\documentclass[twoside]{article} \usepackage{amssymb} % font used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Oscillation of solutions to delay differential equations \hfil EJDE--2000/13} {EJDE--2000/13\hfil El M. Elabbasy, A. S. Hegazi, \& S. H. Saker \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.~{\bf 2000}(2000), No.~13, pp.~1--13. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ftp ejde.math.unt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Oscillation of solutions to delay differential equations with positive and negative coefficients \thanks{ {\em 1991 Mathematics Subject Classifications:} 34K15, 34C10. \hfil\break\indent {\em Key words and phrases:} Oscillation, delay differential equations. \hfil\break\indent \copyright 2000 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted September 6, 1999. Published February 16, 2000.} } \date{} % \author{El M. Elabbasy, A. S. Hegazi, \& S. H. Saker} \maketitle \begin{abstract} In this article we present infinite-integral conditions for the oscillation of all solutions of first-order delay differential equations with positive and negative coefficients. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} Consider the first-order delay differential equation $$ \dot{x}(t)+P(t)x(t-\sigma )-Q(t)x(t-\tau )=0\,, \eqno{(1.1)} $$ where $P(t)$ and $Q(t)$ are positive continuous real functions and $\sigma ,\tau $ are positive constants. Equation (1.1) has the following general form $$ \dot{x}(t)+\sum_{i=1}^{n}P_i(t)x(t-\sigma _i)-\sum_{j=1}^{m}Q_{j}(t)x(t-\tau_j)=0\,, \eqno{(1.2)} $$ where $P_i(t),Q_{j}(t)\in C([t_0,\infty ),R^{+})$ and $\sigma_i,\tau_j\in [0,\infty )$, for $i=1,\dots ,n$ and $j=1,\dots ,m$. By a solution of (1.1) or (1.2), we mean a function $x(t)\in C([t_0-\rho ),R)$ that for some $t_0$ satisfies (1.1) (or (1.2)) for all $t\geq t_0$, where $\rho =\max \{\sigma ,\tau \}$ (or $\rho =\max\{ \max_{1\leq i\leq n} \sigma_i,\max_{1\leq j\leq m}\tau_j\})$. As usual a function $x(t)$ is called oscillatory if it has arbitrarily large zeros. Otherwise the solution is called non-oscillatory. Qian and Ladas [1] obtained for (1.1) the well-known oscillation criterion $$ \liminf_{t\to \infty } \int_{t-\rho}^t[P(s)-Q(s+\tau -\sigma )]\,ds >\frac{1}{e}\,. \eqno{(1.3)} $$ Elabbasy and Saker [32] obtained the oscillation criterion for the generalized equation, $$ \liminf_{t\to \infty } \int_{t-\rho}^t\sum_{i=1}^{p} [P_i(s)-\sum_{k\in J_i}Q_k(s+\tau_k-\sigma_i)]\,ds>\frac{1}{e}\,. \eqno{(1.4)} $$ It is easy to see that (1.3) is given by (1.4) when putting $n=m=1$.\smallskip Many authors have considered the delay differential equation, with positive coefficient, $$ \dot{x}(t)+P(t)x(\tau (t))=0\,. \eqno{(1.5)} $$ The first systematic study of oscillation for all solutions of (1.5) was made by Myshkis [3]. He proved that every solution of (1.5) oscillates if $$ \limsup_{t\to \infty}[t-\tau (t)<\infty ,\quad \liminf_{t\to \infty} [t-\tau (t)] \liminf_{t\to \infty} P(t)>\frac{1}{e}\,. \eqno{(1.6)} $$ In 1972, Ladas, Laksmikatham and Papadakis [4] proved that the same conclusion holds if $$ \limsup_{t\to \infty}\int_{\tau (t)}^tP(s)\,ds>1 \,. \eqno{(1.7)} $$ In 1979, Ladas [5] and, in 1982, Kopltadaze and Canturija [2] replaced (1.7) by $$ \liminf_{t\to \infty} \int_{\tau (t)}^tP(s)\,ds>\frac{1}{e}\,. \eqno{(1.8)} $$ Concerning the constant $1/e$ in (1.8), if the inequality $$ \int_{\tau (t)}^tP(s)\,ds\leq \frac{1}{e} \eqno{(1.9)} $$ holds eventually, then according to a result in [2], (1.5) has a non-oscillatory solution. It is obvious that there is a gap between the conditions (1.7) and (1.8) when the limit $$ \lim_{t\to \infty } \int_{\tau (t)}^tP(s)\,ds \eqno{(1.10)} $$ does not exist. In 1995 Elbert and Stavrolakis [6] established infinite-integral conditions for oscillation (1.5) in the case where $$ \int_{\tau (t)}^tP(s)\,ds\geq \frac{1}{e}\quad\mbox{and}\quad \lim_{t\to \infty}\int_{\tau (t)}^tP(s)\,ds=\frac{1}{e}\,. \eqno{(1.11)} $$ They proved that if $$ \sum_{i=1}^\infty \big[\int_{t_{i-1}}^{t_i}P(s)-\frac{1}{e}\big]\,ds=\infty\,, \eqno{(1.12)} $$ then every solution of (1.5) oscillates. In 1996, Li [7] showed that if $\int_{\tau (t)}^tP(s)\,ds>1/e$ for some $t_0>0$ and $$ \int_{t_0}^\infty P(t)\big[\int_{\tau (t)}^tP(s)\,ds-\frac{1}{e}\big]\,dt=\infty \,, \eqno{(1.13)} $$ then every solution of (1.5) oscillates. Domshlak and Stavrolakis [8] established sufficient conditions for the oscillation, in the critical case where \[ \lim_{t\to \infty}P(t)=\frac{1}{e\tau } \,, \] of the delay differential equation $$ \dot{x}(t)+P(t)x(t-\tau )=0\,. \eqno{(1.14)} $$ Recently Domshlak and Stavrolakis [9] and Jaros and Stavrolakis [10] considered the delay differential equation $$ \dot{x}(t)+a_{1}(t)x(t-\tau )+a_{2}(t)x(t-\sigma )=0 \eqno{(1.15)} $$ and established sufficient conditions for the oscillation of all solutions in the critical state that the corresponding limiting equation admits a non-oscillatory solution. The oscillatory properties of various functional differential equations have been employed by many authors. For some contribution to the oscillation theory of delay differential equations we refer to the articles by Zhang and Goplsamy [11], Gyori and Ladas [12], Li [13], Arino, Ladas and Sficas [14], Ladas and Sficas [15], Ladas, Qian and Yan [16], Arino, Gyori and Jawhari [17], Hunt and Yorke [18], Gyori [19], Cheng [20], Kwang [21], Kulenovic, Ladas and Meimardou [22], Kulenovic and Ladas [23, 24, 25], Goplsamy, Kulenovic and Ladas [26], Ladas and Qian [27, 28], Elabbasy, Saker and Al-Shemas [29], Elabbasy and Saker [30] and Elabbasy, Saker and Saif [31], Elabbasy and Saker [32]. To a large extent, the study of functional differential equations is motivated by having many applications in Physics [33], Biology [34], Ecology [35], and the study of spread of infectious diseases [36]. Our aim in this paper is to give an infinite-integral conditions for oscillation of all solutions of (1.1) and (1.2) by using the generalized characteristic equation and the function of the form $\frac{x(t)}{x(t-\sigma_i)}$. In section 2, we present an infinite-integral condition for oscillation of (1.1) which indicates that condition (1.3) is no longer necessary. In section 3, we extended the results in section 2 to establish infinite sufficient conditions for oscillation of (1.2) which indicates that condition (1.4) is no longer necessary. As far as we known, there are no other results for differential equations with positive and negative coefficients with more than one delay. In the sequel, when we write a functional inequality we will assume that it holds for all sufficiently large values of $t$. \begin{lemma} [{[12]}] % Lemma 1.1 Let $a\in {(-\infty ,0)}$, $\tau \in (0,\infty )$, $t_0\in R$ and suppose that $x(t)\in C[[t_0,\infty ),R]$ satisfies the inequality \[ x(t)\leq a+\max_{t-\tau \leq s\leq t}x(s)\quad\mbox{for }t\geq t_0 \,. \] Then $x(t)$ cannot be a non-negative function. \end{lemma} \begin{lemma}[{[12]}] % Lemma 1.2 Assume that $P_i$ and $\tau_i\in {C[[t_0,\infty ),R^{+}]}$ for $i=1,\dots ,n$. Then the differential inequality $$ \dot{x}(t)+\sum_{i=1}^{n}P_i(t)x(t-\tau_i(t))\leq 0,\,\,t\geq t_0 \eqno{(1.16)} $$ has an eventually positive solution if and only if the equation $$ \dot{y}(t)+\sum_{i=1}^{n}P_i(t)y(t-\tau_i(t))=0,\,\,t\geq t_0 \eqno{(1.17)} $$ has an eventually positive solution. \end{lemma} \begin{lemma}[{[13]}] % Lemma 1.3 Consider the delay differential equation $$ \dot{x}(t)+\sum_{i=1}^{n}R_i(t)x(t-\tau_i)=0,\,\,t\geq t_0 \eqno{(1.18)} $$ and assume that $\limsup_{t\to \infty }\int_{t}^{t+\tau {i}}R_i(s)\,ds>0$ for some $i$ and $x(t)$ is an eventually positive solution of (1.18), then for the same $i$, $$ \liminf_{t\to \infty } \frac{x(t-\tau_i)}{x(t)}<\infty \eqno{(1.19)} $$ \end{lemma} \begin{lemma}[{[13]}] % Lemma 1.4 If (1.18) has an eventually positive solution, then $$ \int_{t}^{t+\tau_i}R_i(s)\,ds<1\,,\;i=1,\dots ,n \eqno{(1.20)} $$ eventually. \end{lemma} \section{Oscillation of solutions to (1.1)} Now we obtain an infinite-integral conditions for oscillation of all solutions of (1.1). We need the following Lemma. \begin{lemma} % Lemma 2.1 Assume that: \begin{enumerate} \item[(h1)] $P, Q \in C([t_0,\infty ),R^{+})$, $\sigma ,\tau \in [0,\infty )$ and $\tau \leq \sigma $ \item[(h2)] $P(t) \geq Q(t+\tau -\sigma )$, for $t\geq t_0+\sigma -\tau $ \item[(h3)] $\int_{t-\sigma }^{t-\tau }Q(s)\,ds\leq 1$ for $t\geq t_0+\sigma $ \end{enumerate} Let $x(t)$ be an eventually positive solution of (1.1) and set $$ z(t)=x(t)-\int_{t-\sigma }^{t-\tau }Q(s+\tau )x(s)\,ds,\; t\geq t_0+\sigma -\tau \,. \eqno{(2.1)} $$ Then $z(t)$ is a non-increasing positive function and satisfies the inequality $$ \dot{z}(t)+[P(t)-Q(t+\tau -\sigma )]\,z(t-\sigma )\leq 0\,. \eqno{(2.2)} $$ \end{lemma} The proof of this lemma can be found as Lemma 2.6.1 in [12]. \begin{theorem} % Theorem 2.2 Assume that (h1), (h2) and (h3) from Lemma 2.1 are satisfied. Also assume that for $R(t)=P(t)-Q(t+\tau -\sigma )$, \begin{enumerate} \item[(h4)] $\int_{t}^{t+\sigma }R(s)\,ds>0$ for $t\geq t_0$ for some $t_0>0$. \item[(h5)] $\int_{t_0}^\infty R(t)\ln \left[ e\int_{t}^{t+\sigma }R(s)\,ds\right] \,dt=\infty $. \end{enumerate} Then every solution of (1.1) oscillates. \end{theorem} \paragraph{Proof.} On the contrary assume that 1.1) has an eventually positive solution $x(t)$. By Lemma 2.1 it follows that the function $z(t)$ is positive and satisfies (2.2). So Lemma 1.2 yields that the delay differential equation $$ \dot{y}(t)+[P(t)-Q(t+\tau -\sigma )]\,y(t-\sigma )=0 \eqno{(2.3)} $$ has an eventually positive solution. Let $\lambda (t)=-\dot{y}(t)/y(t)$. Then $\lambda (t)$ is non-negative and continuous, then there exists $t_{1}\geq t_0$ such that $y(t_{1})>0$ and $y(t)=y(t_{1})\exp {(-\int_{t_{1}}^t\lambda (s)\,ds)}$. Furthermore, if $\lambda (t)$ satisfies the generalized characteristic equation \[ \lambda (t)=R(t)\exp {(\int_{t-\sigma }^t\lambda (s)\,ds)}\,, \] we can show that $$ e^{rx}\geq {x+\frac{ln(er)}{r}}\quad \mbox{for }r>0 \,. \eqno{(2.4)} $$ Define $A(t)=\int_{t}^{t+\sigma }R(s)\,ds$. By using (2.4) we find that \begin{eqnarray*} \lambda (t)&=&R(t)\exp {(A(t)\frac{1}{A(t)}\int_{t-\sigma }^t\lambda (s)\,ds)} \\ &\geq & R(t)[\frac{1}{A(t)}\int_{t-\sigma }^t\lambda (s)\,ds+\frac{\ln (eA(t)} {A(t)}] \end{eqnarray*} or $$ (\int_{t}^{t+\sigma }R(s)\,ds)\lambda (t)-R(t)\int_{t-\sigma }^t\lambda (s)\,ds\geq R(t)(\ln {e\int_{t}^{t+\sigma }}R(s)\,ds) \eqno{(2.5)} $$ Then for $N>T$, $$ \displaylines{ \hfill \int_{T}^{N}\lambda (t)(\int_{t}^{t+\sigma }R(s)\,ds)\,dt-\int_{T}^{N}R(t)\int_{t-\sigma }^t\lambda (s)\,ds\,dt \hfill\llap{(2.6)} \cr \geq \int_{T}^{N}R(t)(\ln {e\int_{t}^{t+\sigma }}R(s)\,ds)\,dt\,. \cr }$$ By interchanging the order of integration, we find that \[ \int_{T}^{N}R(t)(\int_{t-\sigma }^t\lambda (s)\,ds)\,dt\geq \int_{T}^{N-\sigma }(\int_{s}^{s+\sigma }R(t)\lambda (s)dt)\,ds \,. \] Hence \[ \int_{T}^{N}R(t)(\int_{t-\sigma }^t\lambda (s)\,ds)\,dt\geq \int_{T}^{N-\sigma }\lambda (s)(\int_{s}^{s+\sigma }R(t)dt)\,ds \,. \] Then \[ \int_{T}^{N}R(t)(\int_{t-\sigma }^t\lambda (s)\,ds)\,dt\geq \int_{T}^{N-\sigma }\lambda (t)(\int_{t}^{t+\sigma }R(s)\,ds)\,dt \] Hence $$\displaylines{ \hfill \int_{T}^{N}\lambda (t)(\int_{t}^{t+\sigma }R(s)\,ds)\,dt -{\int_{T}^{N-\sigma}\lambda (s)(\int_{s}^{s+\sigma }R(t)dt)\,ds} \hfill\llap{(2.7)}\cr \geq \int_{T}^{N}\lambda (t)(\int_{t}^{t+\sigma }R(s)\,ds)\,dt-\int_{T}^{N}R(t)\int_{t-\sigma }^t\lambda (s)\,ds\,dt\,. \cr}$$ From (2.6) and (2.7), it follows that $$ \int_{N-\sigma }^{N}\lambda (t)(\int_{t}^{t+\sigma }R(s)\,ds)dt\geq { \int_{T}^{N}(R(t))}(\ln {e\int_{t}^{t+\sigma }R(s)\,ds})dt \,. \eqno{(2.8)} $$ On the other hand, by Lemma 1.4, we have $$ \int_{t}^{t+\sigma }R(s)\,ds<1 \eqno{(2.9)} $$ eventually. Then by (2.8) and (2.9), we find \[ \int_{N-\sigma }^{N}\lambda (t)dt\geq \int_{T}^{N}(R(t))\ln (e\int_{t}^{t+\sigma }R(s)\,ds)dt \] or $$ \ln \frac{y(N-\sigma )}{y(N)}\geq \int_{T}^{N}R(t)\ln {(e\int_{t}^{t+\sigma }R(s)\,ds)dt}\,. \eqno{(2.10)} $$ In view of (h5) $$ \lim_{t\to \infty}\frac{y(t-\sigma )}{y(t)}=\infty \,. \eqno{(2.11)} $$ However, by Lemma 1.3, $$ \liminf_{t\to \infty } \frac{y(t-\sigma )}{y(t)}<\infty \eqno{(2.12)} $$ which contradicts (2.11), and this completes the present proof. Therefore, every solution of (1.1) oscillates. \section{Oscillation of solutions to (1.2)} Our objective in this section is to establish infinite-integral conditions for oscillation of all solutions of (1.2). We need the following theorem for the proof of the main results in this section. \begin{theorem} % Theorem 3.1 Assume that: \begin{enumerate} \item[(H1)] $P_i, Q_{j}\in C([t_0,\infty ),R^{+})$, $\sigma_i,\,\tau_j\in [0,\infty )$ for $i=1,\dots ,n$ and $j=1,\dots ,m$ \item[(H2)] There exist a positive number $p\leq n$ and a partition of the set $\{1,\dots ,m\}$ into $p$ disjoint subsets $J_{1}$, J$_{2}$, $J_{3}$, \dots ,$J_{p}$, such that $j\in J_i$ implies that $\tau_{j\leq }\sigma_{i}$ \item[(H3)] $P_i(t)\geq \sum_{k\in J_i}Q_k(t+\tau_k-\sigma_i)$ for $t\geq t_0+\sigma_i-\tau_k$, and $i=1,\dots ,p$, \item[(H4)] $\sum_{i=1}^{p}\sum_{k\in J_{j}}\int_{t-\sigma _i}^{t-\tau_k}Q_k(s)\,ds\leq 1$ for $t\geq t_0+\sigma_i$. \end{enumerate} Let x(t) be an eventually positive solution of (1.2) and set $$ z(t)=x(t)-\sum_{i=1}^{p}\sum_{k\in J_{j}}\int_{t-\sigma _i}^{t-\tau_k}Q_k(s+\tau_k)x(s)\,ds,\quad t\geq t_0+\sigma _i-\tau_k\,. \eqno{(3.1)} $$ Then $z(t)$ is a non-increasing and positive function. \end{theorem} \paragraph{Proof} Assume that t$_{1}\geq t_0+\rho $ is such that $x(t)$ is positive for $t\geq t_{1}-\rho$ $\rho =\max_{1\leq i\leq n}\{\sigma_i\}$. From (2.1) we have \[ \dot{z}(t)=\dot{x}(t)-\sum_{i=1}^{p}\sum_{k\in J_{j}}Q_k(t)x(t-\tau_k)+\sum_{i=1}^{p}\sum_{k\in J_{j}}Q_k(t+\tau_k-\sigma_i)x(t-\sigma_i)\,. \] Hence \[ \dot{z}(t)=\dot{x}(t)-\sum_{j=1}^{m}Q_{j}(t)x(t-\tau _{j})+\sum_{i=1}^{p}\sum_{k\in J_{j}}Q_k(t+\tau_k-\sigma _i)x(t-\sigma_i)\,. \] From (1.2), we have \[ \dot{z}(t)=-\sum_{i=1}^{p}P_i(t)x(t-\sigma _i)+\sum_{i=1}^{p}\sum_{k\in J_{j}}Q_k(t+\tau_k-\sigma _i)x(t-\sigma_i)-\sum_{i=p+1}^{n}P_i(t)x(t-\sigma_i)\,. \] As we know that \[ \sum_{i=p+1}^{n}P_i(t)x(t-\sigma_i)>0\,, \] we have $$ \dot{z}(t)\leq -\left[ \sum_{i=1}^{p}[P_i(t)-\sum_{k\in J_{j}}Q_k(t+\tau _k-\sigma_i)]x(t-\sigma_i)\right] \eqno{(3.2)} $$ By using (H3) we have $$ \dot{z}(t)\leq 0\,\,\,\,for\,\,t\geq t_{1}+\rho \,. \eqno{(3.3)} $$ This implies that $z(t)$ is a non-increasing function. Now we prove that $z(t)$ is positive. For otherwise, there exists a $t_{2}\geq t_{1}$ such that $z(t_{2})\leq 0$. Since $\dot{z}(t)\leq 0$ for $t\geq t_{1}+\rho$ and $\dot{z}(t)\neq 0$ on $[t_{1}+\rho ,\infty )$, there exists a $t_{3}\geq t_{2}$ such that $z(t)\leq z(t_{3})$ for $t\geq t_{3}$. Thus from (2.1) it follows that for $t\geq t_{3}$, \begin{eqnarray*} x(t)&=&z(t)+\sum_{i=1}^{p}\sum_{k\in J_{j}}\int_{t-\sigma _i}^{t-\tau_k}Q_k(s+\tau_k)x(s)\,ds \\ &\leq& z(t_{3})+\sum_{i=1}^{p}\sum_{k\in J_{j}}\int_{t-\sigma_i}^{t-\tau_k}Q_k(s+\tau_k)x(s)\,ds \\ &\leq& z(t_{3})+\sum_{i=1}^{p}\sum_{k\in J_{j}}\int_{t-\sigma_i}^{t-\tau_k}Q_k(s+\tau_k)\,ds( \max_{t-\rho \leq s\leq t} x(s)) \,. \end{eqnarray*} Hence \[ x(t)\leq z(t_{3})+\sum_{i=1}^{p}\sum_{k\in J_{j}}\int_{t-\sigma_i}^{t-\tau_k}Q_k(s+\tau_k)\,ds (\max_{t-\rho \leq s\leq t} x(s))\,. \] Hypothesis (H4) yields \[ x(t)\leq z(t_{3})+\max_{t-\rho \leq s\leq t} x(s)\quad\mbox{for all }t\geq t_{3}\,, \] where $z(t_{3})\leq 0$. Lemma 1.1 implies that $x(t)$ cannot be non-negative function on $[t_{3},\infty )$. Thus contradicting $x(t)>0$. Therefore, $z(t)$ is a non-increasing and positive function. \begin{theorem} % Theorem 3.2 Assume that (H1), (H2), (H3) and (H4) above are satisfied, $\sigma_{p}=\max \{\sigma_{1},\sigma_{2},\sigma_{3},\dots ,\sigma _{p}\}$, $\sum_{i=1}^{p}\int_{t}^{t+\sigma_i}R_i(s)\,ds>0$ for $t\geq t_0$ for some $t_0>0$. Also assume that \begin{enumerate} \item[(H5)] $\limsup_{t\to \infty } \int_{t}^{t+\sigma_{p}}R_{p}(s)\,ds>0$ \item[(H6)] $\int_{t_0}^\infty \left( \sum_{i=1}^{p}R_i(t)\right) \ln \left[ e\sum_{i=1}^{p}\int_{t}^{t+\sigma_i}R_i(s)\,ds\right] \,dt=\infty $ where $R_i(t)=P_i(t)-\sum_{k\in J_i}Q_k(t+\tau_k-\sigma _i)$. \end{enumerate} Then every solution of (1.2) oscillates. \end{theorem} \paragraph{Proof.} On the contrary assume that (1.2) has an eventually positive solution $x(t)$. By Theorem 2.1 it follows that the function $z(t)$ defined by (3.1) is an eventually positive function. Also by (3.2) we have $$ \dot{z}(t)+\sum_{i=1}^{p}[P_i(t)-\sum_{k\in J_{j}}Q_k(t+\tau_k-\sigma_i)]x(t-\sigma_i)\leq 0\,. \eqno{(3.4)} $$ From the fact that eventually $00$ such that $y(t)=y(t_{1})\exp {(-\int_{t_{1}}^t\lambda (s)\,ds)}$. Furthermore, $\lambda (t)$ satisfies the generalized characteristic equation \[ \lambda (t)=\sum_{i=1}^{p}R_i(t)\exp {(\int_{t-\sigma_i}^t\lambda (s)\,ds)} \] with $R_i(t)=P_i(t)-\sum_{k\in J_i}^{p}Q_k(t+\tau_k-\sigma_i)$ Let $B(t)=\sum_{i=1}^{p}\int_{t}^{t+\sigma_i}R_i(s)\,ds$. By using (2.4) we find that \begin{eqnarray*} \lambda (t)&=&\sum_{i=1}^{p}R_i(t)\exp \big(B(t)\frac{1}{B(t)} \int_{t-\sigma_i}^t\lambda (s)\,ds\big) \\ &\geq& \sum_{i=1}^{p}R_i(t)\big[\frac{1}{B(t)}\int_{t-\sigma _i}^t\lambda (s)\,ds+\frac{\ln (eB(t)}{B(t)}\big] \end{eqnarray*} or \[ \sum_{i=1}^{p}\int_{t}^{t+\sigma_i}R_i(s)\,ds\lambda (t)-\sum_{i=1}^{p}R_i(t)\int_{t-\sigma_i}^t\lambda (s)\,ds\geq \sum_{i=1}^{p}R_i(t)(\ln {e\int_{t}^{t+\sigma_i}}R_i(s)\,ds) \] Then for $N>T$, $$ \displaylines{ \int_{T}^{N}\lambda (t)(\sum_{i=1}^{p}\int_{t}^{t+\sigma _i}R_i(s)\,ds)dt-\int_{T}^{N}\sum_{i=1}^{p}R_i(t)\int_{t-\sigma _i}^t\lambda (s)\,ds\,dt \cr \hfill \geq \int_{T}^{N}\sum_{i=1}^{p}R_i(t)(\ln { e\int_{t}^{t+\sigma_i}}R_i(s)\,ds)\,dt\,. \hfill\llap{(3.7)} \cr }$$ Interchanging the order of integration, we find that \[ \int_{T}^{N}\sum_{i=1}^{p}R_i(t)\int_{t-\sigma_i}^t\lambda (s)\,ds\,dt\geq {\int_{T}^{N-\sigma_i}}({\int_{s}^{s+\sigma _i}\sum_{i=1}^{p}R_i(t)\lambda (s)dt)\,ds}\,. \] Hence \[ \int_{T}^{N}(\sum_{i=1}^{p}R_i(t))\int_{t-\sigma_i}^t\lambda (s)\,ds\,dt\geq {\int_{T}^{N-\sigma_i}\lambda (s)}({\int_{s}^{s+\sigma _i}\sum_{i=1}^{p}R_i(t)dt)\,ds}\,. \] Then $$ \int_{T}^{N}(\sum_{i=1}^{p}R_i(t))\int_{t-\sigma_i}^t\lambda (s)\,ds\,dt\geq {\sum_{i=1}^{p}\int_{T}^{N-\sigma_i}\lambda (t)}({% \int_{t}^{t+\sigma_i}R_i(s)\,ds)\,dt}\,. \eqno{(3.8)} $$ From (3.7) and (3.8), it follows that $$\displaylines{ \int_{T}^{N}\lambda (t)(\sum_{i=1}^{p}\int_{t}^{t+\sigma _i}R_i(s)\,ds)dt-\int_{T}^{N-\sigma_i}\lambda (t)\int_{t}^{t+\sigma _i}\sum_{i=1}^{p}R_i(s)\,ds\,dt \cr \hfill \geq \int_{T}^{N}\sum_{i=1}^{p}R_i(t)(\ln { e\sum_{i=1}^{p}\int_{t}^{t+\sigma_i}}R_i(s)\,ds)\,dt\,. \hfill\llap{(3.9)} \cr }$$ Hence $$\displaylines{ \hfill \sum_{i=1}^{p}\int_{N-\sigma_i}^{N}\lambda (t)(\int_{t}^{t+\sigma_i}R_i(s)\,ds)dt \hfill\llap{(3.10)} \cr \geq {\int_{T}^{N}}(\sum_{i=1}^{p}R_i(t))(\ln { e\int_{t}^{t+\sigma_i}\sum_{i=1}^{p}R_i(t)\,ds)}\,dt\,. \cr }$$ On the other hand, by Lemma 1.4, we have $$ \int_{t}^{t+\sigma_i}R_i(s)\,ds<1,\quad i=1,\dots ,p \eqno{(3.11)} $$ eventually. Then by (3.10) and (3.11), we find \[ \sum_{i=1}^{p}\int_{N-\sigma_i}^{N}\lambda (t)dt\geq {\int_{T}^{N}} (\sum_{i=1}^{p}R_i(t)){\ln }({{e\int_{t}^{t+\sigma _i}\sum_{i=1}^{p}R_i(t)\,ds)}\,dt} \] or $$ \sum_{i=1}^{p}\ln \frac{y(N-\sigma_i)}{y(N)}\geq {\int_{T}^{N}} (\sum_{i=1}^{p}R_i(t)){\ln }({{e\int_{t}^{t+\sigma _i}\sum_{i=1}^{p}R_i(t)\,ds)}\,dt}\,. \eqno{(3.12)} $$ In view of (H6) we have $$ \lim_{t\to \infty }\,\prod_{i=1}^{p}\frac{y(t-\sigma_i)}{% y(t)}=\infty \,. \eqno{(3.13)} $$ This implies that $$ \lim_{t\to \infty} \frac{y(t-\sigma_{p})}{y(t)}=\infty\,. \eqno{(3.14)} $$ However by Lemma 1.3, we have $$ \liminf_{t\to \infty} \frac{y(t-\sigma_{p})}{y(t)} <\infty $$ This contradicts (3.14) and completes the present proof. Therefore, every solution of (1.2) oscillates. \begin{thebibliography}{00} \bibitem{1} C. Qian and G. Ladas, {\em Oscillation in differential equations with positive and negative coefficients}, Canad. Math. Bull. 33 (1990), 442--451. \bibitem{2} R. G. 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