195
In (2.35), DR is the Frechet differential of R, y is the solution of (1.25) and ' fJ is solution
of the following linear differential equation (namely the linearized form of (1.25)) :
{
d'fJ
dt + A7] = PDR(y + tJ1(y)). (7] + ~(y)7]),
7](0) = 7]0·
(2.36)
Computations similar to those in Section 2.2 show that if (2.1) holds with
(2.37)
then T", is a contraction with constant 8 (we choose 8 =1/2).
Hence (2.33) and (2.34) are satisfied by f = T and g", = T",; denoting by ~. the
fixed point of T. (and attractive fixed point of
T :Ft,b x 9t -+ :Ft,b x 9t,
(tJ1,~) -+ (TtJ1,T",(~)).
If t.p is any C 1 function in :Ft,b, then it can be checked by elementary calculations that (1 )
T t.p belongs also to C 1 n :Ft,b and
T(t.p,Dt.p) = (Tt.p,DTt.p).
Hence, we obtain recursively,
As n -+ 00, Tnt.p converges to and
D
In (2.35), DR is the Frechet differential of R, y is the solution of (1.25) and ' fJ is solution
of the following linear differential equation (namely the linearized form of (1.25)) :
{
d'fJ
dt + A7] = PDR(y + tJ1(y)). (7] + ~(y)7]),
7](0) = 7]0·
(2.36)
Computations similar to those in Section 2.2 show that if (2.1) holds with
(2.37)
then T", is a contraction with constant 8 (we choose 8 =1/2).
Hence (2.33) and (2.34) are satisfied by f = T and g", = T",; denoting by ~. the
fixed point of T. (and attractive fixed point of
T :Ft,b x 9t -+ :Ft,b x 9t,
(tJ1,~) -+ (TtJ1,T",(~)).
If t.p is any C 1 function in :Ft,b, then it can be checked by elementary calculations that (1 )
T t.p belongs also to C 1 n :Ft,b and
T(t.p,Dt.p) = (Tt.p,DTt.p).
Hence, we obtain recursively,
As n -+ 00, Tnt.p converges to and
D
.
(2.38)
2.4. Proof of Theorem 2.1
We now conclude the proof of Theorem 2.1. We have shown that the graph M of
cI> is positively invariant and that M and cI> are of class Cl; there remains to show that
M is exponentially attracting.
First, we deduce from equation (2.29) concerning an orbit lying on M (z(t)
cI>(y(t))), that
d
dt cI>(y) + A
e) The definition of T", has been arranged for that purpose.