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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 205, 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 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/205\hfil Periodic solutions]
{Periodic solutions for fourth-order $p$-Laplacian functional
 differential equations with sign-variable coefficient}

\author[J. Liu, W. Liu, B. Liu \hfil EJDE-2013/205\hfilneg]
{Jiaying Liu, Wenbin Liu, Bingzhuo Liu}  % in alphabetical order

\address{Jiaying Liu \newline
Department of Mathematics, China University of
Mining and Technology, 
Xuzhou, Jiangsu 221116, China}
\email{ relinaliu@163.com}

\address{Wenbin Liu (corresponding author)\newline
Department of Mathematics, China University of
Mining and Technology, 
Xuzhou, Jiangsu 221116, China}
\email{wblium@163.com}

\address{Bingzhuo Liu \newline
Department of Mathematics, China University of
Mining and Technology, 
Xuzhou, Jiangsu 221116, China}
\email{tuteng3839@163.com}

\thanks{Submitted October 7, 2012. Published September 18, 2013.}
\thanks{Supported by grant 11271364 from the NNSF of China}
\subjclass[2000]{34A12, 34C25}
\keywords{$p$-Laplacian equation; periodic solution;
 multiple deviating argument; \hfill\break\indent 
 Mawhin continuation theorem}

\begin{abstract}
Using the theory of coincidence degree, we show the
 existence of periodic solutions to the fourth-order
 $p$-Laplacian differential equations of Li{\'{e}}nard-type
 \begin{align*}
 &\phi_p(x''))''+f(x(t))x'(t)+\alpha(t)g_1(x(t-\tau_1(t,x(t))))\\
 &+\beta(t)g_2(x(t-\tau_1(t,x(t))))=p(t).
 \end{align*}
 The rate of growth of $g_1(u)$ with respect to the variable
 $u$ is allowed to be greater than $p-1$, and the coefficient
 $\beta (t)$ is allowed to change sign.
\end{abstract}

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

\section{Introduction}

The study of the fourth-order differential equations is of great practical 
significance, whose classical application is to describe the equilibrium
 of elastic beams. The study on periodic oscillations of the fourth-order 
differential equations has gained more and more attention by many researchers, 
and some profound results have been obtained (see \cite{g2,p1,p2,u1}).
However,  the results of periodic solutions to a fourth order $p$-Laplacian delay 
differential equation are relatively rare.

In this article, we consider  the existence of periodic solutions 
to the fourth-order $p$-Laplacian differential equations with multiple 
deviating arguments:
\begin{equation}\label{1.1}
\begin{aligned}
&\phi_p(x''))''+f(x(t))x'(t)+\alpha(t)g_1\big(x(t-\tau_1(t,x(t)))\big)\\
&+\beta(t)g_2\big(x(t-\tau_2(t,x(t)))\big)=p(t)
\end{aligned}
\end{equation}
where $p>1$, $\phi_p(s)=|s|^{p-2}s $ $(s \neq 0)$,
$\phi_p(0)=0$, $\alpha(t),\beta(t),p(t) \in C(\mathbb{R}, \mathbb{R})$,
$\int_0^T p(t)dt=0$, $\int_0^T \beta(t)dt\neq 0$,
$\alpha(t)\geq 0$  $(\leq 0)$ for $t\in \mathbb{R}$,  
$ \int_0^T \alpha(t)dt> 0$  $(< 0)$, $\alpha(t+T)=\alpha(t)$,
$\beta(t+T)=\beta(t)$, $p(t+T)=p(t)$ $\tau_i\in C(\mathbb{R}^2, \mathbb{R})$,
$\tau_i(t+T,x)=\tau_i(t,x)$, $g_i \in C(\mathbb{R}, \mathbb{R})$, $i=1,2$, $T>0$.

In recent years, there have been a number of results on the existence of
 periodic solutions of the second order $p$-Laplacian differential 
equations; see \cite{c1,g1,l1,m1,p3,z1} and the references therein. 
Cheung and Ren \cite{p3} studied the existence of periodic solutions for the 
 $p$-Laplacian delay equation
$$ 
\phi_p(x'))'+f(x'(t))+\beta g(x(t-\tau(t)))=e(t)
$$
where $\beta>0$ is a constant. 
Gao  and Lu \cite{g1}  studied the periodic solutions for 
the $p$-Laplacian Rayleigh differential equation with a delay,
$$ 
\phi_p(x'))'+f(x'(t))+\beta(t)g(x(t-\tau(t)))=e(t)\,.
$$
In 2007, Cheung and Ren \cite{c1} discussed the solvability of periodic
 problems for the Lienard-type $p$-Laplacian delay differential equation
$$ 
\phi_p(x'))'+f(x(t))x'(t)+g(t,x(t-\tau(t)))=e(t)\,,
$$
under the assumption
 $$\lim_{|x|\to \infty}\frac{|g(x)|}{|x|^{p-1}}=r\geq 0. 
$$
Motivated by the above works, we will present the existence of periodic 
solutions for \eqref{1.1}  by using Mawhin's continuation theorem. 
Our main results are different from those results in the literature.
For instance, in our study we allow the growth rate of  $g_1(u)$, 
with respect to $u$, to be greater than $p-1$. Also we allow the  
coefficient $\beta(t)$  to change sign $\mathbb{R}$.

\section{Preliminaries}

For simplicity, we use the following symbols in this article
\begin{gather*}
 C_T=\{ x\in C(\mathbb{R},\mathbb{R}): x(t+T)=x(t) \},\quad
 |x|_\infty=\max_{t\in [0,T]}|x(t)|, \\
C_T^1=\{ x\in C^1(\mathbb{R},\mathbb{R}):x(t+T)=x(t) \},\quad
 \|x\|=\max\{|x|_\infty, |x'|_\infty \}, \\
|x|_p=\Big(\int_0^T|x(t)|^pdt\Big)^{1/p},\quad
  D_p=   \begin{cases}
     1, & 0<p\leq 1,\\
     2^{p-1} ,& p>1.
   \end{cases}
\end{gather*}

To state our main results, we introduce several technical lemmas.

\begin{lemma}[\cite{m1}] \label{lem1} 
Assume that $\Omega$ is an open bounded set in
$ C_{T}^1 $ such that the following three conditions hold:
\begin{itemize}
\item[(1)]  For each $\lambda \in (0,\,1)$, the equation
\begin{equation}\label{2.1}
    (\phi_p(x''))''=\lambda f(t,x(t),x(t-\mu(t)),x'(t)),
\end{equation}
has no $T$-periodic solution on $\partial\Omega$,
 where $ f(t,x,y,z)\in C(\mathbb{R}^4,\mathbb{R})$ and
$ f(t+T,\cdot,\cdot,\cdot))=f(t,\cdot,\cdot,\cdot))$.

\item[(2)]  The equation
\begin{equation*} %\label{2.2}
    F(a)=\frac{1}{2\pi}\int_0^Tf(t,a,a,0)dt=0,
\end{equation*}
has no solution on $\partial\Omega\cap \mathbb{R}$.

\item[(3)] The Brouwer degree satisfies
$\deg_{B}(F,\Omega\cap\mathbb{R},0) \neq 0$.
\end{itemize}
Then \eqref{2.1} has a $T$-periodic solution in $ \bar{\Omega} $ when 
$\lambda=1$.
\end{lemma}

\begin{lemma}[\cite{z1}] \label{lem2} 
 If $ \omega (t)\in c^1(\mathbb{R},\mathbb{R})$ and $ \omega(0)=\omega(T)=0$, 
then there holds
  $$ 
\int_0^T|\omega(t)|^pdt
\leq  \big(\frac{T}{\pi_p}\big)^p\int_0^T|\omega'(t)|^pdt,
 $$
where 
$$ 
\pi_p=\int_0^{(p-1)/p}\frac{ds}{(1-(p-1)^{-1}s^p)^{1/p}}
=\frac{2\pi(p-1)^{1/p}}{p\sin(\pi/p)}.
$$
\end{lemma}

\begin{lemma}[\cite{h1}] \label{lem3}
 Let $ a, b, p>0 $, then there holds
  $$ (a+b)^p\leq D_p(a^p+b^p). $$
\end{lemma}

For the sake of convenience, we list the following assumptions which will
 be used frequently in Section 3.
\begin{itemize}
\item[(H1)] For $i=1,2$, there are positive constants 
$r_i, r_i^\ast, m_i$ with $m_2\leq p-1 $ and $m_1>p-1 $  
such that for $|u|>1$ there hold
\begin{itemize}
\item[(i)] $ r_1|u|^{m_1}\leq |g_1(u)|\leq r_2|u|^{m_1}$ and
    $ r_1^\ast|u|^{m_2}\leq |g_2(u)|\leq r_2^\ast|u|^{m_2}$.

\item[(ii)] $ug_i(u)>0$.
\end{itemize}

\item[(H2)] $ A=D_{\frac{1}{m_1}}
(\frac{r_2^\ast \bar \beta}{\bar \alpha r_1})^{1/m_1}<1$.

\item[(H3)]  There are constants $\gamma,r_3>0$ and $k_0\in Z $ such 
that $m_1=r_3+p-1$ and
  $$
0\leq\tau_1(t,x(t))-k_0T\leq \max \{\frac{\gamma^q}{1+|x|_\infty^{r_3q}},T \},
\quad \forall t\in[0,T],\;x(t)\in C[0,T]. 
 $$
where $q>1: \frac{1}{p}+\frac{1}{q}=1$
\end{itemize}

\section{Main Results} 

\begin{theorem} \label{thm1}
 Suppose that {\rm (H1)--(H3)}. Then  \eqref{1.1} has at least one 
$T$-periodic solution if one of the following two conditions holds
\begin{itemize}
\item[(1)] $m_2=p-1$, $\Delta_1+\Delta_2<1$,

\item[(2)] $ m_2<p-1$, $\Delta_1<1$,
\end{itemize}
where 
\[
\Delta_1= \frac{D_{p-1}\overline{\alpha}r_2 T^{\frac{p-1}{q}}
\gamma}{2^{p-1}(1-A)^{p-1} }\big(\frac{T}{\pi _p}\big)^{p},\quad
\Delta_2=\frac{r_2^*\overline{\beta}D_{m_2+1}
T^{\frac{m_2+1}{q}}}{2^{m_2+1}(1-A)^{m_2+1}}\big(\frac{T}{\pi _p}\big)^{m_2+1}.
\]
\end{theorem}

\begin{proof} 
Without loss of generality, we assume $\alpha(t)\geq 0$,
$t\in \mathbb{R}$, $\int_0^T\alpha (t)dt>0$, and
 $ \int_0^T\beta(t)dt>0$. Consider the  homotopy equation
\begin{equation} \label{3.1}
\begin{aligned}
&\phi_p(x''))''+\lambda f(x(t))x'(t)+\lambda \alpha(t)g_1(x(t-\tau_1(t,x(t))))  \\
&+\lambda \beta(t)g_2(x(t-\tau_2(t,x(t))))=\lambda p(t).
\end{aligned}
\end{equation}
Suppose that $x(t)$  is an arbitrary $T$-periodic solution of \eqref{3.1}.
 Integrating both sides of equation \eqref{3.1} on $[0,T]$  we obtain
$$ 
\int_0^T\alpha(t)g_1(x(t-\tau_1(t,x(t))))dt
=-\int_0^T\beta(t)g_2(x(t-\tau_2(t,x(t))))dt. 
$$
Applying the mean value theorem, then there exists a constant 
$ \xi\in [0,T] $  such that
\begin{equation}\label{3.2}
g_1(x(\xi-\tau_1(\xi,x(\xi))))\int_0^T\alpha(t)dt
=-\int_0^T\beta(t)g_2(x(t-\tau_2(t,x(t))))dt.
\end{equation}
Now, we claim that the inequality
\begin{equation}\label{3.3}
|x(\xi-\tau_1(\xi,x(\xi)))|\leq A|x|_\infty +B
\end{equation}
holds, where
\begin{gather*}
A=D_{\frac{1}{m_1}}(\frac{r_2^\ast \bar \beta}{\bar \alpha r_1})^{1/m_1},
\quad 
B=D_{\frac{1}{m_1}}(\frac{M_{g_2} \bar \beta}{\bar \alpha r_1})^{1/m_1}+1,\\
\bar{\alpha}=\int_0^T\alpha(t)dt,\quad
\bar{\beta}=\int_0^T|\beta(t)|dt,\quad M_{g_2}=\max_{|u|\leq1}|g_2(u)|.
\end{gather*}
In fact, if $|x(\xi-\tau_1(\xi,x(\xi)))|\leq 1$, 
then inequality \eqref{3.3} holds. 
If $|x(\xi-\tau_1(\xi,x(\xi)))|> 1$, we define
\begin{gather*}
 E_1=\{t\in [0,T]: |x(t-\tau_1(t,x(t)))|\leq 1 \}, \\
 E_2=\{t\in [0,T]: |x(t-\tau_1(t,x(t)))|>1 \}.
\end{gather*}
It follows from (H1)(i) that
\begin{align*}
\bar\alpha r_1|x(\xi-\tau_1(\xi,x(\xi)))|^{m_1} 
&\leq \int_0^T\beta(t)g_2(x(t-\tau_2(t,x(t))))dt \\
&=  \int_{E_1}+\int_{E_2}\beta(t)g_2(x(t-\tau_2(t,x(t))))dt \\
&\leq  r_2^\ast \bar {\beta}|x|_\infty^{m_2}+M_{g_2} \bar {\beta}.
\end{align*}
This implies that
\begin{align*}{}
|x(\xi-\tau_1(\xi,x(\xi)))|
&\leq [\frac{1}{\bar\alpha r_1}(r_2^\ast \bar {\beta}|x|_\infty^{m_2}
 +M_{g_2} \bar {\beta})]^{1/m_1}\\
&\leq D_{\frac{1}{m_1}}[(\frac{r_2^\ast 
 \bar {\beta}}{\bar\alpha r_1})^{1/m_1}|x|_\infty^{\frac{m_2}{m_1}}
 +(\frac{M_{g_2}\bar {\beta}}{\bar\alpha r_1})^{1/m_1}]\\
&\leq D_{\frac{1}{m_1}}(\frac{r_2^\ast 
 \bar {\beta}}{\bar\alpha r_1})^{1/m_1}|x|_\infty 
 +D_{\frac{1}{m_1}}(\frac{M_{g_2}\bar {\beta}}{\bar\alpha r_1})^{1/m_1}.
\end{align*}
Thus, it can be easily seen that \eqref{3.3} holds.
Let
\begin{equation}\label{3.4}
  \xi-\tau_1(\xi,x(\xi))=kT+\bar \xi,
\end{equation}
where $ k $ is an integer and $\bar{\xi} \in [0,T]$, thus we have
$$
x(\xi-\tau_1(\xi,x(\xi)))=x(kT+\bar {\xi})=x(\bar{\xi}). 
$$
Noting that
$$ 
|x(t)|\leq |x(\bar{\xi})|+\frac{1}{2}\int_0^T|x'(s)|ds, 
$$
we have
$$ 
|x|_\infty=\max\limits_{t\in[0,T]}|x(t)|\leq A|x|_\infty
+B+\frac{1}{2}\int_0^T|x'(s)|ds,
 $$
which yields
\begin{equation}\label{3.5}
 |x|_\infty \leq \frac{\int_0^T |x'(s)|ds}{2(1-A)}+\frac{B}{1-A}.
\end{equation}

  On the other hand, multiplying both sides of \eqref{3.1} by $x(t)$, 
and integrating  on $[0,T]$, we obtain
\begin{equation} \label{3.6}
\begin{aligned}
\int_0^T|x''(t)|^pdt
&=-\lambda \int_0^Tf(x(t))x'(t)x(t)dt
 -\lambda\int_0^T \alpha(t)g_1(x(t-\tau_1(t,x(t))))x(t)dt  \\
&\quad -\lambda\int_0^T \beta(t)g_2(x(t-\tau_2(t,x(t))))x(t)dt
 +\lambda\int_0^T p(t)x(t)dt   \\
&\leq \lambda\int_0^T \alpha(t)|g_1(x(t-\tau_1(t,x(t))))|\,
|x(t)-x(t-\tau_1(t,x(t)))|dt   \\
&\quad -\lambda\int_0^T \alpha(t)g_1(x(t-\tau_1(t,x(t))))x(t-\tau_1(t,x(t)))dt   \\
&\quad +\int_0^T |\beta(t)g_2(x(t-\tau_2(t,x(t))))x(t)|dt+\bar{p}|x|_\infty,
\end{aligned}
\end{equation}
where $\bar{p}=\int_0^T|p(t)|dt $.

By the condition (H1)(ii), we have
\begin{equation} \label{3.7}
\begin{aligned}
&-\lambda \int_0^T\alpha(t)g_1(x(t-\tau _1(t,x(t))))x(t-\tau _1(t,x(t)))dt \\
&=  -\lambda \int_{E_1} -\lambda \int_{E_2}\alpha(t)g_1(x(t-\tau _1(t,x(t))))
 x(t-\tau _1(t,x(t)))dt \\
&\leq  \int_{E_1}\alpha(t)|g_1(x(t-\tau _1(t,x(t))))x(t-\tau _1(t,x(t)))|dt \\
&\leq  \overline{\alpha}M_{g_1},
\end{aligned}
\end{equation}
where $M_{g_1}=\max _{|u|\leq 1}|g_1(u)|$. Using the condition (H1) again, 
we obtain
\begin{align*}
& \int_0^T\alpha(t)|g_1(x(t-\tau _1(t,x(t))))||x(t)-x(t-\tau _1(t,x(t)))|dt \\
&=  \int_{E_1}+\int_{E_2}\alpha(t)|g_1(x(t-\tau _1(t,x(t))))|
 |x(t)-x(t-\tau _1(t,x(t)))|dt \\
&\leq  \overline{\alpha}M_{g_1}+\overline{\alpha}M_{g_1}|x|_\infty
 +\overline{\alpha}r_2 \max_{t\in[0,T]} |x(t)-x(t-\tau _1(t,x(t)))|\times |x|_\infty^{m_1},
\end{align*}
and
\[
\int_0^T|\beta(t)||g_2(x(t-\tau _2(t,x(t))))x(t)|dt
\leq \overline{\beta}r_2^*|x|_\infty^{m_2+1}+\overline{\beta}M_{g_2}|x|_\infty,
\]
where $M_{g_2}=\max _{|u|\leq 1}|g_2(u)|$ and 
$\overline{\beta}=\int_0^T|\beta(t)|dt$.
So  \eqref{3.6} yields
\begin{equation} \label{3.8}
\begin{aligned}
  \int_0^T|x''(t)|^p dt
&\leq  \overline{\alpha}r_2 \max_{t\in[0,T]}
 |x(t)-x(t-\tau _1(t,x(t)))|\times |x|_\infty^{m_1}
 +\overline{\beta}r_2^*|x|_\infty^{m_2+1}\\
&\quad +(\overline{\alpha}M_{g_1}+\overline{\beta}M_{g_2}
 +\overline{p})|x|_\infty+2\overline{\alpha}M_{g_1} \\
&= \overline{\alpha}r_2 \max_{t\in[0,T]} |x(t)-x(t-\tau _1(t,x(t)))|\times |x|_\infty^{m_1}+\overline{\beta}r_2^*|x|_\infty^{m_2+1}\\
&\quad +\theta|x|_\infty+K,
\end{aligned}
\end{equation}
where $\theta=\overline{\alpha}M_{g_1}+\overline{\beta}M_{g_2}+\overline{p}$ 
and $K=2 \overline{\alpha}M_{g_1}$.

Since $x(0)=x(T)$, there exists a constant $\zeta \in [0,T]$ such that 
$x'(\zeta)=0$.
Let $\omega (t)=x'(t+\zeta)$, then $\omega(0)=\omega(T)=0$. 
By Lemma \ref{lem2}, we have
\[
\int_0^T|x'(t)|^pdt\leq \big(\frac{T}{\pi _p}\big)^p
\Big(\int_0^T|x''(t)|^pdt\Big).
\]
From (H3) and H\"{o}lder's inequality, we have
\begin{equation} \label{3.9}
\begin{aligned}
&  \max_{t\in[0,T]} |x(t)-x(t-\tau _1(t,x(t)))| \\
&=  \max_{t\in[0,T]} |x(t)-x(t-\tau _1(t,x(t))+ k_0T)|\\
&=  \max_{t\in[0,T]}|\int_{t-\tau _1(t,x(t))+k_0T}^{t}x'(s)ds|\\
&\leq  \max_{t\in [0,T]} | \tau _1 {(t,x(t))-k_0T|^{1/q}}
\Big(\int_{t-\tau_1 (t,x(t))+k_0T}^{t} |x'(s)| ^p ds\Big)^{1/p}\\
&\leq  \max_{t\in [0,T]} | \tau _1 {(t,x(t))-k_0T|_\infty^{1/q}}
\Big(\int_{0}^{T} |x'(s)| ^p ds\Big)^{1/p}.
\end{aligned}
\end{equation}
Moreover, from \eqref{3.5} and by H\"{o}lder's inequality, we have
\begin{equation} \label{3.10}
\begin{aligned}
 r_2^*\overline{\beta}|x|_\infty^{m_2+1}
&\leq  r_2^*\overline{\beta}[\frac{\int_{0}^{T} |x'(s)| ds}{2(1-A)}
 +\frac{B}{1-A}]^{m_2+1}\\
&\leq  \frac{r_2^*\overline{\beta}D_{m_2+1}}{2^{m_2+1}(1-A)^{m_2+1}}
 \Big(\int_{0}^{T} |x'(s)|ds\Big)^{m_2+1}
 +\frac{r_2^*\overline{\beta}D_{m_2+1}B^{m_2+1}}{(1-A)^{m_2+1}}\\
&\leq  \frac{r_2^*\overline{\beta}D_{m_2+1}T^{\frac{m_2+1}{q}}}
 {2^{m_2+1}(1-A)^{m_2+1}}\big(\frac{T}{\pi _p}\big)^{m_2+1}
 \Big(\int_0^T|x''(s)|^pds\Big)^{\frac{m_2+1}{p}}\\
&\quad +\frac{r_2^*\overline{\beta}D_{m_2+1}B^{m_2+1}}{(1-A)^{m_2+1}},
\end{aligned}
\end{equation}
and
\begin{equation} \label{3.11}
\begin{aligned}
\theta|x|_\infty
&\leq \theta \big[\frac{\int_{0}^{T} |x'(s)| ds}{2(1-A)}+\frac{B}{1-A}\big]  \\
&\leq \frac{\theta T^{1/q}}{2(1-A)}\big(\frac{T}{\pi _p}\big)
\Big(\int_0^T|x''(s)|^pds\Big)^{1/p}+\frac{\theta B}{1-A}.
\end{aligned}
\end{equation}
By of $m_1=r_3+p-1$  and the condition (H3), and 
combining \eqref{3.9}-\eqref{3.11}, we have
\begin{align}
&\int_0^T|x''(t)|^pdt \nonumber \\
&\leq \overline{\alpha}r_2 \max_{t\in[0,T]} |x(t)
 -x(t-\tau _1(t,x(t)))||x|_\infty^{r_3}|x|_\infty^{p-1}
 +\overline{\beta}r_2^*|x|_\infty^{m_2+1}
 +\theta|x|_\infty+K \nonumber\\
&\leq  \overline{\alpha}r_2 \gamma (\int_0^T|x'(s)|^pds)^{1/p}
 [\frac{\int_{0}^{T} |x'(s)| ds}{2(1-A)}+\frac{B}{1-A}]^{p-1}
 +\overline{\beta}r_2^*|x|_\infty^{m_2+1}
 +\theta|x|_\infty+K \nonumber\\
&\leq  \overline{\alpha}r_2 \gamma (\int_0^T|x'(s)|^pds)^{1/p}\frac{D_{p-1}
\Big(\int_{0}^{T} |x'(s)| ds\Big)^{p-1}}{2^{p-1}(1-A)^{p-1}} \nonumber\\
&\quad + \overline{\alpha}r_2 \gamma D_{p-1}
 \Big(\int_0^T|x'(s)|^pds\Big)^{1/p}\frac{B^{p-1}}{(1-A)^{p-1}}\nonumber \\
&\quad +\frac{\theta T^{1/q}}{2(1-A)}\big(\frac{T}{\pi _p}\big)
 \Big(\int_0^T|x''(s)|^pds\Big)^{1/p} \nonumber\\
&\quad +\frac{r_2^*\overline{\beta}D_{m_2+1}T^{\frac{m_2+1}{q}}}
 {2^{m_2+1}(1-A)^{m_2+1}}\big(\frac{T}{\pi _p}\big)^{m_2+1}(\int_0^T|x''(s)|^pds)
 ^{\frac{m_2+1}{p}} \nonumber\\
&\quad +\frac{r_2^*\overline{\beta}D_{m_2+1}B^{m_2+1}}{(1-A)^{m_2+1}}
 +\frac{B\theta}{1-A}+K  \nonumber\\
&\leq  \frac{D_{p-1}\overline{\alpha}r_2 T^{\frac{p-1}{q}}\gamma}
 {2^{p-1}(1-A)^{p-1} }\Big(\int_0^T|x'(s)|^pds\Big)
 +  \frac{D_{p-1}\overline{\alpha}r_2 B^{p-1}\gamma}{(1-A)^{p-1} }
 \Big(\int_0^T|x'(s)|^pds\Big)^{1/p} \nonumber\\
&\quad +\frac{r_2^*\overline{\beta}D_{m_2+1}T^{\frac{m_2+1}{q}}}
 {2^{m_2+1}(1-A)^{m_2+1}}\big(\frac{T}{\pi _p}\big)^{m_2+1}(\int_0^T|x''(s)|^pds)
 ^{\frac{m_2+1}{p}} \nonumber\\
&\quad +\frac{\theta T^{1/q}}{2(1-A)}\big(\frac{T}{\pi _p}\big)
 \Big(\int_0^T|x''(s)|^pds\Big)^{1/p}+C \nonumber\\
&\leq  \frac{D_{p-1}\overline{\alpha}r_2 T^{\frac{p-1}{q}}
 \gamma}{2^{p-1}(1-A)^{p-1} }\big(\frac{T}{\pi _p}\big)^{p}
 \Big(\int_0^T|x''(s)|^pds\Big) \nonumber\\
&\quad  + \frac{D_{p-1}\overline{\alpha}r_2 B^{p-1}\gamma}{(1-A)^{p-1} }
  \big(\frac{T}{\pi _p}\big)\Big(\int_0^T|x''(s)|^pds\Big)^{1/p} \nonumber\\
&\quad +\frac{r_2^*\overline{\beta}D_{m_2+1}T^{\frac{m_2+1}{q}}}
 {2^{m_2+1}(1-A)^{m_2+1}}\big(\frac{T}{\pi _p}\big)^{m_2+1}\Big(\int_0^T|x''(s)|^pds\Big)
 ^{\frac{m_2+1}{p}} \nonumber\\
&\quad +\frac{\theta T^{1/q}}{2(1-A)}\big(\frac{T}{\pi _p}\big)
 \Big(\int_0^T|x''(s)|^pds\Big)^{1/p}+C \nonumber\\
&=  \Delta_1\Big(\int_0^T|x''(s)|^pds\Big)+  \Delta_2\Big(\int_0^T|x''(s)|^pds\Big)
 ^{\frac{m_2+1}{p}} \nonumber\\
&\quad +  \frac{D_{p-1}\overline{\alpha}r_2 B^{p-1}\gamma}{(1-A)^{p-1} }
 \big(\frac{T}{\pi _p}\big)\Big(\int_0^T|x''(s)|^pds\Big)^{1/p} \nonumber \\
&\quad +\frac{\theta T^{1/q}}{2(1-A)}\big(\frac{T}{\pi _p}\big)
 \Big(\int_0^T|x''(s)|^pds\Big)^{1/p}+C,   \label{3.12}
\end{align}
where
\[
C=\frac{r_2^*\overline{\beta}D_{m_2+1}B^{m_2+1}}{(1-A)^{m_2+1}}
+\frac{B\theta}{1-A}+K.
\]

If $m_2=p-1$ and $\Delta_1+\Delta_2<1$, then
 from \eqref{3.12} it follows that  $\int_0^T|x''(t)|^pdt$ is bounded.
If $m_2<p-1$ and $\Delta_1<1$, then from  $\frac{m_2+1}{p}<1$ and \eqref{3.12} 
we see that $\int_0^T|x''(t)|^pdt$ is also bounded.
Thus, there exists a constant $M>0$ such that
\begin{align*}
\Big(\int_0^T|x''(t)|^pdt\Big)^{1/p}\leq M,
\end{align*}
which shows that there exist positive numbers $M_0$ and $M_1$ such that
\[
|x|_\infty\leq M_0,\quad |x'|_\infty\leq M_1.
\]
Let
\[
\Omega=\{ x(t)\in C_T^1:||x||<\rho  \},
\]
where $\rho>\max \{1,M_0,M_1\}$. Then the homotopy equation \eqref{3.1}
 has no $T$-periodic solution on $\partial\Omega$. In addition,
\begin{align*}
F(\rho)
&= -\frac{1}{T}[\int_0^T\alpha(t)g_1(\rho)dt
 +\int_0^T\beta(t)g_2(\rho)dt-\int_0^Tp(t)dt]\\
&= -\frac{1}{T}g_1(\rho)\int_0^T\alpha(t)dt
 -\frac{1}{T}g_2(\rho)\int_0^T\beta(t)dt.
\end{align*}
It means that the second condition of Lemma \ref{lem1} is satisfied, and
$F(\rho)F(-\rho)<0$ from (H1)(ii).
Consequently, from Lemma \ref{lem1} the equation \eqref{1.1} has at 
least one $T$-periodic solution in $\overline {\Omega}$ .
\end{proof}


\begin{remark} \label{rmk1} \rm
If we replace the conditions $\alpha(t)>0$, $\int_0^T\beta(t)dt>0$ with 
$\alpha(t)<0$, $\int_0^T\beta(t)dt<0$ or $\alpha(t)<0$, $\int_0^T\beta(t)dt>0$ 
or $\alpha(t)>0$, $\int_0^T\beta(t)dt<0$, we can obtain the same conclusion 
as Theorem \ref{thm1}.
\end{remark}

\begin{remark} \label{rmk2} \rm
Condition (H1) can be replaced by
\begin{itemize}
\item[(H1')]  For $i=1,2$, there are positive constants $r_i,r_i^*,m_i,d$  
with $m_2\leq p-1$ and $m_1> p-1$ such that
\begin{itemize}
\item[(i)] $r_1|u|^{m_1}\leq |g_1(u)|\leq r_2|u|^{m_1}$ and
$r_1^*|u|^{m_2}\leq |g_2(u)|\leq r_2^*|u|^{m_2}$
for all $|u|>d \geq 1$,

\item[(ii)] $g_i(u)(\operatorname{sgn}u)>0$ for all $|u|>d\geq1$;
\end{itemize}
\end{itemize}
while the conclusion of Theorem \ref{thm1} is still true.
\end{remark}

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\end{document}