\documentclass[reqno]{amsart}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 30, pp. 1--18.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2004/30\hfil Periodic Duffing equations with delay]
{Periodic Duffing equations with delay}
\author[J.-M. Belley \& M. Virgilio\hfil EJDE-2004/30\hfilneg]
{Jean-Marc Belley \& Michel Virgilio} % in alphabetical order
\address{Jean-Marc Belley \hfill\break
Universit\'{e} de Sherbrooke, Facult\'{e} des sciences\\
Sherbrooke, Qc, Canada J1K 2R1}
\email{Jean-Marc.Belley@USherbrooke.ca}
\address{Michel Virgilio \hfill\break
Universit\'{e} de Sherbrooke, Facult\'{e} des sciences\\
Sherbrooke, Qc, Canada J1K 2R1}
\email{Michel.Virgilio@dmi.usherb.ca}
\date{}
\thanks{Submitted January 22, 2004. Published March 3, 2004.}
\subjclass[2000]{34K13}
\keywords{Duffing equations, periodic solutions, delay equations,
\hfill\break\indent
a priori bounds, contraction principle}
\begin{abstract}
Assuming \textit{a priori} bounds on the mean of
a $T$-periodic function $p$, we show that the
Duffing equation
$$
x''(t) +cx'(t) +g(t-\tau ,x(t-\tau) ,x'(t-\tau)) =p(t),
$$
with delay $\tau$, admits a $T$-periodic solution.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
The existence of $2\pi $-periodic solutions to the Duffing equation
\begin{equation}
x''(t) +g(x(t-\tau ) )=p(t) \label{x''+g(x)=p}
\end{equation}
with delay $\tau \geq 0$ is a challenging problem of current interest. In
\cite{MAwangYU} it is shown that such solutions exist for continuous $2\pi $
-periodic $p:\mathbb{R}\to \mathbb{R}$ of mean $\overline{p}=0$ and
continuous $g:\mathbb{R\to R}$ for which there exist
$A\in [0,1/\pi^2[$ and $C\geq 0$ such that, for all $|x|$ large enough,
one has simultaneously
\begin{gather}
xg(x) >0\,, \label{xg>0}\\
|g(x) |\leq A|x|+C\,. \label{||g(x)|0}) is replaced
by
\begin{equation}
\mathop{\rm sgn}(x) (g(x) -\overline{p}) >0 \label{sign(x)(g(x)-pbar)>0}
\end{equation}
and condition (\ref{||g(x)|0$. In practice though, conditions (\ref{xg>0}) and
(\ref {sign(x)(g(x)-pbar)>0}) are often not met, as in the case of
the classical forced pendulum equation where $\tau =0$ and $g(x)
=a\sin x$ $ (a>0) $. The result presented in
\cite{ZHANGwangYU2000} rests on a complex inequality which is also
not applicable to the forced pendulum equation (since it then
takes the form $0>0$). In \cite{CHENyuYUAN02} it is shown by means
of coincidence degree that equation (\ref{x''+g(x)=p}) with $ \tau
=0$ and $g'<0$ (which also does not hold for the pendulum
equation) possesses a unique $2\pi $-periodic solution if and only
if $ \overline{p}\in g(\mathbb{R}) $. As shown in \cite{ALONSO97},
there are cases where the nonconservative forced pendulum equation
with periodic forcing (and $\tau =0$) admits no periodic solution.
(See also \cite {TARANTELLO89}.) The results obtained here on the
existence of twice continuously differentiable periodic solutions
to equations that generalize ( \ref{x''+g(x)=p}) are applicable to
the forced pendulum equation. As we shall see in Theorem
\ref{thm18}, the contraction principle yields a result which
contains the following:
\begin{theorem} \label{thm1}
Given $A\in ]0,1/\sqrt{2}[$, let $g:\mathbb{R\to R}$ be
a continuous function such that
\begin{equation}
|g(x_2) -g(x_{1}) |0$)
satisfies the Lipschitz condition (\ref{|g(x2)-g(x1)|0$, let $C(T) $ be the class of all continuous
real-valued $T$-periodic functions on $\mathbb{R}$ and $L^{1}(T)
$ the set of all real-valued $T$-periodic functions on $\mathbb{R}$ the
restriction of which to the segment $[0,T] $ are Lebesgue
integrable functions. Let $C^{1}(T) $ be the class of all
continuously differentiable functions in $C(T) $, $C^2(
T) $ the class of all twice continuously differentiable functions in $
C(T) $ and $L^2(T) $ the Hilbert space of all $
x\in L^{1}(T) $ with usual finite norm
\begin{equation*}
\|x\|_2=\big[\frac{1}{T}\int_{[0,T]}|x(t) |^2\,dt\big]^{1/2}.
\end{equation*}
The inner product on $L^2(T) $ associated with this norm is
given by
\begin{equation*}
\langle x,y\rangle_2=\frac{1}{T}\int_{[0,T]
}x(t) y(t) \,dt.
\end{equation*}
For a given $c\in \mathbb{R}$, let $L$ be the linear operator
\begin{equation*}
Lx=x''+cx'.
\end{equation*}
The theorem in the previous section will be extended to the case where $p\in
L^{1}(T) $ and the function $g:\mathbb{R}^{3}\to \mathbb{
R}$ in the Duffing equation
\begin{equation}
Lx(t) +g(t-\tau ,x(t-\tau ) ,x'(t-\tau ) ) =p(t) \label{Lx+g=p}
\end{equation}
with delay $\tau \in \mathbb{R}$ is continuous and such that
$g(t,x,y) $ is $T$-periodic in $t\in \mathbb{R}$ for all
$(x,y) \in \mathbb{R}^2$, and satisfies the Lipschitz condition
\begin{equation}
|g(t,x_2,y_2) -g(t,x_{1},y_{1})
|\leq A|x_2-x_{1}|+B|y_2-y_{1}| \label{|g(t,x2,y2)-g(t,x1,y1)|0})
and so the results in \cite{MAwangYU} and \cite{ZHANGwangYU2000} (as well as
those in \cite{CESARIkannan82}, \cite{FONDAlupo89}, \cite{GOSSEZomari90}
\cite{MAHWINward83} and \cite{OMARIzanolin87} for the case $\tau =0$) do not
apply.
\section{The case $g=g(t,x) $}
In this case condition (\ref{|g(t,x2,y2)-g(t,x1,y1)|0$
for all $t\in \mathbb{R}$ and all $|x|$ large enough.
This is essentially condition (\ref{sign(x)(g(x)-pbar)>0}) found in
\cite{HUANGxiang94}.
\subsection{\textit{A priori} bounds for $\overline{p}$}
Proceeding as in section 3.2, one has, for all $x\in \widetilde{H}$ and all
$n\in \mathbb{N}$,
\begin{equation*}
|\overline{g_{\tau ,r}}[x_{r}] -\overline{g_{\tau ,r}}
[G_{\tau ,r}^{n}(x) ] |\leq \frac{A (\beta')^{n}}{1-\beta '}\|G_{\tau ,r}
(x)-x\|_2\,,
\end{equation*}
where
\begin{equation*}
\beta '=A\sigma '=\frac{A}{\omega }\sqrt{\frac{\omega^2+1}{\omega^2+c^2}}\,.
\end{equation*}
Hence condition (\ref{infgrBAR[xr]1$ and
$\beta'<1$.
\begin{example} \label{ex19}
For $\alpha \in \mathbb{R}$\ the equation with delay $\tau \in \mathbb{R}$
\begin{equation*}
x''(t) +2x'(t) +\sqrt{2}\cos^2(t-\tau ) \ln (1+x^2(t-\tau ) )=\alpha +\sin t-2\cos t
\end{equation*}
is such that $c=2$, $T=2\pi $ (and so $\omega =1$), $\varphi (t)=-\sin t$ and
\begin{equation*}
g(t,x) =\sqrt{2}\cos^2(t) \ln (1+x^2) .
\end{equation*}
Hence
\begin{equation*}
|g(t,x_2) -g(t,x_{1}) |\leq \sqrt{2}|x_2-x_{1}|
\end{equation*}
and so one has (\ref{|g(t,x2)-g(t,x1)|1$),
$\|\varphi \|_2=1/\sqrt{2}$, $\lambda '=2\sqrt{2}/(\sqrt{5}-2) $,
\begin{align*}
\inf_{r\in \mathbb{R}}\overline{g}(t,\varphi +r)
&=\inf_{r\in \mathbb{R}}\frac{\sqrt{2}}{2\pi }\int_{[0,2\pi ] }\cos
^2(t) \ln (1+(r-\sin t)^2) \,dt \\
&\leq \inf_{r\in \mathbb{R}}\frac{\sqrt{2}}{2\pi }\int_{[0,2\pi
] }\cos^2(t) \ln (1+(|r|+|\sin t|)^2) \,dt \\
&<\frac{\sqrt{2}}{2\pi }\ln 2\int_{[0,2\pi ] }\cos^2(t) \,dt \\
&=\frac{1}{\sqrt{2}}\ln 2
\end{align*}
and
\begin{equation*}
\sup_{r\in \mathbb{R}}\overline{g}(t,\varphi +r) =\sup_{r\in
\mathbb{R}}\frac{\sqrt{2}}{2\pi }\int_{[0,2\pi ] }\cos^2(t)
\ln (1+(r-\sin t)^2)\,dt=\infty \,.
\end{equation*}
Hence, by the previous theorem, the delay equation admits a $2\pi $-periodic
solution whenever
\begin{equation*}
\frac{1}{\sqrt{2}}\ln 2+\frac{2}{(\sqrt{5}-2) }\leq \alpha <\infty \,.
\end{equation*}
\end{example}
\section{The case of bounded $g=g(t,x)$}
Let $g:\mathbb{R}^2\to \mathbb{R}$ be a bounded continuous
function such that $g(t,x) $ is $T$-periodic in $t\in \mathbb{R}$
for all $x\in \mathbb{R}$. Condition (\ref{sup(|g(t,x)|:tinR)1$, and so
(\ref{A0$ and $\omega =2\pi /T$, suppose that the nonconservative pendulum
equation
\begin{equation*}
x''(t) +cx'(t) +a\sin x(t) =\alpha +b\sin \omega t
\end{equation*}
with forcing $p(t) =\alpha +b\sin \omega t$ is such that
$| a|<\omega \sqrt{\omega^2+c^2}$. Then
\begin{equation*}
\varphi (t) =-\frac{b\omega }{c^2\omega^2+\omega^4}[
c\cos \omega t+\omega \sin \omega t]
\end{equation*}
is the $T$-periodic solution of mean zero of the linear equation
\begin{equation*}
x''(t) +cx'(t) =b\sin \omega t.
\end{equation*}
One has
\begin{equation*}
\|\varphi \|_2=\frac{|b|}{\sqrt{2}\sqrt{\omega^2+c^2}}
\end{equation*}
and, by Maclaurin's series expansion for trigonometric functions,
\begin{align*}
\overline{\sin }\varphi &=\overline{\sin }(-\frac{be^{i\omega t}}{
2(c\omega +i\omega^2) }-\frac{be^{-i\omega t}}{2(
c\omega -i\omega^2) }) \\
&=\frac{1}{T}\int_{0}^{T}\sin (-\frac{be^{i\omega t}}{2(c\omega
+i\omega^2) }) \cos (\frac{be^{-i\omega t}}{2(
c\omega -i\omega^2) }) \,dt \\
&\quad+\frac{1}{T}\int_{0}^{T}\cos (\frac{be^{i\omega t}}{2(c\omega
+i\omega^2) }) \sin (-\frac{be^{-i\omega t}}{2(c\omega -i\omega^2) }) \,dt \\
&=0\,.
\end{align*}
Thus,
\[
\inf_{r\in \mathbb{R}}a\overline{\sin }(\varphi +r)
=\inf_{r\in \mathbb{R}}a[\overline{\cos }\varphi \sin r+\overline{
\sin }\varphi \cos r]
=\inf_{r\in \mathbb{R}}a\overline{\cos }\varphi \sin r
=-|a\overline{\cos }\varphi |
\]
and
\[
\sup_{r\in \mathbb{R}}a\overline{\sin }(\varphi +r)
=\sup_{r\in \mathbb{R}}a[\overline{\cos }\varphi \sin r+\overline{\sin }
\varphi \cos r]
=\sup_{r\in \mathbb{R}}a\overline{\cos }\varphi \sin r
=|a\overline{\cos }\varphi |
\]
and so, by the previous theorem, one is assured of the existence of a twice
continuously differentiable $T$-periodic solution whenever $\alpha $
satisfies
\begin{equation*}
-|a\overline{\cos }\varphi |+\lambda ''
\frac{|ab|}{\sqrt{2}\sqrt{\omega^2+c^2}}<\alpha
<|a\overline{\cos }\varphi |-\lambda ''
\frac{|ab|}{\sqrt{2}\sqrt{\omega^2+c^2}}
\end{equation*}
where
\begin{equation*}
\lambda ''=\sqrt{\frac{a^2}{\omega \sqrt{\omega^2+c^2}
-|a|}}
\end{equation*}
and
\begin{align*}
\overline{\cos }\varphi &=\overline{\cos }(-\frac{be^{i\omega t}}{
2(c\omega +i\omega^2) }-\frac{be^{-i\omega t}}{2(
c\omega -i\omega^2) }) \\
&=\frac{1}{T}\int_{0}^{T}\cos (\frac{be^{i\omega t}}{2(c\omega
+i\omega^2) }) \cos (\frac{be^{-i\omega t}}{2(
c\omega -i\omega^2) }) \,dt \\
&\quad -\frac{1}{T}\int_{0}^{T}\sin (\frac{be^{i\omega t}}{2(c\omega
+i\omega^2) }) \sin (\frac{be^{-i\omega t}}{2(
c\omega -i\omega^2) }) \,dt \\
&=\sum_{n=0}^{\infty }\frac{(-1)^{n}}{(2^{n}(n!) )^2}
(\frac{b^2}{c^2\omega^2+\omega^4})^{n}.
\end{align*}
\end{example}
In many cases the \textit{a priori} bounds of the previous theorem are never
satisfied. For example, the equation with delay $\tau \in \mathbb{R}$
\begin{equation}
x''(t) +x'(t) +\sqrt{\frac{3
}{2}}\cos (t-\tau ) \sin x(t-\tau ) =\alpha -\sin
t+\cos t \label{x''+x'+(3/2)½cos(t-tau)sinx(t-tau)=alpha-sint+cost}
\end{equation}
is such that $\varphi (t) =\sin t$. Hence
\begin{equation*}
\overline{g}(t,\varphi +r) =\frac{\sqrt{3/2}}{2\pi }
\int_{0}^{2\pi }\cos t\sin (r+\sin t) \,dt=0
\end{equation*}
and so
\begin{equation*}
\inf_{r\in \mathbb{R}}\overline{g}(t,\varphi +r) =\sup_{r\in
\mathbb{R}}\overline{g}(t,\varphi +r) =0.
\end{equation*}
Thus, the \textit{a priori} bounds of the previous theorem are not satisfied
here. So one needs to apply (\ref{aprioriBOUNDwithLAMDA''}) for different
choices than $x=-\varphi $ and/or $n=1$.
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\end{document}