1. Introduction and Preliminaries
Exponential dichotomy is at the heart of the fundamental perturbation results for linear systems of Coppel (see   ) and Palmer (see      ), of the spectral theory of Sacker and Sell  , of the geometric theory of Fenichel , of perturbation results for invariant manifolds , of the fundamental perturbation results for connecting orbits of Beyn and Sandstede (see    ), 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    ). 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  ). Using this concept for nonuniform exponential dichotomies case is presented by L. Barreira, D. Dragicevic and C. Valls (see   ).
Theory of dynamic equations on time scales was introduced by Stefan Hilger  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    ). 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  . 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   ).
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  ). If are matrix-valued function on , then
(1) and ,
(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 ( , pp. 319)). We refer   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
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   .
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 , 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 , 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
can be written as
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
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
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
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,
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 ,
(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  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
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  (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 , 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
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
(2) The associative operator L is Fredholm on .
Proof. Proof of the part (ii) is similar to Palmer . 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
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 .
The first author was supported in part by the VNU Project of Vietnam National University No. QG101-15.