\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 26, pp. 1--5.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/26\hfil Existence of periodic solutions] {Existence of periodic solutions for second-order neutral differential equations} \author[Y. Li\hfil EJDE-2005/26\hfilneg] {Yongjin Li} \address{Yongjin Li \hfill\break Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China} \email{stslyj@zsu.edu.cn} \date{} \thanks{Submitted January 6, 2005. Published March 6, 2005.} \thanks{Supported by grant 10471155 from NNSF of China, by grant 031608 from NSF of Guangdong, \hfill\break\indent and by the Foundation of Sun Yat-sen University Advanced Research Centre.} \subjclass[2000]{34K13, 34K40, 65K10} \keywords{Neutral differential equations; periodic solution; variational method; \hfill\break\indent critical point} \begin{abstract} By means of variational structure and critical point theory, we study the existence of periodic solutions for a second-order neutral differential equation \begin{gather*} (p(t) x' (t - \tau ))' + f(t, x(t), x(t - \tau ), x(t - 2\tau) ) = g(t),\\ x(0) = x(2k\tau), x'(0) = x'(2k\tau). \end{gather*} where $k$ is a given positive integer and $\tau$ is a positive number. \end{abstract} \maketitle \newtheorem{theorem}{Theorem}[section] \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} \section{Results} In this paper we study the existence of periodic solutions of the second order problem $$\begin{gathered} (p(t) x'(t - \tau ))' + f(t, x(t), x(t - \tau ), x(t - 2\tau) ) = g(t),\\ x(0) = x(2k\tau), x'(0) = x'(2k\tau). \end{gathered} \label{e1.1}$$ where $f \in C(\mathbb{R}^4,\mathbb{R}), p, g \in C(\mathbb{R}, \mathbb{R})$, $k$ is a given positive integer and $\tau$ is a positive number. The existence of periodic solutions to \eqref{e1.1} will be studied under the hypotheses: \begin{itemize} \item[(H1)] $f\in C(\mathbb{R}^4, \mathbb{R} )$ \item[(H2)]There exists a continuously differentiable functional $F(t,u, v)$ in $C^1(\mathbb{R}^3, \mathbb{R})$ with $$F'_u (t, x(t - \tau), x(t - 2 \tau)) + F'_v (t, x(t), x(t - \tau)) =f(t, x(t), x(t-\tau), x(t- 2\tau))$$ \item[(H3)] $F(t, u, v)$ is $\tau$-periodic in $t$ \item[(H4)] $p(t)$ is $\tau$-periodic and $0 < m < p(t)$ \item[(H5)] $g(t)$ is $\tau$-periodic and $\overline{g} = \frac {1} {2k\tau} \int^{2k \tau}_{0}|g(t)|^2 dt < m/2$. \end{itemize} In recent years, by using the continuation theorem of coincidence degree theory, the existence of periodic solutions to ordinary equation have been extensively studied. In articles \cite {Belley, Gen, Layton, Lu, Ma}, the following second-order scalar differential equations have been studied: \begin{gather*} x''(t) + ax'(t) + bx(t) + g(x(t - 1 )) = p(t),\\ x''(t) + m^2x(t) + g(x(t - \tau)) = p(t),\\ x''(t) + f(t, x(t), x(t - \tau_0(t))x'(t) + \beta(t) g(x(t - \tau_1(t))) = p(t),\\ x''(t) + c x'(t) + g(t - \tau), x(t - \tau), x'(t -\tau)) = p(t). \end{gather*} However, the study of corresponding problem for second-order neutral differential system with variational structure and critical point theory, to the best of our knowledge, appeared rarely; see \cite {Guo}. In this paper, we study the existence of periodic solutions to \eqref{e1.1} by means of variational technique and critical point theory., For the reader's convenience, we recall some basic definitions. Let $E$ be a real Banach space. A mapping $I$ from $E$ to $\mathbb{R}$ will be called a functional. A critical point of $I$ is a point where $I'(x_0) = \theta$ and a critical value of $I$ is a number $c$ such that $I(x_0) = c$. In applications to differential equations, critical points correspond to weak solution of equations. Indeed this fact makes critical point theory an important existence tool in studying differential equations. A functional $I$ is weakly lower semi-continuous at $x \in C$ if $$x_n \rightharpoonup x \Rightarrow \lim_{n \to \infty} \inf I(x_n) \geq I(x).$$ A functional $I$ is coercive on $C$ means that $$I(x) \to + \infty \quad\mbox{as}\quad \|x\| \to \infty.$$ We will make use of a theorem in \cite {Mawhin} to obtain the critical point of $I$. This theorem is crucial for arriving at our results. \begin{theorem}[\cite {Mawhin}] \label{thm1.1} Let $E$ be a reflexive Banach space, $C$ be weakly closed subset of $E$, and $I: C \to \mathbb{R}$ be weakly lower semi-continuous and coercive. Then $I$ has a minimum on $C$. \end{theorem} The main result of this paper is as follows. \begin{theorem} \label{thm1.2} Under assumptions (H1)-(H5), problem \eqref{e1.1} has at least one $2k\tau$-periodic solution. \end{theorem} \begin{proof} Let $H_0^1(0, 2k\tau)=\{ x(t) \in L^2[0, 2k\tau]: x(0) = x(2k \tau), x'(0) = x'(2k \tau)\}$ denote the Hilbert space with norm and inner product $$\|x\|=\Big(\int^{2k \tau}_{0}|x'(t)|^2 dt\Big)^{1/2}, \quad (x,y)=\int^{2k \tau}_{0} x'(t)y'(t)dt .$$ Since each $x\in E$ can be extended periodically to the whole line, we may do not distinguish $x$ and its extension. A variational method is used for the following functional defined on $E$, $$I(x) = \int^{2k \tau}_{0} [\frac{p(t)}{2}|x'(t)|^2 - F(t, x(t), x(t - \tau)) + g(t)x(t)]dt\,.$$ For $x, y \in E$ and $\alpha \in \mathbb{R}$, we denote by $\varphi (\alpha )$ the function $I(x + \alpha y)$; i.e., \begin{align*} \varphi (\alpha) =& \int^{2k \tau}_{0} \big[\frac{p(t)}{2}(| x'(t) + \alpha y'(t)|^2 )\\ &- F(t, x(t) + \alpha y(t), x(t - \tau) + \alpha y(t -\tau)) + g(t)[x(t) + \alpha y(t)\big] dt\,. \end{align*} Thus \begin{align*} &\varphi '(\alpha) \\ =& \int^{2k \tau}_{0} \big\{ p(t) [x'(t) y'(t) + \alpha y'(t)^2] - [F'_u (t, x(t) + \alpha y(t), x(t - \tau) + \alpha y(t - \tau))y(t)\\ &+ F'_v (t, x(t) + \alpha y(t), x(t - \tau) + \alpha y(t - \tau))y(t - \tau)] + g(t) y(t) \big\}dt \end{align*} So that \begin{align*} \varphi '(0) = & \int^{2k \tau}_{0} \{p(t) x'(t) y'(t) - [F'_u (t, x(t), x(t - \tau))y(t)\\ &+ F'_v (t, x(t), x(t - \tau))y(t - \tau)]\}dt + \int^{2k \tau}_{0} g(t) y(t)dt\\ = & \int^{2k \tau}_{0} p(t) x'(t) d y(t) - \int^{2k \tau}_{0} [F'_u (t, x(t), x(t - \tau))y(t)\\ & + F'_v (t, x(t), x(t - \tau))y(t - \tau)]dt + \int^{2k \tau}_{0} g(t) y(t)dt\\ = & p(t) x'(t) y(t)|^{2k\tau}_0 - \int^{2k \tau}_{0} [p(t) x'(t)]' y(t)dt\\ &- \int^{2k \tau}_{0} [F'_u (t, x(t), x(t - \tau))y(t) \\ & + F'_v (t, x(t), x(t - \tau))y(t - \tau)]dt + \int^{2k \tau}_{0} g(t) y(t)dt\\ =& - \int^{2k \tau}_{0} [p(t) x'(t)]' y(t)dt - \int^{2k \tau}_{0} F'_u (t, x(t), x(t - \tau))y(t)dt\\ & - \int^{(2k - 1)\tau}_{-\tau} F'_v (t, x(t), x(t - \tau))y(t - \tau)dt + \int^{2k \tau}_{0} g(t) y(t)dt\\ = & - \int^{2k\tau}_{0} [p(t) x'(t)]' y(t)dt - \int^{2k\tau}_{0} F'_u (t, x(t), x(t - \tau))y(t)dt\\ & - \int^{2k \tau}_{0} F'_v (t + \tau, x(t + \tau), x(t))y(t)dt + \int^{2k \tau}_{0} g(t) y(t)dt\\ = & - \int^{2k\tau}_{0} \{ [ (p(t) x'(t))' + F'_u (t, x(t), x(t - \tau))\\ &+ F'_v (t, x(t + \tau), x(t)) - g(t)]y(t)\} dt \end{align*} Therefore, the Euler equation corresponding to the functional $I(x)$ is $$(p(t) x'(t))' + F'_u (t, x(t), x(t - \tau)) + F'_v (t, x(t + \tau), x(t)) - g(t) = 0 \label{e1.2}$$ It is easy to see that this equation is is equivalent to \eqref{e1.1}, and that any critical point $x$ of the functional $I$ is a $2k \tau$-periodic solution of \eqref{e1.1}. Since $F(t, u, v) \in C^1(\mathbb{R}^3, \mathbb{R})$, we have $\int^{2k \tau}_{0}F(t,x(t), x(t - \tau)) dt \leq c$; thus \begin{align*} I(x) & = \int^{2k \tau}_{0} [\frac{p(t)}{2}|x'(t)|^2 - F(t, x(t), x(t - \tau)) + g(t)x(t)]dt\\ & = \int^{2k \tau}_{0} \frac{p(t)}{2}|x'(t)|^2 dt - \int^{2k \tau}_{0}F(t, x(t), x(t - \tau)) dt + \int^{2k \tau}_{0} g(t)x(t) dt\\ & \geq \int^{2k \tau}_{0} \frac{p(t)}{2}|x'(t)|^2 dt - \int^{2k \tau}_{0}F(t, x(t), x(t - \tau)) dt - \int^{2k \tau}_{0} |g(t)x(t)| dt\\ & \geq \frac{m}{2} \|x\|^2 - c - ( \int^{2k \tau}_{0}|g(t)|^2 dt)( \int^{2k \tau}_{0}|x(t)|^2 dt)\\ & = \frac{m}{2} \|x\|^2 - c - (2k \tau) \overline {g} ( \int^{2k \tau}_{0}|x(t)|^2 dt)\\ & \geq \frac{m}{2} \|x\|^2 - c - \overline {g} ( \int^{2k \tau}_{0}|x'(t)|^2 dt)\\ & \geq \frac{m - 2\overline{g}}{2} \|x\|^2 - c. \end{align*} It is easy to see the functional $I$ is coercive. If $x_n$ weakly converges to $x$, then by the compact embedding of $H_0^1(0,2k\tau)$ into $C([0, 2k\tau])$, we know the convergence is uniform in $[0, 2k\tau]$. From the trivial inequality $$0 \leq \int^{2k \tau}_{0}p(t)[x'_n(t) - x'(t)]^2dt,$$ we have $$\int^{2k \tau}_{0} p(t) x'^2_n(t) dt \geq 2 \int^{2k \tau}_{0} p(t) x'_ n(t) x' (t) dt - \int^{2k \tau}_{0} p(t) x'^2(t) dt$$ Thus \begin{align*} I(x_n) =& \int^{2k \tau}_{0} [\frac{p(t)}{2}|x'_n(t)|^2 - F(t, x_n(t), x_n(t - \tau)) + g(t)x_n(t)]dt\\ \geq& \int^{2k \tau}_{0} p(t) x'_ n(t) x' (t) dt - \frac{1}{2} \int^{2k \tau}_{0} p(t) x'^2(t) dt\\ & - \int^{2k \tau}_{0} F(t,x_n(t), x_n(t - \tau))dt + \int^{2k \tau}_{0} g(t)x_n(t)dt\,, \end{align*} and hence $$\lim_{n \to \infty} \inf I(x_n) \geq I(x)\,.$$ This implies that $I$ is weakly lower semi-continuous on $H^1_0(0,2k\tau)$, and the existence of a minimum for $I$ follows from Theorem \ref{thm1.1}. Thus \eqref{e1.1} has at least one periodic solution. \end{proof} \subsection*{Example} Let \begin{align*} &f(t, x(t), x(t - \tau), x(t - 2\tau))\\ &= -4( 8 + \sin^2 \frac {2\pi t}{\tau})\big\{[ \frac {1}{1 + x^2(t - \tau)} + \frac {1} {1 + x^2(t - 2\tau)}] \frac {x(t - \tau)}{[1 + x^2(t - \tau)]^2} \\ &\quad + [ \frac {1}{1 + x^2(t)} + \frac {1} {1 + x^2(t - \tau)}] \frac {x(t)}{[1 + x^2(t)]^2}\big\}, \end{align*} $p(t) = 16 + \cos^2 \frac {\pi t}{\tau}$, and $g(t) = 1 + \sin^2 \frac{2\pi t}{\tau}$. Then $F$ can be chosen as $$F(t, u, v) = ( 8 + \sin^2 \frac{2\pi t}{\tau})( \frac {1}{1 + u^2} + \frac {1} {1 + v^2})^2\,.$$ It is easy to see that $F(t, u, v)$ is $\tau$-periodic in $t$, p(t) is $\tau$-periodic with $0 < 15 < p(t)$, $g(t)$ is $\tau$-periodic and $$\overline{g} = \frac {1} {2k\tau} \int^{2k \tau}_{0}|g(t)|^2 dt \leq \frac {1} {2k\tau} \int^{2k \tau}_{0} 4 dt < \frac {15}{2},$$ Since all the assumptions in Theorem \ref{thm1.2} are satisfied, \eqref{e1.1} has at least one periodic solution. \subsection*{Acknowledgement} The authors would like to thank the anonymous referee for his or her suggestions and corrections. \begin{thebibliography}{00} \bibitem{Belley} J. Belley, M. Virgilio; Periodic Duffing equations with delay. \emph{Electron. J. Differential Equations} 2004 (2004), No. 30, 18 pp. (electronic). \bibitem{Gen} Gen Qiang Wang, Ju Rang Yan; Existence of Periodic Solutions for Second Order Nonlinear Neutral Delay Equations (in Chinese), \emph{Acta Mathematica Sinica}, 47(2004), 379-384. \bibitem{Guo} Guo Zhi-ming, Xu Yuan-tong; Existence of periodic solutions to a class of second-order neutral differential difference equations. \emph{Acta Anal. Funct. Appl.} 5 (2003), 13-19. \bibitem{Layton} W. Layton, Periodic solutions of nonlinear delay equations. \emph{J. Math. Anal. Appl.} 77 (1980), 198-204. \bibitem{Lu} Lu Shiping, Ge Weigao; Existence of periodic solutions for a kind of second-order neutral functional differential equation. \emph{Appl. Math. Comput.} 157 (2004), 433-448. \bibitem{Ma} Ma Shiwang, Wang Zhicheng, Yu Jianshe; Coincidence degree and periodic solutions of Duffing equations. \emph{Nonlinear Anal.} 34 (1998), 443-460. \bibitem{Mawhin} J. Mawhin and M. Willem; Critical point theory and Hamiltonian systems. \emph{Applied Mathematical Sciences,} 74. Springer-Verlag, New York, 1989. \bibitem{Rabinowitz} P. H. Rabinowitz; Minimax methods in critical point theory with applications to differential equations. \emph{CBMS Regional Conference Series in Mathematics}, 65. by the American Mathematical Society, Providence, RI, 1986. \end{thebibliography} \end{document}