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

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.