IJDE International Journal of Differential Equations 1687-9651 1687-9643 Hindawi Publishing Corporation 703868 10.1155/2014/703868 703868 Research Article Conjugacy of a Discrete Semidynamical System in a Neighbourhood of the Nontrivial Invariant Manifold Reinfelds Andrejs 1, 2 Candan Tuncay 1 Institute of Mathematics and Computer Science University of Latvia Raiņa bulvāris 29, Rīga 1459 Latvia lu.lv 2 Faculty of Physics and Mathematics University of Latvia Zeļļu iela 8, Rīga 1002 Latvia lu.lv 2014 2522014 2014 21 11 2013 15 01 2014 25 2 2014 2014 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.

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 . For noninvertible mappings in a complete metric space it was extended and generalized by Reinfelds . 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 E and F be Banach spaces, B(a)={rF|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:EE 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|),supxA(x)+2ɛ<1,supx|Φ(x,0)|a(1-supxA(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)(supxA(x)+4ɛ+γsupx|Φ(x,0)|1-supxA(x)-ɛ)×supxDg(x)-11 then there exists a continuous mapping u:EF satisfying the following properties:

u(g(x)+Ψ(x,u(x)))=A(x)u(x)+Φ(x,u(x));

|u(x)-u(x)||x-x|;

usupx  |Φ(x,0)|/(1-supxA(x)-ɛ).

Proof.

The set of continuous mappings u:EF,(4)𝒦={uC(E,F)supx|u(x)|<+} equipped with the norm (5)u=supx|u(x)| is a Banach space. The set (6)𝒦1={u𝒦ua,|u(x)-u(x)||x-x|} is a closed subset of the Banach space 𝒦.

Let us consider the mapping uu, u𝒦1 defined by the equality (7)u(x1)=A(x)u(x)+Φ(x,u(x)), where (8)x1=g(x)+Ψ(x,u(x)). If u𝒦1 then (9)u(supxA(x)+ɛ)u+supx|Φ(x,0)|a. We have (10)|u(x1)-u(x1)|(supxA(x)+ɛ)u-u+(supxA(x)+2ɛ+γu)|x-x|,|x-x|supxDg(x)-1|g(x)-g(x)|=supxDg(x)-1|x1-x1-Ψ(x,u(x))+Ψ(x,u(x))|supxDg(x)-1(|x1-x1|+2ε|x-x|+εu-u). It follows (11)|x-x|supxDg(x)-11-2ɛsupxDg(x)-1|x1-x1|+ɛsupxDg(x)-11-2ɛsupxDg(x)-1u-u. Then (12)|u(x1)-u(x1)|((supxA(x)+2ɛ+γu)supxDg(x)-1ɛ1-2ɛsupxDg(x)-1supxA(x)+ɛ+(supxA(x)+2ɛ+γu)supxDg(x)-1ɛ1-2ɛsupxDg(x)-1)×u-u+(supxA(x)+2ɛ+γu)supxDg(x)-11-2ɛsupxDg(x)-1×|x1-x1|. Let us note that (13)(supxA(x)+2ɛ+γu)supxDg(x)-11-2ɛsupxDg(x)-11. We obtain (14)|u(x1)-u(x1)|(supxA(x)+2ɛ)u-u+|x1-x1|.

We get that 𝒦1 is contraction and consequently we have the invariant manifold r=u(x).

3. Conjugacy of Noninvertible Mappings Definition 2.

Two mappings S,T:XX are conjugate, if there exists a homeomorphism H:XX such that (15)SH(x)=HT(x).

Definition 3.

Two discrete semidynamical systems Sn,Tn:XX(n) are conjugate, if there exists a homeomorphism H:XX such that (16)SnH(x)=HTn(x).

It is easily verified that two discrete semidynamical systems Sn 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:EF such that (17)supxu(x)δ,|u(x)-u(x)||x-x|. Our aim is to find a simpler mapping conjugated with (1).

Theorem 4.

If  supx((Dg(x))-1A(x))+5ɛsupx(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 1.

Proof.

The set of continuous mappings v:E×B(δ)E,(22)={vC(E×B(δ),E)supx,r|v(x,r)||r-u(x)|<+} becomes a Banach space if we use the norm v=supx,r(|v(x,r)|/|r-u(x)|). The set (23)1={vv1,|v(x,r)-v(x,r)||r-r|} is a closed subset of the Banach space .

Let us consider the mapping vv, v1 defined by the equality (24)v(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|vv(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|}=δa, such that (29)supx((Dg(x))-1A(x))+(5ɛ+ω(8δ)+4γδ)supx(Dg(x))-1<1. Then v1, |v(x,r)-v(x,r)||r-r|, the mapping is a contraction, and consequently the functional equation (20) has unique solution in 1.

Next we will prove that the mapping H 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 2.

Proof.

The set (32)2={vv1} is a closed subset of the Banach space .

Let us consider the mapping v1v1, v12 defined by the equality (33)v1(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×|v1Ψ(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 is a contraction and consequently the functional equation (30) has a unique solution in 2.

Consider the mapping G defined by equality G(x,r)=(x+v1(x,r),r).

Lemma 7.

One has GH=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 3, where (38)3={v2v23} is a closed subset of the Banach space . We get (39)|v2(x,r)|(Dg(x))-1v2|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 xx+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 w13 and we have (44)v(x,r)+v1(x+v(x,r),r)=0. We obtain that GH=id.

Lemma 8.

One has HG=id.

Proof.

The set of continuous mappings v3:E×B(δ)×B(δ)E,(45)𝒩={|v3(x,r,z)|max(|r-u(x)|,|z-r|)v3C(E×B(δ)×B(δ),E)xxxxxsupx,r,z|v3(x,r,z)|max(|r-u(x)|,|z-r|)<} becomes a Banach space if we use the norm v3=supx,r,z(|v3(x,r,z)|/max  (|r-u(x)|,|z-r|)). The set (46)𝒩1={zv3𝒩v31,  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 v3v3, v3𝒩1 defined by the equality (49)v3(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))-1max{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 v3𝒩1 and v3𝒩1𝒩2 where (52)𝒩2={v3𝒩supx,|r|δ,|z|δ|v3(x,r,z)|8δ,xxxxx|v3(x,r,z)-v3(x,r,z)||z-z|supx,|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×|v3v3(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-v3max{|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-v3max{|r-u(x)|,|z-r|}. Then v31, |v3(x,r,z)-v3(x,r,z)||z-z|, the mapping 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 mapping w2, 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 w2𝒩2 and we have (60)v1(x,r)+v(x+v1(x,r),r)=0. It follows that HG=id.

Finally we conclude that the mapping H 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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