\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 107, 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 (login: ftp)} \thanks{\copyright 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/107\hfil Periodic solutions] {Periodic solutions for a kind of rayleigh equation with two deviating arguments} \author[Y. Wu, B. Xiao, H.Zhang\hfil EJDE-2006/107\hfilneg] {Yuanheng Wu, Bing Xiao, Hong Zhang} % in alphabetical order \address{Yuanheng Wu \newline College of Continuing Education, Guangdong University of Foreign Studies, Guangzhou 510420, China} \email{wyhcd2006@yahoo.com.cn} \address{Bing Xiao \newline Department of Mathematics, Hunan University of Arts and Science, Changde, Hunan 415000, China} \email{changde1218@yahoo.com.cn} \address{Hong Zhang \newline Department of Mathematics, Hunan University of Arts and Science, Changde, Hunan 415000, China} \email{hongzhang320@yahoo.com.cn} \date{} \thanks{Submitted January 27, 2006. Published September 8, 2006.} \thanks{Supported by the NNSF of China and by project 05JJ40009 from the Hunan Provincial \hfill\break\indent Natural Science Foundation of China} \subjclass[2000]{34C25, 34D40} \keywords{Rayleigh equation; deviating argument; periodic solution; \hfill\break\indent coincidence degree} \begin{abstract} In this paper, we use the coincidence degree theory to establish new results on the existence of $T$-periodic solutions for the Rayleigh equation with two deviating arguments of the form $$x''+f(x(t), x'(t))+g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{2}(t)))=p(t).$$ \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Consider the Rayleigh equation with two deviating arguments $$x''+f(x(t),x'(t)) +g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{2}(t)))=p(t), \label{e1.1}$$ where $\tau_{1}$, $\tau_{2}$, $p:\mathbb{R}\to \mathbb{R}$ and $f$, $g_{1}$, $g_{2}:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions, $f(x, 0)=0$, $\tau_{1}$, $\tau_{2}$ and $p$ are $T$-periodic, $g_{1}$ and $g_{2}$ are $T$-periodic in the first argument, and $T>0$. In recent years, the problem of the existence of periodic solutions of \eqref{e1.1} has been extensively studied in the literature. We refer the reader to \cite{g2,s1,s2,s3,x1} and the references cited therein. Moreover, in the above-mentioned literature, we find the following assumptions: \begin{itemize} \item[(H0)] $g_{1}(t,x) +g_{2}(t,x)=g(x)$, $g(x)\in C(\mathbb{R}, \mathbb{R})$ and there exist constants $k_{1}\geq 0$ and $k_{2}\geq 0$ such that one of the following conditions holds:\\ (1) $xg(x)>0$, for all $|x|>k_{1}$, and $g(x)\geq - k_{2}$, for all $x\leq -k_{1}$, \\ (2) $xg(x)>0$, for all $|x|>k_{1}$, and $g(x)\leq k_{2}$, for all $x\geq k_{1}$; \item[(H1)] $g_{1}(t,x)+g_{2}(t,x)=g(x)$, $g(x)\in C^{1}(\mathbb{R}, \mathbb{R})$ and there exists a constant $K\geq 0$ such that $$|g'(x)|\leq K, \forall x\in \mathbb{R};$$ \item[(H2)] $f(x, y)=f(y)$, and there exist constants $r\geq 0$ and $K>0$ such that $$|f(y)|\leq r|y|+K,\forall y\in \mathbb{R} ;$$ \item[(H3)] $f(x, y)=f(y)$, and there exists constants $n\geq 1$ and $\sigma>0$ such that $$yf(y)\geq \sigma |y|^{n+1},\quad \forall y\in \mathbb{R} \quad\mbox{or}\quad yf(y)\leq -\sigma |y|^{n+1},\quad \forall y\in \mathbb{R}.$$ \end{itemize} These conditions have been considered for the existence of periodic solutions of \eqref{e1.1}. However, to the best of our knowledge, few authors have considered \eqref{e1.1} without the assumptions (H0)--(H3). Thus, it is worth while to continue to investigate the existence of periodic solutions of \eqref{e1.1} in this case. The main purpose of this paper is to establish sufficient conditions for the existence of $T$-periodic solutions of \eqref{e1.1}. The results of this paper are new and they complement previously known results. In particular, we do not use assumptions (H0)--(H3), and we illustrate our results with examples in Section 4. For ease of exposition, throughout this paper we will adopt the following notation: $$|x|_k=\Big(\int^T_0|x(t)|^kdt\Big)^{1/k}, \quad |x|_\infty=\max_{t\in [0,T]}|x(t)|.$$ Let \begin{gather*} X=\{x|x\in C^{1}(\mathbb{R}, \mathbb{R}): x(t+T)=x(t), \mbox{ for all } t\in \mathbb{R}\}, \\ Y=\{x|x\in C(\mathbb{R}, \mathbb{R}), x(t+T)=x(t), \mbox{ for all } t\in \mathbb{R}\} \end{gather*} be two Banach spaces with the norms $$\|x\|_{X}=\max\{|x|_\infty, |x'|_\infty\}, \quad\mbox{and} \quad \|x\|_{Y}=|x|_\infty.$$ Define a linear operator $L: D(L)\subset X\to Y$, with $D(L)=\{x|x\in X:x''\in C(\mathbb{R}, \mathbb{R})\}$ and for $x \in D(L)$, $$Lx=x''. \label{e1.2}$$ We also define the nonlinear operator $N: X\to Y$ by $$Nx=-f(x(t), x'(t)) -g_{1}(t,x(t-\tau_{1}(t))) -g_{2}(t,x(t-\tau_{2}(t)))+p(t).\label{e1.3}$$ It is easy to see that $$\ker L=\mathbb{R},\quad \mbox{and}\quad \mathop{\rm Im}L=\{x : x\in Y , \int^{T}_{0}x(s)ds=0 \}.$$ Thus, the operator $L$ is a Fredholm operator with index zero. Define the continuous projectors $P: X \to \ker L$ \ and $Q:Y \to Y$ by setting $$Px(t)=x(0)=x(T), \quad Qx(t)=\frac{1}{T}\int^{T}_{0}x(s)ds.$$ and let $$L_{P}=L|_{D(L)\cap \ker P}:D(L)\cap \ker P\to \mathop{\rm Im}L$$ Then, according to \cite{s1}, we have that $L_{P}$ has continuous inverse $L^{-1}_{P}$ on $\mathop{\rm Im}L$ defined by $$L^{-1}_{P}y(t)=-\frac{t}{T}\int^{T}_{0}(t-s)y(s)ds+\int^{t}_{0}(t-s)y(s)ds. \label{e1.4}$$ \section{Preliminary Results} \ In view of \eqref{e1.2} and \eqref{e1.3}, the operator equation $Lx=\lambda Nx$ is equivalent to the equation $$x''+\lambda [f(x(t),x'(t))+g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{2}(t)))]=\lambda p(t), \label{e2.1}$$ where $\lambda\in (0, 1)$. For convenience of use, we introduce the Continuation Theorem; see \cite{g1}. \begin{lemma} \label{lem2.1} Let $X$ and $Y$ be two Banach spaces. Suppose that $L:D(L)\subset X\to Y$ is a Fredholm operator with index zero and $N: X\to Y$ is $L$-compact on $\overline{\Omega}$, where $\Omega$ is an open bounded subset of $X$. Moreover, assume that the following conditions are satisfied: \begin{enumerate} \item $Lx\neq \lambda Nx$, for all $x\in \partial \Omega\cap D(L) , \lambda\in (0,1)$; \item $Nx \not \in \mathop{\rm Im}L$, for all $x\in \partial \Omega\cap \ker L$; \item The Brouwer degree, $\deg\{QN, \Omega\cap \ker L , 0 \}$, is not equal to zero. \end{enumerate} Then the equation $Lx= Nx$ has at least one $T$-periodic solution in $\overline{\Omega}$. \end{lemma} The following lemma will be useful for proving our main results in Section 3. \begin{lemma} \label{lem2.2} Assume that the following conditions are satisfied: \begin{itemize} \item[(A1)] One of the following conditions holds: \begin{enumerate} \item $(g_{i}(t,u_{1})-g_{i}(t,u_{2}))(u_{1}-u_{2})>0$, for $i=1,2, u_{i}\in \mathbb{R}$, for all $t\in \mathbb{R}$ and $u_{1}\neq u_{2}$, \item $(g_{i}(t,u_{1})-g_{i}(t,u_{2}))(u_{1}-u_{2})<0$, for $i=1,2, u_{i}\in \mathbb{R}$, for all $t\in \mathbb{R}$ and $u_{1}\neq u_{2}$; \end{enumerate} \item[(A2)] There exists a constant $d>0$ such that one of the following conditions holds: \begin{enumerate} \item $x(g_{1}(t,x)+g_{2}(t,x)-p(t))>0$, for all $t\in \mathbb{R}$, $|x|\ge d$, \item $x(g_{1}(t,x)+g_{2}(t,x)-p(t))<0$, for all $t\in \mathbb{R}$, $|x|\ge d$. \end{enumerate} \end{itemize} If $x(t)$ is a $T$-periodic solution of $\eqref{e2.1}$, then $$|x|_\infty\le d+\sqrt{T}|x'|_2. \label{e2.2}$$ \end{lemma} \begin{proof} Let $x(t)$ be a $T$-periodic solution of \eqref{e2.1}. Set $$x(t_{\max})=\max_{t\in \mathbb{R}} x(t), \quad x(t_{\min})=\min_{t\in \mathbb{R}} x(t),$$ where $t_{\max}, t_{\min} \in \mathbb{R}$. Then $$x'(t_{\max})=0, \quad x''(t_{\max})\le 0, \quad\mbox{and} \quad x'(t_{\min})=0, \quad x''(t_{\min})\ge 0. \label{e2.3}$$ In view of $f(x, 0)=0$ and \eqref{e2.1}, Equation \eqref{e2.3} implies \begin{gather} \begin{aligned} &g_{1}(t_{\max},x(t_{\max}-\tau_{1}(t_{\max}))) +g_{2}(t_{\max},x(t_{\max}-\tau_{2}(t_{\max})))-p(t_{\max})\\ &=-\frac{x''(t_{\max})} {\lambda}\ge 0, \end{aligned} \label{e2.4}\\ \begin{aligned} & g_{1}(t_{\min},x(t_{\min}-\tau_{1}(t_{\min}))) +g_{2}(t_{\min},x(t_{\min}-\tau_{2}(t_{\min})))-p(t_{\min})\\ &=-\frac{x''(t_{\min})}{\lambda}\le 0. \end{aligned}\label{e2.5} \end{gather} Since $g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{2}(t)))-p(t)$ is a continuous function on $\mathbb{R}$, it follows from \eqref{e2.4} and \eqref{e2.5} that there exists a constant $t_{1}\in \mathbb{R}$ such that $$g_{1}(t_{1},x(t_{1}-\tau_{1}(t_{1}))) +g_{2}(t_{1},x(t_{1}-\tau_{2}(t_{1})))-p(t_{1})=0. \label{e2.6}$$ Next we show that the following claim is true. \smallskip \noindent\textbf{Claim:} If $x(t)$ is a $T$-periodic solution of \eqref{e2.1}, then there exists a constant $t_{2}\in \mathbb{R}$ such that $$|x(t_{2})|\leq d . \label{e2.7}$$ \begin{proof} Assume, by way of contradiction, that \eqref{e2.7} does not hold. Then $$|x(t)|> d, \quad \mbox{for all } t\in \mathbb{R}, \label{e2.8}$$ which, together with (A2) and \eqref{e2.6}, implies that one of the following relations holds: \begin{gather} x(t_{1}-\tau_{1}(t_{1}))> x(t_{1}-\tau_{2}(t_{1}))> d;\label{e2.9}\\ x(t_{1}-\tau_{2}(t_{1}))> x(t_{1}-\tau_{1}(t_{1}))> d;\label{e2.10}\\ x(t_{1}-\tau_{1}(t_{1}))< x(t_{1}-\tau_{2}(t_{1}))<- d;\label{e2.11}\\ x(t_{1}-\tau_{2}(t_{1}))< x(t_{1}-\tau_{1}(t_{1}))<- d.\label{e2.12} \end{gather} Suppose that \eqref{e2.9} holds, in view of (A1)(1), (A1)(2), (A2)(1) and (A2)(2), we consider following four cases: \noindent Case (i). If (A2)(1) and (A1)(1) hold, according to \eqref{e2.9}, we obtain \begin{align*} 0 & < g_{1}(t_{1},x(t_{1}-\tau_{2}(t_{1}))) +g_{2}(t_{1},x(t_{1}-\tau_{2}(t_{1})))-p(t_{1}) \\ & < g_{1}(t_{1},x(t_{1}-\tau_{1}(t_{1}))) +g_{2}(t_{1},x(t_{1}-\tau_{2}(t_{1})))-p(t_{1}), \end{align*} which contradicts \eqref{e2.6}. This contradiction implies \eqref{e2.7}. \noindent Case (ii). If (A2)(1) and (A1)(2) hold, according to \eqref{e2.9}, we obtain \begin{align*} 0 & < g_{1}(t_{1},x(t_{1}-\tau_{1}(t_{1}))) +g_{2}(t_{1},x(t_{1}-\tau_{1}(t_{1})))-p(t_{1}) \\ & < g_{1}(t_{1},x(t_{1}-\tau_{1}(t_{1}))) +g_{2}(t_{1},x(t_{1}-\tau_{2}(t_{1})))-p(t_{1}), \end{align*} which contradicts \eqref{e2.6}. This contradiction implies \eqref{e2.7} . \noindent Case (iii). If (A2)(2) and (A1)(1) hold, according to \eqref{e2.9}, we obtain \begin{align*} 0 & > g_{1}(t_{1},x(t_{1}-\tau_{1}(t_{1}))) +g_{2}(t_{1},x(t_{1}-\tau_{1}(t_{1})))-p(t_{1}) \\ & > g_{1}(t_{1},x(t_{1}-\tau_{1}(t_{1}))) +g_{2}(t_{1},x(t_{1}-\tau_{2}(t_{1})))-p(t_{1}), \end{align*} which contradicts \eqref{e2.6}. This contradiction implies \eqref{e2.7}. \noindent Case (iv). If (A2)(2) and (A1)(2) hold, according to \eqref{e2.9}, we obtain \begin{align*} 0 & > g_{1}(t_{1},x(t_{1}-\tau_{2}(t_{1}))) +g_{2}(t_{1},x(t_{1}-\tau_{2}(t_{1})))-p(t_{1}) \\ & > g_{1}(t_{1},x(t_{1}-\tau_{1}(t_{1}))) +g_{2}(t_{1},x(t_{1}-\tau_{2}(t_{1})))-p(t_{1}), \end{align*} which contradicts \eqref{e2.6}. This contradiction implies \eqref{e2.7}. Suppose that \eqref{e2.10} (or \eqref{e2.11}, or \eqref{e2.12}) holds, using the methods similarly to those used in Cases (i)--(iv), we can show that \eqref{e2.7} is true. This completes the proof of the above claim. \end{proof} Let $t_{2}=m T+t_0$, where $t_0\in [0,T]$ and $m$ is an integer. Then, using the Schwarz inequality and the relation $$|x(t)|=|x(t_0)+\int^t_{t_0}x'(s)ds|\le d+\int^T_0|x'(s)|ds, t\in [0, T],$$ we obtain $$|x|_\infty=\max_{t\in[0,T]}|x(t)|\le d+\sqrt{T} |x'|_2.$$ This completes the proof. \end{proof} \section{Main Results} \begin{theorem} \label{thm3.1} Suppose that (A1)(1) and (A2)(1) hold, and there exist nonnegative constants $m_{1}$, $m_{2}$ , $m_{3}$ and $m_{4}$ such that $2m_{1}+4m_{3}<\frac{1}{2T^{2}}$, and one of the following conditions holds: \begin{enumerate} \item $f(x,y)\leq 0$ for all $x\in \mathbb{R}$, $y\in \mathbb{R}$, $|g_{2}(t,x)|\leq m_{3}|x|+m_{4}$ for all $t\in \mathbb{R}$, $x\in \mathbb{R}$, and $$g_{1}(t,x)+g_{2}(t,x)-p(t) \leq m_{1}x+m_{2},\quad \forall t\in \mathbb{R},\; x\geq d;$$ \item $f(x, y)\ge 0$ for all $x\in \mathbb{R}, y\in \mathbb{R}$, $|g_{2}(t,x)|\leq m_{3}|x|+m_{4}$ for all $t\in \mathbb{R}$, $x\in \mathbb{R}$, and $$g_{1}(t,x)+g_{2}(t,x)-p(t) \ge m_{1}x-m_{2},\quad \forall t\in \mathbb{R},\; x\leq - d.$$ \end{enumerate} Then \eqref{e1.1} has at least one $T$-periodic solution. \end{theorem} \begin{proof} We shall seek to apply Lemma \ref{lem2.1}. To do this, it suffices to prove that the set of all possible $T$-periodic solutions of \eqref{e2.1} are bounded. Let $x(t)$ be a $T$-periodic solution of \eqref{e2.1}. Integrating \eqref{e2.1} from $0$ to $T$, we have $$\int^T_0 f(x(t), x'(t))dt+\int^T_0[g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{2}(t)))-p(t)] dt=0. \label{e3.1}$$ Set \begin{gather*} [x(t-\tau_{1}(t))<-d]=\{t|t\in [0, T], x(t-\tau_{1}(t))<-d \},\\ [x(t-\tau_{1}(t))\geq -d]=\{t|t\in [0, T], x(t-\tau_{1}(t))\geq -d \},\\ [x(t-\tau_{1}(t))>d]=\{t|t\in [0, T], x(t-\tau_{1}(t))>d \},\\ [x(t-\tau_{1}(t))\leq d]=\{t|t\in [0, T], x(t-\tau_{1}(t))\leq d \}. \end{gather*} Then, in view of (A2)(1), \eqref{e3.1} implies \begin{align} &\int_{[x(t-\tau_{1}(t))<-d]} |g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)|dt \notag \\ &=-\int_{[x(t-\tau_{1}(t))<-d]} [g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)]dt \notag \\ & =\int_{[x(t-\tau_{1}(t))\geq -d]}[g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)]dt \notag\\ &\quad -\int^T_0[g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)]dt \notag\\ & =\int_{[x(t-\tau_{1}(t))\geq -d]}[g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)]dt \label{e3.2}\\ &\quad -\int^T_0[g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{2}(t)))-p(t)]dt \notag\\ &\quad -\int^T_0g_{2}(t,x(t-\tau_{1}(t)))dt +\int^T_0 g_{2}(t,x(t-\tau_{2}(t)))dt \notag\\ &=\int_{[x(t-\tau_{1}(t))\geq -d]}[g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)]dt \notag \\ &\quad -\int^T_0g_{2}(t,x(t-\tau_{1}(t)))dt +\int^T_0g_{2}(t,x(t-\tau_{2}(t)))dt+\int^T_0f(x(t), x'(t))dt, \notag \end{align} and \begin{aligned} &\int_{[x(t-\tau_{1}(t))>d]} |g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)|dt \\ & =\int_{[x(t-\tau_{1}(t))>d]} [g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)]dt \\ & =-\int_{[x(t-\tau_{1}(t))\leq d]}[g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)]dt\\ &\quad +\int^T_0[g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)]dt\\ &=-\int_{[x(t-\tau_{1}(t))\leq d]}[g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)]dt \\ &\quad+\int^T_0[g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{2}(t)))-p(t)]dt\\ &\quad +\int^T_0g_{2}(t,x(t-\tau_{1}(t)))dt -\int^T_0 g_{2}(t,x(t-\tau_{2}(t)))dt \\ &=-\int_{[x(t-\tau_{1}(t))\leq d]}[g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)]dt \\ &\quad +\int^T_0g_{2}(t,x(t-\tau_{1}(t)))dt -\int^T_0g_{2}(t,x(t-\tau_{2}(t)))dt-\int^T_0f(x(t),x'(t))dt. \end{aligned} \label{e3.3} Now suppose that (1) (or (2)) holds. We shall consider two cases as follows. \noindent\textbf{Case 1:} If (1) holds, it follows from \eqref{e2.2} and \eqref{e3.2} that \begin{align} &\int_{[x(t-\tau_{1}(t))<-d]}|g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)|dt \notag\\ &\le \int_{[x(t-\tau_{1}(t))\geq -d]} |g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)| dt \notag \\ &\quad+\int^T_0|g_{2}(t,x(t-\tau_{1}(t)))|dt +\int^T_0|g_{2}(t,x(t-\tau_{2}(t)))|dt \notag\\ &\le \int_{\{t|t\in [0, T], |x(t-\tau_{1}(t))|\le d \}} |g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)| dt \notag\\ &\quad +\int_{[x(t-\tau_{1}(t))>d]} |g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)| dt \label{e3.4}\\ &\quad +\int_{0}^{T}(m_{3}|x(t-\tau_{1}(t))|+m_{4})dt +\int_{0}^{T}(m_{3}|x(t-\tau_{2}(t))| +m_{4})dt \notag\\ &\le T(\max\{|g_{1}(t,x)+g_{2}(t,x)-p(t)|:t\in \mathbb{R}, |x|\le d\}) \notag\\ &\quad +\int_{0}^{T}(m_{1} |x(t-\tau_{1}(t))|+m_{2})dt +2T(m_{3}|x|_{\infty}+m_{4}) \notag\\ &\le T(\max\{|g_{1}(t,x)+g_{2}(t,x)-p(t)|:t\in \mathbb{R}, |x|\le d\} +m_{2}+2m_{4}) \notag\\ &\quad +T(m_{1}+2m_{3})|x|_{\infty} \notag\\ &\le T(\theta_{1}+m_{2}+2m_{4})+T(m_{1}+2m_{3})(\sqrt{T} |x'|_{2}+d), \notag \end{align} where $\theta_{1}=\max\{|g_{1}(t,x)+g_{2}(t,x)-p(t)|:t\in \mathbb{R}, |x|\le d\}$. Then, \eqref{e3.4} implies \begin{aligned} &\int^T_0|g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)|dt\\ &= \int_{[x(t-\tau_{1}(t))<-d]}|g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)|dt \\ &\quad +\int_{[x(t-\tau_{1}(t))\geq -d]} |g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)|dt\\ &\le 2T(\theta_{1}+m_{2}+m_{4})+2T(m_{1}+m_{3})(\sqrt{T} |x'|_{2}+d), \end{aligned} \label{e3.5} and \begin{aligned} \int^T_0|f(x(t), x'(t))|dt &= -\int^T_0f(x(t), x'(t))dt \\ &= \int^T_0[g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{2}(t)))-p(t)] dt \\ &= \int^T_0[g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)] dt\\ &\quad -\int^T_0g_{2}(t,x(t-\tau_{1}(t))) dt +\int^T_0g_{2}(t,x(t-\tau_{2}(t))) dt \\ &\le \int^T_0|g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)| dt\\ &\quad +\int^T_0|g_{2}(t,x(t-\tau_{1}(t))) |dt +\int^T_0|g_{2}(t,x(t-\tau_{2}(t))) |dt \\ &\le 2T(\theta_{1}+m_{2}+2m_{4})+2T(m_{1}+2m_{3})(\sqrt{T} |x'|_{2}+d). \end{aligned}\label{e3.6} \noindent\textbf{Case 2:} If (2) holds, it follows from \eqref{e2.2} and \eqref{e3.3} that \begin{aligned} &\int_{[x(t-\tau_{1}(t))>d]}|g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)|dt \\ &\le \int_{[x(t-\tau_{1}(t))\leq d]} |g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)| dt \\ &\quad +\int^T_0|g_{2}(t,x(t-\tau_{1}(t)))|dt +\int^T_0|g_{2}(t,x(t-\tau_{2}(t)))|dt\\ &\le \int_{\{t|t\in [0, T],\ |x(t-\tau_{1}(t))|\le d \}} |g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)| dt\\ &\quad +\int_{[x(t-\tau_{1}(t))<-d]} |g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)| dt\\ &\quad +\int_{0}^{T}(m_{3}|x(t-\tau_{1}(t))|+m_{4})dt +\int_{0}^{T}(m_{3}|x(t-\tau_{2}(t))| +m_{4})dt\\ &\le T(\max\{|g_{1}(t,x)+g_{2}(t,x)-p(t)|:t\in \mathbb{R}, |x|\le d\})\\ &\quad+\int_{0}^{T}(m_{1} |x(t-\tau_{1}(t))|+m_{2})dt +2T(m_{3}|x|_{\infty}+m_{4}) \\ &\le T(\theta_{1}+m_{2}+2m_{4})+T(m_{1}+2m_{3})(\sqrt{T} |x'|_{2}+d), \end{aligned} \label{e3.7} which implies \begin{aligned} &\int^T_0|g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)|dt \\ &=\int_{[x(t-\tau_{1}(t))>d]}|g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)|dt \\ &\quad +\int_{[x(t-\tau_{1}(t))\leq d]} |g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)|dt\\ &\le 2T(\theta_{1}+m_{2}+m_{4})+2T(m_{1}+m_{3})(\sqrt{T} |x'|_{2}+d), \end{aligned} \label{e3.8} and \begin{aligned} &\int^T_0|f(x(t), x'(t))|dt\\ &= \int^T_0f(x(t), x'(t))dt \\ &= -\int^T_0[g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{2}(t)))-p(t)] dt \\ &\le \int^T_0|g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)| dt \\ &\quad +\int^T_0|g_{2}(t,x(t-\tau_{1}(t))) |dt +\int^T_0|g_{2}(t,x(t-\tau_{2}(t))) |dt \\ &\le 2T(\theta_{1}+m_{2}+2m_{4})+2T(m_{1}+2m_{3})(\sqrt{T}|x'|_{2}+d). \end{aligned}\label{e3.9} Multiplying \eqref{e2.1} by $x(t)$ and then integrating from $0$ to $T$, by \eqref{e2.3}, \eqref{e3.5} , \eqref{e3.6}, \eqref{e3.8} and \eqref{e3.9}, we have \begin{align} &|x'|_2^2 \notag\\ &= \lambda\int^T_0\{f(x(t), x'(t)) +[g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{2}(t))) -p(t)] \}x(t)dt \notag\\ &= \lambda\int^T_0\{f(x(t), x'(t)) +[g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t) \notag\\ &\quad -g_{2}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{2}(t)))] \}x(t)dt \notag\\ &\le \int^T_0|f(x(t), x'(t))||x(t)|dt \notag\\ &\quad +\int^T_0|g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)||x(t)| dt \notag\\ &\quad +\int^T_0|g_{2}(t,x(t-\tau_{1}(t))) ||x(t)|dt +\int^T_0|g_{2}(t,x(t-\tau_{2}(t))) ||x(t)|dt \label{e3.10}\\ &\le |x|_{\infty} \{\int^T_0|f(x(t),x'(t))|dt \notag\\ &\quad +\int^T_0|g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)| dt +2T(m_{3}|x|_{\infty}+m_{4})\} \notag\\ &\le 2T[(2\theta_{1}+2m_{2}+4m_{4})+(2m_{1}+4m_{3})(\sqrt{T} |x'|_{2}+d)](\sqrt{T} |x'|_{2}+d) \notag\\ &=2(2m_{1}+4m_{3})T^{2}|x'|^{2}_{2}+2T[(2\theta_{1}+2m_{2}+4m_{4}) +2(2m_{1}+4m_{3})d]\sqrt{T} |x'|_{2} \notag\\ &\quad+2Td[(2\theta_{1}+2m_{2}+4m_{4})+(2m_{1}+4m_{3})d]. \notag \end{align} Since $0\leq 2m_{1}+4m_{3}<\frac{1}{2T^{2}}$, \eqref{e3.10} implies that there exists a positive constant $D_1$ such that $$|x'|_2 \le D_1 \mbox{ \ and \ } |x|_{\infty}\le \sqrt{T} |x'|_{2}+d\leq D_1. \label{e3.11}$$ In view of \eqref{e3.5}, \eqref{e3.6}, \eqref{e3.8} and \eqref{e3.9}, it follows from \eqref{e2.1} that \begin{aligned} &\int^T_0|x''(t)|dt \\ &\le \int^T_0|f(x(t), x'(t))|dt +\int^T_0|g_{1}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{1}(t)))-p(t)\\ &\quad -g_{2}(t,x(t-\tau_{1}(t)))+g_{2}(t,x(t-\tau_{2}(t)))|dt\\ &\le \int^T_0|f (x(t), x'(t)) |dt +\int^T_0|g_{1}(t,x(t-\tau_{1}(t))) +g_{2}(t,x(t-\tau_{1}(t)))-p(t)|dt\\ &\quad +\int^T_0|g_{2}(t,x(t-\tau_{1}(t)))|dt +\int^T_0|g_{2}(t,x(t-\tau_{2}(t)))|dt\\ &\le 2T[(2\theta_{1}+2m_{2}+4m_{4})+(2m_{1}+4m_{3})(\sqrt{T} |x'|_{2}+d)]\\ &\le 2T[(2\theta_{1}+2m_{2}+4m_{4})+(2m_{1}+4m_{3})(\sqrt{T} D_{1}+d)] := D_{2}. \end{aligned} \label{e3.12} Since $x(0)=x(T)$, it follows that there exists a constant $\zeta\in [0,T]$ such that $x'(\zeta)=0$ and $$|x'(t)|=|x'(\zeta)+\int^t_{\zeta}x''(s)ds| \leq \int^T_0|x''(t)|dt\le D_2, \quad \forall t\in [0, T],$$ which, together with \eqref{e3.11}, implies $$\|x\|_{X}\leq |x|_\infty+|x'|_\infty\max\{M_{1} , d \} that the conditions (1) and (2) in Lemma \ref{lem2.1} hold. Furthermore, we define a continuous function H(x,\mu) by setting$$ H(x,\mu)=-(1-\mu)x-\mu\cdot \frac{1}{T}\int^{T}_{0}[g_{1}(t,x)+g_{2}(t,x)-p(t)]dt; \quad \mu\in [0,1]. $$In view of (A2)(1), we have$$ xH(x,\mu)\not= 0 \quad \mbox{for all } x\in \partial \Omega\cap \ker L. Hence, using the homotopy invariance theorem, we obtain \begin{align*} \deg\{QN, \Omega\cap \ker L , 0 \} &=\deg\{-\frac{1}{T}\int^{T}_{0}[ g_{1}(t,x)+g_{2}(t,x)-p(t)]dt, \Omega\cap \ker L , 0\}\\ &=deg\{-x, \Omega\cap \ker L , 0 \}\neq 0. \end{align*} In view of the discussions above, from Lemma \ref{lem2.1} we complete the proof of Theorem \ref{thm3.1}. \end{proof} A similar argument leads to the following result. \begin{theorem} \label{thm3.2} Suppose that (A1)(2) and (A2)(2) holds, and there exist nonnegative constants m_{1}, m_{2} , m_{3} and m_{4} such that 2m_{1}+4m_{3}<\frac{1}{2T^{2}}, and one of the following two conditions holds: \begin{enumerate} \item f(x, y)\geq 0 for all x\in \mathbb{R}, y\in \mathbb{R}, |g_{2}(t,x)|\leq m_{3}|x|+m_{4} for all t\in \mathbb{R},  x\in \mathbb{R}, and g_{1}(t,x)+g_{2}(t,x)-p(t) \geq - m_{1}x-m_{2}, for all  t\in \mathbb{R}, x\geq d; \item f(x, y)\leq 0 for all x \in \mathbb{R}, y\in \mathbb{R}, |g_{2}(t,x)|\leq m_{3}|x|+m_{4} for all t\in \mathbb{R}, x\in \mathbb{R}, and g_{1}(t,x)+g_{2}(t,x)-p(t) \leq -m_{1}x+m_{2},\quad \text{for all } t\in \mathbb{R},\; x\leq - d.  \end{enumerate} Then \eqref{e1.1} has at least one $T$-periodic solution. \end{theorem} \section{Examples and Remarks} \begin{example} \label{exa4.1} \rm Let $g(t,x)=x^{13}+\frac{1}{72\pi^{2}} x$ for $t\in \mathbb{R}$, $x\leq 0$, and $g(t,x)=\frac{1}{36\pi^{2}} x$ for $t\in \mathbb{R}$, $x> 0$. Then the Rayleigh equation $$x''- (x')^{4}+g(t,x(t-\sin(t))) =e^{\cos^2 t}, \label{e4.1}$$ has at least one $2\pi$-periodic solution. \end{example} \begin{proof} Let $g_{2}(t,x)=\frac{1}{72\pi^{2}} x$ for $t\in \mathbb{R}$, $x\in \mathbb{R}$, $g_{1}(t,x)=x^{13}$ for $t\in \mathbb{R}$, $x\leq 0$, and $g_{1}(t,x)=\frac{1}{72\pi^{2}} x$ for $t\in \mathbb{R}$, $x> 0$. Then \eqref{e4.1} is equivalent to the equation $$x''- (x')^{4}+g_{1}(t,x(t-\sin(t)))+g_{2}(t,x(t-\sin(t))) =e^{\cos^2 t}. \label{e4.2}$$ From \eqref{e4.2}, we have $f (x, y)=- y^{4}\leq 0$, $\tau_{1}(t)=\tau_{2}(t) =\sin t$, $p(t)=e^{\cos^2 t}$ and $g_{1}(t,x)+g_{2}(t,x)-p(t) =\frac{1}{36\pi^{2}} x -e^{\cos^2 t} \leq \frac{1}{36\pi^{2}} x+e$, for all $t\in \mathbb{R}$, $x>0$. It is straightforward to check that all the conditions needed in Theorem \ref{thm3.1} are satisfied. Therefore, \eqref{e4.2} has at least one $2\pi$-periodic solution. This implies that \eqref{e4.1} has at least one $2\pi$-periodic solution. \end{proof} \begin{remark} \label{rmk4.1} \rm Equation \eqref{e4.1} is a very simple version of Rayleigh equation. Obviously, the conditions (H0)--(H3) are not satisfied. Therefore, the results in \cite{g2,s1,s2,s3,x1} and the references cited therein cannot be applied to \eqref{e4.1}. This implies that the results of this paper are essentially new. \end{remark} \begin{example} \label{exa4.2} \rm Let $g_{1}(t,x)=-\frac{1}{72\pi^{2}} x$ for $t\in \mathbb{R}$, $x\in \mathbb{R}$, $g_{2}(t,x)=-x^{13}$ for $t\in \mathbb{R}$, $x\leq 0$, and $g_{2}(t,x)=-\frac{1}{72\pi^{2}} x$ for $t\in \mathbb{R}$, $x> 0$. Then, the Rayleigh equation $$x''+x^{4}(x')^{6}+g_{1}(t,x(t-\cos(t)))+g_{2}(t,x(t-\sin(t))) =\frac{1}{4} \cos^2 t. \label{e4.3}$$ has at least one $2\pi$-periodic solution. \end{example} \begin{proof} From \eqref{e4.3}, we can obtain $f (x, y)=x^{4}y^{6}$, $\tau_{1}(t)=\cos(t)$, $\tau_{2}(t) =\sin (t)$, $p(t)=\frac{1}{4}cos^2 t$ and $g_{1}(t,x)+g_{2}(t,x)-p(t) =-\frac{1}{36\pi^{2}} x -\frac{1}{4}cos^2 t\geq -\frac{1}{36\pi^{2}} x-\frac{1}{4}$, for $t\in \mathbb{R}$, $x>0$. It is obvious that all the conditions needed in Theorem \ref{thm3.2} are satisfied. Hence, by Theorem \ref{thm3.2}, equation \eqref{e4.3} has at least one $2\pi$-periodic solution. \end{proof} \begin{remark} \label{rmk4.2} \rm In view of \eqref{e4.3}, it is clear that (H0)--(H3), do not hold for \eqref{e4.3}, and so the results obtained in \cite{g2,s1,s2,s3,x1} and the references cited therein cannot be applied to \eqref{e4.3}. \end{remark} \begin{remark} \label{rmk4.3} \rm Using the methods similarly to those used for \eqref{e1.1}, we can study the Rayleigh equation with multiple deviating arguments $$x''+f(x(t), x'(t)) +\sum_{i=1}^{n}g_{i}(t,x(t-\tau_{i}(t))) =p(t), \label{e4.4}$$ where $\tau_{i}(i=1,2,\dots, n)$, $p:\mathbb{R}\to \mathbb{R}$ and $f$, $g_{i}:\mathbb{R}\times \mathbb{R}\to \mathbb{R}$ are continuous functions, $f(x, 0)=0$, $\tau_{i}$ and $p$ are $T$-periodic, $g_{i}$ are $T$-periodic in the first argument, and $T>0$ ($i=1,2,\dots, n$). One may also establish the results similarly to those in Theorems \ref{thm3.1} and \ref{thm3.2} under some minor additional assumptions on $g_{i}(t,x)$ ($i=1,2,\dots, n$). \end{remark} \begin{thebibliography}{00} \bibitem{g1} R. E. Gaines, J. Mawhin; \emph{Coincide degree and nonlinear differential equations}, Lecture Notes in Math.,N0. 568, Spring-Verlag, 1977. \bibitem{g2} Genqiang Wang; \emph{A priori bounds for periodic solutions of a delay Rayleigh equation}, Appl. Math. Lett. 12(1999), 41-44. \bibitem{x1} Xiankai Huang and Z. G. Xiang; \emph{On existence of $2\pi$-periodic solutions for delay Duffing equation $x''+g(t,x(t-\tau(t)))=p(t)$}, Chinese Science Bulletin, 39(1994), 201-203. \bibitem{s1} Shiping Lu, Weigao Ge; \emph{Some new results on the existence of periodic solutions to a kind of Rayleigh equation with a deviating argument}, Nonlinear Analysis, 56(2004), 501-514. \bibitem{s2} Shiping Lu, Weigao Ge, Zuxiou Zheng; \emph{Periodic solutions for neutral differential equation with deviating arguments}, Applied Mathematics and Computation. 152(2004), 17-27. \bibitem{s3} Shiping Lu, Weigao Ge, Zuxiou Zheng; \emph{A new result on the existence of periodic solutions for a kind of Rayleigh equation with a deviating argument} (in Chinese), Acta Mathematica Sinica, 47(2004), 299-304 . \end{thebibliography} \end{document}