Smooth linearization of nonautonomous difference equations with a nonuniform dichotomy

Abstract

There have been obtained many results on smooth linearization for diffeomorphisms, those results cannot be simply applied to nonautonomous difference equations. In this paper we investigate $C^1$ smooth linearization for nonautonomous difference equations with a nonuniform strong exponential dichotomy. Reducing the linear part of the nonautonomous system, defined by a sequence of invertible linear operators on $\mathbb {R}^d$, to a bounded linear operator on a Banach space, we discuss the spectrum and its spectral gaps. Then we obtain a gap condition for $C^1$ linearization of such a nonautonomous difference equation. We finally extend the result to the infinite dimensional case. Our theorems improve known results even in the case of uniform strong exponential dichotomies.

Appendix: Global smooth linearization

In the proof of Theorem 2 we need a result on global smooth linearization, which can be extended from the local $C^1$ linearization theorem given in [36].

Let $(X,\Vert \cdot \Vert )$ be a Banach space and let $F:X\rightarrow X$ be a $C^{1,1}$ diffeomorphism fixing the origin $\mathbf{0}$ and let $\mathbb {A}:=DF(\mathbf{0})$. Recall that F can be $C^1$ linearized if the functional Eq. (4.6) has a solution $\Phi$ which is a $C^1$ diffeomorphism. Moreover, assume that F satisfies

$$\begin{aligned} \Vert DF(x)-\mathbb {A}\Vert \le \eta , \quad \forall x\in X, \end{aligned}$$

(5.1)

where $\eta >0$ is a sufficiently small constant, and that the spectrum $\sigma (\mathbb {A})$ satisfy (3.1). Then, by the Spectral Decomposition Theory (see, e.g., [14, p. 9]) one can further assume that the space X has a direct sum decomposition $X=X_-\oplus X_+$ with $\mathbb {A}$-invariant subspaces $X_-$ and $X_+$, that is,

$$\begin{aligned} \mathbb {A}=\mathrm{diag}(\mathbb {A}_-, \mathbb {A}_+), \end{aligned}$$

(5.2)

where $\mathbb {A}_-:X_-\rightarrow X_-$ and $\mathbb {A}_+:X_+\rightarrow X_+$ are both bounded linear operators such that

$$\begin{aligned} \sigma (\mathbb A_-)&=\sigma _-:=\bigcup _{i=1}^k \{ z\in \mathbb C: a_i \le |z|\le b_i<1\},\\ \quad \sigma (\mathbb A_+)&=\sigma _+:=\bigcup _{j=k+1}^r \{ z\in \mathbb C: 1<a_j \le |z|\le b_j \}. \end{aligned}$$

We have the following result.

Global smooth linearization theoremLetFand$\mathbb {A}$be given above and assume that the numbers$a_i$and$b_i$given in (3.1) satisfy (4.1). Then there exists a$C^1$diffeomorphism$\Phi : X\rightarrow X$such that Eq. (4.6) holds, i.e.,Fcan be$C^1$linearized inX.

Proof

First of all, we give some notations. Let $C^0_b(\Omega ,Z_2)$ consist of all $C^0$ maps h from $\Omega$, an open subset of a Banach space $(Z_1,\Vert \cdot \Vert )$, into another Banach space $(Z_2,\Vert \cdot \Vert )$ such that $\sup _{z\in \Omega }\Vert h(z)\Vert <\infty$. Clearly, $C^0_b(\Omega ,Z_2)$ is a Banach space equipped with the supremum norm $\Vert \cdot \Vert _{C^0_b(\Omega ,Z_2)}$ defined by

$$\begin{aligned} \Vert h\Vert _{C^0_b(\Omega ,Z_2)}:=\sup _{z\in \Omega }\Vert h(z)\Vert , \quad \forall h\in C^0_b(\Omega ,Z_2). \end{aligned}$$

For a constant $\gamma >0$, let $S_\gamma (\Omega ,Z_2)$ consist of all sequences $u:=(u_n)_{n\ge 0}\subset C^0_b(\Omega ,Z_2)$ such that $\sup _{n\ge 0}\{\gamma ^{-n}\Vert u_n\Vert _{C^0_b(\Omega ,Z_2)}\}<\infty .$ Then, $S_\gamma (\Omega ,Z_2)$ is a Banach space equipped with the wighted norm $\Vert \cdot \Vert _{S_{\gamma }(\Omega ,Z_2)}$ defined by

$$\begin{aligned} \Vert u\Vert _{S_{\gamma }(\Omega ,Z_2)}:=\sup _{n\ge 0}\left\{ \gamma ^{-n}\Vert u_n\Vert _{C^0_b(\Omega ,Z_2)}\right\} , \quad \forall u\in S_{\gamma }(\Omega ,Z_2). \end{aligned}$$

$\square$

Let $f:=F-\mathbb {A}$ be the nonlinear term of F and let $\pi _-$ and $\pi _+$ be projections onto $X_-$ and $X_+$ respectively.

Our strategy is firstly to decouple F into a contraction and an expansion by straightening up the invariant foliations. In order to construct the (stable) invariant foliation, we need to study the Lyapunov–Perron equation (cf. [11])

$$\begin{aligned}&q_n(x,y_-)\nonumber \\&\quad =\mathbb {A}_-^n(y_--\pi _- x)+\sum _{m=0}^{n-1}\mathbb {A}_-^{n-m-1}\{\pi _- f(q_m(x,y_-)+F^m(x))-\pi _- f(F^m(x))\} \nonumber \\&\qquad -\sum _{m=n}^{\infty }\mathbb {A}_+^{n-m-1}\left\{ \pi _+ f(q_m(x,y_-)+F^m(x))-\pi _+ f(F^m(x))\right\} , \quad \forall n\ge 0, \end{aligned}$$

(5.3)

where $q_n:X\times X_-\rightarrow X$ is unknown for every integer $n\ge 0$. For our purpose of $C^1$ linearization, we need to find a $C^1$ solution $(q_n)_{n\ge 0}$ of Eq. (5.3), i.e., each $q_n: X\times X_-\rightarrow X$ is $C^1$.

Lemma 4

Let F and $\mathbb {A}$ be given at the beginning of this section. Assume that the numbers $a_{k+1}$, $b_k$ and $b_r$ given in (3.1) satisfy

$$\begin{aligned} b_kb_r<a_{k+1}. \end{aligned}$$

(5.4)

Then, for every neighborhood $\Omega _d\subset \{(x,x_-)\in X\times X_-:\Vert (x,x_-)\Vert < d\}$ of $\mathbf{0}$ with a given constant $d> 0$, Eq. (5.3) has a unique solution

$$\begin{aligned} Q_d:=(q_n)_{n\ge 0}\in S_{\gamma _1}(\Omega _d,X) \end{aligned}$$

such that every $q_n:\Omega _d\rightarrow X$ ($n\ge 0$) is of class $C^1$, where $\gamma _1$ is a positive constant satisfying $b_k<\gamma _1<1.$

We leave the proof after we finish the proof of the theorem. Remind that for every $d>0$, we have obtained a solution $Q_d:=(q_n)_{n\ge 0}\in S_{\gamma _1}(\Omega _d,X)$ of Eq. (5.3). On the other hand, by [11, Theorem 2.1] we know that, for every point $(x,y_-)\in X\times X_-$, Eq. (5.3) has a unique solution $\tilde{Q}(x,y_-):=(\tilde{q}_n(x,y_-))_{n\ge 0}\subset X$ such that

$$\begin{aligned} \sup _{n\ge 0}\big \{\gamma _1^{-n}\Vert \tilde{q}_n(x,y_-)\Vert \big \}<\infty . \end{aligned}$$

By the uniqueness of $(\tilde{q}_n(x,y_-))_{n\ge 0}$ and the fact that $(q_n)_{n\ge 0}\in S_{\gamma _1}(\Omega _d,X)$, we have $\tilde{Q}|_{\Omega _d}=Q_d.$ It means that $\tilde{Q}$ is a global $C^1$ solution of Eq. (5.3). Hence the global (stable) invariant foliation can be constructed by

$$\begin{aligned} \mathcal {M}_s(x):=\{x+\tilde{q}_0(x,y_-):y_-\in X_-\}, \quad \forall x\in X. \end{aligned}$$

The unstable invariant foliation can be obtained by considering the inverse of F under the condition that

$$\begin{aligned} a_1a_{k+1}>b_k. \end{aligned}$$

Therefore, by [31, Theorem 3.1], there exists a homeomorphism $\Psi :X\rightarrow X$, which and its inverse $\Psi ^{-1}:X\rightarrow X$ are both $C^{1}$ such that

$$\begin{aligned} \Psi \circ F&=F_-\circ \pi _-\Psi +F_+\circ \pi _+\Psi , \end{aligned}$$

where $F_-: X_-\rightarrow X_-$ and $F_+:X_+\rightarrow X_+$ are both $C^{1,1}$ diffeomorphisms such that $DF_-(\mathbf{0})=\mathbb {A}_-$ and $DF_+(\mathbf{0})=\mathbb {A}_+$. Recall that $\mathbb {A}_-$ and $\mathbb {A}_+$ are given in (5.2) and have the spectra $\sigma (\mathbb {A}_-)=\sigma _-$ and $\sigma (\mathbb {A}_+)=\sigma _+$ respectively. Then we have the following result.

Lemma 5

Let $F_-$ and $F_+$ be given above. Assume that the numbers $a_i$ and $b_i$ given in (3.1) satisfy

$$\begin{aligned} b_i/a_i<b_k^{-1}, \quad \forall i=1, \ldots , k, \quad b_j/a_j < a_{k+1}, \quad \forall j=k+1, \ldots , r. \end{aligned}$$

Then there exist $C^1$ diffeomorphisms $\psi _-: X\rightarrow X$ and $\psi _+: X\rightarrow X$ that linearize $F_-$ and $F_+$ respectively.

Having found $\psi _-$ and $\psi _+$ in Lemma 5, we finally put

$$\begin{aligned} \Phi =(\psi _-\circ \pi _-+\psi _+\circ \pi _+)\circ \Psi ,\quad \Phi ^{-1}=\Psi ^{-1}\circ (\psi _-^{-1}\circ \pi _-+\psi _+^{-1}\circ \pi _+). \end{aligned}$$

One verifies that $\Phi :X\rightarrow X$ is a $C^1$ diffeomorphism that linearizes F and the proof of the theorem is completed. $\square$

Proof of Lemma 4

Let

$$\begin{aligned} \Vert x\Vert :=\Vert x_-\Vert +\Vert x_+\Vert , \quad \Vert (x,y_-)\Vert :=\Vert x\Vert +\Vert y_-\Vert \end{aligned}$$

for $x=x_-+x_+\in X$ and $y_-\in X_-$. Choose two positive numbers $\gamma _1$ and $\gamma _2$ such that

$$\begin{aligned} b_k<\gamma _1<1<\gamma _2<a_{k+1} \quad \mathrm{and}\quad \gamma _1b_r<\gamma _2, \end{aligned}$$

which is possible because of (5.4). Let

$$\begin{aligned} E_1:=S_{\gamma _1}(\Omega _d,X) \quad \text{ and } \quad E_2:=S_{\gamma _2}(\Omega _d,\mathcal {L}(X\times X_-,X)) \end{aligned}$$

for short, where $\mathcal {L}(X\times X_-,X)$ is the set of all bounded linear operators mapping $X\times X_-$ into X. As mentioned at the beginning of the above proof for the theorem, we understand that $E_1$ and $E_2$ are both Banach spaces equipped the corresponding norms, denoted by $\Vert \cdot \Vert _{E_1}$ and $\Vert \cdot \Vert _{E_2}$ respectively. Define operators $\mathcal {T}: E_1\rightarrow E_1$ and $\mathcal {S}: E_1\times E_2\rightarrow E_2$ by

$$\begin{aligned}&(\mathcal {T} v)_n(x,y_-):=\mathbb {A}_-^n(y_--\pi _- x) \nonumber \\&\quad +\sum _{k=0}^{n-1}\mathbb {A}_-^{n-k-1}\left\{ \pi _- f(v_k(x,y_-)+F^k(x))-\pi _- f(F^k(x))\right\} \nonumber \\&\quad -\sum _{k=n}^{\infty }\mathbb {A}_+^{n-k-1}\left\{ \pi _+ f(v_k(x,y_-)+F^k(x))-\pi _+ f(F^k(x))\right\} \end{aligned}$$

(5.5)

and

$$\begin{aligned}&\mathcal {S}(v,w)_n(x,y_-) :=\Big (\mathrm{diag}(0,-\mathbb {A}_-^n),\mathbb {A}_-^n\Big ) \nonumber \\&\quad +\sum _{k=0}^{n-1}\mathbb {A}_-^{n-k-1}\{D(\pi _- f)(v_k(x,y_-)+F^k(x))(w_k(x,y_-)+DF^k(x)) \nonumber \\&\quad -D(\pi _- f)(F^k(x))DF^k(x)\} \nonumber \\&\quad -\sum _{k=n}^{\infty }\mathbb {A}_+^{n-k-1}\{D(\pi _+ f)(v_k(x,y_-)+F^k(x))(w_k(x,y_-)+DF^k(x)) \nonumber \\&\quad -D(\pi _+ f)(F^k(x))DF^k(x)\} \end{aligned}$$

(5.6)

respectively for all $v:=(v_n)_{n\ge 0}\in E_1$ and all $w:=(w_n)_{n\ge 0}\in E_2$. We claim the following:

• (A1) The operators $\mathcal {T}$ and $\mathcal {S}$ are well defined.

• (A2) The operator $\mathcal {Q}: E_1\times E_2\rightarrow E_1\times E_2$ defined by

  $$\begin{aligned} \mathcal {Q}(v,w):=(\mathcal {T}v, \mathcal {S}(v,w)), \quad \forall (v,w)\in E_1\times E_2, \end{aligned}$$

  (5.7)

  has an attracting fixed point $(v_*,w_*)\in E_1\times E_2$, i.e.,

  $$\begin{aligned} \lim _{n\rightarrow \infty }\mathcal {Q}^n(v,w)=(v_*,w_*), \quad \forall (v,w)\in E_1\times E_2, \end{aligned}$$

  (5.8)

  where $v_*\in E_1$ is the fixed point of $\mathcal {T}$ and $w_*\in E_2$ is the fixed point of $\mathcal {S}(v_*,\cdot )$.

Notice that in [36] the same claims were proved in neighborhoods having sufficiently small diameters $d>0$. The only difference in the present version is that we allow the diameter to be arbitrarily large. This causes a little change in the estimates, i.e., changing $\Vert (x,x_-)\Vert \le 1$ into $\Vert (x,x_-)\Vert \le d$. Thus, we obtain the following inequalities from [36]:

$$\begin{aligned}&\gamma _1^{-n}\Vert (\mathcal {T} v)_n(x,y_-)\Vert \le d+K\eta \Vert v\Vert _{E_1},\\&\gamma _2^{-n}\Vert \mathcal {S}(v, w)_n(x,y_-)\Vert \le 1+K_1\Vert v\Vert _{E_1}+K_2\eta \Vert w\Vert _{E_2},\\&\gamma _1^{-n}\Vert (\mathcal {T} v)_n(x,y_-)-(\mathcal {T} \tilde{v})_n(x,y_-)\Vert \le K\eta \,\Vert v-\tilde{v}\Vert _{E_1},\\&\gamma _2^{-n}\Vert \mathcal {S}(v, w)_n(x,y_-)-\mathcal {S}(\tilde{v}, \tilde{w})_n(x,y_-)\Vert \\&\quad \le (K_1\Vert w\Vert _{E_2}+K_2)\Vert v-\tilde{v}\Vert _{E_1}+K\eta \Vert w-\tilde{w}\Vert _{E_2}, \end{aligned}$$

where $K_1,K_2$ and K are positive constants. The first two inequalities indicate that $\mathcal {T}: E_1\rightarrow E_1$ and $\mathcal {S}:E_1\times E_2\rightarrow E_2$ are well defined, i.e., (A1) holds. The third one means that $\mathcal {T}$ is a contraction since $\eta >0$ is small and therefore has a fixed point $v_*\in E_1$. Moreover, setting $v=\tilde{v}=v_*$ in the last inequality, we see that $\mathcal {S}(v_*,\cdot ):E_2\rightarrow E_2$ is also a contraction and therefore (A2) is proved by the Fiber Contraction Theorem (see e.g. [17] or [12, p. 111]).

Having (A1) and (A2), we choose an initial point $\tilde{v}:=0\in E_1$, whose derivative satisfies $D\tilde{v}=0\in E_2$. Moreover, it is obvious that each $(\mathcal {T}\tilde{v})_n$ is $C^1$ due to (5.5). Then, by the definitions (5.5)–(5.7) of $\mathcal {T}, \mathcal {S}$ and $\mathcal {Q}$, one checks that $\mathcal {Q}(\tilde{v}, D\tilde{v})=(\mathcal {T}\tilde{v}, D(\mathcal {T}\tilde{v})),$ where $D(\mathcal {T}\tilde{v}):=(D(\mathcal {T}\tilde{v}_n))_{n\ge 0}$. This enables us to prove inductively that

$$\begin{aligned} \mathcal {Q}^n(\tilde{v}, D\tilde{v})=(\mathcal {T}^n\tilde{v}, D(\mathcal {T}^n\tilde{v})), \quad \forall n\ge 0. \end{aligned}$$

(5.9)

Combining (5.8) with (5.9) we get $\lim _{n\rightarrow \infty }\mathcal {T}^n\tilde{v}=v_*$ and $\lim _{n\rightarrow \infty }D(\mathcal {T}^n\tilde{v})=w_*$, which implies that $v_*\in E_1$ such that $dv_*=w_*\in E_2$. Since $v_*$ is the fixed point of $\mathcal {T}$, it is a solution of the Lyapunov–Perron Eq. (5.3). Thus, the lemma is proved. $\square$

Proof of Lemma 5

According to [36, Lemma 10] we know that such $\psi _-$ and $\psi _+$ exist in a small neighborhood U of $\mathbf{0}$. We now extend them into the whole space X. In what follows, we only consider the expansion $F_+$ since the case of contraction $F_-$ can be solved by considering its inverse. Choose a sphere $U_0\in U$ such that $U_0\subset \mathrm{int}\, F_+(U_0)\subset U,$ where $\mathrm{int}$ denotes the interior of the set $F_+(U_0)$, and define

$$\begin{aligned} X_i:=F_+^{i+1}(U_0)\backslash F_+^{i}(U_0),\quad V_0:=\psi _+(U_0),\quad Y_i:=\mathbb {A}_+^{i+1} (V_0)\backslash \mathbb {A}_+^i(V_0) \end{aligned}$$

for all $i\in \mathbb {N}\cup \{0\}$. It is clear that

$$\begin{aligned} X_i\cap X_j=\emptyset , \quad \forall i\ne j, \quad X_i\cap U_0=\emptyset , \quad&U_0\cup \bigcup _{i=0}^{\infty }X_i=X, \quad F_+(X_i)=X_{i+1},\\ Y_i\cap Y_j=\emptyset , \quad \forall i\ne j, \quad Y_i\cap V_0=\emptyset , \quad&V_0\cup \bigcup _{i=0}^{\infty }Y_i=X, \quad \mathbb {A}_+(Y_i)=Y_{i+1}. \end{aligned}$$

Then we define

$$\begin{aligned} \psi _0:=\psi _+|_{X_0},\quad \psi _i:=\mathbb {A}_+\circ \psi _{i-1}\circ F_+^{-1},\quad \forall i\in \mathbb {N}, \end{aligned}$$

and define the global solution $\psi _*$ by

$$\begin{aligned} \psi _*(x):= \left\{ \begin{array}{lll} \psi _+(x), &{} \quad \forall x\in U_0, \\ \psi _i(x), &{} \quad \forall x\in X_i, \quad \forall i\in \mathbb {N}, \end{array}\right. \end{aligned}$$

which is $C^1$ in X. Moreover, one verifies that $\psi _*(X_i)=Y_i$ for all $i\in \mathbb {N}\cup \{0\}$ because

$$\begin{aligned} \psi _*\circ F_+^{i+1}(U_0)= \mathbb {A}_+^{i+1}\circ \psi _*(U_0)= \mathbb {A}_+^{i+1}(V_0), \quad \psi _*\circ F_+^{i}(U_0)= \mathbb {A}_+^{i}(V_0). \end{aligned}$$

Thus, $\psi _*:X\rightarrow X$ is one-to-one and therefore it is a $C^1$ diffeomorphism that linearizes $F_+$. Without loss of generality, we still use $\psi _+$ to denote $\psi _*$ and the proof is completed. $\square$