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\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
\begin{equation} \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}
\end{equation}
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
\begin{equation} \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}
\end{equation}
We study \eqref{e2.1} by using variational methods. More precisely,
we investigate the functional $J:H\to \mathbb{R}$, which is defined by
\begin{equation}\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\,,
\end{equation}
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
\begin{equation}
\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}\,.
\end{equation}

\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
\begin{equation} \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}
\end{equation}
Because of Poincare's inequality $\| w \|_2\leq \| w \|$  and
\eqref{e3.1} we have
\begin{equation}   \label{e3.2}
J(w) \ge  \frac{\varepsilon}{2} \| w \|^2-c\,\|R^2 \|_1-c_1\|R f \|_1 \,.
\end{equation}
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
\begin{equation} \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}
\end{equation}

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}