\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2012 (2012), No. 77, pp. 1--8.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2012 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2012/77\hfil Existence of periodic solutions] {Existence of periodic solutions for Rayleigh equations with state-dependent delay} \author[J. O. Alzabut, C. Tun\c{c} \hfil EJDE-2012/77\hfilneg] {Jehad O. Alzabut, Cemil Tun\c{c}} % in alphabetical order \address{Jehad O. Alzabut \newline Department of Mathematics and Physical Sciences, Prince Sultan University \\ P.O. Box 66833, Riyadh 11586, Saudi Arabia} \email{jalzabut@psu.edu.sa} \address{Cemil Tun\c{c} \newline Department of Mathematics, Faculty of Arts and Science, Y\"uz\"unc\"u Yil University\\ 65080 Van, Turkey} \email{cemtunc@yahoo.com} \thanks{Submitted December 22, 2011. Published May 14, 2012.} \subjclass[2000]{34K13} \keywords{Rayleigh Equation; continuation theorem; periodic solutions} \begin{abstract} We establish sufficient conditions for the existence of periodic solutions for a Rayleigh-type equation with state-dependent delay. Our approach is based on the continuation theorem in degree theory, and some analysis techniques. An example illustrates that our approach to this problem is new. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} Lord Rayleigh (John William Strutt: 1842--1919) \cite{s1} introduced the equation $$\label{eq10} x''(t)+f(x'(t))+ax(t)=0$$ to model the oscillations of a clarinet reed. This equation is used for studying problems arising in acoustics, and is referred in the literature as Rayleigh equation. Later on, the Rayleigh equation of the form $$\label{eq11} x''(t)+f(x'(t))+g(t,x(t))=0$$ was studied in the monographs \cite{d1,g1,g4}. In many circumstances, however, it is known that the forces intervening in the system depend depend not only at the current time considered, but also on previous times. Thus, the forced Rayleigh equation with delay $$\label{eq12} x''(t)+f(t,x'(t))+g(t,x(t-\tau))=p(t)$$ has been taken into consideration, see \cite{w1,w2,w3}. Recently, it has been recognized that \eqref{eq12} has widespread applications in many applied sciences such as physics, mechanics and engineering techniques fields. In such applications, it is crucial to know the periodic behavior of solutions for Rayleigh equation. This justifies the intensive interest among researchers in investigating the existence of periodic solutions for this equation in the last decade. Publications \cite{h2,g3,l1,l2,l3,l5,l6,l7,p1,w4,z1,z2,z3,z4} are devoted to various generalizations of equation \eqref{eq12}. Nevertheless, one can realize that all the results obtained in the above mentioned papers have been proved under the assumptions that $\tau$ is a constant, $g$ is bounded and $\int_0^{2\pi}p(t)\,{\rm d} t=0$. However, it is known that the delay may not be only related to time $t$ but also it relates to the current state $x$. Thus, it is worth while to consider a type of Rayleigh equation with state-dependent delay. In this paper, particularly, we consider Rayleigh equation of the form $$\label{eq1} x''(t)+f(t,x(t))+g(x(t-\tau(t,x(t))))=p(t).$$ We shall utilize the continuation theorem of degree theory to obtain sufficient conditions for the existence of periodic solutions of \eqref{eq1}. The main result is proved by bypassing the boundedness of $g$ and the integral condition on $p$. To the best of authors' observations, there exists no paper establishing sufficient conditions for the existence of periodic solutions for \eqref{eq1}. Thus, our result presents a new approach. \section{Preliminaries} Let $$C_{2\pi}=\big\{x:x\in C(\mathbb{R},\mathbb{R}), x(t+2\pi)\equiv x(t),\; \forall\;t \in \mathbb{R}\big\}$$ with the norm $\| x\|_0=\max_{t \in [0,2\pi]} | x(t) |$, for $x \in C_{2\pi}$ and $$C_{2\pi}^{1}=\big\{x:x\in C^{1}(\mathbb{R},\mathbb{R}), x(t+2\pi)\equiv x(t),\; \forall\;t \in \mathbb{R}\big\}$$ with the norm $\| x \|_1=\max_{t \in [0,2\pi]} \{\| x(t) \|_0,\| x'(t)\|_0\}$, for $x \in C_{2\pi}^{1}$. We shall consider \eqref{eq1} under the assumptions that $f \in C_{2\pi}(\mathbb{R}^2,\mathbb{R})$ with $f(\cdot,0)=0$, $g \in C_{2\pi}(\mathbb{R},\mathbb{R})$, $\tau \in C_{2\pi}(\mathbb{R}^2, \mathbb{R}^{+})$ and $p \in C_{2\pi}(\mathbb{R}, \mathbb{R})$. Let $0 \leq \tau(t) \in C_{2\pi}$, then there must exist two integers $k \ge 0$ and $m \ge 1$ such that $$\label{tau} \tau(t) \in [2\pi k,2\pi (k+m)],\quad \tau(t) \notin (0,2\pi k)\cup (2\pi (k+m),\infty).$$ Denote $\Delta_{i}=\big\{t:t \in [0,2\pi],\;\tau(t) \in [2\pi (k+i),2\pi (k+i+1)]\big\}$, $i=0,1,2,\dots,m-1$, \begin{displaymath} \tau_0(t)= \begin{cases} \tau(t), & t \in \Delta_0,\\ 2\pi (k+1), & t \in [0,2\pi]\backslash \Delta_0, \end{cases} \end{displaymath} and \begin{displaymath} \tau_j(t)= \begin{cases} \tau(t), & t \in \Delta_j,\\ 2\pi (k+j), & t \in [0,2\pi]\backslash \Delta_j. \end{cases} \end{displaymath} Then, it is clear that $\cup_{i=0}^{m-1} \Delta_{i}=[0,2\pi]$; $2\pi (k+1)-\tau_0(t) \in [0,2\pi]$ and $\tau_j(t)-2\pi (k+j) \in [0,2\pi]$ for all $t \in [0,2\pi]$, $j=1,2,\dots,m-1$. Let $\delta_0=\sup_{t \in [0,2\pi]}[2\pi(k+1)-\tau_0(t)]$, $\delta_j=\sup_{t \in [0,2\pi]}[\tau_j(t)-2\pi (k+j)]$, then we have $\delta_0, \delta_{m-1}\in [0,2\pi]$, $\delta_j=2\pi,\;j=1,2,\dots,m-2$. The following lemma plays a key role in proving the main result. \begin{lemma}[\cite{d2}] \label{lem2} Let $\tau(t,x(t)) \in C_{2\pi}$ satisfying \eqref{tau} and $x \in C_{2\pi}^{1}$, then $$\int _0^{2 \pi} | x(t-\tau(t,x(t)))-x(t) |^2\,{\rm d} t \leq \Big(\beta_0+\beta_{m-1}+\sum_{j=1}^{m-2}\beta_j \Big) \int_0^{2 \pi}| x'(t)|^2\,{\rm d} t, \;20, d>0 and L \ge 0 such that \begin{itemize} \item[(C1)] | f(t,x)| \leq K for all (t,x)\in \mathbb{R}^2; \item[(C2)] xg(x)<0 and | g(x) | \leq \| p \|_0 implies | x | \leq d; \item[(C3)] | g(x_1)-g(x_2) | \leq L| x_1-x_2|, for all x_1,x_2 \in \mathbb{R}. \end{itemize} If $$\label{condition} 2L \Big(\beta_0+\beta_{m-1}+\sum_{j=1}^{m-2}\beta_j \Big)^{1/2}<1.$$ Then \eqref{eq1} has at least one 2\pi-periodic solution. \end{theorem} \begin{proof} Consider the auxiliary equation $$\label{main313} x''(t)+\lambda f(t,x(t))+\lambda g(x(t-\tau(t,x(t))))=\lambda p(t),\quad \lambda \in (0,1).$$ To complete the proof of this theorem, one can see that it suffices to show that all possible 2 \pi-periodic solutions of \eqref{main313} are bounded. In other words, we shall prove that there exist positive constants M_2 and M_4 independent of \lambda and x such that if x(t) is a 2\pi-periodic solution of equation \eqref{main313} then  \| x \|_0 d\},\quad E_2=\{t \in [0,2\pi]:\;| x(t) | \leq d\}.$$ Multiplying both sides of \eqref{main313} by $x(t)$ and integrating over $[0,2\pi]$, we have \begin{align*} -\int_0^{2\pi}\Big( x'(t) \Big)^2\,{\rm d}t &= -\lambda \int_0^{2\pi}g\Big(x(t-\tau(t,x(t)))\Big)x(t)\,{\rm d}t -\lambda \int_0^{2\pi}f(t,x(t))x(t)\,{\rm d}t\\ &\quad +\lambda \int_0^{2\pi}p(t)x(t)\,{\rm d}t \end{align*} or \begin{align*} \int_0^{2\pi}| x'(t) |^2\,{\rm d}t &= \lambda \int_0^{2\pi}g\Big(x(t-\tau(t,x(t)))\Big)x(t)\,{\rm d}t +\lambda \int_0^{2\pi}f(t,x(t))x(t)\,{\rm d}t\\ &\quad -\lambda \int_0^{2\pi}p(t)x(t)\,{\rm d}t. \end{align*} It follows that \begin{align*} &\int_0^{2\pi}| x'(t) |^2\,{\rm d}t\\ &= \lambda \int_0^{2\pi}\Big[g(x(t-\tau(t,x(t))))-g(x(t))\Big]x(t)\,{\rm d}t +\lambda \int_0^{2\pi}g(x(t))x(t)\,{\rm d}t\\ &\quad +\lambda \int_0^{2\pi}f(t,x(t))x(t)\,{\rm d}t-\lambda \int_0^{2\pi}p(t)x(t)\,{\rm d}t \end{align*} or \begin{align*} &\int_0^{2\pi}| x'(t) |^2\,{\rm d}t\\ &= \lambda \int_0^{2\pi}\Big[g(x(t-\tau(t,x(t))))-g(x(t))\Big]x(t)\,{\rm d}t +\lambda \int_{E_1}g(x(t))x(t)\,{\rm d}t\\ &\quad +\lambda \int_{E_2}g(x(t))x(t)\,{\rm d}t +\lambda \int_0^{2\pi}f(t,x(t))x(t)\,{\rm d}t-\lambda \int_0^{2\pi}p(t)x(t)\,{\rm d}t. \end{align*} This implies \begin{align*} \int_0^{2\pi}| x'(t) |^2\,{\rm d}t &\leq \int_0^{2\pi}\Big | g(x(t-\tau(t,x(t))))-g(x(t))\Big | | x(t) | \,{\rm d}t +g_{d}\int_0^{2\pi}| x(t) | \,{\rm d}t \\ &\quad + K \int_0^{2\pi}| x(t) | \,{\rm d}t+\int_0^{2\pi}| p(t) | | x(t)| \,{\rm d}t, \end{align*} where $g_{d}=\max_{t \in E_2} | g(x(t)) |$. Furthermore, \begin{align*} &\int_0^{2\pi}| x'(t) |^2\,{\rm d}t \\ &\leq L \int_0^{2\pi} \Big | x(t-\tau(t,x(t)))-x(t) \Big| | x(t) | \,{\rm d}t+g_{d} (2\pi)^{1/2}\| x \|_2 + K\| x \|_2+\| p\|_2 \| x \|_2, \end{align*} where $\| x \|_2=\Big( \int_0^{2 \pi} | x(s) |^2 \,{\rm d}s \Big)^{1/2}$. It follows that \begin{align*} \int_0^{2\pi}| x'(t) |^2\,{\rm d}t &\leq L \left(\int_0^{2\pi} \Big| x(t-\tau(t,x(t)))-x(t) \Big|^2 \,{\rm d}t \right)^{1/2}\| x \|_2 +g_{d} (2\pi)^{1/2}\| x \|_2\\ &\quad + K\| x \|_2+\| p \|_2 \| x \|_2. \end{align*} By the consequence of Lemma \ref{lem2}, we obtain \label{es1} \begin{aligned} \int_0^{2\pi}| x'(t) |^2\,{\rm d}t &\leq L \Big(\beta_0+\beta_{m-1}+\sum_{j=1}^{m-2}\beta_j \Big)^{1/2} \Big( \int_0^{2\pi}| x'(t) |^2 \,{\rm d}t \Big)^{1/2} \| x \|_2 \\ &\quad +g_{d} (2\pi)^{1/2}\| x \|_2 + K\| x \|_2+\| p \|_2 \| x \|_2, \end{aligned} Denote $u(t)=x(t)-x(t^{*})$ where $t^{*}$ is defined as in \eqref{main 9}. Then we have $$| x(t) | \leq | x(t^{*}) | + | x(t)-x(t^{*}) | \leq d +| u(t) |.$$ Using the Minkowski inequality \cite{h1}, we obtain $$\label{ewq} \| x \|_2=\Big( \int_0^{2\pi}| x(t) |^2 \,{\rm d}t \Big)^{1/2} \leq (2\pi)^{1/2}d+ \Big( \int_0^{2\pi}| u(t) |^2 \,{\rm d}t \Big)^{1/2}.$$ However, since $u(t^{*})=0,\; u(t+2\pi)=u(t)$ and $u'(t)=x'(t)$ then by the consequence of Lemma \ref{lem3}, we have $$\Big( \int_0^{2\pi}| u(t) |^2 \,{\rm d}t \Big)^{1/2} \leq 2\Big( \int_0^{2\pi}| u'(t) |^2 \,{\rm d}t \Big)^{1/2} = 2 \Big( \int_0^{2\pi}| x'(t) |^2 \,{\rm d}t \Big)^{1/2}.$$ Substituting back in \eqref{ewq}, we obtain $$\label{ews} \Big( \int_0^{2\pi}| x(t) |^2 \,{\rm d}t \Big)^{1/2} \leq (2\pi)^{1/2}d+ 2 \Big( \int_0^{2\pi}| x'(t) |^2 \,{\rm d}t \Big)^{1/2}.$$ It follows from \eqref{es1} and \eqref{ews} that \begin{align*} \int_0^{2\pi}| x'(t) |^2\,{\rm d}t &\leq L \Big( \beta_0+\beta_{m-1}+\sum_{j=1}^{m-2}\beta_j \Big)^{1/2} \Big( \int_0^{2\pi}| x'(t) |^2 \,{\rm d}t \Big)^{1/2}\\ &\quad\times \Big( (2\pi)^{1/2}d+2\Big( \int_0^{2\pi} | x'(t) |^2 \,{\rm d}t \Big)^{1/2}\Big)\\ &\quad +\left(g_{d}(2\pi)^{1/2}+K+\| p \|_2\right)\Big( (2\pi)^{1/2}d+2\Big( \int_0^{2\pi}| x'(t) |^2 \,{\rm d}t \Big)^{1/2}\Big) \\ &= 2L\Big( \beta_0+\beta_{m-1}+\sum_{j=1}^{m-2}\beta_j \Big)^{1/2} \int_0^{2\pi}| x'(t) |^2 \,{\rm d}t\\ &\quad + (2\pi)^{1/2}d L \Big( \beta_0+\beta_{m-1} +\sum_{j=1}^{m-2}\beta_j \Big)^{1/2}\Big( \int_0^{2\pi}| x'(t) |^2 \,{\rm d}t \Big)^{1/2}\\ &\quad +2 \left(g_{d}(2\pi)^{1/2}+K+\| p \|_2\right) \Big( \int_0^{2\pi}| x'(t) |^2 \,{\rm d}t \Big)^{1/2}\\ &\quad +(2\pi)^{1/2}d \left(g_{d}(2\pi)^{1/2}+K+\| p \|_2\right). \end{align*} By \eqref{condition}, we deduce that there exists a constant $M_1>0$ such that $$\int_0^{2\pi}| x'(t) |^2\,{\rm d}t \leq M_1.$$ From \eqref{main 9}, we end up with $$\| x \|_0 \leq d+(2\pi)^{1/2}M_1^{1/2}:=M_2.$$ In view of equation \eqref{eq1}, one can obtain $$\label{son} \| x''(t) \| \leq g_{M_2}+K+\| p \|_0:=M_{3},$$ where $g_{M_2}=\max_{| x | \leq M_2}| g(x) |$. However, since $x(0)=x(2\pi)$ then there exists a constant $\eta \in [0,2\pi]$ such that $x'(\eta)=0$. Therefore, by \eqref{son} we have $$\| x' \|_0 \leq | x'(\eta)| + \int_0^{2\pi} | x''(s) | \,{\rm d}s \leq 2\pi M_{3}:=M_4.$$ Clearly, $M_2$ and $M_4$ are independent of $\lambda$ and $x$. Take \$\Omega=\{x: x \in X, \;\| x \|_0