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Xiaodan Chen, Yonghui Xia, "Topological Conjugacy between Two Kinds of Nonlinear Differential Equations via Generalized Exponential Dichotomy", International Journal of Differential Equations, vol. 2011, Article ID 871574, 11 pages, 2011. https://doi.org/10.1155/2011/871574
Topological Conjugacy between Two Kinds of Nonlinear Differential Equations via Generalized Exponential Dichotomy
Based on the notion of generalized exponential dichotomy, this paper considers the topological decoupling problem between two kinds of nonlinear differential equations. The topological equivalent function is given.
1. Introduction and Motivation
Well-known Hartman’s linearization theorem for differential equations states that a 1 : 1 correspondence exists between solutions of a linear autonomous system and those of the perturbed system , as long as fulfills some goodness conditions, like smallness, continuity, or being Hartman . Based on the exponential dichotomy, Palmer  extended this result to the nonautonomous system. Some other improvements of Palmer’s linearization theorem are reported in the literature. For examples, one can refer to Shi , Jiang , and Reinfelds [5, 6]. Recently, Xia et al.  generalized Palmer’s linearization theorem to the dynamic systems on time scales. Consider the linear system where and is a matrix function.
However, Lin  argued that the notion of exponential dichotomy considerably restricts the dynamics. It is thus important to look for more general types of hyperbolic behavior. Lin  proposed the notion of generalized exponential dichotomy which is more general than the classical notion of exponential dichotomy.
Definition 1.2. System (1.1) is said to have a generalized exponential dichotomy if there exists a projection and such that where is a continuous function with , satisfying .
Example 1.3. Consider the system
Then, system (1.4) has a generalized exponential dichotomy, but the classical exponential dichotomy cannot be satisfied.
For this reason, basing on generalized exponential dichotomy, we consider the topological decoupling problem between two kinds of nonlinear differential equations. We prove that there is a 1 : 1 correspondence existing between solutions of topological decoupling systems, namely, and .
2. Existence of Equivalent Function
Consider the following two nonlinear nonautonomous systems: where , , are matrices.
Definition 2.1. Suppose that there exists a function such that (i)for each fixed , is a homeomorphism of into ;(ii) as , uniformly with respect to ;(iii)assume that has property (ii) too;(iv)if is a solution of system (2.1), then is a solution of system (2.2).If such a map exists, then (2.1) is topologically conjugated to (2.2). is called an equivalent function.
Theorem 2.2. Suppose that has a generalized exponential dichotomy. If fulfill where where are integrable functions and , are positive constants, then the nonlinear nonautonomous system (2.1) is topologically equivalent to the nonlinear nonautonomous system (2.2). Moreover, the equivalent functions fulfill
In what follows, we always suppose that the conditions of Theorem 2.2 are satisfied. Denote that is a solution of (2.2) satisfying the initial condition and that is a solution of (2.1) satisfying the initial condition . To prove the main results, we first prove some lemmas.
Lemma 2.3. For each , system has a unique bounded solution with .
Proof. Let be the set of all the continuous bounded functions with . For each and any , define the mapping as follows:
Simple computation leads to which implies that is a self-map of a sphere with radius . For any ,
Due to the fact that , has a unique fixed point, namely, , and it is easy to show that is a bounded solution of (2.6). Now, we are going to show that the bounded solution is unique. For this purpose, we assume that there is another bounded solution of (2.6). Thus, can be written as follows:
Note that which implies that is convergent; denote it by . That is, Similarly, Therefore, it follows from the expression of that
Noticing that is bounded, is also bounded. So, is bounded. But we see that does not have a nontrivial bounded solution. Thus, ; it follows that Simple calculating shows
Therefore, , consequently . This implies that the bounded solution of (2.6) is unique. We may call it . From the above proof, it is easy to see that .
Lemma 2.4. For each , the system has a unique bounded solution and .
Proof. The proof is similar to that of Lemma 2.3.
Lemma 2.5. Let be any solution of the system (2.1), then is the unique bounded solution of system
Proof. Obviously, is a bounded solution of system (2.19). We show that the bounded solution is unique. If not, then there is another bounded solution , which can be written as follows: By Lemma 2.3, we can get It follows that That is, . Consequently, . This completes the proof of Lemma 2.5.
Lemma 2.6. Let be any solution of the system (2.2), then is the unique bounded solution of system
Proof. Obviously, is a bounded solution of system (2.23). We will show that the bounded solution is unique. If not, then there is another bounded solution . Then, can be written as follows:
By Lemma 2.3, we can get Then, it follows that That is, . Consequently, . This completes the proof of Lemma 2.6.
Now, we define two functions as follows:
Lemma 2.7. For any fixed is a solution of the system (2.2).
Proof. Replace by in (2.6); system (2.6) is not changed. Due to the uniqueness of the bounded solution of (2.6), we can get that . Thus, Differentiating it, noticing that , are the solutions of the (2.1), and (2.6), respectively; therefore, we can obtain It indicates that is the solution of the system (2.2).
Lemma 2.8. For any fixed , is a solution of the system (2.1).
Proof. The proof is similar to Lemma 2.7.
Lemma 2.9. For any .
Proof. Let be any solution of the system (2.2). From Lemma 2.8, is a solution of system (2.1). Then, by Lemma 2.7, we see that is a solution of (2.2), written as . Denote . Differentiating, we have
which implies that is a solution of system (2.23). On the other hand, following the definition of and and Lemmas 2.3 and 2.4, we can obtain
This implies that is a bounded solution of system (2.23). However, by Lemma 2.6, system (2.23) has only one zero solution. Hence, , consequently , that is, . Since is any solution of the system (2.2), Lemma 2.9 follows.
Lemma 2.10. For any .
Proof. The proof is similar to Lemma 2.10.
Now, we are in a position to prove the main results.
Proof of Theorem 2.2. We are going to show that satisfies the four conditions of Definition 2.1.Proof of Condition (i). For any fixed , it follows from Lemmas 2.9 and 2.10 that is homeomorphism and .Proof of Condition (ii). From (2.27) and Lemma 2.3, we derive . So, as , uniformly with respect to .Proof of Condition (iii). From (2.23) and Lemma 2.4, we derive . So, as , uniformly with respect to .Proof of Condition (iv). Using Lemmas 2.7 and 2.8, we easily prove that Condition (iv) is true.
The authors would like to express their gratitude to the editor and anonymous reviewers for their careful reading which improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China under Grant (no. 10901140) and ZJNSFC under Grant (no. Y6100029).
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Copyright © 2011 Xiaodan Chen and Yonghui Xia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.