\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 81, pp. 1--3.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/81\hfil Remark on Duffing equation] {Remark on Duffing equation with Dirichlet boundary condition} \author[P. Tomiczek \hfil EJDE-2007/81\hfilneg] {Petr Tomiczek} \address{Petr Tomiczek \newline Department of Mathematics, University of West Bohemia \\ Univerzitn\'{\i} 22, 306 14 Plze\v{n}, Czech Republic} \email{tomiczek@kma.zcu.cz} \thanks{Submitted April 24, 2007. Published May 29, 2007.} \thanks{Supported by Research Plan MSM 4977751301} \subjclass[2000]{34G20, 35A15, 34K10} \keywords{Second order ODE; Dirichlet problem; variational method; \hfill\break\indent critical point} \begin{abstract} In this note, we prove the existence of a solution to the semilinear second order ordinary differential equation \begin{gather*} u''(x)+ r(x) u'+g(x,u)=f(x)\,,\\ x(0)=x(\pi)=0\,, \end{gather*} using a variational method and critical point theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} We denote $H$ the Sobolev space of absolutely continuous functions $u:(0,\pi)\to \mathbb{R}$ such that $u'\in L^2(0,\pi)$ and $u(0)=u(\pi)=0$. Let us consider the nonlinear problem $$\label{e1.1} \begin{gathered} u''(x)+ r(x)u'+g(x,u)=f(x)\,, \quad x\in[0,\pi]\,,\\ u(0)=u(\pi)=0\,, \end{gathered}$$ where $r\in H$, the nonlinearity $g:[0,\pi]\times\mathbb{R} \to \mathbb{R}$ is Caratheodory's function and $f\in L^1(0,\pi)$. A physical example of this equation is the forced pendulum equation. In articles \cite{a1,a2} the authors assume that the friction coefficient $r$ is nondecreasing and the nonlinearity $g$ satisfies the condition $\frac{g(x,u)-g(x,v)}{u-v}\le k < 1\,.$ They prove the uniqueness of the solution. In this work, we prove the existence of a solution to the problem \eqref{e1.1} under more general condition $G(x,s) \le \frac{1}{2}\bigl(1-\varepsilon+\frac{1}{4} r^2+ \frac{1}{2} r'\bigr) s^2 +c\,, \quad x\in[ 0, \pi]\,, s\in\mathbb{R}\,,$ where $G(x,s)=\int_0^{s}g(x,t)\,dt$, $c>0$, and $\varepsilon\in(0,1)$. \section{Preliminaries} \paragraph{Notation:} We shall use the classical space ${C}^k(0,\pi)$ of functions whose $k$-th derivative %???? AUTHOR: CHANGED $k$-th power to $k$-th derivative ??????????? is continuous and the space $L^p(0,\pi)$ of measurable real-valued functions whose $p$-th power of the absolute value is Lebesgue integrable. We use the symbols $\|\cdot \|$, and $\| \cdot \|_p$ to denote the norm in $H$ and in $L^p(0,\pi)$, respectively. By a solution to \eqref{e1.1} we mean a function $u\in {C}^1(0,\pi)$ such that $u'$ is absolutely continuous, $u$ satisfies the boundary conditions and the equation \eqref{e1.1} is satisfied a.e. in $(0,\pi)$. For simplicity's sake we denote $R(x)=e^{\int_0^x\frac{1}{2} r(\xi)\, d\xi}$ and multiply \eqref{e1.1} by the function $R(x)$. We put $w(x)=R(x) u(x)$ and obtain for $w$ an equivalent Dirichlet problem $$\label{e2.1} \begin{gathered} w''(x)-\bigl(\frac{1}{4}r^2(x)+\frac{1}{2}r'(x)\bigr) w(x) + R(x) g(x,\frac{w}{R(x)})=R(x)f(x)\,, \\ w(0)=w(\pi)=0\,. \end{gathered}$$ We study \eqref{e2.1} by using variational methods. More precisely, we investigate the functional $J:H\to \mathbb{R}$, which is defined by $$\label{e2.2} J(w)=\frac12 \int_0^{\pi}\bigl[(w')^2+\bigl(\frac{1}{4}r^2+ \frac{1}{2} r' \bigr) w^2\bigr]\,dx -\int_0^{\pi }\bigl[ R^2 G(x,\frac{w}{R})-Rfw\bigr] \,dx\,,$$ where $$G(x,s)=\int_0^{s}g(x,t)\,dt\,.$$ We say that $w$ is a critical point of $J$, if $$\langle J'(w), v\rangle = 0 \quad \mbox{for all } v\in H\,.$$ We see that every critical point $w\in H$ of the functional $J$ satisfies $$\int_0^{\pi} \bigl[ w' v'+\bigl(\frac{1}{4}r^2+ \frac{1}{2} r'\bigr) w v \bigr] \,dx - \int_0^{\pi } \bigl[R g(x,\frac{w}{R}) v- R f v\bigr] \,dx=0$$ for all $v \in H$, and $w$ is a weak solution to \eqref{e2.1}, and vice versa. The usual regularity argument for ODE proves immediately (see Fu\v{c}\'{\i}k \cite{f1}) that any weak solution to \eqref{e2.1} is also a solution in the sense mentioned above. We suppose that there are $c>0$ and $\varepsilon\in(0,1)$ such that $$\label{e2.3} G(x,s) \le \frac{1}{2}\bigl(1-\varepsilon+\frac{1}{4} r^2(x)+ \frac{1}{2} r'(x)\bigr) s^2 +c \quad x\in[0, \pi]\,, \, s\in\mathbb{R}\,.$$ \begin{remark} \label{rem1} \rm The condition \eqref{e2.3} is satisfied for example if $g(x,s)=(1-\varepsilon)s$ and $\frac{1}{4} r^2+ \frac{1}{2} r'\ge 0$. It is easy to find a function $r$ which is not nondecreasing on $[0,\pi]$ and which satisfies $\frac{1}{4} r^2+\frac{1}{2} r'\ge 0$. For example $r(x)=-x+\pi+\sqrt{2}$. \end{remark} \section{Main result} \begin{theorem} \label{t3.1} Under the assumption \eqref{e2.3}, Problem (\ref {e2.1}) has at least one solution in $H$. \end{theorem} \begin{proof} First we prove that $J$ is a weakly coercive functional; i. e., $$\lim_{\| w \| \to \infty}J(w)=\infty\quad\mbox{for all } w\in H.$$ Because of the compact imbedding of $H$ into ${C}(0,\pi)$ , $(\| w \|_{{C}(0,\pi)}\leq c_1\| w \|)$, and the assumption \eqref{e2.3} we obtain \label{e3.1} \begin{aligned} J(w)&= \frac12 \int_0^{\pi }\bigl[(w')^2+\bigl(\frac{1}{4}r^2 + \frac{1}{2} r'\bigr) w^2 \bigr]\,dx - \int_0^{\pi }\bigl[R^2 G(x,\frac{w}{R})- R f w\bigr] \,dx \\ &\ge \frac12 \| w \|^2- \frac12 (1-\varepsilon) \| w \|_2^2 -\|R^2 \|_1 c -\|R f \|_1 c_1\| w \|\,. \end{aligned} Because of Poincare's inequality $\| w \|_2\leq \| w \|$ and \eqref{e3.1} we have $$\label{e3.2} J(w) \ge \frac{\varepsilon}{2} \| w \|^2-c\,\|R^2 \|_1-c_1\|R f \|_1 \,.$$ Then \eqref{e3.2} implies $\lim_{\| w \|\to \infty} J(w)= \infty$. Next we prove that $J$ is a weakly sequentially lower semi-continuous functional on $H$. Consider an arbitrary $w_0\in H$ and a sequence $\{w_n\}\subset H$ such that $w_n\rightharpoonup w_0$ in $H$. Due to compact imbedding $H$ into ${C}(0,\pi)$ we have $w_n\to w_0$ in ${C}(0,\pi)$. This and the continuity $g(x,t)$ in the variable $t$ imply $$\label{e3.3} \begin{gathered} \frac12 \int_0^{\pi } \bigl(\frac{1}{4}r^2+ \frac{1}{2} r'\bigr) w_n^2 \,dx -\int_0^{\pi }\bigl[R^2 G(x,\frac{w_n}{R})- R f w_n\bigr] \,dx\to \\ \frac12 \int_0^{\pi }\bigl(\frac{1}{4}r^2+ \frac{1}{2} r'\bigr) w_0^2 \,dx - \int_0^{\pi }\bigl[R^2 G(x,\frac{w_0(x)}{R})- R fw_0\bigr] \,dx\,. \end{gathered}$$ Due to the weak sequential lower semi-continuity of the Hilbert norm $\|\cdot\|$ (i.e. $\liminf_{n\to\infty}\|w_n\|\geq \|w_0\|$) and \eqref{e3.3}, we have $$\liminf_{n\to\infty} J(w_n)\ge J(w_0)\,.$$ The weak sequential lower semi-continuity and the weak coercivity of the functional $J$ imply (see Struwe \cite{s1}) the existence of a critical point of the functional $J$; i.e., a weak solution to the equation \eqref{e2.1} and, consequently, to equation \eqref{e1.1}. \end{proof} \begin{thebibliography}{0} \bibitem{a1} P. Amster: \emph{Nonlinearities in a second order ODE}, Electron. J. Diff. Eqns., Conf. 06, 2001, pp. 13–-21. \bibitem{a2} P. Amster, M. C. Mariani: \emph{A second order ODE with a nonlinear final condition}, Electron. J. Diff. Eqns., Vol. {\bf 2001} (2001), No. 75, pp. 1–-9. \bibitem{f1} S. Fu\v{c}\'{\i}k: \emph{Solvability of Nonlinear Equations and Boundary Value Problems}, D. Reidel Publ. Company, Holland 1980. \bibitem{s1} M. Struwe: \emph{Variational Methods}, Springer, Berlin, (1996). \end{thebibliography} \end{document}