\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)|<A|x|+B}
\end{gather}
This result (like that found in \cite{ZHANGwangYU2000} for a somewhat
different equation with more complicated \textit{a priori} bounds) was
obtained by means of Brouwer degree theory with a continuation theorem based
on Mawhin's coincidence degree (see \cite{mahwin72} and \cite{mahwin79}). It
generalizes, for the case $\overline{p}=0$, a similar result obtained in
\cite{HUANGxiang94} for $\overline{p}\in \mathbb{R}$ where, for all $
|x|$ large enough, condition (\ref{xg>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)|<A|x|+B}) by
\begin{equation}
|g(x) |\leq C  \label{|g(x)<=M}
\end{equation}
for some $C>0$. 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}) |<A|x_2-x_{1}| \label{|g(x2)-g(x1)|<A|x2-x1|}
\end{equation}
for all $x_{1},x_2\in \mathbb{R}$ and let $\varphi $ be the solution of
mean zero of  $x''=p-\overline{p}$,
where $p:\mathbb{R\to R}$ is a continuous $2\pi $-periodic function
of mean $\overline{p}$. If
\begin{equation*}
\inf_{r\in \mathbb{R}}\overline{g}(\varphi +r) +\lambda '
\|\varphi \|_H\leq \overline{p}\leq \sup_{r\in \mathbb{
R}}\overline{g}(\varphi +r) -\lambda '\|
\varphi \|_H
\end{equation*}
where
\begin{gather*}
\overline{g}(\varphi +r) =\frac{1}{2\pi }\int_{[0,2\pi
] }g(\varphi (t) +r) \,dt, \quad
\lambda '=\frac{\sqrt{2}A^2}{1-\sqrt{2}A}, \\
\|\varphi \|_H=\big[\frac{1}{2\pi }\int_{[0,2\pi ] }(\varphi (t)
+\varphi '(t) )^2\,dt\big]^{1/2},
\end{gather*}
then the Duffing equation
\begin{equation*}
x''(t) +g(x(t-\tau ) )=p(t)
\end{equation*}
with delay $\tau \in \mathbb{R}$ admits a twice continuously differentiable $
2\pi $-periodic solution.
\end{theorem}

An equation like that of the conservative forced pendulum
$x''+a\sin x=p$ where $c=\tau =0$ and $g(x) =a\sin x$ ($a>0$)
satisfies the Lipschitz condition (\ref{|g(x2)-g(x1)|<A|x2-x1|}) for $A=a$
and so, by the theorem above, one has the existence of $2\pi $-periodic
solutions of this equation whenever the stated \textit{a priori} bounds on $
\overline{p}$ are respected and $a\in ]0,1/\sqrt{2}[$.

In this paper, one exploits the argument presented in \cite
{BELLEYsaadidrissi2001} for Josephson's equation
\begin{equation*}
x''+cx'+dx'\cos x+a\sin x=p
\end{equation*}
with $a,c,d\in \mathbb{R}$. Note that Josephson's equation does not satisfy
the Lipschitz condition (\ref{|g(t,x2,y2)-g(t,x1,y1)|<A|x2-x1|+B|y2-y1|}) on
which rests this paper. As for results on the existence of almost periodic
solutions to the Duffing equation, one could no doubt employ the techniques
used in \cite{BELLEYsaadidrissi2003} for Josephson's equation. This is left
for future work.

\section{Preliminaries}

For a given $T>0$, 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)|<A|x2-x1|+B|y2-y1|}
\end{equation}
for all $t\in \mathbb{R}$, suitable $A,B\in [0,\infty [$ and
all $(x_{1},y_{1}) ,(x_2,y_2) \in \mathbb{R}^2$. This condition
implies the inequality
\begin{equation}
\sup \{ |g(t,x,y) |:t\in \mathbb{R}\} \leq A|x|+B|y|+C
\label{sup|g(t,x,y)|<A|x|+B|y|+C}
\end{equation}
for all $(x,y) \in \mathbb{R}^2$ and some $C\geq 0$ (put, for
example, $C=\sup \{|g(t,0,0) |:t\in
\mathbb{R}\}$). The inequality (\ref{sup|g(t,x,y)|<A|x|+B|y|+C}) is a
natural generalization of (\ref{||g(x)|<A|x|+B}).

The mean $\overline{x}$ of any $x\in L^{1}(T) $ is given by the
Lebesgue integral
\begin{equation*}
\overline{x}=\frac{1}{T}\int_{[0,T] }x(t) \,dt
\end{equation*}
and $x$ can be identified with its Fourier series
\[
\sum_{n\in \mathbb{Z}}\widehat{x}(n) e^{in\omega t},
\]
where $i=\sqrt{-1}$, $\omega =2\pi /T$ and
\begin{equation*}
\widehat{x}(n) =\frac{1}{T}\int_{[0,T] }x(
t) e^{-in\omega t}\,dt.
\end{equation*}
Hence, $\widehat{x}(0) =\overline{x}$ and, since $x$ is
real-valued, $\widehat{x}(-n) $ is the complex conjugate of $
\widehat{x}(n) $. The class $P(T) $ of real \textit{
trigonometric polynomials} is the subset of all $x\in L^{1}(T) $
with $\widehat{x}(n) =0$ for all but at most finitely many $n\in
\mathbb{Z}$. Given $S\subset L^{1}(T) $, $\widetilde{S}$ denotes
that subset of $L^{1}(T) $ given by
\begin{equation*}
\widetilde{S}=\{x-\overline{x}:x\in S\}
\end{equation*}
and so one has $x=\widetilde{x}+\overline{x}$ for all $x\in L^{1}(T) $.
If $x,y\in \widetilde{L^{1}}(T) $ are such that
\begin{equation*}
\int_{[0,T] }x(t) q'(t)
\,dt=-\int_{[0,T] }y(t) q(t) \,dt
\end{equation*}
for all $q\in \widetilde{P}(T) $, then $y$ is the \textit{weak
derivative} of $x$ (denoted $x'$) and $x$ can be taken continuous
by means of $x=z-\overline{z}$ where
\begin{equation*}
z(t) =\int_{[0,t] }y(t) \,dt
\end{equation*}
for all $t\in \mathbb{R}$. Similarly, if $x,z\in \widetilde{L^{1}}(
T) $ are such that
\begin{equation*}
\int_{[0,T] }x(t) q''(t)
\,dt=\int_{[0,T] }z(t) q(t) \,dt
\end{equation*}
for all $q\in \widetilde{P}(T) $, then $z$ is the \textit{weak
second derivative} of $x$ (denoted $x''$) and $x$ can be
taken continuously differentiable.

Let $H$ be the subspace of $L^2(T) $ consisting of all
$x\in C(T) $ with weak derivative $x'\in \widetilde{L^2}(T) $.
On $H$ one has the inner product
\begin{equation*}
\langle x,y\rangle_H=\langle x,y\rangle_2+\langle x',y'\rangle_2
\end{equation*}
with associated norm
\begin{equation*}
\|x\|_H=[\|x\|_2^2+\|x'\|_2^2]^{1/2}=\|x+x'\|_2
\end{equation*}
and, on $\widetilde{H}$, the well-known Sobolev inequality (see for example
\cite{MAHWINwillem89})
\begin{equation*}
\sup_{0\leq t\leq T}|x(t) |^2\leq \frac{T^2}{12}
\|x'\|_2^2.
\end{equation*}
Consequently, strong convergence in $\widetilde{H}$ implies uniform
convergence (to an element of $\widetilde{C}(T) $). Furthermore,
$\widetilde{H}$ is complete, as is now shown.

\begin{proposition} \label{prop2}
$\widetilde{H}$ is a Hilbert space.
\end{proposition}

\begin{proof}
Let $\{x_{n}\}$ be a Cauchy sequence in $\widetilde{H}$. Then
there exists $y\in \widetilde{L^2}(T) $ such that $\|
y-x_{n}'\|_2\to 0$ (as $n\to \infty $)
and, by above, there exists $x\in \widetilde{C}(T) $ such that $
\|x-x_{n}\|_2\to 0$ (as $n\to \infty $). For
any $q\in \widetilde{P}(T) $, the relation $\langle
x_{n},q'\rangle_2=-\langle x_{n}',q\rangle_2$ holds for all $n$, and so in the limit as $
n\to \infty $, $\langle x,q'\rangle
_2=-\langle y,q\rangle_2$. From this follows that $
y=x'$ in $\widetilde{L^2}(T) $. This shows that $x\in
\widetilde{H}$.
\end{proof}

\begin{remark} \label{rmk3} \rm
Sobolev's inequality yields
\begin{equation*}
|x(t) |\leq \frac{T}{\sqrt{12}}\|x\|_H
\end{equation*}
for all $t\in \mathbb{R}$. Thus, a point evaluation is a bounded linear
functional on $\widetilde{H}$ and so, if a sequence $\{x_{n}\}$
converges weakly in $\widetilde{H}$ to $x_{0}$ (denoted $x_{n}
\rightharpoonup x_{0}$), then it converges pointwise to $x_{0}$. By the
Banach-Steinhaus theorem, $\{x_{n}\}$ is bounded in $\widetilde{
H}$ and so, by the above inequality, $\{x_{n}(t) \}$
is a uniformly bounded sequence of functions.
\end{remark}

For $\psi \in H$, put
\begin{equation*}
g_{\tau }[\psi ] (t) =g(t-\tau ,\psi (
t-\tau ) ,\psi '(t-\tau ) ) .
\end{equation*}

\begin{proposition} \label{prop4}
Given a continuous function $g:\mathbb{R}^{3}\to \mathbb{R}$ for
which $g(t,x,y) $ is $T$-periodic in $t\in \mathbb{R}$ for all $
(x,y) \in \mathbb{R}^2$, let there exist $A,B,C\in [
0,\infty [$ for which condition (\ref{sup|g(t,x,y)|<A|x|+B|y|+C}) is
satisfied for all $t,x,y\in \mathbb{R}$. Then $g_{\tau }[\psi ]
\in L^2(T) $ for all $\psi \in H$.
\end{proposition}

\begin{proof}
One has, for any $\psi \in H$,
\begin{equation*}
\|g_{\tau }[\psi ] \|_2\leq \|A|\psi |+B|\psi '|
+C\|_2\leq C+\sqrt{A^2+B^2}\|\psi \|_H<\infty
\end{equation*}
and so $g_{\tau }[\psi ] \in L^2(T) $.
\end{proof}

One now introduces a subspace of $\widetilde{H}$ used later in section~3.
Let $\mathcal{H}$ be the subspace of $H$ given by
\begin{equation*}
\mathcal{H}=\{x\in C^{1}(T) :x'\in H\}
\end{equation*}
and on which is defined the inner product
\begin{equation*}
\langle x,y\rangle_{\mathcal{H}}=\langle x',y'\rangle_H.
\end{equation*}
This inner product is associated with the norm $\|x\|_{\mathcal{H}}$
on $\widetilde{\mathcal{H}}$ given by
$\|x\|_{\mathcal{H}}=\|x'\|_H$.

\begin{proposition} \label{prop5}
$\widetilde{\mathcal{H}}$ is a Hilbert space.
\end{proposition}

This result is proved like Proposition \ref{prop2} with the
sequences $\{x_{n}\}$ and $\{x_{n}'\}$ replaced by $\{x_{n}'\}$
and $\{x_{n}''\}$ respectively.

\begin{remark} \label{rmk6} \rm
By Wirtinger's inequality $\omega \|x\|_2\leq \|x'\|_2$
on $\widetilde{H}$ (see, for example \cite{MAHWINwillem89}) one obtains
\begin{equation*}
\omega \|x\|_H\leq \|x\|_{\mathcal{H}}
\end{equation*}
for all $x\in \widetilde{\mathcal{H}}$. Furthermore, Sobolev's inequality
yields
\begin{equation*}
|x'(t) |\leq \frac{T}{\sqrt{12}}\|
x'\|_H
\end{equation*}
for all $x\in \widetilde{\mathcal{H}}$ and so if a sequence $\{
x_{n}\}$ converges weakly in $\widetilde{\mathcal{H}}$ to $x_{0}$
(also denoted $x_{n}\rightharpoonup x_{0}$) then $x_{n}'$ converges
pointwise to $x_{0}'$. The sequence $\{x_{n}\}$ also
converges weakly to $x_{0}$\ in $\widetilde{H}$ since, for all $q\in
\widetilde{P}(T) $, one has
\begin{equation*}
\lim_{n\to \infty }\langle x_{n},q '' \rangle_H=-\lim_{n\to \infty }\langle
x_{n},q\rangle_{\mathcal{H}}=-\langle x_{0},q\rangle_{
\mathcal{H}}=-\langle x_{0}',q'\rangle
_H=\langle x_{0},q''\rangle_H
\end{equation*}
and from this follows that $x_{n}$ also converges pointwise (to $x_{0}$).
\end{remark}

Given $p\in L^{1}(T) $, there exists a unique function $\varphi
\in \widetilde{C^{1}}(T) \subset \widetilde{H}$, with weak
second derivative $\varphi ''\in \widetilde{L^{1}}(
T) $, which satisfies the linear differential equation
\begin{equation}
Lx(t) =\widetilde{p}(t) .  \label{Lx=pTILDA}
\end{equation}
Furthermore $\varphi \in \widetilde{C^2}(T) $ whenever $p\in
C(T) $. The substitution in (\ref{Lx+g=p}) of $\varphi +x\in H$
in place of $x\in H$ yields the equivalent equation
\begin{equation}
Lx+g_{\tau }[\varphi +x] =\overline{p}.  \label{Lx+g(phi+x)=pbar}
\end{equation}
The existence of a solution $x\in C^{1}(T) $ of (\ref
{Lx+g(phi+x)=pbar}) with weak second derivative $x''\in
\widetilde{L^2}(T) $ is equivalent to the existence of a
scalar $r\in \mathbb{R}$ and of a function $x_{r}\in \widetilde{\mathcal{H}}$
such that $x=x_{r}$ is a solution of
\begin{equation*}
Lx+g_{\tau }[\varphi +x+r] =\overline{p}
\end{equation*}
in $L^2(T) $. This, in turn, is equivalent to finding a scalar
$r\in \mathbb{R}$ and a function $x_{r}\in \widetilde{\mathcal{H}}$ such
that $x=x_{r}$ is simultaneously a solution of
\begin{equation}
Lx+\widetilde{g_{\tau }}[\varphi +x+r] =0
\label{Lx+gTILDA(phi+r+x)=0}
\end{equation}
in $\widetilde{L^2}(T) $ and of
\begin{equation}
\overline{g_{\tau }}[\varphi +x+r] =\overline{p}.
\label{gBAR(phi+r+xSUBr)=pBAR}
\end{equation}
To ease notation, let $g_{\tau ,r}[x] =g_{\tau }[\varphi
+x+r] $. Then equation (\ref{Lx+gTILDA(phi+r+x)=0}) becomes
\begin{equation}
Lx+\widetilde{g_{\tau ,r}}[x] =0  \label{Lx+grTILDA[x]=0}
\end{equation}
in $\widetilde{L^2}(T) $ while (\ref{gBAR(phi+r+xSUBr)=pBAR})
takes the form
\begin{equation}
\overline{g_{\tau ,r}}[x] =\overline{p}.  \label{grBAR[x]=pBAR}
\end{equation}
This reformulation of the original problem is modeled after that found in
\cite{FOURNIERmahwin85} for the forced pendulum equation.

\section{The case $g=g(t,x,y) $}

For a given $r\in \mathbb{R}$, equation (\ref{Lx+grTILDA[x]=0}) is
equivalent to the system of equations
\begin{equation*}
(n^2\omega^2-in\omega c) \widehat{x}(n) =
\widehat{g_{\tau ,r}}[x] (n)
\end{equation*}
for all $n\in \mathbb{Z\setminus }\{0\}$. Let $G_{\tau ,r}$ be
defined on $\widetilde{\mathcal{H}}$ by
\begin{equation}
G_{\tau ,r}(x) (t) =\sum_{n\in \mathbb{Z\setminus }
\{0\}}\frac{1}{(n^2\omega^2-in\omega c) }
\widehat{g_{\tau ,r}}[x] (n) e^{in\omega t}.
\label{Gtau,r}
\end{equation}
The function
\begin{equation*}
\gamma (t) =\sum_{n\in \mathbb{Z\setminus }\{0\}}
\frac{1}{(n^2\omega^2-in\omega c) }e^{in\omega t}
\end{equation*}
lies in $\widetilde{H}$ and yields
\begin{equation*}
G_{\tau ,r}(x) =\gamma \ast g_{\tau ,r}[x]
\end{equation*}
for all $x\in \widetilde{H}$, where the star represents the convolution
operator.

\subsection{Contraction principle on $\widetilde{\mathcal{H}}$}

The Lipschitz condition
(\ref{|g(t,x2,y2)-g(t,x1,y1)|<A|x2-x1|+B|y2-y1|}) implies
condition (\ref{sup|g(t,x,y)|<A|x|+B|y|+C}) and so, by Proposition \ref{prop4}
, $g_{\tau }[\psi ] \in L^2(T) $ for all $\psi \in H$. Hence
$G_{\tau ,r}:\widetilde{H}\to \widetilde{ \mathcal{H}}\subset
\widetilde{H}$ and so $G_{\tau ,r}$ maps $\widetilde{\mathcal{H}}$ into
itself and $x=x_{r}\in \widetilde{\mathcal{H}}$
is a solution of (\ref{Lx+grTILDA[x]=0}) in the sense of
$\widetilde{L^2}( T) $ whenever it is a fixed point of $G_{\tau,r}$ on
$\widetilde{ \mathcal{H}}$. The existence of such a fixed
point is established by means of the contraction principle in the
following theorem.

\begin{theorem} \label{thm7}
Let $g:\mathbb{R}^{3}\to \mathbb{R}$ be a continuous function such
that $g(t,x,y) $ is $T$-periodic in $t\in \mathbb{R}$ for all $
(x,y) \in \mathbb{R}^2$ and let $A,B\in [0,\infty [
\ $and $c\in \mathbb{R}$ be such that
\begin{equation}
A^2+B^2<\omega^2\big(\frac{\omega^2+\min \{1,c^2\}}{\omega^2+1}\big)
\label{A²+B²<w²(w²+min(1,c²))/(w²+1)}
\end{equation}
and (\ref{|g(t,x2,y2)-g(t,x1,y1)|<A|x2-x1|+B|y2-y1|}) is satisfied for all
$t\in \mathbb{R}$ and all $(x_{1},y_{1}) ,(x_2,y_2) \in \mathbb{R}^2$.
Then, given $p\in L^{1}(T) $ and $r\in \mathbb{R}$, there exists a unique
continuously differentiable $T$-periodic function $x_{r}$ of mean zero,
 with weak second derivative $x_{r}''\in $ $L^2(T) $, which is a
solution of equation (\ref{Lx+grTILDA[x]=0}) in the sense of $L^2(T) $.
Furthermore,
\begin{equation}
x_{r}=\lim_{n\to \infty }G_{\tau ,r}^{n}(x)
\label{xr=limGrn(x)}
\end{equation}
in $L^2(T) $ for all $x\in L^2(T) $, and if $
p\in C(T) $ then $x_{r}\in C^2(T) $.
\end{theorem}

\begin{proof}
For any $x,y\in \widetilde{\mathcal{H}}$ one has
\begin{align*}
\|G_{\tau ,r}(y) -G_{\tau ,r}(x)\|_{\mathcal{H}}^2
&=\sum_{n\in \mathbb{Z\setminus }\{
0\}}\big|\frac{(in\omega -n^2\omega^2) [
\widehat{g_{\tau ,r}}[y] (n) -\widehat{g_{\tau ,r}}
[x] (n) ] }{n^2\omega^2-in\omega c}\big|^2 \\
&\leq \sigma^2\sum_{n\in \mathbb{Z\setminus }\{0\}
}|\widehat{g_{\tau ,r}}[y] (n) -\widehat{g_{\tau ,r}}[x] (n) |^2 \\
&\leq \sigma^2\|g_{\tau ,r}[y] -g_{\tau ,r}[x] \|_2^2\,,
\end{align*}
where
\begin{align*}
\sigma &=\sup_{n\in \mathbb{Z\setminus }\{0\}}\big[\frac{
n^2\omega^2+n^4\omega^4}{n^4\omega^4+n^2\omega^2c^2}\big]^{1/2} \\
&=\begin{cases}
1 & \text{if }c^2\geq 1 \\
\sqrt{\frac{\omega^2+1}{\omega^2+c^2}} & \text{if }c^2<1
\end{cases} \\
&=\sqrt{\frac{\omega^2+1}{\omega^2+\min \{1,c^2\}}}
\end{align*}
and so one obtains
\begin{equation*}
\|G_{\tau ,r}(y) -G_{\tau ,r}(x)
\|_{\mathcal{H}}\leq \frac{\sigma }{\omega }\sqrt{A^2+B^2}
\|y-x\|_{\mathcal{H}}.
\end{equation*}
Hence, whenever (\ref{A²+B²<w²(w²+min(1,c²))/(w²+1)}) holds, $G_{\tau ,r}$
is a contraction on $\widetilde{\mathcal{H}}$ and so admits a unique fixed
point $x_{r}\in \widetilde{\mathcal{H}}$ given by the successive
approximations (\ref{xr=limGrn(x)}) of Banach and Picard.
\end{proof}

Since $\varphi ''\in L^{1}(T) $, the above result
can be reformulated in terms of $z_{r}=\varphi +x_{r}+r$ as follows:

\begin{corollary} \label{coro8}
Let $g:\mathbb{R}^{3}\to \mathbb{R}$ be a continuous function such
that $g(t,x,y) $ is $T$-periodic in $t\in \mathbb{R}$ for all $
(x,y) \in \mathbb{R}^2$. Also, let $c\in \mathbb{R}$ and $
A,B\in [0,\infty [\ $be such that (\ref
{|g(t,x2,y2)-g(t,x1,y1)|<A|x2-x1|+B|y2-y1|}) and
(\ref{A²+B²<w²(w²+min(1,c²))/(w²+1)}) hold for all $t\in \mathbb{R}$.
Then, given
$r\in \mathbb{R}$ and $p\in L^{1}(T) $, there exists a function
$z_{r}\in C^{1}(T) $ of mean $r$, with weak second derivative
$z_{r}''\in L^{1}(T) $, such that for some
constant $k_{r}\in \mathbb{R}$, $x=z_{r}$ is a solution in the sense of
$L^{1}(T) $ of the Duffing equation
\begin{equation*}
x''(t) +cx'(t) +g(
t-\tau ,x(t-\tau ) ,x'(t-\tau ) )=p(t) +k_{r}
\end{equation*}
with delay $\tau \in \mathbb{R}$. Furthermore, if $p\in C(T)$
then $z_{r}\in C^2(T) $.
\end{corollary}

The constant $k_{r}$ is given by
\begin{equation}
k_{r}=\frac{1}{T}\int_{[0,T] }[g(t,z_{r}(t) ,z_{r}'(t) ) -p(t) ]
\,dt.  \label{k=mean[g(t,zr(t),z'r(t))-p(t)]}
\end{equation}
The conservative pendulum equation
\begin{equation}
x''+a\sin x=p(t)  \label{x''+asinx=p(t)}
\end{equation}
with $a\in \mathbb{R}$ and continuous $T$-periodic forcing $p(t)
$, can be used to show that (\ref{Lx+grTILDA[x]=0}) may admit no solution
that also solves (\ref{grBAR[x]=pBAR}) (i.e. for which $k_{r}\neq 0$ for all
$r\in \mathbb{R})$. For example, if the constant term in the Fourier series
of the forcing $p(t) $ is too great, then the pendulum will wind
indefinitely about its fixed point, and so no periodic motion will be
possible. (See also \cite{ALONSO97}.)

When $\overline{p}\in \{\overline{g_{\tau ,r}}[z_{r}]
:r\in \mathbb{R}\}$, equation (\ref{grBAR[x]=pBAR}) is satisfied for
some $r\in \mathbb{R}$ and one obtains the following resolution of the
original equation (\ref{Lx+g=p}).

\begin{corollary} \label{coro9}
If, in the context of the theorem, one has $\overline{p}=\overline{g_{\tau
,r}}[z_{r}] $ for some $r\in \mathbb{R}$, then there exists a
continuously differentiable function with weak second derivative in
$L^{1}(T) $, which is a solution of equation (\ref{Lx+g=p}) in
the sense of $L^{1}(T) $. Furthermore, if $p\in C(T)$ then the solution
is twice continuously differentiable.
\end{corollary}

One now searches for conditions that will permit the existence of some
$r\in\mathbb{R}$ such that $k_{r}=0$ (i.e. such that $x_{r}$ solves not
 only (\ref{Lx+grTILDA[x]=0}) but also (\ref{grBAR[x]=pBAR})).

\begin{lemma} \label{lm10}
In the context of the previous theorem, if $r\to r_{0}$ in $\mathbb{R
}$, then $x_{r}\rightharpoonup x_{r_{0}}$ in $\widetilde{\mathcal{H}}$.
\end{lemma}

\begin{proof}
By the contraction principle, the set $\{x_{r}:r\in \mathbb{R}\}
$ lies in a (weakly compact) ball in $\widetilde{\mathcal{H}}$ and so there
exists a subsequence $\{x_{r_{n}}\}_{n=1}^{\infty }$ such that
$r_{n}\to r_{0}$ and $x_{r_{n}}$ converges weakly in
$\widetilde{\mathcal{H}}$\ to an element $x_{0}$ as $n\to \infty $. Thus, by
Remark \ref{rmk6} both $\{x_{r_{n}}\}_{n=1}^{\infty }$ and
$\{x_{r_{n}}'\}_{n=1}^{\infty }$ are uniformly bounded
sequences which converge pointwise to $x_{0}$ and $x_{0}'$,
respectively. Hence, for all $q\in \widetilde{P}(T) $, one has
by Lebesgue's dominated convergence theorem
\begin{equation*}
\lim_{n\to \infty }\langle Lx_{r_{n}},q\rangle
_2=-\lim_{n\to \infty }\langle x_{r_{n}}'
+cx_{r_{n}},q'\rangle_2=-\langle x_{0}'+cx_{0},q'\rangle_2=\langle Lx_{0},q\rangle
_2
\end{equation*}
and
\begin{equation*}
\lim_{n\to \infty }\langle g_{\tau ,r_{n}}[x_{r_{n}}
] ,q\rangle_2=\langle g_{\tau ,r_{0}}[x_{0}]
,q\rangle_2.
\end{equation*}
Since
$\langle Lx_{r_{n}}+g_{\tau ,r_{n}}[x_{r_{n}}],q\rangle_2=0$
for all $n\in \mathbb{N}$, then in the limit as $n\to \infty $ one
obtains
\begin{equation*}
\langle Lx_{0}+g_{\tau ,r_{0}}[x_{0}] ,q\rangle_2=0
\end{equation*}
for all $q\in \widetilde{P}(T) $. By uniqueness, $x=x_{r_{0}}$
is the only solution in $\widetilde{\mathcal{H}}$ of
\begin{equation*}
\langle Lx+g_{\tau ,r_{0}}[x] ,q\rangle_2=0
\end{equation*}
and so one has $x_{0}=x_{r_{0}}$.
\end{proof}

The intermediate value theorem can now be applied to justify the
existence of a solution of equation (\ref{grBAR[x]=pBAR}) (i.e.
the existence of $r\in \mathbb{R}$ for which $k_{r}=0$). Thus, one
has the following corollary to Theorem \ref{thm7}.

\begin{corollary} \label{coro11}
In the context of the theorem, equation (\ref{Lx+g=p}) admits a continuously
differentiable solution with weak second derivative in $L^{1}(T)$ if
and only if
\begin{equation}
\inf_{r\in \mathbb{R}}\overline{g}(t,\varphi +x_{r}+r,\varphi ' +x_{r}')
\leq \overline{p}\leq \sup_{r\in \mathbb{R}}
\overline{g}(t,\varphi +x_{r}+r,\varphi '+x_{r}') .
\label{infgrBAR[xr]<pBAR<supgrBAR[xr]}
\end{equation}
Furthermore, if $p$ is continuous, then the solution is twice continuously
differentiable.
\end{corollary}

The bounds in (\ref{infgrBAR[xr]<pBAR<supgrBAR[xr]}) being difficult to
calculate in most cases, \textit{a priori} bounds for
$\overline{p}$ that imply condition (\ref{infgrBAR[xr]<pBAR<supgrBAR[xr]})
will now be obtained.

\subsection{\textit{A priori} bounds for $\overline{p}$}

By (\ref{|g(t,x2,y2)-g(t,x1,y1)|<A|x2-x1|+B|y2-y1|}) one has
\begin{align*}
|\overline{g_{\tau ,r}}[x_{r}] -\overline{g_{\tau ,r}}[G_{\tau ,r}^{n}(x) ] |
&\leq \frac{1}{T}\int_{[0,T] }|g_{\tau ,r}[x_{r}] (t)
 -g_{\tau ,r}[G_{\tau ,r}^{n}(x) ] (t) |\,dt \\
&\leq \frac{1}{T}\int_{[0,T] }A|x_{r}(t) -[G_{\tau ,r}^{n}(x) ] (t) |\,dt \\
&\quad +\frac{1}{T}\int_{[0,T] }B|x_{r}'(t) -[G_{\tau ,r}^{n}(x) ] '(t) |\,dt \\
&\leq \sqrt{A^2+B^2}\|x_{r}-G_{\tau ,r}^{n}(x)\|_H \\
&\leq \sqrt{A^2+B^2}\sum_{k=n}^{\infty }\|G_{\tau,r}^{k+1}(x) -G_{\tau ,r}^{k}(x) \|_H \\
&\leq \sqrt{A^2+B^2}\|G_{\tau ,r}(x) -x\|_H\sum_{k=n}^{\infty }
 (\frac{\beta }{\omega })^{k} \\
&=\frac{(\beta /\omega )^{n}}{1-\beta /\omega }\sqrt{A^2+B^2}
\|G_{\tau ,r}(x) -x\|_H\,,
\end{align*}
where
\begin{equation}
\beta =\sigma \sqrt{A^2+B^2}=\sqrt{\frac{(A^2+B^2)
(\omega^2+1) }{\omega^2+\min \{1,c^2\}}}\,.
\label{beta²=(A²+B²)(w²+1)/(w²+min(1,c²))}
\end{equation}
Hence (\ref{infgrBAR[xr]<pBAR<supgrBAR[xr]}) holds whenever, for an $x\in
\widetilde{\mathcal{H}}$ and some $n\in \mathbb{N}$, one has
\begin{align*}
&\inf_{r\in \mathbb{R}}\overline{g_{\tau ,r}}[G_{\tau ,r}^{n}(
x) ] +\lambda_{n}\|G_{\tau ,r}(x)-x\|_H \\
&\leq \overline{p} \\
&\leq \sup_{r\in \mathbb{R}}\overline{g_{\tau ,r}}[G_{\tau
,r}^{n}(x) ] -\lambda_{n}\|G_{\tau ,r}(x) -x\|_H\,,
\end{align*}
where
\begin{equation*}
\lambda_{n}=\frac{(\beta /\omega )^{n}}{1-\beta /\omega }\sqrt{
(A^2+B^2) }\,.
\end{equation*}
For $x=-\varphi $ one has $G_{\tau ,r}(-\varphi ) =0$ and so
(\ref{infgrBAR[xr]<pBAR<supgrBAR[xr]}) holds whenever, for some
$n\in \mathbb{N}$, one has
\begin{equation}
\inf_{r\in \mathbb{R}}\overline{g_{\tau ,r}}[G_{\tau ,r}^{n}(
-\varphi ) ] +\lambda_{n}\|\varphi \|
_H\leq \overline{p}\leq \sup_{r\in \mathbb{R}}\overline{g_{\tau ,r}}[
G_{\tau ,r}^{n}(-\varphi ) ] -\lambda_{n}\|\varphi \|_H\,,  \label{aprioriBOUND}
\end{equation}
where
\begin{equation*}
\overline{g_{\tau ,r}}[G_{\tau ,r}^{n}(-\varphi ) ] =
\frac{1}{T}\int_{[0,T] }g_{\tau ,r}[G_{\tau ,r}^{n}(-\varphi ) ] \,dt\,.
\end{equation*}
The following statement subsumes what has been proved for the case $n=1$.

\begin{theorem} \label{thm12}
Let $p:\mathbb{R}\to \mathbb{R}$ with mean $\overline{p}$ be a $T$-periodic
function which is Lebesgue integrable on $[0,T] $ and, for a
given $c\in \mathbb{R}$, let $\varphi $ be the continuously differentiable
solution of mean zero of the linear equation
$x''+cx'=p-\overline{p}$.
Also let $A,B\in [0,\infty [$ be such that
\begin{equation*}
\beta =\sqrt{\frac{(A^2+B^2) (\omega^2+1) }{
(\omega^2+\min \{1,c^2\}) }}<\omega\,,
\end{equation*}
where $\omega =2\pi /T$ and let $g:\mathbb{R}^{3}\mathbb{\to R}$ be
a continuous function for which $g(t,x,y) $ is $T$-periodic in
$t\in \mathbb{R}$ for all $(x,y) \in \mathbb{R}^2$ and such
that
\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}|
\end{equation*}
for all $t\in \mathbb{R}$ and all $(x_{1},y_{1}) ,(x_2,y_2) \in \mathbb{R}^2$.
If
\begin{equation*}
\inf_{r\in \mathbb{R}}[\overline{g}(t,\varphi +r,\varphi')
+\lambda \|\varphi \|_H] \leq \overline{p}\leq \sup_{r\in \mathbb{R}}
[\overline{g}(t,\varphi+r,\varphi ') -\lambda \|\varphi \|_H]\,,
\end{equation*}
where
\begin{gather*}
\overline{g}(t,\varphi +r,\varphi ') =\frac{1}{T}\int_{
[0,T] }g(t,\varphi (t) +r,\varphi'(t) ) \,dt\,, \\
\|\varphi \|_H=[\frac{1}{T}\int_{[0,T]}|\varphi (t) +\varphi '(t)
|^2\,dt]^{1/2}\,,\\
\lambda =\frac{\beta }{\omega -\beta }\sqrt{(A^2+B^2) }\,.
\end{gather*}
Then the Duffing equation
\begin{equation*}
x''(t) +cx'(t) +g(t-\tau ,x(t-\tau ) ,x'(t-\tau ) )=p(t)
\end{equation*}
with delay $\tau \in \mathbb{R}$ admits a continuously differentiable $T$
-periodic solution with weak second derivative which is Lebesgue integrable
on $[0,T] $. Furthermore, if $p$ is continuous, then the
solution is twice continuously differentiable.
\end{theorem}

\begin{example} \label{ex13} \rm
For $\alpha \in \mathbb{R}$, the Duffing equation with delay $\tau \in
\mathbb{R}$
\begin{equation*}
x''(t) +\frac{1}{3}\cos^2(t-\tau )
\ln (1+x^2(t-\tau ) +(x')^2(t-\tau )) =\alpha +\sin t
\end{equation*}
is such that $c=0$, $T=2\pi $ (and so $\omega =1$),
$\varphi (t)=-\sin t$ and
\begin{equation*}
g(t,x,y) =\frac{1}{3}\cos^2(t) \ln (1+x^2+y^2) .
\end{equation*}
Since
\begin{equation*}
|\ln (1+x_2^2+y_2^2) -\ln (1+x_{1}^2+y_{1}^2) |\leq |x_2-x_{1}|+|
y_2-y_{1}|
\end{equation*}
for all $(x_{1},y_{1}) ,(x_2,y_2) \in \mathbb{R}^2$, then
\begin{equation*}
|g(t,x_2,y_2) -g(t,x_{1},y_{1}) |\leq \frac{1}{3}[|x_2-x_{1}|+|y_2-y_{1}|
]
\end{equation*}
and so one has (\ref{|g(t,x2,y2)-g(t,x1,y1)|<A|x2-x1|+B|y2-y1|}) for
$A=B=\frac{1}{3}$. Furthermore $\beta =2/3<1$, $\|\varphi \|_H=1$,
$\lambda =2\sqrt{2}/3$,
\begin{align*}
\inf_{r\in \mathbb{R}}\overline{g}(t,\varphi +r,\varphi')
&=\inf_{r\in \mathbb{R}}\frac{1}{6\pi }\int_{[0,2\pi ]
}(\cos^2t) \ln (1+(r-\sin t)^2+\cos^2t) \,dt \\
&\leq \inf_{r\in \mathbb{R}}\frac{1}{6\pi }\int_{[0,2\pi ]
}(\cos^2t) \ln (1+(|r|+|\sin t|)^2+\cos^2t) \,dt \\
&=\frac{1}{6\pi }\ln 2\int_{[0,2\pi ] }\cos^2t\,dt \\
&=\frac{1}{6}\ln 2
\end{align*}
and
\begin{equation*}
\sup_{r\in \mathbb{R}}\overline{g}(t,\varphi +r,\varphi')
=\sup_{r\in \mathbb{R}}\frac{1}{6\pi }\int_{[0,2\pi ]
}\ln (1+(r-\sin t)^2+\cos^2t) =\infty .
\end{equation*}
Hence, by the previous theorem, the Duffing equation admits a $2\pi $
-periodic solution whenever
\begin{equation*}
\frac{1}{6}\ln 2+\frac{2\sqrt{2}}{3}\leq \alpha <\infty .
\end{equation*}
\end{example}

The example above does not fulfill condition (\ref{sign(x)(g(x)-pbar)>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)|<A|x2-x1|+B|y2-y1|})
becomes
\begin{equation}
|g(t,x_2) -g(t,x_{1}) |\leq
A|x_2-x_{1}| \label{|g(t,x2)-g(t,x1)|<A|x2-x1|}
\end{equation}
for suitable $A\geq 0$ and all $t,x_{1,}x_2\in \mathbb{R}$, and so (\ref
{sup|g(t,x,y)|<A|x|+B|y|+C}) becomes
\begin{equation}
\sup \{|g(t,x) |:t\in \mathbb{R}\}\leq A|x|+C  \label{sup(|g(t,x)|:tinR)<A|x|+C}
\end{equation}
for some $C\geq 0$ and all $x\in \mathbb{R}$., One can take
$C=\sup \{|g(t,0) |:t\in \mathbb{R}\}$, for
example.

\subsection{Contraction principle on $\widetilde{H}$}

By Proposition \ref{prop4}, one has $G_{\tau ,r}:\widetilde{H}\to
\widetilde{ \mathcal{H}}$ $\subset \widetilde{H}$ and so
$x=x_{r}\in \widetilde{H}$ is a solution of
(\ref{Lx+grTILDA[x]=0}) in the sense of $\widetilde{L^2}( T) $ if
and only if it is a fixed point of $G_{\tau ,r}$ on $
\widetilde{H}$. Thus, the following analog of Theorem \ref{thm7}
for the case $ g=g(t,x) $ does not require the space
$\widetilde{\mathcal{H}}$ for its proof. The less restrictive
space $\widetilde{H}$ is sufficient and this results in a somewhat
different (and more useful) inequality than (\ref
{A²+B²<w²(w²+min(1,c²))/(w²+1)}).

\begin{theorem} \label{thm14}
Let $g:\mathbb{R}^2\to \mathbb{R}$ be a continuous function such
that $g(t,x) $ is $T$-periodic in $t\in \mathbb{R}$ for all
$x\in \mathbb{R}$ and let $A\geq 0\ $and $c\in \mathbb{R}$ be such that
\begin{equation}
A^2<\omega^2\big(\frac{\omega^2+c^2}{\omega^2+1}\big)
\label{A²<w²(w²+c²)/(w²+1)}
\end{equation}
holds and (\ref{|g(t,x2)-g(t,x1)|<A|x2-x1|}) is satisfied for all
$t,x_{1},x_2\in \mathbb{R}$. Then, given $p\in L^{1}(T) $ and
$r\in \mathbb{R}$, there exists a unique continuously differentiable
$T$-periodic function $x_{r}$ of mean zero, with weak second derivative
$x_{r}''\in L^2(T) $, which is a solution of
equation (\ref{Lx+grTILDA[x]=0}) in the sense of $L^2(T) $.
Furthermore,
\begin{equation*}
x_{r}=\lim_{n\to \infty }G_{\tau ,r}^{n}(x)
\end{equation*}
in $L^2(T) $ for all $x\in L^2(T) $, and if
$p\in C(T) $ then $x_{r}\in C^2(T) $.
\end{theorem}

\begin{proof}
Proceeding as in the proof of Theorem \ref{thm7}, one obtains for
all $x,y\in \widetilde{H}$,
\begin{equation*}
\|G_{\tau ,r}(y) -G_{\tau ,r}(x) \|_H\leq \beta '\|y-x\|_H
\end{equation*}
where
\begin{equation*}
\beta '=\sigma 'A=\frac{A}{\omega }\sqrt{\frac{\omega^2+1}{\omega^2+c^2}}<1\,.
\end{equation*}
Hence $G_{\tau ,r}$ is a contraction on $\widetilde{H}$ and so admits a
unique fixed point $x_{r}\in \widetilde{H}$ given by the successive
approximations of Banach and Picard.
\end{proof}

As was done for Corollary \ref{coro8}, one can also reformulate
the result above in terms of $z_{r}=\varphi +x_{r}+r$ where
$\varphi $ is the unique $T$ -periodic solution of mean zero of
(\ref{Lx=pTILDA}) with $\varphi''\in L^{1}(T)$.

\begin{corollary} \label{coro15}
Let $c\in \mathbb{R}$ and $A\geq 0$ be such that (\ref
{|g(t,x2)-g(t,x1)|<A|x2-x1|}) and (\ref{A²<w²(w²+c²)/(w²+1)}) hold and let
$g:\mathbb{R}^2\to \mathbb{R}$ be a continuous function such that
$g(t,x) $ is $T$-periodic in $t\in \mathbb{R}$ for all $x\in
\mathbb{R}$. Then, given $r\in \mathbb{R}$ and $p\in L^{1}(T) $
of mean $\overline{p}$, there exists $z_{r}\in C^{1}(T) $ of
mean $r$ with weak second derivative $z_{r}''\in L^{1}(T) $ such that,
for some $k_{r}\in \mathbb{R}$, $x=z_{r}$ is a
solution in the sense of $L^{1}(T) $ of the Duffing equation
\begin{equation*}
x''(t) +cx'(t) +g(
t-\tau ,x(t-\tau ) ) =p(t) +k_{r}
\end{equation*}
with delay $\tau \in \mathbb{R}$. Furthermore, if $p\in C(T) $
then $z_{r}\in C^2(T) $.
\end{corollary}

The constant $k_{r}$ is again given by (\ref{k=mean[g(t,zr(t),z'r(t))-p(t)]}).
 The following lemma permits one to deduce, under condition
(\ref{infgrBAR[xr]<pBAR<supgrBAR[xr]}), the existence of some $r\in \mathbb{R}$
such that $k_{r}=0$ (i.e. such that $x_{r}$ is also a solution of
(\ref{grBAR[x]=pBAR})).

\begin{lemma} \label{lm16}
In the context of the previous theorem, if $r\to r_{0}$ in
$\mathbb{R}$ then $x_{r}\rightharpoonup x_{r_{0}}$ in $\widetilde{H}$.
\end{lemma}

The proof is like that of Lemma \ref{lm10}. The intermediate value
theorem now yields the following corollary to Theorem \ref{thm12}.

\begin{corollary} \label{coro17}
In the context of the previous theorem, equation (\ref{Lx+g=p}) admits a
continuously differentiable solution with weak second derivative in
$L^{1}(T) $ if and only if (\ref{infgrBAR[xr]<pBAR<supgrBAR[xr]})
is satisfied. Furthermore, if $p$ is continuous, then the solution is twice
continuously differentiable.
\end{corollary}

Clearly condition (\ref{infgrBAR[xr]<pBAR<supgrBAR[xr]}) holds whenever one
has
$\mathop{\rm sgn}(x) (g(t,x) -\overline{p}) >0$
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]<pBAR<supgrBAR[xr]}) holds whenever
\begin{equation*}
\inf_{r\in \mathbb{R}}\overline{g_{\tau ,r}}(G_{\tau ,r}^{n}(x) )
+\lambda_{n}'\|G_{\tau ,r}(x) -x\|_2\leq \overline{p}\leq \sup_{r\in \mathbb{R}}
\overline{g_{\tau ,r}}(G_{\tau ,r}^{n}(x) ) -\lambda_{n}'\|G_{\tau ,r}(x) -x\|_2
\end{equation*}
for some $x\in \widetilde{H}$ and an $n\in \mathbb{N}$, where
\begin{equation*}
\lambda_{n}'=\frac{A(\beta')^{n}}{1-\beta '}\,.
\end{equation*}
The case $x=-\varphi $ and $n=1$ yields the following:

\begin{theorem} \label{thm18}
Let $p:\mathbb{R\to R}$ with mean $\overline{p}$ be a $T$-periodic
function which is Lebesgue integrable on $[0,T] $ and $\varphi $
be the continuously differentiable solution of mean zero of the linear
equation
$x''+cx'=p-\overline{p}$
for a given $c\in \mathbb{R}$. Also let $A\in [0,\infty [$ be
such that
\begin{equation*}
\beta '=\frac{A}{\omega }\sqrt{\frac{\omega^2+1}{\omega^2+c^2}}<1\,,
\end{equation*}
where $\omega =2\pi /T$ and let $g:\mathbb{R}^2\mathbb{\to R}$ be
a continuous function for which $g(t,x) $ is $T$-periodic in $
t\in \mathbb{R}$ for all $x\in \mathbb{R}$ and such that
\begin{equation*}
|g(t,x_2) -g(t,x_{1}) |\leq
A|x_2-x_{1}|
\end{equation*}
for all $t,x_{1},x_2\in \mathbb{R}$. If
\begin{equation*}
\inf_{r\in \mathbb{R}}[\overline{g}(t,\varphi +r) +\lambda
'\|\varphi \|_2] \leq \overline{p}\leq
\sup_{r\in \mathbb{R}}[\overline{g}(t,\varphi +r) -\lambda'\|\varphi \|_2]\,,
\end{equation*}
where
\begin{equation*}
\|\varphi \|_2=[\frac{1}{T}\int_{[0,T]}|\varphi (t) |^2\,dt]^{\frac{1}{2}}
\quad\mbox{and}\quad
\lambda '=\frac{A\beta '}{1-\beta '}
\end{equation*}
then the equation
\begin{equation*}
x''(t) +cx'(t) +g(t-\tau ,x(t-\tau ) ) =p(t)
\end{equation*}
with delay $\tau \in \mathbb{R}$ admits a continuously differentiable
$T$-periodic solution with weak second derivative which is Lebesgue integrable
on $[0,T] $. Furthermore, if $p$ is continuous, then the
solution is twice continuously differentiable.
\end{theorem}

To show that the results of this section can be applied to cases not covered
by section~3.2, consider the following example where $\beta >1$ 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)|<A|x2-x1|}) for $A=\sqrt{2}$.
Furthermore $\beta '=2/\sqrt{5}<1$ (and $\beta =\sqrt{2}>1$),
$\|\varphi \|_2=1/\sqrt{2}$, $\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)<A|x|+C}) then
reduces to
\begin{equation}
\sup \{|g(t,x) |:t\in \mathbb{R}\}\leq C  \label{sup(|g(t,x)|:tinR)<C}
\end{equation}
for some $C\geq 0$ and all $x\in \mathbb{R}$.

\subsection{Contraction principle on $\widetilde{L^2}(T) $}

Suppose that one has simultaneously conditions
(\ref{|g(t,x2)-g(t,x1)|<A|x2-x1|}) and
(\ref{sup(|g(t,x)|:tinR)<C}). Since $g$ is bounded and continuous,
one has $g_{\tau ,r}[x] \in L^2( T) $ for all $x\in L^2(T) $ and
so (\ref{Gtau,r}) defines $G_{\tau ,r}$ as a function from
$\widetilde{L^2}(T) $ to $\widetilde{\mathcal{H}}\subset
\widetilde{L^2}(T) $. Hence a fixed point of $G_{\tau ,r}$ on
$\widetilde{L^2}(T) $ lies in $\widetilde{\mathcal{H}}$.
Proceeding as in the proof of Theorem \ref{thm7}, one obtains for
all $x,y\in \widetilde{L^2}(T) $,
\begin{equation*}
\|G_{\tau ,r}(y) -G_{\tau ,r}(x)
\|_2^2\leq (\beta'')^{2}\|y-x\|
_2^2
\end{equation*}
where
\begin{equation*}
\beta ''=\frac{A}{\omega \sqrt{\omega^2+c^2}}\,.
\end{equation*}
Hence, when $\beta ''<1$, $G_{\tau ,r}$ is a contraction on
$\widetilde{L^2}(T) $ and so admits a unique fixed point
$x_{r}\in \widetilde{\mathcal{H}}$ given by the successive approximations of
Banach and Picard. This proves the following statement.

\begin{theorem} \label{thm20}
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 $x\in \mathbb{R}$ and let $A\geq 0\ $and $c\in \mathbb{R}$ be such
that
\begin{equation}
A<\omega \sqrt{\omega^2+c^2}  \label{A<wROOT(w²+c²)}
\end{equation}
holds and (\ref{|g(t,x2)-g(t,x1)|<A|x2-x1|}) is satisfied for all $
t,x_{1},x_2\in \mathbb{R}$. Then, given $p\in L^{1}(T) $ and $
r\in \mathbb{R}$, there exists a unique continuously differentiable $T$
-periodic function $x_{r}$ of mean zero, with weak second derivative
$x_{r}''\in L^2(T) $, which is a solution of
equation (\ref{Lx+grTILDA[x]=0}) in the sense of $L^2(T) $.
Furthermore,
\begin{equation*}
x_{r}=\lim_{n\to \infty }G_{\tau ,r}^{n}(x)
\end{equation*}
in $L^2(T) $ for all $x\in L^2(T) $, and if $
p\in C(T) $ then $x_{r}\in C^2(T) $.
\end{theorem}

Condition (\ref{A<wROOT(w²+c²)}) is an improvement over (\ref
{A²<w²(w²+c²)/(w²+1)}) which, in turn, is an improvement over (\ref
{A²+B²<w²(w²+min(1,c²))/(w²+1)}) with $B=0$. There are important cases where
$g$ satisfies both the Lipschitz condition (\ref{|g(t,x2)-g(t,x1)|<A|x2-x1|}
) and the boundedness condition (\ref{sup(|g(t,x)|:tinR)<C}). For example,
the nonconservative pendulum equation
\begin{equation}
x''+cx'+a\sin x=p  \label{x"+cx'+asinx=p}
\end{equation}
with $a,c\in \mathbb{R}$ and forcing $p\in L^{1}(T) $ is such
that these two conditions hold for $A=C=|a|$. Hence (
\ref{A<wROOT(w²+c²)}) becomes
\begin{equation*}
|a|<\omega \sqrt{\omega^2+c^2}
\end{equation*}
and so in this manner one obtains the result stated in \cite{TARANTELLO89}\
(and proved in \cite{FOURNIERmahwin85} and \cite{MAHWINwillem89}) to the
effect that the equation
\begin{equation*}
x''+a\sin x=p(t) +(a\overline{\sin }x-
\overline{p})
\end{equation*}
admits a twice continuously differentiable $T$-periodic solution of mean
zero whenever (\ref{A<wROOT(w²+c²)}) holds.

The intermediate value theorem now yields the following statement.

\begin{corollary} \label{coro21}
In the context of the previous theorem, equation (\ref{Lx+g=p}) admits a
continuously differentiable solution with weak second derivative in
$L^{1}(T) $ if and only if (\ref{infgrBAR[xr]<pBAR<supgrBAR[xr]})
is satisfied. Furthermore, if $p$ is continuous, then the solution is twice
continuously differentiable.
\end{corollary}

The pendulum equation (\ref{x"+cx'+asinx=p}) is such that the function
$g(t,x) =a\sin x$ is $2\pi $-periodic in $x$ for all $t\in \mathbb{R}$.
This fact yields results for (\ref{x"+cx'+asinx=p}) that are
stronger than the above corollary and that have been given in
\cite{TARANTELLO89} (and generalized in \cite{BELLEYsaadidrissi2001}). Moreover,
we point out that when (\ref{A<wROOT(w²+c²)}) holds the techniques employed
in \cite{TARANTELLO89} yield the added result that the map $r\to x_{r}$ from
$\mathbb{R}$ to $C^2(T) $ is analytic. On the
other hand, \cite{ORTEGA2000} provides an upper bound, as a function of
$\overline{p}$, for the number of $T$-periodic solutions of
(\ref{x"+cx'+asinx=p}) when $aT^2<18\sqrt{3}$, $c=0$ and
$p\in L^{1}(T) $. No condition other than
(\ref{infgrBAR[xr]<pBAR<supgrBAR[xr]})
is imposed on $p$. Clearly the condition $aT^2<18\sqrt{3}$ in \cite
{ORTEGA2000} is more restrictive than the inequality $aT^2<4\pi^2$
imposed in \cite{BELLEYsaadidrissi2001} and \cite{TARANTELLO89} for the case
$c=0$. Similar results, applicable to the conservative pendulum equation,
are obtained in \cite{SERRAtarallo98} for equation (\ref{Lx+g=p}) with
$\tau=0$, $c=0$ and $g(t,x) =\frac{\partial }{\partial x}V(t,x) $ where
the potential $V(t,x) $ must satisfy certain
subquadraticity conditions. No similar result is obtained here. On the other
hand, the elementary contraction principle allows one in the following
section to replace condition (\ref{infgrBAR[xr]<pBAR<supgrBAR[xr]}) by more
manageable \textit{a priori }bounds.

\begin{example} \label{ex22} \rm
Given a continuous $2\pi $-periodic function $p:\mathbb{R\to R}$,
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 ) =p(t)
\end{equation*}
is such that $c=1$, $T=2\pi $ (and so $\omega =1$),
$g(t,x) =\sqrt{3/2}\cos t\sin x$, and so one has
(\ref{|g(t,x2)-g(t,x1)|<A|x2-x1|})
and (\ref{A<wROOT(w²+c²)}) for $A=\sqrt{3/2}$. One notes that
$\beta''=\sqrt{3}/2<1$ and $\beta '=\sqrt{3/2}>1$, and so
(\ref{A<wROOT(w²+c²)}) is satisfied while (\ref{A²<w²(w²+c²)/(w²+1)}) is not.
Thus, by the previous corollary, the delay equation admits a twice
continuously differentiable $2\pi $-periodic solution if and only if (\ref
{infgrBAR[xr]<pBAR<supgrBAR[xr]}) holds.
\end{example}

\subsection{\textit{A priori bounds for }$\overline{p}$}

Proceeding as in section~4.2, one has
\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*}
for all $x\in \widetilde{L^2}(T) $ and all $n\in \mathbb{N}$.
Hence condition (\ref{infgrBAR[xr]<pBAR<supgrBAR[xr]}) holds whenever
\begin{equation}
\inf_{r\in \mathbb{R}}\overline{g_{\tau ,r}}[G_{\tau ,r}^{n}(x) ]
+\lambda_{n}''\|G_{\tau ,r}(x) -x\|_2\leq \overline{p}\leq \sup_{r\in \mathbb{R}}
\overline{g_{\tau ,r}}[G_{\tau ,r}^{n}(x) ] -\lambda_{n}''\|G_{\tau ,r}(x) -x\|_2
\label{aprioriBOUNDwithLAMDA''}
\end{equation}
for some $x\in \widetilde{L^2}(T) $ and an $n\in \mathbb{N}$,
where
\begin{equation*}
\lambda_{n}''=\frac{A(\beta'')^{n}}{1-\beta''}.
\end{equation*}
Thus the case $x=-\varphi $ and $n=1$ yields the following result.

\begin{theorem}
Let $p:\mathbb{R\to R}$ with mean $\overline{p}$ be a $T$-periodic
function which is Lebesgue integrable on $[0,T] $ and $\varphi $
be the continuously differentiable solution of mean zero of the linear
equation
$x''+cx'=p-\overline{p}$
for a given $c\in \mathbb{R}$. Also let $A\in [0,\infty [$ be
such that
\begin{equation*}
\beta ''=\frac{A}{\omega \sqrt{\omega^2+c^2}}<1
\end{equation*}
where $\omega =2\pi /T$ and let $g:\mathbb{R}^2\mathbb{\to R}$ be
a bounded continuous function for which $g(t,x) $ is $T$
-periodic in $t\in \mathbb{R}$ for all $x\in \mathbb{R}$ and such that
\begin{equation*}
|g(t,x_2) -g(t,x_2) |\leq A |x_2-x_{1}|
\end{equation*}
for all $t,x_{1},x_2\in \mathbb{R}$. If
\begin{equation*}
\inf_{r\in \mathbb{R}}\overline{g}(t,\varphi +r) +\lambda
''\|\varphi \|_2\leq \overline{p}\leq
\sup_{r\in \mathbb{R}}\overline{g}(t,\varphi +r) -\lambda
''\|\varphi \|_2
\end{equation*}
where
\begin{equation*}
\|\varphi \|_2=[\frac{1}{T}\int_{[0,T]}|\varphi (t) |^2\,dt]^{1/2}
\quad\text{and}\quad
\lambda ''=\frac{A\beta ''}{1-\beta''}
\end{equation*}
then the equation
\begin{equation*}
x''(t) +cx'(t) +g(t-\tau ,x(t-\tau ) ) =p(t)
\end{equation*}
with delay $\tau \in \mathbb{R}$ admits a continuously differentiable $T$
-periodic solution with weak second derivative which is Lebesgue integrable
on $[0,T] $. Furthermore, if $p$ is continuous, then the
solution is twice continuously differentiable.
\end{theorem}

The above theorem can now be applied to the forced pendulum equation.

\begin{example} \label{ex24}
Given $T>0$ 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 $n=1$.

\begin{thebibliography}{00}

\bibitem{ALONSO97} J. M. Alonso, ``Nonexistence of periodic
solutions for a damped pendulum equation" , Diff. Int. Eq.
10 (1997), no. 6, 1141-1148.

\bibitem{BELLEYsaadidrissi2001} J.-M. Belley and K. Saadi Drissi,
``Existence of periodic solutions to the forced Josephson
equation" , Acad. Roy.\ Belg. Bull. Cl. Sci. (6) 12 (2001),
no. 7-12, 209-224.

\bibitem{BELLEYsaadidrissi2003} J.-M. Belley and K. Saadi Drissi,
``Almost periodic solutions to Josephson's
equation" , Nonlinearity 16 (2003), no. 1, 35-47.

\bibitem{CESARIkannan82} L. Cesari and R. Kannan, ``Periodic solutions in
the large of Li\'{e}nard systems with forced fording term'', Boll. Un Mat.
Ital., A(6) 1(1982), 217-224.

\bibitem{CHENyuYUAN02} H. B. Chen, L. Yu and X. Y. Yuan, ``Existence and
uniqueness of periodic solution of Duffing's equation'', J.~Math. Res.
Exposition 22(2002), no. 4, 615-620.

\bibitem{FONDAlupo89} A. Fonda and D. Lupo, ``Periodic solutions of second
order ordinary differential equations'', Boll. Un Mat. Ital., A(7) 3(1989),
291-299.

\bibitem{FOURNIERmahwin85} G. Fournier and J. Mahwin, ``On
periodic solutions of forced pendulum-like equations" , J.
Diff. Eq. 60 (1985), 381-395.

\bibitem{GOSSEZomari90} J. P. Gossez and P. Omari, ``Nonresonance with
respect to the Fu\v{c}ik spectrum for periodic solutions of second order
ordinary differential equations'', Nonlinear Anal.(TMA), 14(1990), 1079-1104.

\bibitem{HUANGxiang94} X. K. Huang and Z. G. Xiang, ``$2\pi $-periodic
solutions of Duffing equations with a deviating argument'', Chinese Sci.
Bull., 39(1994), 201-203.

\bibitem{MAwangYU} S. W. Ma, Z.C. Wang and J. S. Yu, \textquotedblleft
Periodic Solutions of Duffing Equations with Delay" ,
Differential Equations and dynamical systems, 8 (2000), no. 3/4, 241-255.

\bibitem{mahwin72} J. Mawhin, ``Equivalence theorems for nonlinear operator
equations and coincidence degree theory for some mappings in locally convex
topological vector spaces'', J. Diff. Eqns.\textit{,} 12(1972), 612-636.

\bibitem{mahwin79} J. Mawhin, ``Topological Degree Methods in Nonlinear
Boundary Value Problems'', CBMS. Vol. 40, Amer. Math. Soc., Providence, RI,
1979.

\bibitem{MAHWINward83} J. Mahwin and J. R. Ward.Jr., ``Periodic solutions of
some forced Li\'{e}nard differential equations at resonance'', Arch. Math.,
41(1983), 337-351.

\bibitem{MAHWINwillem89} J. Mahwin and M. Willem, ``Critical Point Theory
and Hamiltonian Systems'', Springer-Verlag, New York, 1989.

\bibitem{OMARIzanolin87} P. Omari and F. Zanolin, ``On the
existence of periodic solutions of forced Li\'{e}nard differential
equations" , Nonlinear Anal. (TMA), 11(1987), 275-284.

\bibitem{ORTEGA2000} R. Ortega, ``Counting periodic
solutions of the forced pendulum equation" , Nonlinear
Anal. 42 (2000), 1055-1062.

\bibitem{SERRAtarallo98} E. Serra and M. Tarallo, ``A
reduction method for periodic solutions of second-order subquadratic
equations" , Adv. Diff. Eq. 3 (1998), 199-226.

\bibitem{TARANTELLO89} G. Tarantello, ``On the Number of
Solutions for the Forced Pendulum Equation" , J. Diff. Eq.
\textit{,} 80 (1989), 79-93.

\bibitem{ZHANGwangYU2000} Z. Q. Zhang, Z. C. Wang, and J. S. Yu, ``Periodic
solutions of neutral Duffing equations", Electron. J. Qual. Theory Differ.
Equ. 2000, No. 5, 14~pp. (electronic).
\end{thebibliography}

\end{document}