\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 08, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/08\hfil
 Existence and uniqueness of anti-periodic solutions]
{Existence and uniqueness of anti-periodic solutions
 for nonlinear third-order \\ differential inclusions}

\author[T. Haddad, T. Haddad \hfil EJDE-2013/08\hfilneg]
{Touma Haddad, Tahar  Haddad}  % in alphabetical order

\address{Touma Haddad \newline
D\'epartement de Math\'ematiques,
Facult\'e des Sciences,
Universit\'e de Jijel, B.P. 98, Alg\'erie}
\email{touma.haddad@yahoo.com}

\address{Tahar  Haddad \newline
Laboratoire LMPA, Facult\'e des Sciences,
Universit\'e de Jijel, B.P. 98, Alg\'erie}
\email{haddadtr2000@yahoo.fr}

\thanks{Submitted September 28, 2012. Published January 9, 2013.}
\subjclass[2000]{34C25, 34G20, 49J52}
\keywords{Anti-periodic solution; differential inclusions; subdifferential}

\begin{abstract}
 In this article, we study the existence  of anti-periodic
 solutions for the  third-order differential inclusion
 \begin{gather*}
 u'''(t)\in \partial\varphi(u'(t))+F(t,u(t))\quad \text{a.e. on }[0,T]\\
 u(0)=-u(T), \quad u'(0)=-u'(T),\quad u''(0)=-u''(T),
 \end{gather*}
 where $\varphi$ is a proper convex, lower semicontinuous and even
 function, and $F$ is an upper semicontinuous convex compact
 set-valued mapping. Also uniqueness of anti-periodic
 solution is studied.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\allowdisplaybreaks

\section{Introduction}

Existence and uniqueness of anti-periodic solutions for differential
inclusions generated by the subdifferential of a convex lower
semicontinuous even function appear in several articles;  see
\cite{AP1,AP2,AP3,AP4,CH,CP, HA}.
Okochi \cite{ OK1} initiated the study of
anti-periodic solutions of the  differential inclusion
\begin{equation}\label{eq1.1}
\begin{gathered}
f(t)\in u'(t)+\partial\varphi(u(t))\quad\text{a.e. } t\in [0,T]\\
u(0)=-u(T)\quad
\end{gathered}
\end{equation}
in Hilbert spaces, where $\partial\varphi$ is the subdifferential of
an even function $\varphi$ on a real Hilbert space $H$ and
$f\in L^2([0,T],H)$. It was shown in \cite{OK2}, by applying a fixed
point theorem for nonexpansive mapping, that \eqref{eq1.1} has a
unique solution. Later Aftabizadeh and al \cite{AF} studied the
anti-periodic solution of  third-order differential inclusion
\begin{equation}\label{eq1.2}
\begin{gathered}
u'''(t)\in \partial\varphi(u'(t))+f(t)\quad \text{a.e. }t\in [0,T]\\
u(0)=-u(T),\quad  u'(0)=-u'(T), \quad u''(0)=-u''(T),
\end{gathered}
\end{equation}
by using maximal monotone operator theory.

 The  aim of this article is to study the existence of anti-periodic
solutions for the third-order differential inclusion
\begin{equation}\label{eq1.3}
\begin{gathered}
u'''(t)\in \partial\varphi(u'(t))+F(t,u(t))\quad \text{a.e. } t\in [0,T]\\
u(0)=-u(T), \quad u'(0)=-u'(T),\quad u''(0)=-u''(T),
\end{gathered}
\end{equation}
where $\varphi:H\to ]-\infty,+\infty]$ is a convex lower
semicontinuous even function and $F:[0,T]\times H\to 2^{H}$
is an upper semicontinuous convex compact set-valued mapping bounded
above by $L^2$ function. Furthermore, an existence and uniqueness
result when $F$ is single-valued is also studied.

\section{Preliminaries}

Let $H$ be a real Hilbert space with norm $\|\cdot\|$ and inner product
$\langle  \cdot,\cdot \rangle$. The open ball centered at $x$ with radius
$r$ is defined by $ \mathbb{B}_{r}(x)=\{y\in H:\|y-x\|<r\}$, where
$\overline{\mathbb{B}_{r}}(x)$ denotes its closure.
For a proper lower semicontinuous convex function
$\varphi:H\to ]-\infty,+\infty]$, the set-valued mapping
$\partial\varphi: H\to 2^{H}$ defined by
$$
\partial\varphi(x)=\{\xi\in H:\varphi(y)-\varphi(x)\geq\langle\xi,
y-x\rangle,\forall y\in H\}
$$
which is the subdifferential of $\varphi$.
Let us recall a classical closure type lemma from \cite{CR}.

\begin{lemma}\label{lem1}
Let $H$ be a separable Hilbert space. Let $\varphi$ be a convex
lower semicontinuous function defined on $H$ with values in
$]-\infty,+\infty]$. Let $(u_n)_{n\in\mathbb{N}\cup \{\infty\}}$
be a sequence of measurable mappings from $[0,T]$ into $H$ such that
$u_n \to u_{\infty}$ pointwise with respect to the norm
topology. Assume that $(\xi_n)_{n\in\mathbb{N}}$ is a sequence in
$L^2([0,T],H) $ satisfying
$$
\xi_n(t)\in \partial\varphi(u_n(t)) \quad\text{a.e. }t\in [0,T]
$$
for each $n\in \mathbb{N} $ and converging weakly to
$\xi_{\infty}\in L^2([0,T],H)$. Then we have
$$
\xi_{\infty}(t)\in
\partial\varphi(u_{\infty}(t)) \quad\text{a.e. }t\in [0,T].
$$
\end{lemma}

Let us recall a useful result.

\begin{lemma}[\cite{S}] \label{lem2}
Let $H$ be a real Hilbert space. Let
$u\in W_{\rm loc}^{1,2}(\mathbb{R},H)$ be $2T$-periodic and satisfying
$\int_0^{2T} u(t) dt =0$,
then
$$
\|u\|_{L^2([0,2T],H)}\leq\frac{T}{\pi}\|u'\|_{L^2([0,2T],H)}.
$$
\end{lemma}

\section{Main results}

We state and summarize some useful results for anti-periodic
mappings that are crucial for our purpose.

\begin{proposition}\label{prop1}
Let $H$ be a real Hilbert space. Let $u\in W^{3,2}([0,T],H)$
satisfying $u(0)=-u(T)$, $u'(0)=-u'(T)$, $u''(0)=-u''(T)$,
then the following inequalities hold
\begin{itemize}
\item[(A1)] $\|u\|_{\mathcal{C}([0,T],H)
}\leq\frac{\sqrt[]{T}}{2}\|u'\|_{L^2([0,T],H) }$;

\item[(A2)] $\|u\|_{L^2([0,T],H)}\leq\frac{T}{\pi}\|u'\|_{L^2([0,T],H)}$;

\item[(B1)] $\|u'\|_{\mathcal{C}([0,T],H) }\leq\frac{\sqrt[]{T}}{2}
    \|u''\|_{L^2([0,T],H)   }$;

\item[(B2)] $\|u'\|_{L^2([0,T],H)}\leq\frac{T}{\pi}\|u''\|_{L^2([0,T],H) }$;

\item[(C1)]  $\|u''\|_{\mathcal{C}([0,T],H)}\leq\frac{\sqrt[]{T}}{2}
    \|u'''\|_{L^2([0,T],H)}$.
\end{itemize}
\end{proposition}


\begin{proof}
(A1) Since
$u(t)=u(0)+\int_0^{t}u'(s) ds$ and
$u(t)=u(T)-\int_{t}^{T}u'(s)ds$, for all $t\in[0,T]$, by adding
these equalities,  by anti-periodicity, we obtain
$$
2u(t)= \int_0^{t}u'(s) ds-\int_{t}^{T}u'(s)ds, \quad
\forall t\in[0,T].
$$
Hence we have
$$
2 \|u(t)\| \leq \int_0^{t}\|u'(s)\| ds
+\int_{t}^{T}\|u'(s)\|ds
=\int_0^{T}\|u'(s)\|ds, \quad \forall t\in[0,T],
$$
and so, by Holder inequality
$$
\|u\|_{\mathcal{C}([0,T],H)}=\sup_{t\in[0,T]}\|u(t)\|
\leq\frac{\sqrt[]{T}}{2}\|u'\|_{L^2([0,T],H)}.
$$

(A2) For the sake of simplicity, we use the same
 notation $u(t), t\in \mathbb{R}$, to denote the anti-periodic
 extension of $u(t)$, $t\in[0,T]$, such that
$$
u(t+T)=-u(t)\quad \text{and}\quad
u'(t+T)=-u'(t), \quad \text{for } t\in \mathbb{R}.
$$
Then  $u$ is $2T$-periodic function since
$$
u(t+2T)=u(t+T+T)=-u(t+T)=u(t).
$$
Also, since
\[
\int_0^{2T}u(t)dt
=\int_0^{T}u(t)dt+\int_{T}^{2T}u(t)dt
=\int_0^{T}u(t)dt-\int_0^{T}u(t)dt=0,
\]
so, by Lemma \ref{lem2},
$$
\int_0^{2T}\|u(t)\|^2dt\leq \frac{T^2}{\pi^2}\int
_0^{2T}\|u'(t)\|^2dt.
$$
Let us observe that $\|u(t)\|^2$ and
$\|u'(t)\|^2$ are T-periodic because
$\|u(t+T)\|^2=\|-u(t)\|^2=\|u(t)\|^2$ and similarly
$\|u'(t+T)\|^2=\|-u'(t)\|^2=\|u'(t)\|^2.$\\Hence we
deduce that
$$
\int_0^{2T}\|u(t)\|^2dt=2\int _0^{T}\|u(t)\|^2dt\quad
\text{and}\quad
\int_0^{2T}\|u'(t)\|^2dt=2\int_0^{T}\|u'(t)\|^2dt.
$$
 Finally we get the required inequality
$$
\int_0^{T}\|u(t)\|^2dt\leq \frac{T^2}{\pi^2}\int
_0^{T}\|u'(t)\|^2dt.
$$
Similarly, we prove (B1), (B2) and (C1),
 using $u'(0)=-u'(T)$, and $u''(0)=-u''(T)$ respectively.
\end{proof}

The following result deal with convex compact valued perturbations of
a third-order differential inclusion governed by subdifferential
operators of convex lower semicontinuous functions with
anti-periodic boundary conditions. First, to simplify we will assume
that $H=\mathbb{R}^{d}$.


\begin{theorem}\label{Theo1}
Let $H=\mathbb{R}^{d}$, $\varphi:H\to ]-\infty,+\infty]$ be
a proper, convex, lower semicontinuous and even function. Let
$F:[0,T]\times H\to 2^{H}$ be a convex compact set-valued
mapping, measurable on $[0,T]$ and upper semicontinuous on $H$
satisfying:  there is a function $\alpha(\cdot)\in
L^2([0,T],\mathbb{R_{+}})$ such that
$$
F(t,x) \subset \Gamma(t):=\overline{\mathbb{B}}_{\alpha(t)}(0)\quad
\text{for all } (t,x) \in [0,T]\times H.
$$
Then the problem
\begin{gather*}
u'''(t)\in \partial\varphi(u'(t))+F(t,u(t))\quad\text{a.e. }t\in [0,T],\\
u(0)=-u(T), \quad u'(0)=-u'(T), \quad u''(0)=-u''(T),
\end{gather*}
has at least an anti-periodic $W^{3,2}([0,T],H) $ solution.
\end{theorem}


\begin{proof}
 Recall that a $W^{3,2}([0,T],H) $ function $u:[0,T]\to H$ is a solution
of the problem under consideration if there exists a function
$h\in L^2([0,T];H)$ such that
\begin{gather*}
u'''(t)\in \partial\varphi(u'(t))+h(t)\quad\text{a.e. }t\in [0,T],\\
h(t)\in  F(t,u(t))\quad\text{a.e. }t\in [0,T], \\
 u(0)=-u(T),\quad u'(0)=-u'(T),\quad u''(0)=-u''(T).
\end{gather*}
Let us denote by $S^2_{\Gamma}$ the set of all
$L^2([0,T];H)$-selection of $\Gamma$
$$
S^2_{\Gamma}:=\{ f\in L^2([0,T],H): f(t)\in\Gamma(t)\text{ a.e. }t\in
[0,T]\}.
$$
By \cite[Theorem 2.1]{AP1}, for all $f\in S^2_{\Gamma}$, there is a
unique $W^{3,2}([0,T],H) $ solution $u_f$ of
\begin{gather*}
u_f'''(t)\in \partial\varphi(u_f'(t))+f(t)\quad\text{a.e. }t\in [0,T],\\
 u_f(0)=-u_f(T), \quad u_f'(0)=-u_f'(T), \quad u_f''(0)=-u_f''(T),
\end{gather*}
such that
\begin{equation} \label{e*}
\|u_f'''\|_{L^2([0,T],H)}\leq\|f\|_{L^2([0,T],H)}.
\end{equation}
For each $f\in S^2_{\Gamma}$, let us define the set-valued mapping
$$
\Psi(f):=\{g\in L^2([0,T],H); g(t)\in F(t,u_f(t))\quad\text{a.e. }t\in [0,T]\}.
$$
Then it is clear that $\Psi(f)$ is a nonempty
convex weakly compact subset of $S^2_{\Gamma}$, here the
nonemptiness follows from \cite[theorem VI.4]{CV}.
From the above consideration, we need to prove that the convex weakly compact
set-valued mapping
$\Psi:S^2_{\Gamma}\to2^{S^2_{\Gamma}}$ admits a fixed
point. By  Kakutani-Ky Fan fixed point theorem, it is
sufficient to prove that $\Psi$ is upper semicontinuous when
$S^2_{\Gamma}$ is endowed with the weak topology of
$L^2([0,T],H)$.
As $L^2([0,T],H)$ is separable,
$S^2_{\Gamma}$ is compact metrizable with respect to the weak
topology  of $L^2([0,T],H)$. So it turns out to check that the
graph $gph(\Psi)$ is sequentially weakly closed in
$S^2_{\Gamma}\times S^2_{\Gamma}$.
 Let $(f_n,g_n)_n\in gph(\Psi) $ weakly converging to
$(f,g)\in S^2_{\Gamma}\times S^2_{\Gamma}$.
From the definition of $\Psi$, that means
$u_{f_n}$ is the unique $W^{3,2}([0,T],H) $ solution of
\begin{gather*}
u_{f_n}'''(t)\in
\partial\varphi(u_{f_n}'(t))+f_n(t)\quad\text{a.e. }t\in [0,T],\\
 u_{f_n}(0)=-u_{f_n}(T), \quad u_{f_n}'(0)=-u_{f_n}'(T),\quad
 u_{f_n}''(0)=-u_{f_n}''(T),
\end{gather*}
with $f_n\in S^2_{\Gamma}$ and $g_n(t)\in F(t,u_{f_n}(t))$ a.e. $t\in [0,T]$.
Taking into account the
antiperiodicity of $u_{f_n}''$, $u_{f_n}'$ and
$u_{f_n}$,  proposition \ref{prop1} gives
 \begin{gather*}
   \|u''_{f_n}\|_{\mathcal{C}([0,T],H)}\leq
   \frac{\sqrt{T}}{2}\|u'''_{f_n}\|_{L^2([0,T],H)},
   \\
\|u'_{f_n}\|_{\mathcal{C}([0,T],H)}\leq
\frac{T\sqrt{T}}{2\pi}\|u'''_{f_n}\|_{L^2([0,T],H)},
 \\
\|u_{f_n}\|_{\mathcal{C}([0,T],H)}\leq
\frac{T^2\sqrt{T}}{2\pi^2}\|u'''_{f_n}\|_{L^2([0,T],H)}.
 \end{gather*}
for all $n\geq1$.
Using the estimate \eqref{e*}, we have
$$
\|u_{f_n}'''\|_{L^2([0,T],H)}\leq
\|f_n\|_{L^2([0,T],H)}\leq\|\alpha\|_{L^2([0,T],R)}<+\infty.
$$
We may conclude that
\begin{gather*}
  \sup_{n\geq 1}\|u''_{f_n}\|_{\mathcal{C}([0,T],H)}<+\infty,  \\
  \sup_{n\geq 1}\|u'_{f_n}\|_{\mathcal{C}([0,T],H)}<+\infty,\\
  \sup_{n\geq 1}\|u_{f_n}\|_{\mathcal{C}([0,T],H)}<+\infty.
\end{gather*}
By extracting  suitable subsequences, we may assume that
$(u'''_{f_n})$ converges weakly in $ L^2([0,T],H)$ to a
function $\gamma \in L^2([0,T],H)$ and $(u''_{f_n})$
converges pointwise  to a function $w$, namely
\begin{align*}
  w(t)
&:=\lim_{n\to+\infty}u''_{f_n}(t)=\lim_{n\to+\infty} (u''_{f_n}(0)
 +\int_0^{t} u'''_{f_n}(s)ds)\\
&= \lim_{n\to+\infty}u''_{f_n}(0)+\int_0^{t}\gamma(s)ds,\quad \forall
   t\in[0,T].
\end{align*}
Then
\begin{align*}
  v(t) &:=\lim_{n\to+\infty}u'_{f_n}(t)
 =\lim_{n\to+\infty} (u'_{f_n}(0)+\int_0^{t} u''_{f_n}(s)ds)\\
&=\lim_{n\to+\infty}u'_{f_n}(0)+\int_0^{t}w(s)ds,\quad \forall
   t\in[0,T].
\end{align*}
So we have
\begin{align*}
  u(t) &:=\lim_{n\to+\infty}u^{}_{f_n}(t)
=\lim_{n\to+\infty} (u^{}_{f_n}(0)+\int_0^{t} u'_{f_n}(s)ds)\\
&= \lim_{n\to+\infty}u^{}_{f_n}(0)+\int_0^{t}v(s)ds,\quad \forall
   t\in[0,T].
\end{align*}
 We conclude that $u\in W^{3,2}([0,T],H) $ with $u'=v$, $u''=w$ and
$u'''=\gamma$ and satisfying the anti-periodic conditions
$u(0)=-u(T)$,  $u'(0)=-u'(T)$ and
$u''(0)=-u''(T)$. Furthermore, we see that $u'_{f_n}$ converges
pointwise to $u'$ and $u'''_{f_n}$ converges to $u'''$
with respect to the weak topology of $L^2([0,T],H)$.
Combining these facts and applying Lemma \ref{lem1} to the inclusion
$$
u'''_{f_n}(t)-f_n(t)\in\partial\varphi(u'_{f_n}(t))\quad\text{a.e. }t\in [0,T]
$$
it yields
$$
u'''(t)-f(t)\in\partial\varphi(u'(t))\quad\text{a.e. }t\in [0,T].
$$
By uniqueness \cite[Theorem 2.1]{AF}, we have $u=u_f$.
Further using the inclusion
$$
g_n(t)\in F(t,u_{f_n}(t))\quad\text{a.e. }t\in [0,T]
$$
and invoking the closure type Lemma in \cite[theorem VI.4]{CV},  we have
$$
g(t)\in F(t,u_f(t))\quad\text{a.e. }t\in [0,T].
$$
We may then applying the Kakutani-Ky Fan fixed point theorem to the
set-valued mapping $\Psi$ to obtain some $f\in S^2_{\Gamma}$ such
that $f\in \Psi(f)$ or
$$
f(t)\in F(t,u(t))\quad\text{a.e. }t\in [0,T].
$$
This means that
\begin{gather*}
u'''(t)\in \partial\varphi(u'(t))+f(t)\quad \text{a.e. }t\in [0,T],\\
f(t)\in F(t,u(t))\quad \text{a.e. }t\in [0,T],\\
 u(0)=-u(T), \quad u'(0)=-u'(T),\quad  u''(0)=-u''(T).
\end{gather*}
The proof is complete.
\end{proof}

 A more general version of the preceding result is
available by introducing some inf-compactness assumption
\cite{AP1} on the function $\varphi$.

\begin{theorem}\label{Theo2}
Let $H$ be a separable Hilbert space,
$\varphi:H \to [0,+\infty]$ be a proper, convex, lower semicontinuous
and even function satisfying: $\varphi(0)=0$ and for each
$\beta_1,\beta_2>0$, the set
$\{x\in D(\varphi): \|x\| \leq \beta_1, \varphi(x) \leq \beta_2 \}$
is compact. Let
$F:[0,T]\times H\to 2^{H}$ be a convex compact set-valued
mapping, measurable on $[0,T]$ and  upper semicontinuous on $H$
satisfying:
there is $\alpha(\cdot)\in L^2([0,T],\mathbb{R_{+}})$ such that
$$
F(t,x) \subset \Gamma(t):=\overline{\mathbb{B}}_{\alpha(t)}(0)\quad
\forall (t,x) \in [0,T]\times H
$$
Then the problem
\begin{gather*}
u'''(t)\in \partial\varphi(u'(t))+F(t,u(t))\quad\text{a.e. }
t\in [0,T],\\
u(0)=-u(T), \quad u'(0)=-u'(T), \quad u''(0)=-u''(T),
\end{gather*}
has at least an anti-periodic $W^{3,2}([0,T],H)   $ solution.
\end{theorem}


\begin{proof}
 Using the notation of the proof of Theorem \ref{Theo1}, we have
$$
u_{f_n}'''(t)-f_n(t)\in
\partial\varphi(u_{f_n}'(t))\quad\text{a.e. }t\in [0,T],
$$
for every $f_n\in S^2_{\Gamma}$. The absolute continuity of
$\varphi(u_{f_n}'(\cdot))$ and the chain rule theorem
\cite{BZ}, yield
$$
\langle u_{f_n}'''(t), u_{f_n}''(t)\rangle
-\langle f_n(t),u_{f_n}''(t)\rangle=\frac{d}{dt}\varphi(u_{f_n}'(t)),
$$
for every $f_n\in S^2_{\Gamma}$, so that
\[
+\infty > \sup_{n\geq 1}\int_0^{T} |\langle
u_{f_n}'''(t), u_{f_n}''(t)\rangle
-\langle f_n(t),u_n''(t)\rangle| dt
= \sup_{n\geq 1} \int_0^{T}
\big|\frac{d}{dt}\varphi(u_{f_n}'(t))\big|dt.
\]
Further applying the classical definition of the subdifferential to
convex function $\varphi$ yields
$$
0=\varphi(0)\geq
\varphi(u_{f_n}'(t))+\langle0-u_{f_n}'(t),u_{f_n}'''(t)-f_n(t)\rangle
$$
or
$$
0\leq\varphi(u_{f_n}'(t))\leq\langle u_{f_n}'(t),u_{f_n}'''(t)-f_n(t)\rangle.
$$
Hence
$$
\sup_{n\geq 1}|\varphi(u_{f_n}')|_{L^{1}_{\mathbb{R}}([0,T])}<+\infty.
$$
For all $t \in [0,T]$, we have
\[
\varphi(u_{f_n}'(t))
=\varphi(u_{f_n}'(0))+\int_0^{t} \frac{d}{dt}\varphi(u_{f_n}'(s))ds
\leq \varphi(u_{f_n}'(0))+\sup_{n\geq
1}|\varphi(u_{f_n}')|_{L^{1}_{\mathbb{R}}([0,T])}.
\]
Now we assert that $\varphi(u_{f_n}'(t))\leq \beta_2$ for
every $t\in [0,T]$, here $\beta_2$ is a positive constant. Indeed
for all $t\in[0,T]$, we have
\begin{align*}
\varphi(u_{f_n}'(0))
&\leq |\varphi(u_{f_n}'(t))-\varphi(u_{f_n}'(0))|+\varphi(u_{f_n}'(t))\\
&\leq \int_0^{T}|\frac{d}{dt}\varphi(u_{f_n}'(t))|dt+\varphi(u_{f_n}'(t)).
\end{align*}
Hence
$$
\varphi(u_{f_n}'(0))\leq\sup_{n\geq 1} \int_0^{T}
|\frac{d}{dt}\varphi(u_{f_n}'(t))|dt+\frac{1}{T}\sup_{n\geq 1}
\int_0^{T}\varphi(u_{f_n}'(t)) dt <+\infty .
$$
Whence we have
$$
\beta_1:=\sup_{n\geq 1}\sup_{t\in[0,T]}\|u_{f_n}'(t)\|<+\infty ,
\quad
\beta_2:=\sup_{n\geq 1}\sup_{t\in[0,T]}\varphi(u_{f_n}'(t))<+\infty .
$$
So that $(u_{f_n}'(t))$ is relatively compact with respect to the norm
topology of $H$ using the inf-compactness assumption on $\varphi$.
The proof can be therefore achieved as Theorem \ref{Theo1} by
invoking Lemma\ref{lem1} and a closure type lemma in
\cite[Theorem VI-4]{CV}.
\end{proof}

 Here is an existence and uniqueness result related to
Theorem \ref{Theo2} when the perturbation is single-valued.

\begin{theorem}\label{Theo3}
Let $H$ be a separable Hilbert space, $\varphi:H \to
[0,+\infty]$ is a proper, convex, lower semicontinuous and even
function satisfying: $\varphi(0)=0$ and for each $\alpha,\beta>0$,
the set $\{x\in D(\varphi): \|x\| \leq \alpha, \varphi(x) \leq \beta
\}$ is compact and  $f:[0,T]\times H
 \to H$ is a Carath\'eodory mapping satisfying :
\\ \emph{$(\mathcal {H}_1)$} $\| f(t,u)-f(t,v)\|\leq L\|u-v\|$ for
all $(t,u,v)\in[0,T]\times H\times H$, for some positive constant $L>0$.\\
\emph{$(\mathcal {H}_2)$} There is a
$L^2([0,T];\mathbb{R^{+}})$ integrable function
$\alpha:\mathbb{R}\to\mathbb{R^{+}} $ such that
 $\|f(t,u)\|\leq \alpha(t)$ for all $(t,u)\in[0,T]\times H$. If
$0<T<\frac{\pi}{\sqrt[3]{L}}$, then the inclusion
\begin{gather*}
u'''(t)\in \partial\varphi(u'(t))+f(t,u(t))\quad\text{a.e. }t\in [0,T],\\
 u(0)=-u(T),\quad u'(0)=-u'(T),\quad u''(0)=-u''(T),
\end{gather*}
admits a unique $W^{3,2}([0,T];H)$-anti-periodic solution.
\end{theorem}


\begin{proof}
Existence of at least one $W^{3,2}([0,T];H)$-anti-periodic solution
is ensured by Theorem \ref{Theo2}. Indeed, we put
$F(t,u):=\{f(t,u)\}$ for all $(t,u)\in[0,T]\times H$,\
 As $f$ is a Carath\'eodory function, then: $ u\longmapsto f(t,u)$
is continuous, for almost all $t\in [0,T]$  and $t\longmapsto f(t,u)$
is Lebesgue measurable, for  all $u\in H$.
More, by assumption; there is a
$L^2([0,T];\mathbb{R^{+}})$ integrable function $\alpha(\cdot)$
such that
$$
\|f(t,u)\| \leq \alpha(t)\quad \text{for all }(t,u)\in[0,T]\times H.
$$
Therefore, $F$ satisfies hypotheses of Theorem \ref{Theo2}.

 To prove uniqueness, we assume that $(u_1)$ and $(u_2)$ are two
solutions of the inclusion under consideration.
\begin{gather*}
u_1'''(t)\in \partial\varphi(u_1'(t))+f(t,u_1(t))\quad\text{a.e. }t\in [0,T],\\
 u_1(0)=-u_1(T), \quad u_1'(0)=-u_1'(T),\quad  u_1''(0)=-u_1''(T),
\end{gather*}
and
\begin{gather*}
u_2'''(t)\in \partial\varphi(u_2'(t))+f(t,u_2(t))\quad\text{a.e. }t\in [0,T],\\
 u_2(0)=-u_2(T),\quad u_2'(0)=-u_2'(T),\quad
 u_2''(0)=-u_2''(T).
\end{gather*}
For simplicity, let us set
\begin{gather*}
v_1(t)=u_1'''(t)-f(t,u_1(t)),\quad \forall t\in[0,T], \\
v_2(t)=u_2'''(t)-f(t,u_2(t)),\quad \forall t\in[0,T].
\end{gather*}
Then we have
\begin{equation}\label{eq3.1}
v_1(t)-v_2(t)=u_1'''(t)-u_2'''(t)-f(t,u_1(t))+f(t,u_2(t)),\quad
\text{a.e. }t\in [0,T].
\end{equation}
 Multiplying scalarly \eqref{eq3.1} by
$(u'_1-u'_2)$ and integrating on $[0,T]$ yields
\begin{equation}\label{eq3.2}
\begin{aligned}
&\int_0^{T}\langle v_1(t)-v_2(t),u_1'(t)-u_2'(t)\rangle dt\\
&= \int_0^{T}\langle
u_1'''(t)-u_2'''(t),u_1'(t)-u_2'(t)\rangle dt\\
&\quad -\int_0^{T} \langle
f(t,u_1(t))-f(t,u_2(t)),u_1'(t)-u_2'(t)\rangle dt
\end{aligned}
\end{equation}
As $v_1\in\partial\varphi(u'_1)$
and $v_2\in\partial\varphi(u'_2)$ by monotonicity of
$(\partial\varphi)$, \eqref{eq3.2} implies
\begin{equation}\label{eq3.3}
\begin{aligned}
&\int_0^{T}\langle u_1'''(t)-u_2'''(t),u_1'(t)-u_2'(t)\rangle dt\\
&\geq\int_0^{T} \langle f(t,u_1(t))-f(t,u_2(t)),u_1'(t)-u_2'(t)\rangle dt.
\end{aligned}
\end{equation}
 By antiperiodicity, we have
\begin{align*}
&\int_0^{T}\langle u_1'''(t)-u_2'''(t),u_1'(t)-u_2'(t)\rangle dt\\
&=\langle u_1''(T)-u_2''(T),u_1'(T)-u_2'(T)\rangle-\langle
u_1''(0)-u_2''(0),u_1'(0)-u_2'(0)\rangle \\
&\quad -\int_0^{T}\langle u_1''(t)-u_2''(t),u_1''(t)-u_2''(t)\rangle dt\\
&=-\int_0^{T}\| u_1''(t)-u_2''(t)\|^2dt.
\end{align*}
The inequality  \eqref{eq3.3} gives
\begin{align*}
\int_0^{T}\| u_1''(t)-u_2''(t)\|^2dt
&\leq \int_0^{T} \langle f(t,u_2(t))-f(t,u_1(t)),u_1'(t)-u_2'(t)\rangle dt\\
&\leq  L\int_0^{T} \|u_1(t)-u_2(t)\|\,\|u'_1(t)-u'_2(t)\|dt.
\end{align*}
By Holder's inequality, we obtain
\[
\|u_1''-u_2''\|^2_{L^2([0,T],H)}
\leq L\|u_1-u_2\|_{L^2([0,T],H)} \|u'_1-u'_2\|_{L^2([0,T],H)}.
\]
Using the estimates (A1) and (A2) in
proposition \ref{prop1}, we obtain
\[
\frac{\pi^2}{T^2}\|u_1'-u_2'\|^2_{L^2([0,T],H)}
\leq L \frac{T}{\pi} \|u'_1-u'_2\|^2_{L^2([0,T],H)}
\]
or
\[
\|u_1'-u_2'\| ^2_{L^2([0,T],H)}
\leq L\frac{T^{3}}{\pi^{3}}\|u'_1-u'_2\|^2_{L^2([0,T],H)}.
\]
It follows from the choice of T  that
 $\|u_1'-u_2'\| ^2_{L^2([0,T],H)}=0$. By inequality (A2) in
lemma \ref{lem2}, we conclude that $u_1(t)-u_2(t)=0$ for all
$t\in [0,T]$. This completes the proof.
\end{proof}

\section{Applications}
Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$ with a smooth
boundary $\partial \Omega$. Let $\gamma$ be a maximal monotone
operator on $\mathbb{R}$ such that $\gamma=\partial j$, where
 $j :\mathbb{R} \to [0,+\infty]$ is proper, convex, lower
semicontinuous and even with $j(0)=0$.
 We are concerned with the third-order boundary-value problem
\begin{equation} \label{eP}
\begin{gathered}
-u_{ttt}(t,x)-\Delta_{x}u_{t}(t,x) +f(t,u(t,x))=0 \quad
\text{in } [0,T]\times \Omega, \\
\frac{\partial u_{t}}{\partial \nu} (t,x)\in \gamma(u_{t}(t,x))
\quad \text{on }[0,T]\times \partial\Omega,\\
 u(0,x)=-u(T,x), \quad u_{t}(0,x)=-u_{t}(T,x)\quad
u_{tt}(0,x)=-u_{tt}(T,x)\quad \text{in } \Omega,
\end{gathered}
\end{equation}
where $\frac{\partial}{\partial \nu}$ denotes outward normal
derivative, $\Delta_{x}=\sum  ^{n}_{i=1} \frac{\partial
^2}{\partial x_{i}^2}$, and
 $f:[0,T]\times \mathbb{R} \to \mathbb{R}$ is a Carath\'eodory function
 satisfying:
\begin{itemize}
\item[(i)]  $|f(t,u)-f(t,v)|\leq L|u-v|$ for
all $(t,u,v)\in[0,T]\times \mathbb{R}^2$, for some positive
constant $L>0$,

\item[(ii)] There is an
$L^2([0,T];\mathbb{R^{+}})$ integrable function
$\alpha:[0,T]\to\mathbb{R^{+}} $ such that
$| f(t,u)|\leq \alpha(t)$ for all $(t,u)\in[0,T]\times \mathbb{R}$.

\end{itemize}
Let $H= L^2(\Omega)$, and define $\varphi:H\to [0,+\infty]$ by
$$
\varphi (u)= \begin{cases}
\frac{1}{2}\int_{\Omega}|\operatorname{grad} u |^2dx
+\int_{\partial\Omega}j(u)d\sigma,&
\text{if }u\in H^{1}(\Omega)\text{ and }
  j(u) \in L^{1}(\partial \Omega), \\
+\infty, & \text{otherwise}.
\end{cases}
$$
According to Br\'ezis \cite[Theorem 12]{BR}, $\varphi$ is proper,
convex and  lower semicontinuous on $H$, with
$\partial\varphi(u)=-\Delta_{x}u$, and
$D(\varphi)=\{ u \in W^{1,2}(\Omega):
 -\frac{\partial u}{\partial \nu}\in \gamma(u), \text{ a.e. on }
 \partial\Omega\}$.
 We consider $u=u(t,x)=u(t)(x)$ and we rewriter the problem \eqref{eP} in
the abstract form
\begin{gather*}
-u'''(t)+ \partial\varphi(u'(t))+f(t,u(t))\ni 0\quad\text{a.e. }t\in [0,T],\\
 u(0)=-u(T), \quad u'(0)=-u'(T),\quad u''(0)=-u''(T),
\end{gather*}
or
\begin{gather*}
u'''(t)\in \partial\varphi(u'(t))+f(t,u(t))\quad\text{a.e. }t\in [0,T],\\
 u(0)=-u(T), \quad u'(0)=-u'(T),\quad u''(0)=-u''(T).
\end{gather*}
We remark that $\varphi(0)=0$, $\varphi$ is even and that the
inf-compactness condition on $\varphi$ holds because
$W^{1,2}(\Omega)$ is compactly imbedded in $L^2(\Omega)$.
Then, we can applying Theorem \ref{Theo3} to derive the existence of a
solution to \eqref{eP}.  If $0<T<\frac{\pi}{\sqrt[3]{L}}$, then
the solution is unique.

\subsection*{Acknowledgments}
 The authors would like to thank the anonymous
referees for their careful and thorough reading of the original manuscript.

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\end{document}