\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 43, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/43\hfil Interval oscillation criteria] {Interval oscillation criteria for second-order forced delay differential equations under impulse effects} \author[Qiaoluan Li, Wing-Sum Cheung \hfil EJDE-2013/43\hfilneg] {Qiaoluan Li, Wing-Sum Cheung} % in alphabetical order \address{Qiaoluan Li \newline College of Mathematics and Information Science, Hebei Normal University \\ Shijiazhuang 050016, China} \email{qll71125@163.com} \address{Wing-Sum Cheung \newline Department of Mathematics, The University of Hong Kong\\ Pokfulam Road, Hong Kong, China} \email{wscheung@hkucc.hku.hk} \thanks{Submitted July 11, 2012. Published February 8, 2013.} \subjclass[2000]{34K11, 34C15} \keywords{Oscillation; impulse; delay differential equation} \begin{abstract} We establish some oscillation criteria for a forced second-order differential equation with impulses. These results extend some well-known results for forced second-order impulsive differential equations with delay. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} In this article, we consider the second-order impulsive delay differential equation $$\begin{gathered} (p(t)x'(t))'+q(t)x(t-\tau)+\sum_{i=1}^{n}q_i(t)\Phi_{\alpha_i}(x(t-\tau))=e(t), \quad t\geq t_0,\;t\neq t_k,\\ x(t_k^+)=a_kx(t_k),\quad x'(t_k^+)=b_kx'(t_k),\quad k=1,2,\dots, \end{gathered}\label{e1.1}$$ where \begin{gather*} x(t_k^-) :=\lim_{t \to t_k^-}x(t),\quad x(t_k^+) :=\lim_{t \to t_k^+}x(t), \\ x'(t_k^-):=\lim_{h\to 0^-} \frac{x(t_k+h)-x(t_k)}h,\quad x'(t_k^+):=\lim_{h\to 0^+} \frac{x(t_k+h)-x(t_k)}h, \end{gather*} $\Phi_{\alpha}(s):=|s|^{\alpha-1}s$, 0\leq t_0\alpha_2>\dots>\alpha_m>1>\alpha_{m+1}>\dots>\alpha_n>0, the functions p, q, q_i, e are piecewise left continuous at each t_k, more precisely, they belong to the set \begin{align*} PLC[t_0, \infty):=\big\{&h:[t_0,\infty)\to \mathbb R |,\; h \text{ is continuous on each interval }(t_k, t_{k+1}),\\ & h(t_k^{\pm}) \text{ exists, and h(t_k)=h(t_k^-) for all k\in \mathbb{N}} \big\}, \end{align*} and p>0 is a nondecreasing function. A function x\in PLC[t_0,\infty) is said to be a solution of \eqref{e1.1} if x(t) satisfies \eqref{e1.1}, and x(t) and x'(t) are left continuous at every t_k, k\in\mathbb{N}. In the past few decades, there has been a great deal of work on the oscillatory behavior of the solutions of second order differential equations, see \cite{a2,z1} and the references cited therein. Impulsive differential equations are an effective tool for the simulation of processes and phenomena observed in control theory, population dynamics, economics, etc. Research in this direction was initiated by Gopalsamy and Zhang in \cite{g1}. Since then there has been an increasing interest in finding the oscillation criteria for such equations, see \cite{a1,b1,b2,l1,l2,o1,o2} their references. Liu and Xu \cite{l1}, obtained several oscillation theorems for the equation $$\label{e1.2} \begin{gathered} (r(t)x'(t))'+p(t)|x(t)|^{\alpha-1}x(t)=q(t),\quad t\geq t_0,\;t\neq t_k,\\ x(t_k^+)=a_kx(t_k),\quad x'(t_k^+)=b_kx'(t_k),\quad x(t_0^+)=x_0,\quad x'(t_0^+)=x'_0, \end{gathered}$$ which is a special case of \eqref{e1.1}. More recently, Guvenilir \cite{g2} established interval criteria for the oscillation of second-order functional differential equations with oscillatory potentials for the equation $$(k(t)x'(t))'+p(t)x(g(t))+q(t)|x(g(t))|^{\gamma-1}x(g(t))=e(t), \quad t\geq 0.\label{e1.3}$$ We note that when g(t)=t-\tau, this equation is included in \eqref{e1.1}. In this article, some new sufficient conditions for the oscillation of solutions of \eqref{e1.1} are presented, and illustrated by an example. It should be noted that the derivation in this work adopts new estimates which are not a routine extension of the existing techniques used for the non-delay case. As is customary, a solution of \eqref{e1.1} is said to be oscillatory if it has arbitrarily large zeros; otherwise the solution is said to be non-oscillatory. \section{Main Results} We will assume the following three conditions throughout this article. \begin{itemize} \item[(H1)] \tau \geq 0, b_k,a_k> 0, t_{k+1}-t_k>\tau, k=1,2,\dots; \alpha_1>\alpha_2>\dots>\alpha_m>1> \alpha_{m+1}>\dots>\alpha_n>0, (n>m\geq 1). \item[(H2)] p, q, q_i, e \in PLC[t_0, \infty). \item[(H3)] For any T\geq 0, there exist intervals [c_1,d_1] and [c_2,d_2] contained in [T,\infty) such that c_1\dots>\alpha_m>1 >\alpha_{m+1}>\dots>\alpha_n>0, there exists an n-tuple (\eta_1,\dots, \eta_n) satisfying either (a) $\sum_{i=1}^{n}\alpha_i\eta_i=1,\quad \sum_{i=1}^{n}\eta_i<1,\quad 0<\eta_i<1,$ or (b) $\sum_{i=1}^{n}\alpha_i\eta_i=1,\quad \sum_{i=1}^{n}\eta_i=1,\quad 0<\eta_i<1.$ \end{lemma} \begin{theorem} \label{thm2.1} Assume that conditions {\rm (H1)--(H3)} hold, and that there exists \omega\in \Omega such that \begin{aligned} &\int_{c_j}^{d_j}p(t)\omega'^{2}(t)dt - \int_{c_j}^{t_{I(c_j)+1}}Q(t)Q_{I(c_j)}^{j}(t)\omega^2(t)dt\\ &-\sum_{k=I(c_j)+2}^{I(d_j)}\int_{t_{k-1}}^{t_k}Q(t)Q_k^j(t)\omega^2(t)dt -\int_{t_{I(d_j)}}^{d_j}Q(t)Q_{I(d_j)}^j(t)\omega^2(t)dt\\ &\leq L(\omega, c_j,d_j), \end{aligned}\label{e2.1} where L(\omega, c_j, d_j):=0 for I(c_j)=I(d_j), and \begin{align*} &L(\omega, c_j,d_j)\\ &:=\beta_j \big\{\omega^2(t_{I(c_j)+1})\frac{a_{I(c_j)+1}-b_{I(c_j)+1}}{a_{I(c_j)+1} (t_{I(c_j)+1}-c_j)}+\sum_{k=I(c_j)+2}^{I(d_j)}\omega^2(t_k) \frac{a_k-b_k}{a_k(t_k-t_{k-1})}\big\} \end{align*} for I(c_j)0 for all t\geq t_0>0. Define the Riccati transformation $v(t):=\frac{p(t)x'(t)}{x(t)}.$ It follows from \eqref{e1.1} that v(t) satisfies $$v'(t)=-q(t)\frac{x(t-\tau)}{x(t)}-\sum_{i=1}^{n}q_i(t)|x(t-\tau)|^{\alpha_i-1} \frac{x(t-\tau)}{x(t)} +\frac{e(t)}{x(t)}-\frac{v^2(t)}{p(t)},\label{e2.2}$$ for all t\neq t_k,t\geq t_0, and v(t_k^+)=\frac{b_k}{a_k}v(t_k) for all k\in\mathbb{N}. From the assumptions, we can choose c_1,d_1\geq t_0 such that q(t)\geq 0 and q_i(t)\geq 0 for t\in [c_1-\tau, d_1], i=1,2,\dots,n, and e(t)\leq 0 for t\in [c_1-\tau, d_1]. By Lemma \ref{lem2.1}, there exist \eta_i>0, i=1,\dots,n, such that \sum_{i=1}^n \alpha_i\eta_i=1 and \sum_{i=1}^n \eta_i<1. Define \eta_0:=1-\sum_{i=1}^n \eta_i and let \begin{gather*} u_0 :=\eta_0^{-1}\Big|\frac{e(t)x(t-\tau)}{x(t)}\Big|\,x^{-1}(t-\tau),\\ u_i :=\eta_i^{-1}q_i(t)\frac{x(t-\tau)}{x(t)}\,x^{\alpha_i-1}(t-\tau),\quad i=1,2,\dots,n\,. \end{gather*} Then by the arithmetic-geometric mean inequality (see \cite{b3}), we have \sum_{i=0}^{n}\eta_iu_i\geq \prod_{i=0}^{n}u_i^{\eta_i} and so \begin{aligned} v'(t) &\leq -\eta_0^{-\eta_0}\prod_{i=1}^{n}\eta_i^{-\eta_i} q_i^{\eta_i}(t)\frac{x^{\eta_i}(t-\tau)}{x^{\eta_i}(t)} x^{(\alpha_i-1)\eta_i}(t-\tau)|e(t)|^{\eta_0}\\ &\quad \times \frac{x^{\eta_0}(t-\tau)}{x^{\eta_0}(t)}x^{-\eta_0}(t-\tau) -q(t)\frac{x(t-\tau)}{x(t)}-\frac{v^2(t)}{p(t)},\quad t\neq t_k. \end{aligned}\label{e2.3} Since \begin{gather*} \prod_{i=0}^{n}\frac{x^{\eta_i}(t-\tau)}{x^{\eta_i}(t)} =\frac{x^{\eta_0+\eta_1+\dots+\eta_n}(t-\tau)} {x^{\eta_0+\eta_1+\dots+\eta_n}(t)}=\frac{x(t-\tau)}{x(t)}\,, \\ \prod_{i=1}^{n}x^{(\alpha_i-1)\eta_i}(t-\tau)x^{-\eta_0}(t-\tau)=1\,, \end{gather*} we obtain \begin{aligned} v'(t) & \leq -q(t)\frac{x(t-\tau)}{x(t)}-\eta_0^{-\eta_0} \prod_{i=1}^{n}\eta_i^{-\eta_i} q_i^{\eta_i}(t)\frac{x(t-\tau)}{x(t)}|e(t)|^{\eta_0} -\frac{v^2(t)}{p(t)}\\ & = -Q(t)\frac{x(t-\tau)}{x(t)}-\frac{v^{2}(t)}{p(t)},\quad t\neq t_k. \end{aligned} \label{e2.4} Multiply both sides of \eqref{e2.4} by \omega^2(t), with w as prescribed in the hypothesis of the theorem. Then integrate from c_1 to d_1; using integration by parts on the left side, we have \begin{aligned} & \sum_{k=I(c_1)+1}^{I(d_1)}\omega^2(t_k)[v(t_k)-v(t_k^+)]\\ &\leq 2\int_{c_1}^{d_1}\omega(t)\omega'(t)v(t)dt - \int_{c_1}^{d_1}\omega^2(t)Q(t)\frac{x(t-\tau)}{x(t)}dt -\int_{c_1}^{d_1}\frac{v^2(t)\omega^2(t)}{p(t)}dt\\ &= 2\int _{c_1}^{d_1}\omega(t)\omega'(t)v(t)dt - \int_{c_1}^{t_{I(c_1)+1}}\omega^2(t)Q(t)\frac{x(t-\tau)}{x(t)}dt\\ & \quad -\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}} \omega^2(t)Q(t)\frac{x(t-\tau)}{x(t)}dt\\ &\quad -\int_{t_{I(d_1)}}^{d_1}\omega^2(t)Q(t)\frac{x(t-\tau)}{x(t)}dt -\int_{c_1}^{d_1}\frac{v^2(t)\omega^2(t)}{p(t)}dt. \end{aligned} \label{e2.5} To estimate \frac{x(t-\tau)}{x(t)}, we first consider the situation where I(c_1)\tau, there are no impulsive moments in (t-\tau,t). As in the proof of \cite[Lemma 2.4]{a1}, we have x(t)>x(t)-x(t_k^+)=x'(\xi)(t-t_k),\quad \xi\in (t_k,\,\,t). $$Since the function p(t)x'(t) is nonincreasing,$$ x(t)>x'(\xi)(t-t_k)>\frac{p(t)x'(t)}{p(\xi)}(t-t_k). $$From the fact that p(t) is nondecreasing, we have$$ \frac{p(t)x'(t)}{x(t)}<\frac{p(\xi)}{t-t_k}<\frac{p(t)}{t-t_k}. $$We obtain \frac{x'(t)}{x(t)}<\frac{1}{t-t_k}. Upon integrating from t-\tau to t, we obtain \frac{x(t-\tau)}{x(t)}>\frac{t-t_k-\tau}{t-t_k}. (ii) If t\in (t_k, t_k+\tau), then t-\tau\in(t_k-\tau,t_k), and there is an impulsive moment t_k in (t-\tau, t). Similar to (i), we obtain \frac{x'(s)}{x(s)}< \frac{1}{s-t_k+\tau} for s\in (t_k-\tau, t_k]. Upon integrating from t-\tau to t_k, we obtain \frac{x(t-\tau)}{x(t_k)}>\frac{t-t_k}{\tau}. Since x(t)-x(t_k^+)\frac{\tau}{a_k\tau+b_k(t-t_k)}\,.$$ Therefore, $$\frac{x(t-\tau)}{x(t)}>\frac{t-t_k}{a_k\tau+b_k(t-t_k)}.$$ Case (2).t\in [c_1,t_{I(c_1)+1}]$. We consider three sub-cases. (i) If$t_{I(c_1)}>c_1-\tau$,$t\in[t_{I(c_1)}+\tau,t_{I(c_1)+1}]$, then there are no impulsive moments in$(t-\tau, t)$. Making a similar analysis of case 1(i), we obtain$\frac{x(t-\tau)}{x(t)}>\frac{t-\tau-t_{I(c_1)}}{ t-t_{I(c_1)}}$. (ii) If$t_{I(c_1)}>c_1-\tau$,$t\in [c_1,t_{I(c_1)}+\tau)$, then$t-\tau\in [c_1-\tau,\,t_{I(c_1)})$and there is an impulsive moment$t_{I(c_1)}$in$(t-\tau,\, t)$. Similar to case 1(ii), we have $$\frac{x(t-\tau)}{x(t)}>\frac{t-t_{I(c_1)}} {a_{I(c_1)}\tau+b_{I(c_1)}(t-t_{I(c_1)})}\,.$$ (iii) If$t_{I(c_1)}\frac{t-\tau-t_{I(c_1)}}{t-t_{I(c_1)}}\,. $$Case (3). t\in (t_{I(d_1)},\,d_1]. There are three sub-cases to consider: (i) If t_{I(d_1)}+\tau\frac{t-\tau-t_{I(d_1)}}{t-t_{I(d_1)}}\,.$$ (ii) If t_{I(d_1)}+\tau\frac{t-t_{I(d_1)}}{ a_{I(d_1)}\tau+b_{I(d_1)}(t-t_{I(d_1)})}\,. $$(iii) If t_{I(d_1)}+\tau\geq d_1, then there is an impulsive moment t_{I(d_1)} in (t-\tau,\,t). Similar to case 3(ii), we obtain$$ \frac{x(t-\tau)}{x(t)}>\frac{t-t_{I(d_1)}}{a_{I(d_1)}\tau+b_{I(d_1)}(t-t_{I(d_1)})}\,. $$Combining all these cases, we have$$ \frac{x(t-\tau)}{x(t)} >\begin{cases} Q_{I(c_1)}^1(t), & \text{for } t\in [c_1,t_{I(c_1)+1}],\\ Q_k^1(t), & \text{for } t\in (t_k,t_{k+1}],\; k=I(c_1)+1,\dots,I(d_1)-1,\\ Q_{I(d_1)}^1(t), & \text{for } t\in (t_{I(d_1)},d_1]. \end{cases} Hence by \eqref{e2.5}, we have \begin{align*} & \sum_{k=I(c_1)+1}^{I(d_1)}\omega^2(t_k)[v(t_k)-v(t_k^+)]\\ &\leq 2\int _{c_1}^{d_1}\omega(t)\omega'(t)v(t)dt - \int_{c_1}^{t_{I(c_1)+1}}\omega^2(t)Q(t)Q_{I(c_1)}^1(t)dt\\ &\quad -\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}} \omega^2(t)Q(t)Q_{k}^1(t)dt -\int_{t_{I(d_1)}}^{d_1}\omega^2(t)Q(t)Q_{I(d_1)}^1(t)dt\\ &\quad -\int_{c_1}^{d_1}\frac{v^2(t)\omega^2(t)}{p(t)}dt\\ &= -\int_{c_1}^{t_{I(c_1)+1}}\frac{1}{p(t)}[p(t)\omega'(t)-v(t)\omega(t)]^2dt\\ &\quad -\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}}\frac{1}{p(t)} [p(t)\omega'(t)-v(t)\omega(t)]^2dt\\ &\quad -\int_{t_I(d_1)}^{d_1}\frac{1}{p(t)}[p(t)\omega'(t)-v(t)\omega(t)]^2dt\\ &\quad +\int_{c_1}^{t_{I(c_1)+1}}[p(t)\omega'^2(t)-Q(t)Q_{I(c_1)}^1(t)\omega^2(t)] dt\\ &\quad +\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}}[p(t)\omega'^2(t) -Q(t)Q_{k}^1(t)\omega^2(t)]dt\\ &\quad +\int_{t_{I(d_1)}}^{d_1}[p(t)\omega'^2(t)-Q(t)Q_{I(d_1)}^1(t)\omega^2(t)]dt\,. \end{align*} Hence we have \begin{aligned} &\sum_{k=I(c_1)+1}^{I(d_1)}\omega^2(t_k)[v(t_k)-v(t_k^+)]\\ &< \int_{c_1}^{t_{I(c_1)+1}}[p(t)\omega'^2(t)-Q(t)Q_{I(c_1)}^1(t)\omega^2(t)]dt\\ & \quad +\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}}[p(t)\omega'^2(t) -Q(t)Q_k^1(t)\omega^2(t)]dt\\ & \quad +\int_{t_{I(d_1)}}^{d_1}[p(t)\omega'^2(t) -Q(t)Q_{I(d_1)}^1(t)\omega^2(t)]dt\,, \end{aligned}\label{e2.6} for if not, we must have p(t)\omega'(t)=v(t)\omega(t) or x(t)\omega'(t)=x'(t)\omega(t) on [c_1,d_1]. Upon integrating, x(t) will be a multiple of \omega(t), which contradicts the facts that \omega vanishes at c_1 and d_1 while x(t) does not. On the other hand, since \big(p(t)x'(t)\big)'<0 for all t\in (c_1,\,t_{I(c_1)+1}], p(t)x'(t) is nonincreasing in (c_1,\,t_{I(c_1)+1}]. Thus x(t)>x(t)-x(c_1)=x'(\xi)(t-c_1)\geq\frac{p(t)x'(t)}{p(\xi)}(t-c_1),\quad \text{for some } \xi\in (c_1,\,t)\ , and hence \frac{p(t)x'(t)}{x(t)}<\frac{p(\xi)}{t-c_1}. Letting t\to t_{I(c_1)+1}^-, we have $$v(t_{I(c_1)+1})\leq\frac{\beta_1}{t_{I(c_1)+1}-c_1}\,.\label{e2.7}$$ Making a similar analysis on (t_{k-1},t_{k}], k=I(c_1)+2,\,\dots, I(d_1), it is not difficult to see that $$v(t_k)\leq \frac{\beta_1}{t_k-t_{k-1}}\,.\label{e2.8}$$ Here we must point out that \eqref{e2.7} and \eqref{e2.8} play a key role in our method for estimating v(t_j), which is different from the usual techniques for the case without impulses. From \eqref{e2.7} and \eqref{e2.8}, and noting that a_k\le b_k, we have \begin{align*} &\sum_{k=I(c_1)+1}^{I(d_1)}\frac{a_k-b_k}{a_k}\omega^2(t_k)v(t_k)\\ &\geq \beta_1\Big[\sum_{k=I(c_1)+2}^{I(d_1)}\frac{a_k-b_k}{a_k(t_k-t_{k-1})} \omega^2(t_k)+\frac{a_{I(c_1)+1}- b_{I(c_1)+1}}{a_{I(c_1)+1}(t_{I(c_1)+1}-c_1)}\omega^2(t_{I(c_1)+1})\Big]\\ &= L(\omega, c_1, d_1)\,. \end{align*} Since \sum_{k=I(c_1)+1}^{I(d_1)}\omega^2(t_k)[v(t_k)-v(t_k^+)]= \sum_{k=I(c_1)+1}^{I(d_1)}\frac{a_k-b_k}{a_k}\omega^2(t_k)v(t_k)\,, by \eqref{e2.6}, we have \begin{align*} L(\omega, c_1, d_1) & < \int_{c_1}^{t_{I(c_1)+1}}[p(t)\omega'^2(t)-Q(t)Q_{I(c_1)}^1(t)\omega^2(t)]dt\\ & \quad +\sum_{k=I(c_1)+1}^{I(d_1)-1}\int_{t_k}^{t_{k+1}} [p(t)\omega'^2(t)-Q(t)Q_k^1(t)\omega^2(t)]dt\\ & \quad +\int_{t_{I(d_1)}}^{d_1}[p(t)\omega'^2(t) -Q(t)Q_{I(d_1)}^1(t)\omega^2(t)]dt\,, \end{align*} which contradicts \eqref{e2.1}. If I(c_1)=I(d_1), then L(\omega, c_1, d_1)=0, and there are no impulsive moments in [c_1,d_1]. Similar to the proof of \eqref{e2.6}, we obtain $$\int_{c_1}^{d_1}[p(t)\omega'^2(t)-Q(t)Q_{I(c_1)}(t)\omega^{2}(t)]dt>0\,. \label{e2.9}$$ This again contradicts our assumption. Finally, if x(t) is eventually negative, we can consider [c_2,d_2] and reach a similar contradiction. The proof of Theorem \ref{thm2.1} is complete. \end{proof} \begin{theorem} \label{thm2.2} Assume conditions {\rm (H1)--(H3)} hold, a_k\leq b_k and there exists a G\in \Gamma such that \begin{aligned} &\int_{c_j}^{d_j}p(t)g^2(t)dt -\int_{c_j}^{t_{I(c_j)+1}}Q(t)G(t)Q_{I(c_j)}^j(t)dt\\ &-\sum_{k=I(c_j)+1}^{I(d_j)-1} \int_{t_k}^{t_{k+1}}Q(t)G(t)Q_k^j(t)dt -\int_{t_I(d_j)}^{d_j}Q(t)G(t)Q_{I(d_j)}^j(t)dt\\ &\leq R(G,c_j,d_j)\,, \end{aligned}\label{e2.10} where R(G,c_j,d_j):=0 for I(c_j)=I(d_j), j=1,2, and \begin{align*} &R(G,c_j,d_j)\\ &:=\frac{a_{I(c_j)+1}-b_{I(c_j)+1}}{a_{I(c_j)+1}(t_{I(c_j)+1}-c_1)} G(t_{I(c_j)+1})\beta_j + \sum_{k=I(c_j)+2}^{I(d_j)} \frac{a_k-b_k}{a_k}\frac{\beta_j}{t_k-t_{k-1}}G(t_k) \end{align*} for I(c_j)0 for t\geq t_0. If I(c_1)0 for t>s; and \item[(A2)] H has partial derivatives \frac{\partial H}{\partial t} and \frac{\partial H}{\partial s} on D such that \frac{\partial H}{\partial t}=2h_1(t,s)\sqrt{H(t,s)},\quad \frac{\partial H}{\partial s}=-2h_2(t,s)\sqrt{H(t,s)}.$\end{itemize} Similar to \cite[Theorem 2.3]{l1}, we have the following Theorem. \begin{theorem} \label{thm2.3} Assume the conditions {\rm (H1)--(H3)} hold. Suppose that there are$\delta_j\in (c_j, d_j)$,$j=1,2$, and$H\in \mathfrak{H}such that \begin{aligned} & \frac{1}{H(d_j,\delta_j)}\Big[\int_{\delta_j}^{d_j}Q(s)Q_j(s)H(d_j,s)ds -\int_{\delta_j}^{d_j}p(s)h_2^2(d_j, s)ds\Big]\\ &+\frac{1}{H(\delta_j,c_j)}\Big[\int_{c_j}^{\delta_j}Q(s)Q_j(s)H(s,c_j)ds -\int_{c_j}^{\delta_j}p(s)h_1^2(s, c_j)ds\Big]\\ &> P(H, c_j,d_j)\,, \end{aligned}\label{e2.12} whereP(H, c_j, d_j):=0$for$I(c_j)=I(d_j), and \begin{aligned} P(H, c_j, d_j) &:=\frac{\beta_j}{H(d_j,\delta_j)}\Big(H(d_j,t_{I(\delta_j)+1}) \frac{b_{I(\delta_j)+1} -a_{I(\delta_j)+1}}{a_{I(\delta_j)+1}(t_{I(\delta_j)+1} -\delta_j)}\\ &\quad + \sum_{i=I(\delta_j)+2}^{I(d_j)} H(d_j, t_i)\frac{b_i-a_i}{a_i(t_i-t_{i-1})}\Big)\\ & \quad +\frac{\beta_j}{H(\delta_j,c_j)} \Big(H(t_{I(c_j)+1},c_j)\frac{b_{I(c_j)+1} -a_{I(c_j)+1}}{a_{I(c_j)+1}(t_{I(c_j)+1}-c_j)} \\ &\quad + \sum_{i=I(c_j)+2}^{I(\delta_j)}H(t_i, c_j) \frac{b_i-a_i}{a_i(t_i-t_{i-1})}\Big) \end{aligned}\label{e2.13} forI(c_j)\pi/8$,$\alpha_1=5/2$,$\alpha_2=1/2$, and$m$is a positive constant. For any$T>0$, we can choose$k $large enough such that$T\frac{9}{10}m\int_{4k\pi-\frac{\pi}{2}}^{4k\pi- \frac{\pi}{4}}\cos(t/2)\sin^2(8t)dt>16\pi \end{aligned} \label{e2.17} for $m$ large enough. On the other hand, note that $a_k=b_k>0$, so that $L(\omega, c_j, d_j)=0$. It follows from Theorem \ref{thm2.1} that all the solutions of \eqref{e2.14} are oscillatory. \subsection*{Acknowledgments} The authors are very grateful to the anonymous referee for his/her careful reading of the original manuscript, and for the helpful suggestions. W.-S. Cheung was partially supported by grant HKU7016/07P from the Research Grants Council of the Hong Kong SAR, China. Q. Li was partially supported by grants 11071054 from the NNSF of China, A2011205012 from the Natural Science Foundation of Hebei Province, and L2009Z02 from the Main Foundation of Hebei Normal University. \begin{thebibliography}{99} \bibitem{a1} R. P. Agarwal, D. R. Anderson, A. 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