\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{ Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 169, pp. 1--9.\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 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/169\hfil Oscillation criteria] {Oscillation criteria for impulsive dynamic equations on time scales} \author[M. Huang, W. Feng\hfil EJDE-2007/169\hfilneg] {Mugen Huang, Weizhen Feng} \address{Mugen Huang \newline Institute of Mathematics and Information technology, Hanshan Normal University, Chaozhou 521041, China} \email{huangmugen@yahoo.cn} \address{Weizhen Feng \newline School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China} \email{wsy@scnu.edu.cn} \thanks{Submitted September 2, 2007. Published December 3, 2007.} \subjclass[2000]{34A37, 34A60, 39A12, 34B37, 34K25} \keywords{Oscillation; dynamic equations; time scales; impulsive; inequality} \begin{abstract} Oscillation criteria for impulsive dynamic equations on time scales are obtained via impulsive inequality. An example is given to show that the impulses play a dominant part in the oscillations of dynamic equations on time scales. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} In this paper, we are interested in obtaining oscillation criteria for solutions of the second-order nonlinear impulsive dynamic equation on time scales, $$\begin{gathered} y^{\Delta \Delta}(t)+f(t, y^{\sigma}(t))=0, \quad t\in \mathbb{J}_{\mathbb{T}}:=[0, \infty)\cap \mathbb{T},\; t\neq t_k,\; k=1,2,\dots,\\ y(t^{+}_k)=g_k(y(t^{-}_k)),\quad y^{\Delta}(t^{+}_k)=h_k(y^{\Delta}(t^{-}_k)), \quad k=1,2,\dots,\\ y(t^{+}_0)=y_0, \quad y^{\Delta}(t^{+}_0)=y^{\Delta}_0, \end{gathered} \label{e1.1}$$ where $\mathbb{T}$ is a unbounded-above time scale , with $0\in \mathbb{T}$, $t_k\in \mathbb{T}, 0\leq t_00$ ($x\neq 0$) and $f(t, x)/\varphi(x) \geq p(t)$ ($x\neq 0$), where $p(t)\in C_{rd}(\mathbb{T}, \mathbb{R}_+)$ and $x\varphi(x)>0$ ($x\neq 0$), $\varphi'(x)\geq 0$. \item[(H2)] $g_k, h_k\in C(\mathbb{R}, \mathbb{R})$ and there exist positive constants $a_k, a^{*}_k, b_k, b^{*}_k$ such that $$a^{*}_k\leq \frac{g_k(x)}{x}\leq a_k, \quad b^{*}_k\leq \frac{h_k(x)}{x}\leq b_k.$$ \end{itemize} We note that the theory of dynamic equations on time scales are an adequate mathematical apparatus for the simulation of processes and phenomena observed in biotechnology, chemical technology, economic, neural networks, physics, social sciences etc. For further applications and questions concerning solutions of dynamic equations on time scales, see \cite{Aulbach,Bohner2, Bohner3} Recently, impulsive dynamic equations on time scales have been investigated by Agarwal et al. \cite{Agarwal2}, Belarbi et al. \cite{Belarbi}, Benchohra et al. \cite{Benchohra1, Benchohra2, Benchohra3, Benchohra4}, Chang et al. \cite{Chang} and so forth. In \cite{Benchohra4}, Benchohra et al. considered the existence of extremal solutions for a class of second order impulsive dynamic equations on time scales, we can see that the existence of global solutions can be guaranteed by some simple conditions. Based on the oscillatory behavior of the impulsive dynamic equations on time scales, Benchohra et al. \cite{Benchohra1} discuss the existence of oscillatory and nonoscillatory solutions by lower and upper solutions method for the first order impulsive dynamic equations on certain time scales $$\begin{gathered} y^{\Delta}(t)=f(t,y(t)), \quad t\in \mathbb{J}_{\mathbb{T}}:=[0,\infty)\bigcap \mathbb{T},\; t\neq t_k,\; k=1,\dots,\\ y(t^{+}_k)=I_k(y(t^{-}_k)),\quad k=1,\dots. \end{gathered} \label{e1.3}$$ On the other hand, Huang et al. \cite{Huang} considered the second order nonlinear impulsive dynamic equations on time scales $$\begin{gathered} y^{\Delta \Delta}(t)+f(t, y^{\sigma}(t))=0, \quad t\in \mathbb{J}_{\mathbb{T}}:=[0, \infty)\cap \mathbb{T},\; t\neq t_k,\; k=1,2,\dots,\\ y(t^{+}_k)=g_k(y(t^{-}_k)), y^{\Delta}(t^{+}_k)=h_k(y^{\Delta}(t^{-}_k)), \quad k=1,2,\dots,\\ y(t^{+}_0)=y_0, \quad y^{\Delta}(t^{+}_0)=y^{\Delta}_0, \end{gathered} \label{e1.4}$$ extend the well-known results of Chen et al. \cite{Chen} for the impulsive differential equations to \eqref{e1.4}. Motivated by the ideas in \cite{Peng}, we establish the sufficient conditions for the oscillation of all solutions of \eqref{e1.1}, which utilize Riccati transformation techniques and impulsive inequality. Those results extend some well-known impulsive inequality on differential equations to impulsive dynamic equations. Our method is different from most existing ones. An example is given to show that though a dynamic equation on time scales is nonoscillatory, it may become oscillatory if some impulses are added to it. That is, in some cases, impulses play a dominating part in oscillations of dynamic equations on time scales. For the remainder of the paper, we assume that, for each $k=1, 2, \dots,$ the points of impulses $t_k$ are right dense (rd for short). In order to define the solutions of the problem \eqref{e1.1}, we introduce the two spaces: \begin{align*} AC^{i}&=\{y: \mathbb{J}_{\mathbb{T}}\to \mathbb{R} \text{ which is $i$-times $\Delta$-differentiable, and its $i$-th }\\ &\quad \text{delta-derivative $y^{\Delta^{(i)}}$ is absolutely continuous}\}; \\ PC&=\{y: \mathbb{J}_{\mathbb{T}}\to \mathbb{R} \text{ which is rd-continuous expect at $t_k$, for which }\\ &\quad\text{$y(t^{-}_k),y(t^{+}_k), y^{\Delta}(t^{-}_k), y^{\Delta}(t^{+}_k)$ exist with $y(t^{-}_k)=y(t_k)$, $y^{\Delta}(t^{-}_k)=y^{\Delta}(t_k)$}\}. \end{align*} A function $y\in PC\bigcap AC^{2}(\mathbb{J}_{\mathbb{T}}\backslash \{t_1, \dots\}, \mathbb{R})$ is said to be a solution of \eqref{e1.1}, if it satisfies $y^{\Delta \Delta}(t)+f(t, y^{\sigma}(t))=0$ a.e. on $\mathbb{J}_{\mathbb{T}}\backslash \{t_k\}, k=1, 2, \dots$, and for each $k=1, 2, \dots, y$ satisfies the impulsive condition $y(t^{+}_k)=g_k(y(t_k)), y^{\Delta}(t^{+}_k)=h_k(y^{\Delta}(t_k))$ and the initial conditions $y(t^{+}_0)=y_0, y^{\Delta}(t^{+}_0)=y^{\Delta}_0$. A solution $y$ of \eqref{e1.1} is called oscillatory if it is neither eventually positive nor eventually negative; otherwise it is called nonoscillatory. Equation \eqref{e1.1} is called oscillatory if all solutions are oscillatory. \section{Preliminary Results} We will briefly recall some basic definitions and facts from the time scales calculus that we will use in the sequel. For more details see \cite{Agarwal1, Bohner2, Bohner3}. On any time scale $\mathbb{T}$, we define the forward and backward jump operators by $$\sigma(t)=\inf\{s\in \mathbb{T}, {s>t}\}, \quad \rho(t)=\sup\{s\in \mathbb{T}: st. A nonminimal element t\in \mathbb{T} is said to be left-dense if \rho(t)=t and left-scattered if \rho(t)0, there exists a neighborhood \textbf{U} of t satisfying |[f(\sigma(t))-f(s)]-b[\sigma(t)-s]|\leq \varepsilon |\sigma(t)-s|, for all s\in \textbf{U}. We say that f is delta differentiable (or in short: differentiable) on \mathbb{T} provided f^{\Delta}(t) exist for all t\in \mathbb{T}. A function f: \mathbb{T}\to \mathbb{R} is called rd-continuous provided it is continuous at right-dense points in \mathbb{T} and its left-sided limits exist (finite) at left-dense points in \mathbb{T}. The set of rd-continuous functions f: \mathbb{T}\to \mathbb{R} will be denoted by C_{rd}(\mathbb{T}, \mathbb{R}). The derivative and forward jump operator \sigma are related by the formula $$f(\sigma(t))=f(t)+\mu(t)f^{\Delta}(t). \label{e2.1}$$ Let f be a differentiable function on [a,b]. Then f is increasing, decreasing, nondecreasing and nonincreasing on [a, b] if f^{\Delta}>0, f^{\Delta}<0, f^{\Delta}\geq 0 and f^{\Delta}\leq 0 for all t\in [a, b), respectively. We will use the following product and quotient rules for derivative of two differentiable functions f and g: \begin{gather} (fg)^{\Delta}=f^{\Delta}g+f^{\sigma}g^{\Delta}=fg^{\Delta}+f^{\Delta}g^{\sigma}, \label{e2.2} \\ (\frac{f}{g})^{\Delta}=\frac{f^{\Delta}g-fg^{\Delta}}{gg^{\sigma}}, \label{e2.3} \end{gather} where f^{\sigma}=f\circ \sigma, gg^{\sigma}\neq 0. The integration by parts formula reads $$\int^{b}_{a}f^{\Delta}(t)g(t)\Delta t =f(t)g(t)|^{b}_{a}-\int^{b}_{a}f^{\sigma}(t)g^{\Delta}(t)\Delta t. \label{e2.4}$$ Chain Rule: Assume g: \mathbb{T}\to \mathbb{R} is \Delta-differentiable on \mathbb{T} and f: \mathbb{R}\to \mathbb{R} is continuously differentiable. Then f\circ g: \mathbb{T}\to \mathbb{R} is \Delta-differentiable and satisfies $$(f\circ g)^{\Delta}(t)=\{\int^{1}_{0}f'(g(t) +h\mu (t)g^{\Delta}(t))dh\}g^{\Delta}(t). \label{e2.5}$$ A function p: \mathbb{T}\to \mathbb{R} is called regressive if for all t\in \mathbb{T}$$ 1+\mu(t)p(t)\neq 0. $$The set of all rd-continuous function f which satisfy 1+\mu(t)p(t)>0 for all t\in \mathbb{T} will be denoted by \mathcal {R}^{+}. The generalized exponential function e_{p} is defined by$$ e_{p}(t, s)=\exp\big\{\int^{t}_{s}\xi_{\mu(\tau)}(p(\tau))\Delta \tau\big\}, $$with \xi_h(z)=\log(1+hz)/h if h\neq 0 and \xi_h(z)=z if h=0. \begin{lemma}[5, p. 255] \label{lem1} Let y, f\in C_{rd} and p\in \mathcal {R}^{+}. Then$$ y^{\Delta}(t)\leq p(t)y(t)+f(t), $$implies that for all t\in \mathbb{T},$$ y(t)\leq y(t_0) e_{p}(t, t_0)+\int^{t}_{t_0}e_{p}(t, \sigma(s))f(s)\Delta s\,. \end{lemma} \section{Main results} Next, we prove some lemmas, which will be useful for establishing oscillation criteria for \eqref{e1.1}. \begin{lemma}\label{lem2} Assume that m\in PC^{1}[\mathbb{T}, \mathbb{R}] and $$\begin{gathered} m^{\Delta}(t)\leq p(t)m(t)+q(t), \quad t\in \mathbb{J}_{\mathbb{T}}:=[0, \infty)\cap \mathbb{T},\; t\neq t_k,\; k=1,2,\dots,\\ m(t^{+}_k)\leq d_k m(t^{-}_k)+b_k, \quad k=1,2,\dots,\\ \end{gathered} \label{e3.1}$$ then for t\geq t_0, \begin{aligned} m(t) &\leq m(t_0)\prod_{t_01. Then for t\in (t_n, t_{n+1}]_{\mathbb{T}}, it follows from \eqref{e3.1} and Lemma \ref{lem1}, we get m(t)\leq m(t^{+}_n)e_{p}(t, t_n)+\int^{t}_{t_n}e_{p}(t, \sigma(s))q(s)\Delta s\,. Using \eqref{e3.1}, we obtain, from \eqref{e3.2}, \begin{align*} m(t)&\leq [d_n m(t^{-}_n)+b_n]e_{p}(t, t_n) +\int^{t}_{t_n}e_{p}(t, \sigma(s))q(s)\Delta s\\ &\leq d_n e_{p}(t, t_n)\Big[m(t_0)\prod_{t_00, t\geq t'_0\geq t_0 is a nonoscillatory solution of \eqref{e1.1}. If \begin{itemize} \item[(H3)] \int^{\infty}_{t_j}\prod_{t_j0). From \eqref{e1.1} and (H1), for t\in(t_{j+i-1}, t_{j+i}]_{\mathbb{T}}, i=1, 2, \dots, we obtain y^{\Delta \Delta}(t)=-f(t, y^{\sigma}(t))\leq -p(t)\varphi(y^{\sigma}(t))\leq 0; i.e., y^{\Delta}(t) is nonincreasing in (t_{j+i-1}, t_{j+i}]_{\mathbb{T}}, i=1, 2, \dots, then $$\begin{gathered} y^{\Delta}(t^{-}_{j+1})\leq y^{\Delta}(t^{+}_j)=-\alpha<0,\\ y^{\Delta}(t^{-}_{j+2})\leq y^{\Delta}(t^{+}_{j+1})=h_{j+1}\left(y^{\Delta}(t^{-}_{j+1})\right) \leq b^{*}_{j+1}y^{\Delta}(t^{-}_{j+1}) \leq -b^{*}_{j+1}\alpha<0. \end{gathered} \label{e3.3}$$ By induction, we obtain y^{\Delta}(t)\leq -\alpha \prod_{t_jt_j \begin{aligned} y(t)&\leq y(t^{+}_j)\prod_{t_j0, one can find that \eqref{e3.5} contradicts (H3) as t\to \infty. Therefore, y^{\Delta}(t^{-}_k)\geq 0 (t_k\geq t'_0). By condition (H2), we obtain, for any t_k\geq t'_0, y^{\Delta}(t^{+}_k)\geq b^{*}_k y^{\Delta}(t^{-}_k)\geq 0. Since y^{\Delta}(t) is decreasing in (t_k, t_{k+1}]_{\mathbb{T}}, t_k\geq t'_0, we have y^{\Delta}(t)\geq y^{\Delta}(t^{-}_k)\geq 0, t\in (t_k, t_{k+1}]_{\mathbb{T}}, t_k\geq t'_0. The proof of Lemma \ref{lem3} is complete. \end{proof} We remark that when y is eventually negative, under the hypothesis (H1)-(H3), it can be proved similarly that y^{\Delta}(t^{+}_k)\leq 0 and for t\in (t_k,t_{k+1}]_{\mathbb{T}}, y^{\Delta}(t)\leq 0 for t_k\geq t'_0\geq t_0. \begin{theorem}\label{th1} Suppose that {\rm (H1)-(H3)} hold and there exists a positive integer k_0 such that a^{*}_k\geq 1 for k\geq k_0. If \int^{\infty}_{t_0}\prod_{t_00, t\geq t_0 and k_0=1. From lemma \ref{lem3}, we have y^{\Delta}(t)\geq 0, t\in (t_k, t_{k+1}]_{\mathbb{T}}, k=1, 2, \dots. Let $$w(t)=\frac{y^{\Delta}(t)}{\varphi(y(t))}\,. \label{e3.7}$$ Then w(t^{+}_k)\geq 0, k=1, 2, \dots, and w(t)>0, t\geq t_0. Using (H1) and \eqref{e1.1}, when t\neq t_k, \begin{aligned} w^{\Delta}(t) &=-\frac{f(t, y^{\sigma}(t))}{\varphi(y^{\sigma}(t))} -\frac{y^{\Delta}(t)}{\varphi(y(t))\varphi(y^{\sigma}(t))} \int^1_0\varphi'\left(y(t)+h\mu(t)y^{\Delta}(t)\right)dh y^{\Delta}(t)\\ &\leq -p(t)-\frac{\varphi(y(t))}{\varphi(y^{\sigma}(t))} \Big(\frac{y^{\Delta}(t)}{\varphi(y(t))}\Big)^2 \int^1_0\varphi'\left(y(t)+h\mu(t)y^{\Delta}(t)\right)dh\\ &\leq -p(t). \end{aligned} \label{e3.8} Since \varphi'(y(t))\geq 0 and \varphi(y(t))>0, from (H2) and a^{*}_k\geq 1, we obtain $$w(t^{+}_k)=\frac{y^{\Delta}(t^{+}_k)}{\varphi(y(t^{+}_k))} \leq \frac{b_ky^{\Delta}(t^{-}_k)}{\varphi(a^{*}_k y(t^{-}_k))} \leq \frac{b_ky^{\Delta}(t^{-}_k)}{\varphi(y(t^{-}_k))} =b_k w(t^{-}_k), \quad k=1, 2, \dots. \label{e3.9}$$ Applying Lemma \ref{lem2}, we obtain from \eqref{e3.8} and \eqref{e3.9}, \begin{aligned} w(t)&\leq w(t_0)\prod_{t_00. If \int^{\infty}_{t_0}\prod_{t_00, t\geq t_0 be a nonoscillatory solution of \eqref{e1.1}, Lemma \ref{lem3} yields y^{\Delta}(t)\geq 0, t\geq t_0, define w(t) as in \eqref{e3.7}, we get w(t)\geq 0, t\geq t_0, w(t^{+}_k)\geq 0, k=1, 2, \dots, \eqref{e3.8} holds for t\neq t_k and $$w(t^{+}_k)=\frac{y^{\Delta}(t^{+}_k)}{\varphi(y(t^{+}_k))} \leq \frac{b_ky^{\Delta}(t^{-}_k)}{\varphi(a^{*}_ky(t^{-}_k))} \leq \frac{b_ky^{\Delta}(t^{-}_k)}{\varphi(a^{*}_k)\varphi(y(t^{-}_k))} =\frac{b_k}{\varphi(a^{*}_k)}w(t^{-}_k). \label{e3.12}$$ Using Lemma \ref{lem2}, we get from \eqref{e3.8} and \eqref{e3.12} \begin{align*} w(t)&\leq w(t_0)\prod_{t_00 such that $$a^{*}_k\geq 1, \quad \frac{1}{b_k}\geq \Big(\frac{t_{k+1}}{t_k}\Big)^{\alpha},\quad \text{for } k\geq k_0. \label{e3.13}$$ If $$\int^{\infty}t^{\alpha}p(t)\Delta t=\infty. \label{e3.14}$$ Then \eqref{e1.1} is oscillatory. \end{corollary} \begin{proof} Without loss of generality let k_0=1. Then \eqref{e3.6} yields \begin{aligned} &\int^{t}_{t_0}\prod_{t_00. Suppose there exist a positive integer k_0 and a constant \alpha>0 such that \frac{\varphi(a^{*}_k)}{b_k}\geq \Big(\frac{t_{k+1}}{t_k}\Big)^{\alpha}, \quad \text{for } k\geq k_0. $$If \int^{\infty}t^{\alpha}p(t)\Delta t=\infty, then \eqref{e1.1} is oscillatory. \end{corollary} The above corollary can be deduced from Theorem \ref{th2}. Its proof is similar to that of Corollary \ref{cor2}; so we omit it. \section{Example} Consider the second-order impulsive dynamic equation $$\begin{gathered} y^{\Delta \Delta}(t)+\frac{1}{t\sigma^2(t)}y^{\gamma}(\sigma(t))=0, \quad t\geq 1,\; t\neq k,\; k=1, 2, \dots,\\ y(k^{+})=\frac{k+1}{k}y(k^{-}),\quad y^{\Delta}(k^{+})=y^{\Delta}(k^{-}), \quad k=1, 2, \dots,\\ y(1)=y_0, \quad y^{\Delta}(1)=y^{\Delta}_0. \end{gathered} \label{e4.1}$$ where \gamma\geq 3 and \mu(t)\leq ct, where c is a positive constant. Since a_k=a^{*}_k=(k+1)/k, b_k=b^{*}_k=1, p(t)=1/(t\sigma^2(t)), t_k=k and \varphi(y)=y^{\gamma}. It is easy to see that (H1)-(H3) hold. Let k_0=1, \alpha=3, hence$$ \frac{\varphi(a^{*}_k)}{b_k}=(k+1)/ k^{\gamma} =\big(\frac{t_{k+1}}{t_k}\big)^{\gamma} \geq \Big(\frac{t_{k+1}}{t_k}\Big)^{3}, $$and$$ \int^{\infty}t^{\alpha}p(t)\Delta t=\int^{\infty}t^3\frac{1}{t\sigma^2(t)}\Delta t =\int^{\infty}\Big(\frac{t}{\sigma(t)}\Big)^{2}\Delta t. $$Since \mu(t)\leq ct, we get$$ \frac{t}{\sigma(t)}=\frac{t}{t+\mu(t)}\geq \frac{1}{1+c}, $$hence$$ \int^{\infty}\Big(\frac{t}{\sigma(t)}\Big)^{2}\Delta t \geq \frac{1}{(1+c)^2}\int^{\infty}\Delta t=\infty.  By Corollary \ref{cor3}, we obtain that \eqref{e4.1} is oscillatory. But by \cite{Bohner1} we know that the dynamic equation $y^{\Delta \Delta}(t)+\frac{1}{t\sigma^2(t)}y^{\gamma}(\sigma(t))=0$ is nonoscillatory. 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