\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 $$\sum_{j=0}^{k}\sum_{t=1}^{\infty}C_{t,j}(z)(x(q^{j}z))^t=G(z),\quad z\in{\mathbb{C}}, \label{XU}$$ 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 $$\label{1.1} x(az+bzx(z))=H(z)$$ 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 $$\label{1.2} y(y(z))-ay(z)=by(z)H(z).$$ Using the conversion $y(z)=g(\theta(g^{-1}(z)))$, Equation \eqref{1.2} transforms into the equation without functional iteration $$\label{1.3} g(\theta^{2}z)-ag(\theta z)= bg(\theta z)H(g(z)).$$ Suppose $$\label{1.2*} g(z)=\sum_{n=1}^{\infty}a_nz^{n},\quad H(z)=\sum_{n=1}^{\infty}h_nz^{n}.$$ Substituting \eqref{1.2*} into \eqref{1.3}, we obtain $$\label{1.4} \begin{split} &(\theta^{2}-a\theta)a_1z+\sum_{n=1}^{\infty}(\theta^{2(n+1)} -a\theta^{n+1})a_{n+1}z^{n+1} \\ &=b\sum_{n=1}^{\infty}\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2 +\dots+i_{m}=j;}{m=1,2,\dots,j}} \theta^{n+1-j}a_{n+1-j}h_{m}a_{i_1}a_{i_2}\dots a_{i_{m}}z^{n+1}. \end{split}$$ Comparing coefficients, we obtain $$\label{1.5} (\theta^{2}-a\theta)a_1=0,$$ and $$\label{1.6} (\theta^{2(n+1)}-a\theta^{n+1})a_{n+1} =b\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}} \theta^{n+1-j}a_{n+1-j}h_{m}a_{i_1}a_{i_2}\dots a_{i_{m}}.$$ Under $a_1\neq0$, the equality \eqref{1.5} implies that $\theta=a$, then \eqref{1.6} turns into $$\label{1.7} (\theta^{n}-1)\theta^{n+2}a_{n+1} =b\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}} \theta^{n+1-j}a_{n+1-j}h_{m}a_{i_1}a_{i_2}\dots a_{i_{m}}.$$ 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 $$\label{2.1} g(z)=\sum_{n=1}^{\infty}a_nz^{n},\quad a_1\neq 0.$$ \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 $$\label{2.2} \begin{split} & |a_{n+1}|\leq|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}} |\frac{\theta^{n+1-j}}{(\theta^{n}-1)\theta^{n+2}}||a_{n+1-j}||h_{m}||a_{i_1}||a_{i_2}| \dots|a_{i_{m}}| \\ &=|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}} \frac{1}{|\theta^{n}-1|}|a_{n+1-j}||h_{m}||a_{i_1}||a_{i_2}|\dots|a_{i_{m}}| \\ & \leq|b|\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}} \frac{1}{|\theta^{n}-1|}|a_{n+1-j}||a_{i_1}||a_{i_2}|\dots|a_{i_{m}}|. \end{split}$$ 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 $$\label{2.3} G(z)=\sum_{n=1}^{\infty}B_nz^{n}.$$ 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 $$\label{2.4} R(z,\zeta)=\zeta-|a_1|z-|b|\frac{\zeta^{2}-(1-z)\zeta^3 -\zeta^{4}}{(1-z)(1-\zeta)(1-\zeta^{2})}=0.$$ 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 $$\label{2.5} |a_{n+1}|\leq B_{n+1}e^{k(n)},\quad\,n\geq0.$$ In fact, $|a_1|=B_1$, since $k(0)=0$. We assume that $|a_{i+1}|\leq B_{i+1}$, $i0$. 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 $$\label{2.6} \Lambda(vp,\theta)\equiv 0,\ v=1,2,\dots.$$ Then \eqref{1.3} has an analytic solution of the form $$\label{2.7*} g(z)=a_1z+\sum_{n=vp,\ v\in \mathbb{N}}\zeta_{vp}z^{n} +\sum_{n\neq\,vp,\ v\in \mathbb{N}}b_nz^{n},\quad a_1\neq 0,\; \mathbb{N}=\{1,2,\dots\}$$ 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 $$\label{2.7} C_{n+1}=|b|\Gamma\sum_{j=1}^{n}\sum_{\stackrel{i_1+i_2+\dots+i_{m}=j;}{m=1,2,\dots,j}} C_{n+1-j}C_{i_1}C_{i_2}\dots C_{i_{m}},\ n=1,2,\dots,$$ 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 $$\label{2.8} |a_{vp}|\leq C_{vp},\quad v=1,2,\dots.$$ Furthermore, we can prove $$\label{2.9} |a_n|\leq C_n,\quad n\neq vp.$$ In fact, $|a_1|=C_1$. If we suppose that \$|a_{i+1}|\leq C_{i+1},\ i