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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 151, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/151\hfil Periodic solutions]
{Periodic solutions for Li\'enard differential equations with singularities}

\author[S. Li, F. Liao, W. Xing \hfil EJDE-2015/151\hfilneg]
{Shengjun Li, Fang-fang Liao, Wenya Xing}

\address{Shengjun Li \newline
College of Information
Sciences and Technology,
Hainan  University, Haikou 570228, China}
\email{shjli626@126.com}

\address{Fang-fang Liao \newline
Nanjing College of Information Technology,
Nanjing 210046, China}
\email{liaofangfang8178@sina.com}

\address{Wenya Xing \newline
College of Information Sciences and Technology,
Hainan  University, Haikou 570228, China}
\email{wenyaxing@hainu.edu.cn}

\thanks{Submitted November 2, 2014. Published June 10, 2015.}
\subjclass[2010]{34C25}
\keywords{Periodic solution; singular systems; topological degree}

\begin{abstract}
 In this article, we study the second-order forced Li\'enard equation
 $x''+f(x)x'+g(x)=e(t)$. By using the topological degree theory, we prove
 that the  equation has at least one positive periodic solution when $g$
 admits a repulsive singularity near the origin and satisfies some semilinear
 growth conditions near infinity.  Recent results in the literature are
 generalized and complemented.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks


\section{Introduction}

In this work,  we are concerned with the existence of positive  $T$-periodic 
solutions for the Li\'enard equation
\begin{equation} \label{e1.1} 
x''+f(x)x'+g(x)=e(t),
 \end{equation}
where $f:\mathbb{R}\to \mathbb{R}$ is a continuous functions, 
$g: (0,\infty)\to \mathbb{R}$ is continuous and admits a repulsive singularity
near the origin, $e$ is continuous and $T$-periodic.

  As we know, the Li\'enard equation appears in a number of physical models, 
for example, it is  used to describe fluid mechanical and nonlinear elastic
 mechanical phenomena. During the last few decades, the Li\'enard equation
  has attracted many researchers. One important related topic is to look for 
periodic solutions under different conditions.  We refer the reader to 
\cite{ dlu, jma1, jma2, pom, zwa1} and the references therein. 
Here we mention the following results: Fonda et al \cite{fonda2} used 
the Poincare-Birkhoff theorem to obtain
the existence of positive periodic solutions, including all subharmonics, 
for the following special case of \eqref{e1.1}
 \begin{equation}\label{exg}
 x''+g(x)=e(t),
 \end{equation}
 where $e\in C(\mathbb{R},\mathbb{R})$ is $T$-periodic and
$g\in C(\mathbb{R^{+}},\mathbb{R})$ satisfies the following strong force
 condition at $x=0$:
\begin{equation}\label{elim}
 \lim_{x \to 0^{+}}g(x)=-\infty, \quad  \lim_{x \to 0^{+}}G(x)=+\infty ,\quad 
(G(x)=\int^{x}_{1}g(s)ds)
 \end{equation}
and $g$ is superlinear at $x=+\infty$:
   \begin{equation}\label{esf}
   \lim_{x \to +\infty}g(x)=+\infty,
   \end{equation}
Later, Wang \cite{zwa2} used the phase-plane analysis methods proved the 
existence of at least one positive $T$-periodic solution of \eqref{exg} if $g$ 
satisfies \eqref{elim} near $x=0$, and $g$ satisfies semilinear condition at 
$x=+\infty$: there is an integer $k\geq 0$ and a small constant $\varepsilon >0$ 
such that
 \begin{equation}\label{efra}
 \big(\frac{k\pi}{T}\big)^{2}+\varepsilon
\leq \frac{g(x)}{x}\leq (\frac{(k+1)\pi}{T})^{2}-\varepsilon
 \end{equation}
 for all $t$ and all $x\gg1$.
 We note that  the conditions \eqref{efra} are the
standard uniform nonresonance conditions with respect to the Dirichlet boundary
condition, not the periodic boundary condition.


When $f\neq 0$,  equation \eqref{e1.1} is a non-conservative system. 
Lefschetz \cite{lefs} gave the first existence theorem
for equation \eqref{e1.1} under some dissipativity conditions.
 Many researchers tried to improve the results of \cite{lefs}.
  We assume that there exists a constant $d>0$ such that
\begin{equation}\label{e1.3} 
g(x)\operatorname{sgn}(x)>0, \quad | x| \geq d.
 \end{equation}
 Mawhin \cite{jma1} studied the existence of periodic solutions under assumption 
that $g$ satisfies \eqref{e1.3} and the sublinear condition
\begin{equation}\label{e1.4} 
\lim_{| x |\to +\infty}\frac{g(x)}{x}=0.
 \end{equation}
Later, Mawhin and Ward \cite{jma2} improved such a condition, and used the following 
condition
\[ %\label{e} 
\limsup_{x \to +\infty}\frac{g(x)}{x} < (\frac{\pi}{T})^{2}.
\]
instead of \eqref{e1.4}.


 This article is mainly motivated by the work mentioned above and the 
recent papers \cite{zwa2,zhang1,zhang3}. The result is obtained using 
topological degree theory, thanks to a priori estimates on the solutions of 
a suitable family of problems.
Our main result reads as follows:

\begin{theorem} \label{thm1.1} 
Assume that $f, e : \mathbb{R}\to\mathbb{R}, g: (0,\infty)\to \mathbb{R}$
 are continuous functions and $e$ is $T$-periodic. Suppose further that
\begin{itemize} 
\item[(H1)] %\label{ec}
$\lim_{x\to 0^{+}}g(x)=-\infty$ and
$\lim_{x\to 0^{+}}\int^{x}_1 g(r)dr=+\infty$;\\

 \item[(H2)]  There exist  $T$-periodic continuous functions $a, b$
such that
\begin{equation} \label{ee}
 a(t)\leq \liminf_{x\to +\infty}\frac{g(x)}{x}
\leq \limsup_{x\to +\infty}\frac{g(x)}{x}\leq b(t). 
\end{equation}
Also,
\begin{equation}\label{e16} 
\bar{a}>0 \quad  \text{and} \quad \underline{\lambda}_{1}(b)>0,
\end{equation}
here $\bar{a}=\frac{1}{T}\int_{0}^{T}a(t)dt$  and
 $\{\underline{\lambda}_{1}(q)\}$ denotes the first anti-periodic eigenvalues of
\begin{equation}\label{e17} 
x''+(\lambda+q(t))x=0.
\end{equation}
\end{itemize}
Then \eqref{e1.1} has at least one positive $T$-periodic solution.
\end{theorem}

The main novelty in the present paper is represented by the conditions at infinity,
which remind of a situation between the first and the second eigenvalue, but are
more general since the comparison involves the mean and the weighted eigenvalue
associated with the functions $a, b$ controlling the $g(x)/x$.

During the previous few decades, singular differential equations or
singular dynamical systems have been attracted the attention of many 
researchers \cite{az93, dm, fonda, hs, htz, ls, sol, zhang1, zhang2}. 
It is well-known that electrostatic or gravitational forces are the most
important applications of singular interactions. It was also found recently 
that one special singular differential equation which is called 
``Ermakov-Pinney equation'' plays an important role in the studying of the 
stability of periodic solutions of conservative systems with degree of lower 
freedom (see \cite{cz-dcds} and the references therein). Usually, the proof 
is based on either variational approach \cite{az87, az93} or topological methods. 
The proof of the main results in this paper is based on topological methods,
which started with the pioneering paper of Lazer and Solimini \cite{ls}.
From then on, some fixed point theorems in cones for completely continuous 
operators\cite{fw,t04}, the method of upper and lower
solutions \cite{ htz, rtv}, Schauder's fixed point theorem \cite{t07}, 
degree theory \cite{fonda, yz} and a nonlinear alternative principle of 
Leray-Schauder type \cite{ctz, cft, jcz} have been widely applied.


The rest of this article is organized as follows. 
In Section 2, some preliminary results will be given. 
In Section 3, by the use of topological degree theory, we will state
 and prove the main results. To illustrate the new results, some
applications are also given.

\section{Preliminaries}

In this section, we present some results which will be applied in Sections $3$.
Let us first introduce some known results on eigenvalues. Let $q$ be a
 $T$-periodic potential such that $q\in L^{1}(\mathbb{R})$.
Consider the eigenvalue problems of \eqref{e17}
with the $T$-periodic boundary condition :
\begin{equation}\label{e21} 
x(0)-x(T)=x'(0)-x'(T)=0,
\end{equation}
 or, with the anti-$T$-periodic boundary condition :
 \begin{equation}\label{e22} 
x(0)+x(T)=x'(0)+x'(T)=0.
\end{equation}
  We use $\lambda_{1}^{D}(q)<\lambda_{2}^{D}(q)<\dots<\lambda_{n}^{D}(q)<\dots$ 
to denote all eigenvalues of \eqref{e17} with the
Dirichlet boundary condition:
\begin{equation}\label{ea34} 
x(0)=x(T)=0.
\end{equation}

The following are the standard results for eigenvalues. 
See, e.g. Reference \cite{wma}.
\begin{itemize}
\item[(E1)]
there exist two sequences $\{\underline{\lambda}_{n}(q):n\in\mathbb{N} \}$ and 
$\{\bar{\lambda}_{n}(q):n\in \mathbb{Z}^{+}\}$ such that
$$
-\infty <\overline{\lambda}_{0}(q)<\underline{\lambda}_{1}(q)
\leq \overline{\lambda}_{1}(q) <\underline{\lambda}_{2}(q)
\leq \overline{\lambda}_{2}(q)<\dots<\underline{\lambda}_{n}(q)
\leq \overline{\lambda}_{n}(q)<\dots
$$
where $\underline{\lambda}_{n}(q)\to +\infty,
\overline{\lambda}_{n}(q)\to +\infty$ as $n\to +\infty$.
Moreover, $\lambda$ is an eigenvalue of \eqref{e17}-\eqref{e21}
 if and only if $\lambda=\underline{\lambda}_{n}(q)$ or 
$\overline{\lambda}_{n}(q)$ with $n$ is even; and $\lambda$ is an 
eigenvalue of $\eqref{e17}-\eqref{e22}$ if and only if 
$\lambda=\underline{\lambda}_{n}(q)$ or $\overline{\lambda}_{n}(q)$ with $n$ is odd.


\item[(E2)]  
If $q_{1}\leq q_{2}$ then
$$ 
\underline{\lambda}_{n}(q_{1})\geq \underline{\lambda}_{n}(q_{2}),
\overline{\lambda}_{n}(q_{1})\geq \overline{\lambda}_{n}(q_{2}), 
\lambda_{n}^{D}(q_{1})\geq \lambda_{n}^{D}(q_{2}) 
$$
for any $n\geq 1$.


\item[(E3)] For any $n\geq 1$,
$$
\underline{\lambda}_{n}(q)=\min \{ \lambda_{n}^{D}(q_{t_{0}}):
t_{0}\in \mathbb{R}\}, \overline{\lambda}_{n}(q)
=\max \{ \lambda_{n}^{D}(q_{t_{0}}):t_{0}\in \mathbb{R}\} 
$$
where $q_{t_{0}}$ denotes the translation of $q:q_{t_{0}}(t)\equiv q(t+t_{0})$.


\item[(E4)] $\bar{\lambda}_{n}(q), \underline{\lambda}_{n}(q)$ and 
$\lambda_{n}^{D}(q)$ are continuous in $q$
in the $L^{1}$-topology of $L^{1}(0,T)$.

\item[(E5)] It follows from a variational principle that the first eigenvalue 
$\lambda_{1}^{D}$ can be found as
$$
\lambda_{1}^{D}(q)
=  {\inf _{x\in H_{0}^{1}(0,T)\,\, x\neq0}
 \frac{\int^{T}_{0}\big(x'^{2}(t)-q(t)x^{2}(t)\big)dt}{\int^{T}_{0}x^{2}(t)dt}}.
$$
In particular,
              $$
\lambda_{1}^{D}(q)\leq -\bar{q},
$$
where $\bar{q}$ is the mean value. Moreover, the equality holds if and 
only if $q(t)=\bar{q}$ for a.e. $t$.
\end{itemize}

To prove our results, we need the following preliminary results, recall some 
notation and terminology from \cite{cmz}.
Define $L:\text{dom}L\subset X\to Z,~Lx=\dot{x}$, a Fredholm mapping of
index zero, with
$\operatorname{dom}L=\{x\in X:x(\cdot)\text{ is absolutely continuous}\}$,
 where the Banach spaces $X,Z$ are 
\[
X=\{x\in C([0,T],\mathbb{R}^m):x(0)=x(T)\},\quad 
Z=L^1([0,T],\mathbb{R}^m)
\]
with their usual norms. Then $L$ is a Fredholm mapping of index zero \cite{eze}.
Let $M_0$ be the Nemitzky operator from $X$ to $Z$ induced by the map 
$F:X\to \mathbb{R}^m$; that is,
$M_0:X\to Z, x(\cdot)\to F(x(\cdot))$. Consider the equation
\[
Lx=M_0x,\quad x\in \operatorname{dom}L.
\]

\begin{lemma}[{\cite[Theorem 1]{cmz}}] \label{lem2.1} % \label{emawhin} 
Let $\Omega\subset X$ be a bounded open
subset and assume that there is no $x(\cdot)\in\partial_X\Omega$
such that $\dot{x}=f_0(x).$ Then
\[
\deg _{L}(L-M_0,\Omega)=(-1)^m \deg _{B}(f_0,\Omega\cap R^m,0),
\]
where $\deg _{L},~\deg _{B}$ denote the Schauder degree and the Brouwer degree,
 respectively.
\end{lemma}

We refer the reader to \cite{eze} for more details about degree theory.

\section{Proof of Theorem \ref{thm1.1}}\label{esec4}

We will apply Lemma \ref{lem2.1} to the singular problem \eqref{e1.1}. 
To this end, we deform \eqref{e1.1} to a simpler singular autonomous equation
\[ %\label{eg}
 x''+c_{0} x= \frac{1}{x},
\]
where $c_{0}$ for some positive constant satisfy $0<c_{0}<(\pi/T)^{2}$. 
The choice of such a $c_{0}$ implies that the constant functions 
$a(t)=b(t)\equiv c_{0}$ satisfy \eqref{e16}. Consider the  homotopy equation
\begin{equation}\label{eh}
x''+ \tau f(x)x'+g(t,x;\tau)=0, \quad \tau\in[0,1],
\end{equation}
where 
$$
g(t,x;\tau)=\tau (g(x)-e(t))+(1-\tau)(c_{0}x-\frac{1}{x}).
$$
We need to find a priori estimates for the possible positive $T$-periodic 
solutions of \eqref{eh}.

Note that $g(t,x;\tau)$ satisfies the conditions $(\rm{H_{1}})$  uniformly 
with respect to $\tau\in[0,1].$  Moreover, for each $\tau\in[0,1]$,
$g(t,x;\tau)$ satisfies \eqref{ee} with
\begin{gather*}
a=a_{\tau}=\tau a(t)+(1-\tau)c_{0}, \\
b=b_{\tau}=\tau b(t)+(1-\tau)c_{0}.
\end{gather*}
We will prove that  $a_{\tau}$ and $b_{\tau}$ satisfy \eqref{e16} 
uniformly in $\tau\in[0,1]$.
This fact follows from the convexity of the first eigenvalues with 
respect to potentials.


 \begin{lemma} \label{lem3.3}
 Assume $q_{0},q_{1}\in L^{1}(0,T)$. Then, for all $\tau\in[0,1]$,
\begin{equation}\label{ej}
\underline{\lambda}_{1}(\tau q_{1}+(1-\tau)q_{0})
\geq \tau\underline{\lambda}_{1}(q_{1})+(1-\tau)\underline{\lambda}_{1}(q_{0}).
\end{equation}
\end{lemma}

Note that
$$
\bar{a}_{\tau}=\tau \bar{a}+(1-\tau)c_{0}\geq \min(\bar{a},c_{0})>0. 
$$
Applying Lemma \ref{lem3.3} to $q_{1}=b$ and $q_{0}=c_{0}$, we have
\[
 \underline{\lambda}_{1}(b_{\tau})
\geq   \tau\underline{\lambda}_{1}(b)+(1-\tau)\underline{\lambda}_{1}(c_{0})\\
\geq   \min (\underline{\lambda}_{1}(b), \underline{\lambda}_{1}(c_{0}))
  > 0.
\]
Thus $a_{\tau}$ and $b_{\tau}$ defined above satisfy \eqref{e16} uniformly 
in $\tau\in[0,1]$.

For obtaining a priori estimates for all possible positive solutions to 
\eqref{eh}--\eqref{e21}, we simply prove this for all possible positive
solutions to \eqref{e1.1}--\eqref{e21}, because $a_{\tau},b_{\tau}$ 
satisfy \eqref{ee} and also \eqref{e16} uniformly in $\tau\in[0,1]$.


 \begin{lemma} \label{lem3.4} 
Assume that $\underline{\lambda}_{1}(b)>0$ of the equation
 \begin{align*}
 y''+(\lambda+b(t))y=0\,.
 \end{align*}
Then
\[
\|y'\|_{2}^{2} \geq \int^{T}_{0}b(t+t_{0})y^{2}(t)dt
+\lambda_{1}^{D}(b_{t_{0}})\int^{T}_{0}y^{2}(t)dt.
\]
\end{lemma}

\begin{proof}
  By  (E3), we have
\begin{align*}
 \lambda_{1}^{D}(b_{t_{0}})\geq \underline{\lambda}_{1}(b)>0.
\end{align*}
  for all $t_{0}\in \mathbb{R}$.
Then, by the theory of  second order linear differential operators \cite{wne}, 
the eigenvalues of
  \[
  y''+(\lambda+b(t+t_{0}))y=0
   \]
    with Dirichlet boundary conditions form a sequence
\[
\lambda_{1}^{D}(b_{t_{0}})<\lambda_{2}^{D}(b_{t_{0}})<\dots,
\]
which tends $+\infty$, and the corresponding eigenfunctions
  $\psi_{1},\psi_{2},\dots$ are an orthonormal base of $L^{2}(0,T)$. 
Hence, given $c_{i}\in \mathbb{R}$ and $y\in H^{1}_{0}(0,T)$, we can write
\[
 y(t)=\sum _{i\geq 1}c_{i}\psi_{i}(t),
\]
 and
 \begin{align*}
\int^{T}_0((y'(t))^{2}-b(t+t_{0})y^{2}(t))dt 
&= \sum _{i\geq 1}c^{2}_{i}\int^{T}_0((\psi'_{i}(t))^{2}
 -b(t+t_{0})\psi_{i}^{2}(t))dt\\
&= \sum _{i\geq 1}c^{2}_{i}\lambda_{i}^{D}(b_{t_{0}})\int^{T}_0\psi_{i}^{2}(t)dt\\
&\geq   \lambda_{1}^{D}(b_{t_{0}})\int^{T}_0y^{2}(t)dt.
\end{align*}
This completes the proof.
\end{proof}

The usual $L^{p}$-norm is denoted by $\|\cdot\|_{p}$, and the supremum norm 
of $C[0,T]$ is denoted by $\|\cdot\|_{\infty}$.

\begin{lemma} \label{lem3.5}
 Under the assumptions as in Theorem \ref{thm1.1}, there exist $B_{1}>B_{0}>0$ 
such that any positive $T$-periodic solution $x(t)$ of 
\eqref{e1.1}-\eqref{e21} satisfies
\begin{equation}\label{e66}
 B_{0}<x(t_{0})<B_{1},
\end{equation}
for some $t_{0}\in [0,T]$.
\end{lemma}

\begin{proof} 
Let $x(t)$ be a positive $T$-periodic solution of \eqref{e1.1}-\eqref{e21}. 
By (H1), there exists $B_{0}>0$ such that
\[
 g(s)-e(t)<0 \quad \text{for all } 0<s<B_{0}.
\]
Integrate \eqref{e1.1} from $0$ to $T$, we obtain
\[
\int^{T}_0(g(x(t))-e(t))dt= -\int^{T}_0 x''(t)dt-\int^{T}_0f(x)x'dt=0.
\]
Thus $\int^{T}_0(g(x(t))-e(t))dt=0$, there exists $t^{*}\in [0,T]$ 
such that $x(t^{*})>B_{0}$.

Next, noticing \eqref{e16}, we can take some constant 
$\varepsilon_{0}\in \big(0, \min \{ \overline{a},\underline{\lambda}_{1}(b)\}\big)$.
It follows from $(\rm{H_{2}} )$ that there exists a constants $B_{1}(>B_{0})$ 
large enough such that
\begin{equation}\label{e34}
 a(t)-\varepsilon_{0}\leq \frac{g(s)-e(t)}{s}\leq b(t)+\varepsilon_{0}.
\end{equation}
for all $t$ and all $s\geq B_{1}$. We assert that $x(t_{*})<B_{1}$ 
for some $t_{*}$. Otherwise, assume that $x(t)\geq B_{1}$ for all $t$.
Define
\[
 p(t):=\frac{g(x(t))-e(t)}{x(t)}.
\]
By \eqref{e34},
$$
a(t)-\varepsilon_{0}\leq p(t)\leq b(t)+\varepsilon_{0}
$$
for all $t$. Moreover, $x(t)$ satisfies the following differential equation
$$ 
x''+f(x)x'+p(t)x=0.
$$
Write  $x=\tilde{x}+\bar{x}$, where $\bar{x}=\frac{1}{T}\int_{0}^{T}x(t)dt$, 
then $\tilde{x}$ satisfies
\begin{equation}\label{el}
 -\tilde{x}''-f(\tilde{x}+\bar{x})\tilde{x}'=p(t)\tilde{x}+p(t)\bar{x}.
\end{equation}
Integrating \eqref{el} from $0$ to $T$, we have
\begin{equation}
 \label{em} \int^{T}_0 p(t)\tilde{x}(t)dt=-\bar{x}\int^{T}_0 p(t)dt.
\end{equation}
Multiplying \eqref{el} by $\tilde{x}$ and using  integration by parts, we obtain
\begin{equation} \label{eff}
\begin{aligned}
\|\tilde{x}'\|_{2}^{2} 
&= \int^{T}_0 p(t)\tilde{x}^{2}(t)dt+\bar{x}\int^{T}_0 p(t)\tilde{x}(t)dt   \\
&= \int^{T}_0 p(t)\tilde{x}^{2}(t)dt -\bar{x}^{2}(t)\int^{T}_0 p(t)dt   \\
& \leq   \int^{T}_0 p(t)\tilde{x}^{2}(t)dt,
\end{aligned}
\end{equation}
where the fact $\int^{T}_0 p(t)dt>T(\bar{a}-\varepsilon_{0})>0$ is used.

Note that $\tilde{x}(t_{0})=0 $ for some $t_{0}$, $\tilde{x}(t_{0}+T)=0 $, 
so $\tilde{x}(t) \in H_{0}^{1}(t_{0},t_{0}+T)$. We assert that
$\tilde{x} \equiv 0$. On the contrary, assume that $\tilde{x} \not\equiv 0$. 
Now by \eqref{eff}, the first Dirichlet eigenvalue
\[
 \lambda_{1}^{D}(p|_{[t_{0},t_{0}+T]}) 
=   {\inf _{\varphi\in H_{0}^{1}(t_{0},t_{0}+T),\, \varphi\neq0} 
\frac{\int^{t_{0}+T}_{t_{0}}\big(\varphi'^{2}(t)-p(t)\varphi^{2}(t)\big)dt}
{\int^{t_{0}+T}_{t_{0}}\varphi^{2}(t)dt}}\leq0.
\]
So,
\[
\underline{\lambda}_{1}(p)=\min \{ \lambda_{1}^{D}(p)\}\leq0.
\]
On the other hand, $p(t)<b(t)+\varepsilon_{0}$,
\[
\underline{\lambda}_{1}(p)\geq \underline{\lambda}_{1}(b 
+\varepsilon_{0})=\underline{\lambda}_{1}(b) - \varepsilon_{0}>0.
\]
This is a contradiction.

Now it follows from \eqref{em} that $\bar{x}=0$ and $x\equiv 0$, 
a contradiction to the positiveness of $x(t)$. We have proved that 
$x(t^{*})>B_{0}$ for some
 $t^{*}\in [0,T]$ and  $x(t_{*})<B_{1}$ for some $t_{*}\in [0,T]$. 
Thus the intermediate value theorem implies that \eqref{e66} holds. 
\end{proof}

\begin{lemma} \label{lem3.6}
 There exist $B_{2}>B_{1}>0$, $B_{3}>0$ such that any positive $T$-periodic 
solution $x(t)$ of \eqref{e1.1}-\eqref{e21} satisfies
\[
\|x\|_{\infty}<B_{2}, \quad \|x'\|_{\infty}<B_{3}.
\]
\end{lemma}

\begin{proof} From (H2) and \eqref{e34}, we know that there exists
 $h_{0}>0$ such that
 \begin{equation}\label{e386}
g(s)-e(t)\leq (b(t)+\varepsilon_{0})s+h_{0}
\end{equation}
for all $t$ and $s>0$.
Multiplying \eqref{e1.1} by $x$ and then integrating over $[0,T]$, 
using the fact that
$$\int_{0}^{T}f(x(t))x'(t)x(t)dt=0,$$
we obtain
\begin{equation}\label{en}
\begin{aligned}
 \|x'\|_{2}^{2}
 &=  \int^{T}_0 -(xx''+x f(x)x')dt \\
 &= \int^{T}_0 (g(x(t))-e(t))x(t)dt  \\
 &\leq \int^{T}_0 ((b(t)+\varepsilon_{0})x(t)+h_{0}) x(t)dt   \\
 & =   \int^{T}_0 b(t){x}^{2}(t)dt+\varepsilon_{0}\|x\|_{2}^{2}+h_{0}\|x\|_{1}.
\end{aligned}
\end{equation}

 It follows from Lemma \ref{lem3.5} that there exists $t_{0}$ satisfying
 $B_{0}<x(t_{0})<B_{1}$. Let $u(t)=x(t+t_{0})-x(t_{0})$, then 
$u\in H_{0}^{1}(0,T)$. Thus
\begin{align*}
 \int^{T}_0 b(t)x^{2}(t)dt  
&=  \int^{T}_0 b(t+t_{0})x^{2}(t+t_{0})dt\\
&=  \int^{T}_0 b(t+t_{0})\big(x^{2}(t_{0})+2x(t_{0})u(t)+u^{2}(t)\big)dt\\
& \leq   B_{1}^{2}\|b\|_{1}+2B_{1}\|b\|_{2}\|u\|_{2}
 +\int^{T}_0 b(t+t_{0})u^{2}(t)dt.
\end{align*}
The other terms in \eqref{en} by the H\"{o}lder inequality can be estimated 
as follows:
\begin{gather*}
\varepsilon_{0}\|x\|_{2}^{2}
\leq \varepsilon_{0}(TB_{1}^{2}+2B_{1}T^{1/2}\|u\|_{2}+\|u\|_{2}^{2}), \\
h_{0}\|x\|_{1}\leq h_{0}(TB_{1}+T^{1/2}\|u\|_{2}).
\end{gather*}
Thus \eqref{en} reads
\begin{equation}\label{eo}
\|u'\|_{2}^{2}\leq A_{0}+A_{1}\|u\|_{2}+ \varepsilon_{0}\|u\|_{2}^{2}
+\int^{T}_0 b(t+t_{0})u^{2}(t)dt,
\end{equation}
where 
\begin{gather*}
A_{0}=\varepsilon_{0}TB_{1}^{2}+h_{0}TB_{1}+B_{1}^{2}\|b\|_{1},\\
A_{1}=2\varepsilon_{0}B_{1}T^{1/2}+h_{0}T^{1/2}+2B_{1}\|b\|_{2}
\end{gather*}
are positive constants.

On the other hand, using Lemma \ref{lem3.4},
\[
\underline{\lambda}_{1}(b(t))\|u\|_{2}^{2}\leq\lambda_{1}^{D}(b_{t_{0}})
\|u\|_{2}^{2}
\leq\int^{T}_0 \big( u'^{2}(t)-b(t+t_{0})u^{2}(t)\big)dt, 
\]
and we obtain from \eqref{eo} that
\[
(\underline{\lambda}_{1}(b(t))-\varepsilon_{0})\|u\|_{2}^{2}
\leq A_{1}\|u\|_{2}+A_{0}.
\]
Consequently, $\|u\|_{2}<A_{2} $ for some $ A_{2}>0$. By \eqref{eo}, one has
$\|x'\|_{2}=\|u'\|_{2}<A_{3}$
for some $ A_{3}>0.$ From these, for any
$t\in [t_{0},t_{0}+T]$,
\begin{align*}
 |x(t)|  
&\leq  |x(t_{0})|+\big|\int^{t}_{t_{0}}x'(t)dt\big|\\
&\leq   B_{1}+T^{1/2}\|x'\|_{2}\\
&\leq   B_{1}+T^{1/2}A_{3}:=B_{2}.
\end{align*}
Thus $\|x\|_{\infty}<B_{2}$ is obtained.

To prove $\|x'\|_{\infty}<B_{3}$, we write \eqref{e1.1} as
$$
-x''(t)=H(t):=f(x(t))x'(t)+g(x(t))-e(t).
$$
As $\int^{T}_0 H(t)dt=0,$  thus $\|H(t)\|_{1}=2\|H^{+}(t)\|_{1}$.
From \eqref{e386} we have
\begin{align*}
H^{+}(t)
&=  \max (H(t),0)\\
&\leq |f(x(t))|\cdot|x'(t)|+|b(t)+\varepsilon_{0}|x(t)+h_{0}\\
&\leq  C_{1}|x'(t)|+C_{2},
\end{align*}
where $C_{1}=\max _{0\leq y \leq B_{2}} |f(y)|$. 
Since $x(0)=x(T)$, there exists $t_{1}\in [0,T]$ such that $x'(t_{1})=0$. 
Therefore,
\begin{align*}
\|x'\|_{\infty}  
&=   \max _{0\leq t \leq T}|x'(t)|\\
&=   \max _{0\leq t \leq T} \big|\int^{t}_{t_{1}} x''(s)ds \big|\\
&\leq   \int^{T}_{0} \left|H(s)\right|ds \\
&=   2\int^{T}_{0} \left|H^{+}(s)\right|ds \\
&\leq   2\big(C_{3}T^{1/2}\|x'\|_{2}+TC_{4}\big)\\
&\leq   2\big(A_{3}C_{1}T^{1/2}+TC_{2}\big):=B_{3}.
\end{align*}
We have proved is that the $W^{2,1}$ norms of $x$ are bounded.
\end{proof}

Next, we obtain the positive lower estimates for $x(t)$ based on the  
condition (H1).


\begin{lemma} \label{lem3.7} 
There exists a constant $B_{4}\in(0,B_{0})$ such that any positive solution 
$x(t)$ of \eqref{e1.1}--\eqref{e21} satisfies
\[
x(t)>B_{4} \quad \text{for all } t.
\]
\end{lemma}

\begin{proof} 
From (H1), we fix some $A_{4}\in(0,B_{0})$ such that
\begin{align*}
g(s)-e(t)<-B_{3}C_{1}.
\end{align*}
for all $t$ and all $0<s\leq A_{4}$, where $C_{1}$ is the same as above.  
Assume now that
\[
m=\min _{t\in[0,T]}x(t)=x(t_{2})<A_{4}.
\]
By Lemma \ref{lem3.5}, $\max _{t}x(t)>B_{0}$. Let $t_{3}>t_{2}$ be the first time 
instant such that $x(t)=A_{4}$.
Then for any $t\in[t_{2},t_{3}]$, we have $x(t)\leq A_{4}$ and
 $|-f(x(t))x'(t)|\leq B_{3}C_{1}$. Hence, for $t\in[t_{2},t_{3}]$,
\[
x''(t) =   -f(x(t))x'(t)-g(x(t))-e(t)\\
 >   B_{3}C_{1}-f(x(t))x'(t)
 \geq  0.
\]
As $x'(t_{2})=0,x'(t)>0$ for $t\in(t_{2},t_{3}]$. Therefore, 
the function $x:[t_{2},t_{3}]\to\mathbb{R}$ has an inverse,
denoted by $\xi$.

Now multiplying \eqref{e1.1} by $x'(t)$ and integrating over $[t_{2},t_{3}]$, 
we obtain
\begin{align*}
\int^{A_{4}}_{m}-g(\xi(x))dx  
&=  \int^{t_{3}}_{t_{2}}-g(x(t))x'(t)dt  \\
&= \int^{t_{3}}_{t_{2}}(x''(t)x'(t)+f(x(t)(x'(t))^{2}+e(t)x'(t))dt
    \leq A_{5}
\end{align*}
 for some $A_{5}>0$, where  Lemma \ref{lem3.6} is used.
 By (H1),
\begin{equation} \label{ep}
\int^{A_{4}}_{m}-g(\xi(x))dx\to +\infty
\end{equation}
if $m\to 0^{+}$. Thus we know from \eqref{ep} that $m>B_{4}$ for some 
constant $B_{4}>0$. 
\end{proof}

Now we give the proof of Theorem \ref{thm1.1}. Consider the homotopy equation 
\eqref{eh}, we can get a priori estimates as in Lemmas \ref{lem3.5},  \ref{lem3.6}
and \ref{lem3.7}.  That is, any positive $T$-periodic solution of
\eqref{eh} satisfies
\begin{align*}
B'_{4}<x(t)<B'_{2}, \quad \|x'\|_{\infty}<B'_{3}
\end{align*}
for some positive constants $B'_{4},B'_{2},B'_{3}$.

Let $C=\max\{B'_{4}, B'_{2}, B'_{3}\}$ and let the open bounded in $X$ be
\[
\Omega=\{x\in X: \frac{1}{C}<x(t)<C  \text{ and } |x'(t)|<C \text{ for all }
 t\in[0,T]\}.
\]
Obviously, $\Omega$ contains the constant function $x(t)=r_{0}$, where 
$r_{0}>0$ is the solution of
$$
c_{0}x-\frac{1}{x}=0.
$$
Let $X$ be a Banach space of functions such that 
$C^{1}([0,T])\subseteq X\subseteq C([0,T])$, with continuous immersions. Set
 $X_{*}=\{x\in X: \min _{t} x(t)> 0\}$.

Define the space:
\[
D(L)=\{x\in W^{2,1}(0,T): x(0)=x(T), x'(0)=x'(T)\},
\]
 and the following two operators:
\[
L: D(L)\subset X\to L^{1}(0,T), \quad (Lx)(t)=-x''(t),
\]
and
\begin{gather*}
N_{\tau}: X_{*}\to L^{1}(0,T), \\
(N_{\tau}x)(t)=\tau f(x(t))x'(t)+\tau (g(x(t))-e(t))+(1-\tau)(c_{0}x-\frac{1}{x}).
\end{gather*}
Taking $\sigma \in \mathbb{R}$ not belonging to the spectrum of $L$, 
the $T$-periodic for equation \eqref{eh} is thus equivalent to the 
operator equation
\[
 Lx=N_{\tau}x,
\]
which is also can be translated to
\[
x-(L-\sigma I)^{-1} (N_{\tau}-\sigma I)x=0,
\]
since $L-\sigma I$ is invertible.
 By the homotopy invariance of degree and  Lemma \ref{lem2.1},
\begin{align*}
\deg (I-(L-\sigma I)^{-1}(N_{1}-\sigma I),\Omega, 0)
 &=   \deg (I-(L-\sigma I)^{-1}(N_{0}-\sigma I),\Omega, 0)\\
 &=   \deg (c_{0}x-\frac{1}{x}, \Omega\cap\mathbb{R}, 0)=+1.
\end{align*}
Thus \eqref{eh}, with $\tau=1,$ has at least one solution in $\Omega$, 
which is a positive $T$-periodic solution of \eqref{e1.1}.
 The proof of Theorem \ref{thm1.1} is thus complete.


\begin{remark} \label{rmk3.8} \rm
It is known (see (E3), (E5)) that
 \[
\bar{\lambda}_{0}(a)\leq -1/T\int_{0}^{T}a(t)dt<0,
\]
Therefore \eqref{e16} implies
\[
\bar{\lambda}_{0}(a)<0<\underline{\lambda}_{1}(b).
\]
\end{remark}

\begin{remark} \label{rmk3.9} \rm
 Some classes of potentials $q$ for $\underline{\lambda}_{1}(q)>0$ to
hold have been found recently in \cite{caba}. To describe these, 
let $K(\alpha,T)$ denote the best Sobolev constant in the inequality
 \[
C \|u\|_{\alpha}^{2} \le \|u'\|_{2}^{2} \quad
\text{for all } u\in H_0^1(0,T).
\]
 The explicit formula for $K(\alpha,T)$ is
 \[ 
K(\alpha,T) = \begin{cases}
\frac{2\pi}{\alpha T^{1+2/\alpha}} \big( \frac{2}{\alpha+2} \big)^{1-2/\alpha}
 \big(\frac {\Gamma(1/\alpha) }{\Gamma(1/2+1/\alpha)}\big)^{2}, 
& \text{for } 1\le \alpha <\infty, \\[3pt]
4/T, & \text{for } \alpha=\infty, 
\end{cases} 
\] 
and $\Gamma(\cdot)$ is the Euler's Gamma function.
\end{remark}

        Now \cite[Theorem 2.1]{caba} reads as follows: 
let $q\in L^{p}(0,T)$ for some $1\leq p \leq \infty$,  $q_{+}=\max\{q,0\}$ 
is the positive part of $q$ and $p^{*}=\frac{p}{p-1}$ the  conjugate of $p$. If
\begin{align*}
\|q_{+}\|_{p}< K(2p^{*},T),
\end{align*}
 then
\[
\underline{\lambda}_{1}(q)\geq \left(\frac{\pi}{T}\right)^{2}
\Big(1-\frac{\|q_{+}\|}{K(2p^{*},T)}\Big)>0.
\]

\begin{example} \label{examp3.10} \rm
Let $f, \varphi, h\in C (\mathbb{R},\mathbb{R})$, $\varphi (t)\geq 0, \gamma\geq 1$. 
For some $1\leq p \leq \infty$, if
\[
\|\varphi_{+}\|_{p}< K(2p^{*},T),
\]
then the  singular equation
\begin{align*}
x''+f(x)x'+\varphi(t)x-\frac{1}{x^{\gamma}}=h(t)
\end{align*}
has at least one positive $T$-periodic solution.
\end{example}


\subsection*{Acknowledgments}
This work is supported by the National Natural Science  Foundation of China
(Grant No. 11461016), Hainan Natural Science Foundation (Grant No. 113001).

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\end{document}