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\title[]{$1^{st}$ Workshop on Advances in Evolution and Continuum Mechanics (AEPCM'17) ENSA Fez (Morocco). May 22-24, 2017}
\author[R.Bahloul \   \ Docteur en Math\'ematiques Appliqu\'ees ]{\bf  Periodic solutions of partial degenerate differential equations with n delay}
\date{{\color{red}{\bf Rachid Bahloul}}}
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\section{}
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\begin{block}{Introduction}
Ce travail est motiv\'e par l'\'etude des \'equations diff\'erentielles fonctionnelles \`a retard. Le {\color{red}{retard}} est d\'efini comme \'etant la propri \'et\'e d'un syst\`eme physique pour lequel la
r\'eponse \`a une action appliqu\'ee est retard\'ee dans son effet par exemple le retard du {\color{red}{Tonner-eclair}}(visuelle-sonore). Pour un observateur, le tonnerre ne se manifeste pas en m\^{e}me temps que l'\'eclair (retard d'un tonnerre).\\
Si l'orage est trop loin, on n'entend pas le tonnerre (retard infini).
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\section{}
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\begin{block}{}
Les \'equations diff\'erentielles fonctionnelles \`a retard sont apparues dans diff\'erentes disciplines scientifiques (dynamique de population, trafic urbain, physique, biologie,..) \\
{\color{blue}{Mod\`ele en Biologie}}\\
{\colorbox{red}{$\frac{d}{dt}N(t) = -d N(t) + b N(t - {\colorbox{yellow}{r}})$}}\\
o\`u d et b sont respectivement les taux de mortalit\'e et de naissance de la population de cellule N(t).\\
r est la dur\'ee de gestation moyenne de la population.
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\begin{block}{Probl\`eme}
\small{In this work, we study the existence of strong solutions for the following degenerate partial functional
differential equation with n delay}
\begin{eqnarray}\label{e}
\left\{
\begin{array}{ccccc}
\displaystyle{\frac{d}{dt}Mx(t) = Ax(t) + \sum_{j=1}^{n}Bx(t-r_{j}) + f(t)}\\\\
x(0) = x(2\pi).
\end{array}
\right.
\end{eqnarray}
\small{where $f \in L^{p}([-r_{2\pi} ,0], X)$ for some $1 \leq p < \infty, r_{2\pi} = 2\pi N$ ($N \in \mathbb{N}$) and we suppose  A, B and M are a closed linears operators such that $D(A) \subset D(B) \cap D(M)$. 
$u_{t}$ denotes the history function in $C([-r, 0];X)$ defined by {\colorbox{yellow}{$x_{t}(\theta) = x(t + \theta)$}} for $\theta \in [-r, 0]$.}
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\section{}
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\begin{block}{Definition: $R$-bounded}
{\small{Let $X$ and $Y$ be  Banach spaces.
A family of operators $T=(T_{j})_{j \in \mathbb{N}^{\ast}}\subset B(X,Y)$ is called $R$-bounded, if there is a constant $C > 0$ and
$p \in [1, \infty)$ such that for each $n \in N, T_{j} \in $T$, x_{j}\in X$ and for all independent, symmetric, $\left\{-1,1\right\}$-valued random variables $r_{j}$ on a probability space ($\Omega, M, \mu$) the inequality
$$\left\|\sum_{j=1}^{n} r_{j} T_{j} x_{j}\right\|_{L^{p}(0,1;Y)}\leq C \left\|\sum_{j=1}^{n} r_{j} x_{j}\right\|_{L^{p}(0,1;X)}$$
is valid. The smallest $C$ is called $R$-bounded of  $(T_{j})_{j \in \mathbb{N}^{\ast}}$ and it is denoted by $R_{p}$($T$).}}
\end{block}
\begin{block}{Example}
{\small{Let $r > 0$ be fixed, $L \in B(L^{p}([-r, 0], X); X)$ and $k \in \mathbb{Z}$, we define the operator $L_{k}$ on X by: $L_{k}x = L(e_{k}x)$ for all $x \in X$. We have 
$(L_{k})_{k \in \mathbb{Z}}$ is R-bounded and $R_{p}(L_{k}) \leq r^{1/p}$.}}
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\section{}
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\begin{block}{Definition}
Let $f \in L^{p}(\mathbb{T}; X)$. A function $x \in H^{1,p}(\mathbb{T}; X)$ is said to be a $2\pi$-periodic strong $L^{p}$-solution of 
Eq. (\ref{e}) if $x(t) \in  D(A)$ for all $t \geq 0$ and Eq. (\ref{e}) holds almost every where.
\end{block}
{\colorbox{yellow}{Necessary condition}}
\begin{block}{Proposition}
Let $X$ be a {\color{red}{Banach space}}. Suppose that for every $f \in L^{p}(\mathbb{T}; X)$ there exists a unique strong solution of Eq. (\ref{e}) for $1 \leq p <  \infty$. Then
\begin{enumerate}
\item for every $k \in \mathbb{Z}$ the operator $\Delta_{k}=( ikM-A-\sum_{j=1}^{n}B_{j,k})$ has bounded inverse
\item  $\left\{ ik\Delta^{-1}_{k}    \right\}_{k \in \mathbb{Z}}$ is $R$-bounded.
\end{enumerate}
\end{block}
{\colorbox{red}{Condition sufficient}}
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\section{}
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\begin{block}{Definition: UMD Space}
A Banach space $X$ is said to be UMD space if the Hilbert transform is bounded on $L^{p}(\mathbb{R};\  X)$ for all $1 < p < \infty$.
\end{block}
\begin{block}{Example}
1. Any Hilbert space is an UMD space.\\
2. $L^{p}$ (0.1, X) are UMD spaces for every $ 1 < p < \infty $\\
3. $L^{4}$ (0.1, X) is UMD space, the bound $C_{4} \leq 1 + \sqrt{2}$\\
4. $L^{8}$ (0.1, X) is UMD space, the bound $C_{8} \leq 1 + \sqrt{2} + \sqrt{4 + 2\sqrt{2}}$
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\begin{block}{Theorem}
\small{Let X be an UMD space and $A : D(A) \subset X \rightarrow X $ be a closed
linear operator. Then the following assertions are equivalent for $1 < p < \infty.$}
\begin{enumerate}
\item for every $f \in L^{p}(\mathbb{T}, X)$ there exists a unique strong $L^{p}$-solution of Eq.(\ref{e}).
\item $\sigma_{\mathbb{Z}}(\Delta) = \phi$ and $\{ ik \Delta^{-1}_{k} \}_{k \in \mathbb{Z}}$ is R-bounded.\\
$\Delta_{k} = (ikM - A - \sum_{j=1}^{n} B_{j,k})$
\end{enumerate}
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\begin{block}{Reference1}
W.Arend and S.Bu, \emph{The operator-valued Marcinkiewicz multiplier theorem and maximal regularity, Math.Z. 240, (2002), 311-343.}
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\section{}
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\begin{block}{Reference 2}
\small{ {\color{red}{Rachid Bahloul}}, Khalil Ezzinbi and Omar Sidki,  Periodic Solutions in UMD spaces for some neutral partial function differential equations, {\color{blue}{Advances in Pure Mathematics}}, (2016), 6, 713-726, http: dx.doi.org/10.4236/apm.2016.610058.}
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\begin{block}{Reference 3}
\small{ {\color{red}{R.Bahloul}}, M.Bahaj et O.Sidki,Periodic solutions of degenerate equations with finite delay in UMD space, journal of Advances in Dynamical Systems and Applications. ISSN 0973-5321, Volume 10, Number 1, (2015) pp. 23-31.}
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\section{}
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\begin{block}{Reference 4}
{\color{red}{R.Bahloul}}, Periodic Solutions of differential equations  with two variable  in vector-valued space, {\color{blue}{Asian Journal of Mathematics and Computer
Research}}, 12(1): 44-53, 2016, ISSN : 2395-4205 (P), ISSN : 2395-4213 (O).
\end{block}
\begin{block}{Reference 5}
{\color{red}{R.Bahloul}}, M.Bahaj and O.Sidki, \small{Existence and uniqueness of periodic solutions integro-differential equations with two variable}, {\color{blue}{Asian Journal of Mathematics and Computer Research}}, \small{ISSN No. : 2395-4205 (Print), 2395-4213 (Online), Vol.: 17, Issue.: 3}
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\section{}
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\begin{block}{Reference 6}
Sylvain Koumla, Khalil Ezzinbi and {\color{red}{R.Bahloul}}, mild solutions for some partial functional integrodifferential equations with finite delay in Frechet spaces. {\color{blue}{SeMA}}(2016), P 1-13. DOI 10.1007/s40324-016-0096-7
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\begin{block}{Reference 7}
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, 1983.
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\Huge{ \textbf{{\color{blue}{\bf Merci pour votre attention}}}}
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