International Journal of Differential Equations

Volume 2014 (2014), Article ID 703868, 7 pages

http://dx.doi.org/10.1155/2014/703868

## Conjugacy of a Discrete Semidynamical System in a Neighbourhood of the Nontrivial Invariant Manifold

^{1}Institute of Mathematics and Computer Science, University of Latvia, Raiņa bulvāris 29, Rīga 1459, Latvia^{2}Faculty of Physics and Mathematics, University of Latvia, Zeļļu iela 8, Rīga 1002, Latvia

Received 21 November 2013; Accepted 15 January 2014; Published 25 February 2014

Academic Editor: Tuncay Candan

Copyright © 2014 Andrejs Reinfelds. 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.

#### Abstract

The conjugacy of a discrete semidynamical system and its partially decoupled discrete semidynamical system in a Banach space is proved in a neighbourhood of the nontrivial invariant manifold.

#### 1. Introduction

The conjugacy for noninvertible mappings in a Banach space was considered by Aulbach and Garay [1–3]. For noninvertible mappings in a complete metric space it was extended and generalized by Reinfelds [4–9]. In the present paper we consider the case when the linear part of the noninvertible mapping depends on the behaviour of variables in a neighbourhood of the nontrivial invariant manifold.

#### 2. Invariant Manifold

Let and be Banach spaces, , and . Consider the following mapping defined by where the derivative of the diffeomorphism is uniformly continuous ≤ , mappings , , and are Lipschitzian,

At the beginning we will modify the previous results on the existence of invariant manifolds of Neĭmark and Sacker [10, 11] for (1).

Lemma 1. *If
**
then there exists a continuous mapping satisfying the following properties:*(i)*;*(ii)*;*(iii)*.*

*Proof. *The set of continuous mappings **,**
equipped with the norm
is a Banach space. The set
is a closed subset of the Banach space .

Let us consider the mapping , defined by the equality
where
If then
We have
It follows
Then
Let us note that
We obtain

We get that is contraction and consequently we have the invariant manifold .

*3. Conjugacy of Noninvertible Mappings*

*Definition 2. *Two mappings are *conjugate*, if there exists a homeomorphism such that

*Definition 3. *Two discrete semidynamical systems are *conjugate*, if there exists a homeomorphism such that

*It is easily verified that two discrete semidynamical systems and , generated by mappings and , are conjugate if and only if the mappings and are conjugate.*

*Suppose that mapping (1) has an invariant manifold given by Lipschitzian mapping such that
Our aim is to find a simpler mapping conjugated with (1).*

*Theorem 4. If , then there exists a continuous mapping which is Lipschitzian with respect to the second variable such that mappings (1) and
are conjugated in a small neighbourhood of the invariant manifold .*

*We will seek the mapping establishing the conjugacy of (1) and (18) in the form
We get the following functional equation:
or equivalently
The proof of the theorem consists of four lemmas.*

*Lemma 5. The functional equation (20) has a unique solution in .*

*Proof. *The set of continuous mappings **,**
becomes a Banach space if we use the norm . The set
is a closed subset of the Banach space .

Let us consider the mapping , defined by the equality
First we obtain
Here we used Hadamard lemma:
Next we get
In addition,
We choose , where , such that
Then , , the mapping is a contraction, and consequently the functional equation (20) has unique solution in .

*Next we will prove that the mapping is a homeomorphism in the small neighbourhood of the invariant manifold . Let us consider the functional equation
or equivalently
*

*Lemma 6. The functional equation (30) has a unique solution in .*

*Proof. *The set
is a closed subset of the Banach space .

Let us consider the mapping , defined by the equality
We have
We obtain
We get that is a contraction and consequently the functional equation (30) has a unique solution in .

*Consider the mapping defined by equality .*

*Lemma 7. One has .*

*Proof. *Let us consider the functional equation
or equivalently
It is easily verified that the functional equation (36) has the trivial solution. Let us prove the uniqueness of the solution in , where
is a closed subset of the Banach space . We get
It follows that . The mapping , where
also satisfies the functional equation (36). Using the change of variables in (30) we get
Using (20), we obtain
Let us note that
Therefore and we have
We obtain that .

*Lemma 8. One has .*

*Proof. *The set of continuous mappings **,**
becomes a Banach space if we use the norm . The set
is a closed subset of the Banach space .

Let us consider the functional equation
or equivalently
Let us consider the mapping , defined by the equality
We obtain
In addition,
Let and where
We have
Then , , the mapping is a contraction, and consequently the functional equation (47) has a unique solution in . Moreover, this solution is also unique in the closed subset . Let us note that

The mapping , where
satisfies (47). Using the change of variables in (20) we get
Using (30) we obtain
Let us note that
Therefore and we have
It follows that .

Finally we conclude that the mapping is a homeomorphism establishing a conjugacy of the noninvertible mappings (1) and (18).

*Conflict of Interests*

*The author declares that there is no conflict of interests regarding the publication of this paper.*

*Acknowledgment*

*This work was partially supported by Grant no. 345/2012 of the Latvian Council of Science.*

*References*

- B. Aulbach and B. M. Garay, “Linearization and decoupling of dynamical and semidynamical systems,” in
*The Second Colloquium on Differential Equations*, D. Bainov and V. Covachov, Eds., pp. 15–27, World Scientific, Singapore, 1992. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Aulbach and B. M. Garay, “Linearizing the expanding part of noninvertible mappings,”
*Zeitschrift für Angewandte Mathematik und Physik*, vol. 44, no. 3, pp. 469–494, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - B. Aulbach and B. M. Garay, “Partial linearization for noninvertible mappings,”
*Zeitschrift für Angewandte Mathematik und Physik*, vol. 45, no. 4, pp. 505–542, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Reinfelds, “Partial decoupling for noninvertible mappings,”
*Differential Equations and Dynamical Systems*, vol. 2, no. 3, pp. 205–215, 1994. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Reinfelds, “The reduction principle for discrete dynamical and semidynamical systems in metric spaces,”
*Zeitschrift für Angewandte Mathematik und Physik*, vol. 45, no. 6, pp. 933–955, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Reinfelds, “Partial decoupling of semidynamical system,” in
*Mathematics*, vol. 593, pp. 54–61, Latvia University, Riga, Latvia, 1994. View at Google Scholar · View at MathSciNet - A. Reinfelds, “The reduction of discrete dynamical and semidynamical systems in metric spaces,” in
*Six Lectures on Dynamical Systems*, B. Aulbach and F. Colonius, Eds., pp. 267–312, World Scientiffc, Singapore, 1996. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Reinfelds, “Partial decoupling of semidynamical system in metric space,”
*Journal of the Technical University at Plovdiv*, vol. 5, pp. 33–40, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - A. Reinfelds, “Conjugacy of discrete semidynamical systems in the neighbourhood of invariant manifold,” in
*Differential and Difference Equations With Applications*, S. Pinelas, M. Chipot, and Z. Dosla, Eds., vol. 47 of*Springer Proceedings in Mathematics and Statistics*, pp. 571–578, Springer, New York, NY, USA, 2013. View at Google Scholar - Ju. I. Neĭmark, “Existence and structural stability of invariant manifolds of pointwise mappings,”
*Izvestiya Vysshikh Uchebnykh Zavedeniĭ*, vol. 10, pp. 311–320, 1967 (Russian). View at Google Scholar · View at MathSciNet - R. J. Sacker,
*On Invariant Surface and Bifurcation of Periodic Solution of Ordinary Differential Equations*, vol. 333, New York University, 1964. View at MathSciNet

*
*