\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 46, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/46\hfil Formal and analytic solutions]
{Formal and analytic solutions for a quadric iterative functional equation}

\author[P. Zhang \hfil EJDE-2012/46\hfilneg]
{Pingping Zhang} 

\address{Pingping Zhang \newline
Department of Mathematics and Information Science,
Binzhou University, Shandong 256603, China}
\email{zhangpingpingmath@163.com}

\thanks{Submitted December 21, 2011. Published March 23, 2012.}
\subjclass[2000]{39B22, 34A25, 34K05}
\keywords{Iterative functional equation; analytic solution;
small divisor;\hfill\break\indent Brjuno condition}

\begin{abstract}
 In this article, we study a quadric iterative functional equation.
 We prove the existence of formal solutions, and that every formal
 solution yields a local analytic solution when the eigenvalue
 of the linearization for the auxiliary function lying inside the unit
 circle, lying on the unit circle with a Brjuno number, or a root of
 $1$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\allowdisplaybreaks

\section{Introduction}

Solving iterative functional equations is difficult since the
unknown arises in the iteration \cite{MK,ZhJ}.
Using Schauder fixed point theorem, Zhang
\cite{Zh87} proved the existence and uniqueness of solutions for a
general iterative functional equation, the so-called polynomial-like
iterative functional equation,
$$
\lambda_1x(t)+\lambda_2x^{2}(t)+\dots+\lambda_nx^{n}(t)=F(t),\quad
t\in{\mathbb{R}}.
$$
Later various properties of solutions of iterative functional
equations, such as continuity, differentiability, monotonicity,
convexity, analyticity, stability, have received much more attention;
see e.g. \cite{LJM09}--\cite{SiZh}, \cite{Xu1}--\cite{Zhp},
\cite{Zh86}. Among these studies, the existence of analytic
solutions caused  more concerns since it is closely related
to small divisors problem. In \cite{SiJZ02}, analytic invariant
curves for a planar map were obtained by solving the iterative
functional equation
$$
x(z+x(z))=x(z)+G(z)+H(z+x(z)),\ z\in{\mathbb{C}}.
$$
We notice that \cite{SiJZ02} and \cite{RBA} are all based on
eigenvalue of the linearization $\theta$ is inside the unit circle
or a Diophantine number by using Schr\"{o}der conversion and
majorant series. On the other hand,  Reich and his co-authors
\cite{LJM09}-\cite{LJ11} have studied the formal solutions of a
quadric iterative
 functional equation, called the generalized Dhombres functional
equation,
$$
f(zf(z))=\varphi(f(z)),\quad  z\in{\mathbb{C}},
$$
in the ring of formal power series $\mathbb{C}[[z]]$. They described
the structure of the set of all formal solutions when the eigenvalue
$\theta$ of linearization is not a root of $1$, and also showed
every formal solutions yield a local analytic solutions when
$\theta$ is not on the unit circle or a Diophantine number, as well
as represent analytic solutions by infinite products for $\theta$
ling in the unit circle. In 2008,  Xu and  Zhang \cite{Xu2}
studied the analytic solutions of a $q$-difference equation
\begin{equation}
\sum_{j=0}^{k}\sum_{t=1}^{\infty}C_{t,j}(z)(x(q^{j}z))^t=G(z),\quad
z\in{\mathbb{C}}, \label{XU}
\end{equation}
they obtained local analytic solutions under Brjuno condition, and
proved no-existence of analytic solutions when the eigenvalue
$\theta$ of linearization satisfies Cremer condition. Following that,
 Si and Li \cite{SiZh} discussed analytic solutions of the
\eqref{XU} with a singularity at the origin.

In this article, we study the  quadric iterative functional
equation
\begin{equation}\label{1.1}
x(az+bzx(z))=H(z)
\end{equation}
in the complex field, where $x(z)$ is unknown function, $H(z)$ is a
given holomorphic function, $a$ and $b$ are nonzero complex
parameters. It is a more complicated equation than the involutory
function $x^2 (t)=t$, which is the Babbage equation with $n=2$. We discuss
the existence of formal solutions for \eqref{1.1} when $a$ is
arbitrary nonzero complex number. Moreover, every formal solution
yields a local analytic solution when $a$ is lying inside the unit
circle, lying on the unit circle with a Brjuno number or a root of
$1$. Our idea comes from \cite{SiZh}.

Let 
$y(z)=az+bzx(z)$. Then
$$
x(z)=\frac{y(z)-az}{bz}.
$$ 
Therefore,
\begin{equation*}
x(y(z))=\frac{y(y(z))-ay(z)}{by(z)},
\end{equation*}
Then \eqref{1.1} is equivalent to the functional equation
\begin{equation}\label{1.2}
y(y(z))-ay(z)=by(z)H(z).
\end{equation}

Using the conversion $y(z)=g(\theta(g^{-1}(z)))$, Equation \eqref{1.2}
transforms into the equation without functional iteration
\begin{equation}\label{1.3}
g(\theta^{2}z)-ag(\theta z)= bg(\theta z)H(g(z)).
\end{equation}
Suppose
\begin{equation}\label{1.2*}
g(z)=\sum_{n=1}^{\infty}a_nz^{n},\quad
H(z)=\sum_{n=1}^{\infty}h_nz^{n}.
\end{equation}
Substituting \eqref{1.2*} into \eqref{1.3}, we obtain
\begin{equation} \label{1.4}
\begin{split}
&(\theta^{2}-a\theta)a_1z+\sum_{n=1}^{\infty}(\theta^{2(n+1)}
 -a\theta^{n+1})a_{n+1}z^{n+1} \\
&=b\sum_{n=1}^{\infty}\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2
 +\dots+i_{m}=j;}{m=1,2,\dots,j}}
\theta^{n+1-j}a_{n+1-j}h_{m}a_{i_1}a_{i_2}\dots a_{i_{m}}z^{n+1}.
\end{split}
\end{equation}
Comparing coefficients, we obtain
\begin{equation}\label{1.5}
(\theta^{2}-a\theta)a_1=0,
\end{equation}
and
\begin{equation} \label{1.6}
(\theta^{2(n+1)}-a\theta^{n+1})a_{n+1}
=b\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
\theta^{n+1-j}a_{n+1-j}h_{m}a_{i_1}a_{i_2}\dots a_{i_{m}}.
\end{equation}
Under $a_1\neq0$, the equality \eqref{1.5} implies that
$\theta=a$, then \eqref{1.6} turns into
\begin{equation} \label{1.7}
(\theta^{n}-1)\theta^{n+2}a_{n+1}
=b\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
\theta^{n+1-j}a_{n+1-j}h_{m}a_{i_1}a_{i_2}\dots a_{i_{m}}.
\end{equation}
This means the sequence $\{a_n\}_{n=2}^\infty$ can be determined
successively from \eqref{1.7} in a unique manner for any
$a_1\neq0$; that is,  \eqref{1.3} has formal solution for
arbitrary nonzero complex number $a$.  Noticing that the function
$H(z)$ is holomorphic in a neighborhood of the origin, we assume
$$|h_n|\leq1.$$
The reason for this, is that \eqref{1.3} and hypothetic conditions $g(0)=0$,
$g'(0)=a_1$ still hold under the transformations
$$
H(z)=\rho^{-1}F(\rho\,z), \quad\,g(z)=\rho^{-1}G(\rho\,z)
$$
for $|h_n|\leq\rho^{n}$.
We prove analyticity of solutions to \eqref{1.3} under varius hypotheses:
\begin{itemize}
\item[(A1)] (elliptic case) $\theta=e^{2\pi i\alpha}$,
$\alpha\in \mathbb{R}\backslash\mathbb{Q}$ is a Brjuno number;
  i.e., $B(\alpha)=\sum_{k=0}^\infty\frac{\log q_{k+1}}{q_k}<\infty$,
 where $\{\frac{p_k}{q_k}\}$ denotes
 the sequence of partial fraction of the continued fraction expansion of $\alpha$;

\item[(A2)] (parabolic case) $\theta=e^{2\pi i\frac{q}{p}}$ for
some integer $p\in \mathbb{N}$ with $p\geq2,
q\in \mathbb{Z}\backslash\{0\}$, and
$\theta\neq e^{2\pi i\frac{l}{k}}$ for all $1\leq k\leq p-1,
l\in \mathbb{Z}\backslash\{0\}$.

\item[(A3)] (hyperbolic case) $0<|\theta|<1$.

\end{itemize}


\section{Existence of analytic solutions for \eqref{1.3}}

When (A1) is satisfied, that is, $\theta=e^{2\pi i\alpha}$ with
$\alpha$ irrational, small divisors arises inevitably. Since
$(\theta^{n}-1)$ appears in the denominator and the powers of
$\theta$ form a dense subset, there will be $n$ such that
$\frac{1}{\theta^{n}-1}$ is arbitrarily large, see \cite{R04}. In
1942, Siegel \cite{S42} showed a Diophantine condition
 that $\alpha$ satisfies
$$
|\alpha-\frac{p}{q}|>\frac{\gamma}{q^{\delta}}
$$
for some positive $\gamma$ and $\delta$. In 1965, Brjuno \cite{B65}
put forward Brjuno number which satisfies
$$
B(\alpha)=\sum_n\frac{\log q_{n+1}}{q_n}<\infty
$$
and improved Diophantine condition, he showed that as long as
$\alpha$ is a Brjuno number, small divisors is still dealt with
tactfully. In the sequel we discuss the analytic solution of
\eqref{1.3} with Brjuno number $\alpha$. For this purpose, the
Davie's Lemma is necessary.

\begin{lemma}[Davie's Lemma \cite{D94}]
 Assume $K(n)=n\log 2+\sum_{k=0}^{k(n)}g_k(n)\log(2q_{k+1})$, then the
function $K(n)$ satisfies
\begin{itemize}
\item[(a)] There is a universal constant $\tau>0$
(independent of $n$ and of $\alpha$), such that
$$
K(n)\leq n\Big(\sum_{k=0}^{k(n)}\frac{\log q_{k+1}}{q_k}+\tau\Big);
$$

\item[(b)] for all $n_1$ and $n_2$, we have
$K(n_1)+K(n_2)\leq K(n_1+n_2)$;

\item[(c)] $-\log{|\theta^{n}-1|}\leq K(n)-K(n-1)$.
\end{itemize}
\end{lemma}

\begin{theorem} \label{thm2.1}
 Under assumption  {\rm (A1)}, \eqref{1.3} has an analytic solution of the form
\begin{equation}\label{2.1}
g(z)=\sum_{n=1}^{\infty}a_nz^{n},\quad a_1\neq 0.
\end{equation}
\end{theorem}

\begin{proof}
 We prove the formal solution \eqref{2.1} is
convergent in a neighborhood of the origin. From \eqref{1.7}, we
have
\begin{equation} \label{2.2}
\begin{split}
& |a_{n+1}|\leq|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
|\frac{\theta^{n+1-j}}{(\theta^{n}-1)\theta^{n+2}}||a_{n+1-j}||h_{m}||a_{i_1}||a_{i_2}|
\dots|a_{i_{m}}| \\
&=|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
\frac{1}{|\theta^{n}-1|}|a_{n+1-j}||h_{m}||a_{i_1}||a_{i_2}|\dots|a_{i_{m}}| \\
& \leq|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
\frac{1}{|\theta^{n}-1|}|a_{n+1-j}||a_{i_1}||a_{i_2}|\dots|a_{i_{m}}|.
\end{split}
\end{equation}
To construct a majorant series, we define $\{B_n\}_{n=1}^\infty$
by $B_1=|a_1|$ and
\[
B_{n+1}=|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
B_{n+1-j}B_{i_1}B_{i_2}\dots B_{i_{m}},\quad  n=1,2,\dots.
\]
We denote
\begin{equation}\label{2.3}
G(z)=\sum_{n=1}^{\infty}B_nz^{n}.
\end{equation}
Then
\begin{align*}
G(z)&=|a_1|z+\sum_{n=1}^{\infty}B_{n+1}z^{n+1}\\
&=|a_1|z+|b|\sum_{n=1}^{\infty}\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots
+i_{m}=j;}{m=1,2,\dots,j}}
B_{n+1-j}B_{i_1}B_{i_2}\dots B_{i_{m}}z^{n+1}\\
& =|a_1|z+|b|\sum_{n=1}^{\infty}\sum_{j=1}^{n}\frac{G(z)-G^{j+1}(z)}{1-G(z)}
\cdot B_{n+1-j}\cdot z^{n+1-j}\\
& =|a_1|z+|b|\frac{G^{2}(z)-(1-z)G^3(z)-G^{4}(z)}{(1-z)(1-G(z))(1-G^{2}(z))}.
\end{align*}
Let
\begin{equation}\label{2.4}
R(z,\zeta)=\zeta-|a_1|z-|b|\frac{\zeta^{2}-(1-z)\zeta^3
-\zeta^{4}}{(1-z)(1-\zeta)(1-\zeta^{2})}=0.
\end{equation}
We regard \eqref{2.4} as an implicit functional equation, since
$R(0,0)=0$, $R'_{\zeta}(0,0)=1\neq0$. We know that \eqref{2.4} has a unique
analytic solution $\zeta(z)$
 in a neighborhood of the origin
such that $\zeta(0)=0$, $\zeta'(0)=|a_1|$ and $R(z,\zeta(z))=0$, so
 we have $G(z)=\zeta(z)$.
Naturally, there exists constant  $T>0$ such that
 $B_n\leq T^{n},n=1,2,\dots$.
 We now deduce by induction on $n$ that
 \begin{equation}\label{2.5}
|a_{n+1}|\leq B_{n+1}e^{k(n)},\quad\,n\geq0.
\end{equation}
In fact, $|a_1|=B_1$, since $k(0)=0$. We assume that
$|a_{i+1}|\leq B_{i+1}$, $i<n$, $n=1,2,\dots$. Then
\begin{align*}
&|a_{n+1}|\\
&\leq|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
\frac{1}{|\theta^{n}-1|}|a_{n+1-j}||a_{i_1}||a_{i_2}|\dots|a_{i_{m}}|\\
&\leq|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
\frac{1}{|\theta^{n}-1|}B_{n+1-j}e^{k(n-j)}B_{i_1}e^{k(i_1-1)}B_{i_2}e^{k(i_2-1)}
\dots B_{i_{m}}e^{k(i_{m}-1)}\\
& =|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
\frac{1}{|\theta^{n}-1|}B_{n+1-j}B_{i_1}B_{i_2}\dots B_{i_{m}}e^{k(n-m)}\\
& =\frac{1}{|\theta^{n}-1|}B_{n+1}e^{k(n-m)}\\
& \leq\frac{1}{|\theta^{n}-1|}B_{n+1}e^{k(n-1)}\\
& \leq\frac{1}{|\theta^{n}-1|}B_{n+1}e^{\log|\theta^{n}-1|+k(n)}
 =B_{n+1}e^{k(n)},
\end{align*}
by means of Davie's Lemma, thus \eqref{2.5} is proved. Note that
$$
K(n)\leq n(B(\alpha)+\tau)$$ for some universal constant
$\tau>0$. Then
$$
|a_{n+1}|\leq T^{n+1}e^{n(B(\alpha)+\tau)};
$$
that is,
\[
\lim_{n\to\infty}\sup(|a_{n+1}|)^{1/(n+1)}
\leq\lim_{n\to\infty}\sup(T^{n+1}e^{n(B(\alpha)+\tau)})^{1/(n+1)}
=Te^{B(\alpha)+\tau}.
\]
This implies that th radius of convergence for  \eqref{2.1} is at least
$(Te^{B(\alpha)+\tau})^{-1}$, the proof is complete.
\end{proof}

In what follows, we consider the case that the constant $\theta$
is not only on the unit circle, but also a root of unity. Denote the
right side of \eqref{1.7} as
$$
\Lambda(n,\theta)=
b\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
\theta^{n+1-j}a_{n+1-j}h_{m}a_{i_1}a_{i_2}\dots a_{i_{m}}.
$$

\begin{theorem} \label{thm2.2}
 Assume {\rm (A2)} holds and
\begin{equation}\label{2.6}
\Lambda(vp,\theta)\equiv 0,\ v=1,2,\dots.
\end{equation}
Then \eqref{1.3} has an analytic solution of the form
\begin{equation}\label{2.7*}
g(z)=a_1z+\sum_{n=vp,\ v\in \mathbb{N}}\zeta_{vp}z^{n}
+\sum_{n\neq\,vp,\ v\in \mathbb{N}}b_nz^{n},\quad a_1\neq 0,\;
\mathbb{N}=\{1,2,\dots\}
\end{equation}
 in a neighborhood of the origin for some ${\zeta_{vp}}$. Otherwise,
\eqref{1.3} has no analytic solutions in any neighborhood of the
 origin.
\end{theorem}

\begin{proof}
 In this parabolic case
$\theta=e^{2\pi i\frac{q}{p}}$, the eigenvalue $\theta$ is a $p$th root
of unity.

If $\Lambda(vp,\theta)\neq 0$,
for some natural number $v$, then  \eqref{1.7} does not
hold since $\theta^{vp}-1=0$, naturally, \eqref{1.3} has no
formal solutions.

If $\Lambda(vp,\theta)\equiv 0$,
for all natural number $v$, \eqref{1.3} has formal solution
\eqref{2.1}. To prove \eqref{2.1} yields a local analytic solution,
we define the sequence $ \{C_n\}_{n=1}^\infty$ satisfies
$C_1=|a_1|$ and
\begin{equation}\label{2.7}
C_{n+1}=|b|\Gamma\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
C_{n+1-j}C_{i_1}C_{i_2}\dots C_{i_{m}},\ n=1,2,\dots,
\end{equation}
where $\Gamma=\max\{1,|\theta^{i}-1|^{-1}:i=1,2,\dots,p-1\}$. Clearly,
the convergence of series $\sum_{n=1}^{\infty}C_nz^{n}$ can
be proved similar as in Theorem \ref{thm2.1}.

When \eqref{2.6} holds for all natural number $v$, the coefficients
$a_{vp}$ have infinitely
  many choices in $\mathbb{C}$, choose  $a_{vp}= \zeta_{vp}$ arbitrarily such that
  \begin{equation}\label{2.8}
|a_{vp}|\leq C_{vp},\quad  v=1,2,\dots.
\end{equation}
Furthermore, we can prove
\begin{equation}\label{2.9}
|a_n|\leq C_n,\quad n\neq vp.
\end{equation}
 In fact, $|a_1|=C_1$. If we suppose that
$|a_{i+1}|\leq C_{i+1},\ i<n\ (n\neq vp)$, then
\begin{align*}
|a_{n+1}|
&\leq|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
|\frac{\theta^{n+1-j}}{(\theta^{n}-1)\theta^{n+2}}||a_{n+1-j}||h_{m}||a_{i_1}||a_{i_2}|
\dots|a_{i_{m}}|\\
& \leq|b|\Gamma\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
C_{n+1-j}C_{i_1}C_{i_2}\dots C_{i_{m}}\\
& =C_{n+1}.
\end{align*}
From \eqref{2.8}, \eqref{2.9} and the convergence of series
$\sum_{n=1}^{\infty}C_nz^{n}$, the formal solution \eqref{2.1}
yields a local analytic solution \eqref{2.7*} in a neighborhood of
the origin. This completes the proof.
\end{proof}

\begin{theorem} \label{thm2.3}
\ Suppose {\rm (A3)} holds, then
\eqref{1.3} has an analytic solution of the form
\begin{align*}
g(z)=\sum_{n=1}^{\infty}a_nz^{n},\ \ a_1\neq 0.
\end{align*}
\end{theorem}

\begin{proof}
 We prove the formal solution \eqref{2.1} is convergent in
a neighborhood of the origin. Since $0<|\theta|<1$, so
$\lim_{n\to\infty}\frac{1}{\theta^{n}-1}=1$, From
\eqref{1.7}, we have
\begin{equation} \label{2.10}
\begin{split}
|a_{n+1}|
&\leq|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
|\frac{\theta^{n+1-j}}{(\theta^{n}-1)\theta^{n+2}}||a_{n+1-j}||h_{m}||a_{i_1}||a_{i_2}|
\dots|a_{i_{m}}| \\
& \leq|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
\frac{1}{|\theta^{1+j}|}|a_{n+1-j}||a_{i_1}||a_{i_2}|\dots|a_{i_{m}}|.
\end{split}
\end{equation}
Let $\{D_n\}_{n=1}^\infty$ be defined by $D_1=|a_1|$
and
\begin{equation*}
D_{n+1}=|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
\frac{1}{|\theta^{1+j}|}D_{n+1-j}D_{i_1}D_{i_2}\dots D_{i_{m}},\quad
n=1,2,\dots.
\end{equation*}
Denote
\begin{equation}\label{2.11}
F(z)=\sum_{n=1}^{\infty}D_nz^{n}.
\end{equation}
Then
\begin{align*}
 F(z)&=|a_1|z+\sum_{n=1}^{\infty}D_{n+1}z^{n+1}\\
& =|a_1|z+|b|\sum_{n=1}^{\infty}\sum_{j=1}^{n}
 \sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}}
\frac{1}{|\theta^{1+j}|}D_{n+1-j}D_{i_1}D_{i_2}\dots D_{i_{m}}z^{n+1}\\
& =|a_1|z+|b|\sum_{n=1}^{\infty}\sum_{j=1}^{n}\frac{1}{|\theta^{1+j}|}
\cdot\frac{F(z)-F^{j+1}(z)}{1-F(z)}\cdot D_{n+1-j}\cdot z^{n+1-j}\\
& =|a_1|z+|b|\frac{\theta
F^2(z)}{(\theta^{2}-1)(\theta-F(z))}.
\end{align*}
Let
\begin{equation}\label{2.13}
Q(z,\xi)=\xi-|a_1|z-|b|\frac{\theta
\xi^{2}}{(\theta^{2}-1)(\theta-\xi)}=0.
\end{equation}
Since $Q(0,0)=0$, $Q'_{\xi}(0,0)=1\neq0$, then \eqref{2.13} has a
unique analytic solution $\xi(z)$
 in a neighborhood of the origin
such that $\xi(0)=0$, $\xi'(0)=|a_1|$ and $Q(z,\xi(z))=0$, so we
have $F(z)=\xi(z)$. Similar as in Theorem \ref{thm2.2}, we can prove
 \begin{equation}\label{2.14}
|a_n|\leq D_n,\quad  n=1,2,\dots,
\end{equation}
by induction. Then the local analytic solution \eqref{2.1} is
existent in a neighborhood of the origin by means of the convergence
of $\sum_{n=1}^{\infty}D_n$ and inequality \eqref{2.14}.
This completes the proof.
\end{proof}


\section{Formal solutions and analytic solutions of \eqref{1.1}}

In this section we prove the existence of formal solutions and analytic
solutions of \eqref{1.1}.

 \begin{theorem} \label{thm3.1}
Equation \eqref{1.2} has a formal solution
$y(z)=g(\theta\,g^{-1}(z))$ in a neighborhood of the origin, where
$g(z)$ is formal solution of \eqref{1.3}. Under one of the
conditions in Theorems \ref{thm2.1}--\ref{thm2.3}, every formal solution yields an
analytic solution of the form $y(z)=g(\theta\,g^{-1}(z))$, where
$g(z)$ is analytic solution of \eqref{1.3}.
\end{theorem}

\begin{proof}
 Since $g'(0)=a_1\neq0$, the inverse $g^{-1}(z)$
exists in a neighborhood of $g(0)=0$. If we define
$y(z)=g(\theta\,g^{-1}(z))$, then
\begin{equation}
\begin{split}
 y(y(z))-ay(z)
&=g(\theta(g^{-1}(g\theta(g^{-1}(z)))))-ag(\theta(g^{-1}(z))) \\
&=g(\theta^{2}(g^{-1}(z)))-ag(\theta(g^{-1}(z))) \\
&=b(g\theta(g^{-1}(z))H(g^{-1}(z)) \\
&= by(z)H(z).
\end{split}\label{3.1}
\end{equation}
as required, so \eqref{1.2} has a formal solution
$y(z)=g(\theta\,g^{-1}(z))$ in a neighborhood of the origin.

Under one of the conditions in Theorems \ref{thm2.1}--\ref{thm2.3}, the inverse
$g^{-1}(z)$ exists and is analytic in a neighborhood of $g(0)=0$, we
obtain analytic solutions of \eqref{1.2} in a neighborhood of the
origin. The proof is completed.
\end{proof}

Suppose that
$$
y(z)=\theta z+b_2 z^2+b_3 z^3+\dots,
$$
since $ a=\theta$  and $x(z)=\frac{y(z)-az}{bz}$, it follows that
\begin{equation}\label{3.2}
x(z)=\frac{b_2}{b}z+\frac{b_3}{b} z^2+\frac{b_4}{b} z^3+\dots.
\end{equation} 
That is, \eqref{1.1} has a unique formal solution with the form
\eqref{3.2} in a neighborhood of the origin. The formal solution
also is analytic solution when $y(z)$ is analytic in a neighborhood
of the origin.


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\end{document}