\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 137, 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-2013/137\hfil Generalized Bohl-Perron principle] {Generalized Bohl-Perron principle for differential equations with delay in a Banach spaces} \author[M. Gil' \hfil EJDE-2013/137\hfilneg] {Michael Gil'} % in alphabetical order \address{Michael Gil' \newline Department of Mathematics \\ Ben Gurion University of the Negev \\ P.0. Box 653, Beer-Sheva 84105, Israel} \email{gilmi@bezeqint.net} \thanks{Submitted February 27, 2013. Published June 20, 2013.} \subjclass[2000]{34K30, 34K06, 34K20} \keywords{Banach space; differential equation with delay; linear equation; \hfill\break\indent exponential stability} \begin{abstract} We consider a linear homogeneous functional differential equation with delay in a Banach space. It is proved that if the corresponding non-homogeneous equation, with an arbitrary free term bounded on the positive half-line and with the zero initial condition, has a bounded solution, then the considered homogeneous equation is exponentially stable. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \allowdisplaybreaks \section{Introduction and statement of the main result} Recall that the Bohl-Perron principle states that the homogeneous ordinary differential equation (ODE) $dy/dt=A(t)y$ $(t\ge 0)$ with a variable $n\times n$-matrix $A(t)$, bounded on $[0,\infty)$ is exponentially stable, provided the nonhomogeneous ODE $dx/dt=A(t)x+f(t)$ with the zero initial condition has a bounded solution for any bounded vector valued function $f$ \cite{Daleckii}. In \cite[Theorem 4.15]{halan}, the Bohl-Perron principle was generalized to a class of retarded systems with finite delays; also the asymptotic (not exponential) stability was proved. The result from \cite{halan} was a considerable development afterwards, cf. the book \cite{az} and the very interesting papers \cite{ber1, ber2}, in which the generalized Bohl-Perron principle was effectively used for the stability analysis of the first and second order scalar equations. In particular, in \cite{ber1} the scalar non-autonomous linear functional differential equation $\dot{x}(t)+a(t)x(h(t))=0$ is considered. The authors give sharp conditions for exponential stability, which are suitable in the case that the coefficient function $a(t)$ is periodic, almost periodic or asymptotically almost periodic, as often encountered in applications. In \cite{ber2}, the authors provide sufficient conditions for the stability of rather general second-order delay differential equations. In \cite{gi11-del-perron,gi13} a result similar to the Bohl-Perron principle has been derived in terms of the norm of the space $L^p$, which is called the $L^p$-version of the generalized Bohl-Perron principle. In this article, we extend the Bohl-Perron principle to a class of functional differential equations with delay in a Banach space. In Section 3 below, we show that our results can be effectively used for the stability analysis. As it is well-known, the basic method for the stability analysis of functional differential equations is the direct Lyapunov method. By this method very strong results are obtained. But finding Lyapunov's type functionals for nonautonomous vector equations with delay is usually difficult. In Section 3 we suggest explicit sharp stability conditions, which supplement the well-known results on stability of equations with delay in a Banach space; see \cite{Adimy, Arino, Ezzinbi,Matsui, Pao, Wu} and references given therein. Let $X$ be a complex Banach space with a norm $\|\cdot\|_X$ and the unit operator $I$. Denote by $C(\omega)\equiv C (\omega, X)$ the space of continuous functions $u$ defined on a set $\omega\subseteq \mathbb{R}$ with values in $X$ and the finite sup-norm $\|\cdot\|_{C(\omega)}$. For a bounded linear operator $T$ acting from $X$ into a normed space $Y$ we put $\|T\|_{X\to Y}=\sup_{u\in X}\|Tu\|_{Y}/\|u\|_{X}$. Let $A(t)$ be a linear generally unbounded operator in $X$ with a constant dense domain $\operatorname{Dom}(A)$. In $X$, for a positive constant $\eta<\infty$ consider the equation $$\dot y(t)=A(t)y(t)+ \int_0^\eta B(t, s)y(t-s) ds + \sum_1^m B_k(t)y(t-h_k(t)), \label{e1.1}$$ where $\dot y(t)$ is a strong derivative of $y$; $B_k(t)$ $(k=1, \dots, m)$ are bounded continuous operator functions on $[0,\infty)$; $B(t,s)$ is an operator function defined and bounded on $[0,\infty)\times [0,\eta]$, which is continuous in $t$ and integrable in $s$; $0\le h_k(t)\le \eta$ are continuous functions. Let the initial condition be $$y(t)=\phi (t)( -\eta \leq t \le 0) \label{e1.2}$$ for a given $\phi\in C(-\eta, 0)\cap \operatorname{Dom}(A)$. For $w\in C(-\eta, \infty)$, put $$Ew= \int_0^\eta B(t, s)w(t-s) ds + \sum_1^m B_k(t)w(t-h_k(t))\,.$$ Then \eqref{e1.1} takes the form $$\dot y(t)=A(t)y(t)+ Ey(t). \label{e1.3}$$ It is assumed that $A(t)$ generates a strongly continuous evolution family $\{U(t,s)\}$ $(t\ge s\ge 0)$ of bounded operators in $X$. That is, $U(t,s)$ is the evolution operator of the equation $$\dot \zeta(t)=A(t)\zeta(t) \label{e1.4}$$ cf. \cite{Chicone}. Following the Browder terminology \cite{Henry}, a continuous function $y$ satisfying $$y(t)=U(t,0)\phi(0)+\int_0^t U(t,t_1)Ey(t_1)dt_1 \label{e1.5}$$ and \eqref{e1.2} we will be called a \emph{mild solution} to \eqref{e1.1}, \eqref{e1.2}. Consider also the non-homogeneous equation $$\dot x(t)=A(t)x(t)+Ex(t)+f(t),\quad t> 0 \label{e1.6}$$ with a given function $f(t)\in C(0,\infty)$, and the zero initial condition $$x(t)=0, \quad -\eta\le t \le 0. \label{e1.7}$$ Then a continuous function $x$ satisfying $$x(t)=\int_0^t U(t,t_1)(Ex(t_1)+f(t_1))dt_1 \label{e1.8}$$ and \eqref{e1.7} will be called a mild solution to \eqref{e1.6}, \eqref{e1.7}. Below we show that , problems \eqref{e1.1}, \eqref{e1.2} and \eqref{e1.6}, \eqref{e1.7} have unique mild solutions. We will say that \eqref{e1.1} is exponentially stable, if there are positive constants $M_1, \epsilon$, such that $\|y(t)\|\le M_1e^{-\epsilon t}\|\phi\|_{C(-\eta, 0)}$ $(t\ge 0)$ for any mild solution $y(t)$ of \eqref{e1.1}, \eqref{e1.2}. We assume that there are positive constants $\alpha_0$ and $M$, such that \begin{gather} \|U(t,s)\|_X\le Me^{-\alpha_0 (t-s)}\quad \forall t\ge s\ge 0, \label{e1.9}\\ A(t)z\in C(0,\infty)\;\text{for any }z\in \operatorname{Dom}(A). \label{e1.10} \end{gather} \begin{theorem} \label{thm1.1} If conditions \eqref{e1.9} and \eqref{e1.10} hold, and for any $f\in C(0,\infty)$, problem \eqref{e1.6}, \eqref{e1.7} has a bounded mild solution on $[0,\infty)$, then \eqref{e1.1} is exponentially stable. \end{theorem} This theorem is proved in the next section. Suppose $1\le p<\infty$, then for an exponentially bounded and strongly continuous evolution family $U(t, s)$ of bounded linear operators acting in $X$, the following condition is equivalent to \eqref{e1.9}: there exists a constant $M_p > 0$, such that $$\sup_{s\ge 0} \int_s^\infty \|U(t, s)z\|_X^pdt\le M_p\|z\|_X^p, \forall z\in X, \label{e1.11}$$ cf. \cite[p. 75]{Chicone}. Other conditions equivalent to \eqref{e1.9} can be found in \cite[p. 77]{Chicone}. \section{Proofs} It is not difficult to check that for all $\tau>0$, $$\|Ew\|_{C(0, \tau)}\le v_0\|w\|_{C(-\eta,\tau)}\quad\text{for }w\in C(-\eta,\tau), \label{e2.1}$$ where $$v_0=\sup_t \Big(\int_0^\eta \|B(t, s)\|_X ds + \sum_1^m \|B_k(t)\|_X\Big).$$ For brevity, in this section, sometimes we use $\|\cdot\|_{C(0, \tau)}=|\cdot|_\tau$ for $\tau>0$. Let us define the operator $V$ by $$Vw(t)=\int_0^t U(t,t_1)(Ew)(t_1)dt_1$$ for any integrable function $w(t)$ $(t\ge 0)$ with values in $X$. According to \eqref{e1.9} and \eqref{e2.1} it is easy to check that for any finite $T$ and $u\in C(-\eta, T)$ with $u(t)=0$ for $t\le 0$, $V$ satisfies \begin{align*} |V^k u|_{T}&\le Mv_0\int_0^T |V^{k-1}u|_{t} dt\\ & \le(Mv_0)^2 \int_0^T \int_0^t |V^{k-2}u|_{t_1}dt_1 dt\\ &\le \dots \le \frac{(TMv_0)^k}{k!}|u|_{T}. \end{align*} Hence, it follows that \begin{corollary} \label{coro2.1} For any continuous $f$, problem \eqref{e1.6}, \eqref{e1.7} has a unique mild solution $x(t)$, which can be represented as $$x=\sum_1^\infty V^k f_1,\quad \text{where } f_1(t)=\int_0^t U(t,t_1)f(t_1)dt_1. \label{e2.2}$$ \end{corollary} \begin{lemma} \label{lem2.2} Under condition \eqref{e1.10}, if for any $f\in C(0,\infty)$, problem \eqref{e1.6}, \eqref{e1.7} has a bounded mild solution on $[0,\infty)$, then for any $\phi\in C(-\eta, 0)\cap \operatorname{Dom}(A)$ problem \eqref{e1.1}, \eqref{e1.2} has a unique mild solution bounded on $(0,\infty)$. \end{lemma} \begin{proof} Put $$\hat \phi(t)=\begin{cases} \phi(0) &\text{if } t\ge 0, \\ \phi(t) &\text{if }-\eta\le t< 0\,. \end{cases}$$ Then $d\hat \phi(t)/dt=0$ for $t\ge 0$. Consider the equation $$dx(t)/dt=A(t)(x(t) +\phi(0))+E(x(t)+\hat \phi(t ))\;(t>0),$$ with condition \eqref{e1.7}. According to \eqref{e1.10} and \eqref{e2.1}, $A(t)\phi(0)+ E\hat \phi(t)\in C(-\eta,\infty)$. Due to the hypotheses of this lemma, the latter equation has a solution $x\in C(0,\infty)$. Then the function $y(t)=x(t)+\hat \phi(t)\in C(-\eta,\infty)$ and satisfies problem \eqref{e1.1}, \eqref{e1.2}. As claimed. \end{proof} \begin{proof}[Proof of Theorem \ref{thm1.1}] Substituting $$y(t)=y_\epsilon(t) e^{-\epsilon t} \label{e2.3}$$ with an $\epsilon>0$ in \eqref{e1.1}, we obtain $$dy_\epsilon(t)/dt=(A(t)+\epsilon)y_\epsilon(t) +E_\epsilon y_\epsilon(t)\quad (t>0), \label{e2.4}$$ where $$E_\epsilon w(t)=\int_0^\eta B(t,s)e^{\epsilon s}w(t-s) ds + \sum_1^m e^{\epsilon h_k(t)}B_k(t)w(t-h_k(t))$$ for a continuous $w$. It is easy to check that $E_\epsilon\to E$ in the operator norm of $C(0, \infty)$ as $\epsilon\to 0$. Furthermore, due to \eqref{e2.2} we obtain $x=\hat Gf$, where $$\hat G:=(I-V)^{-1} W=\sum_1^\infty V^kW, \quad \text{with } Wf(t)= \int_0^t U(t,t_1)f(t_1)dt_1.$$ By the hypothesis of the theorem, we have $$x=\hat Gf\in C(0,\infty)\quad \text{for any }f\in C(0,\infty).$$ So $\hat G$ is defined on the whole space $C(0,\infty)$. It is closed, since problem \eqref{e1.6}, \eqref{e1.7} has a unique solution. Therefore, $\hat G$ is bounded according to the Closed Graph theorem \cite{Dunford}. Consider now the equation $$\dot x_\epsilon(t)= (A(t)+\epsilon I)x_\epsilon(t) +E_\epsilon x_\epsilon(t)+f(t) \label{e2.5}$$ with the zero initial condition. Its mild solution is defined by $$x_\epsilon(t)=\int_0^t U(t,t_1)(\epsilon x_\epsilon(t_1)+ E_\epsilon x_\epsilon(t_1)dt_1)+f_1 \label{e2.6}$$ where $f_1$ is defined as in \eqref{e2.2}. For solutions $x$ and $x_\epsilon$ of \eqref{e1.8} and \eqref{e2.6}, respectively, we obtain \begin{align*} x_\epsilon(t)-x(t) &= \int_0^t U(t,t_1)(\epsilon x_\epsilon(t_1)+ E_\epsilon x_\epsilon(t_1)-Ex(t_1))dt_1\\ &= V(x_\epsilon(t)-x(t))+f_\epsilon(t), \end{align*} where $$f_\epsilon(t)=\int_0^t U(t,t_1)(\epsilon x_\epsilon(t_1)+ (E_\epsilon-E) x_\epsilon(t_1))dt_1.$$ Consequently, $$x-x_\epsilon=\hat G f_\epsilon. \label{e2.7}$$ However $|\hat G|_T\le \|\hat G\|_{C(0,\infty)}$, and \begin{align*} |(E_\epsilon-E) w|_T &\le \sup_{t\ge 0} (\int_0^\eta \|B(t,s)\|_X |e^{\epsilon s}-1|\:\|w(t-s)\|_X ds \\ &\quad + \sum_1^m |e^{\epsilon h_k(t)}-1|\|B_k(t)w(t-h_k(t))\|_X)\\ & \le v_0(e^{\epsilon \eta}-1)|w|_T. \end{align*} By \eqref{e1.9} and \eqref{e2.1}, $|V|_T\le Mv_0/\alpha_0$. So $|f_\epsilon|_T\le |x_\epsilon|_T (\epsilon + Mv_0\alpha_0^{-1}(e^{\epsilon \eta}-1))$ and $$|x_\epsilon|_T \le |x|_T+ \|\hat G\|_{C(0,\infty)}|x_\epsilon|_T (\epsilon +Mv_0\alpha_0^{-1}(e^{\epsilon \eta}-1)).$$ Thus, for a sufficiently small $\epsilon$, $$|x_\epsilon|_T \le \frac{|x|_T}{1- \|\hat G\|(\epsilon +M\|\hat G\|_{C(0,\infty)}v_0\alpha_0^{-1}(e^{\epsilon \eta}-1)) }.$$ Letting $T\to\infty$, we obtain $x_\epsilon\in C(0,\infty)$. Hence, by Lemma \ref{lem2.2}, a solution $y_\epsilon$ of \eqref{e2.4} is bounded. Now \eqref{e2.3} proves the exponential stability, as claimed. \end{proof} \section{Equations in a Hilbert space} In this section we illustrate Theorem \ref{thm1.1} in a Hilbert space. Let $X=H$ be a Hilbert space with a scalar product $(.,.)$, and the norm $\|\cdot\|_H=\sqrt{(.,.)}$. Let $A(t)$ map $\operatorname{Dom}(A)$ into itself and $$\sup_{z\in \operatorname{Dom}(A)} \frac{Re (A(t)z,z)}{(z,z)} \le -\alpha(t)\le -\alpha_0\quad\forall t\ge 0, \label{e3.1}$$ where $\alpha(t)$ is a positive continuous function and $\alpha_0$ is a positive constant. From \eqref{e1.4} it follows that $$\frac d{dt}(\zeta (t), \zeta (t))= (\dot \zeta (t), \zeta (t))+(\zeta (t), \dot \zeta (t))= 2 Re (\dot \zeta (t), \zeta (t))=2 Re (A(t)\zeta (t), \zeta (t)).$$ Thus $$\frac d{dt}(\zeta (t), \zeta (t))=2 \|\zeta (t)\|_H \frac d{dt}\|\zeta (t)\|_H \le -2\alpha(t)(\zeta (t), \zeta (t)),$$ or $$\frac d{dt}\|\zeta (t)\|_H\le -\alpha(t)\|\zeta (t)\|_H.$$ Solving this inequality with $\zeta (s)\in \operatorname{Dom}(A)$, we obtain $$\|U(t,s)\|_H\le e^{-\int_s^t \alpha (\tau) d\tau}\le e^{-\alpha_0 (t-s)},\quad \text{for }t\ge s\ge 0.$$ Hence, $$\sup_t \int_0^t \|U(t,t_1)\|_H dt_1\le J,$$ where $$J:=\sup_t \int_0^t e^{-\int_{t_1}^t \alpha (\tau) d\tau} dt_1.$$ From \eqref{e1.8} and \eqref{e2.1} it follows that $$\|x\|_{C(0,\infty)}\le v_0J\|x\|_{C(0,\infty)}+ \|f_1\|_{C(0,\infty)}.$$ Consequently, if $$v_0J<1, \label{e3.2}$$ then $$\|x\|_{C(0,\infty)}\le \frac{\|f_1\|_{C(0,\infty)}}{1-v_0J}.$$ Using Theorem \ref{thm1.1} we arrive at the following result. \begin{corollary} \label{coro3.1} Suppose $E$ maps $\operatorname{Dom}(A)$ into itself, and conditions \eqref{e1.10}, \eqref{e3.1}, and \eqref{e3.2} hold. 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