\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 33, 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 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/33\hfil Second-order impulsive DE's] {Asymptotic behavior of second-order impulsive differential equations} \author[H. Liu, Q. Li \hfil EJDE-2011/33\hfilneg] {Haifeng Liu, Qiaoluan Li} \address{Haifeng Liu \newline Department of Science and Technology, Hebei Normal University,\newline Shijiazhuang, 050016, China} \email{liuhf@mail.hebtu.edu.cn} \address{Qiaoluan Li \newline College of Mathematics and Information Science, Hebei Normal University,\newline Shijiazhuang, 050016, China} \email{qll71125@163.com} \thanks{Submitted January 6, 2011. Published February 23, 2011.} \thanks{Supported by grant L2009Z02 from the Key Foundation of Hebei Normal University} \subjclass[2000]{34K25, 34K45} \keywords{Impulsive differential equation; asymptotic behavior; second-order} \begin{abstract} In this article, we study the asymptotic behavior of all solutions of 2-th order nonlinear delay differential equation with impulses. Our main tools are impulsive differential inequalities and the Riccati transformation. We illustrate the results by an example. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Introduction} Consider the impulsive differential equation \begin{gather} \big(r(t)(x'(t))^{\alpha}\big)'+p(t)(x'(t))^{\alpha} +f(t, x(t-\delta))=0, \quad t\ge t_0,\; t\neq t_k, \label{e1.1}\\ x(t_{k}^{+})=J_k(x(t_k)), \quad x'(t_{k}^{+})=I_k(x'(t_k)), \quad k=1,2,3\dots, \label{e1.2} \end{gather} where $\alpha$ is the quotient of positive odd integers. The theory of impulsive differential/difference equations is emerging as an important area of investigation, since it is much richer than the corresponding theory of differential/difference equations without impulsive effects. Moreover, such equations may model several real world phenomena \cite{l1}. There are many papers devoted to the oscillation criteria of differential equations with impulses \cite{h1,l2,l3} and to the asymptotic behavior of all solutions of differential equations without impulses \cite{w1}. Recently, Tang \cite{t1} studied the equation \begin{gather*} (r(t)x'(t))'+p(t)x'(t)+f(t, x(t-\delta))=0, \quad t\neq t_k,\\ x(t_k^+)=J_k(x(t_k)), \quad k=1,2,3\dots,\\ x'(t_{k}^{+})=I_{k}(x'(t_k)), \quad k=1,2,3\dots. \end{gather*} He obtained sufficient conditions of asymptotic behavior of all solutions of the equation. Motivated by \cite{t1}, using impulsive differential inequality and the Riccati transformation, we study the asymptotic behavior of solutions of \eqref{e1.1}, \eqref{e1.2}. \begin{definition} \label{def1} \rm For $\phi\in C([t_0-\delta,t_0], \mathbb{R})$, a function $x:[t_0-\delta, +\infty)\to \mathbb{R}$ is called a solution of \eqref{e1.1}, \eqref{e1.2} satisfying the initial value condition $$x(t)=\phi(t),\quad t\in [t_0-\delta,t_0]$$ if the following conditions are satisfied: \begin{itemize} \item[(i)] $x(t)=\phi(t)$ for $t\in [t_0-\delta, t_0]$, \item[(ii)] $x, x'$ are continuously differentiable for $t>t_0$, $t\neq t_{k}$ ($k=1,2,\dots$) and satisfy \eqref{e1.1}, \item[(iii)] $x(t_k^-)=x(t_k), x'(t_k^-)=x'(t_k)$, $k=1,2,\dots$ and satisfy \eqref{e1.2}. \end{itemize} \end{definition} As is customary, a solution of \eqref{e1.1}, \eqref{e1.2} is said to be non-oscillatory if it is eventually positive or eventually negative. Otherwise, it will be called oscillatory. \section{Main results} In this paper, we assume that the following conditions hold: \begin{itemize} \item[(H1)] $f$ is continuous on $[t_0,+\infty)\times \mathbb{R}$, $xf(t, x)>0$ for $x\neq 0$, and $\frac{f(t,\, x)}{g(x)}\geq h(t)$ for $x\neq 0$, where $g(\gamma x)\geq \gamma g(x)$ for $\gamma>0$, $x'g'(x)>0$, and $h, r'$ are continuous on $[t_0, +\infty)$, $h(t)\geq 0, r(t)>0$. \item[(H2)] $p, J_k, I_k$ are continuous on $\mathbb{R}$ and there exist positive numbers $a_{k}^{*}, a_k, b_{k}^{*}, b_k$ such that $a_{k}^{*}\leq \frac{I_k(x)}{x}\leq a_k, b_{k}^{*}\leq \frac{J_k(x)}{x}\leq b_k$. \item[(H3)] $\lim_{t\to \infty}\int_{t_j}^{t}\prod_{t_j0,t\geq T$. If {\rm (H1)--(H3)} are satisfied, then $x'(t_k)>0$ and $x'(t)>0$ for $t\in (t_k, t_{k+1}]$, where $t_k\geq T$, $k=1,2,\dots$. \end{lemma} \begin{proof} We first prove that $x'(t_k)>0$ for any $t_k\geq T$. If not, there must exist some $j$ such that $x'(t_j)<0$, $t_j\geq T$ and $x'(t_{j}^{+})=I_j(x'(t_j)) \leq a_{j}^{*}x'(t_j)<0$. Let $$x'(t_j)\exp\Big(\int_{t_0}^{t_j}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big) =: \beta <0.$$ From \eqref{e1.1}, it is clear that $$\Big(x'(t)\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big) \Big)'=-\frac{f(t, x(t-\delta))}{\alpha r(t)(x'(t))^{\alpha -1}} \exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big).$$ Since $\alpha$ is the quotient of positive odd integers, $(x'(t))^{\alpha-1}>0$, we obtain $$\Big(x'(t)\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big) \Big)'<0. \label{e2.2}$$ Hence, the function $x'(t)\exp(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds)$ is decreasing on $(t_j, t_{j+1}]$, $$x'(t_{j+1})\exp\Big(\int_{t_0}^{t_{j+1}}\frac{r'(s)+p(s)}{\alpha r(s)} ds\Big) \leq x'(t_{j}^{+})\exp\Big(\int_{t_0}^{t_{j}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big);$$ i.e., $$x'(t_{j+1})\exp\Big(\int_{t_0}^{t_{j+1}}\frac{r'(s)+p(s)}{\alpha r(s)} ds\Big)\leq a_{j}^{*}\beta$$ and \begin{align*} x'(t_{j+2})\exp\Big(\int_{t_0}^{t_{j+2}}\frac{r'(s)+p(s)}{\alpha r(s)} ds\Big) &\leq x'(t_{j+1}^{+})\exp\Big(\int_{t_0}^{t_{j +1}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)\\ &\leq a_{j+1}^{*}a_{j}^{*}\beta. \end{align*} By induction, we obtain $$x'(t_{j+n})\exp\Big(\int_{t_0}^{t_{j+n}}\frac{r'(s)+p(s)}{\alpha r(s)} ds\Big) \leq \prod_{k=0}^{n-1}a_{j+k}^{*}\beta,$$ while for $t\in (t_{j+n}, t_{j+n+1}]$, we have x'(t)\leq \prod_{t_j\leq t_k0$for$t_k\geq T$, one can find that the above inequality contradicts (H3) as$t\to\infty$, therefore,$ x'(t_{k})\geq 0 (t\geq T)$. By condition (H2), we have$x'(t_k^{+})\geq a_{k}^{*}x'(t_k)$for any$t_k\geq T$. Because the function$x'(t)\exp(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds)$is decreasing on$(t_{j+i-1}, t_{j+i}]$, we obtain $x'(t)\exp(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds)>0$ for any$ t\in (t_{j+i-1}, t_{j+i}]$, which implies$x'(t)\geq 0$for$t\geq T$. The proof is complete. \end{proof} \begin{theorem} \label{thm1} If {\rm (H1)-(H3), (H5)} are satisfied, then every solution$x$of \eqref{e1.1}, \eqref{e1.2} satisfies$\liminf_{t\to\infty}|x(t)|=0$. \end{theorem} \begin{proof} Let$x$be a solution of \eqref{e1.1}-\eqref{e1.2}, and by contradiction assume that $\liminf_{t\to\infty}|x(t)|>0.$ Without loss of generality, we may assume that$x(t)>0$on$(t_0, +\infty)$. By Lemma \ref{lem2},$x'(t)>0$for all$t\geq t_0$. We use a Riccati transformation of the form $$V(t)=\frac{r(t)(x'(t))^{\alpha}}{g(x(t-\delta))}.\label{e2.5}$$ Differentiating$V(t), we obtain \begin{align*} V'(t)&= \frac{(r(t)(x'(t))^{\alpha})'g(x(t-\delta))-r(t)(x'(t))^{\alpha}g'(x(t-\delta))x' (t-\delta)}{g^{2}(x(t-\delta))} \\ &=\frac{-p(t)(x'(t))^{\alpha}-f(t, x(t-\delta))}{g(x(t-\delta))} -\frac{x'(t-\delta)g'(x(t-\delta))} {r(t)(x'(t))^{\alpha}}V^2(t)\\ & \leq -p(t)\frac{V(t)}{r(t)}-h(t). \end{align*} From \eqref{e2.5} and (H1), it is clear that \begin{align*} V(t_{k}^{+}) &= \frac {r(t_{k}^{+})(x'(t_{k}^{+}))^{\alpha}} {g(x(t_{k}^{+}-\delta))} \\ &\leq \begin{cases} \frac{r(t_{k})(x'(t_{k}))^{\alpha}a_{k}^{\alpha}} {g(x(t_{k}-\delta))}=a_{k}^{\alpha}V(t_k)=c_kV(t_k), &t_k-\delta \neq t_j, \\[3pt] \frac{r(t_{k})(x'(t_{k}))^{\alpha}a_{k}^{\alpha}}{g(x(t_{j}^{+}))} \leq \frac{a_{k}^{\alpha}}{b_{j}^{*}}V(t_{k})=c_{k}V(t_{k}), &t_k-\delta = t_j, \end{cases} \end{align*} wherec_k's are defined in (H5). Applying Lemma \ref{lem1}, we have \begin{align*} V(t)&\leq \prod_{t_00, $t\geq T$. If (H1), (H2), (H4) are satisfied, then $x'(t_k)>0$ and $x'(t)>0$ for $t\in (t_k, t_{k+1}]$, where $t_k\geq T, k=1,2,\dots$. \end{lemma} \begin{proof} Firstly, for $x(t)>0$, $t\geq T$, we will prove that $x'(t_k)>0$, for any $t_k\geq T$, $T\geq t_0$. If not, there exist some $j$ such that $x'(t_j)<0$, $t_j\geq T$ and $x'(t_{j}^{+})=I_j(x'(t_j)) \leq a_{j}^{*}x'(t_j)<0$. From \eqref{e1.1}, it is clear that $$\Big(x'(t)\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big) \Big)' =-\frac{f(t, x(t-\delta))}{\alpha r(t)(x'(t))^{\alpha -1}} \exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big).$$ Since $\alpha$ is the quotient of positive odd integers, $(x'(t))^{\alpha-1}>0$, we obtain $$\Big(x'(t)\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds \Big)\Big)'<0.$$ Hence, the function $x'(t)\exp\Big(\int_{t_0}^{t}\frac{r'(s)+p(s)}{ \alpha r(s)}ds\Big)$ is decreasing on $(t_j, t_{j+1}]$, $$x'(t_{j+1})\exp\Big(\int_{t_0}^{t_{j+1}}\frac{r'(s)+p(s)} {\alpha r(s)}ds\Big) \leq x'(t_{j}^{+})\exp\Big(\int_{t_0}^{t_{j}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)),$$ i.e., $$x'(t_{j+1})\leq a_{j}^{*}x'(t_j)\exp \Big(-\int_{t_j}^{t_{j+1}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)$$ and $$x'(t_{j+2})\leq a_{j+1}^{*}a_{j}^{*}x'(t_{j})\exp\Big(-\int_{t_j}^{t_{j +2}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big).$$ By induction, we obtain $$x'(t_{j+n}) \leq \prod_{k=0}^{n-1}a_{j+k}^{*}x'(t_j) \exp\Big(-\int_{t_j}^{t_{j+n}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big).$$ Because the function $x'(t)\exp(\int_{t_0}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds)$ is decreasing on $(t_j, t_{j+1}]$, we have $$x'(t)\leq a_{j}^{*}x'(t_j) \exp\Big(-\int_{t_j}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big), \quad t\in (t_j,t_{j+1}].\label{e2.6}$$ Integrating \eqref{e2.6} from $m$ to $t$, we have $$x(t)\leq x(m)+a_{j}^{*}x'(t_j)\int_{m}^{t} \exp\Big(-\int_{t_j}^{u}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)du, \quad t_j0\,(t_k\geq T), we find that the above inequality contradicts condition (H4), therefore x'(t_k)\geq 0 for t\geq T. Further, for t\in (t_j,t_{j+1}], we obtain$$ x'(t)\exp\Big(\int_{t_{0}}^{t}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big) \geq x'(t_{j+1})\exp \Big(\int_{t_{0}}^{t_{j+1}}\frac{r'(s)+p(s)}{\alpha r(s)}ds\Big)>0,  which implies $x'(t)>0$ for $t\geq T$. This completes the proof. \end{proof} Using Lemma \ref{lem3}, we have the following Theorem. \begin{theorem} \label{thm2} If {\rm (H1), (H2), (H4), (H5)} are satisfied, then every solution $x$ of \eqref{e1.1}, \eqref{e1.2} satisfies $\liminf_{t\to \infty}|x(t)|=0$. \end{theorem} \subsection*{Example} Consider \begin{gather*} \Big(t(x'(t))^{3}\Big)'-\big(x'(t)\big)^3 +\frac{1}{t^2}x(t-\frac{1}{3})=0, \quad t\neq k, \; t\geq \frac{1}{2},\\ x'(k^{+})=\frac{k}{k+1}x'(k),\quad x(k^{+})=x(k), \quad k=1,2,\dots. \end{gather*} Comparing with \eqref{e1.1}, \eqref{e1.2}, we see that $r(t)=t$, $p(t)=-1$, $\alpha=3$, $\delta=1/3$, $t_{k+1}-t_k>1/3$ and $a_{k}=a_{k}^{*}=k/(k+1)$, $b_k=b_{k}^{*}=1$. Obviously (H1), (H2) are satisfied, \begin{align*} &\lim_{t\to \infty}\int_{t_j}^{t}\prod_{t_j(j+1)\lim_{t \to \infty}\int_{t_{j}}^{t}\frac{ds}{s+1}=+\infty, \end{align*} and \begin{align*} &\lim_{t\to \infty}\int_{t_0}^{t}\prod_{t_0 \frac{1}{2}\lim_{t\to\infty}\int_{t_0}^{t}ds=+\infty. \end{align*} So (H3) and (H5) are satisfied. By Theorem \ref{thm1}, it is clear that every solution of this equation satisfies $\liminf_{t\to\infty}|x(t)|=0$. \begin{thebibliography}{00} \bibitem{c1} C. Chan, L. Ke; \emph{Remarks on impulsive quenching problems}, Proc. Dyn. Sys. Appl., 1, 1994, 59-62. \bibitem{h1} M. Huang, W. Feng; \emph{Oscillation criteria for impulsive dynamic equations on time scales}, Electronic J. Differ. Equa., 169, 2007, 1-9. \bibitem{j1} J. Jiao, L. Chen and L. Li; \emph{Asymptotic behavior of solutions of second-order nonlinear impulsive differential equations}, J. Math. Anal. Appl., 337, 2008, 458-463. \bibitem{l1} V. Lakshmikantham, D. Bainov, P. 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