1. Introduction
The earliest contributions devoted to linearization can be traced back to Poincaré’s work on complex analytic conjugacy in 1890. Given an analytic diffeomorphism F near a fixed point, Poincaré [Reference Poincaré25] proved that F can be analytically conjugated to its linear part 
$A:=DF(O)$ if A is contractive (i.e., the real parts of all eigenvalues 
$\lambda_1, \ldots, \lambda_p$ of A are less than 1) and satisfies nonresonance conditions of any orders (i.e., 
$\lambda_i \ne \lambda^m$ for all 
$i\in\{1,...,p\},$ where 
$\lambda=(\lambda_1,...,\lambda_p)$, and 
$m=(m_1,...,m_p)$ is a nonnegative integer vector satisfying 
$|m|=\sum_{i=1}^{p} m_i \ge 2$).
Later, Siegel [Reference Siegel29], Brjuno [Reference Brjuno3], and Yoccoz [Reference Yoccoz36] contributed to the case of the eigenvalues on the unit circle. Sternberg [Reference Sternberg30, Reference Sternberg31] initially investigated C k linearization for C n 
$(1\leq k \leq n \leq \infty)$ diffeomorphisms on 
$\mathbb R^n$. Additional conditions for smooth linearization were obtained by Belitskii [Reference Belickii2], Elbialy [Reference ElBialy9], Lu et al. [Reference Lu, Zhang and Zhang18], Rodrigues and Solà-Morales [Reference Rodrigues28], Castañeda and Robledo [Reference Castañeda and Robledo4], Dragičević et al. [Reference Dragičević, Zhang and Zhang8], and Zhang et al. [Reference Zhang, Zhang and Jarczyk37–Reference Zhang and Zhang39] (see also references therein). Observe that all these linearization results involving smoothness of conjugacies require some form of nonresonant or spectral gap conditions.
Without these conditions, the well-known linearization result is the Hartman–Grobman theorem (see [Reference Grobman11, Reference Hartman12]), which answers whether a continuous system or discrete system can be locally linearized by a C 0 conjugacy. For the infinite-dimensional system, Lu [Reference Lu17] presented a classical Hartman–Grobman theorem for the scalar reaction–diffusion equations. By using the methods in Moser’s proof in [Reference Moser20] of the structural stability of Anosov diffeomorphisms, Pugh [Reference Pugh26] gave a global version of Hartman–Grobman theorem on arbitrary Banach spaces. Moreover, the global Hartman–Grobman theorem has been extended by Palmer [Reference Palmer23], Tan [Reference Tan32], Jiang [Reference Jiang15], Hein and Prüss [Reference Hein and Prüss13], Reinfelds and Šteinberga [Reference Reinfelds27], Backes et al. [Reference Backes, Dragičević and Palmer1], Xia et al. [Reference Xia, Wang, Kou and O’Regan35], and others.
We remark that the global C 0 linearization works mentioned above all require that the nonlinear term is globally bounded. A natural question is what happens when the nonlinear term is not necessarily globally bounded. Usually, considering unbounded perturbations can destroy the structure and stability of the system. Hence, it is natural to ask whether unbounded perturbations destroy the topological conjugacy? The answer is negative for systems with uniformly contractive linear part [Reference Lin16]. Recently, Huerta [Reference Huerta14] established a topological conjugacy between a nonlinear system and its linear part by using the crossing time defined by the Lyapunov functions associated with a nonuniformly contractive system, without requiring that the nonlinear part is bounded. The corresponding discrete version is given by Castañeda and Robledo [Reference Castañeda and Robledo5]. Nevertheless, these results are limited to the case when the linear part is contractive. Thus, it is natural to ask whether it is possible to construct conjugacies between a hyperbolic linear system and a nonlinear system obtained by perturbing the linear system by (not necessarily bounded) nonlinearity? Recently, Wu and Xia [Reference Wu and Xia34] attempted to solve this problem for a coupled hyperbolic system. They employed a spectral approach to make a moderate progress by assuming partially unbounded nonlinear perturbations, but failed to solve this problem completely. Moreover, in this situation, the difficulty arises from the fact that the results on the semiflow given by [Reference Huerta14] cannot be extended to a flow.
In the present paper, we investigate a global linearization of autonomous semilinear ordinary differential equations whose linear part is hyperbolic and the nonlinear term is not necessarily bounded. We first decouple a nonlinear hyperbolic system into a contractive system on a stable manifold and an expansive system on an unstable manifold, which is based on a splitting lemma given by [Reference Castañeda and Rosa6, Theorem 4.12]. Then, inspired by [Reference Palmer24, Section 4], we can construct an appropriate transformation to linearize a contractive/expansive system, which is defined by the crossing time with respect to the unit sphere. On the other hand, it is difficult to discuss the regularity of conjugacies in [Reference Castañeda and Robledo5, Reference Huerta14] due to the fact that the construction of conjugacies in [Reference Castañeda and Robledo5, Reference Huerta14] is implicit. However, in our result, we show the Hölder continuity of the conjugacy at the fixed point (see (2.4)).
The rest of this paper is organized as follows. In §2 we state our main result. In §3, we give a decoupling lemma and then prove our main result. In §4, the linearization result for a contractive system on the stable manifold is established. Finally in §5, in order to demonstrate the strength of our result, we present an application to a nonlinear Duffing oscillator without external excitation.
2. Main result
In this section, we state our main result. Consider a nonlinear differential equation on a Banach space 
$X=(X,\|\cdot\|)$ over 
$\mathbb{C}$,
\begin{align}
x'(t)=Ax(t)+f(x(t)), \quad t\in \mathbb{R},
\end{align} where A is a bounded linear operator on X and 
$f:X\to X$ is a continuous mapping with 
$f(O)=O$, where 
$O\in X$ denotes the origin. Provided that f is globally Lipschitz, as in the case when X is finite-dimensional, we have the existence and uniqueness of a global solution of Eq. (2.1) with the prescribed initial condition (see [Reference Deimling7, Section 1] for details). Suppose that O is a hyperbolic fixed point for the linear part of Eq. (2.1), i.e., that the spectrum 
$\sigma(A)$ of A does not intersect the imaginary axis. Then the main result of this paper reads as follows.
Theorem 2.1. Suppose that 
$f: X\to X$ is such that 
$f(O)=O$ and with the property that there exists 
$L_f \gt 0$ such that
\begin{align}
\|f(x)-f(\tilde{x})\|\leq L_f \|x-\tilde{x}\|, \quad \forall x,\tilde{x}\in X.
\end{align} Then, provided that Lf is sufficiently small, there exists a homeomorphism 
$H: X\to X$ which conjugates Eq. (2.1) topologically to its linear part
\begin{align}
x'(t)=Ax(t), \quad t\in\mathbb{R}.
\end{align}More precisely,
\begin{equation*}
H \circ S(t)=e^{At }\circ H, \quad t\in \mathbb R,
\end{equation*} where 
$S(t):X\to X$ is the nonlinear solution operator of (2.1). Moreover, H and its inverse H −1 satisfy
\begin{align}
H(x)=\mathcal{O}(\|x\|^\varrho) \quad \mathrm{and} \quad H^{-1}(x)=\mathcal{O}(\|x\|^\varrho), \quad \mathrm{as} \; x\to O,
\end{align} for some 
$\varrho \in (0,1)$.
Remark 2.2. We point out that the global Hartman–Grobman theorem on Banach spaces was established independently by Pugh [Reference Pugh26] and Palis [Reference Palis22], under an additional assumption that f is bounded, i.e., 
$\sup_{x\in X}\|f(x)\| \lt +\infty$. Huerta [Reference Huerta14] and Castañeda and Robledo [Reference Castañeda and Robledo5] established a C 0-conjugacy between Eq. (2.1) and its linear part, respectively, when f is not necessarily bounded. However, these results are limited to the case when the linear part is exponentially stable, sacrificing the hyperbolicity. Recently, Wu and Xia [Reference Wu and Xia34] addressed this problem for a coupled hyperbolic system by considering partially unbounded nonlinear perturbations, but failed to provide a complete solution due to the difficulties mentioned in the introduction. In this respect, theorem 2.1 firstly formulates a global linearization result on the flow of hyperbolic system where f is not necessarily bounded. On the other hand, we also note that it is difficult to discuss the regularity of conjugacies in [Reference Castañeda and Robledo5, Reference Huerta14] due to the fact that the conjugacies in these works are constructed in an implicit manner. On the other hand, our linearization result gives the Hölder continuity of the conjugacy in a fixed point O (see (2.4)).
Remark 2.3. The proof of theorem 2.1 is partially inspired by the recent work of Dragičević–Zhang–Zhang [Reference Dragičević, Zhang and Zhang8]. Dragičević et al. resolved the open problem proposed by van Strein in [Reference van Strien33]; they demonstrated that the linearization of a 
$C^{1,1}$ hyperbolic system can be locally differentiable and Hölder continuous. Whereas we provide a global linearization result for 
$C^{0,1}$ differential equations, complementing Pugh’s result [Reference Pugh26].
Example 2.4. Notice that theorem 2.1 gives that the following systems on 
$\mathbb R^2$ are topologically conjugated
\begin{equation*}
\begin{cases}
x'=-a x+ f_1(x , y), \\
y'=b y+ f_2(x, y),
\end{cases}
\quad \mathrm{and} \quad
\begin{cases}
x'=-a x, \\
y'=b y,
\end{cases}
\end{equation*} provided that the Lipschitz constant Lf of 
$f=(f_1, f_2)$ is sufficiently small.
3. Proof of our main result
In this section, we give the proof of our main result.
Proof of theorem 2.1
Since A is a hyperbolic linear operator, the spectrum of A is given by 
$\sigma(A):= \sigma_s \cup \sigma_u$ where
\begin{align}
\sigma_s:=\{\varrho \in\sigma(A): \mathrm{Re}\, \varrho \le -\lambda\} \quad and \quad
\sigma_u:=\{\varrho \in\sigma(A): \lambda \le \mathrm{Re}\, \varrho \},
\end{align} and λ > 0 is a constant. It follows from the Spectral Decomposition Theorem (see [Reference Gohberg, Goldberg and Kaashoek10, p. 9]) that the space X has a direct sum decomposition 
$X:=X_s \oplus X_u$ with A-invariant closed subspaces Xs and Xu, where 
$A_s:=A\rvert_{X_s}\colon X_{s}\to X_{s}$ and 
$A_u:=A\rvert_{X_u}\colon X_{u} \to X_{u}$ are bounded linear operators such that 
$\sigma(A_s)=\sigma_s$ and 
$\sigma(A_u)=\sigma_u$, respectively. Then by (3.1), we have that
\begin{align}
\begin{cases}
\|e^{A_s(t-s)}\|\leq K e^{-\lambda (t-s)}, \quad \forall t\geq s,\\
\|e^{A_u(t-s)}\|\leq K e^{\lambda (t-s)}, \quad \forall t\leq s,
\end{cases}
\end{align} where 
$K\ge 1$ is a constant. We note without any loss of generality we may assume that (3.2) holds with K = 1 by passing to a norm which is equivalent to 
$\| \cdot \|$. Indeed, let 
$\pi_s\colon X\to X_s$ and 
$\pi_u \colon X\to X_u$ be the projections associated with the splitting 
$X=X_s\oplus X_u$. For 
$x\in X$, we set
\begin{equation*}
\|x\|_{*}= \max \left \{\sup_{t\ge 0} \left( e^{\lambda t}\|e^{A_s t}\pi_s x\| \right), \ \sup_{t\le 0} \left( e^{-\lambda t}\|e^{A_u t}\pi_u x\| \right) \right \},
\end{equation*} where λ > 0 is such that (3.1) holds. Note that (3.2) implies that 
$\|x\|_* \le K \max \{\|\pi_s\|, \|\pi_u\| \}\|x\|$ for 
$x\in X$. Moreover,
\begin{equation*}
\|x\|_* \ge \max \{\| \pi_s x\|, \|\pi_u x\| \} \ge \frac 12 (\|\pi_s x\|+\|\pi_u x\|)\ge \frac 1 2 \|x\|.
\end{equation*} In addition, for 
$x_s\in X_s$ and 
$t\ge 0$ we have
\begin{equation*}
\|e^{A_s t}x_s\|_* =\sup_{r\ge 0} \left (e^{\lambda r}\|e^{A_s r} e^{A_s t}x_s\| \right )=e^{-\lambda t}\sup_{r\ge 0} \left (e^{\lambda (t+r)}\|e^{A_s (t+r)}x_s\| \right )\le e^{-\lambda t}\|x_s\|_*,
\end{equation*}yielding the first estimate in (3.2) with K = 1. Similarly, one can establish the second estimate in (3.2). Observe that
\begin{equation}
\|x\|_*=\max \{\|x_s\|_*, \|x_u\|_*\}, \quad \text{where}\ x_s=\pi_s x\ \text{and}\ x_u=\pi_u x.
\end{equation} To simplify the notation, we will assume that we passed to this new norm and 
$\|\cdot \|_*$ will be denoted by 
$\|\cdot \|$.
Given an initial value 
$x\in X$, the solution of (2.1) with 
$x(O)=x$ is given by 
$t\mapsto S(t)x$ where 
$S(t): X\to X$ satisfies
\begin{equation}
S(t)x=e^{At}x+\int_0^t e^{A(t-\tau)} f(S(\tau)x) \, d\tau, \quad \text{for}\ t\in\mathbb{R}\ \text{and}\ x\in X.
\end{equation}We note that the integral appearing in (3.4) is the Bochner integral, which represents an extension of the classical Lebesgue integral to functions that take values in a Banach space (see [Reference Mikusinki19]). Then we have the following result.
Lemma 3.1. Suppose that 
$f: X\to X $ is such that 
$f(O)=O$ and that (2.2) holds with 
$2L_f \lt \lambda$. Then, there exist Lipschitz maps 
$h_s\colon X_s \to X_u$ and 
$h_u\colon X_u \to X_s$ satisfying 
$h_s(O)=O$, 
$h_u(O)=O$, 
$\max\{L_{h_s}, L_{h_u}\} \le \frac{L_f}{\lambda-L_f}$, and with the property that Eq. (2.1) is conjugated to the system
\begin{equation}
\begin{cases}
x_s' =A_s x_s+\pi_s f(x_s+h_s(x_s)), \\
x_u' =A_ux_u+\pi_u f(h_u(x_u)+x_u).
\end{cases}
\end{equation} More precisely, there is a homeomorphism 
$\Phi \colon X\to X$ with 
$\Phi(O)=O$ satisfying
\begin{equation}
\Phi \circ S(t)=(S^s(t) \circ \pi_s+ S^u(t)\circ \pi_u)\circ \Phi \quad \forall t\in \mathbb R,
\end{equation}where
\begin{equation*}
S^s(t)x_s=e^{A_s t}x_s+\int_0^t e^{A_s(t-\tau)} \pi_s f( S^s(\tau)x_s+h_s(S^s(\tau)x_s))\, d\tau,
\end{equation*}and
\begin{equation*}
S^u(t)x_u=e^{A_u t}x_u+\int_0^t e^{A_u(t-\tau)} \pi_u f( S^u(\tau)x_u+h_u(S^u(\tau)x_u))\, d\tau.
\end{equation*} Moreover, there exists 
$\tilde{\varrho}\in (0,1)$ and for every bounded set 
$B\subset X$ a constant K > 0 such that
\begin{equation*}
\|\Phi(x_1)-\Phi(x_2)\| \le K\|x_1-x_2\|^{\tilde{\varrho}} \quad \text{and} \quad \|\Phi^{-1}(x_1)-\Phi^{-1}(x_2)\| \le K\|x_1-x_2\|^{\tilde{\varrho}},
\end{equation*} for 
$x_1, x_2\in B$ with 
$\|x_1-x_2\| \le 1$.
Remark 3.2. We notice that lemma 3.1 follows from [Reference Castañeda and Rosa6, Theorem 4.12] and [Reference Castañeda and Rosa6, Proposition 4.16] by considering 
$Z=X_s\times X_u$ (endowed with the norm of the maximum as in [Reference Castañeda and Rosa6]), which can be identified with 
$X=X_s\oplus X_u$ by the canonical isomorphism 
$(x_s, x_u)\to x_s+x_u$. Moreover, 
$C \colon Z\to Z$ and 
$H\colon Z\to Z$ in [Reference Castañeda and Rosa6, Eq. (2.8)] are given by
\begin{equation*}
C (x_s, x_u)=(A_s x_s, A_u x_u) \quad \text{and} \quad H(x_s, x_u)= (\pi_s f(x_s+x_u), \pi_u f(x_s+x_u)).
\end{equation*} Then, note that (3.2) gives [Reference Castañeda and Rosa6, Eq. (2.4)] with 
$\alpha=-\lambda $ and 
$\beta=\lambda$. In addition, observe that
\begin{equation*}
\begin{split}
\|H(x_s, x_u)-H(x_s', x_u')\| &=\max \{\|\pi_s (f(x_s+x_u)-f(x_s'+x_u'))\|,\\
& \quad \|\pi_u (f(x_s+x_u)-f(x_s'+x_u'))\|\} \\
&= \|f(x_s+x_u)-f(x_s'+x_u')\| \\
&\le L_f \|x_s+x_u-x_s'-x_u'\| \\
&\le 2L_f\|(x_s, x_u)-(x_s', x_u')\|,
\end{split}
\end{equation*} where we have taken into account (3.3). Hence, H is Lipschitz and the Lipschitz constant LH satisfies 
$2L_H \lt 2\lambda=\beta-\alpha$. We conclude that [Reference Castañeda and Rosa6, Eq. (2.7)] is verified. Next, observe that Eq. (2.1) can be written as
\begin{equation}
(x_s'(t), x_u'(t))=C(x_s(t), x_u(t))+H(x_s(t), x_u(t)), \quad t\in \mathbb R.
\end{equation} Namely, if 
$x\colon \mathbb R\mapsto X$ is a solution of Eq. (2.1), then 
$t\mapsto (x_s(t), x_u(t)):=(\pi_s(x(t)), \pi_u (x(t)))$ is a solution of Eq. (3.7). Conversely, if 
$t \mapsto (x_s(t), x_u(t))$ is a solution of Eq. (3.7), then 
$t\mapsto x(t):=x_s(t)+x_u(t)$ is a solution of Eq. (2.1). In other words, the relationship between the solution operator S(t) of Eq. (2.1) and of the solution operator 
$\bar S(t)$ of Eq. (3.7) is given by
\begin{equation*}
\bar S(t) (x_s, x_u)=(\pi_s S(t)(x_s+x_u), \pi_u S(t)(x_s+x_u))
\end{equation*}and
\begin{equation*}
S(t)(x)=p_1 \bar S(t)(\pi_s x, \pi_u x)+p_2 \bar S(t)(\pi_s x, \pi_u x),
\end{equation*} where 
$p_1 \colon Z=X_s \times X_u\to X_s$ and 
$p_2\colon Z=X_s\times X_u \to X_u$ are projections on the first and the second coordinate, respectively. The existence of 
$h_s \colon X_s \to X_u$ and 
$h_u \colon X_u \to X_s$ with Lipschitz constants 
$\frac{L_f}{\lambda-L_f}$ and 
$h_s(O)=h_u(O)=O$, together with a homeomorphism 
$\bar{\Phi}\colon Z\to Z$ satisfying
\begin{equation*}
\bar \Phi \circ \bar S(t)=(S^s(t), S^u(t))\circ \bar \Phi \quad \text{for}\
t\in \mathbb R,
\end{equation*} follows from [Reference Castañeda and Rosa6, Theorem 4.12]. We now define 
$\Phi \colon X\to X$ by
\begin{equation*}
\Phi(x)=p_1 \bar \Phi(\pi_s x, \pi_u x)+p_2 \bar \Phi(\pi_sx,\pi_u x).
\end{equation*} It is straightforward to verify that Φ is a homeomorphism satisfying (3.6). The desired Hölder continuity of 
$\bar{\Phi}$ (and consequently of Φ) follows from [Reference Castañeda and Rosa6, Proposition 4.16]. Finally, we note that without any loss of generality we may suppose that the Hölder constant 
$\tilde{\varrho}$ belongs to 
$(0, 1)$ as for 
$0 \lt r_1\le r_2$ and 
$\|z\|\le 1$ we have 
$\|z\|^{r_2}\le \|z\|^{r_1}$.
By applying lemma 3.1, one only needs to linearize the contractive system in the stable manifold (i.e., the first equation of (3.5)), as the linearization of the second equation in (3.5) can be discussed similarly. In what follows, we consider the subsystem
\begin{align}
S^s(t)x_s=e^{A_s t}x_s+\int_0^t e^{A_s(t-\tau)} \pi_s f( S^s(\tau)x_s+h_s(S^s(\tau)x_s))\, d\tau, \quad \mathrm{for} \quad x_s\in X_s.\end{align} Note that Eq. (3.8) describes the flow of Eq. (2.1) on the stable invariant manifold 
$\mathcal{M}_s$ and 
$h_s:X_{s}\to X_{u}$ is Lipschitz continuous such that 
$h_s(O)=O$ and
\begin{align}
\|h_s(x_s)-h_s(\tilde{x}_s)\|\leq L_{h_s} \|x_s-\tilde{x}_s\|, \quad \forall x_s,\tilde{x}_s \in X_{s}.
\end{align}Then we have the following result.
Proposition 3.3. Suppose that 
$f: X\to X$ is continuous such that 
$f(O)=O$ and (2.2) hold. Then there exists a homeomorphism 
$\Psi_s:X_s\to X_s$ such that
\begin{align}
\Psi_s(S^s(t)x_s)=e^{A_s t} \Psi_s(x_s).
\end{align} Moreover, 
$\Psi_s$ and its inverse 
$\Psi_s^{-1}$ satisfy
\begin{align}
\Psi_s(x_s)=\mathcal{O}(\|x_s\|^\alpha) \quad \mathrm{and} \quad \Psi_s^{-1}(x_s)=\mathcal{O}(\|x_s\|^{\alpha}) \quad \mathrm{as} \quad x_s \to O,
\end{align} for some 
$\alpha\in (0,1)$.
We postpone the proof of proposition 3.3 to §4. Similarly to proposition 3.3, the following proposition can be obtained.
Proposition 3.4. Suppose that 
$f:X\to X$ is continuous such that 
$f(O)=O$ and (2.2) hold. Then there exists a homeomorphism 
$\Psi_u:X_u\to X_u$ such that
\begin{equation*}
\Psi_u(S^u(t)x_u)=e^{A_u t} \Psi_u(x_u).
\end{equation*} Moreover, 
$\Psi_u$ and its inverse 
$\Psi_u^{-1}$ satisfy
\begin{align*}
\Psi_u(x_u)=\mathcal{O}(\|x_u\|^\alpha) \quad \mathrm{and} \quad \Psi_u^{-1}(x_u)=\mathcal{O}(\|x_u\|^{\alpha}) \quad \mathrm{as} \quad x_u\to O.
\end{align*} We now proceed to prove theorem 2.1. From propositions 3.3 and 3.4, one knows that the homeomorphism 
$\Psi:X \to X$ defined by
\begin{equation*}
\Psi(x):= \Psi_s(\pi_s x)+\Psi_u(\pi_u x)
\end{equation*}satisfies
\begin{align*}
\Psi \circ (S^s(t)\circ \pi_s+S^u(t)\circ \pi_u)=&~ \Psi_s \circ S^s(t)\circ \pi_s+\Psi_u \circ S^u(t)\circ \pi_u
\\
=&~ e^{A_s t}\circ \Psi_s \circ \pi_s+e^{A_u t}\circ \Psi_u \circ \pi_u
\\
=&~ e^{A t}\circ \Psi_s \circ \pi_s+e^{A t}\circ \Psi_u \circ \pi_u
\\
=&~ e^{A t} \circ \Psi,
\end{align*} for 
$t\in \mathbb R$ as 
$e^{At}\rvert_{X_*}=e^{A_*t}$ for 
$*=s, u$. Moreover,
\begin{align*}
\Psi(x)=\mathcal{O}(\|x\|^\alpha) \quad \mathrm{and} \quad \Psi^{-1}(x)=\mathcal{O}(\|x\|^{\alpha}) \quad \mathrm{as} \quad x\to O.
\end{align*} This result together with lemma 3.1 enable us to define a homeomorphism 
$H:X\to X$ by
\begin{align}
H=\Psi\circ \Phi,
\end{align} where Φ is given by lemma 3.1. Therefore, the homeomorphism H satisfies (2.4) with 
$\varrho:=\alpha\tilde{\varrho}\in (0,1)$. This completes the proof of theorem 2.1.
4. Linearization on the stable manifold
In this section, we give a linearization result for a contractive system on the stable manifold (i.e., proposition 3.3). In order to simplify the notation, in what follows, we will drop the subscript s when denoting an element from Xs, and we write
\begin{equation*}
Y(t,y):=S^s(t)y \, \text{for all}\ t\in\mathbb{R}\ \text{and}\ y\in X_s.
\end{equation*} We note that 
$\{Y(t, \cdot): \ t\in \mathbb R\}$ is a group of (nonlinear) operators, i.e., 
$Y(0, \cdot)=id_{X_s}$ and
\begin{equation}Y(t+s, y)= Y(t, Y(s, y)) \quad \text{for}\ t, s\in \mathbb R\ \text{and}\ y\in X_s.
\end{equation}Now we are ready to prove the proposition.
Proof of proposition 3.3
Firstly, we give the estimate for 
$Y(t,y)$. Given 
$y\in X_s$, it follows from (2.2), (3.2), (3.8), and (3.9) that
\begin{align}
\|Y(t,y)\|\le&~ e^{-\lambda t}\|y\|+\int_0^t e^{-\lambda (t-\tau)}L_f (\|Y(\tau,y)\|+\|h_\tau (Y(\tau,y))\|)\, d\tau
\notag \\
\le&~ e^{-\lambda t}\|y\|+\int_0^t e^{-\lambda (t-\tau)}L_f (1+L_{h_s})\|Y(\tau,y)\|\, d\tau
\end{align} for 
$t\ge 0$. By Gronwall’s inequality, we get
\begin{align*}
\|Y(t,y)\| \le e^{-(\lambda -L_f (1+L_{h_s}))t}\|y\|, \quad t\ge 0, \ y\in X_s.
\end{align*}Set
\begin{equation*}
\mu:=\lambda -L_f (1+L_{h_s}).
\end{equation*}Provided that Lf is sufficiently small, we have that µ > 0. Then,
\begin{align}
\|Y(t,y)\| =\|Y(t-t_0, Y(t_0, y))\| \le e^{-\mu (t-t_0)}\|Y(t_0,y)\|, \quad t\ge t_0, \ y\in X_s,
\end{align} which means that for any given 
$y\ne O$, 
$t\mapsto \|Y(t,y)\|$ is a strictly decreasing function. Furthermore, we obtain from (4.3) that
\begin{equation*}
\lim_{t\to+\infty} \|Y(t,y)\|=0 \, \text{and} \, \lim_{t\to-\infty} \|Y(t,y)\|=+\infty.
\end{equation*}Indeed, the first equality above follows readily from (4.3), while for the second, it is sufficient to note that (4.3) gives (by replacing the roles of t and t 0) that
\begin{equation*}
e^{\mu (t_0-t)}\|Y(t_0, y)\| \le \|Y(t, y)\| \quad t\le t_0,
\end{equation*} yielding the desired claim. Hence, for each 
$y\ne O$, there is a unique time 
$\kappa(y)\in\mathbb{R}$ such that
\begin{align}
\|Y(\kappa(y),y)\|=1.
\end{align}Clearly, when y → O,
\begin{align}
\kappa(y)\to -\infty.
\end{align} Indeed, otherwise there would exist a sequence 
$(y_n)_n$ in 
$X_s\setminus \{O\}$ converging to O such that 
$\kappa(y_n)\nrightarrow -\infty$. Consequently, there exists a subsequence 
$(y_{n_k})_k$ of 
$(y_n)_n$ with the property that 
$\kappa(y_{n_k})\to L\in (-\infty, \infty]$. If L is finite, the continuity of 
$Y(\cdot, \cdot)$ gives that 
$Y(\kappa(y_{n_k}), y_{n_k})\to Y(L, O)=O$, which yields a contradiction since 
$\|Y(\kappa(y_{n_k}), y_{n_k})\|=1$ for each 
$k\in \mathbb N$. Similarly, if 
$L=\infty$ we have
\begin{equation*}
1=\|Y(\kappa(y_{n_k}), y_{n_k})\| \le e^{-\mu \kappa(y_{n_k})}\|y_{n_k}\| \quad \text{for all}\ k\in \mathbb N\ \text{large enough.}
\end{equation*} By passing to the limit when 
$k\to \infty$, we again obtain a contradiction. Further, we claim that 
$y\mapsto \kappa (y)$ is continuous on 
$X_s\setminus \{O\}$. To this end, we fix y ≠ O and ϵ > 0 arbitrary. We may find ϱ > 0 such that 
$\|Y(\kappa(y)-\epsilon, y)\| \gt 1+\varrho$ and 
$\|Y(\kappa(y)+\epsilon, y)\| \lt 1-\varrho$. Since 
$Y(t, \cdot)$ is continuous, there exists δ > 0 such that for each 
$z\in X_s$ with 
$\|y-z\| \lt \delta$,
\begin{equation*}
\|Y(\kappa(y)-\epsilon, z)\| \gt 1+\frac{\varrho}{2} \quad \text{and} \quad \|Y(\kappa(y)+\epsilon, z)\| \lt 1-\frac{\varrho}{2}.
\end{equation*} Consequently, 
$|\kappa(y)-\kappa (z)| \lt \epsilon$ for 
$z\in X_s\setminus \{O\}$ with 
$\|y-z\| \lt \delta$, proving the desired continuity.
Similarly, for any given 
$x\ne O$, 
$t\mapsto \|e^{A_s t}x\|$ is a strictly decreasing function,
\begin{equation*}
\lim_{t\to+\infty}\|e^{A_s t}x\|=0 \, \text{and} \, \lim_{t\to-\infty}\|e^{A_s t}x\|=+\infty.
\end{equation*} Therefore, for each 
$x\ne O$, there is a unique time 
$\sigma(x)\in\mathbb{R}$ such that
\begin{align}
\|e^{A_s\sigma(x)}x\|=1.
\end{align} As for κ, we have that 
$x\mapsto \sigma(x)$ is continuous on 
$X_s\setminus \{O\}$ and
\begin{align}
\sigma(x)\to -\infty, \quad \text{when}\ x\to O.
\end{align}Next, we claim that
\begin{align}
\kappa(Y(t,y))=\kappa(y)-t \, \text{and} \, \sigma(e^{A_s t}x)=\sigma(x)-t.
\end{align} In fact, by using the definition of 
$\kappa(y)$ and (4.1), we have
\begin{equation*}
1=\|Y(\kappa(Y(t,y)),Y(t,y))\|=\|Y(\kappa(Y(t,y))+t,y)\|.
\end{equation*} Due to the uniqueness of 
$\kappa(y)$, we have
\begin{equation*}
\kappa(Y(t,y))+t=\kappa(y).
\end{equation*} Similarly, by using (4.6) and the uniqueness of 
$\sigma(x)$, we obtain that 
$\sigma(e^{A_s t}x)=\sigma(x)-t$.
Now, we can define two mappings 
$\Psi_s \colon X_s \to X_s$ and 
$\tilde{\Psi}_s \colon X_s \to X_s$ by
\begin{align}
\Psi_s(y)=
\left\{
\begin{array}{ll}
e^{-A_s \kappa(y)}Y(\kappa(y),y), & y\neq O, \\
O, & y=O,
\end{array}
\right.
\end{align}and
\begin{align}
\tilde{\Psi}_s(x)=
\left\{
\begin{array}{ll}
Y(-\sigma(x),e^{A_s \sigma(x)}x), & x\neq O,\\
O, & x=O.
\end{array}
\right.
\end{align}Recall that
\begin{equation*}
Y(t,y)=e^{A_s t}y+ \int_0^t e^{A_s(t-\tau)}\pi_s f(Y(\tau,y)+h_s(Y(\tau,y))) \, d\tau.
\end{equation*} Then, when 
$y\ne O$, we can write 
$\Psi_s(y)$ as
\begin{align*}
\Psi_s(y)
=&~ e^{-A_s \kappa(y)} \left(e^{A_s\kappa(y)}y+\int_0^{\kappa(y)} e^{A_s(\kappa(y)-\tau)}\pi_s f(Y(\tau,y)+h_s(Y(\tau,y))) \, d\tau\right)
\\
=&~ y+\int_0^{\kappa(y)} e^{-A_s\tau}\pi_s f(Y(\tau,y)+h_s(Y(\tau,y))) \, d\tau.
\end{align*}Likewise, we have
\begin{equation*}
\tilde{\Psi}_s(x)=x+\int_0^{-\sigma(x)} e^{-A_s(\sigma(x)+\tau)} \pi_s f(Y(\tau,e^{A_s\sigma(x)}x)+h_s(Y(\tau,e^{A_s\sigma(x)}x))) \, d\tau.
\end{equation*} We claim that 
$\Psi_s$ and 
$\tilde{\Psi}_s$ are both continuous. Indeed, due to the continuity of κ and σ on 
$X_s\setminus \{O\}$, we only need to prove their continuity in O. From (4.4) and (4.5), it follows that when y → O, 
$\Psi_s(y)\to O$, which gives the continuity of 
$\Psi_s$ in O. Indeed, for 
$y\in X_s$ with y ≠ O, we have using (4.4) that
\begin{equation*}
\|\Psi_s (y)\|=\|e^{-A_s \kappa (y)}Y(\kappa(y), y)\| \le \|e^{-A_s \kappa(y)}\|,
\end{equation*} while 
$\lim\limits_{y\to O}\|e^{-A_s \kappa (y)}\|=0$ by (4.5) and the first inequality in (3.2). Therefore, 
$\lim\limits_{y\to O}\Psi_s(y)=O$, yielding the continuity of 
$\Psi_s$ in O. Similarly, from (4.3), (4.6), and (4.7), we see that when x → O, 
$\tilde{\Psi}_s(x)\to O$, which implies that 
$\tilde{\Psi}_s$ is continuous in O.
In the following, we claim that
\begin{align}
\sigma(\Psi_s(y))=\kappa(y), \, y\ne O \, \text{and} \, \kappa(\tilde{\Psi}_s(x))=\sigma(x), \, x\ne O.
\end{align}In view of (4.4), we have
\begin{equation*}
\|Y(\kappa(\tilde{\Psi}_s(x)),\tilde{\Psi}_s(x))\|=1, \quad x\ne O.
\end{equation*}By (4.1) and (4.10), we see that
\begin{align*}
1=&~ \|Y(\kappa(\tilde{\Psi}_s(x)), Y(-\sigma(x),e^{A_s\sigma(x)}x))\|
\\
=&~ \|Y(\kappa(\tilde{\Psi}_s(x))-\sigma(x),e^{A_s\sigma(x)}x)\|, \quad x\ne O.
\end{align*} Since 
$t\mapsto \|Y(t,y)\|$ is a strictly decreasing function for each 
$y\ne O$, we assert (see (4.6)) that
\begin{equation*}
\kappa(\tilde{\Psi}_s(x))-\sigma(x)=0, \quad x\ne O.
\end{equation*} Similarly, one can obtain that 
$\sigma(\Psi_s(y))=\kappa(y)$ for each 
$y\ne O$.
Then, we are going to prove that
\begin{equation*}
\Psi_s(\tilde{\Psi}_s(x))=x, \, x\in X_s \, \text{and} \, \tilde{\Psi}_s(\Psi_s(y))=y, \, y\in X_s. \end{equation*} In fact, for y = O, 
$\tilde{\Psi}_s(\Psi_s(y))=\tilde{\Psi}_s(\Psi_s(O))=O=y$. For y ≠ O, by (4.1), (4.9), (4.10), and (4.11), we have
\begin{align*}
\tilde{\Psi}_s(\Psi_s(y))=&~ Y(-\sigma(\Psi_s(y)),e^{A_s \sigma(\Psi_s(y))}\Psi_s(y))
\\
=&~ Y(-\kappa(y),e^{A_s \kappa(y)}e^{-A_s \kappa(y)} Y(\kappa(y),y))
\\
=&~ Y(-\kappa(y),Y(\kappa(y),y))=y.
\end{align*} For x = O, 
$\Psi_s(\tilde{\Psi}_s(x))=\Psi_s(\tilde{\Psi}_s(O))=O=x$. For x ≠ O, we also obtain from (4.1) (4.9), (4.10), and (4.11) that
\begin{align*}
\Psi_s(\tilde{\Psi}_s(x))=&~ e^{-A_s \kappa(\tilde{\Psi}_s(x))} Y(\kappa(\tilde{\Psi}_s(x)),\tilde{\Psi}_s(x))
\\
=&~ e^{-A_s\sigma(x)}Y(\sigma(x),Y(-\sigma(x),e^{A_s\sigma(x)}x))=x.
\end{align*}In this step, we establish (3.10). If y = O, (3.10) is obvious. For y ≠ O, by (4.1), (4.8), and (4.9), we see that
\begin{align*}
\Psi_s(Y(t,y))=&~ e^{-A_s\kappa(Y(t,y))} Y(\kappa(Y(t,y)),Y(t,y))
\\
=&~ e^{-A_s (\kappa(y)-t)} Y(\kappa(y)-t,Y(t,y))
\\
=&~ e^{A_s t} e^{-A_s\kappa(y)}Y(\kappa(y),y)=e^{A_s t}\Psi_s(y).
\end{align*}Finally, it remains to prove (3.11). First, we claim that
\begin{equation}
y\in X_s, \ 0 \lt \|y\| \lt 1 \implies \kappa(y) \lt 0.
\end{equation} In order to prove (4.12) we take 
$y\in X_s$ with 
$0 \lt \| y\| \lt 1$. Suppose that 
$\kappa(y)\ge 0$. Then, (4.4) together with (4.3) yields
\begin{equation*}
1=\|Y(\kappa(y), y)\|\le e^{-\mu \kappa(y)}\|y\| \le \|y\| \lt 1,
\end{equation*} which yields a contradiction. Thus, (4.12) holds. Next, note that 
$\|e^{A_s t}\| \le e^{|t| \|A_s\| }$ for all 
$t\in \mathbb{R}$. In the sequel, we will assume that 
$\lambda \le \|A_s\|$ as if this is not the case, we replace 
$\|A_s\|$ with any number which is greater than λ. Then, similar to (4.2), one can verify that
\begin{equation*}
\|Y(t,y)\| \le e^{(\|A_s\|+L_f (1+L_{h_s}))|t|}\|y\|, \quad \text{for all}\ t\in\mathbb{R}.
\end{equation*}Then, we see from (4.4) that
\begin{equation*}
1=\|Y(\kappa(y),y)\| \le e^{(\|A_s\|+L_f (1+L_{h_s}))|\kappa(y)|}\|y\|,
\end{equation*}which yields that
\begin{align}
e^{-|\kappa(y)|} \le \|y\|^{\frac{1}{\|A_s\|+L_f (1+L_{h_s})}}.
\end{align}Therefore, (4.9) together with (3.2), (4.4), and (4.13) gives
\begin{align*}
\|\Psi_s(y)\| =&~ \|e^{-A_s\kappa(y)} Y(\kappa(y),y)\| \\
\le &~ e^{\lambda\kappa(y)}\|Y(\kappa(y), y)\|
\notag \\
=&~ e^{-\lambda|\kappa(y)|} \le \|y\|^{\frac{\lambda}{\|A_s\|+L_f (1+L_{h_s})}},
\end{align*} which implies that 
$\Psi_s(y)=\mathcal{O}(\|y\|^{\alpha_1})$ as 
$\|y\|\to O$, where
\begin{equation*}
\alpha_1=\frac{\lambda}{\|A_s\|+L_f (1+L_{h_s})}\in (0,1).
\end{equation*}Similar to (4.13), we have
\begin{align}
e^{-|\sigma(x)|} \le \|x\|^{1/\|A_s\|},
\end{align} and 
$\sigma(x) \lt 0$ if 
$0 \lt \|x\| \lt 1$. Then, (4.10) together with (4.3), (4.6), and (4.14) gives
\begin{align*}
\|\tilde{\Psi}_s(x)\|=\|Y(-\sigma(x),e^{A_s\sigma(x)}x)\| \le e^{\mu \sigma(x)}
\le \|x\|^{\mu/\|A_s\|},
\end{align*} which indicates that 
$\tilde{\Psi}_s(x)=\mathcal{O}(\|x\|^{\alpha_2})$ as 
$\|x\|\to O$, where
\begin{equation*}
\alpha_2=\frac{\lambda-L_f(1+L_{h_s})}{\|A_s\|}\in (0,1).
\end{equation*} Therefore, one can take 
$\alpha=\min\{\alpha_1,\alpha_2\}$ such that (3.11) holds. Consequently, the proof of proposition 3.3 is completed.
5. Application to semilinear second-order equations
In this section, in order to demonstrate the strength of our result, we present an application to a nonlinear Duffing oscillator without external excitation.
Example 1.
Consider a nonlinear Duffing oscillator without external excitation
\begin{equation}
\ddot{u}+c\dot{u}+ku+\alpha g(u)=0, \quad u\in\mathbb{R},
\end{equation} where c > 0 is a damping coefficient, 
$k,\alpha$ are constant coefficients with physical significance, respectively, and g(u) is given by
\begin{equation*}
g(u)=
\begin{cases}
3\epsilon u-2 \epsilon, \quad & u \gt 1,
\\
\epsilon u^3, \quad & u\in[-1,1],
\\
3\epsilon u+2\epsilon u, \quad & u \lt -1.
\end{cases}
\end{equation*}Here ϵ > 0 is a constant and
\begin{equation*}
\|g(u)-g(\tilde{u})\|\le 3\epsilon \, \|u-\tilde{u}\|, \quad \forall u,\tilde{u}\in\mathbb{R}.
\end{equation*} In Eq. (5.1), for convenience, we take 
$k=-1$. For 
$k=-1 \lt 0$, the oscillator can be regarded as a system of steel beam that is deflected towards the two magnets, see [Reference Ott21]. In order to linearize Eq. (5.1), as usual, setting 
$v=(u,\dot{u})^T$, one can transform Eq. (5.1) into the first-order problem
\begin{equation}
\dot{v}=Av+f(v),
\end{equation} where 
$f(v)=(0,-\alpha g(u))^T$ and
\begin{equation*}
A=
\left(
\begin{array}{cc}
0 & 1 \\
1 & -c \\
\end{array}
\right).
\end{equation*}We see that there exist two eigenvalues of A:
\begin{equation*}
\lambda_1=-(c/2)-(\sqrt{c^2+4}/2) \lt 0, \quad
\lambda_2=-(c/2)+(\sqrt{c^2+4}/2) \gt 0,
\end{equation*}which implies that A is hyperbolic. Then by applying theorem 2.1, we have the following linearization result for the nonlinear second-order equation (5.1).
Theorem 5.1. Let c > 0 and 
$k=-1$. Provided that ϵ is sufficiently small, the induced first-order problem (5.2) can be completely linearized by means of a homeomorphism H. Moreover, 
$H^{\pm 1}(x)=\mathcal{O}(\|x\|^\varrho)$, as x → 0.
Acknowledgements
The authors would like to express their sincere gratitude to the editors and referees for their valuable suggestions and comments, which led to a significant improvement of the paper. Moreover, we declare that the authors are ranked in alphabetic order of their names and all of them have the same contributions to this paper.
Funding
Y. X. was supported from National Natural Science Foudation of China (No. 12571165, 11671176 and 11931016) and Natural Science Foudation of Zhejiang Province (No. LZ24A010006). D.D was supported in part by the University of Rijeka under the project uniri-iz-25-108.
