\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 89, pp. 1--7.\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-2013/89\hfil Exponential dichotomy and boundedness] {Exponential dichotomy of nonautonomous periodic systems in terms of the boundedness of certain periodic Cauchy problems} \author[D. Lassoued \hfil EJDE-2013/89\hfilneg] {Dhaou Lassoued} % in alphabetical order \address{Dhaou Lassoued \newline Laboratoire SAMM EA4543\\ Universit\'e Paris 1 Panth\'eon-Sorbonne \\ centre P.M.F., 90 rue de Tolbiac \\ 75634 Paris cedex 13, France} \email{Dhaou.Lassoued@univ-paris1.fr, dhaou06@gmail.com} \thanks{Submitted February 1, 2013. Published April 5, 2013.} \subjclass[2000]{47A05, 34D09, 35B35} \keywords{Periodic evolution families; exponential dichotomy; \hfill\break\indent boundedness} \begin{abstract} We prove that a family of $q$-periodic continuous matrix valued function $\{A(t)\}_{t\in \mathbb{R}}$ has an exponential dichotomy with a projector $P$ if and only if $\int_0^t e^{i\mu s}U(t,s)Pds$ is bounded uniformly with respect to the parameter $\mu$ and the solution of the Cauchy operator Problem \begin{gather*} \dot{Y}(t)=-Y(t)A(t)+ e^{i \mu t}(I-P) ,\quad t\geq s \\ Y(s)=0, \end{gather*} has a limit in $\mathcal{L}(\mathbb{C}^n)$ as $s$ tends to $-\infty$ which is bounded uniformly with respect to the parameter $\mu$. Here, $\{ U(t,s): t, s\in\mathbb{R}\}$ is the evolution family generated by $\{A(t)\}_{t\in \mathbb{R}}$, $\mu$ is a real number and $q$ is a fixed positive number. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} Let $\mathbb{C}^n$ be the linear space of all complex vectors and $\mathcal{L}(\mathbb{C}^n)$ the Banach algebra of all linear $\mathbb{C}^n$-valued operators. The norm on $\mathbb{C}^n$ and on $\mathcal{L}(\mathbb{C}^n)$ is denoted by the same symbol, namely $\|\cdot \|$. Consider the nonautonomous $q$-periodic system % $$\label{1} x'(t)=A(t)x(t),\quad x(t)\in \mathbb{C}^n,\; t\in \mathbb{R}.$$ Here, $q>0$ is a given real number and $A(t)$ is a $q$-periodic continuous matrix valued function i.e. $A(t+q)=A(t)$ for all $t\in\mathbb{R}$. Exponential dichotomy is one of the fundamental asymptotic properties of solution of the linear differential system \eqref{1} on $\mathbb{C}^n$, \cite{CP}. In the autonomous case i.e. when $A(t)=A$ is a constant matrix, the exponential dichotomy is equivalent to the fact that the matrix $A$ has no eigenvalues on the imaginary axis $i\mathbb{R}$. Nevertheless, when $A$ depends on time variable, the study of the exponential dichotomy is more difficult. Important results on this topic are obtained in \cite{BZ1} and \cite{Z2}. In this article, we aim to investigate the exponential dichotomy of the evolution family generated by the system \eqref{1}. More exactly, we prove that this evolution family has an exponential dichotomy with respect to a projector $P$ (see the next section for definitions) if and only if the solution of the operator Cauchy Problem % $$\label{AA} \begin{gathered} \dot{X}(t)=A(t)X(t)+ e^{i\mu t}P, \quad X(t)\in {\mathcal{L}}(\mathbb{C}^n),\; t\in \mathbb{R} \\ X(0)=0 \end{gathered}$$ is bounded, uniformly with respect to $\mu$ and the solution $V_{\mu}(\cdot)$ of the Cauchy Problem $$\label{AAA} \begin{gathered} \dot{Y}(t)=-Y(t)A(t)+ e^{i \mu t}Q, \quad Y(t)\in {\mathcal{L}}(\mathbb{C}^n),\; t\geq s \\ Y(s)=0 \end{gathered}$$ has a limit in $\mathcal{L}(\mathbb{C}^n)$ as $s\to -\infty$; i.e. $\int_{-\infty}^t e^{i \mu \tau} QU(\tau,t)d\tau$ exists in $\mathcal{L}(\mathbb{C}^n)$, and $$\sup_{\mu\in \mathbb{R}} \sup_{t\in\mathbb{R}} \big\| \int_{-\infty}^te^{i\mu s}QU(s,t)ds\big\|:=M_2<\infty.$$ Here, $\mu$ is a real number and $Q$ denotes the projector $I-P$. \section{Definitions, notation and preliminary results} For $A\in \mathcal{L}(\mathbb{C}^n)$, we denote by $\sigma(A)$ spectrum of $A$; i.e., the set of all complex scalars $z\in \mathbb{C}$ for which the operator $zI-A$ is not invertible. $I$ denotes the identity linear operator on $\mathbb{C}^n$. As it is well-known, the solution of the operator Cauchy Problem \begin{gather*} \dot{X}(t)=A(t)X(t) ,\quad t\in \mathbb{R} \\ X(0)=I, \end{gather*} denoted by $P(\cdot)$ is called the fundamental matrix associated to the family $\{A(t)\}_{t\in \mathbb{R}}$. For every $t\in \mathbb{R}$, $P(t)$ is invertible and its inverse is the solution of the following operator Cauchy Problem \begin{gather*} \dot{Y}(t)=-Y(t)A(t) ,\quad t\in \mathbb{R} \\ Y(0)=I. \end{gather*} It is not difficult to verify that the family $\mathcal{U}:=\{ U(t,s):=P(t)P^{-1}(s), \; t, s\in \mathbb{R} \}$ satisfies the following properties: \begin{itemize} \item $U(t,t)=I$, for all $t\in \mathbb{R}$. \item $U(t,s)U(s,r)=U(t,r)$, for all $t, s, r \in \mathbb{R}$. \item The map $$(t,s)\mapsto U(t,s): \{(t,s)\in \mathbb{R}^2: \; t\geq s\} \to \mathcal{L}(\mathbb{C}^n)$$ is continuous. \item $U(t+q,s+q)=U(t,s)$ for all $t, s\in \mathbb{R}$. \item $\frac{\partial}{\partial t}U(t,s)=A(t)U(t,s)$ for all $t, s \in \mathbb{R}$. \item $\frac{\partial}{\partial s}U(t,s)=-U(t,s)A(s)$ for all $t, s\in \mathbb{R}$. \item There exist two real constants $\omega \in \mathbb{R}$ and $M\geq 1$ such that $\| U(t,s)\| \leq Me^{\omega \vert t-s\vert}\quad\text{for all } t, s\in \mathbb{R}.$ \end{itemize} \begin{definition}\label{def} \rm The evolution family $\mathcal{U}$ is said to have a uniform exponential dichotomy with respect to the projector $P$ (i.e. $P\in \mathcal{L}(\mathbb{C}^n)$ and $P^2=P$) if there exist positive constants $N_1, \; N_2,\; \nu_1$ and $\nu_2$ such that \begin{itemize} \item[1.] $U(t,s)P=PU(t,s)$, for all $t, s \in \mathbb{R}$ \item[2.] $\| U(t,s)P\| \leq N_1 e^{-\nu_1 (t-s)}$, for all $t\geq s\in \mathbb{R}$ \item[3.] $\| QU(s,t)\| \leq N_2 e^{-\nu_2(t-s)}$, for all $t\geq s\in \mathbb{R}$. \end{itemize} Here, $Q:=I-P$ and $U(s,t)$ is the inverse of $U(t,s)$. \end{definition} It is clear that $Q^2=Q$ and $PQ=QP=0$. \begin{remark} \rm For the special case when $P=I$ (and so $Q=0$), we recognize the uniform exponential stability of the evolution family $\mathcal{U}$. \end{remark} \begin{example} \rm Set two 1-periodic continuous functions $a(\cdot)$ and $b(\cdot)$; i.e., $a(t+1)=a(t)$ and $b(t+1)=b(t)$, for all $t\in \mathbb{R}$, and so, the 1-periodic continuous matrix valued map $t\mapsto A(t)$ given by $$A(t) = \begin{pmatrix} a(t) & 0 \\ 0 & b(t) \end{pmatrix}.$$ The system $X(t)=A(t)X(t)$ leads to the evolution family $\mathcal{U}:=\{P(t)P^{-1}(s), \; s,t \in \mathbb{R}\}$, where for each real $t$, $$P(t) = \begin{pmatrix} e^{\int_0^t a(s)ds} & 0 \\ 0 & e^{\int_0^t b(s)ds} \end{pmatrix}.$$ We take the projectors $P = \begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}$. and $Q =I-P$. It is easy to verify that for all $t, s \in \mathbb{R}$, $PU(t,s)=U(t,s)P$. \end{example} Throughout this article, we assume that there exists a projector $P$ such that $U(t,s)P=PU(t,s)$ for all $s,t \in \mathbb{R}$. Before announcing the main result of this article, we recall some known results which will be used in its proof. State first the following proposition. \begin{proposition}[\cite{ArsBusSai,BCDS}]\label{pro} Let $\mathcal{U}:=\{U(t,s): s,t\in\mathbb{R} \}$ be a strongly continuous and $q$-periodic evolution family acting on the Banach space $X$. Then, the following assertions are equivalent: \begin{itemize} \item[1.] The family $\mathcal{U}$ is uniformly exponentially stable. \item[2.] There are two real constants $N \geq 1$ and $\nu >0$ such that for all $t\geq 0$, we have $\| U(t,0)\| \leq Ne^{-\nu t}$. \item[3.] The spectral radius of $U(q,0)$, $$r(U(q,0)):=\sup\{|\lambda|; \; \lambda \in \sigma(U(q,0))\} =\lim_{n\to \infty} \| U(q,0)^n\|^{1/n},$$ is less than 1. \item[4.] For each $\mu \in \mathbb{R}$, we have $$\sup_{n\in \mathbb{N}} \big\| \sum_{k=1}^n e^{i \mu k}U(q,0)^k \big\|$$ is finite. \end{itemize} \end{proposition} The following technical lemma will be an important ingredient of our proof. \begin{lemma}[\cite{ArsBusSai}]\label{lem} Consider the functions $h_1$ and $h_2$ defined from $[0,q]$ to $\mathbb{C}$, respectively, by \begin{gather*} h_1(t)= \begin{cases} t, & \text{if } t\in [0,q/2] \\ q-t, & \text{if } t\in [q/2,q], \end{cases}\\ h_2(t) = t(q-t)\quad \text{for all } t\in [0,q]. \end{gather*} If we denote $H_j(\mu):=\int_0^q h_j(s)e^{-i\mu s} ds$, for $j=1,2$ and $\mathcal{A}:=\{\frac{4k\pi}{q}: k\in \mathbb{Z}\setminus \{0\}\}$, then \begin{itemize} \item $H_1(\mu)=0$ if and only if $\mu \in \mathcal{A}$, \item $H_2(\mu \neq 0$ for all $\mu \in \mathcal{A}$. \end{itemize} \end{lemma} \section{Main result and its proof} \begin{theorem} The following statements are equivalent: \begin{itemize} \item[i.] The evolution family $\mathcal{U}$ has an exponential dichotomy with respect to the projector $P$. \item[ii.] The following assertions hold: \begin{itemize} \item[1.] $\sup_{\mu\in \mathbb{R}} \sup_{t\in\mathbb{R}} \big\| \int_0^t e^{i\mu s}U(t,s)Pds\big\|:=M_1<\infty$. \item[2.] The solution of the equation \eqref{AAA} has a limit in $\mathcal{L}(\mathbb{C}^n)$ as $s$ tends to $-\infty$ (i.e. $\int_{-\infty}^te^{i\mu s}QU(s,t)ds$ exists) and $$\sup_{\mu\in \mathbb{R}} \sup_{t\in\mathbb{R}} \big\| \int_{-\infty}^te^{i\mu s}QU(s,t)ds\big\|:=M_2<\infty,$$ \end{itemize} where $M_1$ and $M_2$ are two absolutely positive constants. \end{itemize} \end{theorem} \begin{proof} It is not difficult to show that the function $U_{\mu}(t):=\int_0^t e^{i\mu s}U(t,s)Pds$ is the solution of \eqref{AA} and the function $V_{\mu}(t):=\int_{s}^te^{i\mu s}QU(s,t)ds$ is the solution of \eqref{AAA}. $\bullet$ Let us first show that (i) implies (ii). If the evolution family has an exponential dichotomy with respect to the projector $P$, then in view of the Definition \ref{def}, we have $\| U(t,s)P\| \leq N_1 e^{-\nu_1 (t-s)}$, for all $t\geq s\in \mathbb{R}$, for some positive constants $N_1$ and $\nu_1>0$. An easy calculation gives that for all $\mu, t\in\mathbb{R}$, $$\big\| \int_0^t e^{i\mu s}U(t,s)Pds\big\| \leq \frac{N_1}{\nu_1}.$$ Since $\| QU(s,t)\| \leq N_2 e^{-\nu_2(t-s)}$, for all $t\geq s\in \mathbb{R}$, it follows that the improper integral $\int_{-\infty}^t e^{i\mu s}QU(t,s)ds$ is well-defined which implies that the solution of \eqref{AAA},$V_{\mu}(\cdot)$, has a limit in $\mathcal{L}(\mathbb{C}^n)$ as $s$ tends to $-\infty$ in $\mathcal{L}(\mathbb{C}^n)$, and, in addition, $$\big\| \int_{-\infty}^t e^{i\mu s}QU(t,s)ds\big\| \leq \frac{N_2}{\nu_2},$$ $\bullet$ Now, we show the converse. For $j=1,2$, we set the functions $f_j(t):=h_j(t)U(t,0)P$ defined on $[0,q]$ where $h_j$ are those introduced in the Lemma \ref{lem}. We extend them by periodicity on the whole real line. If we put $t=(N+1)q$, where $N$ is a positive integer number, for $j=1,2$, then denoting $$L_k(f_j):=\int_{qk}^{q(k+1)} U((N+1)q,\tau)e^{-i\mu \tau}f_j(\tau)d\tau,$$ we obtain $$\sup_{N\in \mathbb{Z}_+} \big\| \sum_{k=0}^{N} L_k(f_j) \big\| :=M_1(\mu,f_j)<\infty.$$ Moreover, as $U(t+q,s+q)=U(t,s)$ for all $t, s\in \mathbb{R}$, it follows that $U(pq,kq)=U((p-k)q,0)=U(q,0)^{p-k}$ for all $p, k\in\mathbb{Z}_+$ with $p\geq k$. Therefore, since $P$ commutes with $U(t,s)$, and so with every power of $U(t,s)$, we have \begin{align*} L_k(f_j) &= \int_{qk}^{q(k+1)} U((N+1)q,(k+1)q)U((k+1)q,\tau)e^{-i\mu \tau}f_j(\tau)d\tau \\ &= \int_{qk}^{q(k+1)} U((N-k)q,0)U((k+1)q,\tau)e^{-i\mu \tau}f_j(\tau)d\tau \\ &= U(q,0)^{N-k} \int_{qk}^{q(k+1)} U((k+1)q,\tau)e^{-i\mu \tau}f_j(\tau)d\tau \\ &= U(q,0)^{N-k}e^{-i\mu kq} \int_{0}^{q}e^{-i\mu u} U(q,u)f_j(u)du \\ &= (U(q,0)P)^{N-k+1}e^{i\mu (N-k+1)q}e^{-i\mu (N+1)} H_j(\mu). \end{align*} Thanks to the Lemma \ref{lem}, we obtain \begin{gather*} (U(q,0)P))^{N-k+1}e^{i\mu (N-k+1)q} =\frac{e^{i\mu (N+1)}}{H_1(\mu)}L_k(f_1), \quad\text{for all } \mu \notin \mathcal{A}, \\ (U(q,0)P))^{N-k+1}e^{i\mu (N-k+1)q}=\frac{e^{i\mu (N+1)}}{H_1(\mu)}L_k(f_2), \quad \text{for all } \mu \in \mathcal{A}. \end{gather*} Therefore, we have \begin{gather*} \big\| \sum_{k=0}^N (U(q,0)P))^{N-k+1}e^{i\mu (N-k+1)q}\big\| \leq \frac{1}{\vert H_1(\mu)\vert}M_1(\mu, f_1),\quad\text{if } \mu \notin \mathcal{A} \\ \big\| \sum_{k=0}^N (U(q,0)P))^{N-k+1}e^{i\mu (N-k+1)q} \big\| \leq \frac{1}{\vert H_2(\mu)\vert}M_1(\mu, f_2),\text{if } \mu \in \mathcal{A}. \end{gather*} This implies that $$\sup_{N\in \mathbb{Z}_+}\big\| \sum_{k=0}^{N+1} (U(q,0)P))^{k}e^{i\mu kq}\big\| <\infty.$$ Using Proposition \ref{pro}, we deduce that the spectral radius $r(UP)<1$, where $U:=U(q,0)$, which implies that there exist two constants $N_1\geq 0$ and $\nu_1 >0$ such that \begin{equation*} \big\| U(t,s)P\big\| \leq N_1 e^{-\nu_1 (t-s)} \quad \forall t\geq s\in \mathbb{R}\,. \end{equation*} From the second assumption, we have that for all $s\in\mathbb{R}$ large enough, $$\big\| \int_s^t e^{i\mu \tau} QU(\tau,t)d\tau\big\| \leq M_2+1.$$ Consider, for $j=1,2$, the functions $g_j$ defined on $[0,q]$ by $g_j(\tau)=h_j(\tau)U(0,\tau)Q$, where the functions $h_j$ are defined as in the Lemma \ref{lem} and we extend this functions by periodicity on the whole real line. Besides, by derivation, we can show easily that, for $j=1,2$, the function $\int_s^t e^{i\mu \tau}g_j(\tau)U(\tau,t)d\tau$ is the solution of the the differential equation $Z'(t)=-Z(t)A(t)+e^{i\mu t}g_j(t)$, $Z(s)=0$, $t\geq s$. We remark also that this functions are bounded. We proceed as in \cite[Theorem 3.2]{ArsBusSai}. If we put $t=(N+1)q$ and $s=mq$, for $N>m$ two integer numbers, then we have that $$\big\| \int_{mq}^{(N+1)q} e^{-i\mu \tau} g_j(\tau)U(\tau, (N+1)q) d\tau\big\| :=M_2(\mu, g_j)< \infty.$$ It follows that $$\sup_{N\in\mathbb{Z}_+} \sum_{k=m}^{N} \| S_k(g_j) \| =M_2(\mu, g_j)<\infty;$$ where $S_k(g_j)=\int_{kq}^{(k+1)q} e^{-i\mu \tau} g_j(\tau)U(\tau, (N+1)q) d\tau$, for each $k=m,\dots , N$. For each $k=m,m+1,\dots$, similarly to the previous calculation, we have \begin{align*} S_k(g_j) &= \int_{kq}^{(k+1)q} e^{-i\mu \tau} g_j(\tau)U(\tau,(k+1)q)U((k+1)q,(N+1)q) d\tau \\ &= \int_{kq}^{(k+1)q} e^{-i\mu \tau} g_j(\tau)U(\tau,(k+1)q)U(0,(N-k)q) d\tau \\ &= \int_{kq}^{(k+1)q} e^{-i\mu \tau} g_j(\tau)U(\tau,(k+1)q) d\tau U(0,q)^{N-k} \\ &= \int_{0}^{q} e^{-i\mu u} g_j(u)U(u,q) du e^{-i{\mu kq}}U(0,q)^{N-k}\\ &= \int_{0}^{q} e^{-i\mu u} h_j(u)du e^{-i{\mu kq}}(QU(0,q))^{N-k+1}\\ &= H_j(\mu) e^{-i{\mu (N+1)q}}e^{i\mu(N-k+1)}(QU(0,q))^{N-k+1}. \end{align*} By using Lemma \ref{lem}, we can write \begin{gather*} e^{i\mu(N-k+1)}(QU(0,q))^{N-k+1}= \frac{1}{H_1(\mu)}e^{i{\mu (N+1)q}}S_k(g_1), \quad\text{if } \mu \notin \mathcal{A} \\ e^{i\mu(N-k+1)}(QU(0,q))^{N-k+1}= \frac{1}{H_2(\mu)}e^{i{\mu (N+1)q}}S_k(g_2), \quad \text{if } \mu \in \mathcal{A}. \end{gather*} Then \begin{gather*} \big\| \sum_{k=0}^N (QU(0,q)))^{N-k+1}e^{i\mu (N-k+1)q}\big\| \leq \frac{1}{\vert H_1(\mu)\vert}M_2(\mu, g_1), \quad\text{if } \mu \notin \mathcal{A}\\ \big\| \sum_{k=0}^N (QU(0,q)))^{N-k+1}e^{i\mu (N-k+1)q} \big\| \leq \frac{1}{\vert H_2(\mu)\vert}M_2(\mu, g_2), \quad\text{if } \mu \in \mathcal{A}. \end{gather*} By Proposition \ref{pro}, if we denote $V:=U(0,q)$, we deduce that $r(QV)<1$, and then there exist constants $N_2\geq 0$, $\nu_2>0$ such that \begin{equation*} \| QU(s,t)\| \leq N_2 e^{-\nu_2 (t-s)} \quad \forall t\geq s\in\mathbb{R}, \end{equation*} which completes the proof. \end{proof} \subsection*{Acknowledgements} The author would like to thank the anonymous referee for his/her valuable suggestions that helped to improve this article. 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