Let
be a real or a complex Banach space. In this section, we will present a complete study concerning the characterization of uniform exponential trichotomy using a special solvability of an associated integral equation. We introduce a new and natural admissibility concept and we show that the trichotomic behavior of skew-product flows can be studied in the most general case, without any additional assumptions.
Notations
Let
, which is a Banach space with respect to the norm
. We consider the spaces
,
and let
. Then
and
are closed linear subspaces of
. Let
be the space of all continuous functions
with compact support and
.
Let
and let
be the linear space of all Bochner measurable functions
with the property that
, which is a Banach space with respect to the norm
Let
be a metric space and let
be a skew-product flow. For every
we consider the integral equation
with
and
.
Definition 3.1.
The pair
is said to beuniformly admissible for the skew-product flow
if there are
and
such that for every
the following properties hold:
-
(i)
for every
there are
and
such that the pairs
and
satisfy ( );
-
(ii)
if
and
are such that the pair
satisfies ( ), then
;
-
(iii)
if
is such that
, for all
and
has the property that the pair
satisfies ( ), then
.
Remark 3.2.
In the above admissibility concept, the input space is a minimal one, because all the test functions
belong to the space
.
In what follows we will establish the connections between the admissibility and the existence of uniform exponential trichotomy.
The first main result of this paper is as follows.
Theorem 3.3.
Let
be a skew-product flow on
. If the pair
,
is uniformly admissible for
, then
is uniformly exponentially trichotomic.
Proof.
We prove that the pair
is uniformly admissible for
.
Indeed, let
and
be given by Definition 3.1. We consider a continuous function
with the support contained in
and
Let
be such that
, for all
. Since
there is
such that
Let
.
Step 1.
Let
. We consider the function
Then
is continuous and
Since
there is
such that
. Then, from (3.7) it follows that
, so
. According to our hypothesis it follows that there are
and
such that the pairs
and
satisfy ( ).
Then, for every
we obtain that
Let
From (3.8) we have that
so the pair
satisfies ( ). Moreover, since
and
, we deduce that
. Since the pair
satisfies ( ) we obtain that
Taking
we analogously obtain that
and the pair
satisfies ( ).
Step 2.
Let
and let
be such that the pair
satisfies ( ). We consider the functions
, given by
Since
we have that
. Observing that for every
we deduce that
is continuous. Moreover, since
and
, from
we obtain that
.
Let
. We prove that
We set
. If
, then, taking into account the way how
was defined, the relation (3.16) obviously holds. If
then there is
,
such that
. Then, we deduce that
If
then
and
If
then
From relations (3.18) and (3.19) we have that
Then, from (3.17) and (3.20) it follows that
Since the pair
satisfies ( ) we have that
From (3.21) and (3.22) we obtain that the relation (3.16) holds for all
. Since
was arbitrary we deduce that the pair
satisfies ( ). Then according to our hypothesis we have that
In addition, we have that
and that
Taking
, from relations (3.23)–(3.25) and (3.3) it follows that
Observing that
, for all
, from (3.26) and (3.4) we successively deduce that
Setting
, from relation (3.27) we deduce that
Step 3.
Let
be such that
, for all
and let
be such that the pair
satisfies ( ). We consider the functions
, given by
Using analogous arguments with those used in the Step 2 we obtain that
and the pair
satisfies ( ). Moreover, using Lemma 2.7 we have that
, for all
. Then, according to our hypothesis we deduce that
Since
, for all
, using (3.30) and (3.5) we obtain that
Observing that
from (3.31) and (3.32) we deduce that
where
.
Finally, from Steps 1–3 and relations (3.28) and (3.33) we deduce that the pair
is uniformly admissible for the skew-product flow
. By applying Theorem 2.9 we conclude that
is uniformly exponentially trichotomic.
The natural question arises whether the integral admissibility given by Definition 3.1 is also a necessary condition for the existence of the uniform exponential trichotomy. To answer this question, in what follows, our attention will focus on the converse implication of the result given by Theorem 3.3. Specifically, our study will motivate the admissibility concept introduced in this paper and will point out several qualitative aspects. First of all, we prove a technical result.
Proposition 3.4.
Let
be a skew-product flow which is uniformly exponentially trichotomic with respect to the families of projections
. Let
, let
, and let
be such that the pair
satisfies ( ). We denote
and let
denote the inverse of the operator
, for all
. The following assertions hold:
-
(i)
the functions
and
have the following representations
-
(ii)
for every
there is
which does not depend on
or
such that
Proof.
Since
there is
such that
. Let
be given by Definition 2.4 and let
.
-
(i)
Since the pair
satisfies ( ) we have that
Since
from (3.38) we have that
, for all
. Let
. Then we deduce that
For
in (3.39) we obtain that
, for all
. This shows that relation (3.35) holds for all
. For
from (3.38) we have that
so relation (3.35) holds for every
.
For every
from (3.38) we have that
. This implies that
so we deduce that
From relation (3.42) it follows that
, for all
. In particular, we have that relation (3.36) holds for
. For
from (3.38) we obtain that
which implies that
for all
. Thus, we conclude that relation (3.36) holds for every
.
-
(ii)
Let
and let
. Setting
and using Hölder's inequality we deduce that
The second main result of the paper is as follows.
Theorem 3.5.
Let
be a skew-product flow. If
is uniformly exponentially trichotomic, then the pair
,
is uniformly admissible for
.
Proof.
Let
be two constants and let
,
be the families of projections given by Definition 2.4. We set
. For every
and every
we denote by
the inverse of the operator
.
For every
and every
we denote by
Then
, for all
and all
and
, for all
.
Let
and let
be given by Proposition 3.4. We prove that all the properties from Definition 3.1 are fulfilled.
Let
.
Step 1.
Let
,
. We consider the functions
defined by
Since
we have that
and
are correctly defined and continuous. Let
be such that
. Setting
,
and
, we have that
and
, for
.
We observe that
, for all
, which implies that
In addition, we have that
, for all
. This implies that
Since
is continuous from relations (3.48) and (3.49) it follows that
. Using similar arguments we deduce that
. An easy computation shows that the pairs
and
satisfy ( ).
If
, then we take
.
Step 2.
Let
and let
be such that the pair
satisfies ( ).
Suppose that
. From Proposition 3.4 we have that
Let
be such that
. Since
for
we deduce that
. Since
we obtain that
so
. This implies that
, for all
. Moreover, using (3.51), for
we have that
which implies that
Then, we obtain that
Setting
from relations (3.50) and (3.55) it follows that
The case
can be treated using similar arguments with those used above.
Step 3.
Let
be such that
, for all
and let
be such that the pair
satisfies ( ).
Since
and
, for all
, using Lemma 2.7 we deduce that
, for all
. Since the pair
satisfies ( ) we obtain that
Let
. Since
, using relation (3.57) we have that
so
, for all
. This shows that, in this case,
.
If
is given by Step 2, we deduce that
and the proof is complete.
The central result of this paper is as follows.
Theorem 3.6.
A skew-product flow
is uniformly exponentially trichotomic if and only if the pair
is uniformly admissible for
.
Proof.
This follows from Theorems 3.3 and 3.5.
Remark 3.7.
The above result establishes for the first time in the literature a necessary and sufficient condition for the existence of the uniform exponential trichotomy of skew-product flows, based on an input-output admissibility with respect to the associated integral equation. The chart described by our method allows a direct analysis of the asymptotic behavior of skew-product flows, without assuming a priori the existence of a projection families, invariance properties or any reversibility properties. Moreover, the study is done in the most general case, without any additional assumptions concerning the flow or the cocycle.