Exponential Dichotomies and Fredholm Operators of Dynamic Equations on Time Scales ()
1. Introduction and Preliminaries
Exponential dichotomy is at the heart of the fundamental perturbation results for linear systems of Coppel (see [2] [3] ) and Palmer (see [4] [5] [6] [7] [8] ), of the spectral theory of Sacker and Sell [9] [10], of the geometric theory of Fenichel [11], of perturbation results for invariant manifolds [12], of the fundamental perturbation results for connecting orbits of Beyn and Sandstede (see [13] [14] [15] ), and it has proven also a formidable ally to justify and gain insight into the behavior of various algorithmic approaches for solving boundary value problems, for approximating invariant surfaces and for computing traveling waves, among other uses (see [16] [17] [18] ). Hence, it is important to find the conditions for dynamical systems are exponential dichotomy. In 1988, K. J. Palmer presented Fredholm operator concept to show conditions of systems which have exponential dichotomy (see [4] ). Using this concept for nonuniform exponential dichotomies case is presented by L. Barreira, D. Dragicevic and C. Valls (see [19] [20] ).
Theory of dynamic equations on time scales was introduced by Stefan Hilger [21] in order to unify and extend results of differential equations, difference equations, q-difference equations, etc. There are many works concerned with dichotomies of dynamic equations on time scales (see [22] [23] [24] ). The purpose of this paper is to setup and characterize exponential dichotomy in term of Fredholm operators for dynamic equations on time scales.
We now introduce some basic concepts of time scales, which can be found in [25] [26]. A time scale T is defined as a nonempty closed subset of the real numbers. The forward jump operator
is defined by
and the graininess function
for any
. In the following discussion, the time scale
is assumed to be unbounded above and below. We have the following several basis definitions (see [25] [26] ).
Definition 1.1. Let A be an
matrix-valued function on
. We say that A is rd-continuous on
if each entry of A is rd-continuous on
, and the class of all such rd-continuous
matrix-valued funtions on
is denoted by
We say that A is differentiable on
provided each entry of A is differentiable on
, and in this case we put
Definition 1.2. (Regressivity). An
-matrix-valued function A on a time scale
is called regressive (with respect to
) provided
and the class of all such regressive and rd-continuous function is denoted
Throughout this paper we only consider
.
Definition 1.3. Assume A and B are regressive
-matrix-valued functions on
. Then we define
by
and we define
by
Remark 1.1.
is a group.
Definition 1.4. (Matrix Exponential Function). Let
and assume that
is an
-matrix-valued function. The unique matrix-value solution of the IVP
where I denotes as usual the
-identity matrix, is called the matrix exponential function (at
), and it is denoted by
.
We collect some fundamental properties of the exponential function on time scales.
Theorem 1.1. (see [25] ). If
are matrix-valued function on
, then
(1)
and
,
(2)
,
(3)
,
(4)
,
(5)
if
and
commute.
If
, one have the equivalent definition of the exponential function on time scales by
For any
and
, where log is principal logarithm.
Throughout this paper, we assume that the graininess of underlying time scale is bounded on
, i.e.,
. This assumption is equivalent to the fact that there exist positive numbers
such that for every
, there exists
satisfying
(also see ( [27], pp. 319)). We refer [25] [26] for more information on analysis on time scales.
Next, we define several concepts functional analysis which is useful later. The operator
(where
are Banach space), we define
•
is nullspace of T and
,
•
is range of T and
in Y,
•
(if at least one of them is finite).
Definition 1.5. Let
. We say that T is Fredholm operator if
(1)
is closed,
(2)
and
are finite.
If the condition (2) replace either
or
then T is said that semi-Fredholm.
In this paper, we only consider the time scales satisfy
and
. We also denote
,
.
Definition 1.6. The equation
(1)
is said to have an exponential dichotomy or to be exponentially dichotomous on J
if there exist projections matrix
on
such that
for any
and
is an isomorphism for any
and there exist a positive constants
and
, such that
(1)
for all
and any
,
(2)
for all
and any
.
where
and
is fundamental solution matrix of Equation (1) and I is the identity matrix. When previous inequality hold with
. is said to possess an ordinary dichotomy. The definition of exponential dichotomy can be seen in [1] [22] [24].
We denote several Banach spaces which shall be used later.
•
with the norm
•
•
.
•
with the norm
•
•
.
•
with the norm
where
and
or
.
Remark 1.2.
is a closed subspace of
in which
is dense.
With the system (1) we define the bounded associative linear operator
as following
Remark 1.3.
is always finite. Hence the assumption that L is semi-Fredholm means that the range
of L is closed.
Follow [24], we say the pair
is admissible for (1) if for every
there exists a function
such that the pair
satisfies
We say that
is the input space and
is the output space.
The main aim of this paper is to show that the nonautonomous equations have exponential dichotomy on time scales if and only if its associative operator is Fredholm. We now give an outline of the contents of this paper. In Section 2, we use Perron’s method, which was generalized on time scales by J. Zhang, M. Fan, H. Zhu in [1], to show that if the associative operator is semi-Fredholm then the corresponding linear nonautonomous equation has an exponential dichotomy on both
and
. As a consequence, we obtain that Fredholm property implies the admissibility of the pair
. In Section 3, we give the converse of the main theorem of section 2 on the lines. Particularly, the system (1) has an exponential dichotomy on both
and
then the associative operator L is Fredholm on
.
2. The Sufficient for Exponential Dichotomy on Both Two Half Lines
Firstly, we need prove two lemmas that are very useful for the main theorem in this section.
Lemma 2.1. Let
be an
matrix-value function, bounded, rd-continuous and regressive on an interval J, when
. Let
then the following statements are satisfy
(1) If J is a half line then there exist
such that
,
(2) If
then there exist
such that
if and only if
Proof. (1) Let
. Then the solution of the nonhomogenneous equation
(2)
can be written as
(3)
Since f has compact support, so there exist
such that
for all
and
. Then, for
, we obtain
so
has compact support on
if and only if
. This proves the lemma for
. The proof for
is similar.
(2) Let
then (3) is a solution of (2) for all
. Therefore, x has compact support on
if and only if x has compact support on both
and
. It means that
Hence,
or
This completes the proof of the lemma.
We now consider
. Since L is continuous and
is closed in
so
is also a closed subspace. Then we define
to be the restriction of L to
and we have
. In the following lemma, we characterize
, where
is the conjugate operator.
Lemma 2.2. Let
are defined as before. Then
(1) when
or
then
,
(2) when
then
if and only if there exist
such that
(4)
Proof. (1) First, let
and consider
. By Lemma 2.1, the Equation (1) with this f has a solution
. Obviously,
and
, i.e.,
. Therefore, for any
,
Note that
is dense in
. By the continuity of
, we see that
for all
. Thus, as a linear functional on
,
must be zero and
. A similar discussion can be given in the case of
.
(2) We now consider
and take
and
. Let
(5)
where
is a certainly chose function of compact support with
and
.
Clearly,
has compact support and
Thus,
. By Lemma 2.1, it implies
with
. Since
so
From the formula (5) and direct computations, we obtain
For all functions
,
It follows that
. Then
and
are both bounded linear functionals defined
on
and coinciding on the dense subset consisting of the functions of compact support. So (4) holds for all
, as required.
Conversely, suppose there exist
such that (4) is true. Then
so that
has limits as
, hence
is also. On the other hand,
Therefore,
.
Now
defined by (4) is certainly in
. Moreover, if
we have
It means
so the proof is complete.
We now prove the main theorem of this section.
Theorem 2.1. Let the system (1) with
is rd-continuous, bounded and regressive on time scales
. Suppose that the associative operator L of (1) is semi-Fredholm. Then
(1) When
or
then (1) has exponential dichotomy on J,
(2) When
then (1) has exponential dichotomy on both
.
Proof. Since
, the range of the semi-Fredholm operator, is closed. Hence,
is also. Then by Theorem 4.6-C in Taylor [28],
(1) Suppose now
. Then by Lemma 2.2,
. So by the Hahn-Banach theorem,
. That is, for all
then the equation (2) has a solution bounded on J. Then it follows from Theorem 3.6 in [1] that equation (1) has an exponential dichotomy on
. In case
is similar.
(2) We now consider
. By Lemma 2.2 then nul
. Furthermore,
. It follows that
so
. By Lemma 2.2 again,
for some
satisfies
.
Let any
we are going to extend the function f as following
Let
be a basis for subspace
We now choose a function
such that
•
We define
Hence,
and
when
. It means that the equation
has solution on
of the equation
has bounded solution on
. Restricting to
we conclude that equation
has bounded solution for all
. By the results in [22] (Theorem 3.6) used earlier, it follows that Equation (1) has exponential dichotomy on
. A similar argument shows that it has an exponential dichotomy on
. So the proof of the theorem is complete.
By Theorem 3.1 in [24], one has the following corollary about relation between semi-Fredholm property and admissibility.
Corollary 2.1. If the associative operator of (1) is semi-Fredholm operator and
then pair
is admissible for (1).
With the results above, we showed that if the associative operator is semi-Fredholm then the corresponding linear nonautonomous equation has an exponential dichotomy on both
and
. As a consequence, we obtain that Fredholm property implies the admissibility of the pair
.
3. The Sufficient for Fredholm Property on the Line
In this section, we assume that the Equation (1) has exponential dichotomy on both
and
. Then there exist two projections P and Q that satisfy Definition 1.6. Then the adjoint equation
(6)
has exponential dichotomy on
and
with the corresponding propositions
and
. Now the subspace of initial values (at
) of bounded solutions of (1) is
and for (6) is
Theorem 3.1. Let
be an
matrix function bounded, rd-continuous and regressive on
such that the system (1) has an exponential dichotomy on both
and
. Then
(1)
if and only if
(7)
(2) The associative operator L is Fredholm on
.
Proof. Proof of the part (ii) is similar to Palmer [2]. For the part (1), let
so that there exists x in
such that
Then if
we obtained
Conversely, suppose
and satisfy
Note that if
is a vector satisfying
(8)
then the function
satisfies (7). It follows that
for all vectors satisfying (8). This means that the linear algebraic equations
have a solution
. We consider the function
is a bounded solution of nonhomogenneous linear system
so that
as required. The Theorem is proved.
As a consequence of the Theorem 3.1, we obtain that the system (1) has an exponential dichotomy on both
and
if and only if the associative operator L is Fredholm on
.
Acknowledgements
The first author was supported in part by the VNU Project of Vietnam National University No. QG101-15.