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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 168, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2013/168\hfil Maximum number of limit cycles]
{Maximum number of limit cycles for generalized
 Li\'enard differential equations}

\author[S. Badi, A. Makhlouf \hfil EJDE-2013/168\hfilneg]
{Sabrina Badi, Amar Makhlouf}  % in alphabetical order

\address{Sabrina Badi \newline
 Department of Mathematics, University of Guelma \\
 P.O. Box 401, Guelma 24000, Algeria}
\email{badisabrina@yahoo.fr}

\address{Amar Makhlouf \newline
 Department of Mathematics, University of Annaba \\
 P.O. Box 12, Annaba 23000, Algeria}
\email{makhloufamar@yahoo.fr}

\thanks{Submitted February 17, 2013. Published July 22, 2013}
\subjclass[2000]{34C25, 34C29, 54D10, 34G15}
\keywords{Limlit cycle; averaging theory; Li\'enard equation}

\begin{abstract}
 Applying the averaging theory of first and second order to a
 class of generalized polynomial Li\'enard differential equations,
 we improve the known lower bounds for the maximum number of limit
 cycles that this class can exhibit.
\end{abstract}

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

\section{Introduction and statement of the main results}


One of the main problems in the theory of ordinary differential equations
is the study of their limit cycles, their existence, their number and their
stability. A limit cycle of a differential equation is a periodic orbit in
the set of all isolated periodic orbits of the differential equation.
These last years hundreds of papers studied the limit cycles of planar
polynomial differential systems. The Second part of the 16th Hilbert's
problem \cite{D} is related with the least upper bound on the number of
limit cycles of polynomial vector fields having a fixed degree.
The generalized polynomial Li\'enard differential equation
\begin{equation}\label{L}
 \ddot{x}+f(x)\dot{x}+g(x)=0.
\end{equation}
 was introduced in \cite{A}. Here the dot denotes differentiation with respect
to the time $t$, and $f(x)$ and $g(x)$ are polynomials in the variable $x$ of
degrees $n$ and $m$ respectively. The Li\'enard equation, which is often taken
as the typical example of nonlinear self-excited vibration problem, can be used
to model resistor-inductor-capacitor circuits with nonlinear circuit elements.
It can also be used to model certain mechanical systems which contain the
nonlinear damping coefficients and the restoring force or stiffness.
Limit cycles usually arise at a Hopf bifurcation in nonlinear systems with
varying parameters. In mechanical systems, the varying parameter is frequently
 a damping coefficient (see \cite{AA,DingL}). Lots of papers discussed the
possible number of limit cycle of Li\'enard or generalized mixed
Rayleigh-Li\'enard oscillators. Ding and Leung \cite{DingL} investigated
the generalized mixed Rayleigh-Li\'enard oscillator with highly nonlinear terms.
They consider mainly the number of limit cycle bifurcation diagrams of these
systems. For the subclass of polynomial vector fields \eqref{L} we have a
simplified version of Hilbert's problem, see \cite{ADP,SS}.

 Many of the results on the limit cycles of polynomial differential systems
have been obtained by considering limit cycles which bifurcate from a single
degenerate singular point, that are so called \emph{small amplitude limit cycles},
 see \cite{NGL,SL}. We denote by $\hat{H}(m,n)$ the maximum number of small
amplitude limit cycles for systems of the form \eqref{L}. The values of
$\hat{H}(m,n)$ give a lower bound for the maximum number $H(m,n)$
(i.e. the Hilbert number) of limit cycles that the differential equation
\eqref{L} with $m$ and $n$ fixed can have. For more information about the
Hilbert's 16th problem and related topics see {\cite{YI}} and {\cite{JL}}.

 Now we shall describe briefly the main results about the limit cycles on
Li\'enard differential systems as it is described in \cite{LMT}.
 \begin{itemize}
 \item In 1928 Li\'enard {\cite{A}} proved that if $m=1$ and
$F(x)=\int^x_{0}f(s)ds$ is a continuous odd function , which has a unique
root at $x=a$ and is monotone increasing for $x\geq a$, then equation \eqref{L}
has a unique limit cycle.

 \item In 1973 Rychkov \cite{GSR} proved that if $m=1$ and
$F(x)=\int^x_{0}f(s)ds$ is an odd polynomial of degree five, then
equation \eqref{L} has at most two limit cycles.

 \item In 1977 Lins, de Melo and Pugh \cite{ADP} proved that $H(1,1)=0$
 and $H(1,2)=1$.

 \item In 1990, 1996, Dumortier, Li and Rousseau in \cite{DR} and \cite{DL}
 proved that $H(3,1)=1$.

 \item In 1998 Coppel \cite{WAC} proved that $H(2,1)=1$.

 \item In 1997 Dumortier and Chengzhi \cite{DChengzhi}
 proved that $H(2,2)=1$.
 \item In 2010 Chengzhi Li and Llibre \cite{LiLlibrea} proved that $H(1,3)=1$.

 \end{itemize}
 Blows, Lloyd \cite{BLL} and Lynch \cite{LLL,SLynch}
 have used inductive arguments in order to prove the following results.
 \begin{itemize}
 \item If $g$ is odd then $\hat{H}(m,n)=[\frac{n}{2}]$.
 \item If $f$ is even then $\hat{H}(m,n)=n$, whatever $g$ is.
 \item If $f$ is odd then $\hat{H}(m,2n+1)=[\frac{(m-2)}{2}]+n$.
 \item If $g(x)=x+g_{e}(x)$, where $g_{e}$ is even then $\hat{H}(2m,2)=m$.
 \end{itemize}
Christopher and Lynch \cite{CHLY} developed a new algebraic method for
determining the Liapunov quantities of system \eqref{L} and proved
the following:
 \begin{itemize}
 \item $\hat{H}(m,2)=[\frac{(2m+1)}{3}]$,
 \item $\hat{H}(2,n)=[\frac{(2n+1)}{3}]$,
 \item $\hat{H}(m,3)=2[\frac{(3m+2)}{8}]$ for all $1<m\leq50$,
 \item $\hat{H}(3,n)=2[\frac{(3n+2)}{8}]$ for all $1<n\leq50$,
 \item $\hat{H}(4,k)=\hat{H}(k,4)$ for $k=6,7,8,9$ and
$\hat{H}(5,6)=\hat{H}(6,5)$.
\end{itemize}
In 1998, Gasull and Torregrosa \cite{GT} obtained upper bounds for
$\hat{H}(7,6)$, $\hat{H}(6,7)$, $\hat{H}(7,7)$ and $\hat{H}(4,20)$.
In 2006, Yu and Han \cite{YH} proved that $\hat{H}(m,n)=\hat{H}(n,m)$
for $n=4$, $m=10,11,12,13$; $n=5$, $m=6,7,8,9$; $n=6$, $m=5,6$.
In 2009, Llibre, Mereu and Teixeira \cite{LMT} using the averaging theory
studied the maximum number of limit cycles $\tilde{H}(m,n)$
which can bifurcate from the periodic orbits of a linear center
perturbed inside the class of generalized polynomial Li\'enard differential
equations of degrees $m$ and $n$ of the form
\begin{equation}\label{A}
\begin{gathered}
\dot{x}=y, \\
\dot{y}=-x- \sum_{k \geq 1} \epsilon^k (f_{n}^k(x)y+g_{m}^k(x)),
\end{gathered}
\end{equation}
where for every $k$ the polynomials $g_{m}^k(x)$ and $f_{n}^k(x)$ have
degree $m$ and $n$ respectively, and $\varepsilon$ is a small parameter.
In 2011, Badi and Makhlouf {\cite{BM}} using the averaging theory studied
the maximum number of limit cycles $\tilde{H}(m,n)$ which can bifurcate
from the periodic orbits of a linear center perturbed inside the class of
generalized polynomial Li\'enard differential equations of degrees $m$
and $n$ as follows:
\begin{equation}\label{B}
\begin{gathered}
\dot{x}=y, \\
\dot{y}=-x- \sum_{k \geq 1} \epsilon^k (f_{n}^k(x,y)y+g_{m}^k(x)),
\end{gathered}
\end{equation}
where for every $k$ the polynomial $g_{m}^k(x)$ has degree $m$,
 the polynomial $f_{n}^k(x,y)$ has degree $n$ on $x$ and $y$ and
$\varepsilon$ is a small parameter; i.e., the maximal number of
\textit{medium amplitude limit cycles} which can bifurcate from the periodic
orbits of the linear center $\dot{x}=y, \dot{y}=-x$, perturbed as in \eqref{B}.
In fact in {\cite{BM}} the authors computed lower estimations of
 $\tilde{H}(m,n)$. More precisely they compute the maximum number of limit
 cycles $\tilde{H}_{k}(m,n)$ which bifurcate from the periodic orbits of
the linear center $\dot{x}=y, \dot{y}=-x$, using the averaging theory
of order $k$, for $k=1,2$. Of course
$\tilde{H}_{k}(m,n) \leq \tilde{H}(m,n) \leq H(m,n)$.

In this work using the averaging theory we study the maximum number of limit
cycles $\tilde{H}(l,m,n)$ which can bifurcate from the periodic orbits of
 a linear center perturbed inside the class of generalized polynomial
Li\'enard differential equations of degrees $l$, $m$ and $n$ of the form
\begin{equation}\label{C}
\begin{gathered}
\dot{x}=y+\sum_{k \geq 1} \epsilon^k h_{l}^k(x), \\
\dot{y}=-x- \sum_{k \geq 1} \epsilon^k (f_{n}^k(x,y)y+g_{m}^k(x)),
\end{gathered}
\end{equation}
where for every $k$ the polynomials $h_{l}^k(x)$, $g_{m}^k(x)$ and
$f_{n}^k(x,y)$ have degree $l$, $m$ and $n$ respectively and
$\varepsilon$ is a small parameter, i.e. the maximal number of
\textit{medium amplitude limit cycles} which can bifurcate from the
 periodic orbits of the linear center $\dot{x}=y, \dot{y}=-x$,
perturbed as in \eqref{C}.

Let $k$ be a positive integer. We define $E(k)$ as the largest even
integer less than or equal to $k$, and $O(k)$ as the largest odd integer
less than or equal to $k$. The main result that improve the mentioned
previous results is the following.



\begin{theorem} \label{thm1}
If for every $k=1,2$ the polynomials $h_{l}^k(x)$, $g_{m}^k(x)$ and
$f_{n}^k(x,y)$ have degree $l$, $m$ and $n$ respectively, with
$l,m,n \geq 1$, then for $|\varepsilon|$ sufficiently small, the
maximum number of medium limit cycles of the polynomial Li\'enard differential
systems \eqref{C} bifurcating from the periodic orbits of the linear
center $\dot{x}=y, \dot{y}=-x$, using the averaging theory
\begin{itemize}
\item[(a)] of first order
\[
\tilde{H}_{1}(l,m,n)=\Big[\frac{\max\{O(l), O(n+1)\}-1}{2}\Big]
= \max \big\{\big[\frac{l-1}{2}\big], \big[\frac{n}{2}\big]\big\}
\]
\item[(b)] of second order
\begin{align*}
\tilde{H}_{2}(l,m,n)
&= \Big[ \Big(\max\Big\{O(n)+O(m)+1, O(n)+E(l)+1, E(m)+E(l),\\
&\quad 2 O(n)+2, O(l), O(n+1) \Big\}-1\Big)/2 \Big]
\end{align*}
\end{itemize}
\end{theorem}

Of course if $H(l,m,n)$ is the Hilbert number for our polynomial
Li\'enard differential systems \eqref{C}, then
$\tilde{H}_{k}(l,m,n)\neq H(l,m,n)$ for $k=1,2$; i.e. the numbers
$\tilde{H}_{k}(l,m,n)$ provide lower bounds for the Hilbert numbers
of systems \eqref{C}.

This paper is structured as follows. In section 2 we present a summary
of the results on the averaging theory that we we shall need in this paper.
In sections 3 and 4 we prove statements (a) and (b) of Theorem 1 respectively.

\section{The averaging theory of first and second order}

In the proof of our main result we use the averaging theory as it is presented
in \cite{BL}.
Consider the differential system
\begin{equation}\label{1.2}
 x'(t)=\epsilon F_{1}(t,x)+ \epsilon^2 F_{2}(t,x)+ \epsilon^3
R(t,x, \epsilon),
\end{equation}
where $F_{1},F_{2}:\mathbb{R}\times D \to \mathbb{R}^n, R:\mathbb{R}
\times D \times (-\epsilon_{f},\epsilon_{f}) \to \mathbb{R}^n$
are continuous functions, $T$-periodic in the first variable, and $D$ is
an open subset of $\mathbb{R}^n$. Assume that the following hypotheses (i)
and (ii) hold.
\begin{itemize}
\item[(i)] $F_{1}(t,.) \in C^1(D)$ for all $t \in \mathbb{R}$,
$F_{1}, F_{2},R, D_{x}F_{1}$ are locally Lipschitz with respect to $x$,
and $R$ is differentiable with respect to $\epsilon$.
We define
\begin{gather*}
 F_{10}(z)=\frac{1}{T} \int^T_{0} F_{1}(s,z)ds,\\
 F_{20}(z)=\frac{1}{T} \int^T_{0} \big[ D_{z}F_{1}(s,z)
 y_{1}(s,z) + F_{2}(s,z) \big] ds,
 \end{gather*}
 where
 $$
 y_{1}(s,z)=\int^s_{0} F_{1}(t,z)dt.
 $$

 \item[(ii)] For $V \subset D$ an open and bounded set and for each
$ \epsilon \in (-\epsilon_{f},\epsilon_{f})\backslash\{ 0 \} $, there exists
 $a_{\epsilon} \in V$ such that $F_{10}(a_{\epsilon})
+ \epsilon F_{20}(a_{\epsilon})=0$ and $d_{B}(F_{10}
+ \epsilon F_{20},V,a_{\epsilon})\neq 0$.
 \end{itemize}
Then, for $|\epsilon|>0$ sufficiently small there exists a $T$-periodic
solution $\varphi(., \epsilon)$ of the system \eqref{1.2} such that
$\varphi(0, \epsilon)=a_{\epsilon}$.

The expression $d_{B}(F_{10}+ \epsilon F_{20},V,a_{\epsilon})\neq 0$
means that the Brouwer degree of the function
$F_{10}+ \epsilon F_{20}: V \to \mathbb{R}^n$ at the fixed point
 $a_{\epsilon}$ is not zero. A sufficient condition for the inequality to be
true is that the Jacobian of the function $F_{10}+ \epsilon F_{20}$ at
$a_{\epsilon}$ is not zero.

If $F_{10}$ is not identically zero, then the zeros of $F_{10}+ \epsilon F_{20}$
are mainly the zeros of $F_{10}$ for $\epsilon$ sufficiently small.
In this case the previous result provides the
\textit{averaging theory of first order}.

If $F_{10}$ is identically zero and $F_{20}$ is not identically zero,
then the zeros of $F_{10}+ \epsilon F_{20}$ are mainly the zeros of
$F_{20}$ for $\epsilon$ sufficiently small. In this case the previous
result provides the \textit{averaging theory of second order}.
For more information about the averaging theory see \cite{SV,V}.

\section{Proof of statement (a) of Theorem 1}


We shall need the first order averaging theory to prove statement (a) of
Theorem 1.  In order to apply the first order averaging method we
write system \eqref{C} with $k=1$, in polar coordinates $(r, \theta)$
where $x=rcos(\theta), y=rsin(\theta), r>0$. In this way system \eqref{C}
is written in the standard form for applying the averaging theory.
If we write $f_{n}^1(x,y)=\sum^n_{i+j=0}a_{ij}x^iy^j$,
$g_{m}^1(x)=\sum^m_{i=0}b_{i}x^i$ and $h_{l}^1(x)=\sum^l_{i=0}c_{i}x^i$
then system \eqref{C} becomes
\begin{equation}\label{1.3}
\begin{gathered}
\begin{aligned}
\dot{r}&=\epsilon \Big[\sum^l_{i=0}c_{i}r^i \cos^{i+1}(\theta)
 -r \sin^2(\theta)\sum^n_{i+j=0}a_{ij} r^{i+j} \cos^i(\theta) \sin^{j}(\theta)\\
 &\quad - \sin(\theta)\sum^m_{i=0}b_{i} r^i cos^i(\theta)\Big]+O(\epsilon^2),
\end{aligned} \\
\begin{aligned}
\dot{\theta}&=-1- \frac{\epsilon}{r}\Big[r \cos(\theta)\sin(\theta)
 \sum^n_{i+j=0}a_{ij} r^{i+j} \cos^{i}(\theta) sin^{j}(\theta)\\
&\quad  + \cos(\theta)\sum^m_{i=0}b_{i} r^i \cos^{i}(\theta)
 + \sin(\theta) \sum^l_{i=0}c_{i} r^i \cos^{i}(\theta)\Big]
+O(\epsilon^2).
\end{aligned}
\end{gathered}
\end{equation}
Now taking $\theta$ as the new independent variable, this system becomes
\begin{align*}
 \frac{dr}{d \theta}
&= -\epsilon \Big(\sum^l_{i=0}c_{i} r^i \cos^{i+1}(\theta)
 -r \sin^2(\theta)\sum^n_{i+j=0}a_{ij}r^{i+j}\cos^{i}(\theta) \sin^{j}(\theta)\\
&\quad -\sin(\theta) \sum^m_{i=0}b_{i} r^i \cos^i(\theta)\Big) + O(\epsilon^2)
\\
&=\epsilon F_{1}(\theta,r)+O(\epsilon^2).
\end{align*}

Using the notation introduced in section 2 we have
\begin{align*}
 F_{10}(r)&=\frac{-1}{2\pi} \int^{2\pi}_{0}
\Big(\sum^l_{i=0}c_{i} r^i \cos^{i+1}(\theta)
-r \sin^2(\theta)\sum^n_{i+j=0}a_{ij}r^{i+j}\cos^{i}(\theta) \sin^{j}(\theta)\\
&\quad -\sin(\theta) \sum^m_{i=0}b_{i} r^i \cos^i(\theta) \Big) d\theta.
\end{align*}
Since
$$
\int^{2\pi}_{0}\cos^{i+1}(\theta)d\theta=\begin{cases}
0 &\text{if $i$ is even} \\
\alpha_{i}\neq 0 &\text{if $i$ is odd},
\end{cases}
$$
it follows that
\begin{gather*}
\int^{2\pi}_{0}\cos^{i}(\theta)\sin^{j+2}(\theta)d\theta
=\begin{cases}
0 &\text{if $i$ odd  and  $j$ is odd} \\
\beta_{ij}\neq 0 &\text{if $i$ is even  and  $j$  even},
\end{cases}
\\
\int^{2\pi}_{0}\sin(\theta)\cos^{i}(\theta)d\theta=0
\end{gather*}
for $i=0,1,\dots,$ we have 
\begin{align*}
 F_{10}(r)
&=\frac{-1}{2\pi} \int^{2\pi}_{0}
\Big(\sum^l_{i=1,\,i \text{ odd}} c_{i} r^i \cos^{i+1}(\theta)\\
&-\sum^n_{i+j=0,\text{ $i$ even $j$ even}}
a_{ij}r^{i+j+1}\cos^{i}(\theta) \sin^{j+2}(\theta)\Big) d\theta.
\end{align*}
We define
$$
M(l,n)=\begin{cases}
\max\{l,n+1\} &\text{if  $l$ is odd, $n$ is even} \\
\max\{l-1,n+1\} &\text{if $l$ is even, $n$ is even} \\
\max\{l,n\} &\text{if $l$ is odd, $n$ is odd} \\
\max\{l-1,n\} &\text{if $l$ is even, $n$ is odd}.
\end{cases}
$$
Therefore,
$$
M(l,n)=\max\{ O(l), O(n+1) \}
$$
and
$$
\Big[ \frac{M(l,n)-1}{2} \Big]=\Big[\frac{\max\{O(l), O(n+1)\}-1}{2}\Big]
= \max \Big\{\Big[\frac{l-1}{2}\Big], \Big[\frac{n}{2}\Big]\Big\}
$$
finally, we have
$$
F_{10}(r)=\sum^{M(l,n)}_{k=1,\, k \text{ odd}} \sigma_{k}r^k,
$$
with
$$
\sigma_{k}=\frac{-1}{2\pi} \int^{2\pi}_{0}
\left(c_{k} \cos^{k+1}(\theta)-a_{(k-1-j)j}\cos^{k-1-j}(\theta) \sin^{j+2}(\theta)\right) d\theta,
$$
where $k\geq 1$ is an odd integer number and $ j\geq 0$ is an even one.
Since $F_{10}(r)$ is an odd function, it has at most
$[(M(l,n)-1)/2]$ simple positive real roots. From section 2 we obtain
that for $|\epsilon|$ sufficiently small, the maximum number of limit
cycles of system \eqref{C} which can bifurcate from the periodic orbits
of the linear center $\dot{x}=y$, $\dot{y}=-x$ using the averaging theory
of first order is $[(M(l,n)-1)/2 ]$. Hence statement (a) of Theorem 1 is proved.

\section{Proof of statement (b) of Theorem 1}

For proving statement (b) of Theorem 1 we shall use the second order
averaging theory. In this section we consider the differential systems
\begin{equation}\label{D}
\begin{gathered}
\dot{x}=y+\epsilon h_{l}^1(x)+\epsilon^2 h_{l}^2(x)+O(\epsilon^3), \\
\dot{y}=-x-\epsilon(f_{n}^1(x,y)y+g_{m}^1(x))-\epsilon^2(f_{n}^2(x,y)y
 +g_{m}^2(x))+O(\epsilon^3).
\end{gathered}
\end{equation}
where
$$
h_{l}^2(x)=\sum^l_{i=0}\hat{c_{i}}x^i, \quad
f_{n}^2(x,y)=\sum^n_{i+j=0}\hat{a_{ij}}x^iy^j, \quad
g_{m}^2(x)=\sum^m_{i=0}\hat{b_{i}}x^i
$$
Then system \eqref{D} in polar coordinates $(r,\theta), r>0$ becomes
\begin{gather*}
\dot{r}
=\epsilon \frac{xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x)}{r}
+\epsilon^2\frac{xh_{l}^2(x)-y^2f_{n}^2(x,y)-yg_{m}^2(x)}{r}+O(\epsilon^3),
\\
\begin{aligned}
 \dot{\theta}
&=-1- \epsilon \frac{xyf_{n}^1(x,y)+xg_{m}^1(x)+yh_{l}^1(x)}{r^2}
-\epsilon^2 \frac{xyf_{n}^2(x,y)+xg_{m}^2(x)+yh_{l}^2(x)}{r^2}\\
&\quad +O(\epsilon^3).
\end{aligned}
\end{gather*}
Taking $\theta$ as the new independent variable, this system becomes
\begin{align*}
\frac{dr}{d \theta}
&=\epsilon \frac{xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x)}{r}-\epsilon^2
\Big[\frac{xh_{l}^2(x)-y^2f_{n}^2(x,y)-yg_{m}^2(x)}{r}\\
&\quad -\frac{(xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x))(xyf_{n}^1(x,y)
 +xg_{m}^1(x)+yh_{l}^1(x))}{r^3}\Big]
\\
&\quad -\epsilon^3\Big[ \frac{(xh_{l}^1(x)-y^2f_{n}^1(x,y)
 -yg_{m}^1(x))(xyf_{n}^2(x,y)+xg_{m}^2(x)+yh_{l}^2(x))}{r^3}\\
&\quad +  \frac{(xh_{l}^2(x)-y^2f_{n}^2(x,y)-yg_{m}^2(x))(xyf_{n}^1(x,y)
 +xg_{m}^1(x)+yh_{l}^1(x))}{r^3}\\
&\quad - \frac{(xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x))(xyf_{n}^1(x,y)
 +xg_{m}^1(x)+yh_{l}^1(x))^2}{r^5} \Big]+O(\epsilon^4)
\\
&=\epsilon F_{1}(\theta,r)+\epsilon^2 F_{2}(\theta,r)
 +\epsilon^3 F_{3}(\theta,r)+ O(\epsilon^4),
\end{align*}

Now we determine the corresponding function
$$
F_{20}= \frac{1}{2\pi}\int_{0}^{2\pi}
\Big[\frac{d}{dr}F_{1}(\theta,r). \int^{\theta}_{0}F_{1}(\phi,r) d \phi
+ F_{2}(\theta,r) \Big]d\theta.
$$
For this we put $F_{10}\equiv 0$ which is equivalent to
\begin{gather*}
c_{i}=0 \quad \text{for $i$  odd, and} \\
a_{ij}=0 \quad \text{for $i$  even  and  $j$  even}
\end{gather*}
First, we have
\begin{align*}
 \frac{d}{dr}F_{1}(\theta,r)
&=-\sum^l_{i=2, \text{ even}} i c_{i}r^{i-1} \cos^{i+1}(\theta)\\
&\quad + \sum^n_{i+j=2, \text{ $i$ odd  or $j$  odd}}
(i+j+1) a_{ij}r^{i+j} \cos^i(\theta) \sin^{j+2}(\theta)\\
&\quad +\sum^m_{i=1} i b_{i}r^{i-1} \cos^i(\theta)\sin(\theta),
\end{align*}
and
\begin{align*}
\int^{\theta}_{0}F_{1}(\phi,r) d \phi
&=-\sum^l_{i=0,\, i \text{ even}} c_{i}r^i
\int^{\theta}_{0}\cos^{i+1}(\phi)d\phi\\
&\quad + \sum^n_{i+j=1,\text{ $i$  odd  or $j$ odd}}
a_{ij}r^{i+j+1}\int^{\theta}_{0}\cos^i(\phi)\sin^{j+2}(\phi)d\phi
\\
&\quad + \sum^m_{i=0}b_{i}r^i \int^{\theta}_{0}cos^i(\phi)sin(\phi)d\phi
\\
&=-\sum^l_{i=0,\, i \text{ even}}c_{i}r^i A_{i+1}(\theta)
+\sum^n_{i+j=1,\text{ $i$  odd  or $j$  odd}}
a_{ij}r^{i+j+1}A_{i,(j+2)}(\theta)\\
&\quad + \sum^m_{i=0}b_{i}r^i
\Big( \frac{1-cos^{i+1}(\theta)}{i+1} \Big).
\end{align*}
where
\begin{align*}
A_{i}(\theta)
&=\int^{\theta}_{0} \cos^{i}(\phi)d\phi\\
&=\sum^{i-2}_{k=1,\,k \text{ odd}}\frac{(i-k)!}{i!} 
\frac{(i-k)^2.(i-(k-2)))^2\dots(i-1)^2}{(i-k)^2}\sin(\theta)\cos^{i-k}(\theta)\\
&\quad + \frac{(i-1)^2(i-3)^2\dots(2)^2}{i!}\sin(\theta),
\end{align*}
\begin{align*}
&A_{p,(2n+1)}\\
&=\int^{\theta}_{0} \cos^{p}(\phi) \sin^{2n+1}(\phi)d\phi \\
&=\frac{cos^{p+1}(\theta)}{2n+p+1}\Big\{\sin^{2n} 
+ \sum^n_{k=1} \frac{2^k n(n-1)\dots(n-k+1) \sin^{2n-2k}(\theta)}{(2n+p-1)
(2n+p-3)\dots(2n+p-2k+1)} \Big\},
\end{align*}
\begin{align*}
&A_{p,(2n)}\\
&=\int^{\theta}_{0} cos^{p}(\phi) sin^{2n}(\phi)d\phi\\
&=\frac{-cos^{p+1}(\theta)}{2n+p}\Big\{\sin^{2n-1} 
 + \sum^{n-1}_{k=1} \frac{(2n-1)(2n-3)\dots(2n-2k+1) 
\sin^{2n-2k-1}(\theta)}{(2n+p-2)(2n+p-4)\dots(2n+p-2k)} \Big\}\\
&\quad + \frac{(2n-1)!!}{(2n+p).(2n+p-2)\dots(p+2)}
\int^{\theta}_{0} \cos^{p}(\theta)d\theta;
\end{align*}
for more details see \cite{GraRys}.

From the nine main products of
 $\frac{d}{dr}F_{1}(\theta,r)\int^{\theta}_{0}F_{1}(\phi,r) d \phi$, 
only the following five are not zero when we integrate them between 
$0$ and $2\pi$:
\begin{gather*}
\sum^l_{i=2,\,i \text{ even}} 
\sum^m_{k=0,\,k \text{ even}} \frac{i}{k+1} c_{i} b_{k} r^{i+k-1}
\cos^{i+k+2}(\theta), 
\\
-\sum^n_{i+j=2,\text{$i$  even  and $j$  odd}}
\sum^l_{k=0,\,k \text{ even}} (i+j+1) a_{ij}c_{k} r^{i+j+k} 
\cos^i(\theta) \sin^{j+2}(\theta)A_{k+1}(\theta), 
\\
+ \sum^n_{i+j=2} \sum^n_{k+h=1} (i+j+1) a_{ij} a_{kh} 
 r^{i+j+k+h}\cos^i(\theta) \sin^{j+2}(\theta)A_{i,(j+2)},
\end{gather*}
where if $i$ even $j$ is odd, and if $i$ odd $j$ even, and the same 
for $k$ and $h$, with $i+k$ odd and $j+h$ is odd too.
\begin{gather*}
\begin{aligned}
&+\sum^n_{i+j=2,\text{ $i$  odd  and  $j$  even}} 
\sum^m_{k=0,\,k \text{ even}} (i+j+1) a_{ij} b_{k} r^{i+j+k} \cos^i(\theta)\\
&\times \sin^{j+2}(\theta)(\frac{1-cos^{k+1}(\theta)}{k+1}),
\end{aligned} \\ 
-\sum^m_{i=2,\,i \text{ even}} \sum^l_{k=0,\,k \text{ even}} 
i b_{i}c_{k} r^{i+k-1} \cos^i(\theta) \sin(\theta) A_{k+1}(\theta).
\end{gather*}
Then the last five sums are odd polynomial in the variable $r$ of
degree $O(n)+E(l)$, $2O(n)+1$, $O(n)+E(m)$,$E(l)+E(m)-1$, respectively. 
Therefore,
$$
\frac{1}{2\pi}\int_{0}^{2\pi}\big[\frac{d}{dr}F_{1}(\theta,r)
 \int^{\theta}_{0}F_{1}(\phi,r) d \phi \big]d\theta
$$
is an odd polynomial in the variable $r$ and can contribute at most with
$$
\Big[ \frac{\max\{O(n)+E(l), 2O(n)+1, O(n)+E(m),E(l)+E(m)-1 \}-1}{2} \Big]
$$
simple positive real roots to the roots of $F_{20}(r)$.

Now we shall study the contribution of 
$\frac{1}{2\pi} \int^{2\pi}_{0}F_{2}(\theta,r)d\theta$ to $F_{20}(r)$.
The first part,
$$
\frac{xh_{l}^2(x)-y^2f_{n}^2(x,y)-yg_{m}^2(x)}{r},
$$
of $F_{2}(\theta,r)$, contributes at the roots of $F_{20}(r)$ exactly as 
the function $F_{1}(\theta,r)$ contributes to $F_{10}(r)$; i.e. it 
contributes at most with
$$
\Big[\frac{\max\{O(l), O(n+1) \} -1}{2} \Big]
$$
simple positive roots to the roots of $F_{20}(r)$.
Finally we shall study the contribution of the second part
$$
\frac{(xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x))(xyf_{n}^1(x,y)
+xg_{m}^1(x)+yh_{l}^1(x))}{r^3}
$$
of $F_{2}(\theta,r)$ to $F_{20}(r)$, which can be written as
\begin{align*}
&\frac{1}{r^2} 
\Big[\sum^l_{i=0,\,i \text{ even}} c_{i} r^i \cos^{i+1}(\theta)
-\sum^n_{i+j=1,\text{ $i$  odd  or $j$  odd}}a_{ij} r^{i+j+1} 
\cos^i(\theta)\sin^{j+2}(\theta)\\
&-\sum^m_{i=0}b_{i}r^i  \cos^i(\theta)\sin(\theta) \Big],
\end{align*}
\begin{align*}
&\Big[ \sum^n_{i+j=1,\text{ $i$ odd  or  $j$ odd}}a_{ij} r^{i+j+1} 
\cos^{i+1}(\theta)\sin^{j+1}(\theta) 
+ \sum^m_{i=0}b_{i}r^i\cos^{i+1}(\theta)\\
&+\sum^l_{i=0,\, i \text{ even}} 
c_{i} r^i \cos^{i}(\theta)\sin(\theta) \Big].
\end{align*}
From the nine products between the different sums, seven ones will not 
be zero after the integration with respect to $\theta$ between
 $0$ and $2\pi$, and two of these seven are equal.

So the terms which will contribute to $F_{20}(r)$ are
\begin{align*}
&\frac{1}{r^2}\Big[ \sum^l_{k=0,\,k \text{ even}} 
\sum^n_{i+j=1,\text{ $i$ even  and $j$  odd}}c_{k} a_{ij}
 r^{k+i+j+1} \cos^{k+i+2}(\theta) \sin^{j+1}(\theta)\\
&+ \sum^l_{k=0,\,k \text{ even}} \sum^m_{i=0,\, i \text{even}}
 c_{k} b_{i}r^{k+i} \cos^{k+i+2}(\theta) 
\\
&+ \sum^{2n}_{i+j=1, k+h=1,\text{ $i+k$  odd  and  $j+h$  odd}} 
a_{ij}a_{kh}r^{i+j+k+h+2}\cos^{i+k+1}(\theta)\sin^{j+h+3}(\theta)
\\
&+2 \sum^n_{i+j=1,\text{ $i$  odd  and $j$  even }}
\sum^m_{k=0,\,k \text{ even}}a_{ij}b_{k}r^{i+j+k+1} 
\cos^{i+k+1}(\theta)\sin^{j+2}(\theta)
\\
&+\sum^n_{i+j=1,\text{$i$ even  and $j$  odd}}
\sum^l_{k=0,\,k \text{ even}} a_{ij}c_{k} r^{i+j+k+1} 
\cos^{i+k}(\theta)\sin^{j+3}(\theta)
\\
&+\sum^m_{i=0,\,i \text{ even}} 
\sum^l_{k=0,\,k \text{ even}} b_{i} c_{k} r^{i+k} 
\cos^{i+k}(\theta)\sin^{2}(\theta)\Big]
\end{align*}
So the integral between $0$ and $2\pi$ with respect to $\theta$ 
of this last expression is an odd polynomial in the variable $r$ 
of degree $\max \{ O(n)+O(m)+1, O(n)+E(l)+1, E(m)+E(l), 2 O(n)+2 \}$.
 Consequently the contribution of the second part,
$$
\frac{(xh_{l}^1(x)-y^2f_{n}^1(x,y)-yg_{m}^1(x))
(xyf_{n}^1(x,y)+xg_{m}^1(x)+yh_{l}^1(x))}{r^3},
$$
of $F_{2}(\theta,r)$ to the zeros of $F_{20}(r)$ is at most with
$$
\Big[ \frac{\{O(n)+O(m)+1, O(n)+E(l)+1, E(m)+E(l), 2 O(n)+2 \}-1}{2} \Big]
$$
simple positive real roots.

From the above results, we have that $F_{20}(r)$ has at most
$$
\Big[ \frac{\{O(n)+O(m)+1, O(n)+E(l)+1, E(m)+E(l), 2 O(n)+2, O(l),
 O(n+1) \}-1}{2} \Big]
$$
simple positive real roots. So, from the results of section 2 statement (b) 
of Theorem 1 is proved.


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\end{document}