\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 99, pp. 1--5.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/99\hfil Stability of delay differential equations] {Stability of delay differential equations with oscillating coefficients} \author[M. I. Gil'\hfil EJDE-2010/99\hfilneg] {Michael I. Gil'} \address{Michael I. Gil' \newline Department of Mathematics \\ Ben Gurion University of the Negev \\ P.0. Box 653, Beer-Sheva 84105, Israel} \email{gilmi@cs.bgu.ac.il} \thanks{Submitted April 13, 2010. Published July 22, 2010.} \subjclass[2000]{34K20} \keywords{Linear delay differential equation; exponential stability} \begin{abstract} We study the solutions to the delay differential equation equation $$\dot x(t)=-a(t)x(t-h),$$ where the coefficient $a(t)$ is not necessarily positive. It is proved that this equation is exponentially stable provided that $a(t)=b+c(t)$ for some positive constant $b$ less than $\pi/(2h)$, and the integral $\int_0^t c(s)ds$ is sufficiently small for all $t>0$. In this case the $3/2$-stability theorem is improved. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \section{Introduction and preliminaries} This article concerns the equation $$\dot x(t)=-a(t)x(t-h), \label{e1.1}$$ where $\dot x=dx/dt$, the delay $h$ is a positive constant, and $a(t)$ a piece-wise continuous function bounded on $[0,\infty)$. We do not require that $a(t)$ be positive, and therefore, the characteristic function'' $z+a(t)e^{-zh}$ can be unstable for some $t\ge 0$. The sharp stability condition (the so called $3/2$-stability theorem) for first-order functional-differential equations with one variable delay was established by Myshkis \cite{my} (see also \cite{ko}). A similar result was established by Lillo \cite{li}. The $3/2$-stability theorem asserts that \eqref{e1.1} is uniformly stable, provided that $0< h a(t)\le 3/2$ for all $t\ge 0$. The upper bound 3/2 is the best possible. In fact, if $h\sup_t a(t)> 3/2$, then there are equations having unbounded solutions. The $3/2$-theorem was generalized to nonlinear equations and equations with unbounded delays in the very interesting papers \cite{yo1,yo2,yo3}. In this article, under some additional conditions we improve the $3/2$-theorem. We consider \eqref{e1.1} as a perturbation of the equation $$\dot y(t)=-by(t-h) \label{e1.2}$$ with a positive constant $b<\pi /(2 h)$ satisfying a condition stated below. The fundamental solution to \eqref{e1.2} is $$F_b(t)=\frac1{2\pi i}\int_{-i\infty} ^{i\infty} \frac{e^{zt}dz}{z+be^{-zh}}\,.$$ For a function $f$ defined and bounded on $[0,\infty)$ (not necessarily continuous), we introduce the norm $\|f\|_\infty=\sup_{t\ge 0}|f(t)|$. So $\|a\|_\infty=\sup_{t\ge 0}|a(t)|$. In addition, put $$\|f\|_{L^1}=\int_0^\infty |f(t)|dt,$$ if the integral exists. Now we are in a position to formulate our main result. \begin{theorem} \label{thm1.1} Let there be a constant $b\in (0, \pi/(2h))$, such that $$w_b:=\sup_{t\ge 0}|\int_0^t (a(t)-b)dt|$$ is finite and satisfies the inequality $$w_b < \frac{1}{1+ (b+\|a\|_\infty) \|F_b\|_{L^1}}. \label{e1.3}$$ Then \eqref{e1.1} is exponentially stable. \end{theorem} This theorem is proved in the next section. Its assumptions are sharp: if $a(t)\equiv b$, then $w_b=0$ and condition \eqref{e1.3} is automatically fulfilled. Furthermore, let $$ehb<1. \label{e1.4}$$ Then $F_b(t)\ge 0$ and \eqref{e1.2} is exponentially stable, cf. \cite{gi} and references therein. Now, integrating \eqref{e1.2}, we have $$1=F_b(0)=b\int_0^\infty F_b(t-h)dt=b\int_h^\infty F_b(t-h)dt=b\|F_b\|_{L^1}.$$ Thus, Theorem \ref{thm1.1} implies the following result. \begin{corollary} \label{coro1.2} Let \eqref{e1.4} and $$w_b< \frac{b}{2b+\|a\|_\infty} \label{e1.5}$$ hold. Then \eqref{e1.1} is exponentially stable. \end{corollary} Now for a positive constant $\omega$, let $$a(t)=b+u(\omega t), \label{e1.6}$$ where $u(t)$ is a piece-wise continuous function such that $$\nu_u:=\sup_t \big|\int_0^t u(s)ds\big|<\infty.$$ Then $$w_b=\sup_t |\int_0^t u(\omega s)ds|= \nu_u/\omega.$$ For example, when $u(t)=\sin\;(t)$, then $\nu_u=2$. Now Theorem \ref{thm1.1} and \eqref{e1.5} imply our next result. \begin{corollary} \label{coro1.3} Let \eqref{e1.4}, \eqref{e1.6} and $$\omega > \frac{\nu_u(3b+\|u\|_\infty)}{b} \label{e1.7}$$ hold. Then \eqref{e1.1} is exponentially stable. \end{corollary} \begin{example} \label{exa1.4} Consider the equation $$\dot x=-bx(t-1) + c_2\sin (\omega t)x(t-1), \label{e1.8}$$ where $b,c_2$ are positive constant with $b \frac{2c_2(3b+c_2)}{b}. \label{e1.9} \end{example} In summary, for each$c_2$there exists an$\omega$, such that \eqref{e1.8} is exponentially stable. Meanwhile, the$3/2$-stability theorem requires the additional condition$c_2+b<3/2$. Therefore, Theorem \ref{thm1.1} supplements the interesting results obtained in \cite{be2}. \section{Proof of Theorem \ref{thm1.1}} For simplicity, we put$F_b(t)=F(t)$. Due to the Variation of Constants Formula the equation $$\dot x(t) =-bx(t-h) + f(t)\quad (t\geq 0),$$ with a given function$f$and the zero initial condition$x(t)=0$($t \le 0$) is equivalent to the equation $$x(t)=\int_0^t F(t-s)f(s)ds. \label{e2.1}$$ Recall that a function$G(t,s)$, ($t\ge s\ge 0$) differentiable in$t$, is the fundamental solution to \eqref{e1.1} if it satisfies that equation in$t$and the initial conditions $$G(s, s)=1,\quad G(t,s)=0\quad ( t < s,\;s\geq 0).$$ Put$G(t,0)=G(t)$. Subtracting \eqref{e1.2} from \eqref{e1.1} we have $$\frac{d}{dt}(G(t)-F(t))= -b(G(t-h)-F(t-h))+c(t)G(t-h)$$ where$c(t)=-(a(t)-b)$. Now \eqref{e2.1} implies $$G(t)=F(t) +\int_0^t F(t-s)c(s)G(s-h)ds . \label{e2.2}$$ We need the following simple lemma. \begin{lemma} \label{lem2.1} Assume that on each finite segment of the real axis, functions$f(t)$and$v(t)$are boundedly differentiable and$w(t)$is integrable. Then with the notation $$j_w(t, \tau)=\int_\tau^t w(s)ds\quad (t>\tau>-\infty),$$ the equality $$\int_\tau ^t f(s)w(s)v(s)ds=f(t)j_w(t,\tau)v(t) -\int_\tau ^t [f'(s)j_w(s,\tau)v(s)+ f(s)j_w(s)v'(s)]ds$$ is valid. \end{lemma} \begin{proof} Clearly, $$\frac{d}{dt}f(t)j_w(t,\tau)v(t)=f'(t)j_w(t,\tau)v(t) +f(t)w(t)v(t)+f(t)j_w(t,\tau)v'(t).$$ Integrating, this equality and taking into account that$j_w(\tau,\tau)=0$, we arrive at the required result. \end{proof} Put$J(t):=\int_0^t c(s)ds. By the previous lemma, \begin{align*} &\int_0^t F(t-\tau)c(\tau)G(\tau-h)d\tau\\ &=F(0)J(t) G(t-h)- \int_0^t \Big[\frac{dF(t-\tau)}{d\tau} J(\tau)G(\tau-h) + F(t-\tau)J(\tau)\frac{dG(\tau-h)}{d\tau}\Big]d\tau. \end{align*} However, $$\frac{dG(\tau-h)}{d\tau}=-a(\tau-h) G(\tau-2h)\quad\text{and}\quad \frac{dF(t-\tau)}{d\tau}=-\frac{dF(t-\tau)}{dt}=bF(t-\tau-h).$$ Thus, \begin{align*} &\int_0^t F(t-\tau)c(\tau)G(\tau-h)d\tau\\ &=J(t) G(t-h)+ \int_0^t J(\tau)\Big[-b F(t-\tau-h) G(\tau-h)\\ &\quad + F(t-\tau)a(\tau-h)G(\tau-2h)\Big]d\tau. \end{align*} Now \eqref{e2.2} implies the following result. \begin{lemma} \label{lem2.2} It holds that \begin{align*} G(t)&=F(t) + J(t) G(t-h)+ \int_0^t J(\tau)\Big[-b F(t-\tau-h) G(\tau-h)\\ &\quad + F(t-\tau)a(\tau-h)G(\tau-2h)\Big]d\tau\,. \end{align*} \end{lemma} From the previous lemma, $$\|G\|_\infty\le \|F\|_\infty + \|G\|_\infty w_b [1+(b+\|a\|_\infty) \|F\|_{L^1}]\,.$$ If condition \eqref{e1.3} holds, then $$\theta:= w_b [1+ (b+\|a\|_\infty) \|F\|_{L^1}]<1$$ and therefore, $$\|G\|_\infty\le \frac{\|F\|_\infty}{1-\theta}. \label{e2.3}$$ So the stability of \eqref{e1.1} is proved. Substituting $$x_\epsilon(t)=e^{\epsilon t} x(t) \label{e2.4}$$ with\epsilon>0$into \eqref{e1.1}, we have the equation $$\dot x_\epsilon(t)=\epsilon x_\epsilon(t) - a(t)e^{\epsilon h}x_\epsilon(t-h). \label{e2.5}$$ If$\epsilon>0$is sufficiently small, then considering (2.5) as a perturbation of the equation$\dot y(t)=\epsilon y(t) - be^{\epsilon h}y(t-h)$, and applying our above arguments, according to \eqref{e2.3} we obtain$\|x_\epsilon\|_\infty<\infty$for any solution$x_\epsilon$of \eqref{e2.5}. Hence \eqref{e2.4} implies$|x(t)|\le e^{-\epsilon t}\|x_\epsilon\|_\infty$for any solution$x$of \eqref{e1.1}. \begin{thebibliography}{0} \bibitem{be2} Berezansky, L. and Braverman, E.; On stability of some linear and nonlinear delay differential equations, \emph{J. Math. Anal. Appl., 314}, No. 2, 391-411 (2006). \bibitem{gi} Gil, M. I.; On Aizerman-Myshkis problem for systems with delay. \emph{Automatica, 36}, 1669-1673 (2000). \bibitem{li} Lillo, J. 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