\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small 2004 Conference on Diff. Eqns. and Appl. in Math. Biology, Nanaimo, BC, Canada.\newline {\em Electronic Journal of Differential Equations}, Conference 12, 2005, pp. 171--180.\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}} \setcounter{page}{171} \begin{document} \title[\hfilneg EJDE/Conf/12 \hfil Oscillation criteria] {Oscillation criteria for functional differential equations} \author[I. P. Stavroulakis \hfil EJDE/Conf/12 \hfilneg] {Ioannis P. Stavroulakis} \address{Department of Mathematics, University of Ioannina \\ 451 10 Ioannina, Greece} \email{ipstav@cc.uoi.gr} \date{} \thanks{Published April 20, 2005.} \subjclass[2000]{34K11, 34C10} \keywords{Oscillation; delay differential equations} \begin{abstract} Consider the first-order linear delay differential equation $x'(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0},$ and the second-order linear delay equation $x''(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0},$ where $p$ and $\tau$ are continuous functions on $[t_{0},\infty )$, $p(t)>0$, $\tau (t)$ is non-decreasing, $\tau (t)\leq t$ for $t\geq t_{0}$ and $\lim_{t\to \infty }\tau (t)=\infty$. Several oscillation criteria are presented for the first-order equation when $0<\liminf_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds\leq \frac{1}{e} \quad \text{and}\quad \limsup_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds<1,$ and for the second-order equation when $\liminf_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds \leq \frac{1}{e}\quad \text{and}\quad \limsup_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds<1\,.$ \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \section{Introduction} The problem of establishing sufficient conditions for the oscillation of all solutions to the first-order differential equation $$x'(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0}, \label{e1}$$ and to the second-order equation $$x''(t)+p(t)x(\tau (t))=0,\quad t\geq t_{0}, \label{e2}$$ where $p\in C([t_{0,}\infty ),\mathbb{R}^{+})$ (here $\mathbb{R}^{+}=[0,\infty ))$, $\tau \in C([t_{0},\infty )$, $\mathbb{R})$, $\tau (t)$ is non-decreasing, $\tau (t)\leq t$ for $t\geq t_{0}$ and $\lim_{t\to \infty }\tau (t)=\infty$, has been the subject of many investigations. See, for example the references in this article and the references cited therein. By a solution of \eqref{e1} (resp. \eqref{e2}) we understand a continuously differentiable function defined on $[\tau (T_{0}),\infty )$ for some $T_{0}\geq t_{0}$ and such that \eqref{e1} (resp. \eqref{e2}) is satisfied for $t\geq T_{0}$. Such a solution is called \textit{oscillatory} if it has arbitrarily large zeros, and otherwise it is called \textit{nonoscillatory}. In this paper our main purpose is to present the state of the art on the oscillation of all solutions to \eqref{e1} especially in the case where $0<\liminf_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds\leq \frac{1}{e}\quad \text{ and}\quad \limsup_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds<1,$ and for \eqref{e2} when $\liminf_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds \leq \frac{1}{e}\quad \text{and}\quad \limsup_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds<1.$ \section{Oscillation criteria for the first-order equation} In this section we study the delay equation \eqref{e1}. The first systematic study for the oscillation of all solutions to \eqref{e1} was made by Myshkis. In 1950 [42] he proved that every solution of \eqref{e1} oscillates if $$\limsup_{t\to \infty }[t-\tau (t)]<\infty \quad \mbox{and}\quad \liminf_{t\to \infty }[t-\tau (t)]\liminf_{t\to \infty }p(t)> \frac{1}{e}. \label{C1}$$ In 1972, Ladas, Lakshmikantham and Papadakis [33] proved that the same conclusion holds if $$A:=\limsup_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds>1. \label{C2}$$ In 1979, Ladas [32] established integral conditions for the oscillation of \eqref{e1} with constant delay. Tomaras [54-56] extended this result to \eqref{e1} with variable delay. For related results see Ladde [36-38]. The following most general result is due to Koplatadze and Canturija \cite{k5}. If $$\alpha :=\liminf_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds>\frac{1}{e}, \label{C3}$$ then all solutions of \eqref{e1} oscillate; If $$\limsup_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds<\frac{1}{e}, \label{N1}$$ then \eqref{e1} has a nonoscillatory solution. In 1982 Ladas, Sficas and Stavroulakis \cite{l4} and in 1984 Fukagai and Kusano [13] established oscillation criteria (of the type of conditions \eqref{C2} and \eqref{C3}) for \eqref{e1} with \textit{oscillating } coefficient $p(t)$. It is obvious that there is a gap between the conditions \eqref{C2} and \eqref{C3} when the limit $\lim_{t\to \infty} \int_{\tau (t)}^{t}p(s)ds$ does not exist. How to fill this gap is an interesting problem which has been recently investigated by several authors. In 1988, Erbe and Zhang \cite{e6} developed new oscillation criteria by employing the upper bound of the ratio $x(\tau (t))/x(t)$ \ for possible nonoscillatory solutions $x(t)$ of \eqref{e1}. Their result says that all the solutions of \eqref{e1} are oscillatory, if $0<\alpha \leq \frac{1}{e}$ and $$A>1-\frac{\alpha ^{2}}{4}. \label{C4}$$ Since then, several authors tried to obtain better results by improving the upper bound for $x(\tau (t))/x(t)$. In 1991, Jian [20] derived the condition $$A>1-\frac{\alpha ^{2}}{2(1-\alpha )}, \label{C5}$$ while in 1992, Yu and Wang [63] and Yu, Wang, Zhang and Qian [64] obtained the condition $$A>1-\frac{1-\alpha -\sqrt{1-2\alpha -\alpha ^{2}}}{2}. \label{C6}$$ In 1990, Elbert and Stavroulakis \cite{e2} and in 1991 Kwong \cite{k10}, using different techniques, improved \eqref{C4}, in the case where $0<\alpha \leq \frac{1}{e}$, to the conditions $$A>1-(1-\frac{1}{\sqrt{\lambda _{1}}})^{2} \label{C7}$$ and $$A>\frac{\ln \lambda _{1}+1}{\lambda _{1}}, \label{C8}$$ respectively, where $\lambda _{1}$ is the smaller real root of the equation $\lambda =e^{\alpha \lambda }$. In 1994, Koplatadze and Kvinikadze \cite{k6} improved \eqref{C6}, while in 1998, Philos and Sficas \cite{p1} and in 1999, Zhou and Yu [65] and Jaro\v{s} and Stavroulakis \cite{j1} derived the conditions \begin{gather} A>1-\frac{\alpha ^{2}}{2(1-\alpha )}-\frac{\alpha ^{2}}{2}\lambda _{1}, \label{C9} \\ A>1-\frac{1-\alpha -\sqrt{1-2\alpha -\alpha ^{2}}}{2}-(1-\frac{1}{\sqrt{ \lambda _{1}}})^{2}, \label{C10} \\ A>\frac{\ln \lambda _{1}+1}{\lambda _{1}}-\frac{1-\alpha -\sqrt{1-2\alpha -\alpha ^{2}}}{2}, \label{C11} \end{gather} respectively. Consider \eqref{e1} and assume that $\tau (t)$ is continuously differentiable and that there exists $\theta >0$ such that $p(\tau(t))\tau ^{\prime}(t)\geq \theta p(t)$ eventually for all $t$. Under this additional condition, in 2000, Kon, Sficas and Stavroulakis \cite{k2} and in 2003, Sficas and Stavroulakis \cite{s1} established the conditions $$A>\frac{\ln \lambda _{1}+1}{\lambda _{1}}-\frac{1-\alpha -\sqrt{(1-\alpha )^{2}-4\Theta }}{2} \label{e2.1}$$ and $$A>\frac{\ln \lambda _{1}}{\lambda _{1}}-\frac{1+\sqrt{1+2\theta -2\theta \lambda _{1}M}}{\theta \lambda _{1}}\text{ } \label{e2.2}$$ respectively, where $\Theta =\frac{e^{\lambda _{1}\theta \alpha }-\lambda _{1}\theta \alpha -1}{ (\lambda _{1}\theta )^{2}}$ and $M=\frac{1-\alpha -\sqrt{(1-\alpha )^{2}-4\Theta }}{2}.$ \begin{remark}[\cite{k2,s1}] \label{rmk2.1} \rm Observe that when $\theta =1$, then $\Theta =\frac{\lambda _{1}-\lambda _{1}\alpha -1}{\lambda _{1}{}^{2}}$, and \eqref{e2.1} reduces to $$A>2\alpha +\frac{2}{\lambda _{1}}-1,\label{C12}$$ while in this case it follows that $M=1-\alpha -\frac{1}{\lambda _{1}}$ and \eqref{e2.2} reduces to $$A>\frac{\ln \lambda _{1}-1+\sqrt{5-2\lambda _{1} +2\alpha \lambda _{1}}}{\lambda _{1}}. \label{C13}$$ In the case where $\alpha =\frac{1}{e}$, then $\lambda _{1}=e$, and \eqref{C13} leads to $A>\frac{\sqrt{7-2e}}{e}\approx 0.459987065.$ It is to be noted that as $\alpha \to 0$, then all the previous conditions \eqref{C4}-\eqref{C12} reduce to the condition \eqref{C2}, i.e. $A>1$. However, the condition \eqref{C13} leads to $A>\sqrt{3}-1\approx 0.732,$ which is an essential improvement. Moreover \eqref{C13} improves all the above conditions when $0<\alpha \leq 1/e$ as well. Note that the value of the lower bound on $A$ can not be less than $\frac{1}{e}\approx 0.367879441.$ Thus the aim is to establish a condition which leads to a value {\it as close as possible to} $1/e$. For illustrative purpose, we give the values of the lower bound on $A$ under these conditions when $\alpha=1/e$. \begin{center} \begin{tabular}{|cc|} \hline Condition & Lower bound\\ \hline \eqref{C4} & 0.966166179 \\ \eqref{C5} & 0.892951367 \\ \eqref{C6} & 0.863457014 \\ \eqref{C7} & 0.845181878 \\ \eqref{C8} & 0.735758882 \\ \eqref{C9} & 0.709011646 \\ \eqref{C10}& 0.708638892 \\ \eqref{C11}& 0.599215896 \\ \eqref{C12}& 0.471517764 \\ \eqref{C13}& 0.459987065 \\ \hline \end{tabular} \end{center} We see that the condition \eqref{C13} essentially improves all the known results in the literature. \end{remark} \begin{example} [\cite{s1}] \label{exa2.1} \rm Consider the delay differential equation $x'(t)+px\big(t-q\sin ^{2}\sqrt{t}-\frac{1}{pe}\big)=0,$ where $p>0$, $q>0$ and $pq=0.46-\frac{1}{e}$. Then $\alpha =\liminf_{t\to \infty }\int_{\tau (t)}^{t}pds=\liminf_{t\to \infty }p(q\sin ^{2}\sqrt{t}+\frac{1}{pe})= \frac{1}{e}$ and $A=\limsup_{t\to \infty }\int_{\tau (t)}^{t}pds=\limsup_{t\to \infty }p(q\sin ^{2}\sqrt{t}+\frac{1}{pe} )=pq+\frac{1}{e}=0.46\,.$ Thus, according to Remark 2.1, all solutions of this equation oscillate. Observe that none of the conditions \eqref{C4}-\eqref{C12} apply to this equation. \end{example} Following this historical (and chronological) review we also mention that in the case where $\int_{\tau (t)}^{t}p(s)ds\geq \frac{1}{e}\quad \text{and}\quad \lim_{t\to \infty }\int_{\tau (t)}^{t}p(s)ds=\frac{1}{e}$ this problem has been studied in 1995, by Elbert and Stavroulakis \cite{e3}, by Kozakiewicz \cite{k8}, Li \cite{l9,l10}, and in 1996, by Domshlak and Stavroulakis \cite{d5}. \section{Oscillation criteria for the second-order equation} In this section we study the second-order delay equation \eqref{e2}: $x^{\prime\prime}(t)+p(t)x(\tau (t))=0,\;t\geq t_{0},$ For the case of ordinary differential equations, i.e. when $\tau (t)\equiv t,$ the history of the problem began as early as in 1836 by the work of Sturm \cite{s2} and was continued in 1893 by Kneser \cite{k1}. Essential contribution to the subject was made by Hille, Wintner, Hartman, Leighton, Nehari, and others (see the monograph by C. Swanson \cite{s3} and the references cited therein). In particular, in 1948 Hille \cite{h2} obtained the following well-known oscillation criteria. Let $$\limsup_{t\to \infty }t\int_{t}^{+\infty }p(s)ds>1 \label{e3.1}$$ or $$\liminf_{t\to \infty }t\int_{t}^{+\infty }p(s)ds>\frac{1}{4}, \label{e3.2}$$ the conditions being assumed to be satisfied if the integral diverges. Then \eqref{e2} with $\tau (t)\equiv t$ is oscillatory. For the delay differential equation \eqref{e2} earlier oscillation results can be found in the monographs by Myshkis \cite{m2} and Norkin \cite{n1}. In 1968 Waltman \cite{w1} and in 1970 Bradley \cite{b1} proved that \eqref{e2} is oscillatory if $\int^{+\infty }p(t)dt=+\infty .$ Proceeding in the direction of generalization of Hille's criteria, in 1971 Wong \cite{w4} showed that if $\tau (t)\geq \alpha t$ for $t\geq 0$ with $0<\alpha \leq 1$, then the condition $$\liminf_{t\to \infty }t\int_{t}^{+\infty }p(s)ds>\frac{1}{4\alpha } \label{e3.3}$$ is sufficient for the oscillation of \eqref{e2}. In 1973, Erbe \cite{e4} generalized this condition to $$\liminf_{t\to \infty }t\int_{t}^{+\infty }\frac{\tau (s)}{s}p(s)ds> \frac{1}{4} \label{e3.4}$$ without any additional restriction on $\tau$. In 1987, Yan \cite{y1} obtained some general criteria improving the previous ones. An oscillation criterion of different type is given in 1986 by Koplatadze \cite{k3} and in 1988 by Wei \cite{w3}, where it is proved that \eqref{e2} is oscillatory if $$\limsup_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds>1 \label{C2'}$$ or $$\liminf_{t\to \infty }\int_{\tau (t)}^{t}\tau (s)p(s)ds>\frac{1}{e}. \label{C3'}$$ The conditions \eqref{C2'} and \eqref{C3'} are analogous to the oscillation conditions \eqref{C2} and \eqref{C3} respectively, for the first order delay equation \eqref{e1}. The essential difference between \eqref{e3.3}, \eqref{e3.4} and \eqref{C2'}, \eqref{C3'} is that the first two can guarantee oscillation for ordinary differential equations as well, while the last two work only for delay equations. Unlike first-order differential equations, where the oscillatory character is due to the delay only, equation \eqref{e2} can be oscillatory without any delay at all, i.e., in the case $\tau (t)\equiv t$. Figuratively speaking, two factors contribute to the oscillatory character of \eqref{e2}: the presence of the delay and the second order nature of the equation. The conditions \eqref{e3.3}, \eqref{e3.4} and \eqref{C2'}, \eqref{C3'} illustrate the role of these factors taken separately. In what follows it will be assumed that the condition $$\int^{+\infty }\tau (s)p(s)ds=+\infty \label{e3.5}$$ is fulfilled. As it follows from \cite[Lemma 4.1]{k4}, this condition is necessary for \eqref{e2} to be oscillatory. The study being devoted to the problem of oscillation of \eqref{e2}, the condition \eqref{e3.5} does not affect the generality. In this section oscillation results are obtained for \eqref{e2} by reducing it to a first order equation. Since for the latter the oscillation is due solely to the delay, the criteria hold for delay equations only and do not work in the ordinary case. \begin{theorem}[\cite{k7}] \label{thm3.1} Let \eqref{e3.5} be fulfilled and the differential inequality $x'(t)+\Big( \tau (t)+\int_{T}^{\tau (t)}\xi \tau (\xi )p(\xi )d\xi \Big) p(t)x(\tau (t))\leq 0$ have no eventually positive solution. Then \eqref{e2} is oscillatory. \end{theorem} Note that Theorem \ref{thm3.1} reduces the question of oscillation of \eqref{e2} to that of the absence of eventually positive solutions of the differential inequality $$x^{\prime}(t)+\Big( \tau (t)+\int_{T}^{\tau (t)}\xi \tau (\xi )p(\xi )d\xi \Big) p(t)x(\tau (t))\leq 0. \label{e3.6}$$ So oscillation results for first order delay differential equations can be applied since the oscillation of the equation $$u^{\prime}(t)+g(t)u(\delta (t))=0 \label{e3.7}$$ is equivalent to the absence of eventually positive solutions of the inequality $$u^{\prime}(t)+g(t)u(\delta (t))\leq 0. \label{e3.8}$$ This fact is a simple consequence of the following comparison theorem deriving the oscillation of \eqref{e3.7} from the oscillation of the equation $$v^{\prime}(t)+h(t)v(\sigma (t))=0. \label{e3.9}$$ We assume that $g,h:\mathbb{R}^{+}\to \mathbb{R}^{+}$ are locally integrable, $\delta ,\sigma :\mathbb{R}^{+}\to \mathbb{R}$ are continuous, $\delta (t)\leq t,\sigma (t)\leq t$ for $t\in \mathbb{R}^{+}$, and $\delta (t)\to +\infty ,$ $\sigma (t)\to +\infty$ as $t\to +\infty$. \begin{theorem} \label{thm3.2} Let $g(t)\geq h(t)$ and $\delta (t)\leq \sigma (t)$ for $t\in R^{+}$ and let \eqref{e3.9} be oscillatory. Then \eqref{e3.7} is also oscillatory. \end{theorem} \begin{corollary} \label{coro 3.1} Let \eqref{e3.7} be oscillatory. Then the inequality \eqref{e3.8} has no eventually positive solution. \end{corollary} Turning to applications of Theorem \ref{thm3.1}, we will use it together with the criteria \eqref{C2} and \eqref{C3} to get the following result. \begin{theorem}[\cite{k7}] \label{thm3.3} Let $$K:=\limsup_{t\to \infty }\int_{\tau (t)}^{t}\Big( \tau (s)+\int_{0}^{\tau (s)}\xi \tau (\xi )p(\xi )d\xi \Big) p(s)ds>1, \label{C2''}$$ or $$k:=\liminf_{t\to \infty }\int_{\tau (t)}^{t}\Big( \tau (s)+\int_{0}^{\tau (s)}\xi \tau (\xi )p(\xi )d\xi \Big) p(s)ds>\frac{1}{e}. \label{C3''}$$ Then \eqref{e2} is oscillatory. \end{theorem} To apply Theorem \ref{thm3.1} it suffices to note that: (i) \eqref{e3.5} is fulfilled since otherwise $k=K=0$; (ii) since $\tau (t)\to +\infty$ as $t\to +\infty$, the relations \eqref{C2''}, \eqref{C3''} imply the same relations with $0$ changed by any $T\geq 0$. \begin{remark}[\cite{k7}] \label{rmk3.1} \rm Theorem \ref{thm3.3} improves the criteria \eqref{C2'}, \eqref{C3'} by Koplatadze \cite{k3} and Wei \cite{w3} mentioned above. This is directly seen from \eqref{C2''}, \eqref{C3''} and can be easily checked if we take $\tau(t)\equiv t-\tau _{0}$ and $p(t)\equiv p_{0}/(t-\tau _{0})$ for $t\geq 2\tau_{0}$, where the constants $\tau _{0}>0$ and $p_{0}>0$ satisfy $\tau _{0}p_{0}<1/e.$ In this case neither of \eqref{C2'}, \eqref{C3'} is applicable for \eqref{e2} while both \eqref{C2''}, \eqref{C3''} give the positive conclusion about its oscillation. Note also that this is exactly the case where the oscillation is due to the delay since the corresponding equation without delay is non-oscillatory. \end{remark} \begin{remark}[\cite{k7}] \label{rmk3.2} \rm Criteria \eqref{C2''}, \eqref{C3''} look like \eqref{C2}, \eqref{C3} but there is an essential difference between them as pointed out in the introduction. The condition \eqref{C3} is close to be the necessary one, since according to \cite{k5} if $A\leq 1/e$, then \eqref{e3.7} is nonoscillatory. On the other hand, for an oscillatory equation \eqref{e2} without delay we have $k=K=0$. Nevertheless, the constant $1/e$ in Theorem \ref{thm3.3} is also the best possible in the sense that for any $\varepsilon \in (0,1/e]$ it can not be replaced by $1/e-\varepsilon$ without affecting the validity of the theorem. This is illustrated as follows. \end{remark} \begin{example}[\cite{k7}] \label{exa3.1} \rm Let $\varepsilon \in (0,1/e]$, $1-e\varepsilon <\beta <1$, $\tau (t)\equiv \alpha t$ and $p(t)\equiv \beta (1-\beta )\alpha ^{-\beta }t^{-2}$, where $\alpha =\exp(\frac{1}{\beta -1})$. Then \eqref{C3''} is fulfilled with $1/e$ replaced by $1/e-\varepsilon$. Nevertheless \eqref{e2} has a nonoscillatory solution, namely $u(t)\equiv t^{\beta }$. Indeed, denoting $c=\beta (1-\beta )\alpha^{-\beta }$, we see that the expression under the limit sign in \eqref{C3''} is constant and equals $$\alpha c\vert \ln \alpha \vert (1+\alpha c) =(\beta /e)(1+(\beta (1-\beta ))/e)>\beta /e>1/e-\varepsilon .$$ \end{example} Note that there is a gap between conditions \eqref{C2''} and \eqref{C3''} when $0\leq k\leq 1/e$, $kk+\frac{1}{\lambda (k)}-\frac{1-k-\sqrt{1-2k-k^{2}}}{2} \label{C11'} where$\lambda (k)$is the smaller root of the equation$\lambda =\exp (k\lambda )\$. Then \eqref{e2} is oscillatory. \end{theorem} Note that condition \eqref{C11'} is analogous to condition \eqref{C11}. \subsection*{Acknowledgment} The author would like to thank the referee for some useful remarks. \begin{thebibliography}{99} \bibitem{b1} J. S. Bradley; Oscillation theorems for a second order equation. \textit{J. Differential Equations}, \textbf{8} (1970), 397-403. \bibitem{d1} J. Diblik; Positive and oscillating solutions of differential equations with delay in critical case, \textit{J. Comput. Appl. Math}. \textbf{88} (1998), 185-2002. \bibitem{d2} Y. Domshlak; Sturmian comparison method in oscillation study for discrete difference equations, I, \textit{J. Diff. Integr. Eqs}, \textbf{7} (1994), 571-582. \bibitem{d3} Y. Domshlak; Delay-difference equations with periodic coefficients: sharp results in oscillation theory, \textit{Math. Inequal. Appl.}, \textbf{1} (1998), 403-422. \bibitem{d4} Y. Domshlak and A. Aliev; On oscillatory properties of the first order differential equations with one or two retarded arguments, \textit{Hiroshima Math. J. }\textbf{18} (1998), 31-46. \bibitem{d5} Y. Domshlak and I. P. Stavroulakis; Oscillations of first-order delay differential equations in a critical state, \textit{Applicable Anal.}, \textbf{61 } (1996), 359-377. \bibitem{e1} R. Edwards; Functional analysis. Theory and applications. Holt, Rinehart and Winston, Chicago - San Francisco - Toronto, 1965. \bibitem{e2} A. Elbert and I. P. Stavroulakis; Oscillations of first order differential equations with deviating arguments, Univ of Ioannina T.R. No 172 1990, \textit{Recent trends in differential equations}, 163-178, World Sci. Ser. Appl. Anal.,1, World Sci. Publishing Co. (1992). \bibitem{e3} A. Elbert and I. P. Stavroulakis; Oscillation and non-oscillation criteria for delay differential equations, \textit{Proc. Amer. Math. Soc.}, \textbf{123} (1995), 1503-1510. \bibitem{e4} L. H. Erbe; Oscillation criteria for second order nonlinear delay equations, \textit{Canad. Math. Bull.} \textbf{16} (1973), 49-56. \bibitem{e5} L. H. Erbe, Qingkai Kong and B. G. Zhang; Oscillation Theory for Functional Differential Equations, Marcel Dekker, New York, 1995. \bibitem{e6} L. H. Erbe and B. G. Zhang; Oscillation of first order linear differential equations with deviating arguments, \textit{Differential Integral Equations}, \textbf{1} (1988), 305-314. \bibitem{f1} N. Fukagai and T. Kusano; Oscillation theory of first order functional differential equations with deviating arguments, \textit{Ann. Mat. Pura Appl.,} \textbf{136} (1984), 95-117. \bibitem{g1} K. Gopalsamy; Stability and Oscillations in Delay Differential Equations of Population Dynamics, Kluwer Academic Publishers, 1992. \bibitem{g2} I. Gyori and G. Ladas; Oscillation Theory of Delay Differential Equatiosn with Applications, Clarendon Press, Oxford, 1991. \bibitem{1} J. K. Hale; Theory of Functional Differential Equations, Springer-Verlag, New York, 1997. \bibitem{h2} E. Hille; Nonoscillation theorems, \textit{Trans. Amer. Math. Soc. } \textbf{64}(1948), 234-252. \bibitem{i1} A. F. Ivanov, and V. N. Shevelo; Oscillation and asymptotic behavior of solutions of first order differential equations, \textit{Ukrain, Math. Zh.}, \textbf{33} (1981), 745-751, 859. \bibitem{j1} J. Jaro\v{s} and I.P. Stavroulakis; Oscillation tests for delay equations, \textit{Rocky Mountain J. Math.}, \textbf{29} (1999), 139-145. \bibitem{j2} C. Jian; Oscillation of linear differential equations with deviating argument, \textit{Math. in Practice and Theory}, \textbf{1} (1991), 32-41 (in Chinese). \bibitem{k1} A. Kneser; Untersuchungen \"{u}ber die reelen Nulstellen der Integrale linearer Differentialgleichungen. \textit{Math. Ann.} \textbf{42} (1893), 409-435. \bibitem{k2} M. Kon, Y. G. Sficas and I. P. Stavroulakis; Oscillation criteria for delay equations, \textit{Proc. Amer. Math. Soc.,} \textbf{128} (2000), 2989-2997. \bibitem{k3} R. Koplatadze; Oscillation criteria of solutions of second order linear delay differential inequalities with a delayed argument (Russian). \textit{Trudy Inst. Prikl. Mat. I.N.Vekua} \textbf{17} (1986), 104-120. \bibitem{k4} R. Koplatadze; On oscillatory properties of solutions of functional differential equations. \textit{Mem. Differential Equations Math. Phys.} \textbf{3} (1994), 1-177. \bibitem{k5} R. G. Koplatadze and T. A. Chanturija; On the oscillatory and monotonic solutions of first order differential equations with deviating arguments, \textit{Differentsial'nye Uravneniya}, \textbf{18} (1982), 1463-1465. \bibitem{k6} R. G. Koplatadze and G. Kvinikadze; On the oscillation of solutions of first order delay differential inequalities and equations, \textit{Georgian Math. J}. \textbf{1 } (1994), 675-685. \bibitem{k7} R. G. Koplatadze, G. Kvinikadze and I. P. Stavroulakis; Oscillation of second order linear delay differential equations, \textit{Funct. Diff. Equ. } \textbf{7 } (2000), 121-145. \bibitem{k8} E. Kozakiewicz; Conditions for the absence of positive solutions of first order differential inequalities with deviating agruments, 4th Intl. Coll. on Differential Equations, VSP, 1994, 157-161. \bibitem{k9} M. R. Kulenovic and M. K. Grammatikopoulos; First order functional differential inequalities with oscillating coefficients, \textit{Nonlinear Anal. } \textbf{8} (1984), 1043-1054. \bibitem{k10} M. K. Kwong; Oscillation of first order delay equations, \textit{J. Math. Anal. Appl.}, \textbf{156 }(1991), 374-286. \bibitem{k11} M. K. Kwong and W. T. Patula; Comparison theorems for first order linear delay equations, \textit{J. Differential Equations,} \textbf{70} (1987), 275-292. \bibitem{l1} G. Ladas; Sharp conditions for oscillations caused by delay, \textit{Applicable Anal.}, \textbf{9} (1979), 93-98. \bibitem{l2} G. Ladas, V. Laskhmikantham and J. S. Papadakis; Oscillations of higher-order retarded differential equations generated by retarded arguments, \textit{Delay and Functional Differential Equations and Their Applications}, Academic Press, New York, 1972, 219-231. \bibitem{l3} G. Ladas, C. Qian and J. Yan; A comparison result for the oscillation of delay differential equations, \textit{Proc. Amer. Math. Soc.}, \textbf{114} (1992), 939-946. \bibitem{l4} G. Ladas, Y. G. Sficas and I. P. Stavroulakis; Functional-differential inequalities and equations with oscillating coefficients, \textit{Trends in theory and practice of nonlinear differential equations,} (Arlington, Tex., 1982), 277--284, Lecture Notes in Pure and Appl. Math., 90, Marcel Dekker, New York, 1984. \bibitem{l5} G. S. Ladde; Oscillations caused by retarded perturbations of first order linear ordinary differential equations, \textit{Atti Acad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur}., \textbf{63} (1977), 351-359. \bibitem{l6} G. S. Ladde; Class of functional equations with applications, \textit{Nonlinear Anal.}, \textbf{2} (1978), 259-261. \bibitem{l7} G. S. Ladde; Stability and oscillation in single-species processes with past memory, \textit{Int. J. System Sci}., \textbf{10} (1979), 621-647. \bibitem{l8} G. S. Ladde, V. Lakshmikantham and B. G. Zhang; Oscillation Theory of Differential Equations with Deviating Arguments, Marcel Dekker, New York, 1987. \bibitem{l9} B. Li; Oscillations of delay differential equations with variable coefficients, \textit{J. Math. Anal. Appl}., \textbf{192} (1995), 312-321. \bibitem{l10} B. Li; Oscillations of first order delay differential equations, \textit{Proc. Amer. Math. Soc}., \textbf{124} (1996), 3729-3737. \bibitem{m1} A. D. Myshkis; Linear homogeneous differential equations of first order with deviating arguments, \textit{Uspekhi Mat. Nauk}, \textbf{5} (1950), 160-162 (Russian). \bibitem{m2} A. D. Myshkis; Linear differential equations with retarded argument, Nauka, Moscow, 1972 (Russian). \bibitem{n1} S. B. Norkin; Differential equations of the second order with retarded arguments. Nauka, Moscow, 1965 (Russian). \bibitem{p1} Ch. G. Philos and Y. G. Sficas; An oscillation criterion for first-order linear delay differential equations, \textit{Canad. Math. Bull.} \textbf{41} (1998), 207-213. \bibitem{s1} Y. G. Sficas and I. P. Stavroulakis; Oscillation criteria for first-order delay equations, \textit{Bull. London Math. Soc}., \textbf{35} (2003), 239-246. \bibitem{s2} C. Sturm; Sur les \'{e}quations diff\'{e}rentielles lin\'{e}aires du second ordre, \textit{J. Math. Pures Appl.} \textbf{1}(1836), 106-186. \bibitem{s3} C. A. Swanson; Comparison and oscillation theory of linear differential equations, Academic Press, New York and London, 1968. \bibitem{t1} X. H. Tang; Oscillation of delay differential equations with variable coefficients, \textit{J. Math. Study,} \textbf{31} (3)(1998), 290-293. \bibitem{t2} X. H. Tang and J. S. Yu; Oscillations of first order delay differential equations in a critical state, \textit{Mathematica Applicata } \textbf{13} (1)(2000), 75-79. \bibitem{t3} X. H. Tang and J. S. Yu; Oscillation of first order delay differential equations with oscillating coefficients, \textit{Appl Math.-} JCU, \textbf{15} (2000), 252-258. \bibitem{t4} X. H. Tang and J. S. Yu; New oscillation criteria for delay difference equations, \textit{Comput. Math. Appl.}, \textbf{42} (2001), 1319-1330. \bibitem{t5} X. H. Tang, J. S. Yu and Z. C. Wang; Comparison theorems of oscillation of first order delay differential equations in a critical state with applications, \textit{Ke Xue Tongbao}, \textbf{44 }(1999), 26-30. \bibitem{t6} A. Tomaras; Oscillation behavior of an equation arising from an industrial problem, \textit{Bull. Austral. Math. Soc}., \textbf{13} (1975), 255-260. \bibitem{t7} A. Tomaras; Oscillations of a first order functional differential equation, \textit{Bull. Austral. Math. Soc.}, \textbf{17} (1977), 91-95. \bibitem{t8} A. Tomaras; Oscillatory behaviour of first order delay differential equations, \textit{Bull. Austral. Math. Soc}., \textbf{19} (1978), 183-190. \bibitem{w1} P. Waltman; A note on an oscillation criterion for an equation with a functional argument. \textit{Canad. Math. Bull.} \textbf{11}(1968), 593-595 \bibitem{1w2} Z. C. Wang, I. P. Stavroulakis and X. Z. Qian; A survey on the oscillation of solutions of first order linear differential equations with deviating arguments, \textit{Appl. Math. E-Notes}, \textbf{2} (2002), 171-191. \bibitem{w3} J. J. Wei; Oscillation of second order delay differential equation. \textit{Ann. Differential Equations }\textbf{4 }(1988), 473-478. \bibitem{w4} J. S. W. Wong; Second order oscillation with retarded arguments, in Ordinary differential equations'' , pp. 581-596, Washington, 1971; Academic Press, New York and London, 1972. \bibitem{y1} J. Yan, Oscillatory property for second order linear differential equations, \textit{J. Math. Anal. Appl.} \textbf{122 }(1987), 380-384. \bibitem{y2} J. S. Yu and X. H. Tang; Comparison theorems in delay differential equations in a critical state and application, \textit{Proc. London Math. Soc}., \textbf{63} (2001),188-204. \bibitem{y3} J. S. Yu and Z. C. Wang; Some further results on oscillation of neutral differential equations, \textit{Bull. Austral. Math. Soc}., \textbf{46} (1992), 149-157. \bibitem{y4} J. S. Yu, Z. C. Wang, B. G. Zhang and X. Z. Qian; Oscillations of differential equations with deviting arguments, \textit{Panamerican Math. J}., \textbf{2} (1992), 59-78. \bibitem{z1} Y. Zhou and Y. H. Yu; On the oscillation of solutions of first order differential equations with deviating arguments, \textit{Acta Math. Appl. Sinica }\textbf{15}, no.3, (1999), 288-302. \end{thebibliography} \end{document}