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
Terms of use:
© Springer-Verlag, 1979
Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access to
digitized documents strictly for personal use. Each copy of any part of this document must contain
these Terms of use.
This paper has been digitized, optimized for electronic delivery and stamped
with digital signature within the project DML-CZ: The Czech Digital Mathematics
Library http://project.dml.cz
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)<0-,J
J
(with
(I) the inner product
first
eigenvalue
F(t)dt<0)
o
in IR ) so that
x" + ux
x(o)
on any subset
that
=
X(TT) =
of
the second
case of a first
0
(resp.
[ o, IT ] (resp.
condition
=
x(o)
order v e c t o r
has been
replaced
shown
in
[5 ] that
by the mere e x i s t e n c e
is an open
question
to the
problem
0
-
X(2TT)
=
x'(o)
-
X'(2TT)
(1.4)) is v e r i f i e d .
=
0)
such
In the
equation
the first
=
0
condition
in ( 1 . 4 ) could be
of F ( t ) and the second
J 2 7 T FCt)dt
It
linear
[ o , 2tT ] ) of positive measure
in (1.3) (resp.
x' + f(t, x)
it
F ( t ) can be equal
1 (resp. 0 ) of the a s s o c i a t e d
one replaced by
# 0
to Know if (1.4) for (1.1) - ( 1 . 2 ) could be
g e n e r a l i z e d by
7Г
F(t) < a , J 2TT
FCtІdt < 0
2
o
for same a > 0.
conditions
example
Let us finally
distinct
notice
that
of the ones of H a r t m a n ' s
in. [ 3 1 , Chapter V.
Theorem
2 gives
existence
type described for
243
2.
A USEFUL
VERSION
Let X,
holm
OF THE
5- be real normed
ping which
is L-compact
continuous
on X ) .
tions.
shall
for
on bounded
See [ 3 ]
be interested
the abstract
linear
1#
Assume
that
L-completely
(i)
oG n
(ii)
ker
has at
by many a u t h o r s ,
least
For a proof
L-completely
the existence
defini-
of a solution
A =
an open
bounded
: X -* -2-
set Q.cX and a
such that :
= {0}
one
for all
solution
see [5 ] .
0 and
L^ti.
special
cases of which
W and L
then
have
been
a natural
Leray-Schauder
W completely
Xe]o,i[
condition
continuous.
in [5 ] h o w to use this theorem
proofs of some results
of De
and all
in dom
one gets the usual
it is shown
s i m i l a r l y the results
x e dom Ln^U
I f L is i n v e r t i b l e ,
Figueiredo, Brezis and Nirenberg
problems.
(shortly
Fred-
linear) map-
Wx
mapping A
of this theorem,
If L is not i n v e r t i b l e ,
get
exist
of the form I— L
simple
=
(1 - X) Ax + XWx
used
give more
there
(L - A )
(2.1)
for A i s
of X
in proving
continuous
Then
for m a p p i n g s
(not necessary
problem
(iii) Lx *
choice
L : dom' Lc'X-»-^ a linear
subsets
Lx
Theorem
THEOREM
p. 12-13 for the c o r r e s p o n d i n g
(2.1)
De
spaces,
CONTINUATION
of index zero and W : X -*• i- a
mapping
We
LERAY-SCHAUDER
of Cesari and
for abstract
to
Kannan,
e q u a t i o n s . One could
Figueiredo and G o s s e z on
elliptic
244
3.
AN EXISTENCE THEOREM
FOR VECTOR
WITH
CONDITIONS
BOUNDARY
LINEAR
1= [a,b
Let
a mapping
satisfying
Let
Let
a linear
DIFFERENTIAL
of IR / f
: I x IRn x |Rn->|Rn
conditions.
the E u c l i d i a n
in IR
norm
and
(u|v) the inner
now
continuous
(A)
For each
(B)
The
CX
:
(I, IR n ) -
mapping
xGKerA ,
such
IR
n
that
( x ' ( b ) lx(b))
-
( x ' ( a ) lx(a))
= 0
problem
x"+ux
has a n o n t r i v i a l s o l u t i o n
of finite m u l t i p l i c i t y
form a complete
=
0,
if and only
0 < \i1
and
orthogonal
< u2
=
u = u
0
(i = 1, 2,...) with
<
such that
set
A(x)
if
the c o r r e s p o n d i n g
eigenfunctions
in L (I, IR ) with the inner
product
< y,z > = J I ( y(t) I z(t)) dt
(C)
There
exists Y
>
° 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 <
x <= KerA
such
0
relative
to u -_f one
YІx'l
where
|x|
ІУІ2
= max | x ( t ) |
tЄ[ a,b ]
= ( !j
that
, x' = dx/dt
|yct)|2dt)
v>
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)) <
i
(У! + Є) |x|2 + Ő j x |
and all
246
(E)
For
e a c h (p b e l o n g i n g
to
the
eigenspace
J j F ( t ) |(p(t) | 2 d t
(F)
For
lim
S
->•
one
each
s
R > 0 there
cpR ( s )
= 0
< \ii
exists
and
such
|f(t,
x,
relative
for
ux
,
one
has
|<p(t)|2dt
Sj
(p € C ( IR
that
to
, IR * )
a.e.
t^I
nondecreasing
and
all
x with
with
|x|
<R,
oo
has
Then
problem
u)
(3.1-3.2) has at
<
tp
R
(|u|).
least one
solution.
Proof
We shall
apply
Theorem
1 with Ax
(3.3)
which
a
clearly
condition
satisfies
( i i ) . The
X€]0,
the
<
in
Ui
continuity
(iii) becomes
shall
1[
first
and any
show that there
exists
(X ) with
L such
If it
X
G
0
R > 0 such that for
is not
the case,
there will
exist a
(x ) of
that
I
L
n
>
n
loo
and
Using
(3.6)
x"
n
+ ( 1 - X ) a x
+Xf((t,x,x'
)=
n
n
n
n
n
the a s s u m p t i o n
|1 x '
|2
n' 2
any
R
] 0 , 1[ (n £ IN) and a sequence
|x
(3.5)
and
x v e r i f y i n g A ( x ) = 0 and
( 3 . 4 ) is s a t i s f i e d .
sequence
requirement
here
x, x') Ф
(1 - X) ax + Xf(t,
Ix|« >
of dom
and
L-complete
inequality
(3.4)
and we
= ax
= (1
(A) (3.5)
-
XJ a | x l
0
implies
n ' n ' 2
2
+ X
<x
nn n
n
,
f(.,
x .
n
x ' ) >
n
elements
247
Let now e > 0 ; C a r a t h e o d o r y c o n d i t i o n s on f and a s s u m p t i o n
the e x i s t e n c e of 6 G L (I, IR )
(x
n
(D) insure
t^I
(t) |f(t,x ( t ) , x' ( t ) ) < (yi + e)|x ( t ) | 2 + 6 (t)|x ( t ) |
•
n
n
' n
'
e
' n
•
which t o g e t h e r with
|x' | 2
' n'2
(3.7)
Now
such that for a.e.
(3.3) and
(3.6) gives
< (yi
+ e) |x I 2
K1
' n' 2
+ |6 I |x I 0 0
'e'l'n'
let us wгitє
(3.8)
where u
(resp. v ) belongs
a s s o c i a t e d to y i .
eigenspace
Then
|x'
1
(resp. is o r t h o g o n a l ) to the
2
| - yi|x |
K 1 ,
n'2
n'2
2
=
|v- |
' n'2
and we have the W i r t i n g e r ' s inequality
2
2
- yi|v |
^ ' n' 2
(which follows from a s s u m p t i o n
(B) )
U 2 | v |2
'n'2
Those r e s u l t s
KL
and h e n c e
C3.9)
<
together with
YP2
(3.7)
.
and a s s u m p t i o n
[etb - a ) 2 | x j 2
(P 2 - y , ) " '
(C)
+
imply
that
I«JJXJ~1
that
x |
П
Let
|v' | 2
n'2
<
if
|v |
n ,
'oo
n
•>
°°.
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
<
|x|
I
R
I oo
and hence, for a.e. t G I , using assumption ( F ) ,
|x"(t)|
with cpD G
< |a| R
C° CKR
+
tpR(|x'(t)|)
=
, IR*) nondecreasing and
cpR (|x'(t)|)
lim
S-*-oo
s" 2 (pn (s) = 0.
2
249
By a result of Schmitt (see e.g. [ 3 ] p. 69) this implies the exis­
tence of
S = S(R)
> 0 such that any solution
x e dom L of (3.11)
verifies
|x'|
<
S.
Therefore, if
Q = { x G C ^ I , lRn) : | x |TO < R , | x • | ^ < S}
conditions (i) and (iii) of theorem 1 are satisfied and the proof is
complete.
REFEREIЧCES
1.
H. BREZIS and L. NIRENBERG, Characterizations of the ranges of some
nonlinear operators and applications to boundary value problems,
Annali Scuola Norm. Sup., to appear.
2.
D.G. de FIGUEIREDO and J.P. GOSSEZ, Perturbation non linéaire d'un
problème elliptique linéaire près de sa première valeur propre,
C.R. Acad. Sci. Paris (A) 28k
(1977) 163-I66.
3.
R.E. GAINES and J. MAWHIN, "Coincidence Degree and Nonlinear
Differential Equations", Lecture Notes in Math. vol. 568,
Springer, Berlin, 1977.
k.
J.L. KAZDAN and F.W. WARNER, Remarks on some quasilinear elliptic
equations, Comm. Pure Applied Math. 28 (1975) 567-597.
5.
J. MAWHIN, Landesman - Lazer's type problems for nonlinear equations, Conf. Semin. Mat. Univ. Bari n° 'ih'J 9 1977-
Author f s address : Universite de Louvain, Institut Mathématiqцe,
B-1348 Louvain-la-Neuve, Belgique.