Existence and uniqueness of functional differential equations with n delay

TL;DR

This paper establishes necessary and sufficient conditions for the existence and uniqueness of periodic solutions in functional differential equations with multiple delays, using operator-valued Fourier multipliers.

Contribution

It introduces a novel framework based on R-boundedness of Fourier multipliers to analyze solutions of delayed differential equations.

Findings

Derived conditions for existence and uniqueness of solutions.

Connected solution properties to R-boundedness of Fourier multipliers.

Provided a mathematical foundation for analyzing delayed differential equations.

Abstract

In this paper we give a necessary and suffcient conditions for the existence and uniqueness of periodic solutions of functional differential equations with n delay d dt x(t) = Ax(t) + n j=1 Bx(t -- r j) + f (t). The conditions are obtained in terms of R-boundedness of operator valued Fourier multipliers.

Equations

$$\displaystyle\left\{\begin{array}[]{ccccc}\displaystyle{\frac{d}{dt}x(t)=Ax(t)+\sum_{j=1}^{n}Bx(t-r_{j})+f(t)}\\ \\ x(0)=x(2\pi).\end{array}\right.$$

$${\Vert \sum _{j=1}^n{r}_j{T}_j{x}_j\Vert }_{L^p(0,1;Y)}\le C{\Vert \sum _{j=1}^n{r}_j{x}_j\Vert }_{L^p(0,1;X)}$$

$${\Vert \sum _{l=1}^m{r}_l(\sum _{j=1}^n{B}_{j,l})x_l\Vert }_{L^p(0,1;X)}^p$$

$$={\int }_0^1{\Vert B(\sum _{l=1}^m{r}_l(t)\sum _{j=1}^n{e}^{-il{r}_j}{x}_l)\Vert }_X^{p}dt$$

$$\le ||B|{|}^p{\int }_0^1{\Vert \sum _{l=1}^m{r}_l(t)\sum _{j=1}^n{e}^{-il{r}_j}{x}_l\Vert }_X^{p}dt$$

$${\Vert \sum _{l=1}^m{r}_l(\sum _{j=1}^n{B}_{j,l})x_l\Vert }_{L^p(0,1;X)}\le 2^{1/p}n||B||.$$

$$k\left[M_{k+1}-M_k\right]$$

$$=k(C_{k+1}-AD{)}^{-1}[i(k+1)(C_k-AD)-ik(C_{k+1}-AD)](C_k-AD{)}^{-1}$$

$$=k(C_{k+1}-AD{)}^{-1}[ik(C_k-C_{k+1})+i(C_k-A)](C_k-AD{)}^{-1}$$

$$=k(C_{k+1}-AD{)}^{-1}[ik(C_k-C_{k+1})(C_k-AD{)}^{-1}+iI]$$

$$=\frac{-ik}{k+1}{M}_{k+1}(C_k-C_{k+1})M_k+\frac{k}{k+1}{M}_{k+1}.$$

$$C_k-C_{k+1}=-iI+\sum _{j=1}^{n}B{e}^{-ik{r}_j}(1-e^{-i{r}_j}).$$

$$\frac{d}{dt}x(t)=Ax(t)+\sum _{j=1}^{n}Bx(t-r_j)+f(t)$$

$$\widehat{x^{\prime }}(k)=ik\hat{x}(k)=A\hat{x}(k)+\sum _{j=1}^n{B}_{j,k}\hat{x}(k)+\hat{f}(k).$$

$$ik\hat{x}(k)=ik(ikI-A-\sum _{j=1}^n{B}_{j,k}{)}^{-1}\hat{f}(k)$$

$$x(t)=T(t)\phi +{\int }_0^{t}T(t-s)(\sum _{j=1}^{n}Bx(s-r_j)+f(s))ds\text{for}0\le t\le 2\pi .$$

$$A{\int }_0^t{\int }_0^{s}T(t-\xi )g(\xi )d\xi ds={\int }_0^t(T(t-s)-I)g(s)ds\text{mbox}forall0\le t\le a.$$

$$x(t)=\phi +A{\int }_0^{t}x(s)ds+{\int }_0^t(\sum _{j=1}^{n}Bx(s-r_j)+f(s))ds\text{for}0\le t\le 2\pi .$$

$$(ikI-A-\sum _{j=1}^n{B}_{j,k})\hat{x}(k)=\hat{f}(k)\text{for all}k\in \mathbb{Z}.$$

$$x(t)=\phi +A{\int }_0^{t}x(s)ds+{\int }_0^t(\sum _{j=1}^{n}Bx(s-r_j)+f(s))ds$$

$$x(2\pi )=\phi +A{\int }_0^{2\pi }x(s)ds+{\int }_0^{2\pi }(\sum _{j=1}^{n}Bx(s-r_j)+f(s))ds;$$

$$A{\int }_0^{2\pi }x(s)ds+{\int }_0^{2\pi }(\sum _{j=1}^{n}Bx(s-r_j)+f(s))ds=0$$

$$\Rightarrow \frac{1}{2\pi }A{\int }_0^{2\pi }x(s)ds+\frac{1}{2\pi }{\int }_0^{2\pi }(\sum _{j=1}^{n}Bx(s-r_j)+f(s))ds=0$$

$$\Rightarrow \frac{1}{2\pi }A{\int }_0^{2\pi }x(s)ds+\frac{1}{2\pi }{\int }_0^{2\pi }\sum _{j=1}^{n}Bx(s-r_j)ds+\frac{1}{2\pi }{\int }_0^{2\pi }f(s))ds=0$$

$$\Rightarrow \frac{1}{2\pi }A{\int }_0^{2\pi }{e}^{-i0s}x(s)ds+\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{-i0s}\sum _{j=1}^{n}Bx(s-r_j)ds+\frac{1}{2\pi }{\int }_0^{2\pi }{e}^{-i0s}f(s))ds=0$$

$$\Rightarrow (0-A-\sum _{j=1}^n{B}_{j,0})\hat{x}(0)=\hat{f}(0),$$

$$v(t)={\int }_0^{t}x(s)ds$$

$$g(t)=x(t)-\phi -{\int }_0^t(\sum _{j=1}^{n}Bx(s-r_j)+f(s))ds$$

$$\hat{v}(k)$$

$$A\hat{v}(k)$$

$$\hat{g}(k)$$

$$=\hat{x}(k)-\frac{i}{k}{G}_0\hat{x}(0)+\frac{i}{k}{G}_k\hat{x}(k)-\frac{i}{k}\hat{f}(0)+\frac{i}{k}\hat{f}(k)$$

$$f=\underset{n\to +\infty }{\lim }{\sigma }_n(f)$$

$$\hat{x}(k)={\Delta }_k^{-1}\hat{f}(k)=(ikI-A-\sum _{j=1}^n{B}_{j,k}{)}^{-1}\hat{f}(k)$$

$$\widehat{x^{\prime }}(k):=\hat{v}(k)=ik\hat{x}(k)$$

$$x(t-r_j)=\underset{l\to +\infty }{\lim }\frac{1}{l+1}\sum _{m=0}^l\sum _{k=-m}^m{e}^{ikt}{e}^{-ik{r}_j}\hat{x}(k)$$

Full text

Existence and uniqueness of functional differential equations with n delay

Bahloul Rachid

Faculty of Sciences and Technology, Fez, Morocco

bahloulr33@hotmail.com

Mathematics Subject Classification: xxxxx

Keywords: functional differential equations with n delay, R-bounded.

1 Introduction

Let A and B be two closed linear operators defined on a Banach space X with domains D(A) and D(B), respectively such that $D(A)\subset D(B)$. In this paper we show existence and uniqueness of solutions for the following differential equation with n delay

[TABLE]

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 B is bounded. The theory of operator-valued Fourier multipliers has attracted the attention of many papers in recent years. For example, this theory was used in [1] to obtain results about equations $\frac{dx(t)}{dt}=Ax(t)+f(t)$, and in [11] to obtain results about delay equation $\frac{dx(t)}{dt}=Ax(t)+F(x_{t})+f(t)$. In [6], S.Bu studied $L^{p}$-Maximal Regularity of Degenerate delay Equations with Periodic Conditions. We note that in the special case when $B=0$, maximal regularity of Eq. (4) has been studied by Arendt and Bu in $L^{p}$-spaces case and Besov spaces case [[1], [2]], Bu and Kim in TriebelLizorkin spaces case [8]. The corresponding integro-differential equations were treated by Keyantuo and Lizama [[17], [18]], Bu and Fang [7]. In this paper, we characterize the existence and uniqueness for the n delay equation (4) under the condition that X is a UMD space. Here the operator A is not necessarily the generator of a $C_{0}$-semigroup. We use the operator valued multiplier Fourier method. The organisation of this work is as follows : In section 2, we present preliminary results on UMD spaces and $L^{P}$-multiplier. In section 3, we study the existence of periodic strong solution for Eq.(4) with finite delay. In section 4, we give the main abstract result ( theorem [4.2] ) of this work.

1. for every $f\in L^{p}(\mathbb{T};X);1<p<\infty$, there exists a unique $2\pi$-periodic strong $L^{p}$-solution of Eq. (4).

2. $(ikI-A-\sum_{j=1}^{n}B_{j,k})$ has bounded invertible for all $k\in\mathbb{Z}$ and $\{ik(ikI-A-\sum_{j=1}^{n}B_{j,k})^{-1}\}_{k\in\mathbb{Z}}$ is R-bounded.

2 Preliminary Notes

Let $X$ be a Banach Space. Firstly, we denote By $\mathbb{T}$ the group defined as the quotient $\mathbb{R}/2\pi\mathbb{Z}$. There is an identification between functions on $\mathbb{T}$ and $2\pi$-periodic functions on $\mathbb{R}$. We consider the interval $[0,2\pi$) as a model for $\mathbb{T}$.

** Definition 2.1****.**

A Banach space X is said to be UMD space if the Hilbert transform is bounded on $L^{p}(\mathbb{R},X)$ for all $1<p<\infty$.

** Example 2.2****.**

*: [9]

1.Any Hilbert space is an UMD space.

2. $L^{p}(0.1)$ are UMD spaces for every $1<p<\infty$.

3. Any closed subspace of a UMD space is a UMD space.*

** Definition 2.3****.**

[1]

A family of operators $T=(T_{j})_{j\in\mathbb{N}^{\ast}}\subset B(X,Y)$ is called $R$-bounded (* Rademacher bounded or randomized 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*

[TABLE]

is valid. The smallest $C$ is called $R$-bounded of $(T_{j})_{j\in\mathbb{N}^{\ast}}$ and it is denoted by $R_{p}$($T$).

** Definition 2.4****.**

[11]

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 $k\in\mathbb{Z}$.

** Proposition 2.5****.**

[1, Proposition 1.11]** Let $X$ be a Banach space and $\left\{M_{k}\right\}_{k\in\mathbb{Z}}$ be an $L^{p}$-multiplier, where $1\leq p<\infty$. Then the set $\left\{M_{k}\right\}_{k\in\mathbb{Z}}$ is $R$-bounded.

** Theorem 2.6****.**

**(Marcinkiewicz operator-valud multiplier Theorem).

Let $X$, $Y$ be UMD spaces and $\left\{M_{k}\right\}_{k\in\mathbb{Z}}\subset B(X,Y)$. If the sets $\left\{M_{k}\right\}_{k\in\mathbb{Z}}$ and $\left\{k(M_{k+1}-M_{k})\right\}_{k\in\mathbb{Z}}$ are

$R$-bounded, then $\left\{M_{k}\right\}_{k\in\mathbb{Z}}$ is an $L^{p}$-multiplier for $1<p<\infty$.**

We observe that the condition of R-boundedness for $(M_{k})_{k\in\mathbb{Z}}$ is necessary.

** Remark 2.7****.**

[13]

Let $f\in L^{1}(\mathbb{T};X)$. If $g(t)=\int_{0}^{t}f(s)ds$ and $k\in\mathbb{Z},k\neq 0,$ then

$\hat{g}(k)=\frac{i}{k}\hat{f}(0)-\frac{i}{k}\hat{f}(k)$**

3 A criterion for periodic solutions

** Notation 3.1****.**

*Let $k\in\mathbb{Z}$. Denote by $B_{j,k}:=e^{-ikr_{j}}B$,

$\Delta_{k}=(ikI-A-\sum_{j=1}^{n}B_{j,k})$ and $\sigma_{\mathbb{Z}}(\Delta):=\{k\in\mathbb{Z}:\Delta_{k}\text{is not bijective}\}$

$H^{1,p}(\mathbb{T};X)=\left\{u\in L^{p}(\mathbb{T},X):\exists v\in L^{p}(\mathbb{T},X),\hat{v}(k)=ik\hat{u}(k)for\ \ all\ \ k\in\mathbb{Z}\right\}$*

** Definition 3.2****.**

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. (4) if $x(t)\in D(A)$ for all $t\geq 0$ and Eq. (4) holds almost every where.

** Lemma 3.3****.**

[1*, Lemme 2.1]** Let $1\leq p<\infty$ and $u,v\in L^{p}(\mathbb{T};X)$. Then the following assertions are equivalent:

(i) $\displaystyle{\int_{0}^{2\pi}v(s)ds=0}$ and there exists $x\in X$ such that $u(t)=x+\int_{0}^{t}v(s)ds.$

(ii) $\hat{v}(k)=ik\hat{u}(k)$  for any  $k\in\mathbb{Z}$.*

** Definition 3.4****.**

For $1\leq p<\infty$, we say that a sequence $\left\{M_{k}\right\}_{k\in\mathbb{Z}}\subset\mathbf{B}(X,Y)$ is an ($L^{p},H^{1,p}$)-multiplier, if for each $f\in L^{p}(\mathbb{T},X)$ there exists $u\in H^{1,p}(\mathbb{T},Y)$ such that $\hat{u}(k)=M_{k}\hat{f}(k)\ \ \text{ for all}\ \ k\in\mathbb{Z}.$

** Lemma 3.5****.**

*Let $1\leq p<\infty$ and $(M_{k})_{k\in\mathbb{Z}}\subset\mathbf{B}(X)\ \ (\mathbf{B}(X)$ is the set of all bounded linear operators from $X$ to $X$). Then the following assertions are equivalent:

(i) $(M_{k})_{k\in\mathbb{Z}}$ is an ($L^{p},H^{1,p}$)-multiplier.

(ii) $(ikM_{k})_{k\in\mathbb{Z}}$ is an ($L^{p},L^{p}$)-multiplier.*

** Proposition 3.6****.**

*Let $A$ be a closed linear operator defined on an UMD space $X$. Suppose that $\sigma_{\mathbb{Z}}(\Delta)=\phi$ .Then the following assertions are equivalent :

(i) $\{ik(ikI-A-\sum_{j=1}^{n}B_{j,k})^{-1}\}_{k\in\mathbb{Z}}$ is an $L^{p}$-multiplier for $1<p<\infty$

(ii) $\{ik(ikI-A-\sum_{j=1}^{n}B_{j,k})^{-1}\}_{k\in\mathbb{Z}}$ is $R$-bounded.*

Proof.

By [1, Proposition 1.11] it follows that (i) implies (ii). Conversely, define $M_{k}=ik(C_{k}-A)^{-1}$, where $C_{k}=ikI-\sum_{j=1}^{n}B_{j,k}$, By Theorem 2.6 is sufficient to prove that the set $(k(M_{k+1}-M_{k}))_{k\in\mathbb{Z}}$ is R-bounded. We claim first that the set $(\sum_{j=1}^{n}B_{j,k})_{k\in\mathbb{Z}}$ is R-bounded.

since given $x_{j}\in D(A)$ we have :

[TABLE]

By (Lemma 1.7, [1]) we obtain that

$\left\|\sum_{l=1}^{m}r_{l}(\sum_{j=1}^{n}B_{j,l})x_{l}\right\|_{L^{p}(0,1;X)}^{p}\leq 2n^{p}||B||^{p}\displaystyle{\int_{0}^{1}}\left\|\sum_{l=1}^{m}r_{l}(t)x_{l}\right\|_{X}^{p}dt$

We conclude that

[TABLE]

and the claim is proved. Next. We note the following identities

[TABLE]

We have

[TABLE]

Since products and sums of R-bounded sequences is R-bounded [11, Remark 2.2]. Then $\{k(M_{k+1}-M_{k})\}_{k\in\mathbb{Z}}$ is R-bounded and by theorem 2.6, $\{M_{k}\}_{k\in\mathbb{Z}}$ is an $L^{p}$-multiplier. ∎

** Theorem 3.7****.**

Let $X$ be a Banach space. Suppose that for every $f\in L^{p}(\mathbb{T};X)$ there exists a unique strong solution of Eq. (4) for $1\leq p<\infty$. Then

for every $k\in\mathbb{Z}$ the operator $\Delta_{k}=(ikI-A-\sum_{j=1}^{n}B_{j,k})$ has bounded inverse 2. 2.

$\left\{ik\Delta^{-1}_{k}\right\}_{k\in\mathbb{Z}}$* is $R$-bounded.*

Before to give the proof of Theorem (3.7), we need the following Lemma.

** Lemma 3.8****.**

if $(ikI-A-\sum_{j=1}^{n}B_{j,k}(x))=0$ for all $k\in\mathbb{Z}$, then $u(t)=e^{ikt}x$ is a $2\pi$-periodic strong $L^{p}$-solution of the following equation (4) corresponing to the function $f=0$.

Proof.

$(ikI-A-\sum_{j=1}^{n}B_{j,k}(x))=0\Rightarrow ikx=Ax+\sum_{j=1}^{n}B_{j,k}x$.

We have $u(t)=e^{ikt}x$ then

$\displaystyle{u^{\prime}(t)=ike^{ikt}x=e^{ikt}(ikx)=e^{ikt}[Ax+\sum_{j=1}^{n}B_{j,k}x]=Au(t)+\sum_{j=1}^{n}Bu(t-r_{j}).}$

Proof of Theorem 3.7

1. Let $k\in\mathbb{Z}$ and $y\in X$. Then for $f(t)=e^{ikt}y$ , there exists $x\in H^{1,p}(\mathbb{T};X)$ such that:

[TABLE]

Taking Fourier transform, by Lemma 3.3 we have :

[TABLE]

Then we obtain : $(ikI-A-\sum_{j=1}^{n}B_{j,k})\hat{x}(k)=\hat{f}(k)=y\Rightarrow(ikI-A-\sum_{j=1}^{n}B_{j,k})$ is surjective.

If $(ikI-A-\sum_{j=1}^{n}B_{j,k})u=0$, then by Lemma 3.8 $x(t)=e^{ikt}u$ is a 2$\pi$-periodic strong $L^{p}$-solution of Eq. (4) corresponing to the function $f=0$ Hence $x(t)=0$ and $u=0$ then $(ikI-A-\sum_{j=1}^{n}B_{j,k})$ is injective.

2. Let $f\in L^{p}(\mathbb{T},X)$. By hypothesis, there exists a unique $x\in H^{1;p}(\mathbb{T},X)$ such that the Eq. (4) is valid. Taking Fourier transforms, we deduce that $\hat{x}(k)=(ikI-A-\sum_{j=1}^{n}B_{j,k})^{-1}\hat{f}(k)$ for all $k\in\mathbb{Z}$. Hence

[TABLE]

On the other hand, since $x\in H^{1;p}(\mathbb{T},X)$, there exists $v\in L^{p}(\mathbb{T},X)$ such that $\hat{v}(k)=ik\hat{x}(k)=ik(ikI-A-\sum_{j=1}^{n}B_{j,k})^{-1}\hat{f}(k)$ i.e $\{ik\Delta^{-1}_{k}\}_{k\in\mathbb{Z}}$ is an $L^{p}$-multiplier. Then $\{ik\Delta^{-1}_{k}\}_{k\in\mathbb{Z}}$ is R-bounded. ∎

4 Existence of mild solutions of Eq. (4)

It is well known that in many important applications the operator A is the infinitesimal generator of $C_{0}$-semigroup $(T(t))_{t\geq 0}$ on the space X. Let A be a generator of a $C_{0}$-semigroup $(T(t))_{t\geq 0}$.

** Definition 4.1****.**

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. (4) if :

[TABLE]

** Remark 4.2****.**

[14, Remark 4.2]

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 $\displaystyle{\int_{0}^{t}\int_{0}^{s}T(t-\xi)g(\xi)d\xi ds}\in D(A)$ and

[TABLE]

** Lemma 4.3****.**

[10]

Assume that A generates a $C_{0}$-semigroup $(T(t))_{t\geq 0}$ on X, if $x$ is a mild solution of Eq. (4) then

[TABLE]

** Theorem 4.4****.**

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. (4). Then

[TABLE]

Proof.

Let $x$ be a mild solution of Eq. (4). Then by Lemma 4.3, we have

[TABLE]

For $t=2\pi$, we have

[TABLE]

Since: $x(2\pi)=\varphi$, then

[TABLE]

which shows that the assertion holds for $k=0$.

Now, define

[TABLE]

and

[TABLE]

by Remark 2.7 We have:

[TABLE]

and

[TABLE]

∎

** Corollary 4.5****.**

Assume that A generates a $C_{0}$-semigroup $(T(t))_{t\geq 0}$ on X and let $f\in L^{p}(\mathbb{T},X):1\leq p<\infty$ and $x$ be a mild solution of Eq. (4). If $(ikI-A-\sum_{j=1}^{n}B_{j,k})$ has a bounded inverse. Then $(ikI-A-\sum_{j=1}^{n}B_{j,k})$ is an $L^{p}$-multiplier.

Proof.

Let $f\in L^{p}(\mathbb{T},X)$ then from Theorem (4.4) we have:

$\hat{x}(k)=(ikD_{k}-AD_{k}-G_{k})^{-1}\hat{f}(k)$ for all $f\in L^{p}(\mathbb{T};X)$, then

$(ikI-A-\sum_{j=1}^{n}B_{j,k})^{-1}$ is an $L^{p}$-multiplier. ∎

5 Main Result

Our main result in this work is to establish that the converse of theorem (3.7) and corollary (4.5) is true, provided X is an UMD space.

** Theorem 5.1****.**

(Fejer Theorem) : Let $f\in L^{p}(\mathbb{T},X)$. Then

[TABLE]

where $\sigma_{n}(f)=\frac{1}{n+1}\sum^{n}_{m=0}\sum^{m}_{k=-m}e_{k}\hat{f}(k)$, with $e_{k}(t)=e^{ikt}.$

** Theorem 5.2****.**

*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 $1<p<\infty.$

1. for every $f\in L^{p}(\mathbb{T},X)$ there exists a unique strong $L^{p}$-solution of Eq.(4).

2. $\sigma_{\mathbb{Z}}(\Delta)=\phi$ and $\{ik\Delta^{-1}_{k}\}_{k\in\mathbb{Z}}$ is R-bounded.*

Proof.

$1\Rightarrow 2)$ see Theorem 3.7.

$1\Leftarrow 2)$ Let $f\in L^{p}(\mathbb{T};X)$ . Define $\Delta_{k}=(ikI-A-\sum_{j=1}^{n}B_{j,k})$ ,

By Proposition 3.6, the family $\left\{ik\Delta^{-1}_{k}\right\}_{k\in\mathbb{Z}}$ is an $L^{p}$-multiplier it is equivalent to

the family $\left\{\Delta^{-1}_{k}\right\}_{k\in\mathbb{Z}}$ is an $L^{p}$-multiplier that maps $L^{p}(\mathbb{T};X)$ into $H^{1,p}(\mathbb{T};X)$,

namely there exists $x\in H^{1,p}(\mathbb{T},X)$ such that

[TABLE]

In particular, $x\in L^{p}(\mathbb{T};X)$ and there exists $v\in L^{p}(\mathbb{T};X)$ such

that $\hat{v}(k)=ik\hat{x}(k)$

[TABLE]

By Theorem 5.1 we have for $j\in\{1...n\}$

[TABLE]

Then, since B is bounded linear

[TABLE]

By (5) and (6) we have:

[TABLE]

Then using that $A$ and B are closed we conclude that $x(t)\in D(A)$ [[1], Lemma 3.1] and from the uniqueness theorem of Fourier coefficients that

[TABLE]

We have $x\in H^{1,p}(\mathbb{T},X)$ then by lemma 3.3, $x(0)=x(2\pi)$, then the Eq. (1) has a unique 2$\pi$-periodic strong $L^{p}$-solution. ∎

** Theorem 5.3****.**

Let $1\leq p<\infty$. Assume that $A$ generates a $C_{0}$-semigroup $(T(t))_{t\geq 0}$ on $X$. If $\sigma_{Z}(\Delta)=\emptyset$ and $(ikI-A-\sum_{j=1}^{n}B_{j,k})^{-1}$ is an $L^{p}$-multiplier Then there exists a unique mild solution periodic of Eq. (1).

Proof.

For $f\in L^{p}(\mathbb{T};X)$ we define

[TABLE]

By the Fejér Theorem we can assert that $f_{l}\rightarrow f$ as $l\rightarrow\infty$ for the norm in $L^{p}(\mathbb{T};X)$. We have $(ikI-A-\sum_{j=1}^{n}B_{j,k})^{-1}$ is an $L^{p}$-multiplier then there exists $x\in L^{p}(\mathbb{T};X)$ such that $\hat{x}(k)=(ikI-A-\sum_{j=1}^{n}B_{j,k})^{-1}\hat{f}(k)$

put

[TABLE]

Using again the Fejér Theorem we obtain that $x_{n}(t)\rightarrow x(t)$ (as $n\rightarrow\infty)$ and $x_{n}(t)$ is strong $L^{p}$-solution of Eq. (1) and $x_{n}(t)$ verified

[TABLE]

With $t=2\pi$ we obtain

[TABLE]

from which we infer that the sequence $(\varphi_{l})_{n}$ is convergent to some element $\varphi$ as $l\rightarrow\infty$( $\varphi_{l}=x_{l}(0)=x_{l}(2\pi)$). Moreover, $\varphi$ satisfies the condition

[TABLE]

Taking the limit as $l$ goes to infinity in (7), we can write

[TABLE]

$g(2\pi)=T(2\pi)y+\int^{2\pi}_{0}T(2\pi-s)(\sum_{j=1}^{n}Bx(s-r_{j})+f(s))ds\overbrace{=}^{(5)}\varphi=g(0)$

Then $x(2\pi)=\varphi\Rightarrow x(2\pi)=x(0)$, we conclude that $x$ is a $2\pi$- periodic mild solution of Eq. (1). ∎

Acknowledgements. This is a text of acknowledgements.