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 .