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

Authors: Andrejs Reinfelds
Journal: International Journal of Differential Equations
 (2014)

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## 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.


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## Body

## 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

LetE and F be Banach spaces, B(a)={r∈F∣|r|≤a}, and a>0. Consider the following mapping S:E×B(a)→E×B(a) defined by
(1)x1=g(x)+Ψ(x,r)=X(x,r),r1=A(x)r+Φ(x,r)=R(x,r),
where the derivative of the diffeomorphism g:E→E is uniformly continuous ∥Dg(x)-Dg(x′)∥  ≤  ω(|x-x′|), mappings A, Ψ, and Φ are Lipschitzian,(2)∥A(x)-A(x′)∥≤γ|x-x′|,|Ψ(x,r)-Ψ(x′,r′)|≤ɛ(|x-x′|+|r-r′|),|Φ(x,r)-Φ(x′,r′)|≤ɛ(|x-x′|+|r-r′|),sup⁡x∥A(x)∥+2ɛ<1,sup⁡x|Φ(x,0)|≤a(1-sup⁡x∥A(x)∥-ɛ).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(3)(sup⁡x∥A(x)∥+4ɛ+γsup⁡x|Φ(x,0)|1-sup⁡x∥A(x)∥-ɛ)×sup⁡x∥Dg(x)-1∥≤1
then there exists a continuous mapping u:E→F satisfying the following properties:(i)
u(g(x)+Ψ(x,u(x)))=A(x)u(x)+Φ(x,u(x));(ii)
|u(x)-u(x′)|≤|x-x′|;(iii)
∥u∥≤sup⁡x|Φ(x,0)|/(1-sup⁡x∥A(x)∥-ɛ).Proof.
The set of continuous mappingsu:E→F,(4)𝒦={u∈C(E,F)∣sup⁡x|u(x)|<+∞}
equipped with the norm
(5)∥u∥=sup⁡x|u(x)|
is a Banach space. The set
(6)𝒦1={u∈𝒦∣∥u∥≤a,|u(x)-u(x′)|≤|x-x′|}
is a closed subset of the Banach space 𝒦.
Let us consider the mappingu↦ℒu, u∈𝒦1 defined by the equality
(7)ℒu(x1)=A(x)u(x)+Φ(x,u(x)),
where
(8)x1=g(x)+Ψ(x,u(x)).
If u∈𝒦1 then
(9)∥ℒu∥≤(sup⁡x∥A(x)∥+ɛ)∥u∥+sup⁡x|Φ(x,0)|≤a.
We have
(10)|ℒu(x1)-ℒu′(x1′)|≤(sup⁡x∥A(x)∥+ɛ)∥u-u′∥+(sup⁡x∥A(x)∥+2ɛ+γ∥u∥)|x-x′|,|x-x′|≤sup⁡x∥Dg(x)-1∥|g(x)-g(x′)|=sup⁡x∥Dg(x)-1∥|x1-x1′-Ψ(x,u(x))+Ψ(x′,u′(x′))|≤sup⁡x∥Dg(x)-1∥(|x1-x1′|+2ε|x-x′|+ε∥u-u′∥).
It follows
(11)|x-x′|≤sup⁡x∥Dg(x)-1∥1-2ɛsup⁡x∥Dg(x)-1∥|x1-x1′|+ɛsup⁡x∥Dg(x)-1∥1-2ɛsup⁡x∥Dg(x)-1∥∥u-u′∥.
Then
(12)|ℒu(x1)-ℒu′(x1′)|≤((sup⁡x∥A(x)∥+2ɛ+γ∥u∥)sup⁡x∥Dg(x)-1∥ɛ1-2ɛsup⁡x∥Dg(x)-1∥sup⁡x∥A(x)∥+ɛ+(sup⁡x∥A(x)∥+2ɛ+γ∥u∥)sup⁡x∥Dg(x)-1∥ɛ1-2ɛsup⁡x∥Dg(x)-1∥)×∥u-u′∥+(sup⁡x∥A(x)∥+2ɛ+γ∥u∥)sup⁡x∥Dg(x)-1∥1-2ɛsup⁡x∥Dg(x)-1∥×|x1-x1′|.
Let us note that
(13)(sup⁡x∥A(x)∥+2ɛ+γ∥u∥)sup⁡x∥Dg(x)-1∥1-2ɛsup⁡x∥Dg(x)-1∥≤1.
We obtain
(14)|ℒu(x1)-ℒu′(x1′)|≤(sup⁡x∥A(x)∥+2ɛ)∥u-u′∥+|x1-x1′|.
We get thatℒ∈𝒦1 is contraction and consequently we have the invariant manifold r=u(x).


## 3. Conjugacy of Noninvertible Mappings

Definition 2.
Two mappingsS,T:X→X are conjugate, if there exists a homeomorphism H:X→X such that
(15)S∘H(x)=H∘T(x).Definition 3.
Two discrete semidynamical systemsSn,Tn:X→X(n∈ℕ) are conjugate, if there exists a homeomorphism H:X→X such that
(16)Sn∘H(x)=H∘Tn(x).It is easily verified that two discrete semidynamical systemsSn and Tn, generated by mappings S and T, are conjugate if and only if the mappings S and T are conjugate.Suppose that mapping (1) has an invariant manifold given by Lipschitzian mapping u:E→F such that
(17)sup⁡x∥u(x)∥≤δ,|u(x)-u(x′)|≤|x-x′|.
Our aim is to find a simpler mapping conjugated with (1).Theorem 4.
Ifsup⁡x(∥(Dg(x))-1∥∥A(x)∥)+5ɛsup⁡x∥(Dg(x))-1∥<1, then there exists a continuous mapping v:E×B(δ)→E which is Lipschitzian with respect to the second variable such that mappings (1) and
(18)x1=X(x,u(x)),r1=R(x+v(x,r),r)
are conjugated in a small neighbourhood of the invariant manifold r=u(x).We will seek the mapping establishing the conjugacy of (1) and (18) in the form
(19)H(x,r)=(x+v(x,r),r).
We get the following functional equation:
(20)X(x+v(x,r),r)=X(x,u(x))+v(X(x,u(x)),R(x+v(x,r),r))
or equivalently
(21)v(x,r)=(Dg(x))-1((X(x,u(x)),R(x+v(x,r),r))Dg(x)v(x,r)-X(x+v(x,r),r)xxxxxxxxxxx+X(x,u(x))xxxxxxxxxxx+v(X(x,u(x)),R(x+v(x,r),r))).
The proof of the theorem consists of four lemmas.Lemma 5.
The functional equation (20) has a unique solution in ℳ1.Proof.
The set of continuous mappingsv:E×B(δ)→E,(22)ℳ={v∈C(E×B(δ),E)∣sup⁡x,r|v(x,r)||r-u(x)|<+∞}
becomes a Banach space if we use the norm ∥v∥=sup⁡x,r(|v(x,r)|/|r-u(x)|). The set
(23)ℳ1={v∈ℳ∣∥v∥≤1,|v(x,r)-v(x,r′)|≤|r-r′|}
is a closed subset of the Banach space ℳ.
Let us consider the mappingv↦ℒv, v∈ℳ1 defined by the equality
(24)ℒv(x,r)=(Dg(x))-1v(X(x,u(x)),R(x+v(x,r),r))+(Dg(x))-1(Dg(x)v(x,r)-g(x+v(x,r))+g(x)aaaaaaaaaaaaaaaaa-Ψ(x+v(x,r),r)+Ψ(x,u(x))).
First we obtain
(25)|Lv(x,r)|≤∥(Dg(x))-1∥|R(x+v(x,r),r)-u(X(x,u(x)))|+∥(Dg(x))-1∥|Dg(x)v(x,r)-g(x+v(x,r))+g(x)|+∥(Dg(x))-1∥|Ψ(x+v(x,r),r)-Ψ(x,u(x))|≤∥(Dg(x))-1∥×(∥A(x)∥+γ|r|+2ε+ω(|r-u(x)|)+2ε)×|r-u(x)|.
Here we used Hadamard lemma:
(26)g(x′)-g(x)=∫01‍Dg(x+θ(x′-x))dθ(x′-x).
Next we get
(27)|Lv(x,r)-Lv(x,r′)|≤∥(Dg(x))-1∥|R(x+v(x,r),r)-R(x+v(x,r′),r′)|+∥(Dg(x))-1∥|Dg(x)(v(x,r)-v(x,r′))xxxxxxxxxxxxxxx-g(x+v(x,r))+g(x+v(x,r′))|+∥(Dg(x))-1∥×|Ψ(x+v(x,r),r)-Ψ(x+v(x,r′),r′)|≤∥(Dg(x))-1∥(∥A(x)∥+2γmax⁡⁡{|r-u(x)|,|r′|}+2ε)×|r-r′|+∥(Dg(x))-1∥×(ω(max⁡{|r-u(x)|,|r′-u(x)|})+2ε)|r-r′|.
In addition,
(28)|Lv(x,r)-Lv′(x,r)|≤∥(Dg(x))-1∥|R(x+v(x,r),r)-R(x+v′(x,r),r)|+∥(Dg(x))-1∥|v′v(X(x,u(x)),R(x+v(x,r),r))xxxxxxxxxxxxxxx-v′(X(x,u(x)),R(x+v(x,r),r))|+∥(Dg(x))-1∥|Dg(x)(v(x,r)-v′(x,r))xxxxxxxxxxxxxxx-g(x+v(x,r))+g(x+v′(x,r))|+∥(Dg(x))-1∥×|Ψ(x+v(x,r),r)-Ψ(x+v′(x,r),r)|≤∥(Dg(x))-1∥(∥A(x)∥+2γ|r|+3ε)∥v-v′∥|r-u(x)|+∥(Dg(x))-1∥(ω(|r-u(x)|)+ε)×∥v-v′∥|r-u(x)|.
We choose δ>0, where max⁡{|r|,|r′|}=δ≤a, such that
(29)sup⁡x(∥(Dg(x))-1∥∥A(x)∥)+(5ɛ+ω(8δ)+4γδ)sup⁡x∥(Dg(x))-1∥<1.
Then ∥ℒv∥≤1, |ℒv(x,r)-ℒv(x,r′)|≤|r-r′|, the mapping ℒ is a contraction, and consequently the functional equation (20) has unique solution in ℳ1.Next we will prove that the mappingH is a homeomorphism in the small neighbourhood of the invariant manifold r=u(x). Let us consider the functional equation
(30)X(x+v1(x,r),u(x+v1(x,r)))=X(x,r)+v1(X(x,r),R(x,r))
or equivalently
(31)v1(x,r)=(Dg(x))-1(Dg(x)v1(x,r)xxxxxxxxxxxx-X(x+v1(x,r),u(x+v1(x,r)))xxxxxxxxxxxx+X(x,r)+v1(X(x,r),R(x,r))).Lemma 6.
The functional equation (30) has a unique solution in ℳ2.Proof.
The set(32)ℳ2={v∈ℳ∣∥v∥≤1}
is a closed subset of the Banach space ℳ.
Let us consider the mappingv1↦ℒv1, v1∈ℳ2 defined by the equality
(33)ℒv1(x,r)=(Dg(x))-1v1(X(x,r),R(x,r))+(Dg(x))-1×(Dg(x)v1(x,r)-g(x+v1(x,r))+g(x)-Ψ(x+v1(x,r),u(x+v1(x,r)))+Ψ(x,r)).
We have
(34)|Lv1(x,r)|≤∥(Dg(x))-1∥|R(x,r)-u(X(x,r))|+∥(Dg(x))-1∥×|Dg(x)v1(x,r)-g(x+v1(x,r))+g(x)|+∥(Dg(x))-1∥×|Ψ(x+v1(x,r),u(x+v1(x,r)))-Ψ(x,r)|≤∥(Dg(x))-1∥(∥A(x)∥+2ε+ω(|r-u(x)|)+3ε)×|r-u(x)|.
We obtain
(35)|Lv1(x,r)-Lv1′(x,r)|≤∥(Dg(x))-1∥×|v1(X(x,r),R(x,r))-v1′(X(x,r),R(x,r))|+∥(Dg(x))-1∥×|Dg(x)(v1(x,r)-v1′(x,r))xxxxxx-g(x+v1(x,r))+g(x+v1′(x,r))|+∥(Dg(x))-1∥×|v1′Ψ(x+v1(x,r),u(x+v1(x,r)))xxxxxx-Ψ(x+v1′(x,r),u(x+v1′(x,r)))|≤∥(Dg(x))-1∥(∥A(x)∥+2ε+ω(|r-u(x)|)+2ε)×∥v1-v1′∥|r-u(x)|.
We get that ℒ is a contraction and consequently the functional equation (30) has a unique solution in ℳ2.Consider the mappingG defined by equality G(x,r)=(x+v1(x,r),r).Lemma 7.
One hasG∘H=id.Proof.
Let us consider the functional equation(36)X(x+v2(x,r),u(x+v2(x,r)))=X(x,u(x))+v2(X(x,u(x)),R(x+v(x,r),r))
or equivalently
(37)v2(x,r)=(Dg(x))-1×(Dg(x)v2(x,r)-X(x+v2(x,r),u(x+v2(x,r)))xxxxx+X(x,u(x))+v2(X(x,u(x)),R(x+v(x,r),r))).
It is easily verified that the functional equation (36) has the trivial solution. Let us prove the uniqueness of the solution in ℳ3, where
(38)ℳ3={v2∈ℳ∣∥v2∥≤3}
is a closed subset of the Banach space ℳ. We get
(39)|v2(x,r)|≤∥(Dg(x))-1∥∥v2∥|R(x+v(x,r),r)-u(X(x,u(x)))|×∥(Dg(x))-1∥×|Dg(x)v2(x,r)-g(x+v2(x,r))+g(x)|+∥(Dg(x))-1∥×|Ψ(x+v2(x,r),u(x+v2(x,r)))-Ψ(x,u(x))|≤∥(Dg(x))-1∥×(∥A(x)∥+γ|r|+2ε+ω(3|r-u(x)|)+2ε)×∥v2∥|r-u(x)|.
It follows that v2(x,r)≡0. The mapping w1, where
(40)w1(x,r)=v(x,r)+v1(x+v(x,r),r),
also satisfies the functional equation (36). Using the change of variables x↦x+v(x,r) in (30) we get
(41)X(x+w1(x,r),u(x+w1(x,r)))=X(x+v(x,r),r)+v1(X(x+v(x,r),r),R(x+v(x,r),r)).
Using (20), we obtain
(42)X(x+w1(x,r),u(x+w1(x,r)))=X(x,u(x))+v(X(x,u(x)),R(x+v(x,r),r))+v1(X(x,u(x))+v(X(x,u(x)),R(x+v(x,r),r)),xxxxxxxxR(x+v(x,r),r))=X(x,u(x))+w1(X(x,u(x)),R(x+v(x,r),r)).
Let us note that
(43)|w1(x,r)|≤|r-u(x)|+|r-u(x+v(x,r))|≤3|r-u(x)|.
Therefore ∥w1∥≤3 and we have
(44)v(x,r)+v1(x+v(x,r),r)=0.
We obtain that G∘H=id.Lemma 8.
One hasH∘G=id.Proof.
The set of continuous mappingsv3:E×B(δ)×B(δ)→E,(45)𝒩={|v3(x,r,z)|max⁡(|r-u(x)|,|z-r|)v3∈C(E×B(δ)×B(δ),E)∣xxxxxsup⁡x,r,z|v3(x,r,z)|max⁡(|r-u(x)|,|z-r|)<∞}
becomes a Banach space if we use the norm ∥v3∥=sup⁡x,r,z(|v3(x,r,z)|/max⁡(|r-u(x)|,|z-r|)). The set
(46)𝒩1={z′v3∈𝒩∣∥v3∥≤1,xxxxx|v3(x,r,z)-v3(x,r,z′)|≤|z-z′|}
is a closed subset of the Banach space 𝒩.
Let us consider the functional equation(47)X(x,r)+v3(X(x,r),R(x,r),R(x+v3(x,r,z),z))=X(x+v3(x,r,z),z)
or equivalently
(48)v3(x,r,z)=(Dg(x))-1×(Dg(x)v3(x,r,z)-g(x+v3(x,r,z))+g(x)xxxxxx+Ψ(x,r)-Ψ(x+v3(x,r,z),z)xxxxxx+v3(X(x,r),R(x,r),R(x+v3(x,r,z),z))).
Let us consider the mapping v3↦ℒv3, v3∈𝒩1 defined by the equality
(49)ℒv3(x,r,z)=(Dg(x))-1×(Dg(x)v3(x,r,z)-g(x+v3(x,r,z))+g(x)xxxxxx-Ψ(x+v3(x,r,z),z)+Ψ(x,r)xxxxxx+v3(X(x,r),R(x,r),R(x+v3(x,r,z),z))).
We obtain
(50)|Lv3(x,r,z)|≤∥(Dg(x))-1∥max⁡{v3|R(x,r)-u(X(x,r))|,xxxxxxxxxxxxxxxxx|R(x+v3(x,r,z),z)-R(x,r)|}+∥(Dg(x))-1∥×|Dg(x)v3(x,r,z)-g(x+v3(x,r,z))+g(x)|+∥(Dg(x))-1∥|Ψ(x,r)-Ψ(x+v3(x,r,z),z)|≤∥(Dg(x))-1∥(∥A(x)∥+γ|z|+2ε)×max⁡⁡{|r-u(x)|,|z-r|}+∥(Dg(x))-1∥×(ω(max⁡⁡{|r-u(x)|,|z-r|})+2ε)×max⁡⁡{|r-u(x)|,|z-r|}.
In addition,
(51)|Lv3(x,r,z)-Lv3(x,r,z′)|≤∥(Dg(x))-1∥×|Dg(x)(v3(x,r,z)-v3(x,r,z′))xxxxx-g(x+v3(x,r,z))+g(x+v3(x,r,z′))|+∥(Dg(x))-1∥×|Ψ(x+v3(x,r,z),z)-Ψ(x+v3(x,r,z′),z′)|+∥(Dg(x))-1∥×|R(x+v3(x,r,z),z)-R(x+v3(x,r,z′),z′)|≤∥(Dg(x))-1∥×(ω(max⁡{|r-u(x)|,|z-r|,|z′-r|})+2ε)|z-z′|+∥(Dg(x))-1∥×(∥A∥+2ε+2γmax⁡{|r-u(x)|,|z|,|z′-r|})×|z-z′|.
Let v3∈𝒩1 and v3′∈𝒩1∪𝒩2 where
(52)𝒩2={v3′∈𝒩∣sup⁡x,|r|≤δ,|z|≤δ|v3′(x,r,z)|≤8δ,xxxxx|v3′(x,r,z)-v3′(x,r,z′)|≤|z-z′|sup⁡x,|r|≤δ,|z|≤δ}.
We have
(53)|Lv3(x,r,z)-Lv3′(x,r,z)|≤∥(Dg(x))-1∥×|Dg(x)(v3(x,r,z)-v3′(x,r,z))xxxxxx-g(x+v3(x,r,z))+g(x+v3′(x,r,z))|+∥(Dg(x))-1∥×|Ψ(x+v3(x,r,z),z)-Ψ(x+v3′(x,r,z),z)|+∥(Dg(x))-1∥×|v3′v3(X(x,r),R(x,r),R(x+v3(x,r,z),z))xxxxxx-v3′(X(x,r),R(x,r),R(x+v3(x,r,z),z))|+∥(Dg(x))-1∥×|v3′(X(x,r),R(x,r),R(x+v3(x,r,z),z))xxxxxx-v3′(X(x,r),R(x,r),R(x+v3′(x,r,z),z))|≤∥(Dg(x))-1∥(ω(max⁡⁡{|r-u(x)|,|z-r|,8δ})+ε)×∥v3-v3′∥max⁡⁡{|r-u(x)|,|z-r|}+∥(Dg(x))-1∥×max⁡⁡{v3′|R(x,r)-u(X(x,r))|,xxxxxxxxx|R(x+v3′(x,r,z),z)-R(x,r)|}∥v3-v3′∥+∥(Dg(x))-1∥×|R(x+v3(x,r,z),z)-R(x+v3′(x,r,z),z)|≤∥(Dg(x))-1∥(ω(8δ)+ε)∥v3-v3′∥×max⁡⁡{|r-u(x)|,|z-r|}+∥(Dg(x))-1∥(∥A∥+3ε+γ|z|)×max⁡⁡{|r-u(x)|,|z-r|}∥v3-v3′∥=∥(Dg(x))-1∥(∥A(x)∥+4ε+ω(8δ)+γ|z|)×∥v3-v3′∥max⁡⁡{|r-u(x)|,|z-r|}.
Then ∥ℒv3∥≤1, |ℒv3(x,r,z)-ℒv3(x,r,z′)|≤|z-z′|, the mapping ℒ is a contraction, and consequently the functional equation (47) has a unique solution in 𝒩1. Moreover, this solution is also unique in the closed subset 𝒩2. Let us note that
(54)v3(x,r,r)=0.
The mappingw2, where
(55)w2(x,r,z)=v1(x,r)+v(x+v1(x,r),z),
satisfies (47). Using the change of variables (x,r)↦(x+v1(x,r),z) in (20) we get
(56)X(x+w2(x,r,z),z)=X(x+v1(x,r),0)+v(X(x+v1(x,r),0),R(x+w2(x,r,z),z)).
Using (30) we obtain
(57)X(x+w2(x,r,z),z)=X(x,r)+v1(X(x,r),R(x,r))+v(X(x,r)+v1(X(x,r),R(x,r)),R(x+w2(x,r,z),z))=X(x,r)+w2(X(x,r),R(x,r),R(x+w2(x,r,z),z)).
Let us note that
(58)|w2(x,r,z)-w2(x,r,z′)|≤|z-z′|,(59)|w2(x,r,z)|≤|r-u(x)|+|z-r|+|r-u(x+v1(x,r))|≤4max⁡⁡{|r-u(x)|,|z-r|}.
Therefore w2∈𝒩2 and we have
(60)v1(x,r)+v(x+v1(x,r),r)=0.
It follows that H∘G=id.
Finally we conclude that the mappingH is a homeomorphism establishing a conjugacy of the noninvertible mappings (1) and (18).



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