Then t h e r e e x i s t s an o r d i n a r y d i f f e r e n t i a l system:
t h e s o l u t i o n s of which a r e s o l u t i o n s o f ( L ) , ( R ) Ila(t,=)ll < p
Finally, i f t = 0) 0
c
Y(t)
Rn -isomorphisms,
i s , within
(R)
u n i q u e l y determined by t h e c o n d i t i o n s
unique
(L3) :
.
Moreover, t h e system
(with
verifying
(L1), (L2)
(L3).
i s a fundamental system of s o l u t i o n s of
, then f o r every s o l u t i o n x of R~ , such t h a t :
in
and
x ( t ) = Y(t)
(L)
(R)
there exists a
(c + o ( 1 ) )
.
To prove t h e theorem, we o n l y have t o come back t o t h e o r i g i n a l r e s u l t by Ryabov. We f i r s t r e c a l l what Ryabov c a l l s a " s p e c i a l s o l u t i o n " : DEFINITION ( [ 6 ] )
on than
.
A " s p e c i a l s o l u t i o n 1 ' i s a s o l u t i o n of
(L) , defined
R , growing a t most e x p o n e n t i a l l y , with an exponent n o t g r e a t e r l/r
.
Ryabov t h e n proved: LEMMA ( [ l o ] , [ 6 ] ) .
(toyyo) i n
R
through
at
y
o
x
Assume
R"
(L1), (L2)
, there
and
(L3).
Then, f o r each
e x i s t s a unique s p e c i a l s o l u t i o n p a s s i n g
t o . The s e t o f t h e s p e c i a l s o l u t i o n s i s an n-
dimensional space. Each s p e c i a l s o l u t i o n
y(t)
s a t i s f i e s an estimate:
REMARK 2 .
The first part of this lemma means that such a system of
solutions is complete. All we have to do in order to prove the theorem is: (i) observe that a complete family of special solutions is associated to an ordinary equation in R~ ; (ii) that there is uniqueness within isomorphism; (iii) prove the asymptotic formula (end of THM). We will prove (i) , skip (ii) and go very fast on (iii) . To prove (i) , let x be a special solution. It can be expressed as:
Because of the uniqueness property stated in the lemma, we can see that, for each 0
in
terms of x(s),
[-r,01 , s in R
, x (s + 8 ) is uniquely determined in
so that x(s) + x(s+e)
defines a map G(s,e).
Because of the lemma, we have:
Using G, (5) can be written as :
Let: g(s,x)
=
L[s,G(s,-)
*x).
We then have: dx/dt = ~(t,x(t)),
which yields the first part of the theorem. Moreover:
e ?Jr= 11 ; s o , we g e t
Il!l(s,-)il < K
h a s been proved by R.D.
(L3).
The l a s t p a r t o f t h e theorem
Driver [6] u s i n g " s p e c i a l s o l ~ t i o n s . ~ ~
We t r a n s f o r m t h e e q u a t i o n u s i n g t h e r e s o l v e n t ( i n f a c t , we u s e :
Y(t) = Y ( t , O )
: x ( t ) = Y(t)
Y(t,s)
of
(R)
z(t)).
Using t h e Gronwall-type i n e q u a l i t y ( [ 1 ] , [ 3 ] ) we can s e e t h a t : (d/dt)z
is in
1 L (to,+ a ) , so t h a t
each
in
,
c
such t h a t :
4.
W"
z(t)
z
there exists a solution
+
c, t
w ,
and f o r
(and s o a s o l u t i o n
x)
CONCLUDING REMARKS
Rn.
For
system o f
r > 0
(L) -an
o.d.e.'s?
in
R~
, that
0.d.c..
0.d.e.
sub-
c o n t a i n s t h e i n f o r m a t i o n on
Why now do we c o n s i d e r a f o r m u l a t i o n i n terms
I n what way could t h i s concept be more i n t e r e s t i n g tharl
t h e one of s p e c i a l s o l u t i o n s ? be p a r t i a l .
i s an
r =0 , ( L )
for
small ( s e e ( L Z ) ) , t h e r e i s s t i l l an
0.d.e.
asymptotic behaviour. of
z
+
.
+ +
Our theorem i s a p e r t u r b a t i v e r e s u l t : in
has a l i m i t a t
The answer t o t h e s e q u e s t i o n s can o n l y
In ( [ 2 ] ) , we combined t h e
s u l t s on asymptotic i n t e g r a t i o n o f
0.d.e.
f o r m u l a t i o n with r e -
o . d . e T s t o g e t a s y m p t o t i c formulae
f o r f u n c t i o n a l d i f f e r e n t i a l systems. Another i n t e r e s t i n g f e a t u r e i s t h a t n a t u r a l simple a d j o i n t e q u a t i o n i n
(k)
p r o v i d e s u s with a
( R n ) * , which i n f a c t can be used
t o d e s c r i b e t h e l i m i t i n g behaviour o f t h e s o l u t i o n s of c i s e l y , t h e r e e x i s t s a fundamental s o l u t i o n c = l i m t++a
(where
c,x
Y*
of
(L).
(R*)
Pre-
such t h a t :
a r e a s i n t h e theorem).
On t h e o t h e r hand, t h e n o t i o n o f a "subsystem" i s s t i l l " t h e o r e t i c a l , l l i t needs much more work t o be r e a l l y u s e f u l , and
n o t a b l y t h e following q u e s t i o n can be s e t :
I s it p o s s i b l e t o g e t such
subsystems without t h e i n t e r m e d i a r y o f s p e c i a l s o l u t i o n s ?
REFERENCES [l]
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(PubZicatims Math&matiques- Paul and Proc. of Equadiff. ,
Springer Verlag Lect
. Notes,
(1982) .
[2]
Arino, 0. and I . Gyori. Asymptotic i n t e g r a t i o n of f u n c t i o n a l d i f f e r e n t i a l systems (submitted t o Journ. of Math. Anal. and AppZ.), (1984).
[3]
Arino, 0. and I. Gyori. Gronwall-type i n e q u a l i t i e s f o r f u n c t i o n a l d i f f e r e n t i a l systems, i n Proceedings of the Conference on QuaZ&5ative Thsamy of PifferentiaZ Equations, Szeged, Hungary, August (1984)
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[4]
Atkinson, F.V. and J . R . Haddock. C r i t e r i a f o r aysmptotic constancy o f s o l u t i o n s o f f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s . J. Math. Anal. A p p Z . , 91(1983), 410-423.
[5]
Cooke, K.L. Asymptotic t h e o r y f o r t h e d e l a y d i f f e r e n t i a l J. Math. Anal. A p p l . , 19(1967), e q u a t i o n : du/dt = - a * u ( t - r ( t ) ) 160-173.
[6]
D r i v e r , D. Linear d i f f e r e n t i a l systems with small d e l a y s , J. D. E . , 21 (1976), 147-167.
[7]
Gyori, I . On e x i s t e n c e of t h e l i m i t s of s o l u t i o n s o f f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s . CoZZ. Math. Soc. J. BoZyai. 30, QuaZ. Th. of D i f f . Eq., North Holland, (1979)
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Haddock, J . R . and R. Sacker. S t a b i l i t y and asymptotic i n t e g r a t i o n f o r c e r t a i n l i n e a r systems o f f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s . J. Math. Anal. ADDZ., 76 (1980), 328-338.
[9]
Hartman, P . and A. Wintner. Asymptotic i n t e g r a t i o n o f l i n e a r d i f f e r e n t i a l e q u a t i o n s . h e r . J. Math., 77(1955), 45-86.
[lo]
.
Ryabov, J u . A. C e r t a i n asymptotic p r o p e r t i e s o f l i n e a r systems with small time l a g ( i n Russian). Tmdy Sem. Teor. D i f f . Druzby Narodov P. Lumwrrby, 3(1965), 153-165.
[ll]
Slater, G. L.
The differential-difference equation dw/ds , Proc. of Roy. Soc. of Edinburgh,
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=
