\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 124, pp. 1--17.\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-2004/124\hfil Anti-periodic solutions]
{Anti-periodic solutions for second order differential inclusions}

\author[J.-F. Couchouron,  R. Precup\hfil EJDE-2004/124\hfilneg]
{Jean-Fran\c{c}ois Couchouron,  Radu Precup} % in alphabetical order


\address{Jean-Fran\c{c}ois Couchouron \hfill\break
Universit\'{e} de Metz, Math\'{e}matiques INRIA Lorraine \\
Ile du Saulcy, 57045 Metz, France}
\email{Jean-Francois.Couchouron@loria.fr}

\address{Radu Precup \hfill\break
University Babe\c{s}-Bolyai \\
Faculty of Mathematics and Computer Science \\
3400 Cluj, Romania}
\email{r.precup@math.ubbcluj.ro}

\date{}
\thanks{Submitted May 10, 2004. Published October 18, 2004.}
\subjclass[2000]{45N05, 47J35, 34G25}
\keywords{Anti-periodic solution; nonlinear boundary-value problem;
\hfill\break\indent
 dissipative operator; multivalued mapping; fixed point}

\begin{abstract}
 In this paper, we extend the existence results presented in
 \cite{CKP} for $L^{p}$ spaces to operator inclusions of Hammerstein
 type in $W^{1,p}$ spaces. We also show an application of our
 results to anti-periodic boundary-value problems of second-order
 differential equations with nonlinearities depending on $u'$.
\end{abstract}

\maketitle
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\numberwithin{equation}{section}

\section{Introduction}

This paper concerns the second-order boundary-value problem
\begin{gather*}
-u''(t)\in Au(t)+f( t,u( t) ,u'( t) ) \quad\text{for a.e. }t\in [ 0,T] \\
u( 0) =-u( T) ,\quad u'( 0) =-u'( T) ,
\end{gather*}
where $0<T<\infty $, $A$ is an $m$-dissipative multivalued mapping in a
Hilbert space $E$ and $f:[ 0,T] \times E^{2}\to 2^{E}$.
However, in this section, and in Section 2, we shall assume generally
that $E$ is a Banach space.

A function $u\in C^{1}( [ 0,T] ;E) $ is said to be $T$-\textit{anti-periodic}
if $u( 0) =-u( T) $ and  $u'( 0) =-u'( T) $. Note that there
exists a close connection between the anti-periodic problem and the periodic
one. Indeed, if $u\in W^{2,p}( 0,T;E) $ $( 1\leq p<\infty)$ is a
$T$-anti-periodic solution of the inclusion
\[
-u''( t) \in Au( t) +f( t,u(t) ,u'( t) ) \quad\text{a.e. on }[ 0,T]
\]
and $A$, $f$ are odd in the following sense:
\[
A( -x) =-Ax \quad\text{and}\quad f( t,-x,-y) =-f( t,x,y) ,
\]
then the function
\[
\widetilde{u}( t) =\begin{cases}
u( t) , & 0\leq t\leq T \\
-u( t-T) , & T<t\leq 2T
\end{cases}
\]
belongs to $W^{2,p}( 0,2T;E) $, is $2T$-periodic, i.e.,
$\widetilde{u}( 0) =\widetilde{u}( 2T) $, $\widetilde{u}
'( 0) =\widetilde{u}'( 2T) $, and
solves the inclusion
\[
-\widetilde{u}''( t) \in A\widetilde{u}(t) +\widetilde{f}( t,\widetilde{u}( t)
,\widetilde{u}'( t) ) \quad\text{a.e. on }[0,2T]
\]
where
\[
\widetilde{f}( t,x,y) =\begin{cases}
f( t,x,y) , & 0\leq t\leq T \\
f( t-T,x,y) , & T<t\leq 2T.
\end{cases}
\]
The anti-periodic boundary value problem for various classes of
evolution equations has been considered by
Aftabizadeh-Aizicovici-Pavel \cite{AAP}, \cite{AAP2};
Aizicovici-Pavel \cite{AP}, Aizicovici-Pavel-Vrabie \cite{APV},
Cai-Pavel \cite{CPa}, Coron \cite{Coron}, Haraux \cite{H} and
Okochi \cite{O,O2}.

Let us denote by $|\cdot|$ the norm of $E$, by $|\cdot| _{p}$ the
usual norm of $L^{p}(0,T;E) $ and by $| \cdot| _{1,p}$ the norm of
$W^{1,p}(0,T;E) $, $| u| _{1,p}=\max \{| u| _{p},|u'| _{p}\}$. One
of the reasons of working with anti-periodic solutions is given by
the following proposition.

\begin{proposition} \label{prop1.1}
If $u\in W^{1,p}( 0,T;E) $ $( 1\leq p\leq \infty ) $
and $u( 0) =-u( T) $, then
\begin{equation}
| u( t) | \leq \frac{1}{2}T^{\frac{p-1}{p}}| u'| _{p},\;\;\;t\in [
0,T].  \label{eq1.4}
\end{equation}
\end{proposition}

\begin{proof}
Adding $u( t) =u( 0) +\int_{0}^{t}u'( s) ds$
and
$u( t) =u( T) -\int_{t}^{T}u'( s) ds$
we have
\[
2u( t) =\int_{0}^{t}u'( s)
ds-\int_{t}^{T}u'( s) ds.
\]
Hence
\[
2| u( t) | \leq \int_{0}^{t}| u'(
s) | ds+\int_{t}^{T}| u'( s) |
ds=\int_{0}^{T}| u'( s) | ds.
\]
Now H\"{o}lder's inequality gives (\ref{eq1.4}).
\end{proof}

Let us denote
\[
C_{a}^{1}=\{u\in C^{1}( [0,T];E) : u\text{ is $T$-anti-periodic}\}.
\]

In what follows for a subset $K\subset E$, by $P_{a}( K) $ and
$P_{kc}( K) $ we shall denote the family of all nonempty acyclic
subsets of $K$ and, respectively, the family of all nonempty compact convex
subsets of $K$.

Recall that a metric space $\Xi $ is said to be \textit{acyclic} if it has
the same homology as a single point space, and that $\Xi $ is called an
\textit{absolute neighborhood retract} (ANR for short) if for every metric
space $Z$ and closed set $A\subset Z$, every continuous map $f:A\to
\Xi $ has a continuous extension $\widehat{f}$ to some neighborhood of $A$.
Note that every compact convex subset of a normed space is an ANR and is
acyclic.

Our main abstract tools are: The Eilenberg-Montgomery fixed point
theorem \cite{em,hp}; a lemma of Petryshyn-Fitzpatrick \cite{pf};
and strong and weak compactness criteria in $L^{p}( 0,T;E) $ (see
\cite{guo} and \cite{diestel}), where $E$ is a general
(non-reflexive) Banach space.

\begin{theorem} \label{thm1.2}
Let $\Xi $ be acyclic and absolute neighborhood retract, $\Theta $ be a
compact metric space, $\Phi :\Xi \to P_{a}( \Theta ) $ be an upper
semicontinuous map and $\Gamma :\Theta \to \Xi $ be a continuous
single-valued map. Then the
map $\Gamma \Phi :\Xi \to 2^{\Xi }$ has a fixed point.
\end{theorem}

\begin{lemma} \label{lm1.3}
Let $X$ be a Fr\'{e}chet space, $D\subset X$ be closed convex and
$N:D\to 2^{X}$. Then for each $\Omega \subset D$ there exists a
closed convex set $K$, depending on $N$, $D$ and $\Omega $, with $\Omega
\subset K$ and $\overline{\mathop{\rm conv}}( \Omega \cup N(D\cap K) ) =K$.
\end{lemma}

\begin{theorem} \label{thm1.4}
Let   $p\in [ 1,\infty ] $.   Let   $M$
$\subset L^{p}( 0,T;E) $   be countable and suppose that
there exists a $\nu \in L^{p}( 0,T) $   with
  $| u( t) | \leq \nu ( t) $   a.e. on $[ 0,T] $   for all
$u\in M$.   Assume   $M\subset C( [ 0,T];E) $   if   $p=\infty $.   Then
$M$   is relatively compact in   $L^{p}( 0,T;E)
$   if and only if
\begin{itemize}
\item[(i)] $\sup_{u\in M}| \tau _{h}u-u| _{L^{p}(0,T-h;E) }\to 0$ as  $h\to 0$

\item[(ii)] $M( t) $   is relatively compact in
  $E$   for a.e. $t\in [ 0,T] $.
\end{itemize}
\end{theorem}

\begin{theorem} \label{thm1.5}
Let   $p\in [ 1,\infty ] $.   Let
$M\subset L^{p}( 0,T;E) $   be countable and suppose
there exists   $\nu \in L^{p}( 0,T) $   with
    $| u( t) | \leq \nu (t) $   a.e. on $[ 0,T] $   for all  $u\in M$.
If   $M( t) $ is relatively compact in   $E$   for a.e.
$t\in[0,T]$,   then   $M$ is weakly relatively compact in $L^{p}(0,T;E) $.
\end{theorem}

Now, we recall the following definition: A map   $\psi :[
a,b] \times D\to 2^{Y}\setminus \{ \emptyset\}$,
  where   $D\subset X$  and $( X,|\cdot|_{X})$, $( Y,|\cdot| _{Y}) $   are two
Banach spaces, is said to be $(q,p) $\textit{-Carath\'{e}odory}
($1\leq q\leq \infty $,  $1\leq p\leq \infty $)   if
\begin{itemize}
  \item[(C1)] $\psi ( .,x) $   is strongly measurable for each   $x\in D$

\item[(C2)]   $\psi (t,.) $   is upper semicontinuous for a.e. $t\in [ a,b]$

\item[(C3)]  (a) if   $1\leq p<\infty $,   there
exists   $\nu \in L^{q}( a,b;\mathbb{R}_{+}) $
  and   $d\in \mathbb{R}_{+}$   such that
$|\psi ( t,x) | _{Y}\leq \nu ( t) +d|x| _{X}^{p}$   a.e. on $[ a,b] $,
for all   $x\in D$\\
 (b) if   $p=\infty$,  for each   $\rho >0$   there exists
 $\nu _{\rho }\in L^{q}( a,b;\mathbb{R}_{+}) $   such that
 $| \psi( t,x) | _{Y}\leq \nu _{\rho }( t) $ a.e. on $[ a,b] $,   for all
 $x\in D$   with   $| x| _{X}\leq \rho  $.
 \end{itemize}

\section{A General Existence Principle}

The aim of this section is to extend the general
existence principles given in \cite{CP} for inclusions in $L^{p}(
0,T;E) $, to inclusions in $W^{1,p}( 0,T;E) $. Here again
$E$ a Banach space with norm $|\cdot|$. This extension allows us to
consider boundary-value problems for second order differential inclusions
with $u'$ dependence perturbations and, by this, it complements the
theory from \cite{CK}, \cite{CKP} and \cite{CP}.

Let $p\in [ 1,\infty ] $ and $q\in \lbrack 1,\infty \lbrack $. Let $r\in ]1,\infty ]$ be the conjugate exponent of $q$, that is $1/q+1/r=1$. Let $g:[ 0,T] \times E^{2}\to 2^{E}$ and let
$G:W^{1,p}( 0,T;E) \to 2^{L^{q}( 0,T;E) }$ be
the Nemytskii set-valued operator associated to $g$, $p$ and $q$, given by
\begin{equation}
G( u) =\{w\in L^{q}( 0,T;E) : w( s) \in
g( s,u( s) ,u'( s) ) \text{ a.e.
on }[ 0,T] \}.  \label{eq1.6'}
\end{equation}
Also consider a single-valued nonlinear operator
\[
S:L^{q}( 0,T;E) \to W^{1,p}( 0,T;E) .
\]
We have the following existence principle for the operator inclusion
\begin{equation}
u\in SG( u) ,\quad u\in W^{1,p}( 0,T;E) .  \label{eq1.7}
\end{equation}

\begin{theorem} \label{thm2.1}
Let   $K$   be a closed convex subset of
$W^{1,p}( 0,T;E) $, $U$   a convex relatively open
subset of $K$   and $u_{0}\in U$.   Assume
\begin{itemize}
\item[(H1)]   $SG:\overline{U}\to P_{a}( K) $   has closed graph and
maps compact sets into relatively compact sets

\item[(H2)]  $M\subset \overline{U}$, $M$ closed, $M\subset
\overline{\mathop{\rm conv}} ( \{ u_{0}\} \cup SG( M) )$ implies
that $M$ is  compact

\item[(H3)]  $u\notin (1-\lambda )u_{0}+\lambda SG(u)$ for all
 $\lambda \in ]0,1[$  and $u\in \overline{U}\setminus U$.
\end{itemize}
Then \eqref{eq1.7} has a solution in $\overline{U}$.
\end{theorem}

\begin{proof}
Let $D=\overline{\mathop{\rm conv}}( \{ u_{0}\} \cup SG(\overline{U}) ) $.
 Clearly $u_{0}\in D\subset K$. Let $P:K\to \overline{U}$ be given by
$P( u) =u$ if $u\in \overline{U}$ and $P( u) =\overline{u}$ if
$u\in K\setminus \overline{U}$, where
$\overline{u}=( 1-\lambda ) u_{0}+\lambda u\in \overline{U}\setminus U$,
$\lambda \in ]0,1[$. Note $P$ is single-valued, continuous and maps closed
sets into closed sets. Let $\widetilde{N}:D\to P_{a}( K) $,
$\widetilde{N}(u) =SGP( u) $. It is easy to see that
$\widetilde{N}(D) \subset D$, the graph of $\widetilde{N}$ is closed and
$\widetilde{N}$ maps compact sets into relatively compact sets.
Let $D_{0}$ be a closed convex set with
$D_{0}=\overline{\mathop{\rm conv}}( \{ u_{0}\} \cup \widetilde{N}( D_{0}\cap D) ) $
whose existence is
guaranteed by Lemma \ref{lm1.3}. Since $\widetilde{N}( D) \subset D$ we
have $D_{0}\subset D$ and so
$D_{0}=\overline{\mathop{\rm conv}}( \{u_{0}\} \cup \widetilde{N}( D_{0}) ) $.
Using the definition of $P$, we obtain
\[
P( D_{0}) \subset \mathop{\rm conv}( \{u_{0}\} \cup D_{0})
=\overline{\mathop{\rm conv}}( \{u_{0}\} \cup \widetilde{N}( D_{0}) ) \\
=\overline{\mathop{\rm conv}}( \{ u_{0}\} \cup SG(P( D_{0}) ) ) .
\]
In addition, since $D_{0}$ is closed, $P( D_{0}) $ is also
closed. Now (H2) guarantees that $P( D_{0}) $ is compact. Since
$SG$ maps compact sets into relatively compact sets, we have that $\widetilde{
N}( D_{0}) $ is relatively compact. Then Mazur's Lemma guarantees
that $D_{0}$ is compact. Now apply the Eilenberg-Montgomery Theorem with
$\Xi =\Theta =D_{0}$, $\Phi =\widetilde{N}$ and $\Gamma =\,$identity of
$D_{0}$, to deduce the existence of a fixed point $u\in D_{0}$ of $\widetilde{
N}$. If $u\notin \overline{U}$, then $P( u) =( 1-\lambda
) u_{0}+\lambda u=( 1-\lambda ) u_{0}+\lambda SG(
P( u) ) $ for some $\lambda \in ]0,1[$. Since $P(
u) \in \overline{U}\setminus U$, this contradicts (H3). Thus $u\in
\overline{U}$, so $u=SG( u) $ and the proof is complete.
\end{proof}

\begin{remark} \label{rmk2.1} \rm
Additional regularity for the solutions of \eqref{eq1.7} depends on
the values of $S$. In particular if the values of $S$ are in $C_{a}^{1}$
then so are all solutions of \eqref{eq1.7}.
\end{remark}

In what follows $K$ will be a closed linear subspace of $W^{1,p}(0,T;E) $,
 $u_{0}=0$ and $U$ will be the open ball of $K$,
\[
U=\{u\in K:\| u\| <R\}
\]
with respect to an equivalent norm $\| .\| $ on $K$.
For $p\in [ 1,\infty ] $ denote
\[
\mu _{p}:=\sup \{\frac{| u| _{1,p}}{\| u\| }:u \in K,\,u\neq 0\},\quad
\mu _{0}:=\sup \{\frac{| u| _{\infty }}{\|u\| }: u\in K,\,u\neq 0\}.
\]
Note that $\mu _{p}$ and $\mu _{0}$ are finite because of the equivalence of
norms $\|\cdot\|$ and $|\cdot|_{1,p}$ on $K$ and the
continuously embedding of $W^{1,p}( 0,T;E)$ into $C( [0,T];E)$.

Now we give sufficient conditions on   $S$   and
  $g$   in order that the assumptions (H1)-(H2) be
satisfied.
\begin{itemize}
\item[(S1)] There exists a function   $k:[ 0,T]
^{2}\to \mathbb{R}_{+}$   with   $k(
t,.) \in L^{r}( 0,T) $ and a constant $L>0$ such that
\[
| S( w_{1}) ( t) -S( w_{2}) (
t) | \leq \int_{0}^{T}k( t,s) | w_{1}(
s) -w_{2}( s) | ds
\]
for a.e. $t\in [ 0,T] $, and
$| S( w_{1}) '-S( w_{2})'| _{p}\leq L| w_{1}-w_{2}| _{q}$
for all   $w_{1},w_{2}\in L^{q}( 0,T;E) $

\item[(S2)]  $S:L^{q}( 0,T;E) \to K$   and for every
compact convex subset   $C$   of   $E$,   $S$   is sequentially continuous from
$L_{w}^{1}( 0,T;C) $   to   $W^{1,p}(0,T;E) $.   (Here   $L_{w}^{1}( 0,T;C) $
  stands for   $L^{1}( 0,T;C) $
  endowed with the weak topology of   $L^{1}(0,T;E)$)

\item[(G1)] $g:[ 0,T] \times E^{2}\to P_{kc}( E) $

\item[(G2)] $g( .,z) $   has a strongly measurable selection
on   $[ 0,T] $,   for every $z\in E^{2}$

\item[(G3)] $g( t,.) $   is upper semicontinuous for a.e.
$t\in [ 0,T]$

\item[(G4)] If   $1\leq p<\infty $,   then   $|
g( t,z_{1},z_{2}) | \leq \nu ( t) $ for a.e.
$t\in [ 0,T] $ and all   $z_{1},z_{2}\in E$ with $|
z_{1}| \leq \mu _{0}R$; if   $p=\infty $,   then
  $| g( t,z_{1},z_{2}) | \leq \nu (t)$ for a.e. $t\in [ 0,T] $   and all
$z_{1},z_{2}\in E$ with   $| z_{1}| \leq \mu _{\infty }R$
and $| z_{2}| \leq \mu _{\infty }R$.   Here $\nu \in L^{q}( 0,T;\mathbb{R}_{+}) $.

\item[(G5)] For every separable closed subspace   $E_{0}$   of the space
  $E$,   there exists a   $( q,\infty) $-Carath\'{e}odory function
$\omega :[ 0,T]\times [ 0,\mu _{0}R] \to \mathbb{R}_{+}$,
 $\omega (t,0) =0$, such that for almost every   $t\in [ 0,T]$,
\[
\beta _{E_{0}}( g( t,M,E_{0}) \cap E_{0}) \leq \omega
( t,\beta _{E_{0}}( M) )
\]
for every set   $M\subset E_{0}$   satisfying $|
M| \leq \mu _{0}R$, and $\varphi =0$   is the unique solution
in   $L^{\infty }( 0,T;[ 0,\mu _{0}R] ) $
  to the inequality
\begin{equation}
\varphi ( t) \leq \int_{0}^{T}k( t,s) \omega (
s,\varphi ( s) ) ds\;\,\,\;\text{a.e. on   }[
0,T] .  \label{2.3}
\end{equation}
Here   $\beta _{E_{0}}$ is the ball measure of non-compactness on
  $E_{0}$.   (Recall that for a bounded set
$A\subset E_{0}$,   $\beta _{E_{0}}( A) $   is
the infimum of   $\varepsilon >0$   for which $A$
  can be covered by finitely many balls of   $E_{0}$
  with radius not greater than $\varepsilon$)

\item[(SG)]  For every   $u\in \overline{U}$  the set
$SG( u) $  is acyclic in   $K.\medskip $
\end{itemize}

\begin{remark} \label{rm2.2} \rm
If $S$ has values in $C_{a}^{1}$ then a sufficient condition for
(S1) is to exist a function $\theta \in L^{r}( 0,T;\mathbb{R}_{+})
$ such that
\[
| S( w_{1}) '-S( w_{2}) ^{\prime
}| _{p}\leq \int_{0}^{T}\theta ( s) | w_{1}(
s) -w_{2}( s) | ds
\]
for all $w_{1},w_{2}\in L^{q}( 0,T;E)$.
\end{remark}

Indeed, using Proposition 1.1 and H\"{o}lder's inequality, we immediately
see that (S1) is satisfied with
$k( t,s) =\frac{1}{2}T^{\frac{p-1}{p}}\theta ( s) $ and $L=| \theta | _{r}$.

\begin{remark} \rm
In case that $k( t,.) \in L^{\infty }( 0,T) $ for a.e. $t\in [
0,T] $, we may assume that $\omega $ in (G5) is a $( 1,\infty )
$-Carath\'{e}odory function (in order that the integral in
(\ref{2.3}) be defined).
\end{remark}

As in \cite{CP} we can prove the following existence result.

\begin{theorem} \label{thm2.2}
Assume (S1)-(S2), (G1)-(G5) and (SG) hold. In addition assume
(H3). Then \eqref{eq1.7} has at least one solution $u$ in
$K\subset W^{1,p}(0,T;E)$ with $\| u\| \leq R$.
\end{theorem}

The proof is based on Theorem \ref{thm2.1} and consists in showing that conditions
(H1)-(H2) are satisfied. We shall use the following analog of \cite[lemma 4.4]{CP}.

\begin{lemma} \label{lm2.3}
Assume (S1), (S2). Let   $M$   be a
countable subset of   $L^{q}( 0,T;E)$   such
that   $M( t) $   is relatively compact for
a.e. $t\in [ 0,T] $   and there is a function
$\nu \in L^{q}( 0,T;\mathbb{R}_{+}) $   with
$| u( t) | \leq \nu ( t) $   a.e.
on $[ 0,T] $,   for every   $u\in M$. Then
  the set   $S( M) $   is relatively
compact in   $W^{1,p}( 0,T;E) $. In addition $S$ is
continuous from $M$ equipped with the relative weak topology of $L^{q}(
0,T;E) $ to $W^{1,p}( 0,T;E) $ equipped with its strong
topology.
\end{lemma}

\begin{proof}
Let   $M=\{ u_{n}:n\geq 1\} $ and let
$\varepsilon >0$ be arbitrary.   As in the proof of \cite[lemma 4.3]{CP},
we can find functions   $\widehat{u}_{n,k}$
with values in a compact   $\overline{B}_{k}\subset E$
($\overline{B}_{k}$ being a closed ball of a $k$dimensional subspace of $E$)
such that
\[
| u_{n}-\widehat{u}_{n,k}| _{q}\leq \varepsilon
\]
for every   $n\geq 1$.   Then assumption (S1) implies
\begin{gather}
| S( u_{n}) -S( \widehat{u}_{n,k})| _{p}\leq | | k( t,.) | _{r}|
_{p}| u_{n}-\widehat{u}_{n,k}| _{q}\leq \varepsilon | | k( t,.)
|_{r}| _{p},  \label{eq17bis}
\\
| S( u_{n}) '-S( \widehat{u}_{n,k})'| _{p}\leq L|
u_{n}-\widehat{u} _{n,k}| _{q}\leq \varepsilon L.  \label{eq17}
\end{gather}
On the other hand, according to Theorem \ref{thm1.5}, the set
$\{\widehat{u}_{n,k}:n\geq 1\}\subset L^{q}( 0,T;E) $
  is weakly relatively compact in   $L^{q}(0,T;E) $.
Then assumption (S2) guarantees that $\{S(\widehat{u}_{n,k}) :n\geq 1\}$
is relatively compact in   $W^{1,p}( 0,T;E) $.   Hence from
(\ref{eq17bis}) and (\ref{eq17}) we see that $\{ S( \widehat{u}_{n,k}) :n\geq 1\} $
 is a   relatively compact $\varepsilon \varrho $-net of   $S( M) $ with
respect to the norm $|\cdot| _{1,p}$, where $\varrho =\max\{ L,| |
k( t,.) |_{r}| _{p}\} $.   Since   $\varepsilon $
  was arbitrary   we conclude that   $S(M) $   is relatively compact
in $W^{1,p}( 0,T;E)$.

Now suppose that the sequence $( w_{m}) _{m}$ converges weakly in
$L^{q}( 0,T;E) $ to $w$ and $w_{m}\in M$ for all $m\geq 1$. In
view of the relative compactness of $S( M) $, we may assume that
$( S( w_{m}) ) _{m}$ converges in $K$ towards some
function $v\in K$. We have to prove
\[
v=S( w) .
\]
For an arbitrary number $\varepsilon >0$, we have already seen that the
proof of  \cite[lemma 4.3]{CP} provides a compact set $P_{\varepsilon }$
and a sequence $( w_{m}^{\varepsilon }) _{m}$ of $P_{\varepsilon
} $-valued functions satisfying,
\begin{equation}
| w_{m}-w_{m}^{\varepsilon }| _{q}\leq \varepsilon  \label{gngene}
\end{equation}
for every   $m\geq 1$. Now the sequence $( w_{m}^{\varepsilon
}) _{m}$ being weakly relatively compact in $L^{q}( 0,T,E)
, $ a suitable subsequence $( w_{m_{j}}^{\varepsilon }) _{j}$
must be weakly convergent in $L^{q}( 0,T,E) $ towards some
$w^{\varepsilon }$. Then Mazur's Lemma and (\ref{gngene}) provide
\begin{equation}
| w-w^{\varepsilon }| _{q}\leq \varepsilon .  \label{gifge}
\end{equation}
The triangle inequality yields
\begin{equation}
\begin{aligned}
| v-S( w) | _{p}
& \leq | v-S(w_{m_{j}}) | _{p}+| S( w_{m_{j}}) -S(w_{m_{j}}^{\varepsilon }) | _{p} \\
&\quad +| S( w_{m_{j}}^{\varepsilon }) -S( w^{\varepsilon
}) | _{p}+| S( w^{\varepsilon }) -S(w) | _{p}
\end{aligned} \label{eqw}
\end{equation}
and
\begin{equation}
\begin{aligned}
| v'-S( w) '| _{p}
& \leq |v'-S( w_{m_{j}}) '| _{p}+| S(w_{m_{j}}) '-S( w_{m_{j}}^{\varepsilon })'| _{p} \\
&\quad +| S( w_{m_{j}}^{\varepsilon }) '-S(w^{\varepsilon }) '| _{p}
+| S( w^{\varepsilon}) '-S( w) '| _{p}.
\end{aligned} \label{ShifSge}
\end{equation}
Passing to the limit when $j$ goes to infinity in (\ref{eqw}), (\ref{ShifSge})
and using assumption (S2) we obtain
\begin{gather}
| v-S( w) | _{p}\leq \limsup_{j}| S(w_{m_{j}}) -S( w_{m_{j}}^{\varepsilon }) |
_{p}+| S( w^{\varepsilon }) -S( w) | _{p}, \label{eqv}
\\
| v'-S( w) '| _{p}\leq \limsup_{j}| S( w_{m_{j}}) '-S(w_{m_{j}}^{\varepsilon }) '
| _{p}+| S(w^{\varepsilon }) '-S( w) '| _{p}\,.
\label{hSginf}
\end{gather}
According to (\ref{gngene}) and (\ref{gifge}) we deduce from (\ref{eqv}),
(\ref{hSginf}) and assumption (S1) that
\[
| v-S( w) | _{p}\leq 2\varepsilon | |
k( t,.) | _{r}| _{p},\quad | v'-S(w) '| _{p}\leq 2\varepsilon L.
\]
Hence $| v-S( w) | _{1,p}\leq 2\,\varepsilon \varrho $. Since $\varepsilon $
was arbitrary we must have $v=S( w) $ and the proof is complete.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.2}]
(a) First we show that   $G( u) \neq \emptyset
$ \ and so   $SG( u) \neq \emptyset $
for every   $u\in \overline{U}$.   Indeed, since
  $g$   takes nonempty compact values and satisfies
(G2)-(G3), for each strongly measurable function     $u:
[ 0,T] \to E^{2}$   there exists a strongly
measurable selection   $w$   of   $g(
.,u( .) ) $   (see \cite{deimling}, Proof of
Proposition 3.5 (a)). Next, if   $u\in L^{p}( 0,T;E^{2})
$,   (G4) guarantees   $w\in L^{q}( 0,T;E) $. Hence   $w\in G( u) $.
\\
(b) The values of   $SG$   are acyclic according to
assumption (SG).
\\
(c) The graph of   $SG$   is closed. To show this, let
  $( u_{n},v_{n}) \in \,$graph $( SG) $,   $n\geq 1$,   with   $| u_{n}-u|
_{1,p}$, $| v_{n}-v| _{1,p}$ $\to 0$   as
  $n\to \infty $.   Let $\ v_{n}=S(w_{n}) $,
$w_{n}\in L^{q}( 0,T;E) ;w_{n}\in G(u_{n}) $.     Since
$|u_{n}-u| _{1,p}\to 0$, we may suppose that for every
$t\in [ 0,T] $,   there exists a compact   set   $C\subset E^{2}$   with
$\{ ( u_{n}( t) ,u_{n}'( t) ); n\geq 1\} \subset C$.
Furthermore, since   $g$   is upper semicontinuous by (G3) and has compact
values, we have that   $g( t,C) $   is compact. Consequently,
$\{ w_{n}( t) :n\geq 1\} $   is relatively
compact in   $E$.   If we also take into account (G4) we
may apply Theorem \ref{thm1.5} to conclude that (at least for a subsequence)
  $( w_{n}) $   converges weakly in $L^{q}( 0,T;E) $   to some   $w$.
As in \cite[p. 57]{frigon}, since   $g$   has convex values and
satisfies (G3), we can show that   $w\in G( u) $. Furthermore, by using
Lemma \ref{lm2.3} and a suitable subsequence we deduce
$S( w_{n}) \to S( w) $. Thus $v=S( w) $   and so   $( u,v) \in $ graph  $( SG) $.
\\
(d) We show that $SG( M) $   is relatively compact for
each compact   $M\subset \overline{U}$.   Let
$M\subset \overline{U}$   be a compact set and   $(v_{n}) $   be any sequence
of elements of $SG( M) $.   We prove that   $(v_{n}) $   has a convergent
subsequence. Let $u_{n}\in M$   and   $w_{n}\in L^{q}( 0,T;E) $   with
\[
v_{n}=S( w_{n}) \quad\text{and}\quad w_{n}\in G( u_{n}) .
\]
The set   $M$   being compact, we may assume that   $| u_{n}-u| _{1,p}\to 0$
for some  $u\in \overline{U}$.   As above, there exists a
  $w\in G( u) $   with $w_{n}\to w$   weakly in   $L^{q}(0,T;E) $
(at least for a subsequence) and $S( w_{n}) \to S( w) $.   Hence
  $v_{n}\to $ $S( w) $ as we wished. Now (c) and
(d) guarantee (H1).
\\
(e) Finally, we check (H2). Suppose   $M\subset \overline{U}$ is
closed and $M\subset \overline{\mathop{\rm conv}}(\{ 0\}\cup SG(
M) )$. To prove that $M$ is compact it suffices that every
sequence $( u_{n}^{0}) $ of $M$ has a convergent subsequence. Let
$M_{0}=\{ u_{n}^{0}:n\geq 1\} $. Clearly, there exists a countable
subset $M_{1}=\{ u_{n}^{1}:n\geq 1\} $ of $M$, $w_{n}^{1}\in G(
u_{n}^{1}) $ and $v_{n}^{1}=S( w_{n}^{1}) $ with $M_{0}\subset
\overline{\mathop{\rm conv}}( \{ 0\} \cup V^{1}) $, where
$V^{1}=\{ v_{n}^{1}:n\geq 1\} $. Furthermore, there exists a
countable subset $M_{2}=\{ u_{n}^{2}:n\geq 1\} $ of $M$,
$w_{n}^{2}\in G( u_{n}^{2}) $ and $v_{n}^{2}=S(w_{n}^{2}) $ with
$M_{1}\subset \overline{\mathop{\rm conv}}( \{ 0\} \cup V^{2}) $,
where $V^{2}=\{v_{n}^{2}:n\geq 1\} $, and so on. Hence for every
$k\geq 1$ we find a countable subset $M_{k}=\{ u_{n}^{k}:n\geq 1\}
$ of $M$ and correspondingly $w_{n}^{k}\in G( u_{n}^{k}) $ and
$v_{n}^{k}=S( w_{n}^{k}) $ such that $M_{k-1}\subset \overline{
\mathop{\rm conv}}( \{ 0\} \cup V^{k}) $, with $V^{k}=\{
v_{n}^{k}:n\geq 1\} $. Let $M^{\ast }=\bigcup_{k\geq 0}M_{k}$. It
is clear that $M^{\ast }$ is countable, $M_{0}\subset
M^{\ast}\subset M$ and $M^{\ast }\subset \overline{\mathop{\rm
conv}}(\{ 0\} \cup V^{\ast }) $, where $V^{\ast }=\bigcup_{k\geq
1}V^{k}$. Since $M^{\ast }$, $V^{\ast }$ and $W^{\ast
}:=\{w_{n}^{k}:n\geq 1,\,k\geq 1\} $ are countable sets of
strongly measurable functions, we may suppose that their values
belong to a separable closed subspace $E_{0}$ of $E$. Since $|
w_{n}^{k}( t)| \leq \nu ( t) $ where $\nu \in L^{q}( 0,T) $, then
\cite[Lemma 4.3]{CP} guarantees
\[
\beta _{E_{0}}( M^{\ast }( t) )
\leq \beta_{E_{0}}( V^{\ast }( t) )
=\beta _{E_{0}}( S( W^{\ast }) ( t) )
\leq \int_{0}^{T}k( t,s) \beta _{E_{0}}( W^{\ast }(s) ) ds,
\]
while (G5) gives
\begin{equation}
\beta _{E_{0}}( W^{\ast }( s) ) \leq \beta
_{E_{0}}( g( s,M^{\ast }( s) ,E_{0}) \cap
E_{0}) \leq \omega ( s,\beta _{E_{0}}( M^{\ast }(
s) ) ) .  \label{eqAA}
\end{equation}
It follows that
\[
\beta _{E_{0}}( M^{\ast }( t) ) \leq
\int_{0}^{T}k( t,s) \omega ( s,\beta _{E_{0}}( M^{\ast
}( s) ) ) ds.
\]
Moreover the function $\varphi ( t) =\beta _{E_{0}}( M^{\ast }( t))$
belongs to $L^{\infty }(0,T;[ 0,\mu _{0}R] )$. Consequently,
$\varphi \equiv 0$, and so
\[
\varphi ( t) =\beta _{E_{0}}( M^{\ast }( t)
) =0
\]
a.e. on $[ 0,T] $. Let $( v_{i}^{\ast }) $ be any
sequence of $V^{\ast }$ and let $( w_{i}^{\ast }) $ be the
corresponding sequence of $W^{\ast }$, with $v_{i}^{\ast }=S(
w_{i}^{\ast }) $ for all $i\geq 1$. Then, as at step (c), $(
w_{i}^{\ast }) $ has a weakly convergent subsequence in $L^{q}(
0,T;E) $, say to $w$. Also (\ref{eqAA}) together with $\omega (
t,0) =0$ implies that the set $\{w_{i}^{\ast }( t) :i\geq
1\}$ is relatively compact for a.e. $t\in [ 0,T] $. From Lemma
\ref{lm2.3} we then have that the corresponding subsequence of $( S(
w_{i}^{\ast }) ) =( v_{i}^{\ast }) $ converges to
$S( w) $ in $W^{1,p}( 0,T;E) $. Hence $V^{\ast }$ is
relatively compact. Now Mazur's Lemma guarantees that the set $\overline{
\mathop{\rm conv}}( \{ 0\} \cup V^{\ast }) $ is
compact and so its subset $M^{\ast }$ is relatively compact too. Thus $M_{0}$
possesses a convergent subsequence as we wished. Now the result follows from
Theorem \ref{thm2.1}.
\end{proof}

\section{The Anti-Periodic Solution Operator}

For the rest of this paper $E$ will be a real
Hilbert space of inner product $(.,.) $ and norm $|.| $.
Consider the anti-periodic boundary value problem
\begin{equation}
\begin{gathered}
-u''-\varepsilon u'\in Au+g( t,u,u') \quad\text{a.e. on }[0,T] \\
u( 0) =-u( T) ,\quad u'( 0)=-u'( T) ,
\end{gathered} \label{eq2.0}
\end{equation}
in $E$, where $\varepsilon \in \mathbb{R}$ and $A:D( A) \subset
E\to 2^{E}\setminus \{\emptyset \}$ is an odd m-dissipative
nonlinear operator.

Let us consider the \textit{anti-periodic solution operator} associated to
$A$ and $\varepsilon $,
\[
S:L^{2}( 0,T;E) \to H^{2}( 0,T;E) \cap
C_{a}^{1}
\]
defined by $S( w) :=u$, where $u$ is the unique solution of
\begin{equation}
\begin{gathered}
-u''-\varepsilon u'\in Au+w\quad\text{a.e. on }[0,T]\\
u( 0) =-u( T) ,\quad u'( 0)=-u'( T) \,.
\end{gathered} \label{eqB}
\end{equation}
The operator $S$ is well defined as it follows from Theorem \ref{thm3.1} in
Aftabizadeh-Aizicovici-Pavel \cite{AAP}.
It is clear that any fixed point $u$ of $N:=SG$, where $G$ is the Nemytskii
set-valued operator given by (\ref{eq1.6'}) with $p=q=2$, is a solution for
(\ref{eq2.0}).

\begin{theorem} \label{thm3.1}
The above operator $S$ satisfies \emph{(S1)} and \emph{(S2)} for $p=q=2$ and
$K=\overline{C_{a}^{1}}$ in $H^{1}(0,T;E)$ with norm $\| u\|
=| u'| _{2}$. \end{theorem}

\begin{proof}
(I) We first show that $S$ satisfies (S1). Let $w_{1},w_{2}\in L^{2}(0,T;E) $
and denote $u_{i}=S( w_{i}) $, $i=1,2$. Then
$-u_{i}''-\varepsilon u_{i}'=v_{i}+w_{i}$, where
$v_{i}( t) \in Au_{i}( t) $ a.e. on $[ 0,T]$. One has
\[
-( u_{1}-u_{2}) ''( t) -\varepsilon
( u_{1}-u_{2}) '( t) =(
v_{1}-v_{2}) ( t) +( w_{1}-w_{2}) (
t) .
\]
Multiplying by $( u_{1}-u_{2}) ( t) $ and using that $A$
dissipative, we obtain
\begin{equation}
\begin{aligned}
&-( | u_{1}( t) -u_{2}( t) |^{2}) ''+2| u_{1}'( t)
-u_{2}'( t) | ^{2}-\varepsilon ( |u_{1}( t) -u_{2}( t) | ^{2}) ' \\
&\leq 2( w_{1}( t) -w_{2}( t) ,u_{1}(t) -u_{2}( t) ) .
\end{aligned} \label{eq2.10}
\end{equation}
Consequently,
\begin{equation}
| u_{1}( t) -u_{2}( t) | ^{2}
\leq 2\int_{0}^{T}G( t,s) ( w_{1}( s) -w_{2}(s) ,u_{1}( s) -u_{2}( s) ) ds.
\label{eq2.10'}
\end{equation}
Here $G$ is the Green function of the differential operator
$-u''-\varepsilon u'$ corresponding to the anti-periodic
boundary conditions. This yields
\begin{equation}
| S( w_{1}) ( t) -S( w_{2}) (t) | \leq m\int_{0}^{T}| w_{1}( s)
-w_{2}(s) | ds  \label{eq2.11}
\end{equation}
where $m=2\max_{( t,s) \in [ 0,T] ^{2}}G(t,s) $. From
(\ref{eq2.10}) by integration we obtain
\[
\int_{0}^{T}| u_{1}'-u_{2}'| ^{2}ds\leq
\int_{0}^{T}( w_{1}-w_{2},u_{1}-u_{2}) ds.
\]
This together with (\ref{eq2.11}) yields
\[
| S( w_{1}) '-S( w_{2})'| _{2}\leq \sqrt{mT}| w_{1}-w_{2}| _{2}.
\]
(II) The fact that $S$ satisfies (S2) is achieved in several steps:
(1) We first show that the graph of $S$ is sequentially closed in
$L_{w}^{2}( 0,T;E) \times H^{1}( 0,T;E) $. In this
order, let $w_{j}\to w$ weakly in $L^{2}( 0,T;E) $ and
$S( w_{j}) \to u$ strongly in $H^{1}( 0,T;E) $. Then
$( w_{j}-w,S( w_{j}) -S( w) )\to 0$ strongly in $L^{1}( 0,T;\mathbb{R}) $.
Now (\ref{eq2.10'}) implies
\[
| S( w_{j}) ( t) -S( w) (
t) | \to 0\text{ \ \ as }j\to \infty .
\]
Hence $S( w) =u$.\\
(2) For each positive integer $n$ we let
\[
J_{n}=\big( J-\frac{1}{n}A\big) ^{-1},\quad A_{n}=n( J_{n}-J) ,
\]
where $J$ is the identity map of $E$. We also consider the operator
$S_{n}:L^{2}( 0,T;E) $ $\to $ $H^{2}( 0,T;E)\cap C_{a}^{1}$, given by
$S_{n}( w) =u_{n}$,   where
$u_{n}$ is the unique solution of
\begin{equation}
\begin{gathered}
-u_{n}''-\varepsilon u_{n}'=A_{n}u_{n}+w\quad\text{a.e. on }[0,T] \\
u_{n}( 0) =-u_{n}( T) ,\quad u_{n}'(0) =-u_{n}'( T)\, .
\end{gathered}  \label{eqN}
\end{equation}
Then
\[
-| u_{k}''| ^{2}-\varepsilon ( u_{k}',u_{k}'')
=( A_{k}u_{k},u_{k}')'-( ( A_{k}u_{k}) ',u_{k}')+( w,u_{k}'') .
\]
Since $A_{k}$ is dissipative, we have
\[
( ( A_{k}u_{k}) ',u_{k}')
=\lim_{h\to 0}\frac{1}{h^{2}}( A_{k}u_{k}(
t+h) -A_{k}u_{k}( t) ,u_{k}( t+h) -u_{k}(t) ) \leq 0\,.
\]
Hence
\[
| u_{k}''| ^{2}\leq -(A_{k}u_{k},u_{k}') '-( w,u_{k}'')
-\frac{\varepsilon }{2}( | u_{k}'|^{2}) '.
\]
By integration, since $A_{k}$ is odd and $u_{k}$ is anti-periodic, it
follows
\[
| u_{k}''| _{2}^{2}=\int_{0}^{T}|
u_{k}''| ^{2}dt\leq -\int_{0}^{T}( w,u_{k}'') dt\leq \frac{1}{2}( | w| _{2}^{2}
+|u_{k}''| _{2}^{2}) \,.
\]
Consequently,
\begin{equation}
| u_{k}''| _{2}\leq | w| _{2}. \label{eq10}
\end{equation}
Using   $2| u'| ^{2}=( | u|
^{2}) ''-2( u'',u) $ and
$( | u| ^{2}) '=2( u',u) $
we obtain
\begin{equation} \label{eq11}
\begin{aligned}
&2\int_{0}^{T}| u_{k}'-u_{m}'| ^{2}dt\\
&= (| u_{k}-u_{m}| ^{2}) '( T) -(| u_{k}-u_{m}| ^{2}) '( 0)
-2\int_{0}^{T}( u_{k}''-u_{m}'',u_{k}-u_{m}) dt   \\
&= -2\int_{0}^{T}( u_{k}''-u_{m}'' ,u_{k}-u_{m}) dt\,.
\end{aligned}
\end{equation}
On the other hand
\begin{align*}
&( u_{k}''-u_{m}'',u_{k}-u_{m}) \\
&=-\big( A_{k}u_{k}-A_{m}u_{m},u_{k}-u_{m}\big)
 -\varepsilon (u_{k}'-u_{m}',u_{k}-u_{m}) \\
& =-\big( A_{k}u_{k}-A_{m}u_{m},\,J_{k}u_{k}-J_{m}u_{m}+\frac{1}{k}
A_{k}u_{k}-\frac{1}{m}A_{m}u_{m}\big)
 -\varepsilon ( u_{k}'-u_{m}',u_{k}-u_{m})
\end{align*}
and since $A_{k}u_{k}\in AJ_{k}u_{k}$, $A_{m}u_{m}\in AJ_{m}u_{m}$ and $A$
is dissipative, we obtain
\[
-( u_{k}''-u_{m}'',u_{k}-u_{m})
\leq ( A_{k}u_{k}-A_{m}u_{m},\,\frac{1}{k}A_{k}u_{k}-\frac{1}{m}A_{m}u_{m})
+\frac{\varepsilon }{2}( | u_{k}-u_{m}| ^{2})'.
\]
 From (\ref{eqN}) and (\ref{eq10}), also applying Proposition 1.1 to
$u_{k}'$, we see that
\[
| A_{k}u_{k}| _{2}
\leq | u_{k}''|_{2}+| w| _{2}+| \varepsilon | | u_{k}'| _{2}
\leq | u_{k}''| _{2}+| w| _{2}+|\varepsilon | \frac{T}{2}| u_{k}''| _{2}
\leq ( 2+| \varepsilon | \frac{T}{2}) | w| _{2}.
\]
Then
\[
-\int_{0}^{T}( u_{k}''-u_{m}^{\prime \prime
},u_{k}-u_{m}) dt\leq 2( 2+| \varepsilon | \frac{T}{2}
) ^{2}| w| _{2}^{2}( \frac{1}{k}+\frac{1}{m}) .
\]
This together with (\ref{eq11}) shows that
\begin{equation}
\int_{0}^{T}| u_{k}'-u_{m}'| ^{2}dt\leq 2(
2+| \varepsilon | \frac{T}{2}) ^{2}| w|
_{2}^{2}( \frac{1}{k}+\frac{1}{m}) .  \label{eq12}
\end{equation}
Thus there exists $u\in K$ with $u_{k}\to u$ in $K$. From (\ref{eq12}),
letting $m\to \infty $ we have
\begin{equation}
| u_{k}'-u'| _{2}^{2}\leq \frac{2}{k}(
2+| \varepsilon | \frac{T}{2}) ^{2}| w| _{2}^{2}.
\label{eq*}
\end{equation}
Now we show that $u$ is the solution of (\ref{eqB}). Since $(
u_{k}'') $ is bounded in $L^{2}( 0,T;E) $
and $( u_{k}'') $ converges to $w^{\prime
}=u''$ in $\mathcal{D}'( 0,T;E) $, we may
conclude that
\begin{equation}
u_{k}''\to u''\quad \text{weakly in  }L^{2}( 0,T;E) .  \label{eq13}
\end{equation}
Let $\mathcal{A}$ be the realization of $A$ in $L^{2}( 0,T;E) $, i.e., $\mathcal{A}:L^{2}( 0,T;E) \to 2^{L^{2}(
0,T;E) }$,
\[
\mathcal{A}u=\{ v\in L^{2}( 0,T;E) :v( t) \in
Au( t) \text{ a.e. on }[ 0,T] \} .
\]
Then $( \mathcal{A}_{k}u) ( t) =A_{k}u( t) $
a.e. on $[ 0,T] $, so that (\ref{eq13}) implies that
\[
\mathcal{A}_{k}u_{k}\to -u''-\varepsilon u^{\prime
}-w\;\;\,\text{weakly in }L^{2}( 0,T;E) .
\]
Since $u_{k}\to u$ strongly in $L^{2}( 0,T;E) $ and
$\mathcal{A}$ is $m$-dissipative in $L^{2}( 0,T;E) $, this implies
(see Barbu \cite{B}, Proposition II. 3.5) $u\in D( \mathcal{A}) $
and $[ u,-u''-\varepsilon u'-w] \in
\mathcal{A}$. Thus, $u$ is the solution of (\ref{eqB}), i.e., $u=S(
w) $.
Now from (\ref{eq*}) we see that for each bounded set $M\subset L^{2}(
0,T;E) $ and every $\epsilon >0$, there exists a $k_{0}$ such that
\begin{equation}
\| S_{k}( w) -S( w) \| \leq \epsilon \;\;\;
\text{for all   }k\geq k_{0}\text{ and }w\in M.  \label{eq**}
\end{equation}
Hence $S_{k_{0}}( M) $ is an $\epsilon $-net for $S(
M) $.
\\
(3) Now we consider a compact convex subset   $C$   of
  $E$   and a countable set   $M\subset
L^{2}( 0,T;C) $. We shall prove that for each $n$, the set $S_{n}(
M) $ is relatively compact in $K$, equivalently, the set $S_{n}(
M) '$ is relatively compact in $L^{2}( 0,T;E) $. Then, also taking
into account (\ref{eq**}), by Hausdorff's Theorem we shall deduce
that $S( M) $ is relatively compact in $K$ as desired. We shall
apply Theorem \ref{thm1.4} to $S_{n}( M) '$. From (\ref{eq**}) and
assumption (S1) we see that for each   $n$
  and any bounded   $M\subset L^{2}( 0,T;E) $,   the set $S_{n}( M) $   is bounded in
  $K$.   In addition, using
\[
u_{n}( t) =\int_{0}^{T}G( t,s) [ A_{n}u_{n}(
s) +w( s) ] ds
\]
and the Lipschitz property of   $A_{n}$,   we obtain
\begin{align*}
| \tau _{h}u_{n}'-u_{n}'| _{2}^{2}
&\leq \int_{0}^{T}\Big( \int_{0}^{T}| G_{t}( t+h,s) -G_{t}(
t,s) | [ 2n| u_{n}( s) | +|w( s) | ] ds\Big) ^{2}dt \\
&\leq ( 2n| u_{n}| _{2}+| w| _{2})
^{2}\int_{0}^{T}\int_{0}^{T}| G_{t}( t+h,s) -G_{t}(t,s) | ^{2}dsdt.
\end{align*}
This implies
\begin{equation}
\sup_{w\in M}| \tau _{h}S_{n}( w) '-S_{n}(w) '| _{L^{2}( 0,T-h;E) }\to 0\quad
\text{as }h\to 0.  \label{eqA}
\end{equation}
We claim that   $S_{n}( M) '( t) $   is relatively compact in   $E$   for every
  $t\in [ 0,T] $.   Indeed, for any
$w\in M$,   the unique solution   $u_{n}=S_{n}(w) $   of (\ref{eqN}) satisfies
\[
-u_{n}''-\varepsilon u_{n}^{\prime
}+nu_{n}=nJ_{n}u_{n}+w\;\,\,\;\text{a.e. on   }[ 0,T] .
\]
If we denote by   $\widetilde{G}$   the Green function of
the operator   $-u''-\varepsilon u'+nu$
  corresponding to the boundary conditions   $u(
0) =-u( T) $, $u'( 0) =-u^{\prime
}( T) $,   then
\begin{equation}
u_{n}( t) =\int_{0}^{T}\widetilde{G}( t,s) [
nJ_{n}u_{n}( s) +w( s) ] ds.  \label{eq80}
\end{equation}
Using a result by Heinz, the nonexpansivity of   $J_{n}$
and the inclusion   $M( s) \subset C$ a.e. on
$[ 0,T] $,   from (\ref{eq80}), we obtain
\begin{equation}
\beta _{0}( S_{n}( M) ( t) ) \leq
n\int_{0}^{T}\widetilde{G}( t,s) \beta _{0}( S_{n}(
M) ( s) ) ds.  \label{eq81}
\end{equation}
Here   $\beta _{0}$   is the ball measure of
non-compactness corresponding to a suitable separable closed subspace of
  $E$.   Let
\[
\varphi ( t) =\beta _{0}( S_{n}( M) (
t) ) \,,\;\,\,v( t) =\int_{0}^{T}\widetilde{G}(
t,s) \varphi ( s) ds.\,
\]
We have
\[
-v''-\varepsilon v'+nv=\varphi ,\quad v( 0)=-v( T) ,\quad
v'( 0) =-v'(T) .
\]
According to (\ref{eq81}),   $\varphi \leq nv$.   Hence
  $-v''-\varepsilon v'\leq 0$.
Also since $v\geq 0$ we have   $v( 0) =v( T)=0$.
The maximum principle for the operator $-u''-\varepsilon u'$ implies
$v\leq 0$   on   $[0,T] $.   Hence   $v\equiv 0$.   Thus
$\beta _{0}( S_{n}( M) ( t) ) =0$   for all   $t\in [ 0,T] $,   that is
$S_{n}( M) ( t) $   is relatively compact in   $E$.   As a result,
$S_{n}(M) $   is relatively compact in   $C( [ 0,T] ;E) $. Next from (\ref{eq80})
we have
\[
u_{n}'( t) =\int_{0}^{T}\widetilde{G}_{t}(t,s) [ nJ_{n}u_{n}( s) +w( s) ] ds,
\]
whence $S_{n}( M) '( t) $ is relatively
compact in $E$. This together with (\ref{eqA}) via Theorem \ref{thm1.4} implies that
$S_{n}( M) '$ is relatively compact in $L^{2}(0,T;E) $.
\end{proof}

\section{Superlinear Inclusions}

In this section we establish an existence result for
the anti-periodic problem
\begin{equation}
\begin{gathered}
-u''-\varepsilon u'-s( u) \in Au+h(t,u,u') \quad\text{a.e. on }[0,T] \\
u( 0) =-u( T) ,\quad u'( 0)=-u'( T)
\end{gathered} \label{eq2.1}
\end{equation}
in the Hilbert space $E$, where $\varepsilon >0$,
$A:D( A)\subset E\to 2^{E}\setminus \{\emptyset \}$ is odd $m$-dissipative,
$s:E\to E$ is continuous with a possible superlinear growth, and
$h:[0,T]\times E^{2}\to 2^{E}$.
Let $G:H^{1}( 0,T;E) \to 2^{L^{2}( 0,T;E) }$
be the Nemytskii set-valued operator associated with
$g(t,x,y)=s(x)+h( t,x,y)$,
that is
\[
G( u) =\{v\in L^{2}( 0,T;E) :v=s( u) +w,
\text{ }w\in \,\text{sel  }_{L^{2}}h( .,u,u')\},
\]
and let $S$ be the anti-periodic solution operator associated to $A$ and
$\varepsilon $, already defined in Section 3.

The next result concerns condition (H3) and gives sufficient
conditions to obtain a priori bounds of solutions.

\begin{theorem} \label{thm4.1}
Assume that the following conditions hold:
\begin{itemize}
\item[(i)] There exist two even real functions $\phi $, $\psi $ such that
$\psi \in C^{1}( E;\mathbb{R}) $ and
$A=-\partial \phi$ and $s=\psi '$,
where $\partial \phi $ stands for the subdifferential of $\phi $

\item[(ii)] There are $a,b\in \mathbb{R}_{+}$ and $\alpha ,\gamma \in
[1,2[$, $\beta \in [0,2[$ with $\beta +\gamma <2$ such that
\begin{equation}
-( z,y) \leq a| y| ^{\alpha }+b| x| ^{\beta}| y| ^{\gamma }  \label{eq2.4}
\end{equation}
for all $x,y\in E$, $z\in h( t,x,y) $, and for a.e. $t\in [0,T] $.
\end{itemize}
Then there exists a constant $R>0$ such that
$\| u\| =| u'| _{2}<R$ for any solution $u$ of
\begin{equation}
u\in \lambda SG( u)  \label{eq2.5}
\end{equation}
and every $\lambda \in ]0,1[$.
\end{theorem}

\begin{proof}
Let $u$ be any non-zero solution of (\ref{eq2.5}) for some
$\lambda \in ]0,1[$. Let $u_{\lambda }:=\frac{1}{\lambda }u$.
Then $u=\lambda u_{\lambda} $ and
\[
u_{\lambda }=S( w) ,\;\;w\in G( u)
\]
that is
\begin{gather*}
-u_{\lambda }''-\varepsilon u_{\lambda }'\in Au_{\lambda }+w, \\
w=s( u) +v, \\
v\in \text{sel  }_{L^{2}}h( .,u,u').
\end{gather*}
Hence
\[
-u_{\lambda }''-s( u) -\varepsilon u_{\lambda
}'-v\in Au_{\lambda }.
\]
Multiplying by $u'=\lambda u_{\lambda }'$ and using the formula $(
Au_{\lambda },u_{\lambda }') =-( \phi (u_{\lambda }) ) '$ (see
\cite[p. 189]{B}), we obtain
\[
\frac{\lambda }{2}( | u_{\lambda }'| ^{2})
'+( \psi ( u) ) '+\frac{\varepsilon}{\lambda }| u'| ^{2}+( v,u')
=\lambda ( \phi ( u_{\lambda }) ) '.
\]
Thus,
\[
\big( \frac{\lambda }{2}| u_{\lambda }'| ^{2}
+\psi( u) -\lambda \phi (u_{\lambda })\big) '+\frac{
\varepsilon }{\lambda }| u'| ^{2}=-( v,u') .
\]
By integration from $0$ to $T$ and taking into account the anti-periodic
boundary conditions and the fact that $\phi $ and $\psi $ are even, we
deduce
\[
\varepsilon | u'| _{2}^{2}<\frac{\varepsilon }{\lambda }
| u'| _{2}^{2}=-\int_{0}^{T}( v( t),u'( t) ) dt.
\]
Now using (\ref{eq2.4}) and (\ref{eq1.4}) we obtain
\begin{align*}
\varepsilon | u'| _{2}^{2}
&<a| u'|_{\alpha }^{\alpha }+b\int_{0}^{T}| u| ^{\beta }| u'| ^{\gamma }dt \\
&\leq a| u'| _{\alpha }^{\alpha }+b( \frac{1}{2}
| u'| _{1}) ^{\beta }\int_{0}^{T}| u'| ^{\gamma }dt \\
&= a| u'| _{\alpha }^{\alpha }+b\frac{1}{2^{\beta }}| u'| _{1}^{\beta }
| u'| _{\gamma }^{\gamma }.
\end{align*}
Since $\alpha ,\gamma \in \lbrack 1,2[$ there are constants $c_{1}$, $c_{2}$
such that $| u'| _{\alpha }\leq T^{\frac{2-\alpha }{2\alpha }}| u'| _{2}$ and
$| u'|_{\gamma }\leq T^{\frac{2-\gamma }{2\gamma }}| u'| _{2}$.
In addition $| u'| _{1}\leq T^{\frac{1}{2}}| u'| _{2}$. Consequently, one has
\[
\varepsilon | u'| _{2}^{2}<C_{1}| u^{\prime
}| _{2}^{\alpha }+C_{2}| u'| _{2}^{\beta +\gamma },
\]
where the constants $C_{1},C_{2}$ (independent of $u$ and $\lambda $) are:
\[
C_{1}=aT^{\frac{2-\alpha }{2}},\;\;\;C_{2}=b\frac{1}{2^{\beta }}T^{\frac{
2+\beta -\gamma }{2}}.
\]
Now the conclusion follows since $\alpha <2$ and $\beta +\gamma <2$.
\end{proof}

\begin{remark} \label{rmk4.1} \rm
The above result is also true if $\alpha =2$ or $\beta +\gamma =2$ provided
that $a$, respectively $b$, is sufficiently small.
\end{remark}

Now we are ready to state the main result of this section.

\begin{theorem} \label{thm4.2}
Let $E$ be a Hilbert space, $\varepsilon >0$, $s:E\to E$,
$A:E\to 2^{E}$ and $h:[ 0,T] \times E^{2}\to 2^{E}$. Assume:
\begin{itemize}
\item[(i)] $s=\psi '$ for some even function
$\psi \in C^{1}(E;\mathbb{R}) $, and $s$ sends bounded sets into bounded sets

\item[(ii)] $A$ is an m-dissipative mapping with $A=-\partial \phi $ for
some even real function $\phi$

\item[(iii)] $h:[ 0,T] \times E^{2}\to P_{kc}(E) ,$\ $h( .,z) $
has a strongly measurable selection on   $[ 0,T] $   for every $z\in E^{2}$,
 $h( t,.) $   is upper semicontinuous for a.e.
$t\in [ 0,T] $, and for each $\tau >0$ there exists $\nu \in L^{2}( 0,T) $ with
$| h( t,z) | \leq \nu ( t) \;$for a.e. $t\in [ 0,T] $
and all   $z=( z_{1},z_{2}) \in E^{2}$ with $|z_{1}| \leq \tau ;$ in addition
there are $a,b\in \mathbb{R}_{+}$ and
$\alpha ,\gamma \in \lbrack 1,2[$ and $\beta \in \lbrack 0,\infty [$
such that
\[
-( z,y) \leq a| y| ^{\alpha }+b| x| ^{\beta}| y| ^{\gamma }
\]
for all $x,y\in E$, $z\in h( t,x,y) $, and for a.e. $t\in [0,T]$

\item[(iv)] There exists $R>0$ with
\begin{equation}
\varepsilon R^{2}\geq aT^{\frac{2-\alpha }{2}}R^{\alpha }+b\frac{1}{2^{\beta
}}T^{\frac{2+\beta -\gamma }{2}}R^{\beta +\gamma }  \label{eq2.100}
\end{equation}
such that for every separable closed subspace   $E_{0}$
of   $E$,   there exists a   $( 1,\infty) $-Carath\'{e}odory function
$\omega :[ 0,T]\times \mathbb{R}_{+}\to \mathbb{R}_{+}$ such that for
almost every  $t\in [ 0,T]$,
\[
\beta _{E_{0}}( g( t,M,E_{0}) \cap E_{0}) \leq \omega
( t,\beta _{E_{0}}( M) )
\]
(where $g( t,x,y) =s( x) +h( t,x,y) $) for
every bounded set   $M\subset E_{0}$,   and
$\varphi =0$   is the unique solution in   $L^{\infty}( 0,T;\mathbb{R}_{+}) $
to the inequality
\begin{equation}
\varphi ( t) \leq m\int_{0}^{T}\omega ( s,\varphi ( s) ) ds\quad
\text{a.e. on }[ 0,T]  \label{eq2.200}
\end{equation}

\item[(v)] $SG$ has acyclic values.
\end{itemize}
Then \eqref{eq2.1} has at least one solution $u\in H^{2}(0,T;E)
\cap C_{a}^{1}$ with $\| u\| \leq R$.
\end{theorem}

\begin{remark} \label{rmk4.2} \rm
(a) Note that we do not assume $\beta +\gamma <2$, so the
perturbation term $h( t,u,u') $ can have a superlinear
growth in $u;$ inequality \eqref{eq2.100} guarantees that $\|
u\| \neq R$ for each solution of \eqref{eq2.5} and $\lambda \in
]0,1[$. This does not exclude the existence of solutions with $\|u\| >R$.
\\
(b) However, according to Theorem \ref{thm4.1}, if $\beta +\gamma <2$,
 then there exists a sufficiently large constant $R_{0}>0$ such that
\eqref{eq2.100} holds with equality. In this case $R_{0}$ is a bound for all
solutions to \eqref{eq2.5}.
\\
(c) Sufficient conditions for (v) can be found in \cite{CP}.
For example (v) always holds if $A$ is single-valued.
\end{remark}

\section{Applications}

In this section we are concerned with two
applications of Theorem \ref{thm4.2} to partial differential inclusions.

\noindent(I) First we look for a function $u=u( t,x) =u(t) ( x) $ solving
the problem
\begin{equation}
\begin{gathered}
-u_{tt}-\varepsilon u_{t}+\sigma \Delta _{x}^{-1}( | u|
^{p-2}u) +u\in h( t,u,u_{t}) \quad \text{a.e. on }[0,T] \\
u( t,.) \in H_{0}^{1}( \Omega ) \quad \text{for a.e. }t\in [ 0,T] \\
u( 0,x) =-u( T,x) ,\quad u_{t}( 0,x)
=-u_{t}( T,x) \quad \text{a.e. on }\Omega .
\end{gathered} \label{eqPI}
\end{equation}
Here $\Omega $ is a bounded domain of $\mathbb{R}^{n}$, $n\geq 3$,
$2<p<2^{\ast }=\frac{2n}{n-2}$, $\varepsilon >0$, $\sigma \in \mathbb{R}$
and $\Delta _{x}:H_{0}^{1}( \Omega ) \to H^{-1}(\Omega ) $ is the Laplacian.
Also by $|\cdot|$ we mean here the absolute value of a real number.

In this setting we let $E=H_{0}^{1}( \Omega ) $ with the inner
product $( u,v) _{H_{0}^{1}( \Omega ) }=\int_{\Omega
}\nabla u\cdot \nabla vdx$ and norm $| u| _{H_{0}^{1}(
\Omega ) }=( \int_{\Omega }| \nabla u| ^{2}dx) ^{
\frac{1}{2}}$, $A( u) =-u$ with $D( A)
=H_{0}^{1}( \Omega ) $ and $s( u) =-\sigma \Delta
_{x}^{-1}( | u| ^{p-2}u) $. Note that the conditions
(i) and (ii) in Theorem \ref{thm4.2} hold with
\[
\phi ( u) =\frac{1}{2}\int_{\Omega }| \nabla u|
^{2}dx\quad\text{and}\quad \psi ( u) =\frac{\sigma }{p}
\int_{\Omega }| u| ^{p}dx.
\]
Also note that for any bounded $M\subset H_{0}^{1}( \Omega ) $
the set $s( M) $ is relatively compact in $H_{0}^{1}( \Omega
) $, that is $\beta _{H_{0}^{1}( \Omega ) }( s(
M) ) =0$. Here $\beta _{H_{0}^{1}( \Omega ) }$ is the
ball measure of non-compactness in $H_{0}^{1}( \Omega ) $.
Indeed, since $p<2^{\ast }$ we may choose an $\theta >0$ with
$p\leq 2^{\ast}-\frac{\theta }{( 2^{\ast }) '}$, where
$( 2^{\ast}) '=\frac{2n}{n+2}$. This guarantees that
$( 2^{\ast}) '\leq \frac{2^{\ast }-\theta }{p-1}$. Next the embedding
of $H_{0}^{1}( \Omega ) $ into $L^{2^{\ast }-\theta }(\Omega ) $
being compact, we have that $M$ is relatively compact in
$L^{2^{\ast }-\theta }( \Omega ) $. Then the set
$M_{p}:=\{| u| ^{p-2}u:u\in M\} $ is relatively compact in
$L^{\frac{2^{\ast }-\theta }{p-1}}( \Omega ) $ and using the
continuous embeddings
\[
L^{\frac{2^{\ast }-\theta }{p-1}}( \Omega ) \subset L^{(
2^{\ast }) '}( \Omega ) \subset H^{-1}(
\Omega )
\]
we find that $M_{p}$ is relatively compact in $H^{-1}( \Omega ) $. Thus, $s( M) =-\sigma \Delta _{x}^{-1}( M_{p}) $ is
relatively compact in $H_{0}^{1}( \Omega ) $ as desired.

From Theorem \ref{thm4.2} one obtains the following result.

\begin{theorem} \label{thm5.1}
Let $h:[ 0,T] \times H_{0}^{1}( \Omega ) \times
H_{0}^{1}( \Omega ) \to P_{kc}( H_{0}^{1}(\Omega ) ) $ be such that
$h( .,u,v) $ has a strongly measurable selection on   $[ 0,T] $
  for every $u,v\in H_{0}^{1}( \Omega )$, $h(t,.) $   is upper semicontinuous
for a.e. $t\in [ 0,T] $, and for each $\tau >0$ there exists
$\nu \in L^{2}(0,T) $ such that
$| h( t,u,v) |_{H_{0}^{1}( \Omega ) }\leq \nu ( t)$ for a.e.
$t\in [ 0,T] $   and all   $u,v\in H_{0}^{1}(\Omega ) $ with
$| u| _{H_{0}^{1}( \Omega )}\leq \tau $. Assume there are
$a,b,a_{0}\in \mathbb{R}_{+}$ and $\alpha,\gamma \in \lbrack 1,2[$ and
$\beta \in \lbrack 0,\infty \lbrack $ such that
\[
-( w,v) _{H_{0}^{1}( \Omega ) }\leq a| v|_{H_{0}^{1}( \Omega ) }^{\alpha }
+b| u|_{H_{0}^{1}( \Omega ) }^{\beta }| v| _{H_{0}^{1}(\Omega ) }^{\gamma }
\]
for all $u,v\in H_{0}^{1}( \Omega ) $, $w\in h( t,u,v)
$ and for a.e. $t\in [ 0,T] $, and that for each bounded
$M\subset H_{0}^{1}( \Omega ) $,
\[
\beta _{H_{0}^{1}( \Omega ) }( h( t,M,H_{0}^{1}(\Omega ) ) )
\leq a_{0}\beta _{H_{0}^{1}( \Omega) }( M) .
\]
In addition assume that there exists $R>0$ with
\[
\varepsilon R^{2}\geq aT^{\frac{2-\alpha }{2}}R^{\alpha }+b\frac{1}{2^{\beta
}}T^{\frac{2+\beta -\gamma }{2}}R^{\beta +\gamma }.
\]
Then for $a_{0}<\frac{1}{mT}$, \eqref{eqPI} has at least one solution
$u\in H^{2}( 0,T;H_{0}^{1}( \Omega ) ) $ with
\[
| u'| _{2}=( \int_{0}^{T}| u'(t) | _{H_{0}^{1}( \Omega ) }^{2}dt) ^{\frac{1}{2}}
\leq R.
\]
\end{theorem}

\begin{proof}
For any bounded $M$, since $\beta _{H_{0}^{1}( \Omega ) }(s( M) ) =0$, one has
\[
\beta _{H_{0}^{1}( \Omega ) }( g( t,M,H_{0}^{1}(
\Omega ) ) ) \leq a_{0}\beta _{H_{0}^{1}( \Omega) }( M) .
\]
Recall that the space $H_{0}^{1}( \Omega ) $ is separable. It
follows that the unique solution
$\varphi \in L^{\infty }(0,T;\mathbb{R}_{+}) $   of (\ref{eq2.200}) with
$\omega( t,\tau ) =a_{0}\tau $ is $\varphi =0$ provided that $a_{0}mT<1$.
Thus Theorem \ref{thm4.2} applies.
\end{proof}

\begin{corollary} \label{coro5.2}
For every $f\in L^{\infty }( 0,T;H_{0}^{1}( \Omega ) )$ the problem
\begin{gather*}
-u_{tt}-\varepsilon u_{t}+\sigma \Delta _{x}^{-1}( | u|^{p-2}u) +u
=f( t,x) \quad \text{a.e. on }[0,T]\times \Omega \\
u( t,.) \in H_{0}^{1}( \Omega ) \quad \text{for a.e. } t\in [ 0,T] \\
u( 0,x) =-u( T,x) ,\quad u_{t}( 0,x)=-u_{t}( T,x) \quad \text{a.e. on }\Omega .
\end{gather*}
has at least one solution $u\in H^{2}(0,T;H_{0}^{1}( \Omega ) )$
with
\[
| u'| _{2}\leq \frac{| f| _{\infty }\sqrt{T}}{\varepsilon }.
\]
Here $| f| _{\infty }=\mathop{\rm ess\, sup}_{t\in [ 0,T]}
| f( t) | _{H_{0}^{1}( \Omega ) }$.
\end{corollary}

\begin{proof}
In this case $h( t,u,v) =f( t) :=f( t,.) $. Consequently all the assumptions
of Theorem \ref{thm5.1} are satisfied for $a=0$, $b=| f| _{\infty }$, $\alpha =1$,
$\beta =0$, $\gamma =1$, $a_{0}=0$, $\nu ( t) =| f( t) |_{H_{0}^{1}( \Omega ) }$
and $R=\frac{| f| _{\infty }\sqrt{T}}{\varepsilon }$.
 \end{proof}

\noindent (II) For the next application we look for a function
$u=u(t,x) $ solving the problem
\begin{equation}
\begin{gathered}
-u_{tt}-\varepsilon u_{t}+\sigma | u| _{L^{2}( \Omega) }^{p-2}u
-\Delta _{x}u\in h( t,u,u_{t}) \quad  \text{a.e. on }[0,T]\times \Omega \\
u( t,.) \in H_{0}^{1}( \Omega ) \quad \text{for a.e. }t\in [ 0,T] \\
u( 0,x) =-u( T,x) ,\quad u_{t}( 0,x)=-u_{t}( T,x) \quad \text{a.e. on }\Omega .
\end{gathered} \label{eqPII}
\end{equation}
Here again $\Omega $ is a bounded domain of $\mathbb{R}^{n}$, $p>2$,
$\varepsilon >0$ and $\sigma \in \mathbb{R}$, but we need no upper bound for
$p$.
Now we let $E=L^{2}( \Omega ) $, $A=\Delta _{x}$ be the Laplace
operator with $D( A) =H^{2}( \Omega ) \cap H_{0}^{1}( \Omega ) $ and
$s( u) =-\sigma |u| _{L^{2}( \Omega ) }^{p-2}u$. We note that the conditions
(i) and (ii) in Theorem \ref{thm4.2} hold with
\[
\phi ( u) =\begin{cases}
\frac{1}{2}\int_{\Omega }| \nabla u| ^{2}dx, &u\in H^{1}(\Omega ) \\
+\infty ,& \text{otherwise.}
\end{cases}
\]
and $\psi ( u) =-\frac{\sigma }{p}|u| _{L^{2}( \Omega ) }^{p}$.
From Theorem \ref{thm4.2} one obtains the following result.

\begin{theorem} \label{thm5.3}
Let $h:[ 0,T] \times L^{2}( \Omega ) \times
L^{2}( \Omega ) \to P_{kc}( L^{2}( \Omega
) ) $\ be such that $h( .,u,v) $   has a
strongly measurable selection on   $[ 0,T] $
for every $u,v\in L^{2}( \Omega ) ,$\ $h( t,.) $
  is upper semicontinuous for a.e. $t\in [ 0,T] $, and
for every $\tau >0$ there exists $\nu \in L^{2}( 0,T) $ such that
  $| h( t,u,v) | _{L^{2}( \Omega )
}\leq \nu ( t) \;$for a.e. $t\in [ 0,T] $
and all   $u,v\in L^{2}( \Omega ) $ with $|
u| _{L^{2}( \Omega ) }\leq \tau $.   Assume there
are $a,b,a_{0}\in \mathbb{R}_{+}$ and $\alpha ,\gamma \in \lbrack 1,2[$ and
$\beta \in \lbrack 0,\infty \lbrack $ such that
\[
-( w,v) _{L^{2}( \Omega ) }\leq a| v|
_{L^{2}( \Omega ) }^{\alpha }+b| u| _{L^{2}(\Omega ) }^{\beta }| v| _{L^{2}( \Omega )
}^{\gamma }
\]
for all $u,v\in L^{2}( \Omega ) $, $w\in h( t,u,v) $
and for a.e. $t\in [ 0,T] $, and that for each bounded $M\subset L^{2}( \Omega ) $,
\[
\beta _{L^{2}( \Omega ) }( h( t,M,L^{2}( \Omega
) ) ) \leq a_{0}\beta _{L^{2}( \Omega ) }(M) .
\]
In addition assume that there exists $R>0$ with
\[
\varepsilon R^{2}\geq aT^{\frac{2-\alpha }{2}}R^{\alpha }+b\frac{1}{2^{\beta
}}T^{\frac{2+\beta -\gamma }{2}}R^{\beta +\gamma }.
\]
Then for sufficiently small $| \sigma | $ and $a_{0}$
\eqref{eqPII} has a solution $u\in H^{2}( 0,T;L^{2}( \Omega )) $ with
\[
| u'| _{2}=\Big( \int_{0}^{T}| u'(
t) | _{L^{2}( \Omega ) }^{2}dt\Big) ^{1/2}\leq R.
\]
\end{theorem}

\begin{proof}
For any $u,v\in L^{2}( \Omega ) $ with
$| u|_{L^{2}( \Omega ) },| v| _{L^{2}( \Omega )}\leq \eta $, we have
\begin{align*}
| s( u) -s( v) | _{L^{2}( \Omega) }
&=| \sigma | | | u| _{L^{2}( \Omega )
}^{p-2}u-| v| _{L^{2}( \Omega ) }^{p-2}v|_{L^{2}( \Omega ) } \\
&\leq | \sigma | ( | | u| _{L^{2}(\Omega ) }^{p-2}( u-v) | _{L^{2}( \Omega )}
+| ( | u| _{L^{2}( \Omega ) }^{p-2}-|v| _{L^{2}( \Omega ) }^{p-2}) v| _{L^{2}
(\Omega ) }) \\
&\leq | \sigma | ( \eta ^{p-2}| u-v|_{L^{2}( \Omega ) }+( p-2) \eta ^{p-2}|
u-v| _{L^{2}( \Omega ) }) \\
&=| \sigma | ( p-1) \eta ^{p-2}| u-v|_{L^{2}( \Omega ) }.
\end{align*}
Hence for any bounded $M\subset L^{2}( \Omega ) $ one has
\[
\beta _{L^{2}( \Omega ) }( g( t,M,L^{2}( \Omega
) ) ) \leq [ | \sigma | ( p-1)| M| ^{p-2}+a_{0}] \beta _{L^{2}( \Omega )}( M)
\]
where, as above, $g( t,u,v) =s( u) +h(t,u,v) $, and
$| M| =\sup_{u,v\in M}|u-v| _{L^{2}( \Omega ) }$. It is easily seen that the
unique solution $\varphi \in $  $L^{\infty }( 0,T;\mathbb{R}_{+}) $
of (\ref{eq2.200}) with
\[
\omega ( t,\tau ) =[ | \sigma | ( p-1)
\eta ^{p-2}+a_{0}] \tau
\]
where $\eta =R\max \{ 1,\sqrt{T}/2\} $, is $\varphi =0$ provided
that $| \sigma | $ and $a_{0}$ are small enough. Thus Theorem \ref{thm4.2}
applies.
\end{proof}

\begin{corollary} \label{coro5.4}
For every $f\in L^{\infty }( 0,T;L^{2}( \Omega ) ) $, if $| \sigma | $ is
sufficiently small the problem
\begin{gather*}
-u_{tt}-\varepsilon u_{t}+\sigma | u| _{L^{2}( \Omega
) }^{p-2}u-\Delta _{x}u=f( t,x) \quad \text{a.e. on }[0,T]\times \Omega \\
u( t,.) \in H_{0}^{1}( \Omega ) \quad \text{for a.e. }t\in [ 0,T] \\
u( 0,x) =-u( T,x) ,\quad u_{t}( 0,x)=-u_{t}( T,x) \quad \text{a.e. on }\Omega .
\end{gather*}
has at least one solution $u\in H^{2}( 0,T;L^{2}( \Omega )) $ with
$| u'| _{2}\leq \frac{| f|_{\infty }\sqrt{T}}{\varepsilon }$.
Here $| f| _{\infty }= \mathop{\rm ess\,sup}_{t\in [ 0,T] }| f( t) |
_{L^{2}( \Omega ) }$.
\end{corollary}

\begin{thebibliography}{99}
\bibitem{AAP} A. R. Aftabizadeh, S.\ Aizicovici and N.H.\ Pavel,
\textit{Anti-periodic boundary value problems for higher order differential
equations in Hilbert spaces}, Nonlinear Anal. \textbf{18 }(1992),
253-267.

\bibitem{AAP2} A.R. Aftabizadeh, S.\ Aizicovici and N.H.\ Pavel, \textit{On
a class of second-order anti-periodic boundary value problems}, J.\ Math.
Anal.\ Appl. \textbf{171} (1992), 301-320.

\bibitem{AP} S.\ Aizicovici and N.H.\ Pavel, \textit{Anti-periodic solutions
to a class of nonlinear differential equations in Hilbert spaces},
J. Funct. Anal. \textbf{99} (1991), 387-408.

\bibitem{APV} S. Aizicovici, N.H. Pavel and I.I. Vrabie, \textit{
Anti-periodic solutions to strongly nonlinear evolution equations in Hilbert
spaces}, An. \c{S}tiin\c{t}. Univ. Al.I. Cuza Ia\c{s}i Mat. \textbf{44 }(1998),
227-234.

\bibitem{B} V. Barbu, ``Nonlinear Semigroups and Differential Equations in
Banach Spaces'', Ed. Academiei \& Noordhoff International Publishing,
Bucure\c{s}ti-Leyden, 1976.

\bibitem{CPa} Z.\ Cai and N.H.\ Pavel, \textit{Generalized periodic and
anti-periodic solutions for the heat equation in} $R^{1}$, Libertas Math.
\textbf{10} (1990), 109-121.

\bibitem{Coron} J.-M.\ Coron, \textit{Periodic solutions of a nonlinear wave
equation without assumption of monotonicity}, Math. Ann. \textbf{262 }
(1983), 272-285.

\bibitem{CK} J.-F. Couchouron and M.\ Kamenski, \textit{A unified
topological point of view for integro-differential inclusions,\
}in ``Differential Inclusions and Optimal Control'' (eds. J.
Andres, L. G\'{o}rniewicz and P. Nistri)\textit{, }Lecture Notes
in Nonlinear Anal., Vol. 2, 1998, 123-137.

\bibitem{CKP} J.-F. Couchouron, M.\ Kamenski and R.\ Precup, \textit{A
nonlinear periodic averaging principle,} Nonlinear Anal.
\textbf{54} (2003), 1439-1467.

\bibitem{CP} J.-F.\ Couchouron and R.\ Precup, \textit{Existence principles
for inclusions of Hammerstein type involving noncompact acyclic multivalued
maps,} Electronic J.\ Differential Equations \textbf{2002} (2002), No. 04,
1-21.

\bibitem{deimling} K. Deimling, ``Multivalued Differential
Equations'', Walter de Gruyter, Berlin-New York, 1992.

\bibitem{diestel} J. Diestel, W.M. Ruess and\ W. Schachermayer, \textit{Weak
compactness in} $L^{1}(\mu ,\,X)$, Proc. Amer. Math. Soc. \textbf{118}
(1993), 447-453.

\bibitem{em} S. Eilenberg and D. Montgomery, \textit{Fixed point theorems
for multivalued transformations,} Amer. J. Math. \textbf{68 }
(1946), 214-222.

\bibitem{pf} P.M. Fitzpatrick and W.V. Petryshyn, \textit{Fixed point
theorems for multivalued noncompact acyclic mappings,} Pacific J. Math.
\textbf{54 }(1974), 17-23.

\bibitem{frigon} M. Frigon, \textit{Th\'{e}or\`{e}mes d'existence de
solutions d'inclusions diff\'{e}rentielles}, in
``Topological Methods in Differential Equations and
Inclusions'' (eds. A. Granas and M. Frigon), NATO\ ASI\
Series C, Vol. 472, Kluwer Academic Publishers, Dordrecht-Boston-London,
1995, 51-87.

\bibitem{guo} D. Guo, V. Lakshmikantham and X. Liu,
``Nonlinear Integral Equations in Abstract Spaces'', Kluwer
Academic Publishers, Dordrecht-Boston-London, 1996.

\bibitem{H} A. Haraux, \textit{Anti-periodic solutions of some nonlinear
evolution equations,} Manuscripta Math. \textbf{63} (1989), 479-505.

\bibitem{hp} S. Hu and N.S. Papageorgiou, ``Handbook of Multivalued
Analysis, Vol. I: Theory'', Kluwer Academic Publishers,
Dordrecht-Boston-London, 1997.

\bibitem{O} H. Okochi, \textit{On the existence of periodic solutions to
nonlinear abstract parabolic equations,} J. Math. Soc. Japan \textbf{40}
(1988), 541-553.

\bibitem{O2} H.\ Okochi, \textit{On the existence of anti-periodic solutions
to nonlinear evolution equations associated with odd subdifferential
operators,} J.\ Funct.\ Anal. \textbf{91} (1990), 246-258.
\end{thebibliography}

\end{document}