = O there exists a number S e (T,r_), such that for all

x-+ro 2 z

x e ( S , r2)

T

(1.1.28) 0 < u2(x) < { u j ( r2- ) 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_ 1u J u2) ( x ) < e .

Thus

(1.1.29) lim ( W_ 1u ! u - ) ( x ) = K + L M = 0.

x—»r2 l l u

Combining (1.1.23), (1.1.24) and (1.1.29) we obtain

lim ( W_ 1u ' ) ( x ) = L = - K M "1 < 0.

x—r2 l u

This completes the proof of the lemma.

D

Remark. This lemma improves the result of Feller ([Fe,l], Th. 11) which

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

Regular if W e \}{xQ,x2), (aW)"1 G L\XQ,T2),

Exit if (aW)'1 É L ^ X Q , ^ ) , R G L ^ X Q , ^ ) , Entrance if W £ L (xQ,r2), Q G L (xQ,r2),

Natural in all other cases.

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 G L1^ , ^ ) , R G L1(XQ,T2),

Exit if Q £ L ^ X Q , ^ ) , R G L\X0,T2),

Entrance if Q G L (xQ,r2), R £ L ( x0 )r2) ,

Natural if Q £ L V Q , ^ ) , R £ L1^ , ^ ) .

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 r . Regular r . Exit r . Entrance r» Natural (i) (ii) (iii) W G L ^ X Q , ^ ) yes yes no yes no no ( a W r ^ L ^ X Q , ^ ) yes no yes no yes no R S L ^ X Q , ^ ) yes yes no no no no Q € L1( x( ), r2) yes no yes 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 h\x0,T2)

(ii) RGL1(X0,T2)=> W G L ^ X Q , ^ )

(iii) W 6 L \ x0, r2) A (aW)"1 e L1^ , ^ ) => Q,R G L1^ . ^ ) .

Simple 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 u2 with u (x) = sinh(l - x), u.(x) = sinh(l + x) form a fundamental set of solutions of u - u" = 0,

2

lim u . ( x ) = lim u_(x) = 0. x-+l l x—»-l z

2. J = (-7T/2,IT/2), xQ = 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 u . ( x ) = lim u,,(x) = 0.

X^-TT/2 l X-+7T/2 2

3. J = (-ir/2,jr/2), xQ = 0, a(x) = 1, /?(x) = -2 tan x. The points - J T / 2 , TT/2 are

entrance boundary points. The functions u . and u . defined by

u,(x) = (TT/2 - x)/cos x

U4( X ) = (7T/2 + X)/C0S X

form a fundamental set of solutions of

u - u" + 2 tan( . ) u ' = 0,

satisfying

lim u .(x) = lim u,(x) = 1.

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

/ \ x Uj(x) = e u2(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) = 0, lim (W'1 u\)(x) = -1

x—r2 z x^r2 z

and a positive decreasing solution « , of (1.1.6) satisfying

lim uJx) = l, lim (W~1 u\)(x) = 0.

x-*r2 J x->r2 i

Each solution of (1.1.6) is bounded on (xn,r?), but a solution u of (1.1.6) is positive and decreasing 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. of (1.1.6) for which lim W u\(x) < 00.

1 x-+r2 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,

p > O, not both zero. Then (W_ 1u'Xx) = p2( W_ 1u p ( x ) + p3(W_ 1u^)(x) < 0 for x G (xft,r_), so u is decreasing. Conversely, if u is a decreasing solution of (1.1.6) then there are p - , p , such that u = P2 UT + P iui - **y taking the limits for x -* r . we see

lim u(x) = p . . lim . u,(x) = p . > 0

x—>T2 x—*r2

and

lim (W"lu')(x) = p lim (W_ 1u»)'(x) = p > 0. x-+r2 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? and u - , lim (W u)(x) is finite.

z ■* 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, lim (W'1 u\)(x) = -1,

x-+r2 l x-*r2 z

and a bounded positive increasing solution u. with lim (W u \)(x) = oo.

1

x^r

2 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 u. is bounded.

_1 l

From Lemma 1.1.12 it follows that lim (W u')(x) = oo. x—»ro

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

lim u(x)=l, lim (W~1 u\)( x) = O,

x-+r2 5 x-^r2 J

and an unbounded positive increasing solution u. of (1.1.6) with

lim (W~1u))(x)<oo.

x^r2 1

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

fol-x ^ 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) = 0, lim (W~}u\)(x) = 0,

x—>r2 z x-*r2 l

and an unbounded positive increasing solution u. of (1.1.6) with

lim (W~1u\)(x) = oo.

x—r2 i

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 u')(x) = L,

x—►ror2 z as;

(1.1.6) follows from Proposition 1.1.1 and Lemma 1.1.4. Assume L < 0, then then L < 0. The existence of an unbounded positive increasing solution u. of

lim (-W"1u»)(x).u.(x) = oo,

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)

r- Regular r_ Exit r- Entrance r_ Natural I yes yes no no II yes no yes no III yes yes no yes IV no yes no 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

-1 2 2 State diagram for lim (W u ' )fx), where u - aD u - 0Du = 0, u € C (J)

x^r2 r„ Regular r . Exit r» Entrance r„ Natural I yes no yes no II yes yes no no III yes no yes yes IV no no yes yes I II III IV

For each positive increasing solution u lim (W u ')(x) is finite. x - r2

There exists a positive decreasing solution u-with lim (W u l ) ( x ) < 0 . ^ x—>r*> ^ There exists a positive decreasing solution u , with lim (W ul)(x) = 0.

■* X—>f) J

For each positive decreasing solution u lim (W 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.

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.

Lemma 1.1.18

Let the continuous bisection <f>: J —► J satisfy the following conditions:

<t> e C2(J), '<j> *(t) > 0 for all t e J, <f>(tQ) = xQ, <j>d.) = r. (i - 1,2) and let <f> 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 expression

(1.1.30) 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) W(t) = exp{-\ (ior1)(s)ds].

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 tfa^Xfls^'feJds J t0 J t0

"

It,

*

(sW(s))

- 1

ds

v0 • *(t) f YK ' - 1 J xQ

So by (1.1.31) we have

W(t) = (W o <t>)(t)-<t> *(0 and it follows that

Moreover,

hence

W e L

1

^ ) <=* W e

L\X0,T2).

(«W)"1(t) = ((aW)"1 o 0(t).* "(t),

(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,

°

R(t):= W(t) f (aW)_1(s)ds. 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!( t0, r2) ^ * 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.

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 - (aD2u + 0Du) = f, f e C(J), u e C(J) n C2(J),

denoted by uf.

Then the general solution of (1.1.32) in C(J) is given by

u = uf + c . u . + c^u2

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 D2U + £Du)(x) = 0, i = 1,2.

x->r i

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

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:

(1.1.34)

T(x,s) = ( K0Q W ) "1( S ) U1( S ) U2( X ) f o r r j < s < x < r2

= ( 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

KQ = u j(xQ).u2(x0) - U ^ X Q X U ^ X Q ) = U J U Q ) - u2'(x0). For each f e C(J) we define the function uf by

(uf)(x) = | ^

(1.1.35) (uf)(x)= | 2 r(x,s)f(s)ds. 1

Then the mapping f - » uf 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|| f 2 r(x,s)ds J rl = llfll {u2(x) J ((K0aW)"1u1)(s)ds Uj(x) J 2( K0a W ) '1u2) ( s ) d s } 1 +

^ j ' l l f H { u2( x ) J (W"1up'(s)ds

Uj(x) ƒ 2( W_ 1u p ' ( s ) d s } . 1

+

Since W u ! is positive increasing and W uJ is negative increasing both in­ tegrals exist. It follows that for each x € ( r . , r . ) uf(x) is well-defined. We have for all x e (r. ,r-)

uf(x) - u2(x) I ((KQaW)"J u j f)(s)ds + Uj(x) J 2((K( )aW)"1u2f)(s)ds. -Xhen-u^-is-twice-continuously-dif-ferentiable-with Y u£(x) = u ^ x ) J ((K0aW)_ 1 U lf)(s)ds

H{W f

x 2((K0aW)"1u2f)(s)ds and Y u"{(x) = u2(x) I ((K0aW)"l u j f)(s)ds u"(x) J 2((KnaW)"1u0f)(s)ds ' 0 " " ' - 21 ( ( u J u2- u ^ u1X K0a W ) "1f ) ( x ) .

From Lemma 1.1.11 we have

( u j u2 - u^u1)(x) = K0.W(x).

uf(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. < < a . < a < < a. < b . < < b < b . < < r„.

1 n+1 n 1 1 n n+1 2 Moreover, let (f ) be a sequence of continuous functions such that

fn(x) = f(x) x e [an,bn],

0 < f (x) < f(x) x e (a , , a ) u ( b , b . ) , nv ' v ' v n+1 n n n + 1 " f (x) = 0 x € (r.,a , ) u (b , , r . ) .

nv ' v 1 n+1' v 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 '

uf ( x ) - ( a u " )(x)-(/Ju!. )(x) = 0,

rn ln ln

and for x e (r. ,a .) we have by the definition of ur

v 1 n+1 ' f

uf (x) = Uj(x) . | 2 ((KoaW)_1u2fnXs)ds > 0 n+1

and similarly for x e (b .,r_)

Thus uf is positive increasing on (r.,a .) and positive decreasing on (b ,,r_). Hence u,. has a positive maximum at a point x e (a , ,b .).

v n+1 2' fn m n+1 n+1

Then u ' (x ) = 0 and u" (x ) < 0. So fnv m; fnv m' u , (x ) = f (x ) + (au" )(x ) fnv m' nv nv v fn' nv < f (x ) < ||f||. - n\ m / ii ii Thus certainly (1.1.36) l | uf| | < | | f | | ' n for all n e l .

Because of~the posiïiVity of-KT~the sequence (uf (x))_is~increasing for each n

x G J, and since

[

2

r(x,s).(f .

J r j n

(s) - f(s)) ds

converges to zero, for each x G (r.,r_) the sequence (ur (x)) converges from

l Z tn

below to uf(x). Thus also

(1.1.37) ||uf || < ||Uf||.

Applying Lebesgue's Monotone Convergence Theorem on the integral repre­ sentation (1.1.35) for uf and using (1.1.36) we obtain

' n

(1.1.38) ||Kf|| = ||uJ| = sup uf(x) = sup lim uf (x) < sup sup \xf (x)

1 XGJ l XGJ n-+oo ln XGJ nQN ^n < sup sup ||uf || = sup ||uf || < sup ||f|| = ||f||

XGJ n&N In nelT rn 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).

The boundedness of uf follows from (1.1.38), but this does not guarantee that uf e C(J). So we have to prove that lim uf(x) exists for i = 1,2. We will

1 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) uf(x) = af(x) Uj(x) + bf(x) u2(x) with af(x)= f 2( ( K a W ) " ' u f)(s)ds and bf( x ) = J ((K0aW)_ 1 Ulf)(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) = lim u/x) = 0. x-+r2 -x-r2 I

Proof

If r„ is a regular or an exit boundary point, then lim u.(x) = 0, lim u.(x)

z x-*i"2 ~l x—>*2

is finite and lim af(x) = 0. Moreover, if r» is regular, then also lim bf(x)

x-+r2 * l x-»r2 f

is finite. In view of (1.1.39) we then have lim uf(x) = 0. x-+r2 *

If r_ is an exit boundary point, then clearly lim af(x) u.(x) = 0, so it

re-z x-»r2 '

mains to show that

(1.1.41) lim bf(x) u-(x) = 0.

we have

|bf(x)|.u2(x) < K Q1 ||f||.u2(x) ƒ ((aW)"1u1)(s)ds

= K Q1| | f | | . u2( x ) | ( W ^ u j V W d s

= KQ1||fl|.u2(x){(W"1u1')(x)- lim ( W_ 1u ; X t ) }

s K ^ l l f l l . u ^ x M W ^ u j X x ) . From (1.1.29) and the proof of Lemma 1.1.13 we have

lim u (x).(W"1u')(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) = lim u/x) = lim f(x). x—*r2 x-*r2 ' 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 (xQ,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) uf(x) = f(x) f 2 r(x,s)ds + g(x) J rl

with g(x) = | 2 r(x,s).(f(s) - f(x))ds. Thus *

2'

By the definition of V (1.1.34) we have

(1.1.44) |g(x)| < e f 2 r(x,s)ds, x Ê ( b , r , ) . J rl

2 (W"1uj)'(s)ds (by 1.1.8)

r Y

f 2 r(x,s)ds = u2(x) ƒ (K0aW)"1(s)u1(s)ds

+ Uj(x) J 2(K0aW)_ 1(s)u2(s)ds

= U2( X ) . K Q1. | ( W ' ^ V ^ d s

+ u1( x ) . K -0 1. ^2

= u ^ . K ^ . ^ W ^ U j ' X x ) - ( W ^ ' X r , * ) }

+ u ^ x J - K J ^ . / o - (W_ 1u^)(x)\ (by Lemma 1.1.17)

= 1 - KQ1.u2(x).(W"1u1l)(r1+) (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

-P

»r-> J r .

(1.1.45) lim I 2 T(x,s)ds= 1 x-+r2 J r j

if r_ is a natural boundary point. Finally it follows from (1.1.43) - (1.1.45) that

lim u.(x) - f(x) f 2 T(x,s)ds = 0

i-*2 f J rl

and with (1.1.45) we obtain (1.1.42).