1. Introduction
In this paper, we consider the general cohomological equation
\begin{equation}
\psi(x)=M(x)\varphi(F(x))-\varphi(x),~~x\in \mathbb{R}^n,
\end{equation}proposed in [Reference Banyaga, de la Llave and Wayne3] to solve the normal form problem, where 
$\psi: \mathbb{R}^n\to \mathbb{R}^m$, 
$M: \mathbb{R}^n\to {\cal M}^{m\times m}$ (
${\cal M}^{m\times m}$ is denoted by the space of 
$m\times m$ real valued matrices), 
$F: \mathbb{R}^n\to \mathbb{R}^n$ are given and 
$\varphi: \mathbb{R}^n\to \mathbb{R}^m$ is unknown.
Cohomological equation and its generalized forms play an important role in many areas of mathematics, such as group representations [Reference Kirillov13, Reference Moore and Schmidt19], ergodic theory [Reference Anosov1, Reference Katok and Robinson10], dynamical systems [Reference Katok and Hasselblatt11, Reference Livšic15, Reference Lyubich18], singularity theory [Reference Arnol’d, Gusein-Zade and Varchenko2] etc. Specifically, if 
$M$ is a constant matrix and 
$\psi\equiv0$, equation (1.1) becomes
\begin{equation*}M\varphi\circ F=\varphi,\end{equation*}which is the fundamental and important problem, called linearization [Reference Banyaga, de la Llave and Wayne3, Reference Zhang and Zhang21]. For a general non-constant matrix 
$M$, if 
$\psi\equiv0$, equation (1.1) is concerned with the problem of normal form, which is also a basic and important tool in bifurcation theory [Reference Chow, Li and Wang5, Reference Guckenheimer and Holmes8]. As a kind of auxiliary equation, cohomological equation (1.1) arises in various important problems in dynamical systems, for example, the study of the existence of invariant measures, conjugacy problems, reparametrization of flows, rigidity and stability questions in dynamical systems etc.[Reference Belitskii and Tkachenko4, Reference Damjanović and Katok6, Reference Damjanović and Katok7, Reference Katok and Hasselblatt11, Reference Katok and Kononenko12].
The general cohomology equation can also induce a cocycle. Let 
$G$ denote either 
$\mathbb{N}$, 
$\mathbb{Z}$ or 
$\mathbb{R}$, 
$\rho: G\to GL(k,\mathbb{R})$ a homeomorphism, that is a linear representation of 
$G$, and 
$T: G\times X\to X$ a dynamical system with phase space 
$X$ and time 
$G$. A one-cocycle twisted by 
$\rho$ is a map 
$\eta: G\times X\to \mathbb{R}^k$ such that
\begin{equation}
\eta (g_2+g_1,x)=\rho(g_1)\eta(g_2,T(g_1)x)+\eta(g_1,x),
\end{equation}where 
$k\in \mathbb{N}$, 
$g_1,g_2\in G$ and 
$T(g)x:=T(g,x)$ (see [Reference Katok and Hasselblatt11, Definition 2.9.1]). If 
$\rho$ is the identity representation then such an 
$\eta$ is called an untwisted cocycle. A cocycle can also be defined via a function 
$\varphi: X\to \mathbb{R}^k$ as
\begin{equation}
\eta(g,x):=\rho(g)\varphi(T(g)x)-\varphi(x),
\end{equation}which is called coboundary. Two cocycles are cohomologous if their difference is a coboundary. In the discrete-time case, every cocycle is determined by the function 
$\psi(x)=\eta(1,x)$. Let 
$F(\cdot)=T(1)$ and 
$M=\rho(1)$. Equation (1.3) becomes the vector form of (1.2)
\begin{equation*}
\psi(x)=M\varphi(F(x))-\varphi(x),\quad x\in X.
\end{equation*} The following fact is a special solvability criterion for equation (1.1) in the case when 
$M={\rm id}$. Equation (1.1) is solvable if and only if for every integer 
$p\geq 1$ and every periodic point 
$x$ of the period 
$p$ the sum 
$\sum_{k=0}^{p-1}\psi(F^k(x))$ vanishes [Reference Katok and Hasselblatt11, Reference Lyubich18]. As a consequence, if the mapping 
$F$ has no periodic points, then with any 
$\psi$ equation (1.1) is solvable [Reference Katok and Hasselblatt11, Reference Lyubich18]. Concretely, with 
$\psi \gt 0$ (in particular for the Abel equation) equation (1.1) is solvable if and only if the mapping 
$F$ has no periodic points. Concerning the case of hyperbolic 
$\rho$, i.e. 
$\rho(1)=M$ is hyperbolic, the most celebrated results on this kind are the Livšic-type theorems for hyperbolic diffeomorphisms and flows [Reference Guillemin and Kazhdan9, Reference Katok and Hasselblatt11, Reference Livšic14–Reference de la Llave, Marco and Moriyon17], which reads that for any bounded 
$\psi=\eta(1,.)$ equation (1.3) has a unique bounded solution 
$\varphi$. If 
$\psi$ is continuous then the solution of (1.3) is also continuous [Reference Katok and Hasselblatt11, Theorem 2.9.2]). In addition, as shown in [Reference Katok and Hasselblatt11], it is worth mentioning that even if 
$\psi, F$ and 
$\rho$ are smooth, the solution 
$\varphi$ cannot be expected to be very regular. If 
$\eta(n,x)$ is bounded uniformly in 
$n$ and 
$x$, it is proved that 
$\varphi(x)={\rm sup}_{n\in \mathbb{N}}-\sum_{i=0}^n\psi(F^i(x))$ is a solution of (1.3) [Reference Katok and Hasselblatt11], also see [Reference Lyubich18]. For the partially hyperbolic case, the first attempt was addressed by Katok and Kononenko [Reference Katok and Kononenko12], who introduced a new method to solve (1.3) when 
$F$ is partially hyperbolic. By introducing the notion of periodic cycles functional, they [Reference Katok and Kononenko12] proved that the subspace of cocycles cohomologous to a constant is the common zero set of the periodic cycles functionals under some transitive condition. With an additional hypothesis of accessibility on 
$F$, Wilkinson [Reference Wilkinson20] obtained the existence and higher regularity of solutions to equation (1.3).
In 1996, by adding the smooth assumption on 
$\psi$ and 
$F$, Banyaga et al. [Reference Banyaga, de la Llave and Wayne3] considered the local smooth solutions to equation (1.1). The main result for a contraction 
$F$ is given as follows.
Theorem 1 [Reference Banyaga, de la Llave and Wayne3, Theorem 5.1]
Assume that in (1.1) we have (i) 
$F, \psi, M$ are 
$C^r$ (
$r\geq 2$) and 
$F(O)=O$, where 
$O$ is the origin; (ii) 
$D^{i}\psi(O)=O$ for 
$i\leq k \lt r$; (iii) 
$\|DF(O)\|\leq \tilde{\alpha} \lt 1$; (iv) 
$\|M(O)\|\leq \tilde{\beta}$; (v) 
$\tilde{\alpha}^{k+1}\tilde{\beta} \lt 1$. Then there exists a 
$C^r$ solution 
$\varphi$ of equation (1.1) defined in a sufficiently small neighborhood of zero.
In this paper, we continue to consider cohomological equation (1.1) for planar contractions near the origin. Since 
$M(O)$ plays an important role in the study of equation (1.1), we first consider the case that the mapping 
$M$ is a real constant matrix. By using the idea of invariant manifold and the estimations of linearization, given in [Reference Zhang and Zhang21], we present new criteria on the existence of 
$C^1$ solutions to equation (1.1), that is, conditions on the eigenvalues of 
$M$ (
$M(O)$ in Theorem 1) and 
$DF(O)$. Furthermore, as mentioned in Remark 1, the linearization result [Reference Zhang and Zhang21, Theorem 1] is a special case of our Theorem 4. In the case when 
$M$ is a matrix mapping, we also give sufficient conditions for the existence of 
$C^1$ solutions to equation (1.1) in the Poincaré domain, which requires lower smoothness on the given mappings 
$F,\gamma$ than Theorem 1.
Throughout this paper, let 
$F: U\to \mathbb{R}^2$ be a diffeomorphism such that
\begin{equation}
F(O)=O,\quad DF(O)=S:={\rm diag}(\lambda_1,\lambda_2),
\end{equation}where 
$U\subseteq \mathbb{R}^2$ is a neighborhood of 
$O$ and 
$\lambda_1,\lambda_2\in \mathbb{C}$ satisfy
\begin{equation}
0 \lt |\lambda_1|\leq|\lambda_2| \lt 1.
\end{equation} For 
$x:=(x_1,x_2)\in U$ we denote the norm 
$\|.\|$ as 
$\|x\|:=\max \{|x_1|,|x_2|\}$.
2. 
$M$ is a constant matrix
Before presenting the main results in this section, we give some necessary conditions on 
$\psi$ if equation (1.1) has a 
$C^1$ diffeomorphic solution 
$\varphi$ with 
$\varphi(O)=O$.
Lemma 1. Let 
$F: U\to \mathbb{R}^2$ be a 
$C^1$ mapping such that 
$F(O)=O$ and 
$M$ is a constant matrix. Assume that equation (1.1) has a 
$C^1$ diffeomorphic solution 
$\varphi$ in 
$U$ with 
$\varphi(O)=O$ and 
$D\varphi(O)={\rm id}$. Then 
$\psi$ is also 
$C^1$ with 
$\psi(O)=O$ and 
$D\psi(O)=M\cdot DF(O)-{\rm id}$, where 
${\rm id}$ denotes the identity mapping.
Proof. It is easy to prove that 
$\psi$ is 
$C^1$ in 
$U$ and 
$\psi(O)=O$ because 
$\varphi,F$ are 
$C^1$ and 
$\varphi(O)=F(O)=O$. Differentiating both sides of equation (1.1), we obtain
\begin{equation*}
D\psi(O)=M\cdot D\varphi(F(O))DF(O)-D\varphi(O)=M\cdot DF(O)-{\rm id}.
\end{equation*} Under the conditions of Lemma 1 and putting 
$\varphi={\rm id}+\Phi$, equation (1.1) becomes
\begin{equation}
M\Phi(F(x))-\Phi(x)=\gamma(x),\;\;x\in U,
\end{equation}where
\begin{equation}
\gamma:=\psi-MF+{\rm id},
\end{equation} Clearly, 
$D\gamma(O)=D\Phi(O)=O.$
As mentioned in our introduction, for the special case of 
$\psi\equiv0$, equation (1.1) becomes the linearization problem 
$M\varphi\circ F-\varphi=0$, which can be seen as the homogeneous form of (1.1). Actually, using Lemma 1, we have 
$M=\{DF(O)\}^{-1}$. In this case, the mapping 
$\gamma$ in (2.1) becomes 
$\gamma=-MF+{\rm id}$.
When 
$M$ is a constant matrix, we can easily check that equation (1.1) has the formal solution
\begin{equation}
\varphi(x)=\sum_{n=0}^{\infty}(M^{n+1}F^{n+1}(x)-M^{n}F^{n}(x))+x-\sum_{n=0}^{\infty}M^n\psi(F^n(x)),\quad x\in U.
\end{equation} Substituting 
$\psi=\gamma+MF-{\rm id}$ into (2.3), we get
\begin{eqnarray*}
\varphi(x)=&&\sum_{n=0}^{\infty}(M^{n+1}F^{n+1}(x)-M^{n}F^{n}(x))+x-\sum_{n=0}^{\infty}M^n\gamma(F^n(x))
\\
&&-\left(\sum_{n=0}^{\infty}M^{n+1}F^{n+1}(x)-\sum_{n=0}^{\infty}M^nF^n(x)\right)\\
=&&-\sum_{n=0}^{\infty}M^n\gamma(F^n(x))+x,
\end{eqnarray*}which means that 
$-\sum_{n=0}^{\infty}M^n\gamma(F^n(x))=\varphi(x)-x=\Phi(x)$ is a solution of equation (2.1) and it is easy to check the fact. So we only need to consider the existence of 
$C^1$ solution to the equivalent equation (2.1), that is to prove the convergence of the series 
$-\sum_{n=0}^{\infty}M^n\gamma(F^n(x))$.
Based on the relationship between the eigenvalues of matrix 
$M$ and 
$\lambda_i,$ 
$i=1,2$ (see condition (iv) in Theorem 1), we may assume that 
$M\in {\cal M}^{2\times 2}$ is of the diagonal form, i.e.
\begin{equation}
M:={\rm diag}(M_1,M_2),
\end{equation}where 
$M_1,M_2\in \mathbb{R}$ such that 
$|M_1|\geq |M_2|$. Due to the different techniques in the proofs, we will show our results in the following two cases: 
$|M_1\lambda_2| \lt 1$ and 
$|M_1\lambda_2|\geq 1$.
The following Theorems 2 and 3 are our main results in this section.
Theorem 2. Let 
$F,\psi: U\to \mathbb{R}^2$ be 
$C^1$ mappings fixing 
$O$ and 
$D\psi(O)=M\cdot DF(O)-{\rm id}$. Assume that two eigenvalues 
$\lambda_1,\lambda_2$ of 
$DF(O)$ satisfy (1.5), and 
$M_1,M_2$ in (2.4) satisfy 
$|M_1\lambda_2| \lt 1$. Then there exist a neighborhood 
$\Omega\subset U$ of 
$O$ and a 
$C^1$ diffeomorphism 
$\varphi:\Omega\rightarrow\mathbb{R}^2$ such that equation (1.1) holds.
Theorem 3. Let 
$F,\psi: U\to \mathbb{R}^2$ be 
$C^{1,\alpha}$ 
$(0 \lt \alpha\leq 1)$ mappings fixing 
$O$ and 
$D\psi(O)=M\cdot DF(O)-{\rm id}$. Assume that two eigenvalues 
$\lambda_1,\lambda_2$ of 
$DF(O)$ satisfy (1.5), and 
$M_1,M_2$ in (2.4) satisfy 
$|M_1\lambda_2|\geq1$. If 
$\alpha \gt \alpha_3:=-{\rm log}|M_1|/{\rm log}|\lambda_2|-1$, then there exist a neighborhood 
$\Omega\subset U$ of 
$O$ and a 
$C^1$ diffeomorphism 
$\varphi:\Omega\rightarrow\mathbb{R}^2$ such that equation (1.1) holds.
In order to prove Theorems 2 and 3, we need the following useful lemma, which helps us to find a suitable transformation for 
$F$.
Lemma 2 (Reference Zhang and Zhang21, Lemma 1)
Suppose that 
$F: U\to \mathbb{R}^2$ is a 
$C^{1,\alpha}$ 
$(0 \lt \alpha\leq 1)$ contraction and that the two eigenvalues 
$\lambda_1$ and 
$\lambda_2$ of 
$DF(O)$ satisfy (1.5). Then there exists a closed disk 
$V\subset U$ centered at 
$O$ such that 
$F$ has a 
$C^{1,\alpha}$ invariant curve
\begin{equation*}
\Gamma:=\{(x_1,x_2)\in V: x_1=g(x_2)\},
\end{equation*}where 
$g: V\cap \mathbb{R}\to \mathbb{R}$ is 
$C^{1,\alpha}$ and 
$g(0)=g'(0)=0$, if the constant 
$\alpha$ satisfies that 
$0 \lt \alpha \lt \alpha_1$ where
\begin{equation*}
\alpha_1:={\rm log}|\lambda_1|/{\rm log}|\lambda_2|-1.
\end{equation*} The existence of an invariant curve guaranteed in Lemma 2 allows us to define a 
$C^{1,\alpha}$ transformation 
$\Theta: V\to \mathbb{R}^2$ by
\begin{equation}
\Theta(x):=(x_1-g(x_2),x_2),\;\;x\in V.
\end{equation} Clearly, 
$\Theta(O)=O,D\Theta(O)={\rm id}$ and its inverse mapping 
$\Theta^{-1}$ is also 
$C^{1,\alpha}$ since 
$g$ is 
$C^{1,\alpha}$, that is, there exists a constant 
$L_1 \gt 0$ such that
\begin{equation*}\|(D\Theta^{-1})(x)-(D\Theta^{-1})(y)\|\leq L_1\|x-y\|^{\alpha}\end{equation*}for all 
$x,y$ in a neighborhood of the origin. So we can consider the mapping
\begin{equation*}\tilde{F}:=\Theta\circ F\circ \Theta^{-1}\end{equation*}instead of 
$F$, which satisfies 
$\tilde{F}(O)=O,D\tilde{F}(O)=S$ and has a straight line 
$x_2-$axis as its invariant curve, i.e. 
$\tilde{F}_1(0,x_2)=x_2,\; x_2\in V\cap \mathbb{R},$ where 
$\tilde{F}_1:=\pi_1\tilde{F}$ and 
$\pi_1$ denotes the projection onto the 
$x_1$-axis.
Furthermore, for each 
$n\in \mathbb{N}$ and a sufficiently small closed disk 
$\Omega\subset V$ centered at 
$O$, let
\begin{equation}
D\tilde{F}^n(x):=\left[\begin{array}{cc}
a_n(x)~~~~~~b_n(x)\\
c_n(x)~~~~~~d_n(x)
\end{array}\right],
\end{equation}where 
$a_n,b_n,c_n,d_n : \Omega\to \mathbb{R}$ are functions. The following result provides useful estimations of 
$a_n,b_n,c_n, d_n$, which were given in [Reference Zhang and Zhang21].
Lemma 3 (Reference Zhang and Zhang21, Lemma 3)
The mapping 
$\tilde{F}$ is 
$C^{1,\alpha}$ 
$(0 \lt \alpha \lt \alpha_1)$ in 
$U$ and
\begin{eqnarray*}
&|a_n(x)|\leq K_1|\lambda_1|^n,~~~~~~~~~~~~~~ |b_n(x)|\leq K_1|\lambda_1|^n,
\\
&|c_n(x)|\leq K_1|\lambda_2|^n,~~~~~~~~~~~~~~ |d_n(x)|\leq K_1|\lambda_2|^n
\end{eqnarray*}for all 
$n\in \mathbb{N}$ and all 
$x\in \Omega$, where 
$K_1 \gt 0$ is a constant independent of 
$n$ and 
$x$.
Let 
$(\tilde{F}^n)_1:=\pi_1\tilde{F}^n$ and 
$\psi_1:=\pi_1 \psi$. From Lemma 3 and its proof in [Reference Zhang and Zhang21], we have
\begin{equation}
\|\tilde{F}^n(x)\|\leq K_1|\lambda_2|^n,\quad\|D\tilde{F}^n(x)\|\leq K_1|\lambda_2|^n,\quad |(\tilde{F}^n)_1(x)|\leq K_2|\lambda_1|^n
\end{equation}for all 
$x\in \Omega$, where 
$K_2 \gt 0$ is a constant independent of 
$n$ and 
$x$.
Case 1, when 
$|M_1\lambda_2| \lt 1$.
For this case, we consider equation (2.1) directly and get that (1.1) has a 
$C^1$ solution under the assumption that the known mappings 
$F,\psi$ are 
$C^1$.
Proof of Theorem 2
As proved in [Reference Zhang and Zhang21, p. 9], there exist a closed disk 
$\Omega\subset U$ centered at 
$O$ and a constant 
$K_3 \gt 0$ independent of 
$n$ and 
$x$ such that
\begin{equation}
\|F^n(x)\|\leq K_3 |\lambda_2|^n,\quad \|DF^n(x)\|\leq K_3 |\lambda_2|^n,~~x\in \Omega.
\end{equation} Since 
$F,\psi$ are 
$C^1$ and 
$\gamma=\psi-MF+{\rm id}$, we conclude that 
$\gamma$ is also 
$C^1$ with 
$\gamma(O)=O$ and there exists a constant 
$L_2 \gt 0$ such that
\begin{equation}
\|D\gamma(F^n(x))\|\leq L_2,\;\; x\in \Omega,~ n\in \mathbb{N}.
\end{equation} In what follows, we only need to prove the uniform convergence of the series 
$-\sum_{n=0}^{\infty}M^nD\gamma(F^n(x))DF^n(x)$ near 
$O$. For this purpose, let us note that
\begin{eqnarray*}
&& \|{M}^nD\gamma(F^n(x))DF^n(x)\|\leq |M_1|^n \cdot L_2 K_3 |\lambda_2|^n,\quad x\in \Omega,
\end{eqnarray*}which implies the uniform convergence of the series 
$-\sum_{n=0}^{\infty}M^nD\gamma(F^n(x))DF^n(x)$ in 
$\Omega$, because 
$|M_1\lambda_2| \lt 1$ in the considered case. It follows that the series 
$-\sum_{n=0}^{\infty}M^n\gamma(F^n(x))$ also converges uniformly near 
$O$ by the fact that
\begin{equation*}\|M^n\gamma(F^n(x))\|=\|M^n\gamma(F^n(x))-M^n\gamma(F^n(O))\|
\leq \|{M}^nD\gamma(F^n(\tau_1x))DF^n(\tau_1x)\|\cdot \|x\|,\end{equation*}where 
$\tau_1\in (0,1)$ is a number depending on 
$x$. Moreover, we have
\begin{equation*}
D\varphi(O)=-\sum_{n=0}^{\infty}M^nD\gamma(F^n(O))DF^n(O)+{\rm id}={\rm id},
\end{equation*}implying that 
$\varphi$ is a 
$C^1$ diffeomorphism near O.
Case 2, when 
$|M_1\lambda_2|\geq 1.$
For this case, we introduce another number
\begin{equation*}
\alpha_3=-{\rm log}|M_1|/{\rm log}|\lambda_2|-1,
\end{equation*}as given in Theorem 3. Clearly, 
$\alpha_3\geq 0$.
Proof of Theorem 3
Since 
$F,\psi$ are 
$C^{1,\alpha}$ and 
$\gamma=\psi-MF+{\rm id}$, we obtain that 
$\gamma: U\to \mathbb{R}^2$ is also 
$C^{1,\alpha}$, that is, there exists a constant 
$L_3 \gt 0$ such that
\begin{equation*}\|D\gamma(x)-D\gamma(y)\|\leq L_3\|x-y\|^{\alpha},\quad x,y \in U.\end{equation*}By the above inequality and (2.8), we have
\begin{equation}
\|D\gamma(F^n(x))\|=\|D\gamma(F^n(x))-D\gamma(F^n(O))\|
\leq L_3K_3^\alpha|\lambda_2|^{n\alpha},
\end{equation}and then
\begin{eqnarray*}
&&\|{M}^nD\gamma(F^n(x))DF^n(x)\|\cdot \|x\|\leq |M_1|^n \cdot L_3 K_3^{\alpha}|\lambda_2|^{n\alpha}K_3 |\lambda_2|^n, \;x\in \Omega.
\end{eqnarray*} For all 
$\alpha \gt \alpha_3$, we have 
$|M_1||\lambda_2|^{\alpha+1} \lt 1$ and thus 
$-\sum_{n=0}^{\infty}M^nD\gamma(F^n(x))DF^n(x)$ is uniformly convergent in 
$\Omega$. As in the previous proof of Theorem 2, we conclude that equation (1.1) has a 
$C^1$ diffeomorphic solution 
$\varphi$.
Theorems 2-3 show that equation (1.1) has a 
$C^1$ solution in the cases where 
$|M_1\lambda_2| \lt 1$ or 
$|M_1\lambda_2|\geq 1$ and 
$|M_1||\lambda_2|^{\alpha+1} \lt 1$. In order to estimate the smoothness on 
$F$, we continue to consider the case of 
$|M_1\lambda_2|\geq 1$ on the assumption that 
$|M_1\lambda_1| \lt 1$, which enable us to define a new number
\begin{equation*}
\alpha_2:=\max\left\{-\frac{\log|\lambda_2|+ \log| M_1|}{\log |\lambda_1|},-\frac{\log|\lambda_2|+\log| M_2|}{\log |\lambda_2|}\right\}.
\end{equation*} One can check that 
$0\leq\alpha_2\leq\alpha_3\leq\alpha_1$. Clearly, equation (1.1) has a 
$C^1$ diffeomorphic solution when 
$\alpha \gt \alpha_3$. In what follows, it suffices to discuss equation (1.1) for all 
$\alpha\in (\alpha_2,\alpha_3]$.
Theorem 4. Let 
$F,\psi: U\to \mathbb{R}^2$ be 
$C^{1,\alpha}$ 
$(0 \lt \alpha\leq 1)$ mappings fixing 
$O$ and 
$D\psi(O)=M\cdot DF(O)-{\rm id}$. Assume that two eigenvalues 
$\lambda_1,\lambda_2$ of 
$DF(O)$ satisfy (1.5), and 
$M_1,M_2$ in (2.4) satisfy 
$|M_1\lambda_2|\geq1$ and 
$|M_1\lambda_1|\leq1$. If 
$\alpha \gt \alpha_2$ and 
$\psi_1$ vanishes on an invariant curve 
$\Gamma$ of 
$F$, then there exist a neighborhood 
$\Omega\subset U$ of 
$O$ and a 
$C^1$ diffeomorphism 
$\varphi:\Omega\rightarrow\mathbb{R}^2$ such that equation (1.1) holds.
Proof. For 
$\alpha \gt \alpha_3$, the desired conclusion has been obtained in Theorem 3, so we only need to consider the case of 
$\alpha_2 \lt \alpha\leq \alpha_3$. Recall that 
$\tilde{F}=\Theta\circ F\circ \Theta^{-1}$ and put
\begin{equation}
\Phi:=\tilde{\Phi}\circ \Theta+\Theta-{\rm id},
\end{equation}where 
$\Theta$ is defined in (2.5). Now equation (2.1) takes the form
\begin{equation}
M\tilde{\Phi}\circ \Theta\circ F+M\Theta\circ F-MF-\tilde{\Phi}\circ \Theta-\Theta+{\rm id}=\gamma.
\end{equation} By compositing both sides of (2.12) with 
$\Theta^{-1}$ on the right, it becomes
\begin{equation*}
M\tilde{\Phi}\circ \tilde{F}-\tilde{\Phi}=\gamma\circ \Theta^{-1}-M\tilde{F}+MF\circ \Theta^{-1}+{\rm id}-\Theta^{-1}.
\end{equation*}Thus, equation (2.1) can be rewritten as
\begin{equation}
M\tilde{\Phi}(\tilde{F}(x))-\tilde{\Phi}(x)=\eta(x),~~~x\in U
\end{equation}where, by (2.2), 
$\eta:=\psi\circ \Theta^{-1}-M\tilde{F}+{\rm id}$.
In order to make use of the properties of 
$\tilde{F}$, we may consider equation (2.13) instead of (1.1) and similar to the above proof, it suffices to prove the convergence of 
$\sum_{n=0}^{\infty}M^nD\eta(\tilde{F}^n(x))D\tilde{F}^n(x)$.
Note that 
$\tilde{F},\psi$ and 
$\Theta^{-1}$ are 
$C^{1,\alpha}$, by Lemma 1 we claim that 
$\eta$ is also 
$C^{1,\alpha}$, satisfying 
$\eta(O)=D\eta(O)=O$. Actually, we have
\begin{eqnarray*}
&&\|D\eta(x)-D\eta(y)\|
\\
&=&\|D\psi(\Theta^{-1}(x))D\Theta^{-1}(x)-D\psi(\Theta^{-1}(y))D\Theta^{-1}(y)-M(D\tilde{F}(x)-D\tilde{F}(y))\|
\\
&\leq &\|D\psi(\Theta^{-1}(x))D\Theta^{-1}(x)-D\psi(\Theta^{-1}(x))D\Theta^{-1}(y)\|
\\
&+&\|D\psi(\Theta^{-1}(x))D\Theta^{-1}(y)-D\psi(\Theta^{-1}(y))D\Theta^{-1}(y)\|+|M_1|\|D\tilde{F}(x)-D\tilde{F}(y)\|
\\
&\leq &\|D\psi(\Theta^{-1}(x))\|\cdot\|D\Theta^{-1}(x)-D\Theta^{-1}(y)\|\\
&+&\|D\psi(\Theta^{-1}(x))-D\psi(\Theta^{-1}(y))\|\cdot\|D\Theta^{-1}(y)\|+|M_1|\|D\tilde{F}(x)-D\tilde{F}(y)\|
\\
&=& L_4\|x-y\|^\alpha
\end{eqnarray*}for all 
$x,y$ in a neighborhood of the origin, where 
$L_4$ is a positive constant independent of 
$x$ and 
$y$. Since 
$\tilde{F}_1(0,x_2)=0$ and 
$\psi_1$ vanishes on the invariant curve 
$\Gamma$ defined in Lemma 2, that is, 
$\psi_1(g(x_2),x_2)=0$ for all 
$x_2\in U\cap \mathbb{R}$, then 
$\eta(x)=\psi(\Theta^{-1}(x))-M\tilde{F}(x)+{\rm id}=\psi(x_1+g(x_2),x_2)-M\tilde{F}(x_1,x_2)+(x_1,x_2),x\in U,$ which leads to
\begin{equation}
\eta_1(0,x_2)=0~~~{\rm and} ~~~\frac{\partial\eta_1(0,x_2)}{\partial x_2}=0,\;\; x_2\in U\cap \mathbb{R}.
\end{equation}Let
\begin{equation}
D\eta(\tilde{F}^n(x)):=\left[\begin{array}{cc}
\tilde{a}_n(x)~~~~~~\tilde{b}_n(x)\\
\tilde{c}_n(x)~~~~~~\tilde{d}_n(x)
\end{array}\right],
\end{equation}where 
$\tilde{a}_n,\tilde{b}_n,\tilde{c}_n,\tilde{d}_n : U\to \mathbb{R}$ are functions.
Concerning (2.15), we have
\begin{eqnarray}
|\tilde{b}_n(x)|&=&\left|\frac{\partial \eta_1(\tilde{F}^n(x))}{\partial x_2}\right|=\left|\frac{\partial \eta_1(\tilde{F}^n(x))}{\partial x_2}-\frac{\partial \eta_1(0,(\tilde{F}^n)_2(x))}{\partial x_2}\right|\nonumber\\
&\leq& L_4|(\tilde{F}^n)_1(x)|^\alpha\leq L_4K_2^\alpha|\lambda_1|^{n\alpha}\end{eqnarray}due to (2.14) and the last inequality of (2.7).
On the other hand, making use of the first inequality of (2.7), we get
\begin{eqnarray*}
\|D\eta(\tilde{F}^n(x))\|&=&\|D\eta(\tilde{F}^n(x))-D\eta(\tilde{F}^n(O))\|
\leq L_4\|\tilde{F}^n(x)\|^\alpha\leq L_4K_1^\alpha|\lambda_2|^{n\alpha},
\end{eqnarray*}which implies that
\begin{equation}
|\tilde{a}_n(x)|,|\tilde{c}_n(x)|,|\tilde{d}_n(x)|\leq L_4K_1^\alpha|\lambda_2|^{n\alpha},~~ x\in \Omega.
\end{equation}Therefore, we obtain
\begin{align*}
&\|{M}^nD\eta(\tilde{F}^n(x))D\tilde{F}^n(x)\|\\
&\quad=\left\|\left[\begin{array}{cc}
M_1^n~~~~0\\
0~~~~ M_2^n
\end{array}\right]\cdot
\left[\begin{array}{cc}
\tilde{a}_n(x)~~~~~~\tilde{b}_n(x)\\
\tilde{c}_n(x)~~~~~~\tilde{d}_n(x)
\end{array}\right] \cdot\left[\begin{array}{cc}
a_n(x)~~~~~~b_n(x)\\
c_n(x)~~~~~~d_n(x)
\end{array}\right]\right\|
\\
&\quad\leq\left\|\left[\begin{array}{cc}
M_1^n\tilde{a}_n(x)a_n(x)+M_1^n\tilde{b}_n(x)c_n(x)~~~~~ M_1^n\tilde{a}_n(x)b_n(x)+M_1^n\tilde{b}_n(x)d_n(x)\\
M_2^n\tilde{c}_n(x)a_n(x)+M_2^n\tilde{d}_n(x)c_n(x)~~~~~M_2^n\tilde{c}_n(x)b_n(x)+M_2^n\tilde{d}_n(x)d_n(x)
\end{array}\right]\right\|
\end{align*}for all 
$x\in \Omega$. According to the inequalities of Lemma 3 and formulas (2.16) and (2.17), we get
\begin{equation}
\max\{|M_1^n\tilde{a}_n(x)a_n(x)|,
|M_1^n\tilde{a}_n(x)b_n(x)|\}
\leq |M_1|^nL_3K_1^\alpha|\lambda_2|^{n\alpha}K_1|\lambda_1|^n,
\end{equation}
\begin{equation}
\max\{|M_1^n\tilde{b}_n(x)c_n(x)|,|M_1^n\tilde{b}_n(x)d_n(x)|\}
\leq |M_1|^nL_3K_2^\alpha|\lambda_1|^{n\alpha}K_1|\lambda_2|^n,
\end{equation}
\begin{equation}
\max\{|M_2^n\tilde{c}_n(x)a_n(x)|,|M_2^n\tilde{c}_n(x)b_n(x)|\}
\leq |M_2|^nL_3K_1^\alpha|\lambda_2|^{n\alpha}K_1|\lambda_1|^n,
\end{equation}
\begin{equation}
\max\{|M_2^n\tilde{d}_n(x)c_n(x)|,|M_2^n\tilde{d}_n(x)d_n(x)|\}
\leq |M_2|^nL_3K_1^\alpha|\lambda_2|^{n\alpha}K_1|\lambda_2|^n.
\end{equation} The assumption of 
$\alpha \gt \alpha_2$ confirms that 
$|\lambda_i|^{\alpha}|\lambda_2||M_i| \lt 1$ for 
$i=1,2$ and thus the series 
$\sum_{n=0}^{\infty}M^nD\eta(\tilde{F}^n(x))D\tilde{F}^n(x)$ and 
$\sum_{n=0}^{\infty}M^n\eta(\tilde{F}^n(x))$ are both uniformly convergent in 
$\Omega$. Therefore, the mapping 
$\tilde{\Phi}(x):=-\sum_{n=0}^{\infty}M^n\eta(\tilde{F}^n(x))$ is a 
$C^1$ solution of equation (2.13) with 
$\tilde{\Phi}(O)=D\tilde{\Phi}(O)=O$, and then 
$\varphi=(\tilde{\Phi}+{\rm id})\circ \Theta$ is a 
$C^1$ diffeomorphism solution to equation (1.1).
Remark 1. In our introduction, we mentioned that when 
$\psi\equiv0$, that is, 
$\gamma=-MF+{\rm id}$, equations (1.1) and (2.1) become the problem of linearization. Since 
$M\cdot DF(O)={\rm id}$, it means that 
$|\lambda_1M_1|=1$. Consequently, we have 
$\alpha_2 \lt \alpha_3=\alpha_1$, where
\begin{equation*}\alpha_2=1-\log |\lambda_2|/\log |\lambda_1|,\end{equation*}which is exactly the number 
$\alpha_0$ given in [Reference Zhang and Zhang21, Theorem 1]. Furthermore, the mapping 
$\eta$ in (2.13) becomes 
$\eta=-M\tilde{F}+{\rm id}$, implying that 
$\eta_1(0,x_2)=0$ by the straightening property of 
$\tilde{F}$. Therefore, our Theorem 4 is a generalization of the linearization result [Reference Zhang and Zhang21, Theorem 1] in the Poincaré domain.
Remark 2. Theorem 4 gives a sufficient condition for the existence of 
$C^1$ solutions to equation (1.1) in the case 
$|\lambda_1M_1|\leq1$. When 
$|\lambda_1M_1| \gt 1$, we have two subcases, that is 
$0 \lt \alpha_2 \lt \alpha_1 \lt \alpha_3$ and 
$0 \lt \alpha_1\leq\alpha_2 \lt \alpha_3$. By the proofs in Theorems 3-4, equation (1.1) can be solved similarly for all 
$\alpha_2 \lt \alpha \lt \alpha_1$ (resp. 
$\alpha_1 \lt \alpha \lt \alpha_2$) and 
$\alpha \gt \alpha_3$. Since 
$\alpha_2$ is the sharp estimate to 
$\alpha$, as proved in [Reference Zhang and Zhang21], the problem for 
$\alpha_1\le\alpha\le \alpha_3$ is still unsolved.
3.
$M$ is a matrix mapping
According to the proof of Lemma 1, if 
$\varphi$ is a 
$C^1$ diffeomorphic solution of equation (1.1) satisfying 
$\varphi(O)=O$, then 
$\psi$ is also 
$C^1$ with 
$\psi(O)=O$ and 
$D\psi(O)=M(O)DF(O)-{\rm id}$. Consequently, by putting 
$\varphi={\rm id}+\Phi$ and 
$D\varphi(O)={\rm id}$, equation (1.1) becomes
\begin{equation}
M(x)\Phi(F(x))-\Phi(x)=\gamma(x),\;\;x\in U,
\end{equation}where
\begin{equation*}\gamma(x):=\psi(x)-M(x)F(x)+x\end{equation*}and 
$D\gamma(O)=D\Phi(O)=O.$
Theorem 5. Let 
$F,\psi: U\to \mathbb{R}^2$ be 
$C^{1,\alpha}$ 
$(0 \lt \alpha\leq 1)$ mappings fixing 
$O$, 
$D\psi(O)=M(O)DF(O)-{\rm id}$ and 
$M: U\to {\cal M}^{2\times 2}$ be a 
$C^{1,\alpha}$ matrix mapping. Assume that two eigenvalues 
$\lambda_1,\lambda_2$ of 
$DF(O)$ satisfy (1.5) and 
$M(O)={\rm diag}(M_1,M_2)$ satisfying 
$|M_1|\geq |M_2|$. If
(i)
$|M_1\lambda_2| \lt 1$ and
$\alpha\in (0,1]$
or
(ii)
$|M_1\lambda_2|\geq 1$ and
$\alpha \gt \alpha_3$,
then there exist a neighborhood 
$\Omega\subset U$ of 
$O$ and a 
$C^1$ diffeomorphism 
$\varphi:\Omega\rightarrow\mathbb{R}^2$ such that equation (1.1) holds.
Proof. It is easy to verify that the function
\begin{eqnarray*}
\Phi(x)=-\sum_{n=0}^{\infty}\left(\left[\prod_{j=0}^{n-1}M(F^j(x))\right]\cdot\gamma(F^n(x))\right)
\end{eqnarray*}is a formal solution of equation (3.1).
Since the inequality
\begin{equation*}
\|F(x)\|\leq (\|DF(O)\|+\|DF(\tau_2 x)-DF(O)\|)\cdot\|x\|\leq(|\lambda_2|+L\|x\|^{\alpha})\cdot\|x\|
\end{equation*}is true for all 
$x\in U$ and some 
$\tau_2\in (0,1)$, we can prove inductively that there exists a closed disk 
$V\subset U$ centered at 
$O$ such that
\begin{equation}
\|F^j(x)\|\leq (|\lambda_2|+\delta)^j,\quad j\in \mathbb{N},\; x\in V,
\end{equation}where 
$\delta:=(1-|\lambda_2|)/2$ (see the proof in [Reference Zhang and Zhang21, p. 9]). Note that the matrix mapping 
$M$ is 
$C^{1,\alpha}$, there exists a real constant 
$K_4 \gt 0$ such that
\begin{equation*}
\|M(x)\|=\|M(x)-M(O)+M(O)\|\leq \|M(O)\|+\|DM(\tau_3 x)\|\cdot \|x\|\leq |M_1|+K_4\|x\|
\end{equation*}for 
$x\in U$ and some 
$\tau_3\in (0,1)$, whence
\begin{equation}
\left\|\prod_{j=0}^{n-1}M(F^j(x))\right\|\leq K_5|M_1|^n,
\end{equation}where
\begin{equation*}K_5:=\prod_{j=0}^{\infty}\left\{1+\frac{K_4}{|M_1|}\left(\frac{1+|\lambda_2|}{2}\right)^j\right\} \lt \infty.\end{equation*} Since 
$F,\psi$ are 
$C^{1,\alpha}$ and 
$\gamma(x)=\psi(x)-M(x)F(x)+x$, we get that 
$\gamma: U\to \mathbb{R}^2$ is also 
$C^{1,\alpha}$, that is, there exists a constant 
$L_5 \gt 0$ such that
\begin{equation*}\|D\gamma(x)-D\gamma(y)\|\leq L_5\|x-y\|^{\alpha},\quad x,y \in U.\end{equation*}By the above inequality and (2.8), we have
\begin{equation}
\|D\gamma(F^n(x))\|
\leq L_5\|F^n(x)\|^\alpha\leq L_5K_3^\alpha|\lambda_2|^{n\alpha},
\end{equation}and thus
\begin{eqnarray*}
&&\|\gamma(F^n(x))\|\leq\|D\gamma(F^n(\tau_4 x))DF^n(\tau_4 x)\|\cdot \|x\|\leq L_5 K_3^{\alpha}|\lambda_2|^{n\alpha}K_3 |\lambda_2|^n, \;x\in \Omega
\end{eqnarray*}for some 
$\tau_4\in (0,1)$.
To prove the uniform convergence of the function series
\begin{equation}
\sum_{n=0}^{\infty}D\left(\left[\prod_{j=0}^{n-1}M( F^j(x))\right]\cdot\gamma(F^n(x))\right),
\end{equation}note that by (2.8) and (3.3) we have
\begin{eqnarray*}
&&\left\|D\left(\left[\prod_{j=0}^{n-1}M( F^j(x))\right]\cdot\gamma(F^n(x))\right)\right\|\\
&=& \left\|\sum_{k=0}^{n-1}\left[\prod_{i=0}^{k-1}M( F^i(x))\right]DM(F^k(x))DF^k(x)\cdot\prod_{i=k+1}^{n-1}M( F^i(x))\cdot\gamma(F^n(x))\right.\\
&&+\left.\prod_{k=0}^{n-1}M( F^k(x))\cdot D\gamma(F^n(x))DF^n(x)\right\|\\
&\leq&K_5 |M_1|^n\cdot\left(\sum_{k=0}^{n-1}K_3|\lambda_2|^k+1\right)\|D\gamma(F^n(x)) DF^n(x))\|
\\
&\leq&K_5 K_6 L_5 K_3^{1+\alpha}|\lambda_2|^{n\alpha}|M_1\lambda_2|^n,
\end{eqnarray*}where
\begin{equation*}K_6:=K_3\sum_{k=0}^{\infty}|\lambda_2|^k+1 \lt \infty.\end{equation*} The assumptions in condition (i) (resp. (ii)) guarantee 
$|M_1||\lambda_2|^{\alpha+1} \lt 1$, and then the series (3.5) is uniformly convergent in 
$\Omega$. Arguing similarly as in the proof of Theorem 2, we conclude that equation (1.1) has a 
$C^1$ diffeomorphic solution 
$\varphi$.
Theorem 6. Let 
$F,\psi: U\to \mathbb{R}^2$ be 
$C^{1,\alpha}$ 
$(0 \lt \alpha\leq 1)$ mappings fixing 
$O$, 
$D\psi(O)=M(O) DF(O)-{\rm id}$ and 
$M: U\to {\cal M}^{2\times 2}$ be a 
$C^{1,\alpha}$ matrix mapping. Assume that two eigenvalues 
$\lambda_1,\lambda_2$ of 
$DF(O)$ satisfy (1.5) and 
$M(O)={\rm diag}(M_1,M_2)$ satisfying 
$|M_1|\geq |M_2|$, 
$|M_1\lambda_2|\geq 1,\;|M_1\lambda_1|\leq1.$ If
\begin{equation*}
\alpha \gt \alpha_4:=\max\left\{\alpha_2,-\log q/\log|\lambda_2|-1\right\}, ~q:=\|DM(O)\|
\end{equation*}and 
$\psi_1$ vanishes on an invariant curve 
$\Gamma$ of 
$F$, then there exist a neighborhood 
$\Omega\subset U$ of 
$O$ and a 
$C^1$ diffeomorphism 
$\varphi:\Omega\rightarrow\mathbb{R}^2$ such that equation (1.1) holds.
Proof. Firstly, the formal solution of equation (3.1) can be decomposed into the following two parts
\begin{eqnarray*}
\Phi(x)&=&-\sum_{n=0}^{\infty}\left[\prod_{j=0}^{n-1}M(F^j(x))\right]\cdot\gamma(F^n(x))
\\
&=&-\sum_{n=0}^{\infty}M^n(O)\gamma(F^n(x))-\sum_{n=0}^{\infty}\left[\prod_{j=0}^{n-1}m( F^j(x))\right]\cdot\gamma(F^n(x))
\\
&=&\Phi_1(x)+\Phi_2(x),
\end{eqnarray*}where 
$m(x):=M(x)-M(O)$ and 
$\Phi_1(x):=-\sum_{n=0}^{\infty}M^n(O)\gamma(F^n(x))$, 
$\Phi_2(x):=-\sum_{n=0}^{\infty}\left[\prod_{j=0}^{n-1}m( F^j(x))\right]\cdot\gamma(F^n(x))$ for 
$x\in U$. According to the proof of Theorem 4, we confirm the convergence of function 
$\Phi_1$ because 
$\alpha \gt \alpha_2$. In what follows, it suffices to prove the convergence of 
$\Phi_2$.
Clearly, the mapping 
$m$ is 
$C^{1,\alpha}$ by its definition. So for all 
$x,y\in U$, there exists a constant 
$L_6 \gt 0$ such that
\begin{equation*}
\|Dm(x)-Dm(y)\|\leq L_6\|x-y\|^{\alpha}.
\end{equation*}Consequently, we have
\begin{eqnarray}
\|m(x)\|&=&\|m(x)-m(O)\|\leq\|m'(\tau_5 x)\|\cdot \|x\|\nonumber
\\ &=&\|m'(O)+m'(\tau_5 x)-m'(O)\|\cdot \|x\|
\leq (\|m'(O)\|+L_4\| x\|^{\alpha})\end{eqnarray}for all 
$x\in U$ and some 
$\tau_5\in(0,1)$. Furthermore, by (2.8) and the definition of 
$q$ we obtain
\begin{equation*}
\|m(F^j(x))\|\leq (\|m'(O)\|+L_4\|F^j(x)\|^{\alpha})\leq (q+L_4 K_3^{\alpha} |\lambda_2|^{\alpha j}),\quad x\in U.
\end{equation*}Therefore, we get
\begin{equation}
\left\|\prod_{j=0}^{n-1}m( F^j(x))\right\|\leq K_7 q^n,
\end{equation}where
\begin{equation*}K_7:=\prod_{j=0}^{\infty}\left\{1+\frac{L_4 K_3^{\alpha}}{q}|\lambda_2|^{\alpha j}\right\} \lt \infty.\end{equation*}In order to prove the convergence of the function series
\begin{equation}
\sum_{n=0}^{\infty}\left[\prod_{j=0}^{n-1}m( F^j(x))\right]\cdot\gamma(F^n(x)),
\end{equation}note that, by (2.8) and (3.7), for all 
$x\in U$ we have
\begin{eqnarray*}
&&\left\|D\left(\left[\prod_{j=0}^{n-1}m( F^j(x))\right]\cdot\gamma(F^n(x))\right)\right\|\\
&=& \left\|\sum_{k=0}^{n-1}\left[\prod_{i=0}^{k-1}m( F^i(x))\right]Dm(F^k(x))DF^k(x)\cdot\prod_{i=k+1}^{n-1}m( F^i(x))\cdot\gamma(F^n(x))\right.\\
&&+\left.\prod_{k=0}^{n-1}m( F^k(x))\cdot D\gamma(F^n(x))DF^n(x)\right\|\\
&\leq&K_7 q^n\cdot\left(\sum_{k=0}^{n-1}K_3|\lambda_2|^k+1\right)\|D\gamma(F^n(x)) DF^n(x)\|
\\
&\leq&K_7 K_8 L_2 K_3^{1+\alpha}q^n|\lambda_2|^{(1+\alpha)n},
\end{eqnarray*}where
\begin{equation*}K_8:=K_3\sum_{k=0}^{n-1}|\lambda_2|^k+1 \lt \infty.\end{equation*} Applying the condition 
$\alpha \gt -\log |q|/\log |\lambda_2|-1$ in this theorem, we get 
$0 \lt q\leq|\lambda_2|^{-(\alpha+1)}$ and thus 
$q|\lambda_2|^{\alpha+1} \lt 1$. So the series (3.8) is uniformly convergent in 
$U$, which admits the convergence of 
$\Phi_2$. Moreover, one checks
\begin{equation*}
\Phi_2(O)=-\sum_{n=0}^{\infty}\left[\prod_{j=0}^{n-1}m( F^j(O))\right]\cdot\gamma(F^n(O))=O~~
{\rm and}~~
D\Phi_2(O)=O.
\end{equation*} Therefore, equation (1.1) has a 
$C^1$ solution 
$\varphi$ near the origin 
$O$ with 
$\varphi(O)=O$ and 
$D\varphi(O)={\rm id}$.
Remark 3. Compared with Theorem 1 (also see [Reference Banyaga, de la Llave and Wayne3, Theorem 5.1]), our Theorem 6 requires lower smoothness on the given mappings 
$F$ and 
$\gamma$, where 
$F, \gamma$ are 
$C^r,r\geq2$ in Theorem 1 and here we only need 
$C^{1,\alpha}$ for 
$\alpha \gt \alpha_2$.
Acknowledgments
The authors are very grateful to the editor and reviewer for their careful checking and helpful suggestions.
Funding statement
This work was supported by the National Science Foundation of China (#12571180), Sichuan Provincial Natural Science Foundation of China (#2023NSFSC0064) and Jiaxing Public Welfare Research Program of China (2025CGW024,2024AD30076).
