Solvability of Nonlinear Equations and Boundary Value Problems 9027710775, 9789027710772

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Solvability of Nonlinear Equations and Boundary Value Problems

Mathematics and Its Applications

Managing Editor: M. HAZEWINKEL

Department of Mathematics, Erasmus University, Rollerdam, Holland Editorial Board: R. W. BROCKETT, Hamard University, Cambndge, Mass., U.SA. Yu. I. MANIN, Stek.lov Institute of Mathematics, Moscow, US.S.R. G.-C. ROTA, M.I.T., Cambridge, Mass., U.SA .

Volume 4

Professor SVATOPLUK FUCiK 21. 10. 1944- 18.5. 1979

Svatopluk Fucik Proftssor of Mathematical Analysis Charles University, Prague, Czechoslovakia

Solvability of Nonlinear Equations and Boundary Value Problems

D. REIDEL PUBLISHING COMPANY Dordrecht: Holland/ Boston: U.S.A./ London: England

Library of Congress Cataloging in Publication Data

Fucik, Svatopiuk Solvability of Nonlinear Echt, Holland in rn-edition with Societv of Czt>choslovak Mathematicians and Phvsicists Pra1-,rue - Husova .'>, Czt>choslovakia. Sold and distributt>d in Albania, Bulgaria, Chinesr People's Rt>public, Cuba, Czechoslovakia, Gt>rman Ot>mocratic Rt>public, Hungary, Kort>an Pt>oplt>'s Ot>mocratic Republic, Mongolia, Poland, Rumania, the U.S.S.R., Vietnam, and Yugoslavia by Artia: Prague. Sold and distributed in the U.S.A. and Canada by Kluwer Boston Inc., Lincoln Building. JliO Old Derby Street, Hingham, MA 0:.104:-l, U.S.A. In all other countries, sold and distributed by Kluwn Academic Publishers Group. P.O. Box

:-1:n. :noo

AH Oordrecht, Holland.

D. Reidel Publishing Company is a mt>mber of the Kluwer Group. All Rights Reserved Copyright © 1980 bv Society of Czechoslovak Mathematicians and Physicists. Prague. Czechoslovakia. No part of tht> material protected bv this copyright notict> mav be reproduced or utilized in anv form or bv anv means. electronic or mechanical, including photocopving, recording or bv any informational storage and retrieval system, without written permission from the copvright owner. Printed in Czechoslovakia

C O N T E N T S Preface Chapter

1: Notation

1

Chapter

2: Introduction

5

PART I

: BOUNDARY VALUE PROBLEMS FOR ORDINARY DIFFERENTIAL EQUATIONS

10

Chapter

3: Sobolev spaces, Nal

11

Chapter

4: Weak solutions of boundary value problems for ordinary differential equations

13

Chapter

5: Bounded nonlinear perturbations of linear invertible operators

26

Chapter

6: Bounded nonlinear perturbations of linear noninvertible operators

30

PART II

: NONLINEAR PERTURBATIONS OF LINEAR INVERTIBLE OPERATORS

39

Chapter

7: Abstract results

40

Chapter

8: Solvability of boundary value problems for ordinary differential equations

44

PART III : NONLINEAR PERTURBATIONS OF LINEAR NONINVERTIBLE OPERATORS

53

Chapter

55

9: Alternative lemma

Chapter 10: Abstract results of the Landesman-Lazer type

59

Chapter 11: Sublinear nonlinearities

67

Chapter 12: Nonlinear operators with a linear growth

71

Chapter 13: Solvability of boundary value problems for ordinary differential equations with bounded nonlinearities

73

Chapter 14: Solvability of boundary value problems for ordinary differential equations with sublinear nonlinearities

86

Chapter 15: Solvability of boundary value problems for ordinary differential equations whose nonlinearities are of a linear growth

95

PART IV

BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS

103

Chapter 16: Sobolev spaces (N~l)

104

Chapter 17: weak solutions of boundary value problems for partial differential equations

107

Chapter 18: Nonlinear perturbations of linear partial differential operators

110

PART V

: PERIODIC PROBLEM FOR ORDINARY DIFFERENTIAL EQUATIONS (INTRODUCTION) Chapter 19: Linear differential operators with constant coefficients in spaces of T-periodic functions; some examples PART VI METHOD OF A PRIORI ESTIMATES Chapter 20: Definition and properties of the Leray-Schauder degree Chapter 21: Abstract reeult Chapter 22: Existence of periodic solutions of generalized Lienard vector equations with forcing terms APPENDIX A: Historical and biblipgraphical notes PART VII : METHOD OF TRUNCATED EQUATIONS Chapter 23: Introduction Chapter 24: Dirichlet problem for nonlinear second order ordinary differential equations Chapter 25: Boundary value problem with bounded nonlinearity and general null-space of the linear part Chapter 26: Periodic solutions of ordinary differential equations PART VIII: POTENTIAL EQUATIONS Chapter 27: Elementary critical point principle Chapter 28: Abstract result Chapter 29: Nonlinear elliptic boundary value problems Chapter 30: Variational approach PART IX : EQUATIONS WITH SUPERLINEAR NONLINEARITIES Chapter 31: Application of the alternative lemma Chapter 32: Dirichlet problem for nonlinear second order ordinary differential equations Chapter 33: Abstract result of Kazdan-Warner type Chapter 34: Classical solutions of Dirichlet problem for second order partial differential equations Chapter 35: The method of P. Hess PART X : EQUATIONS WITH RAPID NONLINEARITIES Chapter 36: Introduction Chapter 37: Auxiliary lemmas Chapter 38: Proof of Theorem 36.8 Chapter 39: Proof of Theorem 36.10

117

118 132 134 140 144 157 163 163 169 175 187 196 197 203 211 221 233 234 240 247 253 265 276 276 280 295 299

PART XI : EQUATIONS WITH JUMPING NONLINEARITIES Chapter 40: Ranges of perturbations of positively homogeneous operators Chapter 41: Results of the Ambrosetti-Prodi type Chapter 42: Second order ordinary differential equations Chapter 43: Results of E. N. Dancer Chapter 44: Further examples PART XII : EPILOGUE Chapter 45: Periodic solutions of boundary value problems for nonlinear heat equation Chapter 46: Periodic solutions of the nonlinear telegraph equation Chapter 47: Existence of multiple solutions of nonlinear equations

303

REFERENCES

358

APPENDIX B: Supplementary references

373

SUBJECT INDEX

382

LIST OF SYMBOLS

389

304 314 322 326 332 335 336 343 347

F O R E WO R D svatopluk Fucik died prematurely on May 18,1979. He was 34 years old and knew since 1973 that his time was severely limited. 1973 is also the year when Fucik wrote the first of twenty-one papers devoted to nonlinear noncoercive problems, the main topic of this monograph. The final version of the monograph was produced in hospital, in April 1979. Having had the privilege of knowing Svatopluk Fucik personally and of working with him, I could easily add many comments about his optimistic and attractive personality, but I think that the above short description of his last period of mathematical activity tells more than anything else about his positive and courageous attitude in the face of inescapable reality. This monograph is devoted to the solvability of equations of the form

Lum Su

(1)

in Banach spaces, with operator and

S

L

a linear (in general non-invertible)

a nonlinear mapping; and to applications to

ordinary and partial differential equations. Finding its roots in the work of Lyapunov and Schmidt on nonlinear integral equations, the study of this type of equation has became an extremely active domain of research in the seventies, thanks mainly to the pioneering work of Cesari in the case of large nonlinearities

S

and to that of Landesman and Lazer on ne-

cessary and sufficient solvability conditions for classes of bounded nonlinearities

S . The dramatic growth of interest

in those questions is reflected in the bibliography of this monograph as well as in the various concurrent terminologies used to describe situations covered by (1). In addition to the oldest expressions such as 'Lyapunov-Schmidt method' and 'branching or bifurcation theory'

(which is used today in a

more specific sense), one has used such expressions as 'noncoercive nonlinear equations', 'semicoercive nonlinear problems•, 'Landesman-Lazer problems', 'coincidence degree',

'alternative problems',

'nonlinear equations with linear asymp-

totes',

'nonlinear problems at resonance',

-Schmidt methods',

'global Lyapunov-

'weakly nonlinear equations',

'abstract

semilinear equations'. ReasQns for this recent interest in problems of type (1) lie in the number and the variety of occurring phenomena as well as in the diversity of tools used in their study. Fixed point theory, degrees type arguments, variational methods, monotone operator techniques - as well as many specific aspects of the theory of ordinary and partial differential equations have been successful in attacking those problems and, in many cases, refined or generalized versions of these basic tools have emerged from the researche. A characteristic feature of Fucik's work in this area lies in his energetic ability to use and combine those abovementioned approaches in th~ study of (1) and in the nwnber of new specific situations he discovered by a sharp analysis of simple-looking ordinary or partial differential equations. All those characteristics are present in this monograph, which gives a thorough account of the state of the art in the study of the range of semilinear operator equations and of applications to noncoercive boundary value problems. Since the main concepts and results of nonlinear analysis which are used are carefully recalled, this book can serve as an introduction into this field for readers having a reasonable background in functional analysis and differential equations. On the other hand,

the exhaustive and up-to-date character of its content

makes this monograph indispensable for specialists in current nonlinear problems. Finally, the numerous open questions listed by Fucik in every chapter will serve as landmarks for the future development of this part of nonlinear functional analysis and boundary value problems. This is the way in which Svatopluk Fucik will continue to contribute to the development of a domain of analysis which already owes so much to his remarkable ingeniousness and formidable energy. September 1979

Jean Mawhin

P R E F A C E These notes formed the basis of a course entitled 'Nonlinear Problems' given at the Department of Mathematical Analysis, Charles University, Prague, during the years 1976 and 1977. The subject of interest is that part of nonlinear functional analysis which deals with the solvability of nonlinear operator equations. The exposition proceeds so that before general results are presented, the method is illustrated by applying it to the Dirichlet problem for a simple ordinary differential equation of the second order. Where possible, the general results are applied to boundary value problems for nonlinear ordinary differential equations as well as for nonlinear partial differential equations and they are also used to prove the existence of periodic solutions of nonlinear ordinary differential equations (in the Epilogue applications to nonlinear partial differential equations both of parabolic and hyperbolic types are also mentioned). The text is not intended to serve as a textbook. Proofs of the auxiliary facts about linear differential equations and about function spaces (e.g. about the Sobolev spaces) are not included. The properties of the Leray-Schauder degree and the Schauder fixed point theorem as well as the main results about the surjectivity of monotone coercive oper~tors are used without proofs. The proofs and a detailed investigation of the above mentioned facts have been included in the study program at Charles University which was very convenient for both the instructor and the students since before the lecture 'Nonlinear Problems' almost all preliminaries had been prepared. In order for the text of these notes to be selfcontained, these preliminaries are collected without proofs with references to literature. The problems of solvability of noncoercive nonlinear

equations have been very popular recently. Therefore, it was the author's desire to give a survey of results concerning these problems which are known at present, as well as to show the methods which have been used to obtain them and some of their consequences to individual concrete problems. It should be mentioned that due to a lack of time some examples were not calculated in all their details but solved merely by intuition. The original intention was to provide a 'complete' list of references. Of course, to fulfil this intention has proved almost impossible. Thus, it is possible that some of the mentioned 'open problems' are solved in literature unknown to me. Thanks are due to all participants of the course who suggested many valuable improvements. The text of the lecture was mimeographed during the year 1977 and this mimeographed version became the basis of the present lecture notes. I obtained many valuable and useful comments on the original version from many mathematicians and I thank mainly H. Amann, L. Cesari, E. N. Dancer, P. Drabek, J. K. Hale, P. Hess, G. Hetzer, K. Iseki, J. Mawhin, J. Milota, K. R. Schneider - each of them shares merit for the fact that the lecture notes have come into existence. Finally, I thank J. Jarnik who contributed to the quality of the English translation. I shall be very grateful for any remarks, comments and suggestions concerning the present lecture notes. December 1978

Svatopluk Fucik Department of Math. Analysis Faculty of Mathematics and Physics Charles University Sokolovska 83 18600 Prague Czechoslovakia

C H A P T E R

l

N O T A T I O N

1.1. Denote by ~ the set of all positive integers. Let n be a fixed bounded domain in a Euclidean N-space RN , with the boundary an and the closure n = n U-an • A general point of ~N will be denoted by x = (x 1 , ••. ,~) . All functions defined on n will be assumed real-valued. If u is a function defined on n then denote u+

x ,-.. max {u(x),O}

(the positive part of the function u

u) and

: x I-+ max {-u(x},O}

(the negative part of the function Integration over

n

u ).

will be considered always with res-

pect to the Lebesgue N-dimensional measure. 1.2.

For any N-tuple of nonnegative integers Cl

=

( Cl l •... '"N)

(called a multiindex), the corresponding differential operator

of the order

Ia I

"1 +

• • • + "N

is written as

o" . 1.3.

Let

X

and

Z

be two vector spaces. If

ping defined on the set Z

Dom[TJ C X

is a map-

(we write

C X

T : Dom[TJ if

T

with values in the space

Dom[TJ

X

we write only -

T l

-

-+

z

x-+ Z ), denote by

Im[TJ

the ~et of all values of the mapping Im[TJ - T

T. i.e.

(Dom[TJ).

linear mapping. In this X -+ z be a -is a vector subspace of that case we shall suppose Dom[L] 1.4.

Let

L : Dom[L]

C

the vector space

X

the operator

i.e.

L

•

and denote by

e}

Ker[L] = {u € Dom[L]; Lu., 1.5.

Let

X

X

*

X

*

and

the symbols in

u

ex

Let

C

X -+

T : Dom[T]

each bounded set

1.7.

COMPLETE CONTINUITY.

X

z

M

of

T

is continuous on

Dom[T] , T{M)

1.8.

IMBEDDING.

then

X

(!)

•

We shall use

Z

be two Banach spaces.

is said to be bounded if for is a bounded set in

C

•

X and Z are real z • Then T is said to

X -+

Dom[T]

if for each bounded sub-

is a relatively compact set in Dom[T]

Let

Z

Suppose that

be completely continuous on set

and

T (M)

T : Dom[T]

and

The pairing between

X, respectively.

MC Dom [TJ

Banach spaces. Let

•

< .u * ,u >

to denote the strong convergence

ll~tl

"-+"

BOUNDED MAPPINGS.

A mapping

X

is denoted by

X and weak convergence in

1.6.

I Iul Ix , e

denotes the adjoint (dual) space of

all bounded linear functionals on €

.

be a real Banach space with a norm

its zero element;

u*

the null-space of

Ker [L]

X

Z

.

and

Z

is continuously imbedded into

be two Banach spaces:

z

X

(we write

4

Z)

if

u € X

{a)

every element

(b)

every convergent sequence in in

is also an element of

Z , and

X

is also convergent

X

and

z

Thus the imbedding operator Z

i

: X

i

: u f----+ u

----+-

defined by is a linear continuous mapping between

Z

it is bounded, i.e. there exists a positive number -

2 -

and hence

k

such

that

for all

u

(ii)

z

ex. The space

(we shall write

z

i: X 1.9.

X X

is said to be compactly imbedded into

GG

if the imbedding operator

Z )

is completely continuous.

(!) Ck(n) is the colu and its derivatives are defined and continuous on n •

SPACES OF CONTINUOUS FUNCTIONS.

lection of all functions of order·~ k (ii)

ck (O)

u

such that

is the collection of all functions

u

from

Ck(O) --:uch that Dau for !al ~ k are uniformly continuous on n • Ck(O) is Banach space with respect to the norm

I lul I e

k - C (O)

r

sup joau(x)I • lal!_k x f n

then u and all its derivatives up to the In the are uniquely continuously extensible on k shall tacitly assume that u EC (0) and its desequel, we rivatives Dau for !al ~ k are defined and continuous on If

u

order

Ck(O)

n.

k

0 • (iii) Let k CE N U {O} and A e (0,1] • Denote by Ck'A(O-)-the collection of all functions u in ck(O) such that

D0 u

K ~ o

for

lal • k

are hQlderian, i.e. there exists

such that jo 0 u(x)-D 0 u(y)j < Kj jx-yl IAN, -

ck,A(O)

x, YE n •

R

is. a Banach space with respect to the norm

I I u I I c k • A(O) -

-

I Iu I I k -

c

(O)

+ I I

r

a •k

sup

x,y, n x~y

Ck':\(O) and C~(O) we denote the set of all 0 respectively, whose defunctions from Ck'A(O) and Ck(O) vanish on the boundary ao k-1 the order rivatives up to (!Y)

By

.

-

3 -

LEBESGUE SPACES. (!) Lp(n) is the collection of all equivalence classes (almost everywhere) of functions u(x) defined on n and such that 1.10.

Jlu(x)IPdx

< •

D

for a fixed p, l ~ p respect to the norm

< •.

LP(n)

is a Banach space with

I

1/p

I lul IL (n) • ( lu(x) !Pax) P n

•

(ii) For 1 < p < • , Lp(n) is a separable reflexive Banach space with the dual space Lq(n) , where p-1 + q-1 • l •

u f: LP (O) and V E Lq (a) 1 < p < •) then by the Holder inequality, over n and (iii)

If

J lu(x)v(x) ldx n (iv) product

L2 (n)

~

(p-l ♦q-1 • l

u•v

,

is integrable

I lul IL (n) I lvl IL (n) • P q

is a Hilbert space with respect to the inner


L (n) • Ju- u 0 in X it is 2.4.

EXAMPLES.

(1)

- *

Tun_,_ Tu0

in

x*

Let there exist a mapping t : Xx X - X* such that (c) t is bounded, (d) -t(u,u) • Tu for each u e X ; for all u, w, h EX and for an arbitrary sequence of real numbers such that tn-+ o it is t(u+tnh,w) _.. t(u,w) (f) (g)

for all if

un -

u, w ~ X it is < t(u,u)-t{w,u), u-w > u

in

X

~

o

and

lim < t(un,un) - t(u,un), un - u n ..... then for an arbitrary w E X it is t(w,un) _.. t(w,u) (h)

if

in

x*

> •

< z, u > •

> •

o ,

w E X

then lim < t(w,un) , un n-+-o,

The above conditions imply that the operator T is pseudomonotone. The main result concerning the surjectivity of pseudomonotone operators is the following theorem.

2.5,

THEOREM.

Let X be a reflexive and separable Banach T: X-+ X be a pseudomonotone operator which is coercive.

-

space and let Then 2.6.

T

REMARKS.

is surjective.

The concept of monotonicity was probably in-

troduced first by GOLOMB

[1J to be used for an investigation

of nonlinear integral equations (see MINTY

[2]);

an explosion

of papers involving monotone operators and their generalizations started with the papers by VAJNBERG-KACUROVSKIJ

[6] , MINTY [7j . (8] , MINTY [9] •

ZARANTONELLO SKIJ

[4], [s],

For survey papers, see KACUROV-

The first application of Theorem 2.2 to some boundary value problems for nonlinear elliptic partial differential

[10] and [11]), For applications of the theory of monotone operators

equations is due to BROWDER (see his survey papers

to the solvability of nonlinear integral equations see e.g. VAJNBERG [12],

[13], DOLPH-MINTY [14], FIGUEIREDO-GUPTA [15],

CARROLL [16] , BROWDER [1 7] • Elliptic partial differential operators satisfying the conditions introduced in Example 2.4 (ii) were investigated by VISIK [18], a generalization to the solvability of abstract operator equations was given by LERAY-LIONS [19]. Definition 2.3 of the pseudomonotonicity weakens the definition due to BREZIS [20]

(by avoiding an explicit condition on fil-

ters rather than sequences and dropping the boundedness hypothesis). The theory of monotone operators and the generalization as well as the applications are explained, for example, in the books by LIONS [21]

(where the proofs of assertions from Sec-

tions 2. 4 and 2. 5 are also given), DUBINSKIJ [22], SKRYPNIK [23] ,

[24] , PASCAL! [25], ... Note that t.he assumption of separability in Theorem 2.5

may be omitted and that operators on nonreflexive Banach space have been considered (see e.g. GOSSEZ

[26]).

The theory of monotone operators plays an essential role -

7 -

not qnly in the solvability of nonlinear differential equations, but also in quite different fields of mathematics, for example in the theory of games - see e.g. ROCKAFELLAR [27], BENSOUSSAN 2.7.

[28] ,

[29] .

LEMAIRE

Since 1967 the assumptions of coercivity and monotoni-

city have been replaced (see e.g. POCHOZAJEV [30], NECAS [31]) by a weaker assumption

(2)

.

"'

This is the case with the Fredholm alternative for nonlinear operators presented in FUCfK-NECAS-SOUCEK [32, Chapter II]. 2.8.

Here we shall consider a special case of the equation

(1), namely, we shall assume the operator

T

to be of the

form Tu = LU - Su

(3)

where

L

is linear and

S

is a (generally) nonlinear ope-

rator. The most interesting case will be that of an operator L which is not necessarily continuous and invertible so that the equation (1) obtained will involve T satisfying neither the coercivity condition nor the condition (2).

A typical example of an operator equation (1) with (3) is the following: Consider a second order ordinary differential equation

(4)

- u'' (x) - g(u(x)) = f(x)

,

x E: (0,1r)

and the boundary conditions U(O) = U(1r) • 0 ,

(5)

where f

g

is a continuous function on the real line

R1

and

is a given right-hand side. We shall suppose that there

exist limits (6)

JI

-

lim E;-+oo

.

filil E;

\)

= lim E,+-=

.s.fil E;

and we shall distinguish the following types of the nonlinea·-

8 -

rity

I.

g µ

for example, the differential equation - u'' - u + arctan u •

µ;

II. JUMPING NONLINEARITY:

v

f

•

for example, the differen-

tial equation

- u'' - µu + + vu with

µ,

- f

v € ~l , or the equation

- u'' - AU+ eU

III. RAPID NONLINEARITY:

µ

•

a

~

v •

f

, for example, the diffe-

rential equation - u'' - u 3 2.9.

f

.

The purpose of these notes is to present some results

about the solvability of the above types nonlinear equation. We shall consider not only second order nonlinear ordinary differential equations but also higher order and partial differential equatio~s. Moreover, the boundary conditions of the Dirichlet type will be in some cases generalized to general boundary value problems. ror ordinary differential equations and their systems, periodic problems will also be considered.

-

9 -

P A R T B O U N D A R Y

0 R D I N A R Y

V A L U E

I P R O B L E MS

D I F F E R E N T I A L

E

F O R

QUA T I O N S

This part is nothing other than a continuation of the Introduction. First we recall the definition and basic properties of the Sobolev spaces of functio~s of one variable (Chapter 3) and define the notion of a weak solution of the boundary value problem for an ordinary differential equation (Chapter 4). In Chapter 4 we shall also apply the theorems about the surjectivity of monotone and pseudomonotone operators which were mentioned in Chapter 2 (see Sections 2.2 and 2.5). We also prove a theorem on the regularity of weak solutions of boundary value problems for ordinary differential equations (i.e., sufficient conditions are given under which the weak solution is the classical one) which will be used in the sequel. In Chapter 5 we shall explain the main idea of weakly coercive equations in which the operator has the form of the sum of a linear invertible ordinary differential operator of the second order and a nonlinear compact perturbation (an abstract theorem will be given in Part II). The same problem, but in the case of a linear noninvertible ordinary differential operator is considered in Chapter 6; the abstract theory is given in Part III.

-

10 -

C H A P T E R

S O B O L E V

3.1.

DEFINITION.

wk,P(a,b)

Let

3

N = 1

S P A C E S

k e

~

and

p €

the set of all functions

which are absolutely continuous on

k-1

derivatives up to the order

[1 ,oo)

• Denote by

u on the interval [a,b] [a,b] together with their

and whose derivatives

(which exist almost everywhere) are elements of 3.2.

It is easy to see that the function k

(1)

I lul lwk,Pca,b) •

(i)

(i~ol lu

p 1/p I ILP((a,b)))

(u E wk•Pca,b)) is a norm on the linear space

3.3.

DEFINITION.

wk•Pca,b)

3.4.

wk•Pca,b)

The Sobolev space is a normed linear space

with the norm defined by the relation (1),

THEOREM.

(!)

wk·Pca,b)½½ ck-l([a,b])

(ii)

wk•Pca,b)

is a Banach space;

(iii}

wk•P(a,b)

is separable;

(iv)

wk•Pca,b)

5.s a reflexive space if and only if

E (l ,"')

(~)

wk•P(a,b)

is a Hilbert space with the inner product k

w·• (a,b)

E

p e

;

k 2

(u, v

u(k)

Lp((a,b)) •

wk• 2 (a,b)).

•

l

i=0

L 2 ((a,b))

(For the proofs of the abovementioned main

properties of the Sobolev spaces see e.g. KUFNER-JOHN-FUC!K

[33]; Czech readers can use also FUC!K-MILOTA [34].) 3.5.

NOTATION.

Put

~•P(a,b) • {u £ wk•P(a,b); u(a) • ••• • u(k-l)(a) • = u(b) s u(k-l)(b) = o} . -

11 -

3.6.

The function llull_k p

Wei'

•

l:u X

IR 1

and a constant

c ~ o

x

such that for all

we have

IRk+l

~ K(x;t 0 ,

(lo}

1Ak(x;t 0 , . . . ,tk>I

..• ,(k-l) + cJtklp-l

(11)

JA1(x:to, ... ,1;k>I !. K(x;to•• .. ,tk-1> + cj,;kjP

,

(1 = O, ••• ,k-1).

Let

♦ e wk•Pca,b) • Let

f E L 1 ((a,b)) ,

subspace of

wk,,P(a,b)

v

which contains the space

be a closed ~•P(a,b)

as a closed subspace, i.e.

~•P(a,b) (; V (; wlt•P(a,b) .

(12)

Finally, let (13)

Ge

v*

G(~•P(a,b)) •

A function

u

e

V

dary value problem

be such that

(e} .

is said to be a weak solution of the boun{V,G, ♦}

for the differential equation (9)

if the integral identity

k

(14)

b JA 1 (x;u(x)+~(x),u'(x)+~'(x), ••. ,u o

if

>. < o

if

A• O

Thus an arbitrary nontrivial function (6) satisfying the boundary conditions (5) has the form

(7) >. = n 2

where 5.5.

a IO,

u(x) • a sin f i x .

for a positive integer

n.

From the so-called Fredholm alternative for linear equa-

tions involving a completely contin-.1ous operator (see e.g. DUNFORD-SCHWARTZ [39], KOLMOGOROV-FOMIN [40], LJUSTERNIK-SOBOLEV [4(1, •.• ) it follows that L maps the space w~• 2 co,11) onto itself if

>.

I

n2

n

5.6.

IN

€

tinuous linear inverse

and, moreover, there exists a con-

L-l

THE PROPERTIES OF

S •

w~• 2 co,11)

(!)

The imbedding

½ c;c 0 c[o,ir]>

implies immediately that the mapping

w~• 2 co,11) -,. W~• 2 (o,ir)

S\

is completely continuous. (ii)

Since

exists a constant

:

c

I I Sui I

(8)

for any 5,7.

1j,

u E

!R 1 -,. 1R 1 >

O 1 2

w0 • (0,11)

w~• 2 co,11)

is bounded and continuous there

such that ~ c

•

The solvability of the equation (2) for

>.

I n2

is equi-

valent to the solvability of the equation

u • L- 1 su

(9)

in the space

w~• 2 co,n) . To prove that the equation (9) has

at least one solution we use Schauder fixed point theorem. 5. 8.

SCHAUDER FIXED POINT THEOREM.

Let

::J(,

be a convex

bounded closed nonempty subset of a Banach space

- 28 -

X. Let

F : '){.,

C

Dom [F]

-+

X

be a completely continuous mapping. Then there exists at least one Fu 0 5.9.

REMARK.

to SCHAUDER

1C

u0 E

such that

u0

=

The original proof of the above theorem is due

[42], [43]. The list of references containing the

proof of the fixed point theorem for completely continuous operators (with various generalizations) is large (see e.g. BROW-

[11], FUCfK-NECAs-souCEK-SOUCEK [32], FUCfK-MILOTA [34], [35], FUCfK-KUFNER [38], AMANN [44], DEIMLING [45], VAN DER WALT [46], CRONIN [47], BERGER-BERGER [48], SCHWARTZ [49], ZEIDLER [so],.,.).

DER

KRASNOSEL'SKIJ

Put

5,10,

u e w~• 2 co,rr) •

Fu• L- 1su, Then

F

is completely continuous (since

- see Section 5.5 - and

S

2:

tion 5.6) and there exiscs e w~• 2 (o,rr} we have

11 Fu I I 1 2

w0 •

(0,11)

Thus the operator

F

L-l

is continuous

is completely continuous - see Sec>

o

such that for any

~ I IL- l 11 11 Su 11 ~ I IL - l 11 c

=

u e

2:

maps the closed ball

{ u E W~ • 2 ( o, rr) ;

I Iu I I

1 2

w0 •


o

VE

define an operator

on the

by

x Im[L]

(w-e:PS(w+KPCS(w+v)), KPCS(w+v)) Ve

maps the space

for any

e:

>

Ker[L]

x

Im[L]

o

Now we are ready to prove the following theorem. THEOREM. Let 1j, : /Rl -,. !Rl function on IRl such that 6. 4.

(17)

lim ij,(t) • lj,("')

be a bounded and continuous

E

IR 1

/;;-+"'

(18)

'

lim "' ( i:;) - lj,(-a,) f: 11 o and a nonempty convex closed bounded set 'X C Ker [L] x Im [L] such that ( ,C)

V

C 'JC •

EQ

Denote w

If

VE

sup Ker I):,] Im I):,] sup

I IS(w+v)I I 1 2 WO,

• c
0 0

lim

t-+-m

- 34 -

W~' 2 (O,1r)

•

11

11

• Jf(x) sin nx

dx - •J)I 1 2 1 2 For

11w11 2 1 2

c

(O,,r)

1 2

w0 •

(0,11)

such that r ;,;_ t 0 /2c 2

it is

- 35 -

with .;;,_ t 0 / 2

+ t 2 c;


IIIllx !. a+ bp6;

inf

(5)

I lwl

a~ 0,

• y >

~

O ,

where

~ - {H; 11 w 11 H • 1 •

w

11 v 11 X ,::. ak

E

J

£

kpo •

Po ]

m2

m2

o

satisfying

and

(w,v)

E

J(er(L] x Im[P 0 ]

satisfying

11w11 8

•

11 v 11 x ~

P ,

a + bp ~

it is (8)

For

e Ker[L] )( :rm[P°] '

(W,V)

~a+ bp~

we have

I lw -

(9)

I lvl Ix~

cwOS(w+KQc(w+v))

kpo < -

-~

+

cm 1 ~

I IOI I (11

1lu

~ 6 4

a(p 0 +a+bp 0 )

+

Po

-~

) < -

provided

The inequalities (8) and (9) prove the validity of the implication I-2 for

10.s.

Put {

l

2

• 6-1

-

' 63 -

6 >

l

(A+B) 6 ~ m(6)(A 6 +s 6 )

Then V

-

A, B. Denote

I !KOCI I • vf

If

I lwl Ix

a+bp 6

w

and

~ Ker[L] ,

~ p , then

+ v am< o) a

I lvl Ix .s,

Im[PcJ ,

I IKOcS(w+v)

As

for any positive

I Ix

I Iv I Ii

6 , 6 , v

,

,:;. vllS(w+v>llz ·~ va + vBm(

a~

o,

b

~

0

such that

r(p,b) !. z(a) for each

p E [_o , p 1 )

Studying the functions

r

and

z

we obtain the proof of the

following lemma.

10.6.

LEMMA.

a~ 0 ,

b

~

Let 0

a~ O, p1 > 0

~

6

0 ,

~

6

O

be fixed. Choose

such that one of the following con-

ditions is fulfilled:

- o, -

I.

6

II.

0 < 6
Jw-(x)dx

0

( w(x) • :!:

!n

0

Ji

sin nx )

11

is an asymptote of the operator S with respect to Ker[L] (see (6.5)). Compare with Theorem 6.4, inequalities (6.19)! (ii) Let L and s be the same as in the previous example while the function + is supposed to be only bounded and continuous. Then the function 11

11

, :. w

1-+

Jf(x)w{x)dx - lim sup "1(0 Jw+(x)dx + 0

0

(+oo 11

+ lim

inf

•

o

6 •

1

a

and




o

tc

and

-

>

71 -

o

such that

for all

t ~ tc ,

w

E

Ker [LJ

, I Iw I IH •

1

and

v

E

Im [Pc] ,

llvllx.ir Then the equation (1) has at least one solution in

Dom[L] • P r o o f According to Theorem 10.10, Lemma 10.6 and Lemma 10.4 it is sufficient to choose Po> 0 and k E (0,1) such that 1 ~ -1 -1 (3) va(l-v~)- {rk - 1!v~) kpo~ ~ tc. ~ Po

.

If Po> o and k E (0,1) fulfil (3) then the assumption (10.5) of Lemma 10.4 is satisfied and this fact together with the just mentioned Theorem 10.10 implies the assertion. 12.3. REMARK. Note that the operator s : X -+- X satisfies the assumption (10.3) with a - 1 and a < V -1 if it is both a bounded and a quasibounded mapping such that -1 < V

Isl

(see Definition 7.5).

- 72 -

CH APTER SOLVABILITY

OF

P R O B L E MS

B O U N D E D

BOUNDARY

F O R

DIFFER EN.TI AL

13 VALUE

O R D I N A R Y

EQUATIONS

WITH

N O N L I N E A R I T I E S

13.1. We shall apply the results from Chapter 11 to obtain the existence of a solution of the general boundary value problem {V,h, ♦} for the differential equation (l)

k }

(-1)

i,J-0

i di (J') -i(Ai.(x)u (x)) dx J }:

(-l)j

where

M

~ dxJ

j EM

lb (u(j) (x)) • f(x) j

,

is the set of nonnegative integers less than or equal

to k-1 • The solvability of analogous equations was investigated in Chapter 8. In contradistinction to Chapter 8 we shall assume that the boundary value problem {V,e,e} for the equation (2)

has a nontrivial solution. 13.2.

THE OPERA/l'OR

L.

(!)

We shall impose essentially the

same assumptions on the coefficients

Aij(x)

as in Section 8.1.

Precisely: (3)

(4)

there exists a constant most all

x t: (a,b)

c0

we have

- 73 -

>

O

such that for al-

Moreover, let (S)

Aij(x) • Aji(x) X

e

(a,b)

i, j • o, •••

for

,k

and almost all

.

Let

~• 2 ca,b) ~ v(; In the space

wk• 2 ca,b)

consider the inner product

V

b (6)

b

JAkk(x)u(k)(x)v(k)(x)dx + Ju(x)v(x)dx

V •

a

a

(u, v e V), and the norm 1/2 v •

1lul Iv· Define the operator

V-+ V

L

r

V •

JAij(x)u(j)(x)v(i)(x)dx

i,j•0 for all (ii)

a

u, VE V.

L

The properties of

reover,

as in Section 8.1, i.e.

b

k

(7)

u EV •

L

are collected in Section 8.2. Mo-

is self-adjoint (this follows from the assumption

(S)) and we shall suppose that Ker[L]

(8) The properties of

L

r

{8}

•

imply that the dimension of

finite. The self-adjointness of

L

Ker[L]

is

implies that the correspon-

ding projections have the favorable properties required in Section 10.1. 13.3.

THE SPACES.

13.4.

THE OPERATOR

Put

(!)

S •

Let

M

be a nonempty set of

nonnegative integers less than or equal to k-1 • For let +j be a continuous function on ~ 1 satisfying (9)

l

lim+,(O•+J· o and (35)

c,

n,+ _r (!j,j(ao)f [w(j)(x)]+dx ~ J EM a b

-

1j,j(- 00

)f

[w(j)(x)]-dx)

Cl

(or

b

- lj,j(-.,>f [w(j)(x)]-dx)

a for each nontrivial classical solution

w E c 2 k([a,b])

the equation (37)

~l

i,j-o

(-l)i -di ( Aij(x)u (j) (x) ) • O dxi - 84 -

of

~

satisfying the boundary conditions (36), 13.13.

OPEN PROBLEM.

In this chapter some additional assump-

tions on the nontrivial solutions of the boundary value problem

{V,e,e}

for the linear differential equation (37) are

introduced, which are not formulated in terms of the coefficients

Aij(x)

(see the assumptions (23) and (29)). It would

be interesting to study this problem and to determine the spaces

V

and conditions upon coefficients

that the assumption (23)

Aij(x)

guaranteeing

(or the assumption (29)) is fulfilled.

(Note: E. N. Dancer has informed me that the condition (23) always holds,)

- 85 -

CH APTER

S O L V A B I L I T Y

V A L U E

S U B L I N E A R 14.1.

O F

P R O B L E MS

D I F F E R E N T I A L

14

B O U N D A R Y

F O R

O R D I N A R Y

E Q U A T I O N S

WI T H

N O N L I N E A R I T I E S

This chapter deals with the solvability of the general

boundary value problem

{V,h,~}

for the equation

(1)

We shall apply Theorem 11.3. The difference between the present problem and that in Chapter 13 is that the functions wj will be unbounded with different infinite limits at plus and minus infinity. In the case of bounded functions ter 13 the solvability of right hand side

f

ding the subspace of

{V,h,~}

wj

considered in Chap-

for (1) is obtained if the

is an element of a certain strip surrounL1

((a,b))

consisting of the right hand

sides g for which the boundary value problem the linear equation

{V,0,0}

for

~l (-1) i di (Aij(X)U ( J' ) (X) ) • g(x) i,j•O dxi has at least one solution. Now we shall prove that if the functions

~j

are unboun-

ded then (under certain additional assumptions) the strip will be the whole space 14.2.

THE OPERATOR

L 1 ((a,b)) • L.

In this chapter we shall suppose - 86 -

that the coefficients

Aij

of the operator

L

defined by the

relation (13.7) satisfy the assumptions (13.3) -

(13.5),

(13.8).

14.3.

THE OPERATOR

S .

M be a nonempty subset of nonk-1 • For each j e

Let

negative integers less than or equal to

w,

€ M, let

J

be a continuous odd and monotone (nonincrea1 IR • Let there exist constants c 1 > O and

sing) function on 6 e

(O,l)

such that

lwjI ~ c 1 (1

(2)

for each

i;

ltl 6 >

+

e IR 1 , and let

(3)

lim w,(1;) J

i; .....,

= - "'.

Suppose

(4)

f E L 1 ((a,b))

,

(5)

~

,

(6)

h EV

wk• 2 (a,b)

E

is such that

0

k 2

w·• (a,b)

u e ~• 2 (a,b)

for all

Define the operator

(for

S: V-... V

V see Section

13.2) by the relation b (7)

v =

ff(x)v(x)dx

- _~ 2

w-• (a,b)

a

-

b

k

JAij(x)~(j)(x)v(i)(x)dx -

l

i,j-o

a b

l j

for

e

fwj(u(j)(x)+~(j)(x))v(j)(x)dx M

a

u, v e V • According to our assumptions, the operator

pletely continuous map of

V

into

- 87 -

V

S

is a com

and there exists a

constant

a> O

such that

I !Sul Iv~

(8)

+

a(l

I lul It> ,

u 6 V •

To apply Theorem 11.3 to the weak solvability of the boundary value problem

{V,h,t}

for the nonlinear ordinary diffe-

rential equation (1) (i.e. to the solvability of the operator equation

Lu• Su,

(9) where

L

is defined by (13.7) and

is defined by (7)), we

S

shall prove that every constant is a subasymptote of the operator

S

with respect to

14.4.

LEMMA.

Ker[L]

Let the operator

• Precisely: L

satisfy the following con-

dition: w e Ker[LJ ,

(10)

for some Let

K

>

i

meas {x e (a,b); w(i) (x) = o} > o E' M ~

w = o •

0

Then (11)

: w I-+

'I'

w e Ker [L]

K ,

is a subasymptote of the ·operator Proof.

Let

w

o

>

S

with respect to

Ker [L]

be such a number that

for all x e [a,bJ and any j • O, ••• ,k-1 • (The existence __k 2 of w follows from the irnbedding w-• (a,b) ½ C k-1 ( [ a,b] ) .) Suppose that there exist quences

t

n

E JR 1

>

o and

n n

Let us take a fixed l

o and se-

r1 > with

,

v < K-£

(13)

I lwnl Iv=

E1

n n

n

-

j e M • We have

and thus we can consider

- 88 -

l

wn

=

K1 .

£

Ker [L]

wn-+ w

in

with V

since

.

the space

Ker [L]

For

µ

o

>

is finite-dimensional. denote

- {x

.73t_2µ , .. )

Since

E (a,b); W(j) (X) ~ 2µ}

~ {-.. ,-2µ]

~

{x e (a,b); W(j) (X)

93 j(-2µ,2µ)

-

{x E (a,b);

w(j)

As

-2µ}

~

lw(j) (x) I

.

2µ}


o , then

'

w.(t w(j)(x)+ ♦ (j)(x)+t 0 v(j)(x))w(j)(x) ~ J nn nn n
o

such that

d 1 ~ wj(x;t 0 , ••• ,tk-l) ~ d 2 for almost all

x E [a,b]

and all

(t 0 , ••• ,tk_ 1 ) e ~k.

Moreover, more complicated cases can be considered, where the nonlinearities depend on all ti , i • o, ••• ,k-1. In the following example we have a set M consisting of only

- 93 -

j and the nonlinearity depends only on two va~j , ~i . It is clear that the same reasoning can be

one element riables

used if the set

M

consists of many indices, and each non-

linearity depends on many variables.

(More lucid conditions

in the case of potential operators may be found in Section 29.11.) 14.10.

EXAMPLE.

Let the operator

L

satisfy all assumptions

from Section 13.2 and from Lemma 14.4. Let n

let

G

L~((a,b))

d1

a1

where i,

and

a2

j e {o, •••• k-1} •

f E L 1 ((a,b))

and

be such that ~

n(x)

~

d2

are two fixed positive numbers, Let

o1 • oj e (0,1) •

Let

oi + oj < 1 .

Define a function

and put

b v • Jf(x}v(x)dx - k 2

w-•

a

(a,b)

b - Jg(x;u(i)(x)+~(i)(x),u(j)(x)+~(j)(x))v(i)(x)dx a

for

u, v e v Then the equation Lu• Su

has at least one solution. This assertion follows from Theorem 11.3 (with o s o1 + + oj ) - the form of a subasymptote can be calculated in the same way as in Lemma 14.4.

-

94 -

CH APTER S O L V A B I L I T Y P R O B L E M S R E N T I A L

F O R

B O U N D A R Y

O R D I N A R Y

£ Q U A T I O N S

N E A R I T I E S 15.. 1.

O F

15

A R E

O F

D I F F E -

WH O S E

A

V A L U E

N O N L I -

L I N E A R

G R O WT H

In this chapter the results from Chapter 14 will be

generalized considering the growth condition (14.2) upon the nonlinear terms of ordinary differential operators with •

l

o

D

• The results will be obtained by virtue of the abstract

Theorem 12.2. 15.2.

THE OPERATOR

cients

Aij

L •

We shall suppose that the coeffi-

of the operator

L

defined by the relation

(13.7) satisfy the asswnptions (13.3) 15.3.

THE OPERATOR

S .

Let

M

(13.5),

~13.8).

be a nonempty subset of the

set of nonnegative integers less than or equal to j e

~

k-1 • For

be a continuous odd and monotone (nonincreas~na) function on IR 1 . Let there exist constants c 1 ~ ~

let

n

c2

wJ(()

>

o

such that

( l)

t E ~l , and

for each

"'

(2)

Suppose (14.4) -

.

(14.6) and define the operator

S

V -+ V

by the relation (14.7). According to our asswnptions, the operator pletely continuous map of

V

into

norm of the identity mapping from by

d*

V. Denoting by V

into

the norm of the identity map from

w1t• 2 ca,b)

, we have - 95 -

S

is a comc•

the

Ck-l([a,b])

and

V

into

b

I ISul

If f(x)v(x)dx

sup

Iv•

vE V

I lvl lv•l k

- _k 2

w·• (a,b)

-

a

b

- 1,j•O r aIA ij (X) ♦ (j)(x)v< 1 >cx)dx

-

b

- jGr M J~.(u(j)(x)+♦ (j)(x))v(j)(x)dxl J

~

a

~ c*I lfl IL ((ab)) 1

+c*ll¢ll_k,2

w·

sup

JI

v&V

I I vj lv•l

i_Jc

2

w·• (a,b)

+

ma~ IIAijllL((a,b))+ (a,b) 1,J•O,.,.,k 00

b

+

+ a*I lhl

•

r

~.(u(j)(x)+♦ {j)(x))v(j)(x)dxl ~

jeMJ

a

~ c*llfllL ((ab))+ a*llhll k 2 + 1 • w·• (a,b) + c"I

l♦I

I

ma~

1c

w·• 2 (a,b) i,J•O, ••• ,k

I IA 1 jl IL ((a,b)) + oo

b

J(cl+c21u1 2ax]

½

+

a

b

+ c*c 2 (b-a)½

r ( Ji ♦ (j)(x)l 2 ax) jE M

~

a

c*I lfl IL ((ab)) + d"I lhl 1

½

•

- 96 -

I.Jc 2 w·• (a,b)

+

+ c"'I 1+11 k 2

I IAijl IL_((a,b)) + -

max

w-• (a,b) i,j-o, ••• ,k

+ c *c 1 (b-a) card M + + c"'c 2 (b-a)~d*I lul Iv+ c*c 2 (b-a)~I l+I l __1r 2

w-• (a,b)

(where

card M is the nwnber of points in the set

(3)

I !Sul Iv~ a+ al lul Iv ,

u e

M ). Thus

V ,

where a• c"'llfllL 1 ((a,b)) + d"'llhll __1r 2 + w-• (a,b)

(4)

+ c:"I l+l l __1c 2 max I IAijl IL..,((a,b)) + w-• (a,b) i,j•O, ••• ,k

+ c"'c 1 {b-a) card M + c"'c 2 (b-a)~I l+I I

_1c

,

2

w-• (a,b)

(5)

LEMMA. Suppose, moreover, that the operator fies the following condition:

L

15.4.

meas {x

we Ker[L],

(6)

for some For

w

E

i

E

M ~ w

Ker[L] ,

(a,b); w(i)(x) • o}

€

a

j e: M

(a,b); lw(j)

e

E

o

O •

I lwl Iv • 1 ,

Bj (w) • {x

>

sa~ls-

Cj(w) • {x E (a,b); o £


o

put

I ~ £} ,

lw(j>cx>I


O;

inf

(meas Bj(w)

w E Ker[L]

£

I lwl lv•l -

£

•

> 0

•

p r o o f . Suppose e:"' • o . Then there exist monotone sequence of positive numbers e: 0 with - 97 -

j0 € M , a

and a sequence wn £ Ker [L] , 11 wn 11 V • 1 such that wn converges to w0 in the space V (since the subspace Ker[LJ is finite-dimensional), w(i)(x) converges uniformly to n W6i) (x) on [a, b] for i ~ o, ••• , k-1 (since the inlbedding

wk• 2 (a,b)

from

into

Ck-l([a,b])

is compact) so that

jo

j

meas B 0 cw) 0 be arbitrary but fixed, Then there exists n 0 e IN such that for each n.::. n 0 and X t: [a,b] we have (jo)

lwn

(jo) (x)

- wo

(x)

I

< n

and so (8) (j

C {x e (a,b); lwo for

n

~

n0

o>

n + en}

(x) I


o , we have

Consequently,

w0

tradiction with

= I

o

by the assumption (6). This is a con-

jw0 1 iv

•

1 •

Now, we are ready to apply Theorem 12.2. 15.5. THEOREM. Let the assumptions and the notation introduced in Sections 15.2 - 15.4 be observed. Let

* ls * -1 (I jKj jc (b-a) d ) ,

(10)

where

K is the right inverse of the operator Then the boundary value problem

{V,h,,}

L. for the equa-

tion

Y.

(11)

(-1) 1

1,J•O

¾

(A .(x)u(j)(x)) -

dx

}:

(-l)j

j e M

iJ

~

,i,.

dxJ

(u(j) (x)) • f(x)

J

has at least one weak solution

u e V,

.

.

the assumption (10) together As V • I IKI I P r 0 0 f -1 Let £ E (O, 1/) and with (5) · implies that 0 < B < \)

r

E

(o •

-¾) 2c

be fixed and such that 13 < rv- 1 (l+r)-l

To obtain Theorem 15.5 we shall apply Theorem 12. 2. It is sufficient to verify the condition (C) with r . Suppose that "v this condition is not fulfilled. Then there exist K > o and sequences lim t n

n-+oo

such that "v

(12)

v

~

K

V

-

Bj(W) E

- j IEM

(x)+♦ (j) (x))w(j) (x)dx

-

IIA1jllL.,.((a,b))-

--+-

This is a contradiction.

as

00

n --+

00

•

(Note that the estimates in the above

calculatfon are the same as those in Section 14.4.) 15.6. REMARXS. (!) In the same way it is possible to prove an existence theorem for the weak solvability of {V ,h, } for (11) provided the functions

tions of Theorem 14.6 with (ii)

,,.j 6 -

.

j e M

1

and ( 10).

satisfy the assump-

As in Remark 14.10 it is possible to consider more

general nonlinearities in the equation (11). (iii)

Also in the case of boundary value problems for

ordinary differential equations whose nonlinearities have a linear growth it is possible to apply the regularity result (see Theorem 4.19) which yields the existence of a classical solution (the result would be analogous to Theorem 14.7). 15.7. (13)

PROBLEM.

We consider the Dirichlet problem

Au''(x) + u(x) + ;(u(x)) • f(x)

{

,

X

€

(0,1r)

u(O) • u(,r) • o (i.e.

v •

w01 • 2 co,ir))

, where

monotone function on

,p

is a continuous, odd and

~l , lim

l/lCO • "' ,

f;~m

For the sake of simplicity we consider • 4/3 ,

E

*

~

11/4 ,

c

*

., ,r ½

d

*

• l

A• 1 • Then

I !Kl I -

• The above theory

yields a sufficient condition for the weak solvability of (14) for any right hand side

c2


IR1

be a continuous function satisfying

- 123 -

ilil

lim

E;

I t; 1.... .,

Then for an arbitrary right-hand side

f E

c 0 ( [o, l]

)

there

exists at least one solution of (9) with U(O) - u(l)

•

u'(O) - u'(l)

(see also Part X). 19.5.

(!)

EXAMPLE.

g: ~ 2 -,. R 1

Let

be a continuous func-

tion such that

lim

(18)

I ;; 1 .... .,

uniformly in

x E ~l ;

g(x+T;F;) ~g(x;r;),

(19)

Suppose that there exist

M

>

XEIR 1 ,

O,

m > O

·;;£R1 •

such that

(20)

if

I ;; I ~

M

Let

and

f E c0

T

x

E

IR 1 •

satisfy T

ff(x)dx

(21)

• o.

0

Then the equation u''(X} + g(X;U(X)) = f(X)

(22)

has at least one T-periodic solution. Hint.

The condition (18) iml!llies that the mapping

S

X

Iii SI

• o

-+

Z

defined by S: U(X) I-+- f(X) - g(X;U)X)) is bounded and quasibounded with the quasinorm

(for definition see Section 7.5). For the proof of the existence of a T-periodic solution of the equation (22) use Theorem 12.2 together with Remark 12.3. It is sufficient to verify the condition (C) of Theorem 12.2 (this condition is easily

- 124 -

satisfied). (ii)

The assumption (20) may be replaced by

(2Q I) (iii)

In Section 22.5 (iii) we shall show that it is even

m = o in the condition (20) (or (20')) for

possible to admit

the equation (22) to be T-periodically solvable. (iv)

Analogous equations are considered in CHANG [89]. The

case g(x; 0 • g(O is studied in HARVEY [90], LEACH [91], LOUD [92], OPIAL [93] , SEIFERT [94] , . . . . Further references to the problem considered will be given in Part VI.

(!)

EXAMPLE. tions such that 19.6.

(23)

lim

(24)

sup

Let

g_iQ

E

IR 1

be continuous func-

-

0

•

jhI


0

rn >

o

such that

i;g(O ~ - rn

(25)

provided

> M •

IC I

f f.

Let

co

satisfy

T

T

ft(x)dx

(26)

-

0

0

Then the equation u''(x) + g(u(x)) + h(u(x))u'(x) • f(x)

(27)

has at least one T-periodic solution. Hint.

Put

s : u(x) The mapping

I-+ f(x)-g(u(x))-h[u(x))u'(x)

S : X -

T

,

u EX.

is quasibounded with a quasinorm

l)sl


H -

where

Therefore

Put R0 •

1 + max{y 1 ,Y 2 } •

Steps 2 and 3 prove that the assumption 21.1 (v) is valid. Step 4.

It is easy to see that the function - 150 -

r

defined in

Section 21.1 (vi) has a form

Define

by

.

(r)

r, ,_ l

(t

1, ..• ,m)

ni (r)

-r.

(£

m+ 1, ••• , N)

n•

l ~

]_ £

£

Using the assumption 22.1 (v) we have

a

N >

!R

for

I Ir! IN~ r 0 , which implies together with the basic pro!R

perties of the Brouwer degree (see Section 20.2) that d[r; B(mlRO),o]

d[n; B(mlRO),oJ

=

=

+

-

l

This completes the proof. 22.5.

REMARKS.

Cl)

For proofs of the assertions in Step 2

and 3, the procedure from ROUCHE-MAWHIN [lo8J is used. (ii)

Steps 2 and 3 imply immediately that (under the assumpu e Dom[L:

tion of Theorem 22.4) every solution

of the equa-

tion (5) satisfies (18)

sup x E iRl

llu(x)II N IR

(19)

sup 11 u, (x) 11 N x € IR 1 [R

N½r 0 + 2Tw- 2 k(M + llfll 0 1 N ) , CT(IR ,IR )


- .art C {G,g.}

c?l"\1e-> -m. = Ill

• G

then

G

g(-"')

g(-"')

g(O


= G, g( .. ) P r o o f

(_!)


n

for

meas y(x) • o

n

> O

for

such that

o ,

g(E;} >

g(-~)


-O+

Let

n

1

.

2M TT

> -

E;

Then for each


a

i

Let

be continuous

f E L1 ((O,n)) • Suppose

o

>

and denote

r

r(a,i) = 2 inf i

o

sufficiently small and such

Put ea n

-1

-1

arc sin (c 3 -a)

Then 11

IJf(x) sin nx

dxl

- 11

sup E; E IRl

0

r cos ne - 2ne


l

sup

sin ne =

E; E IR.l

n-1

inf

ico L

E;~a

k=O

nyldy 11

n

- 2n.

1ics> I

sup E; E

k+e

sin ne


c 1 (,i,,f,n) + c 2 (a,1j,,f,n)

ic

i

00 ) = ,i,(b 1 ) and is odd, according to the inequality (9) and with respect to Theorem 6,4 there exists at least one solution u E w~• 2 co,rr) of the problem (6). From Sections

Since

24.2 and 24.3 it follows that

Thus the function

u

is also a weak solution of the Dirichlet

problem (1). The assertions of the following two theorems are obtained immediately by verifying the complex assumption (8) in Section 24.4. 24.5.

THEOREM.

~unction on

~l ,

Let

be a continuous, bounded and odd

wlF,)

a> O . Suppose lim ; 2 (-+oo

min T €

[a, i,;]

,i,(,)

Then the Dirichlet problem (1) has at least one weak solution for ~very

f E L 1 ((O,rr))

-

173 -

with

Tl

o •

ff(x) sin nx dx 0

(Hint: Calculate the limit of the right hand side in (8) multiplied by b. ) 24.6.

DEFINITION.

function

ljJ

A bounded, continuous, non-trivial and odd

is said to be expansive if for each o

p < sup

~

there exist sequences

ljJ (

IRl

F; €

0 < ak < bk

0

p ,

,

.

-1

lim bkak k-+oo such that

lim min ,j,(0 > p k+oo r;. e: [ak,bk]

24.7.

EXAMPLE.

The function

given by the relation (3) is

ljJ

expansive. 24.8.

THEOREM.

Let

be an expansive function. Then

ljJ 7[

fE L 1 ((0,n))

,

If

f (X) sin nx dxl


o

O _ •l C (0)

u EV

of the boundary value

satisfies

;;. O _ C (n)

c 3 (a,i,f) • c 3

,

where

t-1

(---c2___ ] inf

i (0

1;,;:,a

Proof.

+ sup 1 r; e IR

Ii (0 I

The positiveness of the constant

c2

is an easy

consequence of the assumption (12), the continuity of the variable sional.

w

and the fact that

Ker[L]

u EV

Suppose that there exists a weak solution boundary value problem (Au)(x) + i(u(x)) = f(x) (13)

Bru = o

on

an ,

- 180 -

r

in

is finite-dimen-

in

n ,

r = o, ..• ,k-1

of the

such that

I I Pu I I C

Then there exists

O _

(l'l)

w E Ker [L]

> c3 •

,

I lwl I C

O _

=

such that

1

(n)

u • tw + Qu and t

where

a>

o

>

is sufficiently small. Set E

=

Then IJ f(x)w(x)dxl J!w(x)jdx - lff(x)w(x)dxj (~a n n -

(E)[

supl 1ico1 + inf ico) + I Jf(x)w(x)dxl s.;,a 11

i; e= IR

inf icuf lw(x)jdx - t E;::_a n


K

const. El+p

o • Probably, the proof of (9) with

41

given

by (18) might be based on the investigation of the set of nodal lines of

w E Ker[LJ

and on some version of the strong

maximum principle (of course, if

k • l

and for example in

the case of Dirichlet problem). Unfortunately, no correct result from this field is known to us.

25.18.

OPEN PROBLEM.

Is it possible to choose the function

(18) so that (9) is valid in the case of

k • 1

and in the

case of Dirichlet problem? (According to a personal communication, some results in this direction were obtained by V. A. KONDRATJEV and E. N. DANCER.)

25.19.

REMARK.

Consider now the homogeneous Dirichlet prob-

lem. In this case the condition (17) is necessary and sufficient for the weak solvability of (5) provided pansive.

~(t)

(It is easy to see that (17) with the sign~

is exis a

necessary condition and it is possible to prove by contradiction that the equality does not occur.)

25.20.

REMARK.

Routine and tedious calculations would play

an essential role in generalizing the above results by considering the nonlinearities

L jjj~k-1

c-1)ljloj(w.(Dju>) J

are bounded continuous functions) instead of

- 185 -

in

25.21.

REMARK.

Suppose that

let problem. Let

Ker[L]

by

w0 which is positive in bounded set MC

there exist constants aw 0

a

v


= .,

is a generalized expansiv~ function.

THEOREM.

Suppose (1). The equation (2) h~s at least

one T-periodic solution provided q

(5)

~

T 1 T- Jf(x)dx

~

p,

0

T

T

(6)

2Tw -2k ( sup f; E

f

lg(;;)-T- 1 f(x)dxl +

IR 1

proof.

0

0


n2

g (oo) > p , g (-oo) < q and thus (with respect p,q p,q to Corollary 22.6) there exists at least one T-periodic solu2k tion u 0 E CT of the equation

Obviously

-(-l)ki/2k) (x) + a2k-lu (2k-l) (x) +

+ h(u(x))u' (x) + g p,q fu(x)} = f(x) Moreover, by (22.23) we have

I !u0 j I

O CT

~

T

n 1 + 2Tw- 2 k( sup

lg(E;)-T- 1 Jf(x)dxl +

I; e IR 1

0

T +

llf-T- 1 ff(x)dxjj 0 ) 0 CT




IR

F(i,y,z) • F(t(T(y),8,y,0))

>

F(t(0,0,y,0)) = F(0,y,0)

according to (10), which contradicts (6). Let

t(t,e,y,e) •

tk(t,y} E IRk, -

(i,y,z)

(tn(t,y),tm(t,y),tq(t,y))

k • n, m, q . If

q

I Iii I n ~ ro • IR

with

>

0

and

I IYI I m ~ co IR

with

t(T(y),e,y,0) and

I Iii I q IR

m

c 0 , then

aF - - - ,z> d I tq I 12q 2

o

such that

o(y)


.

Be,. •

>.e). -

the set of all eigen-

a

o ta and let A be a setaken with respect to their

values of the operator B. Let quence of all eigenvalues of B

Ao

and with a norm

is a linear completely continuous

selfadjoint operator and denote by

Choose

28

=

"o -

inf a ,

W be the null-space of the operator

(2)

(i.e.

w~Ker[L]).

28.2.

STRONG CONTINUITY.

(~)

Suppose tbat

X

and

Z

are

real Banach spaces. Let T : Dom[TJ C X --+ Z • Then

T

is said to be strongly continuous on

u € (ii)

C ,

un

~

u ~Tun

CC Dom [T]

if

Tu .

The concept of strong continuity is close to the concept

of complete continuity (see Section 1.7). Namely, it holds: Let

C

X

--+

z

X

be a reflexive Banach space, let

be strongly continuous on -

203 -

CC Dom[T]

T : Dom[T] C , where

C

is

a closed convex set. Then (iii)

T

is also completely continuous on

C.

For a linear mapping from a reflexive Banach space into

a (not necessarily reflexive) Banach space the two concepts, i.e. the complete continuity and the strong continuity, coincide. (iv)

Generally, there exists a strongly continuous operator

which is not completely continuous and, on the other hand, there exists a completely continuous operator whic~ is not strongly continuous. (For the proofs and for examples see e.g. FUCfK-NECAs-souCEK-SOUCEK [32] • ) 28.3.

DIFFERENTIABILITY.

(!)

An operator

said to be Frechet differentiable at

u EX

a bounded linear operator

X

T'(u}

on

T: X-,. Z

is

if there exists

into

Z

such that

T(u+h) - Tu• T'(u)(b) + R(u,h) for all

h € X

where lim h+e

The operator

T'(u)

is uniquely determined and is called the

Frechet derivative of (ii)

T

at

u.

T: X-+ X*

An operator

there exists a functional

is said to be potential if

9'": X-+ R 1

'r (u) • Tu ,

u

€.

such that X

The functional

'T with this property is called the potential

of the operator

T

for

If

"

ul' u 2 e X '{' ul -

(iv)

X -+ IR 1

is Frechet differentiable on

there exists

l

T

X -+ X*

tential operator with a potential

then

.

is a strongly continuous po'{': X -+ IR.l •

5"" is strongly continuous. - 204 -

X

such that

ff' u2 • < 'I"' (~ul + (l-l)u 2 }, ul-u2>

Suppose that

Then

E (O,l}

Proof.

Let

un EX,

exists a subsequence

un

{un };. 1

~

u. Suppose that there

of

{un}~=l

and

£

>

o

such

k

that ( 4)

Then there exists a subsequence

'Ir

nk

'Ir ,

-+

'Ir

R.

of

nk };=l

€

nk

and

j Un }:. l k

R.

such that

(O, 1)

9,

5""u

- q-u •




o

e [o, "')

d , 2 - a, - BT 1+61I { 2

be the (unique) solution of the algebraic

equation

(16) If

o • 8

= O

•

then - 206 -

(17) Let

c(r) 6 E (0,1]

od-l

s

.

• Obviously

(18)

and thus ( 1 9)

li111 c(r) = r+m

m

•

The Implicit Function Theorem implies that the derivative c'(r) exists and c'(r)(d-2aoc 6 - 1 (r)) -

(20)

ac1r 6 -l = O .

Thus £.:...W_ ~ lim

(21)

sor 6-l

l

W

m

d-2ac 0 - 1 (r)

r+m

(cS)

,

where if

6

if

c5 •

€

(0,1)

(22) 1

,

0

f (0,1)

and the l'Hospital rule yields c(r) • w(cS) + sr 0

lim

(23)

o

•

This implies (24)

-

1

2

\) ( o)

>

p ( o)

\) ( 0)

>

l -2 + ~ l 2d + 2 11d d

2\JW

( 0)

+ 2w(o)

if

and (25)

According to the assumptions (10),

~ + .!.11d-2 + 2 0

2

2

(11) there exists

such that >

-

c(r 0 )

+ 2 --6 + 4a

o+er 0

- 207 -

a

cl+6(r) 0

r0

>

0

i.e._. if (26)

c(r 0 )

m

K(r 0 )

c0 > -

then A(6)

+

½µc~

+ 2c 0 (a+Brg) + 4Bc~+ 6

1, 2

(iii) Denote by t t and tj the partial Frechet derivatives of t with respect to the first, second and third variable, respectively. Now, the following inequalities hold: If

then (28)

for

~ di lvl I~ - al lvl 18

-

Bl lwl

- Bllzll~ llvll 8 ~ c 0 (dc 0

I~ -

I lvl 18 - Bl lvl l~+o (a+Brg) - 2Bcg) • o

If

then (30)

for

~

-di lzl I~+ cil lzl

Bl lwl I!

18 ·•

1 lzl

la+ Bl lvl I~

I

lzl la+

+ Bl lzl 1~+ 6 ~ c 0 (-dc 0 + (ci+Brg) + 2acg} • o • If (31)

llwll 8

•

r0

,

llvll 8 ~c 0

,

llv*ll 8 ~c 0

then (32)

t(w,v,z) -

~

t(e,v * ,e)

208 -

,

llzllH,:;,c 0

for di lzl I~ - H 21 u - I Iv*;

* = ~(A,v * = 2l H - :f'(v)

0

1~+6 -

+ al lvl I~ I lzl IH + Bl lzl I~ I !vi IH + al lzl

~

t

1t0


l

(~)

~N

(N

1)

Let

n

be a bounded do-

with a Lipschitzian boundary

ao

if

(see Section 16.3). Let

'i'f• 2 (o)

be a closed subspace of

V

such that

~• 2 (o)CVCif• 2 (n)

(l) (ii)

~

B O U N D A R Y

P R O B L E MS

V A L U E 29.1.

29

Let

(2)

c > o

Suppose that there exists

such that

(3)

for all

~i

c

IR 1

(Ii I = k)

and almost all

x e

fl

• Let

Put (5)

~ (v ,u)

I ii •

t

J'l,

,

if and only if

such that (9)

x en

and

satisfy the Carath~odory condition on c1 ~ o ,

be the number of all

p

N whose length is less or equal

o

(10)

l

\~-16

fj[~k-1

J

[bi(x;U\~,j,2(x)+c2

for almost all

x En

and all

~ E ~P

Let

(11)

and define a functional

by (12)

:/

u EV .

u-+ fb(x;9k_ 1 u(x)+9k_ 1 $(X))dx Q

(iii)

According to Section 4.3,

(iv)

The functional

!:I

u e V

where

for all

u EV

possesses the Frechet derivative

:!' for any

Y'(u) E ~l

(u)

= Su

S : V-+ V

is given by

(13)

u, VE V • (~)

It follows easily from

w1 o

such that

, ~ a . From (26) we have

g(F;M) ~ - n 1 a + R(F;M) - £\F.:MI - an 2 + at for all

F;M. where

- 218 -

Then

rf f(x)w(x)dx ~

Jb(x;ri;rMw(x))dx = Jg(r,;rMw(x))dx -

n

n

n

n 1 a meas n + rJR(i;rMw(x))dx -

> -

£rf 1vMw(x)ldx -

n

n

- n 2 a meas n + Ea meas n - rf f(x)w(x)dx

~

n

~

r ( f-E

J\vMw(x) \ dx)

- an 1 meas

n -

a,1 2 meas

n

+

(l

The above calculation implies the validity of the condition (17)

if

29.lo.

E

>

O

is sufficiently small.

REMARK.

Theorem 29.9 extends the results from Theorem

18.6 and Remark 13.8 (ii): the nonlinearity considered in the equation (18) with (24) depends on many variables. Analogously as in the proof of Theorem 29.9, we can prove the following result on the basis of Theorem 28.5.

29.11. g

THEOREM.

Suppose (1) -

(4),

(7),

(11) and

(20). Let

be an even continuously differentiable function in the va-

riables

~i ,

i EM,

g(O)

o • Suppose that the derivati-

z

ves

satisfy a growth condition (30)

where

cs> o

and

6

E (O,l)

- 219 -

• Let

(31)

lim inf¾, T-+•

T

Then the equation (18) (with solvable in 29.12.

b(x;t)

given by (24)) is

V.

REMARKS.

(!)

The above theorem extends the results

of Theorems 14.5 and 18.3 in the potential case mainly in the following respects: a)

no monotonicity assumptions are imposed upon the functions

gj b)

the nonlinearities

depend on many derivatives

Dju,

j f: M

Thus a partial answer to open problems 14.8 and 18.10 is given.

(!!.) In the same way it is possible to consider boundary value problems whose nonlinearities have a linear growth. If the assumptions of Theorem 29.11 hold with

6

z

1

and if

sufficiently small then equation (18) is solvable in

- 220 -

c5 V.

is

CH APTER

V A R I A T I ON A L 30.1.

INTRODUCTION.

30

A P P R OA C H

Theorem 28.5, whose application yields

some generalizations of the results from the previous parts, concerns potential equations. Unfortunately, its proof has not been variational. It is very important (mainly for the purposes of numerical methods) and also adequate to its formulation to provide the variational proof of Theorem 28.5. For simplicity, we shall deal only with the situation described in Remark 28,7 (ii).

30.2.

THE FIRST VERS~ON OF VARIATIONAL THEOREM 28.5.

Let the

assumptions from Sections 28.l and 28.4 be satisfied. Moreover, suppose (1)

In this case (for notation see Section 28.6 (i)) we have

'X .. { >.

E A; >- > >. 0

Z and on

W

x

V

)

•

(II ,

{0} ,

=

let us consider a functional

~ : w

X

V -

IR 1

defined by

(2) Denote 0

(r) •

Y (w)

sup

I lwl ltt•r w E Ker [L]

and suppose (3)

lim o(r) • -

r .. m

Then the equation - 221 -

m

•

(4)

Lu• Su

has at least one solution

u 0 • w0 +v 0 & H • W x V

t(w0 ,v 0 ) •

(S)

t(w,v) •

min

(w,v)

such that

fi W>

uo(x)-u(i) tex;ii-c'i» + ---....--- + t+

a

4>_

= sz 0

(constant mappings). From 33.4 we have s_w0

~

z0

~

s+, s_ e fR1

such that

s+wo

The conclusion follows from Theorem 33.15 since Assumption 33.11 is obviously satisfied and z 0 ~ s+wo ~ u ➔ su ~ S(s+w0 ) ~ sz 0 = 4>+

(That means Assumption 33.9 holds.)

- 252 -

CH APTER C L A S S I C A L

S O L U T I

P R O B L E M F O R

P A R T I A L 34. 1.

O N

34 S

O F

S E C O N D

D I F F E R E N T I A L

INTRODUCTION.

D I R I C H L E T

E

O R D E Q

R

U A T I O N S

In this cha~ter we shall discuss the

use of abstract results of Kazdan-Warner type in dealing with the classical solvability of nonlinear elliptic boundary value problems of the type Au= f(x;u)

{

(l)

where

A

u = 0

"

""

is a second order uniformly elliptic operator

I

N

(2)

on

in

Au

-

~

i, =l The boundary coefficients of

an A

a .. (x) 1 J

a2 u axiaxj

N

+

l

i=l

bi(x)

au axi

is supposed to be smooth and also the

are supposed to be smooth. say

(3)

with

(4) where

µ > O

is a constant.

For the sake of simplicity we shall suppose that the function

f(x;,)

is continuously differentiable in

der continuous in

~

and a-Hol-

x.

Our intention being the application of Chapter 33, we put (5)

with the norm of

c2 •a(n)

; -

253 -

(6) and C • {u ~ Y; u ~ o

(7)

o} .

on

First we shall verify Assumption 33.13. In Section 34.7 we shall present the partial differential analogy of Theorem 32.7. After this we shall give sufficient conditions for the validity of Asswnption 33.4. The main result of this chapter will be stated in Theorem 34.13. This Theorem is due to KAZDAN-WARNER [154]. The verification of the asswnptions from Chapter 33 is based on some fundamental results on linear partial differential equations the proofs of which are omitted - the reader is requested to consult the quoted literature. The first result of the type just mentioned is the following maximum principle. c(x) E CO'a(O)

34.2. MAXIMUM PRINCIPLE. Let for X 6 0 Let Au+ c{x)u

.

in

~ 0

.

c(x)

in

Q

~ 0

Q

If U(X) ~ M

and

X fi Q

•

u(x 0 ) • M at an interior point

XO

of

.

Q

then

=M

u

.

(The proof of this assertion is included in many basic textbooks on partial differential equations - see e.g. PROTTER-WEINBERGER [1 76], BERS-JOHN-SCHECHTER [1 77], . . . . ) 34.3.

Now consider the Dirichlet problem

Au

We assume

f(x}

·in

on

an

{ u• o

(8)

f e

c0

a

Q

•a(O} •

Under the above assumptions the Dirichlet problem (8) is uniquely solvable and satisfies the a priori estimates - 254 -

(9)

llull 2 a_ c

where the constant

c

•

(n)

~ cllfll o a_ c

•

is independent of

(n)

f.

The following estimate also holds:

I Iu I l w2 • P (n)

(10)

!. c I If I I.L

p

en> •

(For the proofs see e.g. AGMON-DOUGLIS-NIRENBERG [157, Theorems 7.3 and 15.2]. The a priori estimate (10) was used also in Chapter 25.) The following Fredholm alternative is of great importance. 34.4.

FREDHOLM ALTERNATIVE.

Au + c(x)u • f(x)

{

(11)

Consider the problem

u = o on where

c(x} € CO'a(n) ,

n

in

an

f E CO,a(O) • The problem (11) is

either boundedly invertible for an arbitrary f

=

o

c(x}

~

o

trivial solution when 34.5.

REMARK.

34.5.

UPPER AND LOWER SOLUTIONS.

or has a non-

everywhere in n then (11) is 0 uniquely solvable for an arbitrary f € c •a(n) and, moreover, the mapping (A+c)-l fro~ Z into X is bounded.

(!)

Consider the problem (l),

An uoper solution to this problem is a function

EC2,a(fi)

u+ €

satisfying

{

( 12)

(ii}

If

f

Au+

~

f(x;u+)

u+

>

0

on

in

n

an

A lower solution of (1) is a function'

u

E c 2 • 0 (n)

sa-

tisfying (13)

{

Au

-

u-

~

in

f(x;u_)

< 0

on

an

n

.

The terms SU[lersolution and subsolution are sometimes -

255 -

used instead of those of upper and lower solutions, respectively (see Section 32.6).

34.7.

THEOREM.

Let

u+

be an upper solution of {I) and

u

a lower solution of (1), (14)

Then there exists a solution problem {l} which satisfies

u

of the boundary value

(15}

Proof.

af at

(16) for

X € Q

w > o

Choose a number

t

and

(X;t} + w

E [min_ u_ (X)

{

(17)

T

>

0

, max_ U+ (x)]

xe1n We define an operator

such that

•

x~n by

v • Tu

if

Av+ wV = f(x;u} + wu

an •

on

V • 0

(The definition of the mapping

in

T

is correct as follows from

Remark 34.5.) Let us show that Tu~ Tv

in

n.

T

is monotone: if

Tu• Tv •

~

v

in

n

We have ATu + wTu • f(x;u) + wu ATv + wTV • f(x;v} + wV

and

u

o

an.

on

n ,

in in

n

Putting w = TV - Tu

we get {

Aw + ww • (f(x;v)-f(x;u}) + w(v-u) w • o on an •

in

n

However, the function F

is increasing in

t

(x,t) i -

f(x;t) + "'t

by (16), hence for -

256 -

v

~

u

we have

then

0 ~ F(x;v) - F(x;u) •

(f(x;v)-f(x;u)) + w(v-u}

m

Aw+ ww.

Therefore Aw

{

w •

w

By the maximum principle Now let

ul

~

.

Tu+

+ wW ~ o on

in

0

O , so

~

n

an Tv ~Tu.

Let us prove that

{ Au 1 + wu 1 • f(x;u+) + wU+ u1 = o

an

on

It is

Ul !. u+

n

in

'

therefore

and

By the maximum principle it is

u 1 -u+

~

O

n , i.e.

in

u 1 =Tu+~ u+. Now define

u2

#

Tu 1 • It is

u 1 !. u+

and thus

Tu 1

~

~Tu+, i.e. u 2 ~ u 1 • By induction we get a monotonically decreasing sequence of iterations

Similarly, starting from

~

we get a monotonically increa-

sing sequence

u_ where

= Tu 1 Moreover, V

and

vn

~

v1

=

v1

~

v2

~

Tvn-1 .

u_ !. u+ i.e.

~

u 1 • By induction

➔

Tu_ ,:;, Tu+ ,

vn !. un

- 257 -

for all

n

~ ~

, and

monotone, the pointwise limits and

~(X) - lim Vn(X) n+m

'\,

both exist. Let us show that and

'\,

u

and

V

are fixed points of

il • Til •

In fact, taking into account (10) and the continuity of f(x;t)

we see that

takes bounded pointwise convergent

T

sequences into sequences which converge in

w2 •Pcn)

for

1 < p N

(;

c0 • 0

(n)

we may conclude that the iterations converge in

for

a• 1 -

~ p

(see Section 16.3). Returning to (10)

c2 • 0

we see that the iterations converge in

i'i •

cn) .

lim un .-- lim Tun-l • T(lim un_ 1 ) • n~w n~m

Therefore

Ti'i •

n~~

Since

i'i

E c2 •0

cn) ,

it is a classical solution of the Dirich-

let problem (1). Analogously we prove the assertion for 34.8.

REMARK.

REMARK.

V

The above theorem was first proved by AMANN

[178]. See also SATTINGER [179], 34.9.

'\,

[180].

Now we shall study the validity of the assump-

tion 33.4. First of all we shall show that there exists E ~l

(18)

Al€

such that the Dirichlet problem {

=

Au

u

=

o

n

on

AU

in

an

has a one-dimensional space of solutions for

A= Al

which

is generated by a nonnegative function. such problems are studied generally for abstract linear operators in semiordered spaces. The main result is the so-called Krein-Rutman theorem (see KREIN-RUTMAN [181] and also KRASNOSELSKII [182]) which asserts that the so-called 'strongly - 258 -

positive' linear operator has one and only one normed eigenfunction which is positive (in the sense of partial ordering), the corresponding eigenvalue being real and simple. We shall not explain the theory of positive solutions of linear equations but we state here without proof only the corollary concerning the Dirichlet problem for second order partial differential equations. 34.lO~

THEOREM.

Let the coefficients

aij , bi

satisfy the

assumptions from Section 34.1. Then the problem (18) has a

n

unique normed positive solution in an eigenvalue

which corresponds to

Al> O. The eigenvalue

Al

problem (18) has a nontrivial solution for 1. 1 < .>.

is simple. If the A€ ~l

then

•

The following very useful theorem also facilitates the verification of Assumption 33.4. 34.11.

MAXIMUM PRINCIPLE AT THE BOUNDARY. Au

< O

in•

fl

Let

,

let u(x)

~

M,

x e fl

and let u(x 0 ) • M for some

x0

tangent to

u

If

e an . Assume that there exists a sphere an at x0 • is continuous at

and

au

av

KC fl

exists (wherE>

L

av

is any outward directional derivative) then :~ (XO)

unless

u

=M

in

fl

> 0

•

(For the proof see again e.g. the book by PROTTER-WEINBERGER

[1 76] . ) 34.12.

Denote by

~

the positive solution of

- 259 -

{ Au •

u •

in

.>. 1u

0

ll

on

with

f+ 2 (x)dx

• l

n

the existence of which follows from Section 34.10. Since has its minimum on an, we have (see Section 34.11)

*

Therefore, assuming statement:

< const.
. 1 u.

on

in

n

an

Notice that the adjoint problem to {

Lu• o in u • o on

n

an

has also only a one-dimensional space of solutions generated by a positive function ; on g with

f•

2 (x)dx - 1

n and that {

Lu• f(x)

in

n

u • o on an f £ c0 • 0

with

co)

has a solution if and only if

ff(X)lj,(X)dx • 0 . n Let

g

n

X

IR1 ......

IR1

(g: {x,t) .,_.. g(x;t)) - 260 -

be a-Holderian

in

x

and continuously differentiable in

S • s1 : u

x e

g{x;u{x)) ,

1-+

t . Define

o.

Obviously

S : X -,. Z and S is completely continuous with respect to the Arzel~-Ascoli th,eorem. 34.13.

THE KAZDAN-WARNER THEOREM.

tisfies both Condition

A+

Condition A+

There is a number

nuous function

h+(x;t) :

in

x , Lipschitzian in

Suppose that

and Condition

n t

x

s+

A

g(x;~)

sa-

, where

and a bounded conti-

R 1 ---., ~l

which is o-Holderian

and such that if

then and Jh+(x;u(x))w(x)dx

~

o •

n

Condition A_

There is a number

nuous function

h_(x;~) : n • ~ 1 ---., ~ 1

in

x , Lipschitzian in

~

u Ex,

u

s_

and a bounded contiwhich is a-Holderian

and such that if
o 0

and lim inf g(x;t)

t+-m

uniformly in 34.15.

>

h_(x)

x.

THEOREM.

Assume

ft

(x;t} ~ o •

Then a solution of (19) exists if and only if there exists z 0 ~ X such that fg(x;z 0 (x))w(x)dx = 0 0

(Proof

follows immediately from Corollary 33.16.)

34.16. REMARK. Theorems 34.13 and 34.15 are included in XAZDAN-WARNER [154]. Moreover, Theorem 34.15 is proved under some additional assumptions in KLINGELHOFFER [183]. For - 262 -

analogous results see also the papers of E. N. Dancer.

34.17. if

An application of Theorem 34.15 shows that

EXAMPLE.

h ~ CO'a(n) ,

h(x) ~ 0

and

h(x) ; 0 , then one can

solve {

-~u - ~ 1 u • f(x) - h(x)eu u •

for

f € CO'a(n)

o

in

o

an

on

if and only if

Jf(X) ♦ (X)dx

> O.

0

34.18.

(!)

REMARKS.

DAN-WARNER

The following result is proved in KAZ-

[1s4_] in an analogous way:

There exist ~onstants f E CO,a(O)

u • t_


0,

xn

UC

L 1 ((0,l))

~ [o,1] , · zn E IR 2

and

- 285 -

a boun-

such

op

Since

sequence

is continuous (in p, x, y) we conclude that the (x z ))}~n=l is bounded which contradicts n n• f n n' n

{• (Wp up

(18).

37.10.

REMARK.

Put

.,

M , f E L 1 ((O,l)) and· z e IF"' • Then (12), (13) and Lemma 37.5 imply illUllediately the following inequality: for

p

~

~ lz 1 1

lnp,f(z)l 2

1 2 + 2jz 1 1 J y 2 (s)ds + 2k} + (1 + 2£~ 1 (1-£ 0 )- 1 )(J!t(s)lds) • 0 ta As usual, denote for an arbitrary nonzero complex number , ~ E

q:

37.11. LEMMA. Let X be a topological space and let a continuous complex valued function defined on [o, l] and satisfying F(x,u) ~ o for all (x,u) ~ [0,1] x X •

Then for any lued function +u

F

be

x X

u EX there exists a continuous real vaon [o, 1] such that

1'iu(x) E Arg F(x,u) , Moreover, for an arbitrary

u e

X

x E [o, 1] • the value

is independent of the choice of a continuous function [o,1] with wu (x) e Arg F (x, u) , x e [o, 1] , - 286 -

wu

on

and the mapping Proof •

l

is continuous on

X.

The assertions of Lemma, except perhaps the conti-

nuity of the function

f

,

are well-known from the classical

complex analysis. Now we shall prove the continuity of Let

u0 f X

l

•

be fixed and let a= min {IF(x,u 0 )i; x E [0,1]} .

c

Let

>

o

n

be arbitrary and choose

i

2 arcsin

Then there exists a neighborhood

€

e:




f

p, ,z

(x))

such

O

37.13.

LEMMA.

Let

f E Ll{(0,1)) • Then lim p, f(z) 11 z 11 2~ ... IR

uniformly with respect to Proof.

Let

f

n

E C(

p EM

[o, 1] )

be such a sequence that

lim I I fn-f I ILl ((O, 1)) - 0 n-.m Thus l

sup JI fn(s) ids - q n Ii IN 0


0

d

(28)

sin 2 1j, + ICP f z (lj,) • n' E (0,2,r); jcos ti < n}

I

(♦

for

-

llzll II 2

~ rl •

p E

M

n

!. -

C

tl

I

be

E > 0 >

such

to

.!. d,P C


sup {

1R

for

Hj

I Iu I I

e [0,1] •

t

N; u e K}

(j

• 1 , 2)

IR

and, moreover, v 1 (i) • v 2 (1) = V.

Let us define a function by

h

[0,1]

x

t

e [o, 1]

,

t

e [o, 1]

from

h(t,u)

Then h mapping

au

into

u e

H1

u e

H2

!RN - {8}

is a homotopy in IRN - {0} between the identity I and the mapping I - v and hence by the homotopy

property of Brouwer's degree (see Section 20.2) it is d[I;U,e] • d[I-v;U,8] • 0 which contradicts the assumption

0 e U.

Let us suppose for the sake of brevity that 8 t G~(K 1 ) (the other case may be proved in the same way). Let V be the component of the set IRN - K1 with e e v • Let us define a mapping h 1 from [o, 1] x av into ir,N - { 0} by h 1 (t,u) = t'.l -

(1-t)

(u - F(u)) •

It is easy to see that h 1 is a homotopy in IRN - {8} ween the identity and the mapping F - I and hence

bet-

d[F-I;v,e] • d[I;V,e] • 1 Since Brouwer's degree of the mapping u0 e V

there exists

F - I

is nonzero,

such that F(u 0 ) • u 0 •

39.3.

Let

g

1R1

-+

IR 1

satisfy locally tne Lipschitz

- 300 -

condition. Let us denote by defined by

F

F

tR 2

is continuous on

there exists

~2

[R 2

into

w9 ,f(l,z)

F(z) • The mapping

the mapping from

Since

such that 211 .

Further, let us set 2

I Iz I I

[R

It is clear that

K

and

2 ,:;. P

----'II.'-'--

F(z) •

2

z} •

IR

IR 2

is a compact subset of

and ~he

assumptions (a) and (c) of Lemma 39.2 are satisfied. Let us show that 0 ( G=(K) • Let function from [o, 1] into v(O)

v be an arbitrary continuous IR 2 such that and

.. 0

I I V(l) 11

>

[R2

p

Let us set tl

sup {t € [o, 1];

I I v 11

t2 = inf {t € [o, 1]; I IV (t) 11 Then for

t

E [t 1 , t 2J

ro}

IR2 IR2

=

.

p}

it is I

lv

11

IR

2




o

Y

be a fixed number.

(iii) Suppose that J : X -+ z is a mapping with the following properties: (J 1)

J

is positively a-homogeneous, i.e. u EX,

t > O °"J(tu) • taJu

(J 2)

J

is a homeomorphism from

(J 3)

J

is odd, i.e. u E X

(iv)

Let

S

Y-+ z

~

J (-u) •

X

onto

-

Ju •

Z

be a mapping satisfying the following

conditions: (S 1)

S

is positively a-homogeneous and odd;

(S 2)

S

is continuous; - 304 -

(S 3)

the mappings u 1-+ Su+

(y)

Let

G

(vi)

Let

JI

40.2.

.

u

-

a,-+

Su

are completely continuous from

X

X ---+Z

."

into

z

.

be a completely continuous mapping.

be real parameters.

NOTATION.

Let

be defined by

u

T(µ,v)

EX •

Denote A-1 -

{(µ,v)

€

JR2;

AO

• ~2 - A-1

A1

= { (µ,v) e:A0 ; d [ z-T(µ,v)(J -1 z); Kz(l),ez]

~

o} ,

where Kz (r) •

{z

E

z; I I z I I z


0 such that + + I IY 1-Y2l la ~ Pl IY 1-Y2I 18 ,

.!L..!.:_

I ly~-y;I la for each

~

Pl IY1-Y2I la

y1 , y2 E H .

Let S: H-+ H be a linear selfadjoint completely continuous operator and suppose that J is the identity operator. Let T(µ,v) be the operator defined in Section 40.2. It is easy to see that all results from Chapter 40 are applicable. Now we shall consider a situation which is not included in Chapte~ 40. Choose

such that

Suppose, moreover, that and 41.2.

dim Ker [L] • 1 is the linear hull of h 0

Ker [L] LEMMA.

Let

a• B

6(0,B) >

H ,

I Ih 0 I IH - 1 •

be real numbers anq suppose

(1)

Then

e

o, where

ll(o,B) •

- 314 -

P r

=

sup {6 > O;

0

0

and

V

f

Suppose that there exists

e Im[L]

n

.

I lvn 11 H ,:;. 6n

lim 0

lim on n-+oo

.

0

such that

'

13S(h 0 +vn)

- . ho>H

.

~ 0

and the assumption (Y 2)' imply H

~

O

which contradicts (1). 41.3.

THEOREM.

a, 13 E IR 1

Let

and let the assumptions sta-

ted in Section 41.1 and (1) be fulfilled. Let

I IGu for each

u, v EH

-

Gv

I IH

~ c

G: H - H be such that

c > o

a Lipschitzian mapping, i.e., there exists

I Iu-v I I H

Suppose sup u~H

I IGu I IH

< ..,

•

If (2) (3)

where of L

k

= I IK I I f P I IS I I ( Ia I+ I i3 I } + c) k I-k < o(a,a)

< 1

,

,

11 K 11 and 11 S 11 are the norms of the right inverse and the operator S , respectively, then there exists

a lower Semicontinuous function r : Im[LJ -

IR 1

bounded from below on bounded subsets of following properties:

- 315 -

Im[LJ

and with the

.

(a)

The equation

(4)

Lu+ T(a,B)u +Gu~ y

has a solution for the right hand side

ye H

if and only if

(bl The equation (4) has at least two solutions for the right hand side ye H provided

41.4. PROOF OF THEOREM 41.3. Denote by projection from H onto Im[L] , i.e.

pc: y Step

1 •

For fixed F t,y

I-+

Pc

y - HhO.

t c

m1

and

ye H

the orthogonal

y ~ H. define a mapping

Im[L] -+ Im[L]

by

With respect to (2), the mapping Ft,y is Lipschitzian with a constant k < 1 and thus according to the Banach contraction mapping principle there exists a unique fixed point Vt

of

,Y

E

Im[L]

s t e p 2 • For all t 1 , t 2 e m1 (by an easy calculation)

Ilvt1,y 1 Step

3 •

2

vt 2 ,y 1 IH

~

and

y 1 , y 2 EH

l~k lt1-t2I + 1

we obtain

!~~I IIY1-Y2i IH ·

Define ty: t

1-+

P(T(a,fl)(th 0 +vt,y> + G(th 0 +vt,y>) ,

where P: y 1-+ y-Pcy, ye H. The equation (4) with the right hand side y EH has a solution u 0 6 H if and only if there exists t ~ m1 such that 0

- 316 -

(5)

Py.

Step

4 •

Since V

and Ker[L] lent to

= V

t,y

t,Pcy

is one-dimensional, the equation (5) is equiva4>

C

p y

(t)

where

Step

5 .

For fixed

nuous on

y EH, the function

and, moreover, lim ltl~m

4>

C

4>

c P y

is conti-

-.

(t)

p y

{This follows from the inequalities 4>

Pcy

{t)

~

t 0 , V

¢

Pcy

(t)

>

-t - sup I !Gui (h 0 + _hy) t ' 0 .H U€H if

t
0

ted) such that for arbitrary


a

+

n

a

a

a,

and a

1


1

(d)

µ

> 1

(e)

µ

> 1

~

is arbitrary;

µ

is arbitrary,

Proof . of (2).

1

,

. .

Let

V

> 1

V

> 1

V

> 1

u0

E

µ = 1

.~ .

µ½+v ½

€ IN

v½(l!½_q e IN µ½+v½

,

l!½(v½-ll µ½+}~

W~' 2 (o,n)

E IN

be a nontrivial weak solution

c;, virtue of the regularity assertion (see Section

4.19) it is u 0 € c 2 c[o,n]) and u 0 is a nontrivial classical solution of (2). According to the Uniqueness Theorem for ordinary differential equations the function

u 0 has a fjnite (O,n) • If u 0 has no then we obtain either (a) or (b). If u 0 (O,n) then v > o , µ > o and the

number of zero points in the interval zero point in

(O,n)

has a zero point in function

u0

is periodic with the period

n(µ-½ + v-½) • (This follows from the fact that on the interval where the function u 0 is positive there exist constants € JR 1 such that

- 323 -

m

>

o ,

a E

and. analogously, u 0 (x) • rt sin v~(x-b) with suitable constants n < o, b £ ~ 1 on the interval where u 0 (x) < o .) Hence one of the equations k(v-~ +

1 ,

µ-~) •

k(v-~ + µ-½) + µ-½ • 1 , k(v-½ + µ-½) + v-½ • 1 must have a positive integer k for a solution. Thus one of the conditions (c) - (e) is fulfilled. Conversely, if one of the conditions (a)-(e) is fulfilled then it is easy to construct a nontrivial classical solution of (2) and thus also a nontrivial weak s~lution. Now we are ready to present the following existence theorem. 42.3.

THEOREM.

ilt- {(µ,v)

Put E

...

u U { (µ,v) k•O

...

U

U {(µ,v)

IR2; II




k•l

< vI:; < nk+l (µ I:; ) } U

k, nk(µl:i) < vi:;< e'k(/s)}

where

{

(k+l)T T - k

{

T

E

(k, 2k+1]

kt T - (k+l)

.

t

E

~2k+l,oo)

kT T - (k+l)

•

T

lo

(k+l,2k+l]

(k+l) T T - k kT

~

- 324 -

t € (2k+l,"') T E (k,m)

.

•

If

e ]ft

(µ,v)

and

f E L 1 ((0,w)) u e

has at least one weak solution P

w~•

2

then the problem

(1)

co,w) •

r o o f • With respect to Section 40. 6 (iv) we have m:, C • The assertion of Theorem follows from Section 40.6 (vi).

CA 1

In the same way we obtain the following result. 42.4.

Put

THEOREM.

where

Let If

~ : ~1 (µ,v)

'€

R1 ~

be a bounded and continuous function. then the periodic problem

X

(6) u(O)

has for any

f

Proof.

Put

X •

{u

y ..

Z

E

=

€

•

u'

C([0,211])

u

s

Ut-+ U

G

u

-

~

(O) •

u' (211)

at least one solution.

c 2 c[o,2w]); U(0)

J

I-+

(0,2w)

u(2w) ,

C ( [o, 211])

C • {u e Z·, u(x) I-+

€

u(2w), u' (0) • u' (211)} ,

; ~ 0

X

e [0,211]}

u' •

(u(x))

and apply Chapter 40.

- 325 -

CH APTER RESULTS

OF

43

E. N. DANCER

E. N. DANCER [205] studied the solvability of the problem µU+ + vu - f {-u" 0 u(O} = u(11)

(1)

also in the case

(µ,v)

~

in

(O • 11}

1dt,. The following theorem holds.

43.1. THEOREM. Let (µ,v} f: IR 2 - ~ , where the set m, defined in Section 42. 3. Then there exists f e c"' ( [o, 11]) such that the problem (1} has no solution. 43.2. that if

REMARK.

As

(µ,v) E R 2 -

R 2- ~

C A2

is

(see Theorem 43.1), it follows

~ the problem (42.1) with a bounded

continuous function w : IR.1 -+ IR1 is not weakly solvable for a certain right hand side f E L1 ((0,11}} and there exists also f E C([0,11]} such that (42.1) is not solvable. (This result follows immediately from Theorem 43.1 and from S~tion 40. 7 (i) • ) 43.3. REMARK. Note that Theorem 41.3 implies immediately th a t i n every componen t T Of IR. 2- nv """ t h ere ex i sts ( µ,v ) e T near to ((2n+l) 2 , (2n+l) 2 ) such that (1) is not weakly solvable for an arbitrary right hand side f E L1 ((0,11)) • Theorem 43.1 gives the same result 'globally' for the points ( µ, v) from the components of IR 2 - 'fl, . Thus it foJ;_lows from Chapter 42 and Theorem 43.1 that

43.4. THE PROOF OF THEOREM 43.1. (i) First of all we notice that according to Section 40.5 (iii) and the density of - 326 -

c"' ( [o, 11])

in L 1 ((O, 11)) it is sufficient to prove the exisf E L 1 ((0,11)) for which (1) has no solution.

tence of (ii)

e

.

µ < 1

Let

e L 1 ((0,11)) W~' 2 (0,11)

or 1J > l V > l V < 1 Let f ' ' and suppose that (1) has a weak solution

E.

u0 E

. Then

1T

Jf(x) sin x dx • 0

11

11

(1-µ)Ju;(x) sin x dx + (v-1)fu~(x) sin x dx,

•

0

0

and thus II

11

Jf(x) sin x dx

~

o

~

Jf(x) sin x dx

or

0

0

0

is a necessary condition for the weak solvability of (1). This proves the assertion of Theorem 43.l in the above mentioned special case. (iii) by

For

~(µ,v) a

(µ,v) €

m2 ,

> l

µ

denote

µU + + vu - 0 u' (O) 0 , a

=

By an elementary calculation we obtain that ~(µ,v)(,r) > o ~-1

-1

a e ~1

(µ,v) € IR2- ?tt'.- if and only if

P r o o f

~(µ,v)

and

the solution of the initial value problem

{ -:::)-= Then

v > l

and the functions

have at least one zero point in

(0,11)

~(µ,v) +l

'

if and only if

and both the

Analogously,

- 327 -

functions

t{µ,v) +1

t(µ,v)

'

-1

have at least one zero point in

if and only if

(O,w}

1

(/ -l)v½ .....,..½-~ri;µ

+

> k ,

V

The proof of Theorem 43.l will be complete if we prove the following assertion. Let

(iv)

E (0,11)

µ > 1 , v > 1 • Suppose that there exists such that t(µ,v)(t)•t(µ,v)(t) +1

for (1)

-1

t e [t0 ,11] Then there exists has no weak solution.

Proof.

Put for simplicity

t

±

>

t0 E

O

f E L1 ((0,11))

such that

• t(µ,v) • It is necessary ±1

to distinguish the following cases:

II.

and the analogous possibilities for the derivatives

•~,

•:

if

The reasoning in all cases is al.most the same. Thus we shall prove only the case I. Without any restriction of generality we suppose that t 0 > 11-211µ-½ Let

f

be defined as follows:

- 328 -

f

{

X f-+

0

-1

,

[o,t 0]

X

e

X

E ( t 0 , n]

'

f , the boundary va-

We shall prove that for the above given lue problem (1) has no weak solution. Suppose to the contrary that (1).

u

is a weak solution of

Then (according to the regularity result - see Section u E C 1 ([0,n]) C 2 ([o,t ]) ii C 2 ([t 0 ,n]) ,

n

4.19 and 4.21) If

a • u'(O} > O

0

then the Uniqueness Theorem implies

Let

there exists

In the interval

such that

(2)

for, in the opposite case, (~ ) '('r)

~ o ,

-r

+

E (t 0 , t 1 )

,

we have

and thus

u(,) for

t E (t 0 ,t 1]

~

at+(T) > o

, which is impossible. From (2) we have

Denote

Obviously, an interval

FE c 1 ([t 0 ,n]) . We shall prove that there exists

./l C (t 0 ,t 1 ) F' (t)

with the following property: < O ,

- 329 -

t

E ,fi, ,

u'(t) < o

Suppose

for all t e (t 0 ,T 1 ) • As F(t0 ) • o and we obtain the existence of vt as a consequence of the continuity of F , F' and the fact that the implication F(T 1 )

o


0 and F(Tl) < 0 we obtain the existence of the interval .It; analogously as in the previous

step.

..It we have

On the interval 0 >

F'(t) = (u't+ - Ut~)(t) =

=

u''(t)t+(t) + u'(t)t~(t) - u'(t)t~(t) - u(t)t~'(t)

=

uu(t)t+(t) - f(t)t+(t) + uu(t)t+(t) • f(t)t+(t)

= t+(t)

> 0

which is a contradiction. If with

n = u'(O) < O

t+ If

replaced by u'(O) • o

the proof is quite analogous as above t_

then· u(t) • O

for

t

Denote tl • inf {t E (t 0 , 11); u(t} .. 0} As (

u''(t 0 +) - 1 we have tl > to and Thus u is a solution of

(tO,tl)

.

i

-u" =

µU -

1

in

=

.

u(t) > 0

(t 0 ,t 1 ) 0

- 330 -

,

u(t 1 ) - 0 •

for

t E

From this fact we see that

u

is a solution of

and thus we obtain

This implies

so

t 1 • 2wµ-½ + t 0

>

w ,

which is a contradiction.

43. 5.

REMARK.

DANCER [2o5] considers not only ordinary dif-

ferential equations with constant coefficients but also with variable coefficients (his paper includes also some results for partial differential equations).

_4~3~·-6~•__REMA___;_~RK-~. Some sufficient conditions for the solvability of (1) and (42.6) in the case of (µ,v) e A_ 1 are given in DANCER [2 l 7] •

43.7.

REMARK.

Further results from this field are announced

in DANCER [218]. It is worth noticing also that some results which are connected with the OPEN PROBLEM 26.12 are announced there.

- 331 -

CH APTER

44

F U R T H E R E X AMP L E S 44.1. (p-1)-HOMOGENEOUS ORDINARY DIFFERENTIAL EQUATIONS. DRABEK [219] , [220] applied the abstract results from Chapter 40 to obtain the weak solvability of the boundary value problem

-[Ju'(x)IP- 2 u'(x))' - µju+(x)jP- 2 u+(x) + (1)

{ +vJu-(x)Jp- 2u-(x) + g(x,u(x)) • f(x) •

x E (0 1 ,r)

u(O) • u(rr) • O . Using essentially the same method as in Chapter 43 (but involving more complicated technical de.tails) he proved essentially the same results as in the case p • 2 (considered in Chapters 42 and 43). Notice that the spectral problems for (1) with µ • v • ~ are investigated in NECAS [221] , [222], see also FUCfK-NECAS-SOUCEK-SOUCEK [32] • The results frpm Chapter 40 are applicable also to the case of nonlinear ordinary equations with nonconstant coefficients and of the type (1). 44.2.

J (sin nx)-dx

t+m

O ff

0 ff

lim inf w(t}f (sin nx}+dx - lim sup ~(t} J (sin nx}-dx C+-m

Q

t+m

- 338 -

0


IJ 1wn(X)ldx +

1

E

max g(t) J W~(X)dX • [-an-cl ,-aJ n-nn

min g(c)Jw:(x)dx [a,an +c1J n E; E

max g(c>Jw~(x)dx [:-an-cl ,-aJ n

- [ supl lgif 1wn(x) ldx + c e IR nn +

min g(c>J w:(x}dx t E [a,an +c 1J nn - 351 -

i

i.e.

sup lgIJ lwn(x)!dx + min g(c)J w:(x)dx Cf IRl nn CE [a,an +cl] nn

~

max

i;

[-a 0 -c 1 ,-aJ

ni~ I; Ii

[a,an +c 1] +

~

g(c>Jw~(x)dx 0

(g(F;)-g+) fw~(x)dx + n

min

~ E [-an -c 1 ,-a]

,Vn~IN,

(g_-g(C)) fw~(x)dx 11

and -1 (an+c ) l+p meas 3(a+c 1 )an 1

nn

sup ~

€

Ig(F;) I

~

lRl

(g(~)-g+) Jw~(x)dx + il

Vn e

fll

Since

• meas

.i

n

sup lg(Ol

2Ky-l ,

(a +c )l+p n 1

n e lN ,

for all K

>

~

n

n 0 • Thus

2Ky,- 1 fw~(x)dx + 2Ky-lfw~(x)dx

n

2Ky- 1 f 1wn(x) ldx

n

>

2K ,

n

which is a contradiction. 47.5.

REMARK.

From the above proof, it is iITL~ediately appa-

rent that instead of (G 2) we can suppose that the corresponding limits are sufficiently large but not infinite. 47.6.

THEOREM.

SuppO$e (Ll)- (L4),

=

g+

(G 3)

lim g(C) ,

g_

z

s+oo

Then the equation

(1)

f = fl+f2

E

fJ

if

(Gl), (G2), (LG) and

lim

g(s) •

(+-oo

has at least two different solutions for ':lf_-Y • .l

Proof.

Let

fEZ

besuchthat

f 1 -Qfl!i'.ff-J'. 2

From the proof of Theorem 47.4 it follows that there exists a rectangle

-13 n

in

such that

X

0

d[_~(J,.); ,2,n,

0]

=

1.

0

Further, there is

H

(10)

l . such that

w E Ker[LJ

>

f (g+;+(x)-g_w-(x))dx Q

Since the integral on the right hand side in (lo) is nonnegative,

f 1 /. o

and thus

f ¢ Im~LJ • Let a c-onstant

Y ,

such that the equation Lu+ Gu= (l+K)f has no solution in

X

(note that -

353 -

G

has a bounded range

0

be

in

z ). We consider the homotopy mapping (t,u)

Y( I

t E [o,KJ ,

I-+

u + (L+jP)- 1 (Gu - jPu - (l+t)f)

u E X . There exists a constant 7' ( t, u) •

(11)

e for some t

➔ I IPcu I IX For

n "

[o, K] ,

u

such that

e x ->

c •


b

I IPcul Ix < c} •

such that

[o,K] , 'v'u e acn

E

• 1

Indeea, let us suppose, to the contrary, that to each n > n 0 there exist tn € [o,KJ and un € acn such that J{(tn,un) • • e . Then

tn--+- t

We may assume

• n. Put

(n-,. •) • By (11) we have

un •aw + Pcun , where n n We may assume that

(L

1)

I IPunl Ix•

--+- w n and (L 3)). According to (13) we have w

J

g(a nwn (x) + (Pcun )(x))w(x)dx • (l+t n )H -~ n ?;.

Choose

(l+tn.)H > (l+tn>J(g+;+(x)-g_;-(x))dx n

£0 > O

sup

so that for all lg(O I meas {x

f,; € Rl

< 2

-1 -

H - 2-

lJ

£

l:

e (0,£ 0 )

n; lw(x) I

we have ~

d

-+ (x)-g_w-(x))dx (g+w

n - 354 -




Jw(x) I >E}

~[

I

{x

€ O;

+ Jw{x) J >d {x

f E .i;

sup Jg(O IR 1

)g(anwn(x)+(Pcun)(x))w(x)dx

lw(x) J~d

I

meas {x

E .i;

Jw(x)

I

~ d

>

E; E

> fg(a n wn (x)+(Pcun )(x)J.w(x)dx n

2- 1 ,fi,w>H + 2-lJ (g+;+(x)-g_:-(x))dx

n =

-+ (x)-g_w-(x))dx (2 -1 +tn)H + 2- lf (g+w n

i.e.

J

g(a w (x)+(Pcu )(x))w(x)dx > n n

n

·

{xen;lw(x}l>c) +

2-lJ

(g+;+,x)-g_;-(x))dx

n

If n ~ n 0 (n 0 sufficiently large) then Jwn(x)J > ,/2 for any x E {x en; lw(x)J > E} . Assuming n--.. ~ and using the Lebesgue Dominated Convergence Theorem we obtain

J ~ (2

-1

J

g+w(x)dx .+

{x,n; w(x)>d

-

+t)

40

!BFI

41

U

I

k

ll

.. 389 -

2

*

Ck(n)

3

2

Ck(ll)

3

V

111

c~ c(i)

3

L2(0)

4

ck' .l. (IT)

3

wk 2

11

Ck, .l. (ii)

3

c.Jo>

337

ck

118

C~(IR 1 ,IRN)

llB

0

T

~ (Q)

104

HO(Q)

336

Hl(Q)

337

w

w

' (a,b)

wk 2 •

104 (Q)

~ z 0. (Q)

104

Dom[T]

l

Im[T]

1

Ker [L]

2

';C(v,u)

110

~(v, u)

111

0

Hl(Q)

337

meas

8 1,2(Q)

337

det.

127, 129, 134

L

4, 104

Jae [f (x)]

134

d[f;D,p]

134

{v,f, ♦}

108

w

w

p

(Q)

L,. (Q)

4

L2,T

llB

l N L2 'T (IR , IR )

llB

wk•P(a,b)

11

~•P(a,b)

11

wk•Pcn>

104

~·Pun 0

104

- 390 -

M

4