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\usetheme{Berlin}
\title[]{Workshop {\bf CIMPA-Tanger 05-17 mai 2014} '' EDPs Non Lin\'eaires et Applications: Etude Th\'eorique et Num\'erique''}
\author[R.Bahloul \   \ Docteur en Math\'ematiques Appliqu\'ees ]{\bf  Periodic solutions of abstract neutral functional  differential equations and applications}
\date{{\color{red}{\bf Rachid Bahloul, Ezzinbi Khalil and Sidki Omar}}}
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\begin{block}{Probl\`eme}
In this work, we study the existence of periodic solutions for the following neutral partial functional differential equations of the following form
\begin{equation}\label{e}
\displaystyle{\frac{d}{dt}[x(t) - L(x_{t})]= A[x(t)- L(x_{t})]+G(x_{t})+f(t)},
\end{equation}
where  $A : D(A) \subseteq  X \rightarrow X$ is a linear closed operator on  Banach space ($X, \left\|.\right\|$) and
$f \in L^{p}(\mathbb{T}, X)$ for all $p \geq 1$. For $r_{2 \pi} := 2\pi N$  ( some $N \in \mathbb{N}$) $L$ and $G$ are in $B(L^{p}([- r_{2 \pi} ,0],\ \ X);\ \ X)$  is the space of all bounded linear operators  and  $x_{t}$ is an element of $L^{p}([- r_{2 \pi}\ \ ,0],\ \ X)$  which is defined as follows {\colorbox{yellow}{$x_{t}(\theta) = x(t + \theta)$}} for $\theta \in [-r, 0]$.
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\begin{block}{Definition}
For $1\leq p < \infty$ , a sequence $\left\{M_{k}\right\}_{k \in \mathbb{Z}} \subset \mathbf{B}(X,Y)$ is said to be an $L^{p}$-multiplier if for each $f \in L^{p}(\mathbb{T}, X)$, there exists $ u\in$ $L^{p}(\mathbb{T}, Y)$ such that $\hat{u}(k) = M_{k}\hat{f}(k)$ for all $k \in \mathbb{Z}$.
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\begin{block}{Theorem (Neumann Expansion)} Let $A \in B(X,X)$, where X is a Banach space.
If $\parallel A \parallel < 1$ then $I - A$ is invertible, moreover 
$$(I - A)^{-1} = \sum_{k=0}^{\infty}A^{k}.$$
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\begin{block}{Definition}
Assume that $A$ generates a $C_{0}$-semigroup $(T(t))_{t \geq 0}$ on $X$. A function $x$ is called a mild solution of Eq. ($1$) if:
$$Dx_{t} =T(t)D\varphi + \int^{t}_{0}T(t-s)(Gx_{s} + f(s))ds\;\mbox{  for}\; \; 0 \leq t \leq 2\pi.$$
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\begin{block}{Remark}
Let $(T(t))_{t \geq 0}$ be the $C_{0}$-semigroup generated by $A$. If $g: [0, a] \rightarrow X$ is a continuous function, then $\displaystyle{\int_{0}^{t}\int_{0}^{s}T(t-s)g(\xi)d\xi ds} \in D(A)$ and
$$A\int_{0}^{t}\int_{0}^{s}T(t - s)g(\xi)d\xi ds = \int_{0}^{t}( T(t - s) - I)g(s)ds; 0 \leq t \leq a.$$
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\begin{block}{Lemma} 
Assume that $A$ generates a $C_{0}$-semigroup $\left\{T(t)\right\}_{t \geq 0}$ on $X$, if $x$ is a mild solution then 
$$Dx_{t} = D\varphi + A\int^{t}_{0}Dx_{s}ds + \int^{t}_{0}(Gx_{s} + f(s))ds \;\; \mbox{for}\;\;  0 \leq t \leq 2\pi.$$
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\begin{block}{Theorem} 
Assume that $A$ generates a $C_{0}$-semigroup $(T(t))_{t \geq 0}$ on $X$ and  $f \in L^{p}(\mathbb{T}; X)$ \\
For some $1 \leq p < \infty$ ; if $x$ is a mild solution of Eq. (\ref{e}). Then\\
$$(ikD_{k}-AD_{k}- G_{k})\hat{x}(k)= \hat{f}(k)\;\;\text{ for all} \;\;k \in \mathbb{Z}.$$
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\begin{block}{Theorem} Let $ 1 \leq p < \infty $. Assume that $A$ generates a $C_{0}$-semigroup $(T(t))_{t \geq 0}$ on $X$. If $\sigma_{Z}(\Delta)=\phi$ and  $ (ikD_{k}-AD_{k} - G_{k})^{-1}$ is an $L^{p}$-multiplier Then there exists a unique mild periodic solution  of Eq. ($1$).
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\begin{frame}{Application}
Let $A$ be a closed \textbf{linear} operator on a Hilbert space $H$ and suppose that $i\mathbb{Z}\subset \rho(A)$ and $ \displaystyle \sup_{k\in\mathbb{Z}}\left\|k(ik D_{k} - AD_{k})^{-1}\right\|=:M < \infty$.\\
If $\displaystyle{\left\|G\right\| < \frac{1}{(2r_{2\pi})^{1/p}M }}$ then for every $f \in L^{p}(\mathbb{T}; X),$ there exists a unique strong $L^{p}$-solution of Eq. ($1$).\\
From the identity  
$$ikD_{k} - AD_{k} - G_{k} = (ikD_{k} - AD_{k})(I - G_{k}(ikD_{k} - AD_{k})^{-1})$$
it follows that  \\$ikD_{k} - AD_{k} - G_{k}$ is invertible whenever
$\left\|G_{k}(ikD_{k} - AD_{k})^{-1}\right\| < 1.$, we observe that  $\left\|G_{k}\right\| \leq (2r_{2\pi})^{1/p} \left\|G\right\|$.\\
Hence, \\
$\left\|G_{k}(ikD_{k} - AD_{k})^{-1}\right\|=\left\|G_{k}(ikD_{k} - AD_{k})^{-1}\right\|\leq (2r_{2\pi})^{1/p}\left\|G\right\|M:=\alpha < 1.$\\
Then $\sigma_{Z}(\Delta)=\phi$ and  we deduce that 
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\begin{align*}
(ikD_{k} - AD_{k} - G_{k})^{-1}&=(ikD_{k}-AD_{k})^{-1}(I-G_{k}(ikD_{k}-AD_{k})^{-1})^{-1}\\
&=(ikD_{k}-AD_{k})^{-1}\sum_{n=0}^{\infty}[G_{k}(ikD_{k}-AD_{k})^{-1}]^{n} 
\end{align*}
Moreovery
\begin{align*}
\left\|ik(ikD_{k}-AD_{k}-G_{k})^{-1}\right\| &\leq \left\|ik(D_{k}-AD_{k}\right\| \sum_{n=0}^{\infty}[G_{k}(ikD_{k}-AD_{k})^{-1}]^{n}\\
&\leq \frac{1+M}{1-\alpha}
\end{align*}

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and
$$\sup_{k \in \mathbb{Z}} \left\|ik(ikD_{k}-AD_{k}-G_{k})^{-1}\right\| < \infty.$$
we conclude that there exists a unique strong $L^{p}$-solution of Eq. (1). 
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\section{}
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\begin{block}{Reference 1}
\small{W. Arendt, S. Bu, Operator-valued Fourier multipliers on periodic Besov spaces and applications, Proc. Edinburgh Math. Soc. 47, (1), 15-33 (2004)}
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\begin{block}{Reference 2}
\small{C.Lizama, Fourier multipliers and perodic solutions of delay equatons in Banach spaces, J.Math.Anal.Appl. 324,(2), 921-933 (2006)}
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\section{}
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\begin{block}{Reference 3}
J.Wu, Theory and Applications of Partial Differential Equations, Appl, Math .Sci. 119, Springer-verlag, (1969).
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\begin{block}{Reference 4}
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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