\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 39, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/39\hfil Positive periodic solutions] {Positive periodic solutions for Li\'enard type $p$-Laplacian equations} \author[J. Meng\hfil EJDE-2009/39\hfilneg] {Junxia Meng} \address{Junxia Meng \newline College of Mathematics and Information Engineering, Jiaxing University, Jiaxing\\ Zhejiang 314001, China} \email{mengjunxia1968@yahoo.com.cn} \thanks{Submitted February 17, 2009. Published March 19, 2009.} \thanks{Supported by grant 20070605 from Scientific Research Fund of Zhejiang Provincial \hfill\break\indent Education Department} \subjclass[2000]{34K15, 34C25} \keywords{$p$-Laplacian; positive periodic solutions; Li\'enard equation; \hfill\break\indent topological degree} \begin{abstract} Using topological degree theory, we obtain sufficient conditions for the existence and uniqueness of positive periodic solutions for Li\'enard type $p$-Laplacian differential equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} In recent years, the existence of periodic solutions for the Duffing equation, Rayleigh equation and Li\'enard type equation has received a lot of attention. We refer the reader to \cite{WGQ,lsp,lsp2,zmg,lbw,zfx} and the references cited therein. However, as far as we know, fewer papers discuss the existence and uniqueness of positive periodic solutions for Li\'enard type $p$-Laplacian differential equation. In this paper we study the existence and uniqueness of positive $T$-periodic solutions of the Li\'enard type $p$-Laplacian differential equation of the form: $$\label{e1} (\varphi_p(x'(t)))'+f(x(t))x'(t )+g(x(t ))=e(t),$$ where $p>1$ and $\varphi_p:\mathbb{R} \to \mathbb{R}$ is given by $\varphi_p(s)=|s|^{p-2}s$ for $s\neq0$ and $\varphi_p(0)=0$, $f$ and $g$ are continuous functions defined on $\mathbb{R}$. $e$ is a continuous periodic function defined on $\mathbb{R}$ with period $T$, and $T>0$. By using topological degree theory and some analysis skill, we establish some sufficient conditions for the existence and uniqueness of $T$-periodic solutions of \eqref{e1}. The results of this paper are new and they complement previously known results. \section{Preliminaries} For convenience, let us denote $$C_{T}^1:=\{x\in C^1(\mathbb{R}, \mathbb{R}): x \text{ is T-periodic} \},$$ which is a Banach space endowed with the norm $\|x\|=\max\{|x|_{\infty}, |x'|_{\infty}\}$, and $$|x|_\infty=\max_{t\in [0,T]}|x(t)|, \quad |x'|_\infty=\max_{t\in [0,T]}|x'(t)|, \quad |x|_k=\Big(\int^T_0|x(t)|^kdt\Big)^{1/k}.$$ For the periodic boundary-value problem $$\label{e2} (\varphi_p(x'(t)))'=\widetilde{f}(t,x,x'),\quad x(0)=x(T),\quad x'(0)=x'(T)$$ where $\widetilde{f}$ is a continuous function and T$-$periodic in the first variable, we have the following result. \begin{lemma}[\cite{MJM}] \label{lem1} Let $\Omega$ be an open bounded set in $C_T^1,$ if the following conditions hold \begin{itemize} \item[(i)] For each $\lambda \in(0,1)$ the problem $(\varphi_p(x'(t)))'=\lambda \widetilde{f}(t,x,x') ,\quad x(0)=x(T),\quad x'(0)=x'(T)$ has no solution on $\partial\Omega$; \item[(ii)] The equation $F(a):=\frac{1}{T}\int_0^T \widetilde{f}(t,a,0)\,dt=0$ has no solution on $\partial \Omega \cap \mathbb{R}$; \item[(iii)] The Brouwer degree of $F$ satisfies $\deg (F,\Omega \cap \mathbb{R}, 0)\neq 0,$ \end{itemize} Then the periodic boundary value problem \eqref{e2} has at least one T$-$periodic solution on $\overline \Omega$. \end{lemma} Set $$\label{e3} \Psi (x)=\int_{0}^{x}f(u)du, \quad y (t)= \varphi_p(x '(t))+\Psi (x (t)).$$ We can rewrite \eqref{e1} in the form $$\label{e4} \begin{gathered} { x' (t)} = |y(t)-\Psi (x(t))|^{q-1}\mathop{\rm sign}(y(t)-\Psi (x(t))), \\ { y' (t)}=- g( x (t ))+e(t), \end{gathered}$$ where $q>1$ and $\frac{1}{p}+\frac{1}{q}=1$. \begin{lemma} \label{lem2} Suppose that the following condition holds. \begin{itemize} \item[(A1)] $g$ is a continuously differentiable function defined on $\mathbb{R}$, and $g_{x} '(x)<0$. \end{itemize} Then \eqref{e1} has at most one $T$-periodic solution. \end{lemma} \begin{proof} Suppose that $x_{1}(t)$ and $x_{2}(t)$ are two $T$-periodic solutions of \eqref{e1}. Then, from \eqref{e4}, we obtain $$\label{e5} \begin{gathered} { x_{i}' (t)} = |y_{i}(t)-\Psi (x_{i}(t))|^{q-1} \mathop{\rm sign}(y_{i}(t)-\Psi (x_{i}(t))), \\ { y_{i}' (t)}=- g( x_{i} (t ))+e(t), \quad i=1,2. \end{gathered}$$ Set $$\label{e6} v(t)=x_{1}(t)-x_{2}(t), \quad u(t)=y_{1}(t)-y_{2}(t),$$ it follows from \eqref{e5} that \label{e7} \begin{aligned} { v' (t)} &= |y_{1}(t)-\Psi (x_{1}(t))|^{q-1}\mathop{\rm sign} (y_{1}(t)-\Psi (x_{1}(t)))\\ &\quad -|y_{2}(t)-\Psi (x_{2}(t))|^{q-1}\mathop{\rm sign}(y_{2}(t)-\Psi (x_{2}(t))), \\ { u' (t)}&=- [g( x_{1} (t ))-g( x_{2} (t ))], \end{aligned} Now, we prove that $u(t)\leq 0$ for all $t\in \mathbb{R}$. Contrarily, in view of $u\in C^{2}[0, T]$ and $u(t+T)=u(t)$ for all $t\in \mathbb{R}$, we obtain $$\max_{t\in \mathbb{R}} u(t)>0.$$ Then, there must exist $t^{*}\in \mathbb{R}$ (for convenience, we can choose $t^{*}\in (0, T)$) such that $$u(t^{*})=\max_{t\in [0, \ T]} u(t)=\max_{t\in \mathbb{R}} u(t)>0,$$ which, together with $g '(x)<0$, implies that \label{e8} \begin{gathered} u'(t^{*}) =- [g( x_{1} (t^{*} ))-g( x_{2} (t^{*} ))]=0, \quad x_{1} (t^{*} )= x_{2} (t^{*} ), \\ \begin{aligned} u''(t^{*}) &= (- (g( x_{1} (t ))-g( x_{2} (t ))))' |_{t=t^{*}}\\ &=-[g_{x}'( x_{1} (t^{*} ))x'_{1} (t^{*} )-g_{x}'( x_{2} (t^{*} ))x'_{2} (t^{*} )]\leq 0. \end{aligned} \end{gathered} Then \label{e9} \begin{aligned} u''(t^{*}) & = -g_{x}'( x_{1} (t^{*} ))[x'_{1} (t^{*} )-x'_{2} (t^{*} )] \\ & = -g_{x}'( x_{1} (t^{*} ))[|y_{1}(t^{*})-\Psi (x_{1}(t^{*}))|^{q-1}\mathop{\rm sign}(y_{1}(t^{*})-\Psi (x_{1}(t^{*}))) \\ & \quad -|y_{2}(t^{*})-\Psi (x_{2}(t^{*}))|^{q-1}\mathop{\rm sign}(y_{2}(t^{*})-\Psi (x_{2}(t^{*})))] \\ & = -g_{x}'( x_{1} (t^{*} ))[|y_{1}(t^{*})-\Psi (x_{1}(t^{*}))|^{q-1}\mathop{\rm sign}(y_{1}(t^{*})-\Psi (x_{1}(t^{*}))) \\ & \quad -|y_{2}(t^{*})-\Psi (x_{1}(t^{*}))|^{q-1}\mathop{\rm sign}(y_{2}(t^{*})-\Psi (x_{1}(t^{*})))]. \end{aligned} In view of $$\label{e10} -g_{x}'( x_{1} (t^{*} ))>0, \quad u(t^{*})=y_{1}(t^{*})-y_{2}(t^{*})>0,$$ and \begin{align*} & |y_{1}(t^{*})-\Psi (x_{1}(t^{*}))|^{q-1}\mathop{\rm sign}(y_{1}(t^{*}) -\Psi (x_{1}(t^{*}))) \\ &-|y_{2}(t^{*})-\Psi (x_{1}(t^{*}))|^{q-1}\mathop{\rm sign}(y_{2}(t^{*}) -\Psi (x_{1}(t^{*})))>0 . \end{align*} It follows from \eqref{e9} that \label{e11} \begin{aligned} u''(t^{*}) & = -g_{x}'( x_{1} (t^{*} ))[|y_{1}(t^{*}) -\Psi (x_{1}(t^{*}))|^{q-1}\mathop{\rm sign}(y_{1}(t^{*}) -\Psi (x_{1}(t^{*}))) \\ & \quad -|y_{2}(t^{*})-\Psi (x_{1}(t^{*}))|^{q-1}\mathop{\rm sign}(y_{2}(t^{*})-\Psi (x_{1}(t^{*})))]>0, \end{aligned} which contradicts the second equation of \eqref{e8}. This contradiction implies that $$u(t)=y_{1}(t)-y_{2}(t)\leq 0 \quad \text{ for all } t\in \mathbb{R}.$$ By using a similar argument, we can also show that $$y_{2}(t)-y_{1}(t)\leq 0 \quad \text{for all } t\in \mathbb{R}.$$ Therefore, we obtain $y_{2}(t)\equiv y_{1}(t)$ for all $t\in \mathbb{R}$. Then, from \eqref{e7}, we get $$g( x_{1} (t ))-g( x_{2} (t )) \equiv0 \quad \text{for all } t\in \mathbb{R},$$ again from $g_{x} '(x)<0$, which implies that $x_{2}(t)\equiv x_{1}(t)$ for all $t\in \mathbb{R}$. Hence, \eqref{e1} has at most one $T$-periodic solution. The proof is complete. \end{proof} \section{Main Results} Using Lemmas \ref{lem1} and \ref{lem2}, we obtain our main results: \begin{theorem} \label{thm1} Let {\rm (A1)} hold. Suppose that there exists a positive constant $d$ such that \begin{itemize} \item[(A2)] $g( x)-e(t) <0$ for $x >d$ and $t \in \mathbb{R}$, $g( x)-e(t) >0$ for $x \leq 0$ and $t \in \mathbb{R}$. \end{itemize} Then \eqref{e1} has a unique positive $T$-periodic solution. \end{theorem} \begin{proof} Consider the homotopic equation of \eqref{e1} as follows: $$\label{e12} (\varphi_p(x'(t)))'+\lambda f(x(t ))x'(t ) +\lambda g( x(t ))=\lambda e(t),\quad \lambda \in(0,1)$$ By Lemma \ref{lem2}, and (A1), it is easy to see that \eqref{e1} has at most one positive $T$-periodic solution. Thus, to prove Theorem \ref{thm1}, it suffices to show that \eqref{e1} has at least one $T$-periodic solution. To do this, we shall apply Lemma \ref{lem1}. Firstly, we will claim that the set of all possible $T$-periodic solutions of \eqref{e12} is bounded. Let $x(t)\in C_{T}^1$ be an arbitrary solution of \eqref{e12} with period $T$. By integrating two sides of \eqref{e12} over $[0,T]$, and noticing that $x'(0)=x'(T)$, we have $$\label{e13} \int_0^{T} (g( x(t))-e(t)) \,dt=0.$$ As $x(0)=x(T)$, there exists $t_0\in[0,T]$ such that $x'(t_0)=0$, while $\varphi_p(0)=0$ we see \label{e14} \begin{aligned} |\varphi_p(x'(t)) | &= |\int_{t_0}^t(\varphi_p(x'(s)))'\,ds|\\ &\leq \lambda\int_0^{T}|f(x(t ))\|x'(t ) |\,dt + \lambda\int_0^{T}|g( x(t )) |\,dt + \lambda\int_0^{T}|e(t )|\,dt, \end{aligned} where $t \in [t_{0}, t_{0}+T]$. From \eqref{e13}, there exists a $\bar{\xi}\in [0,T]$ such that $g( x(\bar{\xi} ))-e(\bar{\xi} ) =0$. In view of (A2), we obtain $| x(\bar{\xi} )| \leq d$. Then, we have $$|x(t)|=|x(\bar{\xi})+\int^t_{\bar{\xi}}x'(s)ds|\le d+ \int^t_{\bar{\xi}}|x'(s)|ds, \ t\in [\bar{\xi}, \quad \bar{\xi}+T],$$ and $$|x(t)|=|x(t-T)|=|x(\bar{\xi})-\int^{\bar{\xi}}_{ t-T }x'(s)ds|\le d+\int^{\bar{\xi}}_{ t-T }|x'(s)|ds, t\in [\bar{\xi}, \quad \bar{\xi}+T].$$ Combining the above two inequalities, we obtain \label{e15} \begin{aligned} |x|_\infty &=\max_{t\in [0,T]}|x(t)| =\max_{t\in[\bar{\xi}, \ \bar{\xi}+T]}|x(t)| \\ & \le \max_{t\in[\bar{\xi}, \ \bar{\xi}+T]} \{ d+\frac{1}{2}(\int^t_{\bar{\xi}}|x'(s)|ds +\int^{\bar{\xi}}_{ t-T }|x'(s)|ds) \} \\ & \le d+ \frac{1}{2}\int^{T}_{0 }|x'(s)|ds. \end{aligned} Denote $E_1 = \{t: t\in[0,T],\,|x(t )|>d\}, \quad E_2 = \{t:t\in[0,T], |x(t)| \leq d\} .$ Since $x (t)$ is $T$-periodic, multiplying $x (t)$ and \eqref{e12} and then integrating it from $0$ to $T$, in view of (A2), we get \label{e16} \begin{aligned} \int^T_0 |x '(t)|^{p} dt & = -\int^T_0 (\varphi_p(x '(t)) )' x (t) dt \\ & = \lambda\int_{E_1} [g ( x (t ))-e (t )] x (t) dt+\lambda\int_{E_2} [g ( x (t ))-e (t )] x (t) dt \\ & \leq \int^T_0 \max\{|g ( x (t ))-e (t )|:t\in \mathbb{R}, |x(t)| \leq d \} |x (t)| dt \\ & \leq D T |x |_{\infty}, \end{aligned} where $D=\max\{|g ( x )-e (t )|: |x | \leq d , \; t\in \mathbb{R} \}$. For $x(t)\in C(\mathbb{R}, \mathbb{R})$ with $x(t+T)=x(t)$, and $0< r\le s$, by using H\"older inequality, we obtain \begin{align*} \Big(\frac{1}{T}\int^T_0|x(t)|^rdt\Big)^{1/r} &\le \Big(\frac{1}{T}(\int^T_0(|x(t)|^{r})^{s/r}dt)^{r/s} (\int^T_01dt)^{\frac{s-r}{s}} \Big)^{1/r}\\ &=\Big(\frac{1}{T}\int^T_0|x(t)|^s dt\Big)^{1/s}, \end{align*} this implies that $$\label{e17} |x|_{r}\leq T^\frac{s-r}{rs}|x|_{s}, \quad \text{for } 0< r\le s.$$ Then, in view of \eqref{e15}, \eqref{e16} and \eqref{e17}, we can get \label{e18} \begin{aligned} (\int^{T}_{0 }|x'(t)|dt) ^{p} &\leq T^{ p-1 }|x'(t)|^{p}_{p} = T^{p-1} \int^T_0 |x '(t)|^{p} dt \\ & \leq T^{p-1}D T |x |_{\infty} \\ & \leq T^{p }D (d+ \frac{1}{2}\int^{T}_{0 }|x'(s)|ds). \end{aligned} Since $p>1$, the above inequality allows as to choose a positive constant $M_1$ such that $\int^{T}_{0 }|x'(s)|ds\leq M_1, \quad |x|_{\infty} \leq d+ \frac{1}{2}\int^{T}_{0}|x'(s)|ds \leq M_1.$ In view of \eqref{e14}, we have \label{e19} \begin{aligned} | x' |_{\infty}^{p-1} &=\max_{t\in[0,T]}\{|\varphi_p(x'(t)) |\} \\ & = \max_{t\in[t_{0},t_{0}+T]}\{|\int_{t_0}^t(\varphi_p(x'(s)))'\,ds|\} \\ & \leq \int_0^{T}|f(x(t ))\|x'(t ) |\,dt + \int_0^{T}|g( x(t )) |\,dt + \int_0^{T}|e(t ) |\,dt \\ &\leq [\max\{|f ( x )|: |x | \leq M_1 \} ]M_1+ T[\max\{|g ( x )|: |x | \leq M_1 \}+|e|_{\infty}] . \end{aligned} Thus, we can get some positive constant $M_2> M_1+1$ such that for all $t\in \mathbb{R}$, $|x'(t)|\leq M_2$. Set $\Omega=\{x\in C_{T}^1:\|x\|\leq M_2+1\}$, then we know that \eqref{e12} has no solution on $\partial \Omega$ as $\lambda \in (0,1)$ and when $x(t)\in\partial\Omega\cap \mathbb{R}$, $x(t)=M_2+1$ or $x(t)=-M_2-1$, from $(A_{2})$ , we can see that \begin{gather*} \frac{1}{T}\int_0^{T}\{-g( M_2+1)+e(t)\}\,dt = -\frac{1}{T}\int_0^{T}\{ g( M_2+1)-e(t)\}\,dt>0 , \\ \frac{1}{T}\int_0^{T}\{-g( -M_2-1)+e(t)\}\,dt = -\frac{1}{T}\int_0^{T}\{g( -M_2-1)-e(t)\}\,dt<0, \end{gather*} so condition (ii) is also satisfied. Set $H(x,\mu)=\mu x-(1-\mu)\frac{1}{T}\int_0^{T}\{g( x)-e(t)\}\,dt,$ and when $x\in\partial\Omega\cap \mathbb{R}$, $\mu\in[0,1]$ we have $xH(x,\mu)=\mu x^2-(1-\mu)x\frac{1}{T}\int_0^{T}\{g( x)-e(t)\}\,dt>0 .$ Thus $H(x,\mu)$ is a homotopic transformation and $$\deg \{F,\Omega\cap \mathbb{R},0\} = \deg \{-\frac{1}{T}\int_0^{T}\{g(x)-e(t)\}\,dt,\Omega\cap \mathbb{R},0\} =\deg \{x,\Omega\cap \mathbb{R},0\}\neq 0.$$ so condition (iii) is satisfied. In view of the previous Lemma \ref{lem1}, there exists at least one solution with period $T$. Suppose that $x(t)$ is the $T$-periodic solution of \eqref{e1}. Let $\bar{t}$ be the global minimum point of $x (t)$ on $[0, T ]$. Then $x '(\bar{t} ) = 0$ and we claim that $$(\varphi_p(x'(\bar{t})))' =(|x'(\bar{t})|^{p-2} x'(\bar{t}))' \geq 0.\label{e20}$$ Assume, by way of contradiction, that \eqref{e20} does not hold. Then $$(\varphi_p(x'(\bar{t})))' =(|x'(\bar{t})|^{p-2} x'(\bar{t}))' <0,$$ and there exists $\varepsilon> 0$ such that $(\varphi_p(x'(t)))' =(|x'(t)|^{p-2} x'(t))' <0$ for $t\in (\bar{t}-\varepsilon, \bar{t} + \varepsilon)$. Therefore, $\varphi_p(x'(t)) = |x'(t)|^{p-2} x'(t)$ is strictly decreasing for $t\in (\bar{t}-\varepsilon, \bar{t} + \varepsilon)$, which implies that $x'(t)$ is strictly decreasing for $t\in (\bar{t}-\varepsilon, \bar{t} + \varepsilon).$ This contradicts the definition of $\bar{t}$. Thus, \eqref{e20} is true. From \eqref{e1} and \eqref{e20}, we have $$\label{e21} g( x(\bar{t} ))-e(\bar{t})\leq 0.$$ In view of (A2), \eqref{e21} implies $x(\bar{t} )>0$. Thus, $$x(t) \geq \min_{t\in [0, \ T]} x(t)=x(\bar{t} )>0, \quad\text{for all }t \in \mathbb{R},$$ % \eqno (23) which implies that \eqref{e1} has at least one positive solution with period $T$. This completes the proof. \end{proof} \section{An Example} As an application, let us consider the following equation $$\label{e24} (\varphi_p x'(t))'+ e^{x(t)} x'(t ) - (x^{9}(t )+x(t )-12) =\cos^{2}{t},$$ where $p=\sqrt{5}$. We can easily check the conditions (A1) and (A2) hold. By Theorem \ref{thm1}, equation \eqref{e24} has a unique positive $2\pi$-periodic solution. 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