International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article
Special Issue

Recent Advances in Oscillation Theory 2011

View this Special Issue

Research Article | Open Access

Volume 2011 |Article ID 871574 | https://doi.org/10.1155/2011/871574

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

Academic Editor: Yuri V. Rogovchenko
Received03 Jul 2011
Accepted15 Aug 2011
Published15 Oct 2011

Abstract

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 Hartmanā€™s linearization theorem for differential equations states that a 1ā€‰:ā€‰1 correspondence exists between solutions of a linear autonomous system Ģ‡š‘„=š“š‘„ and those of the perturbed system Ģ‡š‘„=š“š‘„+š‘“(š‘„), as long as š‘“ fulfills some goodness conditions, like smallness, continuity, or being Hartman [1]. Based on the exponential dichotomy, Palmer [2] extended this result to the nonautonomous system. Some other improvements of Palmerā€™s linearization theorem are reported in the literature. For examples, one can refer to Shi [3], Jiang [4], and Reinfelds [5, 6]. Recently, Xia et al. [7] generalized Palmerā€™s linearization theorem to the dynamic systems on time scales. Consider the linear systemĢ‡š‘„=š“(š‘”)š‘„,(1.1) where š‘„āˆˆā„š‘› and š“(š‘”) is a š‘›Ć—š‘› matrix function.

Definition 1.1. System (1.1) is said to possess an exponential dichotomy [8] if there exists a projection š‘ƒ and constants š¾>0,š›¼>0 such that ā€–ā€–š‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1ā€–ā€–(š‘ )ā‰¤š¾š‘’āˆ’š›¼(š‘”āˆ’š‘ )ā€–ā€–š‘ˆ[]š‘ˆ,forš‘ ā‰¤š‘”,š‘ ,š‘”āˆˆā„,(š‘”)š¼āˆ’š‘ƒāˆ’1ā€–ā€–(š‘ )ā‰¤š¾š‘’š›¼(š‘”āˆ’š‘ ),forš‘”ā‰¤š‘ ,š‘ ,š‘”āˆˆā„(1.2) hold, where š‘ˆ(š‘”) is a fundamental matrix of linear system Ģ‡š‘„=š“(š‘”)š‘„.

However, Lin [9] 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 [9] 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 š‘ƒ and š¾ā‰„0 such that ||š‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1(||ī‚µāˆ’ī€œš‘ )ā‰¤š¾expš‘”š‘ ī‚¶||š›¼(šœ)š‘‘šœ,(š‘”ā‰„š‘ ),š‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1(||ī‚µī€œš‘ )ā‰¤š¾expš‘”š‘ ī‚¶š›¼(šœ)š‘‘šœ,(š‘”ā‰¤š‘ ),(1.3) where š›¼(š‘”) is a continuous function with š›¼(š‘”)ā‰„0, satisfying limš‘”ā†’+āˆžāˆ«š‘”0š›¼(šœ‰)š‘‘šœ‰=+āˆž,limš‘”ā†’āˆ’āˆžāˆ«0š‘”š›¼(šœ‰)š‘‘šœ‰=+āˆž.

Example 1.3. Consider the system āŽ›āŽœāŽœāŽĢ‡š‘„1Ģ‡š‘„2āŽžāŽŸāŽŸāŽ =āŽ›āŽœāŽœāŽœāŽœāŽāˆ’13āˆš001|š‘”|+13āˆš|āŽžāŽŸāŽŸāŽŸāŽŸāŽ āŽ›āŽœāŽœāŽš‘„š‘”|+11š‘„2āŽžāŽŸāŽŸāŽ .(1.4) 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 1ā€‰:ā€‰1 correspondence existing between solutions of topological decoupling systems, namely, Ģ‡š‘„(š‘”)=š“(š‘”)š‘„(š‘”)+š‘“(š‘”,š‘„) and Ģ‡š‘„(š‘”)=š“(š‘”)š‘„(š‘”)+š‘”(š‘”,š‘„).

2. Existence of Equivalent Function

Consider the following two nonlinear nonautonomous systems:Ģ‡š‘„=š“(š‘”)š‘„+š‘“(š‘”,š‘„),(2.1)Ģ‡š‘„=š“(š‘”)š‘„+š‘”(š‘”,š‘„),(2.2) where š‘„āˆˆā„š‘›, š“(š‘”),ā€‰ā€‰šµ(š‘”) are š‘›Ć—š‘› matrices.

Definition 2.1. Suppose that there exists a function š»āˆ¶ā„Ć—ā„š‘›ā†’ā„š‘› such that (i)for each fixed š‘”, š»(š‘”,ā‹…) is a homeomorphism of ā„š‘› into ā„š‘›;(ii)š»(š‘”,š‘„)ā†’āˆž as |š‘„|ā†’āˆž, uniformly with respect to š‘”;(iii)assume that šŗ(š‘”,ā‹…)=š»āˆ’1(š‘”,ā‹…) has property (ii) too;(iv)if š‘„(š‘”) is a solution of system (2.1), then š»(š‘”,š‘„(š‘”)) is a solution of system (2.2).If such a map š» exists, then (2.1) is topologically conjugated to (2.2). š» is called an equivalent function.

Theorem 2.2. Suppose that Ģ‡š‘„=š“(š‘”)š‘„ has a generalized exponential dichotomy. If š‘“(š‘”,š‘„),š‘”(š‘”,š‘„) fulfill ||||||š‘“ī€·š‘“(š‘”,š‘„)ā‰¤š¹(š‘”),š‘”,š‘„1ī€øī€·āˆ’š‘“š‘”,š‘„2ī€ø||||š‘„ā‰¤š‘Ÿ(š‘”)1āˆ’š‘„2||,||||||š‘”ī€·š‘”(š‘”,š‘„)ā‰¤šŗ(š‘”),š‘”,š‘„1ī€øī€·āˆ’š‘”š‘”,š‘„2ī€ø||||š‘„ā‰¤š‘Ÿ(š‘”)1āˆ’š‘„2||,š‘š‘(š‘”,š¹,šŗ)ā‰¤šµ,(š‘”,š‘Ÿ)ā‰¤šæ<1,(2.3) where ī€œš‘(š‘”,š¹,šŗ)=š‘”āˆ’āˆžī‚µāˆ’ī€œš¾expš‘”š‘ ī‚¶(+ī€œš›¼(šœ‘)š‘‘šœ‘š¹(š‘ )+šŗ(š‘ ))š‘‘š‘ ,š‘”+āˆžī‚µī€œš¾expš‘”š‘ ī‚¶(ī€œš›¼(šœ‘)š‘‘šœ‘š¹(š‘ )+šŗ(š‘ ))š‘‘š‘ ,š‘(š‘”,š‘Ÿ)=š‘”āˆ’āˆžī‚µāˆ’ī€œš¾expš‘”š‘ ī‚¶ī€œš›¼(šœ‘)š‘‘šœ‘š‘Ÿ(š‘ )š‘‘š‘ +š‘”+āˆžī‚µī€œš¾expš‘”š‘ ī‚¶š›¼(šœ‘)š‘‘šœ‘š‘Ÿ(š‘ )š‘‘š‘ ,(2.4) where š¹(š‘”),šŗ(š‘”),š‘Ÿ(š‘”)ā‰„0 are integrable functions and šµ, šæ are positive constants, then the nonlinear nonautonomous system (2.1) is topologically equivalent to the nonlinear nonautonomous system (2.2). Moreover, the equivalent functions š»(š‘”,š‘„),šŗ(š‘”,š‘¦) fulfill ||||||||š»(š‘”,š‘„)āˆ’š‘„ā‰¤šµ,šŗ(š‘”,š‘¦)āˆ’š‘¦ā‰¤šµ.(2.5)

In what follows, we always suppose that the conditions of Theorem 2.2 are satisfied. Denote that š‘‹(š‘”,š‘”0,š‘„0) is a solution of (2.2) satisfying the initial condition š‘‹(š‘”0)=š‘„0 and that š‘Œ(š‘”,š‘”0,š‘¦0) is a solution of (2.1) satisfying the initial condition š‘Œ(š‘”0)=š‘¦0. To prove the main results, we first prove some lemmas.

Lemma 2.3. For each (šœ,šœ‰), system š‘§ī…ž=š“(š‘”)š‘§āˆ’š‘“(š‘”,š‘‹(š‘”,šœ,šœ‰))+š‘”(š‘”,š‘‹(š‘”,šœ,šœ‰)+š‘§)(2.6) has a unique bounded solution ā„Ž(š‘”,(šœ,šœ‰)) with |ā„Ž(š‘”,(šœ,šœ‰))|ā‰¤šµ.

Proof. Let š”¹ be the set of all the continuous bounded functions š‘„(š‘”) with |š‘„(š‘”)|ā‰¤šµ. For each (šœ,šœ‰) and any š‘§(š‘”)āˆˆš”¹, define the mapping š‘‡ as follows: ī€œš‘‡š‘§(š‘”)=š‘”āˆ’āˆžš‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1([]āˆ’ī€œš‘ )š‘”(š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§)āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ š‘”+āˆžš‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1[](š‘ )š‘”(š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§)āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ .(2.7)
Simple computation leads to ||||ā‰¤ī€œš‘‡š‘§(š‘”)š‘”āˆ’āˆž||š‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1(||(+ī€œš‘ )š¹(š‘ )+šŗ(š‘ ))š‘‘š‘ š‘”+āˆž||š‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1||ā‰¤ī€œ(š‘ )(š¹(š‘ )+šŗ(š‘ ))š‘‘š‘ š‘”āˆ’āˆžī‚µāˆ’ī€œš¾expš‘”š‘ ī‚¶+ī€œš›¼(šœ‘)š‘‘šœ‘(š¹(š‘ )+šŗ(š‘ ))š‘‘š‘ š‘”+āˆžī‚µī€œš¾expš‘”š‘ ī‚¶š›¼(šœ‘)š‘‘šœ‘(š¹(š‘ )+šŗ(š‘ ))š‘‘š‘ ā‰¤šµ,(2.8) which implies that š‘‡ is a self-map of a sphere with radius šµ. For any š‘§1(š‘”),š‘§2(š‘”)āˆˆš”¹, ||š‘‡š‘§1(š‘”)āˆ’š‘‡š‘§2(||ā‰¤ī€œš‘”)š‘”āˆ’āˆž||š‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1(||ī€·š‘§š‘ )š‘Ÿ(š‘ )1(š‘ )āˆ’š‘§2(ī€ø+ī€œš‘ )š‘‘š‘ š‘”+āˆž||š‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1||ī€·š‘§(š‘ )š‘Ÿ(š‘ )1(š‘ )āˆ’š‘§2ī€øā‰¤ā€–ā€–š‘§(š‘ )š‘‘š‘ 1āˆ’š‘§2ā€–ā€–ī‚øī€œš‘”āˆ’āˆžī‚µāˆ’ī€œš¾expš‘”š‘ ī‚¶+ī€œš›¼(šœ‘)š‘‘šœ‘š‘Ÿ(š‘ )š‘‘š‘ š‘”+āˆžī‚µī€œš¾expš‘”š‘ ī‚¶ī‚¹ā€–ā€–š‘§š›¼(šœ‘)š‘‘šœ‘š‘Ÿ(š‘ )š‘‘š‘ ā‰¤šæ1āˆ’š‘§2ā€–ā€–.(2.9)
Due to the fact that šæ<1, š‘‡ has a unique fixed point, namely, š‘§0(š‘”), and š‘§0(ī€œš‘”)=š‘”āˆ’āˆžš‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1(ī€ŗš‘”ī€·š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§0(ī€øī€»āˆ’ī€œš‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ š‘”+āˆžš‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§0ī€øī€»(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ ,(2.10) it is easy to show that š‘§0(š‘”) 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 š‘§1(š‘”) of (2.6). Thus, š‘§1(š‘”) can be written as follows: š‘§1(š‘”)=š‘ˆ(š‘”)š‘ˆāˆ’1(0)š‘„0+ī€œš‘”0š‘ˆ(š‘”)š‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ =š‘ˆ(š‘”)š‘ˆāˆ’1(0)š‘„0+ī€œš‘”0[]š‘ˆš‘ˆ(š‘”)š‘ƒ+(š¼āˆ’š‘ƒ)āˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ =š‘ˆ(š‘”)š‘ˆāˆ’1(0)š‘„0+ī€œš‘”āˆ’āˆžš‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»āˆ’ī€œ(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ 0āˆ’āˆžš‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»+ī€œ(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ 0+āˆžš‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»āˆ’ī€œ(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ š‘”+āˆžš‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ .(2.11)
Note that ī€œ0āˆ’āˆžš‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ =š‘ˆ(š‘”)š‘ˆāˆ’1ī€œ(0)0āˆ’āˆžš‘ˆ(0)š‘ƒš‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ ā‰¤š‘ˆ(š‘”)š‘ˆāˆ’1(||||ī€œ0)0āˆ’āˆžš‘ˆ(0)š‘ƒš‘ˆāˆ’1(ī€ŗš‘”ī€·š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1(ī€øī€»||||š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ ā‰¤š‘ˆ(š‘”)š‘ˆāˆ’1ī€œ(0)0āˆ’āˆžī‚µāˆ’ī€œš¾exp0š‘ š›¼ī‚¶(šœ‘)š‘‘šœ‘(š¹(š‘ )+šŗ(š‘ ))š‘‘š‘ ,(2.12) which implies that āˆ«0āˆ’āˆžš‘ˆ(0)š‘ƒš‘ˆāˆ’1(š‘ )[š‘”(š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1(š‘ ))āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))]š‘‘š‘  is convergent; denote it by š‘„1. That is, ī€œ0āˆ’āˆžš‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ =š‘ˆ(š‘”)š‘ˆāˆ’1(0)š‘„1.(2.13) Similarly, ī€œ0+āˆžš‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ =š‘ˆ(š‘”)š‘ˆāˆ’1(0)š‘„2.(2.14) Therefore, it follows from the expression of š‘§1(š‘”) that š‘§1(š‘”)=š‘ˆ(š‘”)š‘ˆāˆ’1ī€·š‘„(0)0āˆ’š‘„1+š‘„2ī€ø+ī€œš‘”āˆ’āˆžī€ŗš‘”ī€·š‘ˆ(š‘”)š‘ƒš‘ˆ(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»āˆ’ī€œ(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ š‘”+āˆžš‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ .(2.15)
Noticing that š‘§1(š‘”) is bounded, āˆ«š‘”āˆ’āˆžš‘ˆ(š‘”)š‘ƒš‘ˆ(š‘ )[š‘”(š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1āˆ«(š‘ ))āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))]š‘‘š‘ āˆ’š‘”+āˆžš‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1(š‘ )[š‘”(š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1(š‘ ))āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))]š‘‘š‘  is also bounded. So, š‘ˆ(š‘”)š‘ˆāˆ’1(0)(š‘„0āˆ’š‘„1+š‘„2) is bounded. But we see that š‘§ī…ž=š“(š‘”)š‘§ does not have a nontrivial bounded solution. Thus, š‘ˆ(š‘”)š‘ˆāˆ’1(0)(š‘„0āˆ’š‘„1+š‘„2)=0; it follows that š‘§1(ī€œš‘”)=š‘”āˆ’āˆžš‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1(ī€ŗš‘”ī€·š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1(ī€øī€»āˆ’ī€œš‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ š‘”+āˆžš‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ .(2.16) Simple calculating shows ||š‘§1(š‘”)āˆ’š‘§0(||ā‰¤ī€œš‘”)š‘”āˆ’āˆž||š‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1(||||š‘§š‘ )š‘Ÿ(š‘ )1(š‘ )āˆ’š‘§0(||+ī€œš‘ )š‘‘š‘ š‘”+āˆž||š‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1||||š‘§(š‘ )š‘Ÿ(š‘ )1(š‘ )āˆ’š‘§0||ā‰¤ā€–ā€–š‘§(š‘ )š‘‘š‘ 1āˆ’š‘§0ā€–ā€–ī‚øī€œš‘”āˆ’āˆžī‚µāˆ’ī€œš¾expš‘”š‘ ī‚¶+ī€œš›¼(šœ‘)š‘‘šœ‘š‘Ÿ(š‘ )š‘”+āˆžī‚µī€œš¾expš‘”š‘ ī‚¶ī‚¹ā€–ā€–š‘§š›¼(šœ‘)š‘‘šœ‘š‘Ÿ(š‘ )ā‰¤šæ1āˆ’š‘§0ā€–ā€–.(2.17)

Therefore, ā€–š‘§1āˆ’š‘§0ā€–ā‰¤šæā€–š‘§1āˆ’š‘§0ā€–, consequently š‘§1(š‘”)ā‰”š‘§0(š‘”). This implies that the bounded solution of (2.6) is unique. We may call it ā„Ž(š‘”(šœ,šœ‰)). From the above proof, it is easy to see that |ā„Ž(š‘”,(šœ,šœ‰))|ā‰¤šµ.

Lemma 2.4. For each (šœ,šœ‰), the system š‘§ī…ž=š“(š‘”)š‘§+š‘“(š‘”,š‘‹(š‘”,šœ,šœ‰)+š‘§)āˆ’š‘”(š‘”,š‘‹(š‘”,šœ,šœ‰))(2.18) has a unique bounded solution Ģƒš‘”(š‘”,(šœ,šœ‰)) and |Ģƒš‘”(š‘”,(šœ,šœ‰))|ā‰¤šµ.

Proof. The proof is similar to that of Lemma 2.3.

Lemma 2.5. Let š‘„(š‘”) be any solution of the system (2.1), then š‘§(š‘”)=0 is the unique bounded solution of system š‘§ī…ž=š“(š‘”)š‘§+š‘“(š‘”,š‘„(š‘”)+š‘§)āˆ’š‘“(š‘”,š‘„(š‘”)).(2.19)

Proof. Obviously, š‘§ā‰”0 is a bounded solution of system (2.19). We show that the bounded solution is unique. If not, then there is another bounded solution š‘§1(š‘”), which can be written as follows: š‘§1(š‘”)=š‘ˆ(š‘”)š‘ˆāˆ’1(0)š‘§1(ī€œ0)+š‘”0š‘ˆ(š‘”)š‘ˆāˆ’1(ī€ŗš‘“ī€·š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1(ī€øī€»š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ .(2.20) By Lemma 2.3, we can get š‘§1(ī€œš‘”)=š‘”āˆ’āˆžš‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1(ī€ŗš‘“ī€·š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1(ī€øī€»āˆ’ī€œš‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ š‘”+āˆžš‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1ī€ŗš‘“ī€·(š‘ )š‘ ,š‘‹(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»(š‘ )āˆ’š‘“(š‘ ,š‘‹(š‘ ,šœ,šœ‰))š‘‘š‘ .(2.21) It follows that ||š‘§1(||ā‰¤ī€œš‘”)š‘”āˆ’āˆž||š‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1(||||š‘§š‘ )š‘Ÿ(š‘ )1(||+ī€œš‘ )š‘‘š‘ š‘”+āˆž||š‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1||||š‘§(š‘ )š‘Ÿ(š‘ )1||ā‰¤ī€œ(š‘ )š‘‘š‘ š‘”āˆ’āˆžī‚µāˆ’ī€œš¾expš‘”š‘ ī‚¶||š‘§š›¼(šœ‘)š‘‘šœ‘š‘Ÿ(š‘ )1||+ī€œ(š‘ )š‘‘š‘ š‘”+āˆžī‚µī€œš¾expš‘”š‘ ī‚¶||š‘§š›¼(šœ‘)š‘‘šœ‘š‘Ÿ(š‘ )1||||š‘§(š‘ )š‘‘š‘ ā‰¤šæ1||.(š‘”)(2.22) That is, ā€–š‘§1ā€–ā‰¤šæ|š‘§1|. Consequently, š‘§1(š‘”)ā‰”0. This completes the proof of Lemma 2.5.

Lemma 2.6. Let š‘¦(š‘”) be any solution of the system (2.2), then š‘§(š‘”)=0 is the unique bounded solution of system š‘§ī…ž=š“(š‘”)š‘§+š‘”(š‘”,š‘¦(š‘”)+š‘§)āˆ’š‘”(š‘”,š‘¦(š‘”)).(2.23)

Proof. Obviously, š‘§ā‰”0 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 š‘§1(š‘”). Then, š‘§1(š‘”) can be written as follows: š‘§1(š‘”)=š‘ˆ(š‘”)š‘ˆāˆ’1(0)š‘§1(ī€œ0)+š‘”0š‘ˆ(š‘”)š‘ˆāˆ’1(ī€ŗš‘”ī€·š‘ )š‘ ,š‘Œ(š‘ ,šœ,šœ‰)+š‘§1(ī€øī€»š‘ )āˆ’š‘”(š‘ ,š‘Œ(š‘ ,šœ,šœ‰))š‘‘š‘ .(2.24)
By Lemma 2.3, we can get š‘§1(ī€œš‘”)=š‘”āˆ’āˆžš‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1(ī€ŗš‘”ī€·š‘ )š‘ ,š‘Œ(š‘ ,šœ,šœ‰)+š‘§1(ī€øī€»āˆ’ī€œš‘ )āˆ’š‘”(š‘ ,š‘Œ(š‘ ,šœ,šœ‰))š‘‘š‘ š‘”+āˆžš‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1ī€ŗš‘”ī€·(š‘ )š‘ ,š‘Œ(š‘ ,šœ,šœ‰)+š‘§1ī€øī€»(š‘ )āˆ’š‘”(š‘ ,š‘Œ(š‘ ,šœ,šœ‰))š‘‘š‘ .(2.25) Then, it follows that ||š‘§1(||ā‰¤ī€œš‘”)š‘”āˆ’āˆž||š‘ˆ(š‘”)š‘ƒš‘ˆāˆ’1(||||š‘§š‘ )š‘Ÿ(š‘ )1(||+ī€œš‘ )š‘‘š‘ š‘”+āˆž||š‘ˆ(š‘”)(š¼āˆ’š‘ƒ)š‘ˆāˆ’1||||š‘§(š‘ )š‘Ÿ(š‘ )1||ā‰¤ī€œ(š‘ )š‘‘š‘ š‘”āˆ’āˆžī‚µāˆ’ī€œš¾expš‘”š‘ ī‚¶||š‘§š›¼(šœ‘)š‘‘šœ‘š‘Ÿ(š‘ )1||+ī€œ(š‘ )š‘‘š‘ š‘”+āˆžī‚µī€œš¾expš‘”š‘ ī‚¶||š‘§š›¼(šœ‘)š‘‘šœ‘š‘Ÿ(š‘ )1||||š‘§(š‘ )š‘‘š‘ ā‰¤šæ1||.(š‘”)(2.26) That is, ā€–š‘§1ā€–ā‰¤šæā€–š‘§1ā€–. Consequently, š‘§1(š‘”)ā‰”0. This completes the proof of Lemma 2.6.

Now, we define two functions as follows:š»(š‘”,š‘„)=š‘„+ā„Ž(š‘”,(š‘”,š‘„)),(2.27)šŗ(š‘”,š‘„)=š‘¦+Ģƒš‘”(š‘”,(š‘”,š‘¦)).(2.28)

Lemma 2.7. For any fixed (š‘”0,š‘„0),š»(š‘”,š‘‹(š‘”,š‘”0,š‘„0)) is a solution of the system (2.2).

Proof. Replace (šœ,šœ‰) by (š‘”,š‘‹(š‘”,šœ,šœ‰)) in (2.6); system (2.6) is not changed. Due to the uniqueness of the bounded solution of (2.6), we can get that ā„Ž(š‘”,(š‘”,š‘‹(š‘”,š‘”0,š‘„0)))=ā„Ž(š‘”,(š‘”0,š‘„0)). Thus, š»ī€·ī€·š‘”,š‘‹š‘”,š‘”0,š‘„0ī€·ī€øī€ø=š‘‹š‘”,š‘”0,š‘„0ī€øī€·ī€·š‘”+ā„Žš‘”,0,š‘„0ī€øī€ø.(2.29) Differentiating it, noticing that š‘‹(š‘”,š‘”0,š‘„0), ā„Ž(š‘”,(š‘”0,š‘„0)) are the solutions of the (2.1), and (2.6), respectively; therefore, we can obtain ī€ŗš»ī€·ī€·š‘”,š‘‹š‘”,š‘”0,š‘„0ī€øī€øī€»ī…žī€·=š“(š‘”)š‘‹š‘”,š‘”0,š‘„0ī€øī€·ī€·+š‘“š‘”,š‘‹š‘”,š‘”0,š‘„0ī€·ī€·š‘”ī€øī€ø+š“(š‘”)ā„Žš‘”,0,š‘„0ī€·ī€·ī€øī€øāˆ’š‘“š‘”,š‘‹š‘”,š‘”0,š‘„0ī€·ī€·ī€øī€ø+š‘”š‘”,š‘‹š‘”,š‘”0,š‘„0ī€øī€·ī€·š‘”+ā„Žš‘”,0,š‘„0ī€·ī€·ī€øī€øī€ø=š“(š‘”)š»š‘”,š‘‹š‘”,š‘”0,š‘„0ī€·ī€·ī€øī€ø+š‘”š‘”,š»š‘”,š‘”0,š‘„0.ī€øī€ø(2.30) It indicates that š»(š‘”,š‘‹(š‘”,š‘”0,š‘„0)) is the solution of the system (2.2).

Lemma 2.8. For any fixed (š‘”0,š‘¦0), šŗ(š‘”,š‘Œ(š‘”,š‘”0,š‘¦0)) is a solution of the system (2.1).

Proof. The proof is similar to Lemma 2.7.

Lemma 2.9. For any š‘”āˆˆā„,š‘¦āˆˆā„š‘›,š»(š‘”,šŗ(š‘”,š‘¦))ā‰”š‘¦.

Proof. Let š‘¦(š‘”) be any solution of the system (2.2). From Lemma 2.8, šŗ(š‘”,š‘¦(š‘”)) is a solution of system (2.1). Then, by Lemma 2.7, we see that š»(š‘”,šŗ(š‘”,š‘¦(š‘”))) is a solution of (2.2), written as š‘¦1(š‘”). Denote š½(š‘”)=š‘¦1(š‘”)āˆ’š‘¦(š‘”). Differentiating, we have š½ī…ž(š‘”)=š‘¦ī…ž1(š‘”)āˆ’š‘¦ī…ž(š‘”)=š“(š‘”)š‘¦1ī€·(š‘”)+š‘”š‘”,š‘¦1ī€ø(š‘”)āˆ’š“(š‘”)š‘¦(š‘”)āˆ’š‘”(š‘”,š‘¦(š‘”))=š“(š‘”)š½(š‘”)+š‘”(š‘”,š‘¦(š‘”)+š½(š‘”))āˆ’š‘”(š‘”,š‘¦(š‘”)),(2.31) which implies that š½(š‘”) is a solution of system (2.23). On the other hand, following the definition of š» and šŗ and Lemmas 2.3 and 2.4, we can obtain ||||=||||ā‰¤||||+||||=||||+||||š½(š‘”)š»(š‘”,šŗ(š‘”,š‘¦(š‘”)))āˆ’š‘¦(š‘”)š»(š‘”,šŗ(š‘”,š‘¦(š‘”)))āˆ’šŗ(š‘”,š‘¦(š‘”))šŗ(š‘”,š‘¦(š‘”))āˆ’š‘¦(š‘”)ā„Ž(š‘”,(š‘”,šŗ(š‘”,š‘¦(š‘”))))Ģƒš‘”(š‘”,(š‘”,š‘¦))ā‰¤2šµ.(2.32)
This implies that š½(š‘”) is a bounded solution of system (2.23). However, by Lemma 2.6, system (2.23) has only one zero solution. Hence, š½(š‘”)ā‰”0, consequently š‘¦1(š‘”)ā‰”š‘¦(š‘”), that is, š»(š‘”,šŗ(š‘”,š‘¦))=š‘¦(š‘”). Since š‘¦(š‘”) is any solution of the system (2.2), Lemma 2.9 follows.

Lemma 2.10. For any š‘”āˆˆā„,š‘„āˆˆā„š‘›,šŗ(š‘”,š»(š‘”,š‘„))ā‰”š‘„.

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 š»(š‘”,ā‹…) satisfies the four conditions of Definition 2.1.Proof of Condition (i). For any fixed š‘”, it follows from Lemmas 2.9 and 2.10 that š»(š‘”,ā‹…) is homeomorphism and šŗ(š‘”,ā‹…)=š»āˆ’1(š‘”,ā‹…).Proof of Condition (ii). From (2.27) and Lemma 2.3, we derive |š»(š‘”,š‘„)āˆ’š‘„|=|ā„Ž(š‘”,(š‘”,š‘„))|ā‰¤šµ. So, |š»(š‘”,š‘„)|ā†’āˆž as |š‘„|ā†’āˆž, uniformly with respect to š‘”.Proof of Condition (iii). From (2.23) and Lemma 2.4, we derive |šŗ(š‘”,š‘¦)āˆ’š‘¦|=|Ģƒš‘”(š‘”,(š‘”,š‘¦))|ā‰¤šµ. So, |šŗ(š‘”,š‘¦)|ā†’āˆž as |š‘„|ā†’āˆž, uniformly with respect to š‘”.Proof of Condition (iv). Using Lemmas 2.7 and 2.8, we easily prove that Condition (iv) is true.

Hence, systems (2.1) and (2.2) are topologically conjugated. This completes the proof of Theorem 2.2.

Acknowledgments

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).

References

  1. P. Hartman, ā€œOn the local linearization of differential equations,ā€ Proceedings of the American Mathematical Society, vol. 14, pp. 568ā€“573, 1963. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  2. K. J. Palmer, ā€œA generalization of Hartman's linearization theorem,ā€ Journal of Mathematical Analysis and Applications, vol. 41, pp. 753ā€“758, 1973. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  3. J. Shi, ā€œAn improvement of Harman's linearization thoerem,ā€ Science in China A, vol. 32, pp. 458ā€“470, 2002. View at: Google Scholar
  4. L. P. Jiang, ā€œA generalization of Palmer's linearization theorem,ā€ Mathematica Applicata, vol. 24, no. 1, pp. 150ā€“157, 2011 (Chinese). View at: Google Scholar
  5. A. Reinfeld, ā€œA generalized Grobman-Hartman theorem,ā€ LatviÄ­skiÄ­ MatematicheskiÄ­ Ezhegodnik, vol. 29, pp. 84ā€“88, 1985. View at: Google Scholar
  6. A. Reinfelds, ā€œA reduction theorem for systems of differential equations with impulse effect in a Banach space,ā€ Journal of Mathematical Analysis and Applications, vol. 203, no. 1, pp. 187ā€“210, 1996. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. Y. Xia, J. Cao, and M. Han, ā€œA new analytical method for the linearization of dynamic equation on measure chains,ā€ Journal of Differential Equations, vol. 235, no. 2, pp. 527ā€“543, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. W. A. Coppel, Dichotomies in Stability Theory, vol. 629 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1978.
  9. M. Lin, ā€œGeneralized exponential dichotomy,ā€ Journal of Fuzhou University, vol. 10, no. 4, pp. 21ā€“30, 1982 (Chinese). View at: Google Scholar

Copyright Ā© 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.


More related articles

 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views4227
Downloads625
Citations

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.