## Qualitative Theory of Differential, Difference, and Dynamic Equations

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

**Academic Editor:**Tuncay Candan

#### 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.

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#### Copyright

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.