In this section we will give a condition for solving (28) for near the solution of the same equation with . Writing
we look for solutions of
in the Banach space of -functions on ℝ, bounded together with their derivatives and with small norms. We observe that and equation reads
In Section 3 (see also [1, 19]) we have seen that (36) has an exponential dichotomy on both and with projections . So the only bounded solution of is . In other words . So we are lead to prove the following.
Theorem 4.1 Let Y, X be Banach spaces, be a small parameter and . Let , , be -functions such that
there exist such that .
and suppose there exists such that and the derivative is invertible. Then there exist and unique -function , defined in a neighborhood of such that
and for , , (35) has a unique solution satisfying
Moreover, for -functions and we have
Proof We look for solutions of (35) that are close to . Let be the projection such that . Note . From the implicit function theorem, we solve the projected equations
for unique such that
provided is sufficiently small and η in a fixed closed ball with . Note that are -smooth. Setting , we need to solve the bifurcation equations:
Then and so
uniformly with respect to η. We conclude that (40) can be written as
Note that are -functions of and that uniformly with respect to η, so
is in . By (41), system (42) is equivalent to
Because of the assumptions we can apply the implicit function theorem to (43) to obtain a -function defined in a neighborhood of satisfying (43) and such that (38) holds. Setting
we see that , are bounded -solutions of (35) with such that , . Then we can write , ) for continuous where
Clearly (39) follows differentiating the above equality at . The proof is complete. □
Remark 4.2 Note that, because of , (39) is equivalent to
which has the unique solution
Now we apply Theorem 4.1 to (28) with , , as in (33), (34) and
where is the Banach space of -functions bounded together with their derivatives with the usual sup-norm.
We already observed that and . Moreover,
where have been defined in the previous Section 3. So and . We recall, from , that where are solutions of the adjoint equation of (16):
and . Hence (37) reads
or passing to time :
A direct application of Theorem 4.1 gives the following.
Theorem 4.3 Let and be given as in (46) where , are two independent bounded solutions (on ℝ) of the adjoint equation (45). Suppose that and exist so that
Then there exist , , unique -functions and with and , defined for , and a unique solution of (24) with , , , such that
Remark 4.4 (i) Equation (48) implies
for -functions with . Then we have
with , so . Hence
which gives a first order approximation of . Next, can be computed using (44) adapted to this case. Hence are bounded solutions of
Since (36) has exponential dichotomies on both (with projection ) and (with projection ) it follows that
is the fundamental solution of (36). Note formulas (50) are well defined at , i.e., , due to the first assumption of (47). Next, passing to time and taking , we get
for . Note that solves
with , and is a fundamental solution of (16).
(ii) Using (49), the functions are bounded solutions of (7) in the interval such that
Summarizing, we obtain the following corollary of Theorem 4.3.
Corollary 4.5 Let and be given as in (46) where , are two independent bounded solutions (on ℝ) of the adjoint equation (45). Suppose that and exist so that
Then there exist , , unique -functions and with and , defined for , and a unique solution of (7) with , , , such that