Exponential Dichotomies and Fredholm Operators of Dynamic Equations on Time Scales

For time-varying non-regressive linear dynamic equations on a time scale with bounded graininess, we introduce the concept of the associative operator with linear systems on time scales. The purpose of this research is the characterizations of the exponential dichotomy obtained in terms of Fredholm property of that associative operator. Particularly, 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 −  . Moreover, we also give the converse result that the linear systems have an exponential dichotomy on both +  and −  then the associative operator is Fredholm on  .


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 : the graininess function ( ) ( ) 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 m n × 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 m n × matrix-valued funtions on  is de- noted by ( ) ( ) We say that A is differentiable on  provided each entry of A is differentia- ble on  , and in this case we put ( ) ( ) where and the class of all such regressive and rd-continuous function is denoted Throughout this paper we only consider ( ) is a group.
We collect some fundamental properties of the exponential function on time scales.
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 1 2 , m m such that for every t + ∈  , there exists ( ) m c t m ≤ − < (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 : T X Y → (where , X Y are Banach space), we define  ( ) ) . We say that T is Fredholm operator if (1) ( ) R T is closed, (2) NulT and def T are finite.If the condition (2) replace either nulT < +∞ or def T < +∞ then T is said that semi-Fredholm.
In this paper, we only consider the time scales satisfy sup = +∞  and inf = −∞  .We also denote Definition 1.6.The equation ( ) is said to have an exponential dichotomy or to be exponentially dichotomous on J ( ) , A e t s is fundamental solution matrix of Equation ( 1) and I is the identity matrix.When previous inequality hold with 1 2 0 α α = = .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. ( ) { } : | is bounded and rd -continuous ( ) With the system (1) we define the bounded associative linear operator ( ) ( ) NullL is always finite.Hence the assumption that L is semi-Fredholm means that the range ( ) Follow [24], we say the pair x BC J ∈ such that the pair ( ) ( ) BC J 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 ( ) ( ) ( ) 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  .

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.

( )
A t be an n n × matrix-value function, bounded, rd-continuous and regressive on an interval J, when , , J then the following statements are satisfy (1) If J is a half line then there exist ( ) Then the solution of the nonhomogenneous equation ( ) ( ) can be written as x t e t e t f t Since f has compact support, so there exist 0 r ≥ such that ( )  (2) Let J =  then (3) is a solution of (2) for all t ∈  .Therefore, x has compact support on  if and only if x has compact support on both +  and , 0 and , 0 . Proof.

( )
f R T ∈ .Therefore, for any ( ) By the continuity of α , we see that (2) We now consider J =  and take ( ) where φ is a certainly chose function of compact support with ( ) Clearly, f  has compact support and From the formula (5) and direct computations, we obtain For all functions ( ) are both bounded linear functionals defined on ( ) BC J and coinciding on the dense subset consisting of the functions of compact support.So (4) holds for all ( ) Conversely, suppose there exist n ξ ∈  such that (4) is true.Then has limits as t → +∞ , hence is also.
On the other hand, ( A so the proof is complete. We now prove the main theorem of this section.Theorem 2.1.Let the system (1) with ( ) A t is rd-continuous, bounded and regressive on time scales  .Suppose that the associative operator L of ( 1) is semi-Fredholm.Then (2) We now consider J =  .By Lemma 2.2 then nul * T < +∞ .Furthermore, We now choose a function ( ) , p BC J L J 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 ( ) ( ) ( )

The Sufficient for Fredholm Property on the Line
In this section, we assume that the Equation (1) has exponential dichotomy on

L
. H. Tien, L. D. Nhien DOI: 10.4236/am.2019.10100440 Applied Mathematics so  is also a closed subspace.Then we define restriction of L to  and we have →  is the conjugate operator.Lemma 2.2.Let ( ), , A t J T are defined as before.Then (

=
 or −  then(1) has exponential dichotomy on J, (2) When J =  then (1) has exponential dichotomy on both , the range of the semi-Fredholm operator, is closed.Hence, ( ) R T is also.Then by Theorem 4.6-C in Taylor[28], 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 J − =  is L. H. Tien, L. D. Nhien DOI: 10.4236/am.2019.10100446 Applied Mathematics similar.
on  .Restricting to +  we conclude that equation 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 J solution of nonhomogenneous linear system Lx f = so that ( ) f R L ∈ 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  .
3. Assume A and B are regressive n n × -matrix-valued functions on  .Then we define A B A by