\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small Tenth MSU Conference on Differential Equations and Computational Simulations,\newline \emph{Electronic Journal of Differential Equations}, Conference 23 (2016), pp. 21--33.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2016 Texas State University.} \vspace{9mm}} \begin{document} \setcounter{page}{21} \title[\hfilneg EJDE-2016/Conf/23 \hfil d'Alembert's formula] {d'Alembert's formula and periodic mild solutions to iterated higher-order differential equations in Hilbert spaces} \author[N. Iraniparast, L. Nguyen \hfil EJDE-2016/conf/23 \hfilneg] {Nezam Iraniparast, Lan Nguyen} \address{Nezam Iraniparast \newline Department of Mathematics, Western Kentucky University, Bowling Green KY 42101, USA} \email{Nezam.Iraniparast@wku.edu} \address{Lan Nguyen \newline Department of Mathematics, Western Kentucky University, Bowling Green KY 42101, USA} \email{Lan.Nguyen@wku.edu} \thanks{Published March 21, 2016} \subjclass[2010]{34G10, 34K06, 47D06} \keywords{Keywords and phrases: Abstract higher-order differential equations; \hfill\break\indent Fourier series; periodic mild solutions; operator semigroups} \begin{abstract} We give necessary and sufficient conditions for the periodicity of solutions of mild solutions to the iterated higher-order differential equation $$ \prod_{j=1}^{n}(\frac{d}{dt} -A_j)u(t)=f(t), \quad 0 \le t\le T, $$ in a Hilbert space. Our results are illustrated with examples and applications. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction} In this article we study the periodicity of solutions of the iterated higher-order differential equation \begin{equation}\label{high} \prod_{j=1}^{n}\big(\frac{d}{dt} -A_j\big)u(t)=f(t), \quad 0 \le t\le T, \end{equation} where $A_j$ are linear, closed and mutually commuting operators on a Hilbert space $E$, and $f$ is a function from $[0,T]$ to $E$. The asymptotic behavior and, in particular, the periodicity of solutions of the higher-order differential equation \begin{equation}\label{n} u^{(n)}(t)=Au(t)+f(t), \quad 0 \le t\le T , \end{equation} has been a subject of intensive study for recent decades. When $n=1$, it is well known \cite{hale} that, if $A$ is an $n\times n$ matrix on $\mathbb{C}^n$, then \eqref{n} admits a unique $T$-periodic solution for each continuous $T$-periodic forcing term $f$ if and only if $\lambda_k=2k\pi /T$, $k\in\mathbb{Z}$, are not eigen-values of $A$. That result was extended by Krein and Dalecki \cite{dakr} to the Cauchy problem in an abstract Banach space. It was shown \cite[Theorem II 4.3]{dakr} that, if $A$ is a linear, bounded operator on $E$, then \eqref{n} admits a unique $T$-periodic solution for each $f\in C[0,T]$ if and only if $2k\pi i/T \in \varrho(A)$, $k \in \mathbb{Z}$. Here $\varrho(A)$ denotes the resolvent set of $A$. For an unbounded operator $A$, the situation changes dramatically and the above statement generally fails. When $A$ generates a strongly continuous semigroup, periodicity of solutions of \eqref{first} has intensively been studied recently (see e.g. \cite{haraux, lm, pruss, vusc}). Corresponding results on the periodic solutions of the second order differential equation were obtained in \cite{cili, sc}, when $A$ is the generator of a cosine family. Related results on the periodicity of solutions of \eqref{n}, when $A$ is a closed operator, can be found in \cite{eiti, hakr, minh, ng, vusc2} and the references therein. Unfortunately, for the complete higher-order differential equations, we have little consideration about the regularity of their solutions, mainly because of the complexity of the structure of the equation. In \cite{san} and \cite{gosan}, the authors studied the iterated higher-order Cauchy problem of the type \begin{equation}\label{sa} \begin{gathered} \prod_{j=1}^{n}\big(\frac{d}{dt} -A_j\big)u(t)=0, \quad t > 0,\\ u^{(j)}(0) = x_j\in E \quad (j=0,1, \dots , n-1) \end{gathered} \end{equation} and stated that, under some certain conditions, \eqref{sa} is well posed if and only if $A_i$ are generators of $C_0$-semigroups. Moreover, they found the formula of solutions in the form $u(t)=\sum_1^n u_i(t)$, where $(d/dt -A_j)u_i =0$. That result suggests that \eqref{sa} is in some sense the correct way to consider higher-order Cauchy problems. Later, in \cite{gogoob}, the nonautonomous version of iterated evolution equation \eqref{sa} was studied, where a nice structure of the solutions was found. In this paper we investigate the periodicity of mild solutions of the iterated higher-order differential equation \eqref{high} when $A_j$, $j= 0, 1, \dots , n-1$, are linear and closed operators on a Hilbert space $E$. The main tool we use here is the Fourier series method. For an integrable function $f(t)$ from $[0,T]$ to $E$, the Fourier coefficient of $f(t)$ is defined by $$ f_k = \frac{1}{T}\int_0^Tf(s)e^{-2k\pi is/T} ds, \quad k \in Z. $$ Then $f(t)$ can be represented by Fourier series \[ f(t) \approx \sum_{k = -\infty}^{\infty}e^{2k\pi it/T} f_k. \] We first give the definition of mild solution to \eqref{high}, when $n=1$. \begin{definition} \label{def1.1} \rm (i) A continuous function $u(\cdot)$ is a mild solution of the differential equation \begin{equation}\label{first} u'(t)=Au(t)+f(t), \quad t\in[0,T] , \end{equation} if $\int_0^tu(s)ds\in D(A)$ and $$ u(t)=u(0)+A\int_0^tu(s)ds +\int_0^tf(s)ds $$ for all $t\in [0, T]$. (ii) Suppose $f$ is a continuous function. A function $u(\cdot)$ is a classical solution of \eqref{first} if $u(t)$ is continuously differentiable, $u(t)\in D(A)$, and \eqref{first} holds for all $t\in [0, T]$. \end{definition} It is not hard to see that, if a mild solution of \eqref{first} is continuously differentiable, then it is a classical solution. Furthermore, if $u(t)$ is a mild solution on $[0, T]$ with $u(0)=u(T)$, then $u(t)$ can be continuously extended to a $T$-periodic mild solution of \eqref{first} on $\mathbb{R}$, provided $f(t)$ has been extended $T$-periodically, too. Therefore, we call a mild solution of \eqref{first} $T$-periodic if $u(0)=u(T)$. We now consider the iterated differential equation \eqref{high} and employ the substitution (see also \cite{san}) by defining $U(\cdot):=(u_1(\cdot), u_2(\cdot),\dots , u_n(\cdot))^T$ with \begin{align*} u_1(\cdot)&=u(\cdot)\\ u_2(\cdot)&=u_1(\cdot)' -A_1u_1(\cdot) \\ &\dots\\ u_n(\cdot)=u_{n-1}(\cdot)' -A_{n-1}u_{n-1}(\cdot). \end{align*} Then we have \begin{align*} u_1(\cdot)'&=A_1u_1(\cdot) +u_2(\cdot);\\ u_2(\cdot)'&=A_2u^1(\cdot) +u^3(\cdot);\\ &\dots \\ u_{n-1}(\cdot) '&=A_{n-1}u^{n-1}(\cdot) +u_{n}(\cdot);\\ u_n(\cdot)'&= A_nu_n(\cdot)+f(\cdot). \end{align*} That can be written in matrix form as \begin{equation}\label{big} U' (t)= \mathcal{C} U(t) +F(t), \quad t\in \mathbb{R}, \end{equation} on the product space $E^{n}$, where $F(t)= (0, 0,\dots ,0, f(t))^T$ and \begin{equation}\label{mat} \mathcal{C}:= \begin{pmatrix} A_1 &I &0& \cdots & \cdots &0\\ 0 &A_2 & I&\ddots & &\vdots \\ \vdots & & \ddots & \ddots & \ddots &\vdots \\ \vdots & & & \ddots & \ddots &\vdots \\ \vdots & & & \ddots & A_{n-1}& I\\ 0&\cdots &&\cdots &0 & A_n \end{pmatrix} \end{equation} with $D(\mathcal{C}):= D(A_1)\times D(A_2) \times \dots \times D(A_n)$. Note that the product space $E^n$ is again a Hilbert space with the norm $\|(x_1, x_2, \dots ,x_n)^T\|:= \sqrt{\sum_1^n\|x_i\|^2}$. In \cite{san}, is was stated that $\mathcal{C}$ is generator of a $C_0$-semigroup in $E^n$ if and only if $A_i$ ($i=1, 2, \dots , n$) are generators of $C_0$ semigroups in $E$. That suggests the following definition of mild (classical) solutions for iterated higher-order differential equation. \begin{definition} \label{def1.2} \rm A continuous function $u(\cdot)$ is a mild (classical) solution of the higher-order differential equation \eqref{high} if $u$ is the first component of a mild (classical) solution of the first-order differential equation \eqref{big}. \end{definition} We next establish the relationship between the Fourier coefficients of the periodic solutions of \eqref{high} and those of the inhomogeneity $f$. Then, as the main result, we give an equivalent condition so that \eqref{high} admits a unique periodic solution for each inhomogeneity $f$ in a certain function space. Our result generalizes some well-known ones, as in Section 3 we present several particular cases, among which, $A$ generates a $C_0$ semigroup and a cosine family. Throughout this article, if not otherwise indicated, we assume that $E$ is a complex Hilbert space and $A_i$, $i=1, \dots , n$, are linear, closed and mutually commuting operators on $E$ with $D= D(A_j)$, $j=1, 2, \dots , n$, dense in $E$. The spectrum and resolvent set of $A$ are denoted by $\sigma(A)$ and $\varrho(A)$, respectively and $(\lambda -A)^{-1}$ is denoted by $R(\lambda, A)$. Two unbounded operators $A$ and $B$ are said to commute if for each $\lambda_1\in \varrho(A)$ and $\lambda_2\in \varrho(B)$ we have $(\lambda_1-A)^{-1}(\lambda_2 -B)^{-1} =(\lambda_2-B)^{-1}(\lambda_1 -A)^{-1} $. That definition is equivalent to the fact that $AB=BA$ as the following simple lemma shows. \begin{lemma}\label{12} Suppose $A$ and $B$ are two commuting operators. Then for each $x\in D$ with $Bx\in D$ we have $Ax\in D$ and $BAx=ABx$. \end{lemma} \begin{proof} Let $\alpha\in \varrho(A)$ and $\beta\in \varrho(B)$ and put $y=ABx$. Then $$ (\alpha -A) (\beta -B) x =\alpha \beta x -\beta A x - \alpha B x +y $$ or \begin{align*} x&= (\beta -B)^{-1}(\alpha -A)^{-1}( \alpha \beta x -\beta A x - \alpha B x +y)\\ &=(\alpha -A)^{-1}(\beta -B)^{-1}( \alpha \beta x -\beta A x - \alpha B x +y), \end{align*} which implies $$ (\beta -B)(\alpha -A) x =\alpha \beta x -\beta A x - \alpha B x +y $$ or $BAx=y=ABx$. \end{proof} Let $J=[0, T]$. For the sake of simplicity (and without loss of generality) we assume $T = 1$. For $p\ge 1$, $L_p(J)$ denotes the space of $E$-valued $p$-integrable functions on $J$ with $\|f\|_{L_p(J)}=(\int_0^1\|f(t)\|^pdt )^{1/p}<\infty$ and $C(J)$ the space of continuous functions on $J$ with $\|f\|_{C(J)}= \max\limits_{J}\|f(t)\|$. Moreover, if $m\ge 1$, we define the function space \[ W_2^m(J): =\{f\in L_2(J): f', f'', \dots , f^{(m)}\in L_2(J)\} \] which is a Hilbert space with the norm \[ \|f\|_{W_2^m}:=\Big(\sum_{j=0}^m\|f^{(j)}\|^2_{L_2(J)}\Big)^{1/2}. \] We will use the following simple lemma. \begin{lemma}\label{22} If $F$ is an absolutely continuous function on $J$ such that $f= F' \in L_p(J)$, then for $k\neq 0$ we have $$ F_k=\frac{1}{2k\pi i}f_k +\frac{F(0)-F(1)}{2k\pi i}, $$ where $f_k$ and $F_k$ are the Fourier coefficients of $f$ and $F$, respectively. \end{lemma} Finally, a continuous function $u(\cdot)$ is said to be a 1-periodic solution of \eqref{high} (or to be a solution in $W_2^m(J)$) if the corresponding mild solution $\mathcal{U}(\cdot)$ of \eqref{big} is 1-periodic (or in $W_2^m(J, E^n)$) respectively. \section{Periodic mild solutions of higher-order differential equations} \begin{proposition}\label{propp} Suppose $f\in L_p(J)$ and $u$ is a mild solution of the first-order differential equation \begin{equation}\label{first2} u'(t)=Au(t)+f(t), \quad t\in J. \end{equation} Then \begin{equation}\label{formula} (2k\pi i -A)u_k-f_k = u(0)-u(1) \end{equation} for $k\in \mathbb{Z}$. \end{proposition} \begin{proof} Let $u$ be a mild solution of \eqref{first}, i.e., \begin{equation} \label{n-1} u(t) = u(0) +A\int_0^tu(s)ds+\int_0^tf(s)ds. \end{equation} First, if $k=0$, then using \eqref{n-1} for $t=1$ we have $u(1) = u(0) +Au_0+f_0$, from which \eqref{formula} holds for $k=0$. Next, if $k\neq 0$, taking the $k^{th}$ Fourier coefficient on both sides of \eqref{n-1}, we obtain \begin{align*} u_k &= A\int_0^1e^{-2k\pi i s}\int_0^su(\tau)d\tau ds +\int_0^1e^{-2k\pi i s}\int_0^sf(\tau)d\tau ds\\ &=AU_k +F_k, \end{align*} where $U_k$ is the $k^{th}$ Fourier coefficient of $U(t)=\int_0^tu(\tau)d\tau $ and $F_k$ is the $k^{th}$ Fourier coefficient of $F(t)=\int_0^tf(\tau)d\tau $. Using now Lemma \ref{22} for $U(t)= \int_0^tu(\tau)d\tau $ and $F(t)= \int_0^tf(\tau)d\tau $ we obtain \[ u_k=\frac{A(u_k -U(1))}{2k\pi i}+\frac{f_k-F(1)}{2k\pi i}, \] from which we have \begin{align*} (2k\pi i-A)u_k &=f_k -(AU(1)+F(1))\\ &=f_k -(A\int_0^1u(s)ds+\int_0^1f(s)ds)\\ &=f_k+(u(0)-u(1)). \end{align*} Hence, \eqref{formula} holds. Here we used the fact that $u$ is a mild solution of \eqref{first}, implying $ u(1)=u(0) +A\int_0^1u(s)ds +\int_0^1 f(s)ds$. \end{proof} If $u$ is a 1-periodic solution of \eqref{first}, then we have a nice relationship between Fourier coefficients of $u$ and those of $f$, as the following result shows. \begin{corollary}\label{key} Suppose $f\in L_p(J)$ and $u$ is a continuous mild solution of \eqref{first}. Then $u$ is 1-periodic if and only if \begin{equation}\label{fourier} (2k\pi i-A) u_k = f_k \end{equation} for every $k\in \mathbb{Z}$. \end{corollary} Next we give a sufficient condition for the existence of 1-periodic mild solutions of \eqref{high}. \begin{proposition}\label{key2} Suppose $f\in L_p(J)$. Then the iterated differential equation \eqref{high} admits a continuous, 1-periodic mild solution if and only if there is a sequence $(u_k)_{k=-\infty}^{\infty}\subset E$, such that \begin{itemize} \item[(i)] For each $m$, $0\le m\le n-1$, the function \begin{equation}\label{conti} v_m(t):=\sum_{k=-\infty}^{\infty}e^{-2k\pi it}[\prod_{j=1}^m(2k\pi i-A_j)]u_k \end{equation} is continuous on $[0,1]$ and \item[(ii)] The equality \begin{equation}\label{fourier-n} \prod_{j=1}^{n}(2k\pi i-A_j) u_k = f_k \end{equation} holds for every $k\in \mathbb{Z}$. \end{itemize} \end{proposition} \begin{proof} Suppose \eqref{high} admits a 1-periodic mild solution $u$. By the definition of solution $u$, there is a 1-periodic mild solution $U(t)=(u_1(t), u_2(t), \dots , u_n(t))^{T}$ of \eqref{big} with $u=u_1$. By Corollary \ref{key}, we have $$ (2k\pi i-\mathcal{C})U_k=(0, 0, \dots , f_k)^T $$ or $$ \begin{pmatrix} (2k\pi i-A_1) &-I &0& \cdots & \cdots &0\\ 0 &(2k\pi i-A_2) & -I&\ddots & &\vdots \\ \vdots & & \ddots & \ddots & \ddots &\vdots \\ \vdots & & & \ddots & \ddots &\vdots \\ \vdots & & & &\ddots &-I \\ 0&\cdots &&\cdots &0 & (2k\pi i-A_n) \end{pmatrix} \begin{pmatrix} (u_1)_k\\ (u_2)_k\\ \vdots \\ (u_n)_k\\ \end{pmatrix} =\begin{pmatrix} 0\\ 0\\ \vdots \\ f_k\\ \end{pmatrix}, $$ which implies \begin{equation}\label{aa} \begin{aligned} (u_2)_k&=(2k\pi i -A_1)(u_1)_k=(2k\pi i -A_1)u_k;\\ (u_3)_k&=(2k\pi i -A_2)(u_2)_k=(2k\pi i -A_2)(2k\pi i -A_1)u_k;\\ &\dots \\ (u_n)_k&=(2k\pi i -A_{n-1})(u_{n-1})_k =\prod_{j=1}^{n-1}(2k\pi i-A_j)u_k;\\ f_k&=(2k\pi i -A_n)(u_n)_k=\prod_{j=1}^{n}(2k\pi i-A_j)u_k. \end{aligned} \end{equation} Hence, for each $j$, $0\le j\le n-1$, the function $$ v_j(t):=\sum_{k=-\infty}^{\infty}e^{2k\pi it}[\prod_{z=1}^j(2k\pi i-A_z)]u_k $$ is the same as $u_j(t)$, which is continuous on $[0,1]$. Moreover, \eqref{fourier-n} follows from \eqref{aa}. Conversely, suppose for each $j$, $0\le j\le n-1$, the function \eqref{conti} is continuous on $[0,1]$ and\eqref{fourier-n} holds. We show that there exists a mild solution $U$ of \eqref{big}, which is 1-periodic. To this end, for each $k\in \mathbb{Z}$ we define \begin{align*} u_1(t)&:=\sum_{k=-\infty}^{\infty}e^{2k\pi it}u_k;\\ u_2(t)&:=\sum_{k=-\infty}^{\infty}e^{2k\pi it}(2k\pi i-A_1) u_k;\\ &\dots \\ u_n(t)&:=\sum_{k=-\infty}^{\infty}e^{2k\pi it}\prod_{j=1}^{n-1}(2k\pi i-A_j)u_k. \end{align*} Then, by the assumption, $U(t):=(u_1(t), u_2(t), \dots , u_n(t))^T$ is a continuous function with the following Fourier coefficients: \begin{align*} (u_2)_k&=(2k\pi i -A_1)u_k;\\ (u_3)_k&=(2k\pi i -A_2)(u_2)_k =(2k\pi i-A_2)(2k\pi i-A_1)u_k\\ &\dots \\ ( u_n)_k&=(2k\pi i-A_{n-1})(u_{n-1})_k=\prod_{j=1}^{n-1}(2k\pi i-A_j)u_k \end{align*} and by \eqref{fourier-n}, \[ (2k\pi i-A_n)(u_n)_k=\prod_{j=1}^{n}(2k\pi i-A_j)u_k=f_k \] Hence, $$ \begin{pmatrix} (2k\pi i-A_1) &-I &0& \cdots & \cdots &0\\ 0 &(2k\pi i-A_2) & -I&\ddots & &\vdots \\ \vdots & & & \ddots & \ddots &\vdots \\ \vdots & & & \ddots & \ddots &\vdots \\ \vdots & & & \ddots & \ddots &-I\\ 0&\cdots &&\cdots &0 & (2k\pi i-A_n)\\ \end{pmatrix} \begin{pmatrix} (u_1)_k\\ (u_2)_k\\ \vdots \\ (u_n)_k\\ \end{pmatrix} =\begin{pmatrix} 0\\ 0\\ \vdots \\ f_k\\ \end{pmatrix} $$ or $(2k\pi -\mathcal{C})U_k=(0,0, \dots , f_k)^T$. By Corollary \ref{key}, $U(t)=(u_1(t), u_2(t), \dots , u_n(t))^T$ is a 1-periodic mild solution of \eqref{big} and hence, $u(t)$ is a 1-periodic mild solution of \eqref{high}. \end{proof} Note that Proposition \ref{propp}, Corollary \ref{key} and Proposition \ref{key2} also hold if $E$ is a Banach space. We now can state the main result of the paper. \begin{theorem}\label{main} Suppose $E$ is a Hilbert space. Then the following are equivalent \begin{itemize} \item[(i)] For each function $f\in W_2^1(J)$, Equation \eqref{high} admits a unique 1-periodic mild solution in $W_2^1(J)$; \item[(ii)] For each $k\in \mathbb{Z}$ and $1\le j\le n$, $2k\pi i \in \varrho(A_j)$ and there is a number $M>0$ such that \begin{equation}\label{h} \sup_{k\in \mathbb{Z}}\|(2k\pi i-A_j)^{-1} (2k\pi i-A_{j+1})^{-1}\cdots (2k\pi i-A_n)^{-1}\| =M< \infty. \end{equation} \end{itemize} \end{theorem} \begin{proof} (i) $\Rightarrow$ (ii): Suppose for each function $f\in W_2^1(J)$, Equation \eqref{high} admits a unique 1-periodic mild solution $u\in W_2^1(J)$. By the definition of solution $u$ in $W_2^1(J)$, the corresponding solution $U$ of \eqref{big} belongs to $W_2^1(J, E^n)$. We prove that $U$ is the only mild solution of \eqref{big} corresponding to $f$ by showing that $U\equiv 0$ is the only mild solution of \eqref{big}corresponding to $f\equiv 0$. Indeed, if $f\equiv 0$, then $u\equiv 0$. Hence, its Fourier coefficients $u_k=0$ for all $k\in \mathbb{Z}$. In the proof of Theorem \ref{key2} we have $(u_2)_k=(2k\pi i -A_1)u_k=0$ for all $k\in \mathbb{Z}$. Hence, $u_2(t)\equiv 0$. Similarly, we have $u_j(t)\equiv 0$ for all $1\le j\le n$ and thus, $U(t)\equiv 0$.\\[5pt] Define the operator: $$ G:f\in W_2^1(J)\mapsto Gf\in W_2^1(J, E^n) $$ as follows: $(Gf)(t)$ is the unique solution of \eqref{big} corresponding to $f$. Then $G$ is a linear, everywhere defined operator. We will prove its boundedness by showing $G$ is a closed operator. To this end, suppose $\{f_m\}_{m=1}^{\infty}$ is a sequence of functions in $F_1= W_2^1(J)$ such that $f_m\to f$ in $F_1$ and $Gf_m$ approaches some function $V=(V_1, V_1, \dots ,V_n)^T$ in $F_2 =W_2^1(J, E^n)$ as $m\to \infty$. We show that $f\in D(G)$ and $Gf=V$. Since $Gf_m$ is a mild solution of \eqref{big} corresponding to $f_m$, we have $$ Gf_m(t)=Gf_m(0)+\mathcal{C}\int_0^tGf_m(s)ds+\int_0^tF_m(s)ds. $$ Hence, \begin{equation}\label{ind} \begin{gathered} (Gf_m)_1(t) =(Gf_m)_1(0)+A_1\int_0^t(Gf_m)_1(s)ds+\int_0^t(Gf_m)_2(s)ds\\ (Gf_m)_2(t) =(Gf_m)_2(0)+A_2\int_0^t(Gf_m)_2(s)ds+\int_0^t(Gf_m)_3(s)ds\\ \dots \\ (Gf_m)_{n-1}(t) =(Gf_m)_{n-1}(0)+A_{n-1}\int_0^t(Gf_m)_{n-1}(s)ds +\int_0^t(Gf_m)_n(s)ds\\ (Gf_m)_n(t) =(Gf_m)_n(0)+A_n\int_0^t(Gf_m)_n(s)ds+\int_0^tf_m(s)ds. \end{gathered} \end{equation} Consider now the sequence $\{x_m\}_{m\ge 1}$ in $E$, where $x_m=\int_0^t(Gf_m)_1(s)ds$. We have $$ x_m= \int_0^t(Gf_m)_1(s)ds \to \int_0^tV_1(s)ds $$ as $m\to \infty$, and from \eqref{ind}, \begin{align*} A_1x_m= A_1\int_0^t(Gf_m)_1(s)ds &=(Gf_m)_1(t) -(Gf_m)_1(0)-\int_0^t(Gf_m)_2(s)ds \\ &\to V_1(t) -V_1(0)-\int_0^tV_2(s)ds \end{align*} as $m\to \infty$. Since $A_1$ is a closed operator, we have $\int_0^tV_1(s)ds \in D(A_1)$ and $$ A_1\int_0^tV_1(s)ds =V_1(t) -V_1(0)-\int_0^tV_2(s)ds, $$ which implies $$ V_1(t)=V_1(0)+ A_1\int_0^tV_1(s)ds +\int_0^tV_2(s)ds. $$ In the same manner, we can show that \begin{gather*} V_2(t)=V_2(0)+ A_2\int_0^tV_2(s)ds +\int_0^tV_3(s)ds,\\ \dots \\ V_{n-1}(t) =V_{n-1}(0)+A_{n-1}\int_0^tV_{n-1}(s)ds+\int_0^tV_n(s)ds,\\ V_n(t) =V_n(0)+A_n\int_0^tV_n(s)ds+\int_0^tf(s)ds, \end{gather*} i.e., $V$ is a mild solution of \eqref{big} corresponding to $f$ and consequently, $Gf=V$. So, $G$ is a bounded operator from $F_1$ to $F_2$. Next we show that $2k\pi i \in \varrho(A_j)$ for each $k\in \mathbb{Z}$ and each $1\le j\le n$. Let $x$ be any vector in $ E$, $k\in \mathbb{Z}$ and take $f(t)=e^{2k\pi i t}x$ and $V=(V_1, V_2, \dots , V_n)^T$ be the unique mild solution of \eqref{big} corresponding to $f$. From Fourier coefficient Identity \eqref{aa} we have $$ \prod_{j=1}^{n}(2k\pi i-A_j)(V_n)_k =f_k =x, $$ which shows $\prod_{j=1}^{n}(2k\pi i-A_j)$ and hence, $(2k\pi i-A_n)$, is surjective. Using Lemma \ref{12} we have $(2k\pi i-A_j)$ is surjective for each $1\le j\le n$. Assume now that for some $1\le j\le n$, $(2k\pi i-A_j)$ contrarily is not injective. Without loss of generality we can assume that $A_j$ is the first operator with non-injective $(2k\pi i-A_j)$, i.e., $(2k\pi i-A_l)$ are injective for $1\le l 0$ (since $t\mapsto \mathcal{T}(t)x$ is differentiable at $t_0$ if and only if $\mathcal{T}(t_0)x\in D(\mathcal{C})$). Hence, $U(1)$, and thus, $U(0) $ (the same as $U(1)$) belongs to $ D(\mathcal{C})$. So $U$ is a classical solution. The uniqueness of the 1-periodic classical solution is obvious. \end{proof} If $n=1$, then Theorem \ref{p2} becomes Gearhart theorem in \cite{ge} (See also \cite{pruss}). We see clearly that statement (iv) in Theorem \ref{p2} holds if for each $j$, $0\le j\le n$, we have $2k\pi i \in \varrho(A_j)$ and \begin{equation}\label{hhh} \sup_{k\in \mathbb{Z}}\|(2k\pi i-A_j)^{-1}\| < \infty. \end{equation} But in general, condition \eqref{hhh} is stronger than \eqref{hh} (they are equivalent if $n=1$). Hence, unless $n=1$, the existence and uniqueness of 1-periodic mild solution of \eqref{high} does not imply \eqref{hhh}. The next example shows that in some special cases the two conditions are equivalent. \subsection*{Example} Suppose $B=A^2$, where $A$ generates a $C_0$ group on $E$. Consider the second-order differential equation \begin{equation}\label{2} u''(t)=Bu(t)+f(t), \quad 0\le t\le 1. \end{equation} We can rewrite \eqref{2} as $$ (\frac{d}{dt}-A)(\frac{d}{dt}+A)u(t)=f(t). $$ Hence, from Theorem \ref{p2} we have the following result. \begin{theorem} The following statements are equivalent: \begin{itemize} \item[(i)] For each function $f\in W_2^1(J)$, Equation \eqref{2} admits a unique 1-periodic mild solution in $W_2^1(J)$; \item[(ii)] For each function $f\in W_2^1(J)$, Equation \eqref{2} admits a unique 1-periodic classical solution; \item[(iii)] For each $k\in \mathbb{Z}$, $2k\pi i \in \varrho(B)$ and \begin{equation}\label{hhhh} \sup_{k\in \mathbb{Z}}\|(2k\pi i-A)^{-1}\| < \infty. \end{equation} \item[(iii)] For each $k\in \mathbb{Z}$, $-4k^2\pi^2 \in \varrho(B)$ and \begin{equation}\label{hhhhh} \sup_{k\in \mathbb{Z}}\|(4k^2\pi^2 i+B)^{-1}\| < \infty. \end{equation} \end{itemize} \end{theorem} \begin{proof} Let $A_1=-A$ and $A_2=A$. 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