\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 18, pp. 1--12.\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 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/18\hfil Oscillation for higher order ODEs] {Oscillation for higher order nonlinear ordinary differential equations with impulses} \author[C. Zhang, W. Feng\hfil EJDE-2006/18\hfilneg] {Chaolong Zhang, Weizhen Feng} \address{Chaolong Zhang \hfill\break Department of Computation Science,\\ Zhongkai University of Agriculture and Technology, Guangzhou, 510225, China} \email{zhcl88@126.com} \address{Weizhen Feng \hfill\break School of Mathematical Sciences \\ South China Normal University, Guangzhou 510631, China} \email{wsy@scnu.edu.cn} \date{} \thanks{Submitted October 19, 2005. Published February 2, 2006.} \thanks{Supported by grant 011471 from the Natural Science Foundation of Guangdong, \hfill\break\indent grant 0120 from the Natural Science Foundation of Guangdong Higher Education, \hfill\break\indent and grant Z03052 from the Natural Science Foundation of Guangdong Education Bureau.} \subjclass[2000]{34A37, 34K06, 34K11, 34K25} \keywords{Higher order; impulses; oscillation; ODE} \begin{abstract} In this paper, we study the oscillation of solutions to higher order nonlinear ordinary differential equations with impulses. Several criteria for the oscillations of solutions are given. We find some suitable impulse functions such that all solutions are oscillatory under the impulse control. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \allowdisplaybreaks \section{Introduction} There are many publication on the oscillation of solutions to classical second order nonlinear ordinary differential equations; see for example \cite{a1,b1,b2,k1,k2,k3,t1,w1,w2,w3}. There are also some publications on the oscillation of second order ODEs with impulses \cite{c1,h1,l2}, and some on higher order \cite{c2,f1}. In this paper, we study higher order nonlinear ODEs with impulses. Under conditions (A) (B) (C) stated below, we can always find some suitable impulse functions such that all the solutions of the equation become oscillatory under the impulse control. We believe that this oscillation result, under the impulse control, is significant both for the theory and the applications. \section{Main results} We consider the system \begin{equation} \label{e1} \begin{gathered} x^{(2n)}(t)+f(t,x(t))=0, \quad t\geq t_0,\; t\neq t_k,\\ x^{(i)}(t_k^+)=g_{k(i)}(x^{(i)}(t_k)),\quad i=0,1,\dots,2n-1,\; k=1,2\dots,\\ x^{(i)}(t^+_0)=x^{(i)}_0, \end{gathered} \end{equation} where \begin{gather*} x^{(i)}(t_k)=\lim_{h\to 0^{-}}\frac{x^{(i-1)}(t_k+h)-x^{(i-1)}(t_k)}{h}, \\ x^{(i)}(t^+_k)=\lim_{h\to 0^{+}}\frac{x^{(i-1)}(t_k+h)-x^{(i-1)}(t^+_k)}{h}\,, \end{gather*} $ 00$ for $x\neq0$; $\frac{f(t,x)}{\varphi(x)}\geq p(t)$ for $x\neq 0$, where $p(t)$ is positive and continuous on $[t_0,+\infty)$; $x\varphi(x)>0$ for $x\neq 0$; $\varphi'(x)\geq0$. \item[(B)] $g_{k(i)}(x)$ is continuous on $(-\infty,+\infty)$, and there exist positive numbers $a^{(i)}_k,b^{(i)}_k$ such that $$ a^{(i)}_k\leq \frac{g_{k(i)}(x)}{x}\leq b^{(i)}_k,i=0,1,\dots,2n-1. $$ \item[(C)] \begin{equation} \label{e2} \begin{aligned} &(t_1-t_0)+\frac{a^{(i)}_1}{b^{(i-1)}_1}(t_2-t_1)+\frac{a^{(i)}_1a^{(i)}_2} {b^{(i-1)}_1b^{(i-1)}_2}(t_3-t_2)\\ &+\dots+\frac{a^{(i)}_1a^{(i)}_2\dots a^{(i)}_m} {b^{(i-1)}_1b^{(i-1)}_2\dots b^{(i-1)}_m}(t_{m+1}-t_m)+\dots=+\infty, \end{aligned} \end{equation} \end{itemize} \begin{definition} \label{def1} \rm A function $x:[t_0,t_0+\alpha)\to \mathbb{R}$, $t_0>0$, $\alpha>0$ is said to be a solution of \eqref{e1}, if \begin{itemize} \item[(i)] $x^{(i)}(t^+_0)=x^{(i)}_0$, $i=0,1,\dots 2n-1$ \item[(ii)] for $t\in[t_0,t_0+\alpha)$ and $t\neq t_k$, $x(t)$ satisfies $ x^{(2n)}(t) +f(t,x(t))=0$ \item[(iii)] $x^{(i)}(t)$ is left continuous on $t_k\in[t_0,t_0+\alpha)$, and $x^{(i)}(t^+_k)=g_{k(i)}x^{(i)}(t_k)$, $i=0,1,\dots 2n-1$. \end{itemize} \end{definition} \begin{definition} \label{def2} \rm A solution of \eqref{e1} is said to be non-oscillatory if it is eventually positive or eventually negative. Otherwise,this solution is said to be oscillatory. \end{definition} Since \eqref{e1} can be transformed into a first-order impulsive differential system, theorems on the existence of solutions, the uniqueness of solutions and the existence of global solutions can be seen in \cite{l1}. In the following, we always assume the solutions of \eqref{e1} exists on $[t_0,+\infty)$. \begin{lemma} \label{lem1} Let $x(t)$ be a solution of \eqref{e1}, and conditions (A), (B), (C) be satisfied. Suppose that there exists an $i\in\{{1,2,\dots,2n-1\}}$ and some $T\geq t_0$, such that $x^{(i)}(t)>0$ $(<0)$, $x^{(i+1)}(t)\geq0$ $(\leq0)$ for $t\geq T$. Then there exists some $T_1\geq T$, such that $x^{(i-1)}(t)>0$ $(<0)$, for $t\geq T_1$. \end{lemma} \begin{proof} Without loss of generality, let $T=t_0$, $x^{(i)}(t)>0$, $x^{(i+1)}(t)\geq 0$ for $t\geq T$. Assume that for any $t_k>T$, $x^{(i-1)}(t_k)<0$. By $x^{(i+1)}(t)\geq0$, $x^{(i)}(t)>0$, $t\in(t_k,t_{k+1}]$, we have that $x^{(i)}(t)$ is monotonically nondecreasing on $(t_k,t_{k+1}]$. For $t\in(t_1,t_2]$, we have $$ x^{(i)}(t)\geq x^{(i)}(t^+_1) $$ Integrating the above inequality, we have \begin{equation} \label{e3} x^{(i-1)}(t_2)\geq x^{(i-1)}(t^+_1)+x^{(i)}(t^+_1)(t_2-t_1) \end{equation} Similarly, \begin{equation} \label{e4} x^{(i-1)}(t_3)\geq x^{(i-1)}(t^+_2)+x^{(i)}(t^+_2)(t_3-t_2) \end{equation} From $x^{(i)}(t_2)\geq x^{(i)}(t^+_1)$ and \eqref{e3}, \eqref{e4}, we have \begin{align*} &x^{(i-1)}(t_3)\geq x^{(i-1)}(t^+_2)+x^{(i)}(t^+_2)(t_3-t_2)\\ &\geq b^{(i-1)}_2x^{(i-1)}(t_2)+a^{(i)}_2x^{(i)}(t_2)(t_3-t_2)\\ &\geq b^{(i-1)}_2[x^{(i-1)}(t^+_1)+x^{(i)}(t^+_1)(t_2-t_1)]+a^{(i)}_2 x^{(i)}(t_2)(t_3-t_2)\\ &\geq b^{(i-1)}_2[x^{(i-1)}(t^+_1)+x^{(i)}(t^+_1)(t_2-t_1)+\frac {a^{(i)}_2}{b^{(i-1)}_2}x^{(i)}(t^+_1)(t_3-t_2)]\\ \end{align*} Applying induction, we have that for any natural number $m$, \begin{equation} \label{e5} \begin{aligned} x^{(i-1)}(t_m) &\geq b^{(i-1)}_{m-1}\dots b^{(i-1)}_3b^{(i-1)}_2\big\{x^{(i-1)}(t^+_1) +x^{(i)}(t^+_1)[(t_2-t_1)\\ &\quad +\frac {a^{(i)}_2}{b^{(i-1)}_2}(t_3-t_2)+\dots+\frac {a^{(i)}_2a^{(i)}_3 \dots a^{(i)}_{m-1}} {b^{(i-1)}_2b^{(i-1)}_3\dots b^{(i-1)}_{m-1}}(t_m-t_{m-1})]\big\} \end{aligned} \end{equation} By condition (C) and $a^{(i)}_k>0$, $b^{(i-1)}_k>0$, for all sufficiently large $m$, we have $x^{(i-1)}(t_m)>0$. Which is contrary to the assumption. Hence, there exists some $j$ such that $t_j>T$ and $x^{(i-1)}(t_j)\geq0$. Then $$ x^{(i-1)}(t^+_j)\geq a^{(i-1)}_jx^{(i-1)}(t_j)\geq0. $$ Note that $x^{(i)}(t)>0$ yields $x^{(i-1)}(t)$ being monotonically increasing on $(t_j,t_{j+1}]$. For $t\in(t_j,t_{j+1}]$, we have $$ x^{(i-1)}(t)> x^{(i-1)}(t^+_j)\geq0. $$ Especially, $$ x^{(i-1)}(t_{j+1})>x^{(i-1)}(t^+_j)>0. $$ Similarly, for $t\in(t_{j+1},t_{j+2}]$, we have $$ x^{(i-1)}(t)>x^{(i-1)}(t^+_{j+1})\geq a^{(i-1)}_{j+1}x^{(i-1)}(t_{j+1})>0. $$ By induction,for $t\in(t_{j+m-1},t_{j+m}]$, we have $x^{(i-1)}(t)>0$. So for $t\geq t_{j+1}$, we have $$ x^{(i-1)}(t)>0. $$ Summing up the above discussion, there exists some $T_1\geq T$ such that $x^{(i-1)}(t)>0$, $t\geq T_1$. The proof of the other case in this theorem is similar; so we omit it. The proof of Lemma \ref{lem1} is complete. \end{proof} \begin{lemma} \label{lem2} Let $x(t)$ be a solution of \eqref{e1} and conditions (A), (B), (C) be satisfied. Suppose that there exist an $i\in\{1,2,\dots,2n\}$ and some $T\geq t_0$ such that $x(t)>0$, $x^{(i)}(t)\leq0$, for $t\geq T$, and $x^{(i)}(t)$ is not always equal to 0 in $[t,+\infty)$. Then $x^{(i-1)}(t)>0$ for all sufficiently large $t$. \end{lemma} \begin{proof} Without loss of generality, let $T=t_0$. We claim that $x^{(i-1)}(t_k)>0$ for any $t_k\geq T$. If it is not true, then there exists some $t_j\geq T$, such that $x^{(i-1)}(t_j)\leq0$. Since $x^{(i)}(t)\leq0$, $x^{(i-1)}(t)$ is monotonically non-increasing in $(t_k,t_{k+1}]$ for $k\geq j$. Also because $x^{(i)}(t)$ is not always equal to 0 in $[t,+\infty)$, there exists some $t_l\geq t_j$ such that $x^{(i)}(t)$ is not always equal to 0 in $(t_l,t_{l+1}]$. Without loss of generality, we can assume $l=j$, that is, $x^{(i)}(t)$ is not always equal to 0 in $(t_j,t_{j+1}]$. So we have $$ x^{(i-1)}(t_{j+1})0$ ($t\geq T$). Hence, we have $x^{(i-1)}(t_k)>0$ for any $t_k\geq T$. So we have $x^{(i-1)}(t)>0$ for all sufficiently large $t$. The proof of Lemma \ref{lem2} is complete. \end{proof} \begin{lemma} \label{lem3} Let $x(t)$ be a solution of \eqref{e1} and conditions (A), (B), (C) be satisfied. Suppose $T\geq t_0$, $x(t)>0$ for $t\geq T$. Then there exist some $T'\geq T$ and $l\in\{1,3,\dots,2n-1\}$ such that for $t\geq T'$, \begin{equation} \label{e6} \begin{gathered} x^{(i)}(t)>0,\quad i=0,1,\dots,l;\\ (-1)^{i-1}x^{(i)}(t)>0,\quad i=l+1,\dots,2n-1;\\ x^{(2n)}(t)\leq 0. \end{gathered} \end{equation} \end{lemma} \begin{proof} Let $T=t_0$. Since $x(t)>0(t\geq t_0)$, by \eqref{e1} and that $p(t)$ is nonnegative and is not always equal to 0 in any $(t,+\infty)$, we have $$ x^{(2n)}(t)=-f(t,x(t))\leq-p(t)\varphi(x(t))\leq0 $$ and $x^{(2n)}(t)$ is not always equal to 0 in $(t,+\infty)$. By Lemma \ref{lem2}, we have $x^{(2n-1)}(t)>0$. Without loss of generality, let $x^{(2n-1)}(t)>0$ for $t\geq t_0$. So $x^{(2n-2)}(t)>0$ is monotonically nondecreasing on $(t_k,t_{k+1}]$. If for any $t_k$, $x^{(2n-2)}(t_k)<0$, then $x^{(2n-2)}(t)<0(t\geq t_0)$. If there exists some $t_j$ such that $x^{(2n-2)}(t_j)\geq0$, by that $x^{(2n-2)}(t)$ is monotonically increasing and $a^{(2n-2)}_k>0$,we get $x^{(2n-2)}(t) >0$ for $t>t_j$. So there exists some $T_1\geq T$, such that one of the following statements hold \begin{gather} x^{(2n-1)}(t)>0,\quad x^{(2n-2)}(t)>0,\quad\mbox{for } t\geq T_1 \label{A1}\\ x^{(2n-1)}(t)>0, \quad x^{(2n-2)}(t)<0,\quad\mbox{for } t\geq T_1 \label{B1} \end{gather} When \eqref{A1} holds, Lemma \ref{lem1} yields that $x^{(2n-3)}(t)>0$ for all sufficiently large $t$. Using Lemma \ref{lem1} repeatedly, for all sufficiently large $t$,we can conclude that $$ x^{(2n-1)}(t)>0,\quad x^{(2n-2)}(t)>0,\dots ,x'(t)>0,\quad x(t)>0. $$ When \eqref{B1} holds, by Lemma \ref{lem2}, we have $x^{(2n-3)}(t)>0$, for all sufficiently large $t$. Hence,there exists some $T_2\geq T_1$ such that \begin{gather} x^{(2n-3)}(t)>0,\quad x^{(2n-4)}(t)>0,\quad\mbox{for }t\geq T_2 \label{A2} \\ x^{(2n-3)}(t)>0,\quad x^{(2n-4)}(t)<0,\quad\mbox{for }t\geq T_2 \label{B2} \end{gather} Repeating the discussion above, we can get, eventually, that there exist some $T'\geq T$ and $l\in\{1,3,\dots,2n-1\}$, such that for $t\geq T'$, \begin{gather*} x^{(i)}(t)>0,\quad i=0,1,\dots ,l;\\ (-1)^{i-1}x^{(i)}(t)>0,\quad i=l+1,l+2,\dots,2n-1;\\ x^{(2n)}(t)\leq 0. \end{gather*} The proof of Lemma \ref{lem3} is complete. \end{proof} We remark that if $x(t)$ is an eventually negative solution of \eqref{e1}, then there are conclusions similar to Lemma \ref{lem2} and Lemma \ref{lem3}. \begin{theorem} \label{thm1} If conditions (A),(B),(C) hold, $a^{(0)}_k\geq1$ and \begin{equation} \label{e7} \begin{aligned} &\int^{t_1}_{t_0}p(t)dt+\frac{1}{b^{(2n-1)}_1}\int^{t_2}_{t_1}p(t)dt+ \frac{1}{b^{(2n-1)}_1b^{(2n-1)}_2} \int^{t_3}_{t_2} p(t)dt+\dots\\ &+\frac{1}{b^{(2n-1)}_1b^{(2n-1)}_2 \dots b^{(2n-1)}_{m}} \int^{t_{m+1}}_{t_{m}}p(t)dt+\dots=+\infty \end{aligned} \end{equation} then every solution of \eqref{e1} is oscillatory. \end{theorem} \begin{proof} Let $x(t)$ be a non-oscillatory solution of \eqref{e1}. Without loss of generality, let $x(t)>0(t\geq t_0)$, By Lemma \ref{lem3} and \eqref{e1}, there exists $T'\geq t_0$ such that, for $t\geq T'$, we have $$ x^{(2n)}(t)\leq 0,\quad x^{(2n-1)}(t)>0,\quad x'(t)>0,\quad x(t)>0. $$ So $x^{(2n-1)}(t)$ is monotonically non-increasing on $(t_k,t_{k+1}]$ and $x(t)$ is monotonically increasing on $(t_k,t_{k+1}]$. Let $$ u(t)=\frac{x^{(2n-1)}(t)}{\varphi(x(t))}. $$ Then $u(t^+_k)\geq0$ ($k=1,2,\dots$), $u(t)\geq0$ ($t\geq t_0$). Since $\varphi'(x)\geq0$, for $t\neq t_k$, \begin{gather} u'(t)=-\frac{f(t,x(t))}{\varphi(x(t))}-\Big[\frac{x^{(2n-1)}(t)x'(t)} {\varphi^{2}(x(t))}\Big]\varphi'(x(t)) \leq-p(t) \label{e8} \\ u(t^+_k)=\frac{x^{(2n-1)}(t^+_k)}{\varphi(x(t^+_k))} \leq\frac{b^{(2n-1)}_kx^{(2n-1)}(t_k)}{\varphi(a^{(0)}_kx(t_k))} \leq\frac{b^{(2n-1)}_kx^{(2n-1)}(t_k)}{\varphi(x(t_k))} \leq b^{(2n-1)}_ku(t_k) \label{e9} \end{gather} Integrating \eqref{e8} from $t_0$ to $t_1$ we have \begin{gather} u(t_1)\leq u(t^+_0)-\int^{t_1}_{t_0}p(t)dt \,,\label{e10}\\ u(t^+_1)\leq b^{(2n-1)}_1u(t_1)\leq b^{(2n-1)}_1[u(t^+_0) -\int^{t_1}_{t_0}p(t)dt] \,. \label{e11} \end{gather} Similar to the above inequality, we have \begin{equation} \label{e12} \begin{aligned} u(t^+_2)&\leq b^{(2n-1)}_2u(t_2)\\ &\leq b^{(2n-1)}_2[u(t^+_1)-\int^{t_2}_{t_1}p(t)dt] \\ &\leq b^{(2n-1)}_2[b^{(2n-1)}_1u(t^+_0)-b^{(2n-1)}_1\int^{t_1}_{t_0}p(t)dt -\int^{t_2}_{t_1}p(t)dt]\\ &\leq b^{(2n-1)}_1b^{(2n-1)}_2[u(t^+_0)-\int^{t_1}_{t_0}p(t)dt- \frac{1}{b^{(2n-1)}_1}\int^{t_2}_{t_1}p(t)dt] \end{aligned} \end{equation} By induction, for any natural number $m$, we have \begin{equation} \label{e13} \begin{aligned} u(t^+_m) &\leq b^{(2n-1)}_1b^{(2n-1)}_2\dots b^{(2n-1)}_m [u(t^+_0)-\int^{t_1}_{t_0}p(t)dt-\frac{1}{b^{(2n-1)}_1} \int^{t_2}_{t_1}p(t)dt\\ &-\dots-\frac{1} {b^{(2n-1)}_1b^{(2n-1)}_2\dots b^{(2n-1)}_{m-2}}\int^{t_{m-1}}_{t_{m-2}}p(t)dt\\ &-\frac{1}{b^{(2n-1)}_1b^{(2n-1)}_2\dots b^{(2n-1)}_{m-2}b^{(2n-1)}_{m-1}} \int^{t_m}_{t_{m-1}}p(t)dt] \end{aligned} \end{equation} By \eqref{e7} and \eqref{e13}, for all sufficiently large $m$, $u(t^+_m)<0$. This contradicts $u(t^+_m)\geq0$. So every solution of \eqref{e1} is oscillatory. The proof of Theorem \ref{thm1} is complete. \end{proof} \begin{theorem} \label{thm2} If conditions (A), (B), (C) hold, $b_k^{(i)}\leq1$, $a_k^{(0)}\geq1$, $b_k^{(0)}\geq1$ ($i=1,2,\dots,2n-1,\; k=1,2,\dots$) and $\int^{+\infty}t^{2n-1}p(t)dt=+\infty$, then every bounded solution of \eqref{e1} is oscillatory. \end{theorem} \begin{proof} Let $x(t)$ be a non-oscillatory solution of \eqref{e1}. Without loss of generality, let $x(t)>0$ for $t\geq t_0$. By Lemma \ref{lem3}, we can divided \eqref{e6} into two cases:\\ Case (i): If $l=1$, then $x(t)>0$, $x'(t)>0$, $x''(t)<0$, $x'''(t)>0$, $x^{(4)}(t)<0$, \dots, $x^{(2n-1)}(t)>0$, $x^{(2n)}(t)\leq0$.\\ Case (ii): If $l\geq3$, then $x(t)>0$, $x'(t)>0$, $x''(t)>0$, $x'''(t)>0$, \dots, $x^{(l)}(t)>0$, $x^{(l+1)}(t)<0$, \dots, $x^{(2n-1)}(t)>0$, $x^{(2n)}(t)\leq0$.\\ Both cases tells us that $x'(t)>0$, $t\in(t_k,t_{k+1}]$, $k=1,2,\dots.$ So $x(t)$ is monotonically increasing on $(t_k,t_{k+1}]$. Since $a^{(0)}_k\geq1$, $x(t)$ is monotonically increasing on $[t_0,+\infty)$, that is, $x(t)\geq x(t_0)$ for $ t\geq t_0$. By \eqref{e1}, we have \begin{equation} \label{e14} x^{(2n)}(s)=-f(s,x(s))\leq-p(s)\varphi(x(t_0))=-cp(s),\quad s\in(t_k,t_{k+1}] \end{equation} where $c=\varphi(x(t_0))>0$. Multiplying \eqref{e14} by $s^{2n-1}$ and then integrating it from $t_k$ to $t$, we have \begin{equation} \label{e15} \int^{t}_{t_k}s^{2n-1}x^{(2n)}(s)ds<-c\int^{t}_{t_k}s^{2n-1}p(s)ds,\quad t\in(t_k,t_{k+1}]\,. \end{equation} We will consider the following two cases: \noindent{\bf (a)} if the case (i) holds, then for $t\in(t_k,t_{k+1}]$ we have, \begin{align*} &\int^{t}_{t_k}s^{2n-1}x^{(2n)}(s)ds\\ &=\int^{t}_{t_k}s^{2n-1}dx^{(2n-1)}(s)\\ &=t^{2n-1}x^{(2n-1)}(t) -t^{2n-1}_{k}x^{(2n-1)}(t^+_k)-(2n-1)\int^{t}_{t_k}s^{2n-2}x^{(2n-1)}(s)ds\\ &=\dots\\ &=\sum^{2n-1}_{i=0}(-1)^{i+1}\frac{(2n-1)!}{i!}t^{i}x^{(i)}(t)+ \sum^{2n-1}_{i=0}(-1)^{i}\frac{(2n-1)!}{i!}t^{i}_{k}x^{(i)}(t^+_k). \end{align*} Especially, for any natural number $k$, \begin{align*} &\int^{t_{k+1}}_{t_k}s^{2n-1}x^{(2n)}(s)ds\\ &=\sum^{2n-1}_{i=0}(-1)^{i+1}\frac{(2n-1)!}{i!}t^{i}_{k+1}x^{(i)}(t_{k+1}) +\sum^{2n-1}_{i=0}(-1)^{i}\frac{(2n-1)!}{i!}t^{i}_{k}x^{(i)}(t^+_k). \end{align*} No matter if $i$ is odd or even, for $i=1,2,\dots2n-1$, $$ (-1)^{i}(x^{(i)}(t^+_k)-x^{(i)}(t_k)) \geq(-1)^{i}(b^{(i)}_k-1)x^{(i)}(t_k)\geq0. $$ For any natural number $m$ and $t\in(t_m,t_{m+1}]$, we have \begin{align*} &\int^{t}_{t_1}s^{2n-1}x^{(2n)}(s)ds\\ &=\int^{t_2}_{t_1}s^{2n-1}x^{(2n)}(s)ds+\int^{t_3}_{t_2}s^{2n-1}x^{(2n)}(s)ds\\ &\quad+\dots+\int^{t_m}_{t_{m-1}}s^{2n-1}x^{(2n)}(s)ds+\int^{t}_{t_m}s^{2n-1}x^{(2n)}(s)ds\\ &=\sum^{2n-1}_{i=0}(-1)^{i+1}\frac{(2n-1)!}{i!}t^{i}x^{(i)}(t) +\sum^{2n-1}_{i=0}(-1)^{i}\frac{(2n-1)!}{i!}t^{i}_{1}x^{(i)}(t^+_1)\\ &\quad+\sum^{m}_{k=2}\sum^{2n-1}_{i=0}(-1)^{i}\frac{(2n-1)!}{i!}t^{i}_{k} (x^{(i)}(t^+_k)-x^{(i)}(t_k))\\ &\geq-(2n-1)!x(t)+\sum^{2n-1}_{i=0}(-1)^{i}\frac{(2n-1)!}{i!}t^{i}_{1}x^{(i)}(t^+_1)\\ &\quad+\sum^{m}_{k=2}\sum^{2n-1}_{i=0}(-1)^{i}\frac{(2n-1)!}{i!}t^{i}_{k} (b^{(i)}_k-1)x^{(i)}(t_k)\\ &\geq-(2n-1)!x(t)+\sum^{2n-1}_{i=0}(-1)^{i}\frac{(2n-1)!}{i!} t^{i}_{1}x^{(i)}(t^+_1). \end{align*} Combining the inequality above and \eqref{e15}, we have $$ -(2n-1)!x(t)+\sum^{2n-1}_{i=0}(-1)^{i}\frac{(2n-1)!}{i!}t^{i}_{1}x^{(i)} (t^+_1) \leq-c\int^{t}_{t_1}s^{2n-1}p(s)ds. $$ So $x(t)\to +\infty,$ as $t\to +\infty$. This contradicts that $x(t)$ is bounded. \noindent{\bf (b)} If the case (ii) holds, then $x(t)$ is non-negative and strictly increasing on $t\in[t_1,+\infty)$. Hence, for any natural number $m$, we have \begin{gather*} x(t)=x(t^+_m)+\int^{t}_{t_{m}}x'(s)ds,\quad t\in(t_m,t_{m+1}],\\ x(t_m)=x(t^+_{m-1})+\int^{t_m}_{t_{m-1}}x'(s)ds, \\ \dots\\ x(t_2)=x(t^+_1)+\int^{t_2}_{t_1}x'(s)ds \end{gather*} and \begin{equation} \label{e16} x(t)=\sum^{m}_{k=2}(x(t^+_k)-x(t_k))+x(t^+_1)+\sum^{m-1}_{k=1} \int^{t_{k+1}}_{t_k} x'(s)ds+\int^{t}_{t_m}x'(s)ds \end{equation} Since $x''(t)>0,t\in(t_k,t_{k+1}],k\geq1$, we can get \begin{gather*} x'(t)>x'(t^+_1)\geq a^{(1)}_1x'(t_1),\quad t\in(t_1,t_2]\\ x'(t)>x'(t^+_2)\geq a^{(1)}_2x'(t_2)>a^{(1)}_2a^{(1)}_1x'(t_1), \quad t\in(t_2,t_3]\,. \end{gather*} Applying induction, for any natural number $k$, $$ x'(t)>x'(t^+_k)\geq a^{(1)}_{k}a^{(1)}_{k-1}\dots a^{(1)}_1x'(t_1), \quad t\in(t_k,t_{k+1}]\,. $$ Combining \eqref{e16} and $a^{(0)}_k\geq1$, we have $$ x(t)>x'(t_1)\sum^{m-1}_{k=1}a^{(1)}_{k}a^{(1)}_{k-1}\dots a^{(1)}_1(t_{k+1}-t_k),\quad t\in(t_m,t_{m+1}] $$ From the condition (C) and $b^{(0)}_k\geq1$, we have $$ \sum^{+\infty}_{k=1}a^{(1)}_{k}a^{(1)}_{k-1}\dots a^{(1)}_1(t_{k+1}-t_k) =+\infty $$ Then $x(t)\to +\infty$ ($t\to +\infty$), which contradicts that $x(t)$ is bounded. Therefore, every solution of \eqref{e1} is oscillatory. The proof of Theorem \ref{thm2} is complete. \end{proof} \begin{theorem} \label{thm3} If conditions (A), (B), (C) hold, $\prod^{m}_{k=1}a^{(0)}_{k}>b>0$ ($m=1,2,\dots$), $b_k^{(2n-1)}\leq1$, and for any $\delta>0$, \begin{equation} \label{e17} \big|\int^{+\infty}\inf_{\delta\leq|x|<+\infty} f(t,x)\,dt\big|=+\infty \end{equation} then every solution of \eqref{e1} is oscillatory. \end{theorem} \begin{proof} Let $x(t)$ be a non-oscillatory solution of \eqref{e1}. Without loss of generality, let $x(t)>0$, $t\geq t_0$. By Lemma \ref{lem3}, $x'(t)\geq0$, $t\geq t_0$. So $x(t)$ is monotonically nondecreasing on $(t_0,+\infty)$. \begin{gather*} x(t_1)\geq x(t^+_0),x(t_2)\geq x(t^+_1) \geq a^{(0)}_1x(t_1)\geq a^{(0)}_1x(t^+_0),\\ x(t_3)\geq x(t^+_2)\geq a^{(0)}_2x(t_2)\geq a^{(0)}_2a^{(0)}_1x(t^+_0) \end{gather*} By induction, we have $$ x(t_{m+1})\geq x(t^+_{m})\geq a^{(0)}_{m}x(t_{m}) \geq\dots\geq a^{(0)}_1a^{(0)}_2\dots a^{(0)}_{m} x(t^+_0)>bx(t^+_0). $$ We can assume that $x(t)\geq bx(t^+_0)$, $t\in(t_0,+\infty)$. By \eqref{e17}, as $t\to +\infty$, we have $$ \int^{t}_{t_0} f(s,x(s))ds\geq\int^{t}_{t_0}\inf_{bx(t^+_0)\leq|x| <+\infty} f(s,x)ds\to +\infty\,; $$ that is, $\int^{t}_{t_0} f(s,x(s))ds\to +\infty$. Integrating \eqref{e1} from $t_0$ to $t_1$, we have $$ x^{(2n-1)}{(t_1)}+\int^{t_1}_{t_0} f(s,x(s))ds=x^{(2n-1)}{(t_0^+)} $$ Similar to the above formula, for any natural number integrating \eqref{e1} from $t_{k-1}$ to $t_k$, we have $$ x^{(2n-1)}{(t_k)}+\int^{t_k}_{t_{k-1}} f(s,x(s))ds=x^{(2n-1)}{(t_{k-1}^+)}$$ So, we have \begin{gather*} x^{(2n-1)}{(t_1)}+\int^{t_1}_{t_0} f(s,x(s))ds=x^{(2n-1)}{(t_0^+)},\\ x^{(2n-1)}{(t_2)}+\int^{t_2}_{t_1} f(s,x(s))ds=x^{(2n-1)}{(t_1^+)},\\ \dots\\ x^{(2n-1)}{(t_m)}+\int^{t_m}_{t_{m-1}} f(s,x(s))ds=x^{(2n-1)}{(t_{m-1}^+)},\\ x^{(2n-1)}{(t)}+\int^{t}_{t_m} f(s,x(s))ds=x^{(2n-1)}{(t_m^+)}\,. \end{gather*} For $t\in(t_m,t_{m+1}]$, we have $$ x^{(2n-1)}{(t)}+\sum^{m}_{i=1}x^{(2n-1)}{(t_i)}+\int^{t}_{t_0} f(s,x(s))ds =\sum^{m}_{i=0}x^{(2n-1)}{(t_i^+)}. $$ Then $$ x^{(2n-1)}{(t)}+\sum^{m}_{i=1}\big(x^{(2n-1)}{(t_i)} -x^{(2n-1)}{(t_i^+)}\big)+\int^{t}_{t_0} f(s,x(s))ds=x^{(2n-1)}{(t^+_0)}\,. $$ Lemma \ref{lem3} shows that $x^{(2n-1)}(t)>0$ for sufficiently large $t$. Hence, \begin{equation} \label{e18} x^{(2n-1)}{(t)} \leq-\sum^{m}_{i=1}\Big((1-b_k^{(2n-1)})x^{(2n-1)}{(t_i)}\Big) -\int^{t}_{t_0}f(s,x(s))ds+x^{(2n-1)}{(t^+_0)} \end{equation} By condition $b_k^{(2n-1)}\leq1$ and \eqref{e18}, we have $x^{(2n-1)}{(t)}\leq -\int^{t}_{t_0} f(s,x(s))ds+x^{(2n-1)}{(t^+_0)}\to -\infty$ as $t\to +\infty$. So, for all sufficiently large $t$, $x^{(2n-1)}(t)<0$. This contradicts that $x^{(2n-1)}(t)>0$. So every solution of \eqref{e1} is oscillatory. The proof of Theorem \ref{thm3} is complete. \end{proof} \begin{corollary} \label{coro1} Assume the conditions (A), (B), (C) hold, and $a_k^{(0)}\geq1$, $b_k^{(2n-1)}\leq1$. If $\int^{+\infty}p(t)dt=+\infty$, then every solution of \eqref{e1} is oscillatory. \end{corollary} \begin{proof} By $b_k^{(2n-1)}\leq1$, we have \begin{align*} &\int^{t_1}_{t_0}p(t)dt+\frac{1}{b^{(2n-1)}_1}\int^{t_2}_{t_1}p(t)dt+ \frac{1}{b^{(2n-1)}_1b^{(2n-1)}_2}\int^{t_3}_{t_2} p(t)dt+\dots\\ &+\frac{1}{b^{(2n-1)}_1b^{(2n-1)}_2\dots b^{(2n-1)}_{m}} \int^{t_{m+1}}_{t_{m}}p(t)dt\\ &\geq\int^{t_1}_{t_0}p(t)dt+\int^{t_2}_{t_1}p(t)dt+\int^{t_3}_{t_2} p(t)dt +\dots +\int^{t_{m+1}}_{t_{m}}p(t)dt\\ &=\int^{t_{m+1}}_{t_0}p(t)dt\\ \end{align*} and $\int^{t_{m+1}}_{t_0}p(t)dt\to +\infty$ as $m\to +\infty$. Then \eqref{e7} holds. By Theorem \ref{thm1}, every solution of \eqref{e1} is oscillatory. \end{proof} \begin{corollary} \label{coro2} Assume conditions (A), (B), (C) hold, and that there exists a positive number $\alpha>0$, such that $a^{(0)}_k\geq1$, $\frac{1}{b_k^{(2n-1)}}\geq(\frac{t_{k+1}}{t_k})^{\alpha}$. If $\int^{+\infty}t^{\alpha}p(t)dt=+\infty$, then every solution of \eqref{e1} is oscillatory. \end{corollary} \begin{proof} By $\frac{1}{b_k^{(2n-1)}}\geq(\frac{t_{k+1}}{t_k})^{\alpha}$, we have \begin{align*} &\int^{t_1}_{t_0}p(t)dt+\frac{1}{b^{(2n-1)}_1}\int^{t_2}_{t_1}p(t)dt+ \frac{1}{b^{(2n-1)}_1b^{(2n-1)}_2}\int^{t_3}_{t_2} p(t)dt+\dots\\ &+\frac{1}{b^{(2n-1)}_1b^{(2n-1)}_2\dots b^{(2n-1)}_{m}} \int^{t_{m+1}}_{t_{m}}p(t)dt\\ &\geq\frac{1}{b^{(2n-1)}_1}\int^{t_2}_{t_1}p(t)dt+ \frac{1}{b^{(2n-1)}_1b^{(2n-1)}_2}\int^{t_3}_{t_2} p(t)dt+\dots\\ &\quad +\frac{1}{b^{(2n-1)}_1b^{(2n-1)}_2\dots b^{(2n-1)}_{m}} \int^{t_{m+1}}_{t_{m}}p(t)dt\\ &\geq\frac{1}{t^{\alpha}_{1}}[\int^{t_2}_{t_1}t^{\alpha}_{2}p(t)dt +\int^{t_3}_{t_2}t^{\alpha}_{3}p(t)dt +\dots+\int^{t_{m+1}}_{t_{m}}t^{\alpha}_{m+1}p(t)dt]\\ &\geq\frac{1}{t^{\alpha}_{1}}[\int^{t_2}_{t_1}t^{\alpha}p(t)dt +\int^{t_3}_{t_2}t^{\alpha}p(t)dt +\dots+\int^{t_{m+1}}_{t_{m}}t^{\alpha}p(t)dt]\\ &=\frac{1}{t^\alpha_1}\int^{t_{m+1}}_{t_1}t^{\alpha}p(t)dt \end{align*} and $\int^{t_{m+1}}_{t_1}p(t)dt\to +\infty$ as $m\to +\infty$. Then \eqref{e7} holds. By Theorem \ref{thm1}, we every solution of \eqref{e1} is oscillatory. \end{proof} \section{Examples} subsection*{Example 3.1} Consider the equation \begin{equation} \label{e19} \begin{gathered} x^{(2n)}(t) +\frac{1}{4t}x^{3}=0, \quad t\geq \frac{1}{2},\; t\neq k,\; k=1,2,\dots\\ x(k^+)=\frac{k+1}{k}x(k),\quad x^{(i)}(k^+)=x^{(i)}(k),\quad i=1,\dots,2n-1,\\ x(\frac{1}{2})=x_0,x^{(i)}(\frac{1}{2})=x^{(i)}_0,\ \end{gathered} \end{equation} where $a^{(0)}_{k}=b^{(0)}_{k}=\frac{k+1}{k}>1$, $a^{(i)}_{k}=b^{(i)}_{k}=1$, $i=1,2,\dots,2n-1$, $p(t)=\frac{1}{4t}$, $\varphi(x)=x^{3}$, $f(t,x)=\frac{1}{4t}x^{3}$, $t_k=k$, $t_0=\frac{1}{2}$. It is obvious that the conditions (A) and (B) are satisfied. For condition (C),we have: For $i>1$, $a^{(i)}_{k}=b^{(i-1)}_{k}=1$, \begin{align*} &(t_1-t_0)+(t_2-t_1)+(t_3-t_2)+\dots+(t_{m+1}-t_m)+\dots\\ &=\frac{1}{2}+1+\dots+1+\dots=+\infty. \end{align*} For $i=1$, $a^{(1)}_{k}=1$, $b^{(0)}_{k}=\frac{k+1}{k}$, \begin{align*} &(t_1-t_0)+\frac{1}{2}(t_2-t_1)+\frac{1}{3}(t_3-t_2)+\dots +\frac{1}{m+1}(t_{m+1}-t_m)+\dots\\ &=\frac{1}{2}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{m+1}+\dots=+\infty. \end{align*} Therefore, condition $(C)$ holds. Since $b^{(2n-1)}_k=1$, we have \begin{align*} &\int^{t_1}_{t_0}p(t)dt+\frac{1}{b^{(2n-1)}_1}\int^{t_2}_{t_1}p(t)dt+ \frac{1}{b^{(2n-1)}_1b^{(2n-1)}_2} \int^{t_3}_{t_2} p(t)dt+\dots\\ &+\frac{1} {b^{(2n-1)}_1b^{(2n-1)}_2 \dots b^{(2n-1)}_{m}} \int^{t_{m+1}}_{t_{m}}p(t)dt\\ &=\int^{t_1}_{t_0}p(t)dt+\int^{t_2}_{t_1}p(t)dt+\int^{t_3}_{t_2} p(t)dt+\dots +\int^{t_{m+1}}_{t_{m}}p(t)dt\\ &=\int^{t_{m+1}}_{t_0}p(t)dt=\int^{t_{m+1}}_{t_0}\frac{1}{4t}dt \\ &=\frac{1}{4}\ln t|^{t_{m+1}}_{t_0}=\frac{1}{4}(\ln t_{m+1}-\ln t_0) \end{align*} Since $\ln t_{m+1}\to +\infty$ as $m\to +\infty$, we get that the condition of Theorem \ref{thm1} hold. So every solution of \eqref{e19} is oscillatory. \subsection*{Example 3.2} Consider the sub-linear system \begin{equation} \label{e20} \begin{gathered} x^{(2n)}(t) +\frac{1}{t^2}x^{\frac{1}{3}}=0, \quad t\geq \frac{1}{2},\; t\neq k,k=1,2,\dots\,,\\ x(k^+)=x(k), x^{(i)}(k^+)=\frac{k}{k+1}x^{(i)}(k),\quad i=1,\dots,2n-1\,,\\ x(\frac{1}{2})=x_0,\quad x^{(i)}(\frac{1}{2})=x^{(i)}_0, \end{gathered} \end{equation} where $a^{(0)}_{k}=b^{(0)}_{k}=1$, $a^{(i)}_{k}=b^{(i)}_{k}=\frac{k}{k+1}$, $i=1,2,\dots,2n-1$, $p(t)=\frac{1}{t^2}$, $t_k=k$, $\varphi(x)=x^{\frac{1}{3}}$, $f(t,x(t))=\frac{1}{t^2}x^{\frac{1}{3}}(t)$, $t_0=\frac{1}{2}$. It is obvious that the condition (A) and (B) hold. For condition (C), we have: For $i>1$ and $a^{(i)}_{k}=b^{(i-1)}_{k}=\frac{k}{k+1}$, $$ (t_1-t_0)+(t_2-t_1)+(t_3-t_2)+\dots+(t_{m+1}-t_m)+\dots =\frac{1}{2}+1+\dots+1+\dots=+\infty. $$ For $i=1$ and $a^{(1)}_{k}=\frac{k}{k+1},b^{(0)}_{k}=1$, \begin{align*} &(t_1-t_0)+\frac{1}{2}(t_2-t_1)+\frac{1}{3}(t_3-t_2) +\dots+\frac{1}{m+1}(t_{m+1}-t_m)+\dots\\ &=\frac{1}{2}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{m+1}+\dots=+\infty. \end{align*} So, condition $(C)$ holds. Let $\alpha=1$. Then $$ \frac{1}{b^{(2n-1)}_k}=\frac{k+1}{k}\geq \frac{t_{k+1}}{t_k}=\frac{k+1}{k} \int^{+\infty}tp(t)dt=\int^{+\infty}t\frac{1}{t^2}dt =\int^{+\infty}\frac{1}{t}dt=+\infty\,. $$ Therefore, the conditions of Corollary \ref{coro2} are satisfied. Then every solution of $(3.2)$ is oscillatory. \subsection*{Acknowledgements} The authors are grateful to the anonymous referee for his (her) suggestions and comments on the original manuscript. \begin{thebibliography}{99} \bibitem{a1} F. V. Atkinson, \emph{On second-order nonlinear oscillations}, Pacific J. Math., 5 (1955), 643-647. \bibitem{b1} S. Belhohorex, \emph{Oscillation solutions of certain nonlinear differential equations of the second order}, Mat. Fyz. Casopis Sloven. Akad. Vied., 11 (1961), 250-255. \bibitem{b2} G. J. Buther, \emph{Integral averages and the oscillation of second order ordinary diferential equations}, SIAM J. Math. Anal., 11 (1980), 190-200. \bibitem{c1} Chen, Yongshao; Feng, Weizhen; \emph{Oscillation of second order nonliner ODE with impulses}, J. Math. Annal. Appl., 210 (1997), 150-169. \bibitem{c2}Chen, Yongshao; Feng, Weizhen; \emph{Oscillation of higher order liner ODE with impulses}, Journal of South China Normal University (Natural Science), 2003, No.3, 14-19. \bibitem{f1} Feng, Weizhen; \emph{Oscillations of fourth order ODE with Impulses}, Ann. of Diff. Eqs., 19 (2003), no. 2, 136-145. \bibitem{h1} Huang, Chunchao; \emph{Oscillation and nonoscillation for second order liner impulsive differential equation}, J. Math. Annl. Appl., 214(1997), 378-394. \bibitem{k1} I. V. Kamenev, \emph{Oscillation of solutions of second order nonlinear equations with signvariable coefficients}, Differential'nye Uravneniya, 6 (1970), 1718-1712. \bibitem{k2} I. V. Kamenev, \emph{A criterion for the oscillation of solutions of second order ordinary differential equations}, mat. Zametki, 6 (1970), 773-776. \bibitem{k3} I. V. Kamenev, \emph{Some specifically nonlinear oscillation theorem}, Mat. Zametki, 10 (1971), 129-134. \bibitem{l1}V. Lakshmikantham, D. D. Bainov, and P. S. Simeonov. \emph{Theory of Impulsive Differential Equations}, World Scientific, Singapore/New Jersey/London, 1989. \bibitem{l2} Luo, Jiaowan; Jokenath Debnath; \emph{Oscillation of second order nonliner ODE with impulses}, J.Math.Annl.Appl., 240 (1999), 105-114. \bibitem{t1} C. C. Travis, \emph{A note on second order nonlinear oscillation}, Math. Japan. 18 (1973), 261-264. \bibitem{w1} J. W. Wacki and J. S. W. Wong, \emph{Oscillation of solutions of second order nonlinear differential equations}, Pacific J. Math., 24 (1968), 111-117. \bibitem{w2} J. S. W. Wong, \emph{On second order nonlinear oscillation}, Funkcial. Ekavac., 11 (1968), 207-234. \bibitem{w3} J. S. W. Wong, \emph{Oscillation theorems for second order nonlinear differential equations}, Bull. Inst. Math. Acad. Sinica , 3 (1975), 283-309. \end{thebibliography} \end{document} \end{document}