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General exponential dichotomies on time scales and parameter dependence of roughness
Advances in Difference Equations volume 2013, Article number: 339 (2013)
Abstract
This paper focuses on a new notion called the general exponential dichotomy on time scales, which is more general and contains as special cases most versions of dichotomies on the continuous systems and discrete systems. We establish the existence of parameter dependence of roughness for the general exponential dichotomy on time scales under sufficiently small linear perturbation. Moreover, we also show that the stable and unstable subspaces of general exponential dichotomies for the perturbed system are Lipschitz continuous for the parameters.
MSC:34N05, 34D09.
1 Introduction
The notion of exponential dichotomies extends the idea of hyperbolicity from autonomous systems to nonautonomous systems and gives a direct sum of the stable and unstable subspaces for the splitting of the state space [1, 2]. The exponential dichotomy together with its variants and extensions has been widely studied and discussed and plays a central role in the study of the nonautonomous systems [3–15]. In particular, the roughness of dichotomies states that the behavior of a dichotomy does not change much under sufficiently small linear perturbations and has been extensively studied for the continuous systems [1, 5, 6, 9, 14, 16–20] and the discrete systems [6, 21–23].
The theory of dynamic equations on time scales, which originates from [24, 25], is related not only to the set of real numbers (continuous systems) and the set of integers (discrete systems) but also to more general time scales (an arbitrary nonempty closed subset of the real numbers ℝ) [26, 27]. The concept of exponential dichotomies on time scales is a very important method and tool to explore the dynamic behavior of nonautonomous dynamic systems on time scales [28–38]. However, we note that there exist various different notions of dichotomies and different kinds of dichotomic behavior in the continuous systems and the discrete systems. It is of great interest to look for more general types of dichotomies on time scales in order to unify the notions of dichotomies in the continuous and discrete case. The main novelty of our work is that we introduce a new notion called the general exponential dichotomy on time scales, which includes and extends the existing notions of dichotomies for the continuous systems and the discrete systems usually found in the literature. Moreover, we also discuss parameter dependence of roughness for the general exponential dichotomy on time scales under sufficiently small linear perturbation.
The content of this paper is as follows. In Section 2, we define a new notion called the general exponential dichotomy on time scales for the linear dynamical system on time scales. Then we establish the existence of parameter dependence of roughness for the general exponential dichotomy on time scales in Section 3. Particularly, the stable and unstable subspaces of general exponential dichotomies for the perturbed system are Lipschitz continuous for the parameters.
2 General exponential dichotomy on time scales
In this section, we first introduce some basic knowledge and definitions on time scales, which can be found in [24, 25].
Let be a time scale, i.e., an arbitrary nonempty closed subset of the real numbers ℝ. is the forward jump operator of and is a graininess function. Throughout this paper, the time scale is assumed to be unbounded above and below. denotes the set of rd-continuous functions . is the space of positively regressive functions.
Define
for a given and for any , . For any , define the exponential function by
for , where Log is the principal logarithm.
Let
for any bounded function and define
Then we have
where .
Let be a Banach space and be the space of bounded linear operators defined on X. We consider the linear system on time scales
where . Let be the evolution operator satisfying for and any solution of system (2.1). Moreover, we also assume that and for any , which imply that is invertible.
Now we introduce a new notion called the general exponential dichotomy on time scales.
Definition 2.1 System (2.1) is said to admit a general exponential dichotomy on a time scale if there exist projections such that
and there exist a constant , and bounded functions with , such that, for ,
hold and for ,
hold, where is the complementary projection of .
In order to facilitate the reader’s understanding, we now consider some specific examples of general exponential dichotomies on different time scales.
Example 2.1 Let , then , .
If functions a, b are positive constants, then the general exponential dichotomy on time scales reduces to the exponential dichotomy ( and are positive constants) [1] by
and the nonuniform exponential dichotomy ( and ε is a positive constant) [5] by
If functions , then we get the generalized exponential dichotomy (, are positive constants) [10, 11] by
Let , and let the functions , be positive constants. If , , then we obtain the -dichotomy [14] by
If and , , are positive constants, then Definition 2.1 agrees with the nonuniform polynomial dichotomy (, and ) [7] by
the ρ-nonuniform exponential dichotomy (, , ) [6] by
and the nonuniform -dichotomy (, , ) [9] by
Example 2.2 If , then , .
Carrying out similar arguments as those in Example 2.1, we conclude that the general exponential dichotomy on time scales includes the existing dichotomies for the linear discrete system as special cases such as the uniform exponential dichotomy [1], -dichotomy [23], nonuniform exponential dichotomy [5], nonuniform polynomial dichotomy [21], ρ-nonuniform exponential dichotomy [6], nonuniform -dichotomy [8, 22].
Example 2.3 Let , , and let the functions a, b be positive constants.
We have , . Let c be a positive constant and . Then (2.2) and (2.3) reduce to
Example 2.4 Let , , and let the functions a, b be positive constants, where .
We get and . If , where c is a positive constant, then (2.2) reduces to
3 Parameter dependence of roughness
The section focuses on parameter dependence of roughness for the general exponential dichotomy on time scales under the sufficiently small linear perturbation. We consider the linear perturbed system
where , is an open subset of a Banach space (the parameter space). In the rest of the section, we let be the evolution operator associated to system (3.1) for each .
To obtain our conclusion, we let
and assume that the following conditions hold:
-
(a1) there exist a positive constant c and a function such that
and
where ;
-
(a2) there exist positive constants , such that
-
(a3) and for each fixed ;
-
(a4) is a decreasing function and is an increasing function.
Now we state our main result in this section.
Theorem 3.1 Assume that system (2.1) admits a general exponential dichotomy on a time scale with and conditions (a1)-(a4) hold with sufficiently small c. Then system (3.1) also admits a general exponential dichotomy on the time scale , i.e., for each , there exist projections such that
and
where are the complementary projections of ,
Moreover, if Y is a finite-dimensional space and is Lipschitz continuous for the parameter λ, then the stable subspace and the unstable subspace are Lipschitz continuous for the parameter λ.
The proof of Theorem 3.1 is nontrivial and is achieved in several steps:
-
(i)
construct some bounded solutions of perturbed system (3.1) (Lemmas 3.1, 3.2);
-
(ii)
semigroup properties of the bounded solutions of system (3.1) (Lemma 3.3);
-
(iii)
construction of the projections in (3.2) (Lemmas 3.4, 3.5 and (3.15));
-
(iv)
norm bounds for the evolution operator (Lemmas 3.6, 3.7, 3.8);
-
(v)
and are Lipschitz continuous for the parameter λ (Lemma 3.9).
We set
where
It is not difficult to show that and are both Banach spaces.
Lemma 3.1 For each , there exists a unique bounded solution satisfying
and is Lipschitz continuous for the parameter λ.
Proof Direct calculation shows that satisfying (3.6) is a solution of (3.1). For each , we define an operator on by
By (2.2), (2.3), (a1) and (a2), we have
for and
for . For , we get
Then
and
that is,
This implies that . Similarly, for each , we get
If c is sufficiently small, then is a contraction and for each , there exists a unique such that and (3.6) holds.
Next we show that is Lipschitz continuous for the parameter λ. For any , there exist bounded solutions satisfying (3.6). It follows from (a2) that
for and
for . Moreover, we also have
for . It follows from (2.2), (2.3), (a1) and (a2) that
for and
for . For , we get
Then
The proof of the lemma is complete. □
Similarly, we have the following lemma.
Lemma 3.2 For each , there exists a unique bounded solution satisfying
and is Lipschitz continuous for the parameter λ.
Lemma 3.3 For each , the bounded solutions and of system (3.1) satisfy
Proof From (3.6), we get
for , and let . We define the operator by
for any and each , where is obtained from replacing s by l. Obviously, . Carrying out similar arguments to the proof of Lemma 3.1, we have
and
for any . This means that there exists a unique such that . On the other hand, we also note that and . Therefore, . Similarly, the second identity of (3.9) holds. □
Now we construct the projections for each . We first set
Lemma 3.4 For each , we have
-
(b1) , are projections for each and ;
-
(b2) , , ;
-
(b3) , , , ;
-
(b4) , .
Proof It follows from Lemma 3.3 that (b1) and (b2) hold. By (3.6) and (3.8), we get
which imply that (b3) holds. By Lemma 3.1 and Lemma 3.2, we have and since satisfies identity (3.6) with and satisfies identity (3.8) with . Therefore, (b4) holds. □
Lemma 3.5 For each , is invertible.
Proof For each , combining (b3) and (b4) together gives
It follows from (3.11) and (3.12) that
Note that
and
Then
and
Then
which means that is invertible if c is sufficiently small. □
We set
for each and . It is not difficult to show that , , . Then , are projections and (3.2) is valid.
Lemma 3.6 We have
Proof We first prove that for each , if is a bounded solution of (3.1), then
A straightforward calculation shows that
and
It follows from (3.20), (2.2) and (2.3) that
and
We note that
for and
for . Then
when letting in (3.21). Consequently,
which means that (3.18) holds.
For each , we let , , . It is clear that and are solutions of (3.1) with the same initial value . Then
is a bounded solution of (3.1) with the initial value since is bounded for . It follows from (3.18) that
By (2.2), (2.3), (a1) and (a2), we get
and
for . Then
For , we have
and
For , we get
and
Then
for . Combining (3.22) and (3.23) together gives (3.16). Similarly, (3.17) holds. □
Lemma 3.7 We have
Proof According to Lemma 3.4, we get
Let , , . By using (3.15), we obtain
On the other hand,
Then and since is invertible. It follows from Lemma 3.6 that
and
which yield the desired inequalities. □
Lemma 3.8 For each , we have
Proof For each and any , we set
It follows from Lemma 3.7 that and are bounded solutions of (3.1). Combining (3.6) and (3.8) together gives
and
Taking leads to
and
By using Lemma 3.7 and (a2), for , we have
and
For , we get
and
Then
and
For , one has
and
since and for .
Similarly, (3.25) holds for . □
It follows from Lemma 3.7 and Lemma 3.8 that (3.3) and (3.4) hold. Next we show that the stable subspace and the unstable subspace are Lipschitz continuous for the parameter λ.
Lemma 3.9 and are Lipschitz continuous for the parameter λ.
Proof By using Lemma 3.1 and Lemma 3.2, and are Lipschitz continuous for the parameter λ. It follows from (3.10) that and are Lipschitz continuous since is Lipschitz continuous for the parameter λ. If Y is a finite-dimensional space, then and are both Lipschitz continuous for the parameter λ. Equation (3.15) implies that the conclusion of lemma holds. □
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Acknowledgements
The authors are grateful to the anonymous referee for carefully reading the manuscript and for important suggestions and comments. This research is supported by the National Natural Science Foundation of China (No. 11201128) and (No. 11126269) and the youth scientific funds of Heilongjiang University (QL201007).
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Zhang, J., Song, Y. & Zhao, Z. General exponential dichotomies on time scales and parameter dependence of roughness. Adv Differ Equ 2013, 339 (2013). https://doi.org/10.1186/1687-1847-2013-339
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DOI: https://doi.org/10.1186/1687-1847-2013-339