\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 10, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/10\hfil An asymptotic property for delay equations] {An asymptotic property of solutions to linear nonautonomous delay differential equations} \author[J. G. Dix, Ch. G. Philos, \& I. K. Purnaras\hfil EJDE-2005/10\hfilneg] {Julio G. Dix, Christos G. Philos, Ioannis K. Purnaras} % in alphabetical order \address{Julio G. Dix \hfill\break Department of Mathematics\\ Texas State University\\ 601 University Drive, San Marcos, TX 78666, USA} \email{julio@txstate.edu} \address{Christos G. Philos \hfill\break Department of Mathematics\\ University of Ioannina\\ P. O. Box 1186\\ 451 10 Ioannina, Greece} \email{cphilos@cc.uoi.gr} \address{Ioannis K. Purnaras \hfill\break Department of Mathematics\\ University of Ioannina\\ P. O. Box 1186\\ 451 10 Ioannina, Greece} \email{ipurnara@cc.uoi.gr} \date{} \thanks{Submitted July 9, 2004. Published January 12, 2005.} \subjclass{34K25, 34C10, 35K15} \keywords{Delay differential equation; asymptotic behavior; \hfill\break\indent characteristic equation} \begin{abstract} We study first order linear delay differential equations with variable coefficients and constant delays. Using solutions to a characteristic equation, we show asymptotic properties of solutions to the delay equation. To illustrate the hypothesis of the main theorem, we present an example. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} We study the asymptotic behavior of solutions to the delay differential equation \begin{equation} x'(t)=a(t)x(t)+\sum_{j=1}^kb_j(t)x(t-\tau_j), \quad \text{for } t\geq 0 \label{E}\\ \end{equation} with initial condition \begin{equation} x(t)=\phi (t),\quad \text{for } \min_{1\leq j\leq k}\{-\tau_j\} \leq t \leq 0, \, \label{C} \end{equation} where the coefficients $a$ and $b_j$ are continuous real-valued functions on $[0,\infty)$, the delays $\tau_j$ are positive real numbers ($j=1,2,\dots, k$), $k$ is a positive integer, and $\phi$ is a given continuous function. The work on this paper is motivated by the publication of the following interesting results. Driver, Sasser and Slater \cite{d5} obtained significant results on asymptotic behavior, non-oscillation, and stability of solutions to first order linear delay differential equations with constant coefficients and one constant delay. Driver \cite{d3} obtained similar results for first order linear autonomous delay differential equations with infinitely many distributed delays. The results in \cite{d5} have been improved and extended by Philos \cite{p1} for first order linear delay differential equations with coefficients that are periodic having a common period, and delays that are constants multiples of this period. These results have been extended and improved by Kordonis, Niyianni and Philos \cite{k1} for first order linear autonomous neutral delay differential equations. The results in \cite{k1,p1} have been extended and slightly improved by Philos and Purnaras in \cite{p2}. There the authors study first order linear neutral delay differential equations with periodic coefficients having a common period and constant delays that are multiples of this period. Graef and Qian \cite{g1} obtained results closely related to the ones above for first order forced delay differential equations. Driver \cite{d4} and Arino and Pituk \cite{a1} obtained important results for linear differential systems with small delays. In the present paper, we define a characteristic equation and then utilize its solution to state asymptotic results for solutions of the delay equation. Also we obtain a non-oscillation result, Remark \ref{rmk1}. Our main result is stated as Theorem \ref{thm1} and proved in the next section. The limit obtained in Theorem \ref{thm1} is found explicitly when the solution to the characteristic equation is a constant. An application of Theorem \ref{thm1} provides a necessary and sufficient condition for all solutions of \eqref{E} to be bounded, and a necessary and sufficient condition for all solutions of \eqref{E} to tend to zero at $\infty $. The last section contains an example and discussions on the results of the paper. \section{Statement of Results} We shall assume that the delays are positive and denote \[ \tau =\max\{ \tau_j : 1\leq j \leq k\},\quad \sigma= \min\{ \tau_j : 1\leq j \leq k\}\,. \] Let $C([-\tau ,0],\mathbb{R})$ denote the set of continuous real-valued functions on $[-\tau ,0]$. By a solution $x$ to the delay differential equation \eqref{E}, we mean a continuous real-valued function, defined on $[-\tau,\infty )$, which is continuously differentiable on $[0,\infty )$ and satisfies \eqref{E}. It is well-known that for each given $\phi \in C([-\tau ,0],\mathbb{R})$, problem \eqref{E}-\eqref{C} has a unique solution; see for example \cite{d1,h1,h2}. With the delay equation \eqref{E}, we associate the integral equation \begin{equation} \label{ast} \begin{gathered} \lambda (t)=a(t)+\sum_{j=1}^kb_j(t) \exp\Big[ -\int_{t-\tau_j}^t\lambda (s)ds\Big],\quad \mbox{for } t\geq 0\,;\\ \lambda(t)= \lambda_0(t), \quad \mbox{for } -\tau\leq t \leq 0 \, \end{gathered} \end{equation} which is called the (generalized) \emph{characteristic equation} of \eqref{E}. This equation is obtained when looking for solutions of the form \[ x(t)=\phi(0)\exp \Big[ \int_0^t\lambda (s)ds\Big]\,. \] Note that this solution can not change sign; i.e., $x(t)$ is either positive or negative or identically zero. By a solution $\lambda $ to the characteristic equation, we mean a continuous real-valued function, defined on $[-\tau,\infty )$, which satisfies \eqref{ast}. \begin{lemma} \label{lem1} For each $\lambda_0$ in $C([-\tau ,0],\mathbb{R})$, the characteristic equation has a unique global solution. \end{lemma} \begin{proof} Let $u(t)=\exp \big[ \int_0^t\lambda (s)ds\big]$ for $t\geq 0$, and $w(t)=\exp \big[ \int_0^t\lambda_0(s)ds\big]$ for $ -\tau\leq t\leq 0$. Then using the characteristic equation, for $0\leq t\leq \sigma$, we obtain the linear differential equation \[ u'(t)=\lambda(t)u(t)=a(t)u(t)+\sum_{j=1}^k b_j(t) w(t-\tau_j) \] with $u(0)=1$. The solution to this equation is \[ u(t)=\Big[1+\int_0^t \sum_{j=1}^k b_j(s) \exp\Big[\int_0^{s-\tau_j}\lambda_0(r)\,dr -\int_0^s a(r)\,dr\Big] \,ds\Big] \exp\Big[\int_0^t a(r)\,dr\Big] \] which allows defining $\lambda(t)=u'(t)/u(t)$ on $[0,\sigma]$. For the next interval, let $w(t)=u(t)$ on $[-\tau,\sigma]$. Then for $\sigma\leq t \leq 2\sigma$, we obtain the differential equation \[ u'(t)=a(t)u(t)+\sum_{j=1}^k b_j(t) w(t-\tau_j) \] whose solution is \begin{equation} \label{linsol} u(t)=\Big[1+\int_0^t \sum_{j=1}^k b_j(s) \exp\Big[\int_0^{s-\tau_j}\lambda(r)\,dr -\int_0^s a(r)\,dr\Big] \,ds\Big] \exp\Big[\int_0^t a(r)\,dr\Big] \end{equation} which allows defining $\lambda(t)$ on $[\sigma,2\sigma]$. Proceeding in this manner, we define $\lambda(t)$ for all $t\geq -\tau$, which completes the proof. \end{proof} \begin{remark} \label{rmk1} \rm If the solution to \eqref{E}-\eqref{C} does not have zeros on some interval $[t^*-\tau,t^*]$, then the solution does not have zeros on $[t^*,\infty)$; i.e., the solution can not change sign on $[t^*,\infty)$. To show this claim let $t^*$ be the initial time for the characteristic equation and $\lambda_0(t)$ be given implicitly by $x(t)=x(t^*)\exp\big[\int_{t^*}^t\lambda_0(s)\,ds\big]$, with $t^*-\tau\leq t \leq t^*$. Then, by the uniqueness of solutions to \eqref{E}, \[ x(t)=x(t^*)\exp\Big[\int_{t^*}^t\lambda(s)\,ds\Big],\quad \mbox{for } t\geq t^*\,. \] Therefore, $x(t)$ can not have zeros on $[t^*,\infty)$. \end{remark} Our main result is the following theorem. \begin{theorem} \label{thm1} Assume that \begin{equation} \label{H} \sup_{t\geq \tau }\sum_{j=1}^k| b_j(t)| \tau_j\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]<1 \,. \end{equation} Then for each solution $x$ of \eqref{E}-\eqref{C} there exists a constant $L_{\phi,\lambda_0}$ such that \[ \lim_{t\to \infty } x(t)\exp \Big[ -\int_0^t\lambda(s)ds\Big] = L_{\phi,\lambda_0} \,, \] and \[ \lim_{t\to \infty }\Big\{ x(t)\exp \Big[ -\int_0^t\lambda(s)ds\Big] \Big\} '=0. \] \end{theorem} \begin{remark} \label{rmk2} \rm Under the conditions of Theorem \ref{thm1}, a solution to \eqref{E} can not grow faster than the exponential function determined by the characteristic equation; i.e., there exists a constant $M$ such that \[ \big| x(t)\big| \leq M \exp \Big[ \int_0^t\lambda(s)ds\Big], \quad \text{for }t\geq 0\,. \] \end{remark} \begin{remark} \label{rmk3} \rm When the solution to \eqref{ast} is a constant $\lambda_0$ satisfying \eqref{H}, \[ \lim_{t\to\infty} x(t)\exp (-t\lambda_0) =L_{\phi,\lambda_0}\,. \] In particular when zero is the solution to \eqref{ast}, $\lim_{t\to\infty} x(t)=L_{\phi,0}$. \end{remark} Note that if $\lambda$ is a solution of \eqref{ast}, then \[ x(t)=\phi(0)\exp \Big[ \int_0^t\lambda(s)ds\Big] \] is a solution of \eqref{E} with initial function $\phi(t)=\phi(0)\exp \big[ \int_0^t\lambda(s)ds\big]$. Then we obtain easily the following results. \begin{remark} \label{rmk4} \rm Under the assumptions of Theorem \ref{thm1}, we have: \begin{enumerate} \item Every solution of \eqref{E} is bounded if and only if $\limsup_{t\to \infty } \int_0^t\lambda(s)ds<\infty$. \item Every solution of \eqref{E} tends to zero at $\infty $ if and only if \\ $\lim_{t\to \infty } \int_0^t\lambda(s)ds=-\infty$. \end{enumerate} \end{remark} \section{Proof of main result} \begin{proof}[Proof of Theorem \ref{thm1}] For solutions $x$ of \eqref{E}-\eqref{C} and $\lambda$ of \eqref{ast}, we define \[ y(t)=x(t)\exp \Big[ -\int_0^t\lambda(s)ds\Big], \quad t\geq -\tau. \] Differentiating in this function, and using \eqref{E}, \eqref{ast}, we obtain \begin{align*} &y'(t)\\ &=\Big(x'(t)-x(t)\lambda(t)\Big) \exp\Big[-\int_0^t\lambda(s)ds\Big]\\ &= \Big(\sum_{j=1}^kb_j(t)x(t-\tau_j) -x(t)\sum_{j=1}^kb_j(t)\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]\Big) \exp\Big[-\int_0^t\lambda(s)ds\Big]\,. \end{align*} Using that $x(t-\tau_j)=y(t-\tau_j)\exp \big[ \int_0^{t-\tau_j}\lambda(s)ds\big]$, the above equality yields \[ y'(t)=-\sum_{j=1}^kb_j(t)[ y(t)-y(t-\tau_j)] \exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]\quad \mbox{for }t\geq0\,. \] From this equation and the fundamental theorem of calculus, \begin{equation} y'(t)=-\sum_{j=1}^kb_j(t) \Big[ \int_{t-\tau_j}^ty'(s)ds\Big] \exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]\quad \mbox{for }t\geq \tau\,. \label{e3.1} \end{equation} If all $b_j$'s are identically zero on $[\tau,\infty)$, from \eqref{e3.1}, $y'=0$ and $y$ is constant on $[\tau ,\infty )$ which would complete the proof. Therefore, we assume that at least one $b_j$ is not identically zero on $[\tau ,\infty )$ . Let \[ \mu_{\lambda_0}=\sup_{t\geq \tau }\sum_{j=1}^k| b_j(t)| \tau_j\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big] \,. \] Then, by \eqref{H}, \begin{equation} 0<\mu_{\lambda_0}<1\,. \label{e3.2} \end{equation} Note that the maximum of $|y'|$ on $[0,\tau]$ depends on $x$ and $\lambda $; hence, on the initial functions $\phi $ and $\lambda_0$. Let \[ M_{\phi,\lambda_0} =\max\big\{| y'(t)|: 0\leq t\leq \tau \big\} \,. \] We shall show that $M_{\phi,\lambda_0} $ is also a bound of $|y'|$ on the whole interval $[0,\infty )$; i.e., \begin{equation} |y'(t)| \leq M_{\phi,\lambda_0} \quad \mbox{for all } t\geq 0\,. \label{e3.3} \end{equation} On the contrary, assume that there exist $\epsilon>0$ and $t\geq 0$ such that $|y'(t)| > M_{\phi,\lambda_0}+\epsilon$. Since $|y'(t)| \leq M_{\phi,\lambda_0}$ for $0\leq t\leq \tau$, by the continuity of $y'$, there exists $t^{\ast }>\tau$ such that \[ |y'(t)| < M_{\phi,\lambda_0}+\epsilon\,, \quad\mbox{for }0\leq t0$. In view of \eqref{e3.1} and \eqref{e3.3}, \begin{align*} | y'(t)| &\leq \sum_{j=1}^k| b_j(t)| \Big[\int_{t-\tau_j}^t| y'(s)|ds\Big] \exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big] \\ &\leq M_{\phi,\lambda_0} \sum_{j=1}^k| b_j(t)| \tau_j \exp \Big[-\int_{t-\tau_j}^t\lambda(s)ds\Big] \\ &\leq M_{\phi,\lambda_0} (\mu_{\lambda_0}) \quad \mbox{for } t\geq \tau \,. \end{align*} Using this inequality, we can show by induction that \begin{equation} | y'(t)| \leq M_{\phi,\lambda_0}(\mu_{\lambda_0})^n \quad \text{for }t\geq n\tau \quad (n=0,1,\dots ). \label{e3.6} \end{equation} For an arbitrary $t\geq 0$, we set $n=\lfloor t/\tau \rfloor$ (the greatest integer less than or equal to $t/\tau$). Then $t\geq n\tau $ and $\frac{t}{\tau }-1T\geq0$, from \eqref{e3.7}, we have \begin{align*} | y(t)-y(T)| &\leq \int_T^t| y'(s)| ds \leq \int_T^t M_{\phi,\lambda_0}(\mu_{\lambda_0})^{\frac s{\tau} -1} \,ds\\ &=M_{\phi,\lambda_0}\frac \tau{\ln(\mu_{\lambda_0})} \Big[ (\mu_{\lambda_0})^{\frac s{\tau} -1}\Big]_{s=T}^{s=t} \\ &=M_{\phi,\lambda_0}\frac \tau{\ln(\mu_{\lambda_0})} \Big[ (\mu_{\lambda_0})^{\frac t{\tau} -1}-(\mu_{\lambda_0})^{\frac T{\tau} -1}\Big]\,. \end{align*} As $T\to \infty$, we have $t\to \infty$ and, by \eqref{e3.2}, the two right-most terms above approach zero. Therefore, $\lim_{T\to \infty }| y(t)-y(T)|=0$ which by the Cauchy convergence criterion implies the existence of $\lim_{t\to \infty }y(t)$. We call this limit $L_{\phi,\lambda_0} $ because it depends on $y$ which in turn depends on the initial functions $\phi$ and $\lambda_0$. This shows the first limit in Theorem \ref{thm1} and completes the proof. \end{proof} \section{Discussion} To illustrate the hypothesis in Theorem \ref{thm1}, we provide an example of a non-autonomous (and non-periodic) delay differential equation of the form \eqref{E}, for which \eqref{ast} has a explicit solution and satisfies \eqref{H}. \medskip \noindent\textbf{Example.} Let $k=1$, $\tau_1=2$, and $a(t)=1/(2(t+3))$, $b_1(t)=1/(2(t+1))$ for $t\geq 0$. It is easy to verify that \[ \lambda (t)=\frac{1}{t+3} \] is a solution of \eqref{ast} and satisfies \eqref{H}. Indeed, we can easily check that \[ \sup_{t\geq \tau_1}| b_1(t)|\tau_1 \exp \Big[-\int_{t-\tau_1}^t\lambda(s)ds\Big] =\sup_{t\geq 2}\frac{1}{t+3}=\frac{1}{5}<1 . \] \begin{remark} \label{rmk5} \rm Finding conditions on $a$ and $b_j$ that guarantee hypothesis \eqref{H} remains an open question. To imply this hypothesis, we can use for example the stronger condition \[ \sup_{t\geq \tau }\sum_{j=1}^k| b_j(t)| \exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]<\frac{1}{\tau } \,. \] Furthermore, if, for each $t \geq 0$, it holds $b_j(t)\geq 0$ for all $j$'s or $b_j(t)\leq 0$ for all $j$'s, from the characteristic equation, it follows that \[ |\lambda(t)-a(t)| = \sum_{j=1}^k| b_j(t)| \exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]\,,\quad\mbox{for }t\geq0\,. \] Note that this equality is obvious when $k=1$. Under the above assumptions, the condition \eqref{H} is implied by $\sup_{t\geq 0} |\lambda(t)-a(t)|<1/\tau$ . This is the strategy in the next lemma. \end{remark} \begin{lemma} \label{lem2} Assume that the coefficients $a$, $b_j$ and the initial function of the characteristic equation satisfy the following conditions for all $t\geq 0$: $b_j(t)\geq 0$ and, for some $c$ with $0 \leq c<\frac{1}{\tau}$, \begin{equation} \label{H2} \begin{aligned} &\sum_{j\in J(t)} b_j(t)\exp\Big[\int_0^{t-\tau_j}\lambda_0(s)\,ds -\int_0^t a(s)\,ds\Big] \\ &+ \sum_{j\not\in J(t)} b_j(t) \exp\Big[\int_0^{t-\tau_j}c\,ds- \int_{t-\tau_j}^t a(s)\,ds\Big] \leq c, \end{aligned} \end{equation} where $J(t)$ consists of those indices $j$ for which $t-\tau_j\leq 0$, ($j=1,2,\dots k$). Then \[ \sup_{t\geq 0} \sum_{j=1}^k| b_j(t)| \tau_j\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big]<1 \,, \] which implies the hypothesis for Theorem \ref{thm1}. \end{lemma} \begin{proof} Since $b_j(t)\geq 0$, the definitions of $\tau$ and of $\lambda$ imply \[ \sum_{j=1}^k b_j(t) \tau_j\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big] \leq \tau \sum_{j=1}^k b_j(t)\exp \Big[ -\int_{t-\tau_j}^t\lambda(s)ds\Big] =\tau |\lambda(t)-a(t)|\,. \] The statement of this lemma follows if we show that $|\lambda(t)-a(t)|\leq c$ for $t\geq 0$. As in the proof of Lemma \ref{lem1}, let $u(t)= \exp\big[\int_0^t \lambda(s)\,ds\big]$ for $t\geq -\tau$, with $\lambda$ defined by \eqref{ast}. From the characteristic equation, \[ \lambda(t)-a(t)=\sum_{j=1}^k b_j(t) \exp \Big[ -\int_{t-\tau_j}^t \lambda(s)\,ds\Big] =\frac{1}{u(t)}\sum_{j=1}^k b_j(t) \exp \Big[ \int_0^{t-\tau_j} \lambda(s)\,ds \Big]\,. \] Since $u(t)$ is the solution given by \eqref{linsol}, for $t\geq 0$, \[ \lambda(t)-a(t) =\frac{\sum_{j=1}^k b_j(t)\exp\big[\int_0^{t-\tau_j}\lambda(s)\,ds\big] \exp\big[ -\int_0^t a(s)\,ds\big]} {1+\int_0^t \sum_{j=1}^k b_j(s) \exp\big[\int_0^{s-\tau_j}\lambda(r)\,dr -\int_0^{s} a(r)\,dr\big] \,ds}\,. \] Since $b_j(t)\geq 0$, the denominator in the above expression is greater than or equal to 1 and \begin{equation} \label{lu} |\lambda(t)-a(t)| \leq \sum_{j=1}^k b_j(t)\exp\Big[\int_0^{t-\tau_j}\lambda(s)\,ds -\int_0^t a(s)\,ds\Big]\,. \end{equation} When $0\leq t \leq \sigma$, we have $t-\tau_j\leq 0$, then all $j$'s are in the class $J(t)$ and $t-\tau_j\leq s\leq 0$. So we use $\lambda(s)=\lambda_0(s)$ in \eqref{lu}. Therefore, \eqref{H2} implies \begin{equation} \label{part1} |\lambda(t)-a(t)|\leq c \quad \mbox{for all $t$ in $[0,\sigma]$}\,. \end{equation} For each fixed $t$ in $[\sigma,2\sigma]$, we have two possible cases:\\ Case 1: $j\in J(t)$. Here $t-\tau_j\leq 0$ and $t-\tau_j\leq s\leq 0$; so we use $\lambda(s)=\lambda_0(s)$ in \eqref{lu}. Then, for this case, each summand in \eqref{lu} is equal to \[ b_j(t)\exp\Big[\int_0^{t-\tau_j}\lambda_0(s)\,ds -\int_0^t a(s)\,ds\Big]\,. \] Case 2: $j\not\in J(t)$. Here $0\tau$, and the second summation needs to be less than or equal to $c$ for all large $t$. This is very restrictive. In particular, it requires $\int_{t-\tau_j}^ta(s)\,ds\to \infty$ as $t\to\infty$. Note that in the example above $a$, $b_j$ do not satisfy the conditions of Lemma \ref{lem2}. \smallskip The real number $L_{\phi,\lambda_0} $, in Theorem \ref{thm1}, has been given explicitly in two special cases: For linear autonomous delay differential equations and for linear delay differential equations with periodic coefficients having a common period and constant delays that are multiples of this period. See \cite{d3,d5,g1,p1} (and \cite{k1,p2} for linear neutral delay differential equations). The proof of Theorem \ref{thm1} is based on an integral representation of $y'$. Meanwhile, in the autonomous case, and in the case where the coefficients are periodic with a common period and the delays are multiples of this period, the proof is based on an integral representation of $y$. See \cite{d3,d5,g1,p1} (and \cite{k1,p2} for the neutral case). We would be interested in generalizing our theorem to linear delay differential equations with variable coefficients and variable delays. 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