Boundary value problems at resonance for vector second order ...

EQUADIFF 4

Jean Mawhin Boundary value problems at resonance for vector second order nonlinear ordinary differential equations In: Jiří Fábera (ed.): Equadiff IV, Czechoslovak Conference on Differential Equations and Their Applications. Proceedings, Prague, August 22-26, 1977. Springer-Verlag, Berlin, 1979. Lecture Notes in Mathematics, 703. pp. [241]--249. Persistent URL: http://dml.cz/dmlcz/702225

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BOUNDARY FOR

VALUE

VECTOR

PROBLEMS

SECOND

ORDINARY

1.

ORDER

DIFFERENTIAL

J. M a w h i n ,

AT RESONANCE NONLINEAR EQUATIONS

Louvain-la-Neuve

INTRODUCTION The

aim of this paper

continuation

theorem

tions for second linear boundary given

in section

is to use a version

(see section

The

and

[ 1 ] for semilinear

elliptic

and dealing with consider linear

linear

resonance boundary

differential

G o s s e z [ 2 ] , and

=

dence

in x'.

Moreover

case of a

and

Nirenberg

eigenvalue

for systems

=

of t . We

of ordinary

non-

0

conditions,

above and

considered

of proof differs

there, as i l l u s t r a t e d

x " + f ( t , x ) = 0

conditions

x(o)

allowing for f a depenfrom the

could be s u c c e s s f u l l y

system

(1.1)

with boundary

at the first

x, x')

the technique

quoted

some of the problems special

and

of the form

the C a r a t h e o d o r y

used in the papers

conditions

equations of the form

problems

x"+f(t,

with f v e r i f y i n g

Brizis

soluwith

f(x, u ) ,

problems

value

of

equations

by the recent work of K a z d a n

Warner [4 ] , de Figueiredo

Lu

differential

type of sharp sufficient

3 are motivated

scalar

Leray-Schauder

2) to prove the existence

order v e c t o r ordinary conditions.

of the

= X(TT)

=

0

ones

applied

to

in [5 ] . In the

242

(1.2)

(resp.

the obtained

x(o)

conditions

-

X(2TT)

=

for e x i s t e n c e

x'(o)

-

X'(2TT)

=

0

of a solution are TT

(1.3)

2

2

F(t) : = lim | x r ( x | f ( t , x) < 1 , J F ( t ) sin tdt < •-2TT

(1.4)

(resp.

F(t)

° such that

< x,u for

EQUATIONS

v.

A

be

internal

the C a r a t h e o d o r y

(1.1) denote

of u and

product

] be a compact

SECOND ORDER

all u b e l o n g i n g

to the

>

for each

=

eigenspace

|xL


and

has

245 Important special cases of mappings A satisfying (A) - (B) (C) are given by Ax

=

(x(a) , x(b))

( D i r i c h l e t o r P i c a r d boundary c o n d i t i o n s )

Ax

=

(x'(a)

(Neumann boundary c o n d i t i o n s )

Ax

=

( x ( a ) - x ( b ) , x ' ( a ) - x ' ( b ) ) ( p e r i o d i c boundary c o n d i t i o n s )

Ax

=

(x(a),

, x'(b))

We s h a l l for

the boundary

x'(b)) be i n t e r e s t e d value

in proving the existence

of

solutions

problem

(3.1)

x " + f ( t , x , x ' ) = 0

(3.2)

A(x)

i.e. the existence of mapping

=

0,

x : I-MR

having an absolutely conti­

nuous first derivative and verifying (3.1) a.e. in I and (3.2). By taking X = C

1

n

(I, |R ) with the norm

IxIoo = max C l x L ' l x ' loo ] ' =t = L 1

(I, |Rn) with the norm |x|x= J-Jxtt)!

absolutely continuous on I and

dt, dom L = [xGX : x' is

A(x) = 0 }, L : dom LcK -+i,

x •-> -x",

W : X-*-?-, x i-> f(., x, x ' ) , it is routine to check that L is Fredholm of index zero and closed and that. W is L-completely continuous. The problem is then equivalent to an abstract one of type (2.1) and we shall prove the following existence theorem. Theorem 2 . and (C).

(D)

Let f be like above and let A satisfy conditions (A), (B)

Assume that the following conditions hold :

F(t) : =

Ti^ f i x i - oo ^

sup n u G , R

( X

| f C t

' X' |X|2

g ) )

^ ;

uniformly in t^I, i.e. for each e > 0, there exists p 6

< y

> 0 and

e L X (I, IR ) such that for a.e. tel, all x with |x|> p

(x|f(t, X, U))
•

one

each

s

R > 0 there

cpR ( s )

= 0

< \ii

exists

and

such

|f(t,

x,

relative

for

ux

,

one

has

|

°°.

oo

now y = ^n ,

n = 1, 2,

x /1 x I n ' n'

...

U s i n g ( 3 . 9 ) , the finite d i m e n s i o n of the e i g e n s p a c e a s s o c i a t e d to yi and going to s u b s e q u e n c e s , one can assume that (for the norm y

->

y

,

X

with y in the e i g e n s p a c e a s s o c i a t e d to y i ,

-*•

l-l^)*

X

| y | oe> = 1 and

X €

[0,1

]

248 From

(3.6)and Wirtinger's inequality we then obtain, if X

the characteristic I

n

=

function of the set {tel :

|x ( t ) | * 0} ' n ' [x

X. J T X ( t ) | y ( t ) | l

n

n

denotes

2

|y ( t ) | * 0} , ' n '

{tel :

(t)|f (t, x ( t ) , x' (t)1

L-H

n

D

>

L dt

_

|x ( t ) | 2

"

Hence, by assumption

=

[W. - t 1 - X n ) a

(D) and by Fatou's

]|yj>

lemma,

fx (t) | f(t, X ( t ) , X' (t)l X

X (t) ly ( t ) |

J 7 lin o J I _

J

' n

n

2

^ n

'

'

n

n

dt + (1-X )a|y| 2

J

O

|x(t)|2 (3.10)

Уi IУ ľ

> But, on I , |x ( t ) | = n ' n ' X

-+• 1

a.e. in

|y ( t ) | |x | -+ 9° ' n ' ' n' °°

if n + » and, as y -+• y, n (3.10) and

I, which implies together with

assump­

tion (D) that \JT

F(t) | y ( t ) | 2 dt >

(1 - X_) a J j | y ( t ) | 2 dt

+

VH J I | y ( t ) | 2 dt ,

a contradiction with assumption (3.11) for

x"

someX£]0,1[,

+ ( 1 - X ) a x one

( E ) . Thus, if +

x^domL

Xf(t,x,x')

=

satisfies

0

has


0 such that any solution

x e dom L of (3.11)

verifies |x'|