\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 167, 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 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/167\hfil Comparison theorems] {Comparison theorems for second-order neutral differential equations of mixed type} \author[T. Li\hfil EJDE-2010/167\hfilneg] {Tongxing Li} \address{Tongxing Li \newline School of Control Science and Engineering, Shandong University, Jinan, Shandong 250061, China} \email{litongx2007@163.com} \thanks{Submitted October 22, 2010. Published November 24, 2010.} \subjclass[2000]{34K11} \keywords{Oscillation; neutral functional differential equations; \hfill\break\indent mixed type; second-order; comparison theorem} \begin{abstract} Three comparison theorems are established for the oscillation of the second-order neutral differential equations of mixed type $$\big(r(t)[x(t)+p_1(t)x(t-\sigma_1)+p_2(t)x(t+\sigma_2)]'\big)' +q_1(t)x(t-\sigma_3) +q_2(t)x(t+\sigma_4)=0.$$ Our results are new even when $p_2(t)=q_2(t)=0$. An example is provided to illustrate the main results. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} This article concerns the oscillatory behavior of the second-order linear neutral differential equation of mixed type $$\label{lh1.1} \left(r(t)[x(t)+p_1(t)x(t-\sigma_1)+p_2(t)x(t+\sigma_2)]'\right)' +q_1(t)x(t-\sigma_3) +q_2(t)x(t+\sigma_4)=0,$$ for $t\geq t_0$. We will use the following conditions: \begin{itemize} \item[(H1)] $r\in C^1([t_0,\infty),\mathbb{R})$, $r(t)>0$ for $t\geq t_0$; \item[(H2)] $p_i\in C([t_0,\infty),[0,a_i])$, where $a_i$ are constants for $i=1$, $2$; \item[(H3)] $q_j\in C([t_0,\infty),[0,\infty))$, and $q_j$ are not eventually zero on any half line $[t_*,\infty)$ for $t_*\geq t_0$, $j=1$, $2$; \item[(H4)] $\sigma_i\geq0$ are constants, for $i=1, 2, 3, 4$. \end{itemize} We put $z(t)=x(t)+p_1(t)x(t-\sigma_1)+p_2(t)x(t+\sigma_2)$. By a solution of \eqref{lh1.1}, we mean a function $x\in C([T_x,\infty), \mathbb{R})$ for some $T_x\geq t_0$ which has the properties that $z\in C^1([T_x,\infty), \mathbb{R})$ and $rz'\in C^1([T_x,\infty), \mathbb{R})$ and satisfying \eqref{lh1.1} on $[T_x,\infty)$. We consider only those solutions $x$ of \eqref{lh1.1} which satisfy $\sup\{|x(t)|:t\geq T\}>0$ for all $T\geq T_x$. We assume that \eqref{lh1.1} possesses such a solution. As is customary, a solution of \eqref{lh1.1} is called oscillatory if it has arbitrarily large zeros on $[t_0,\infty)$; otherwise, it is called non-oscillatory. Equation \eqref{lh1.1} is said to be oscillatory if all its solutions are oscillatory. Recently, there has been much research activity concerning the oscillation and non-oscillation of solutions of varietal types of differential equations. We refer the reader to \cite{ag2,bacu1,bacu2,dzurina,dzurina1,erbz,hasan,ladde,xu,zafer,zhang} and the references cited therein. D\v{z}urina \cite{dzurina1} presented sufficient conditions for the oscillation of the second-order differential equation with mixed argument $$\big(\frac{1}{r(t)}u'(t)\big)'+p(t)u(\tau(t))+q(t)u(\sigma(t))=0,\quad t\geq t_0.$$ Some oscillation results for the second-order neutral differential equation $$(r(t)|z'(t)|^{\gamma-1}z'(t))'+q(t)|x(\sigma(t))|^{\gamma-1}x(\sigma(t)) =0,$$ where $z(t)=x(t)+p(t)x(\tau(t))$ and $t\geq t_0$ were obtained by \cite{dzurina2,dong,han2,liu}. Regarding the oscillatory behavior of neutral differential equations with mixed arguments; see e.g., the papers \cite{AgG,dzurina3,gracei,grace1,grace2,yan1,yan}. Agarwal and Grace \cite{AgG} studied the oscillation of the even-order equation $$(x(t)+ax(t-\tau)-bx(t+\tau))^{(n)}+q(t)x(t-g)+p(t)x(t+h)=0.$$ D\v{z}urina et al. \cite{dzurina3} established some oscillation criteria for the mixed neutral equation $$\left(x(t)+p_1x(t-\tau_1)+p_2x(t+\tau_2)\right)''=q_1(t)x(t-\sigma_1) +q_2(t)x(t+\sigma_2).$$ Grace and Lalli \cite{gracei} examined the oscillatory behavior for the second-order equation $$(x(t)+\lambda x(t-\tau))''=q(t)x(t-\sigma)+p(t)x(t+\beta).$$ Grace \cite{grace1} obtained some oscillation theorems for the odd-order neutral differential equation $$\left(x(t)+p_1x(t-\tau_1)+p_2x(t+\tau_2)\right)^{(n)}=q_1x(t-\sigma_1) +q_2x(t+\sigma_2).$$ Grace \cite{grace2} and Yan \cite{yan1} established several sufficient conditions for the oscillation of solutions of odd-order neutral functional differential equation $$(x(t)+cx(t-h)+Cx(t+H))^{(n)}+qx(t-g)+Qx(t+G)=0.$$ Yan \cite{yan} considered the oscillation of even-order mixed neutral differential equation $$\left(x(t)-c_1x(t-h_1)-c_2x(t+h_2)\right)^{(n)}+qx(t-g_1)+px(t+g_2)=0.$$ To the best of our knowledge, there are only few results on the oscillation of \eqref{lh1.1}. It is interesting to study \eqref{lh1.1} since it has some applications in the study of vibrating masses attached to an elastic bar (see \cite{hale}). The aim of this paper is to establish some oscillation results for \eqref{lh1.1}. The organization of this paper is as follows: In Section 2, we reduce the problem of the oscillation of \eqref{lh1.1} to the oscillation of the first-order inequalities under the case when $$\label{lh1.3} \int_{t_0}^\infty\frac{1}{r(t)}\,{\rm d}t=\infty.$$ In Section 3, we give an example and a remark to illustrate our results. Below, when we write a functional inequality without specifying its domain of validity we assume that it holds for all sufficiently large $t$. \section{Main results} In the following, we will establish some oscillation criteria for \eqref{lh1.1}. Throughout this paper, we denote \begin{gather*} Q(t)=Q_1(t)+Q_2(t),\\ Q_1(t)=\min\{q_1(t),q_1(t-\sigma_1),q_1(t+\sigma_2)\},\\ Q_2(t)=\min\{q_2(t),q_2(t-\sigma_1),q_2(t+\sigma_2)\}. \end{gather*} \begin{theorem}\label{lth3.1} Assume that \eqref{lh1.3} holds. Further, assume that $$\label{mh1} [y(t)+a_1y(t-\sigma_1)+a_2y(t+\sigma_2)]'+Q(t) \Big(\int_{t_1}^{t-\sigma_3}\frac{1}{r(s)}\,{\rm d}s\Big)y(t-\sigma_3) \leq 0$$ has no eventually positive solution for all sufficiently large $t_1$, $t_1\geq t_0$. Then \eqref{lh1.1} is oscillatory. \end{theorem} \begin{proof} Let $x$ be a non-oscillatory solution of \eqref{lh1.1}. Without loss of generality, we assume that there exists $t_1\geq t_0$ such that $x(t)>0$, $x(t-\sigma_1)>0$, $x(t+\sigma_2)>0$, $x(t-\sigma_3)>0$ and $x(t+\sigma_4)>0$ for all $t\geq t_1$. Then $z(t)>0$ for $t\geq t_1$. In view of \eqref{lh1.1}, we obtain $$\label{lcj} (r(t)z'(t))'=-q_1(t)x(t-\sigma_3)-q_2(t)x(t+\sigma_4)\leq0,\quad t\geq t_1.$$ Thus, $r(t)z'(t)$ is non-increasing function. Consequently, it is easy to conclude that there exist two possible cases of the sign of $z'(t)$, that is, $z'(t)>0$ or $z'(t)<0$ eventually. If there exists $t_2\geq t_1$ such that $z'(t_2)<0$, then from \eqref{lcj}, we see that $$r(t)z'(t)\leq r(t_2)z'(t_2)<0,\quad t\geq t_2.$$ Integrating the above inequality from $t_2$ to $t$, we obtain $$z(t)\leq z(t_2)+ r(t_2)z'(t_2)\int_{t_2}^t \frac{1}{r(s)}\,{\rm d}s.$$ Letting $t\to\infty$, we obtain $\lim_{t\to\infty}z(t)=-\infty$ due to \eqref{lh1.3}, which is a contradiction. Thus, there exists a $t_2\geq t_1$ such that $$\label{118} z'(t)>0$$ for $t\geq t_2$. Using \eqref{lh1.1}, for all sufficiently large $t$, we have \begin{align*} &(r(t)z'(t))'+q_1(t)x(t-\sigma_3)+q_2(t)x(t+\sigma_4) +a_1(r(t-\sigma_1)z'(t-\sigma_1))'\\ &+a_1q_1(t-\sigma_1)x(t-\sigma_1-\sigma_3) +a_1q_2(t-\sigma_1)x(t+\sigma_4-\sigma_1)\\ &+a_2(r(t+\sigma_2)z'(t+\sigma_2))' +a_2q_1(t+\sigma_2)x(t+\sigma_2-\sigma_3)\\ &+a_2q_2(t+\sigma_2)x(t+\sigma_2+\sigma_4) =0. \end{align*} Thus \label{259} \begin{aligned} &(r(t)z'(t))'+a_1(r(t-\sigma_1)z'(t-\sigma_1))' +a_2(r(t+\sigma_2)z'(t+\sigma_2))'\\ &+Q_1(t)z(t-\sigma_3) +Q_2(t)z(t+\sigma_4)\leq 0. \end{aligned} By \eqref{118}, we have $z(t+\sigma_4)\geq z(t-\sigma_3)$. Then, from \eqref{259}, we obtain $$\label{jl2} (r(t)z'(t))'+a_1(r(t-\sigma_1)z'(t-\sigma_1))'+a_2(r(t+\sigma_2)z'(t+\sigma_2))'+Q(t)z(t-\sigma_3)\leq0.$$ It follows from \eqref{lcj} that $$\label{xj1} z(t)=z(t_2)+\int_{t_2}^t\frac{r(s)z'(s)}{r(s)}\,{\rm d}s\geq r(t)z'(t)\int_{t_2}^t\frac{1}{r(s)}\,{\rm d}s.$$ Set $y(t)=r(t)z'(t)>0$. From \eqref{jl2} and \eqref{xj1}, we see that $y$ is an eventually positive solution of $$[y(t)+a_1y(t-\sigma_1)+a_2y(t+\sigma_2)]'+Q(t)y(t-\sigma_3)\int_{t_2}^{t-\sigma_3}\frac{1}{r(s)}\,{\rm d}s\leq 0.$$ This completes the proof. \end{proof} \begin{theorem}\label{lth3.2} Assume that \eqref{lh1.3} holds and $$\label{ssmh1} u'(t)+Q(t)\frac{\int_{t_1}^{t-\sigma_3} \frac{1}{r(s)}\,{\rm d}s}{1+a_1+a_2}u(t+\sigma_1-\sigma_3)\leq0$$ has no eventually positive solution for all sufficiently large $t_1$, $t_1\geq t_0$. Then \eqref{lh1.1} is oscillatory. \end{theorem} \begin{proof} Let $x$ be a non-oscillatory solution of \eqref{lh1.1}. Without loss of generality, we assume that there exists $t_1\geq t_0$ such that $x(t)>0$, $x(t-\sigma_1)>0$, $x(t+\sigma_2)>0$, $x(t-\sigma_3)>0$ and $x(t+\sigma_4)>0$ for all $t\geq t_1$. Then $z(t)>0$ for $t\geq t_1$. Proceeding as in the proof of Theorem \ref{lth3.1}, we obtain that $y(t)=r(t)z'(t)>0$ is non-increasing and satisfies inequality \eqref{mh1}. Define $$u(t)=y(t)+a_1y(t-\sigma_1)+a_2y(t+\sigma_2)>0.$$ Then $$u(t)\leq (1+a_1+a_2)y(t-\sigma_1).$$ Substituting the above formulas into \eqref{mh1}, we find $u$ is an eventually positive solution of $$u'(t)+Q(t)\frac{\int_{t_1}^{t-\sigma_3} \frac{1}{r(s)}\,{\rm d}s}{1+a_1+a_2}u(t+\sigma_1-\sigma_3)\leq0.$$ The proof is complete. \end{proof} From Theorem \ref{lth3.2} and \cite[Theorem 2.1.1]{ladde}, we establish the following corollary. \begin{corollary}\label{xlth3.2} Assume that \eqref{lh1.3} holds, $\sigma_1-\sigma_3<0$ and $$\label{1mh1} \liminf_{t\to\infty}\int_{t+\sigma_1-\sigma_3}^tQ(u) \Big(\int_{t_1}^{u-\sigma_3}\frac{1}{r(s)}\,{\rm d}s\Big) \,{\rm d}u>\frac{1+a_1+a_2}{{\rm e}}$$ for all sufficiently large $t_1$, $t_1\geq t_0$. Then \eqref{lh1.1} is oscillatory. \end{corollary} \begin{theorem}\label{lth3.3} Assume that \eqref{lh1.3} holds and $$\label{xxmh1} w'(t)-\frac{Q(t+\sigma_1)}{1+a_1+a_2}\Big(\int_{t_1}^{t+\sigma_1} \,\frac{{\rm d}u}{r(u-\sigma_1)}\Big) w(t+\sigma_1-\sigma_3)\geq0$$ has no eventually positive solution for all sufficiently large $t_1$, $t_1\geq t_0$. Then \eqref{lh1.1} is oscillatory. \end{theorem} \begin{proof} Let $x$ be a non-oscillatory solution of \eqref{lh1.1}. Without loss of generality, we assume that there exists $t_1\geq t_0$ such that $x(t)>0$, $x(t-\sigma_1)>0$, $x(t+\sigma_2)>0$, $x(t-\sigma_3)>0$ and $x(t+\sigma_4)>0$ for all $t\geq t_1$. Then $z(t)>0$ for $t\geq t_1$. Proceeding as in the proof of Theorem \ref{lth3.1}, we obtain \eqref{lcj}--\eqref{jl2} for $t\geq t_2\geq t_1$. Integrating \eqref{jl2} from $t$ to $\infty$ yields $$\label{123} r(t)z'(t)+a_1r(t-\sigma_1)z'(t-\sigma_1)+a_2r(t+\sigma_2)z'(t+\sigma_2)\geq \int_t^\infty Q(s)z(s-\sigma_3)\,{\rm d}s.$$ Since $r(t)z'(t)$ is non-increasing, we get $$\label{234} r(t)z'(t)+a_1r(t-\sigma_1)z'(t-\sigma_1)+a_2r(t+\sigma_2)z'(t+\sigma_2) \leq (1+a_1+a_2)r(t-\sigma_1)z'(t-\sigma_1).$$ In view of \eqref{123} and \eqref{234}, we have $$\label{345} z'(t-\sigma_1)\geq \frac{1}{(1+a_1+a_2)r(t-\sigma_1)}\int_t^\infty Q(s)z(s-\sigma_3)\,{\rm d}s.$$ Integrating \eqref{345} from $t_2$ to $t$, we see that \begin{align*} z(t-\sigma_1) &\geq\int_{t_2}^t\frac{1}{(1+a_1+a_2)r(u-\sigma_1)}\int_u^\infty Q(s)z(s-\sigma_3)\,{\rm d}s\,{\rm d}u \\ &\geq \int_{t_2}^t\frac{1}{1+a_1+a_2}Q(s)z(s-\sigma_3)\int_{t_2}^s \frac{1}{r(u-\sigma_1)} \,{\rm d}u \,{\rm d}s. \end{align*} Thus $$z(t)\geq\frac{1}{1+a_1+a_2}\int_{t_2}^{t+\sigma_1} Q(s)z(s-\sigma_3)\int_{t_2}^s \frac{1}{r(u-\sigma_1)} \,{\rm d}u \,{\rm d}s.$$ Let $$w(t)=\frac{1}{1+a_1+a_2}\int_{t_2}^{t+\sigma_1}Q(s)z(s-\sigma_3)\int_{t_2}^s \frac{1}{r(u-\sigma_1)} \,{\rm d}u \,{\rm d}s>0.$$ Then $z(t)\geq w(t)$ and \begin{align*} w'(t)&=\frac{1}{1+a_1+a_2}Q(t+\sigma_1)z(t+\sigma_1-\sigma_3)\int_{t_2}^{t+\sigma_1} \frac{1}{r(u-\sigma_1)} \,{\rm d}u \\ & \geq \frac{1}{1+a_1+a_2}Q(t+\sigma_1) w(t+\sigma_1-\sigma_3)\int_{t_2}^{t+\sigma_1} \frac{1}{r(u-\sigma_1)} \,{\rm d}u. \end{align*} Hence, we find $w$ is an eventually positive solution of $$w'(t)-\frac{Q(t+\sigma_1)}{1+a_1+a_2}\Big(\int_{t_2}^{t+\sigma_1} \,\frac{{\rm d}u}{r(u-\sigma_1)}\Big) w(t+\sigma_1-\sigma_3)\geq0.$$ This completes the proof. \end{proof} Due to Theorem \ref{lth3.3} and \cite[Theorem 2.4.1]{ladde}, we obtain the following corollary. \begin{corollary}\label{jlth3.2} Assume that \eqref{lh1.3} holds, $\sigma_1-\sigma_3>0$ and $$\label{891mh1} \liminf_{t\to\infty}\int_t^{t+\sigma_1-\sigma_3}Q(u+\sigma_1) \Big(\int_{t_1}^{u+\sigma_1}\frac{1}{r(s-\sigma_1)}\,{\rm d}s\Big)\,{\rm d}u>\frac{1+a_1+a_2}{{\rm e}}$$ for all sufficiently large $t_1$, $t_1\geq t_0$. Then \eqref{lh1.1} is oscillatory. \end{corollary} \section{Example and remark} For an application of our results, we will give the following example. Consider the equation $$\label{y1} [x(t)+a_1x(t-\sigma_1)+a_2x(t+\sigma_2)]'' +\frac{\alpha}{t}x(t-\sigma_3) +\frac{\beta}{t}x(t+\sigma_4)=0,\quad t\geq t_0,$$ where $a_1$, $a_2$, $\alpha$ and $\beta$ are positive constants. Let $r(t)=1$, $p_1(t)=a_1$, $q_1(t)=\alpha/t$ and $q_2(t)=\beta/t$. Then $Q_1(t)=\alpha/(t+\sigma_2)$, $Q_1(t)=\beta/(t+\sigma_2)$ and $Q(t)=(\alpha+\beta)/(t+\sigma_2)$. Assume that $\sigma_3>\sigma_1$. Since $$\liminf_{t\to\infty}\int_{t+\sigma_1-\sigma_3}^tQ(u) \Big(\int_{t_1}^{u-\sigma_3}\frac{1}{r(s)}\,{\rm d}s\Big)\,{\rm d}u =(\alpha+\beta)(\sigma_3-\sigma_1),$$ we conclude that \eqref{y1} is oscillatory if $$(\alpha+\beta)(\sigma_3-\sigma_1)>\frac{1+a_1+a_2}{{\rm e}}$$ due to Corollary \ref{xlth3.2}. Suppose that $\sigma_3<\sigma_1$. Since $$\liminf_{t\to\infty}\int_t^{t+\sigma_1-\sigma_3}Q(u+\sigma_1) \Big(\int_{t_1}^{u+\sigma_1}\frac{1}{r(s-\sigma_1)}\,{\rm d}s\Big) \,{\rm d}u\\ =(\alpha+\beta)(\sigma_1-\sigma_3),$$ we conclude that \eqref{y1} is oscillatory if $$(\alpha+\beta)(\sigma_1-\sigma_3)>\frac{1+a_1+a_2}{{\rm e}}$$ due to Corollary \ref{jlth3.2}. \begin{remark} \label{rmk3.1} \rm The equation $$\label{mmm1} [x(t)+a_1x(t-\sigma_1)]''+q_1(t)x(t-\sigma_3)=0,\quad \sigma_1<\sigma_3,\; t\geq t_0$$ is a special case of \eqref{lh1.1}. Applying results of \cite[Theorem 2]{zafer} and \cite[Corollary 1]{zhang}, we obtain a sufficient condition for \eqref{mmm1} to be oscillatory, that is, if $a_1<1$ and $$\label{cc1} \liminf_{t\to\infty}\int_{t-\sigma_3}^t q_1(s)(s-\sigma_3)\,{\rm d}s>\frac{1}{(1-a_1){\rm e}},$$ then \eqref{mmm1} is oscillatory. 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