\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 24, pp. 1--13.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/24\hfil Nonlinear dynamic systems] {Oscillation and nonoscillation criteria for two-dimensional time-scale systems of first-order nonlinear dynamic equations} \author[D. R. Anderson\hfil EJDE-2009/24\hfilneg] {Douglas R. Anderson} \address{Douglas R. Anderson \newline Concordia College, Department of Mathematics and Computer Science, Moorhead, MN 56562, USA} \email{andersod@cord.edu} \thanks{Submitted January 20, 2009. Published January 29, 2009.} \subjclass[2000]{34B10, 39A10} \keywords{Nonoscillation; nonlinear system; time scales} \begin{abstract} Oscillation criteria for two-dimensional difference and differential systems of first-order linear difference equations are generalized and extended to nonlinear dynamic equations on arbitrary time scales. This unifies and extends under one theory previous linear results from discrete and continuous systems. An example is given illustrating that a key theorem is sharp on all time scales. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \section{prelude} \label{prelude} Jiang and Tang \cite{jt} establish sufficient conditions for the oscillation of the linear two-dimensional difference system \begin{equation}\label{diffsys} \Delta x_n = p_n y_n, \quad \Delta y_{n-1} = -q_n x_n, \quad n\in\mathbb{Z}, \end{equation} where $\{p_n\}$, $\{q_n\}$ are nonnegative real sequences and $\Delta$ is the forward difference operator given via $\Delta x_n=x_{n+1}-x_n$; see also Li \cite{li}. The system \eqref{diffsys} may be viewed as a discrete analogue of the differential system \begin{equation}\label{ctssys} x'(t) = p(t) y(t), \quad y'(t) = -q(t) x(t), \quad t\in\mathbb{R}, \end{equation} investigated by Lomtatidze and Partsvania \cite{lp}. Oscillation questions in difference and differential equations are an interesting and important area of study in modern mathematics. Furthermore, within the past two decades, these two related but distinct areas have begun to be combined under a powerful, more robust and general theory titled dynamic equations on time scales, a theory introduced by Hilger \cite{hilger}. We wish to generalize \eqref{diffsys} and \eqref{ctssys} to the nonlinear time-scale system of the form \begin{equation} \label{system} x^\Delta(t) = p(t)f\big(y(t)\big), \quad y^\Delta(t) = -q(t)g\big(x(t)\big), \quad t\in \mathbb{T}, \end{equation} where $\mathbb{T}$ is an arbitrary time scale (any nonempty closed set of real numbers) unbounded above, with the special cases of $\mathbb{T} =\mathbb{Z}$ and $\mathbb{T} =\mathbb{R}$ yielding systems related to \eqref{diffsys} and \eqref{ctssys}, respectively, as important corollaries. In this general time-scale setting, $\Delta$ represents the delta (or Hilger) derivative \cite[Definition 1.10]{bp1}, \[ z^\Delta(t):=\lim_{s\to t}\frac{z(\sigma(t))-z(s)}{\sigma(t)-s} =\lim_{s\to t}\frac{z^\sigma(t)-z(s)}{\sigma(t)-s}, \] where $\sigma(t):=\inf\{s\in\mathbb{T} : s > t\}$ is the forward jump operator, $\mu(t):=\sigma(t)-t$ is the forward graininess function, and $z\circ\sigma$ is abbreviated as $z^\sigma$. In particular, if $\mathbb{T} =\mathbb{R}$, then $\sigma(t)=t$ and $x^\Delta=x'$, while if $\mathbb{T} =h\mathbb{Z}$ for any $h>0$, then $\sigma(t)=t+h$ and \[ x^\Delta(t)=\frac{x(t+h)-x(t)}{h}. \] A function $f:\mathbb{T}\to\mathbb{R}$ is right-dense continuous provided it is continuous at each right-dense point $t\in\mathbb{T}$ (a point where $\sigma(t)=t$) and has a left-sided limit at each left-dense point $t\in\mathbb{T}$. The set of right-dense continuous functions on $\mathbb{T}$ is denoted by $\mathop{\rm C_{rd}}(\mathbb{T})$. It can be shown that any right-dense continuous function $f$ has an antiderivative (a function $\Phi:\mathbb{T} \to\mathbb{R}$ with the property $\Phi^\Delta(t)=f(t)$ for all $t\in\mathbb{T}$). Then the Cauchy delta integral of $f$ is defined by \[ \int_{t_0}^{t_1} f(t)\Delta t = \Phi(t_1)-\Phi(t_0), \] where $\Phi$ is an antiderivative of $f$ on $\mathbb{T}$. For example, if $\mathbb{T} =\mathbb{Z}$, then \[ \int_{t_0}^{t_1} f(t)\Delta t=\sum_{t=t_0}^{t_1-1}f(t), \] and if $\mathbb{T}=\mathbb{R}$, then \[ \int_{t_0}^{t_1} f(t)\Delta t = \int_{t_0}^{t_1}f(t)dt. \] Throughout we assume that $t_00$ on $[t_0,\infty)_\mathbb{T}$, and $q:\mathbb{T} \to\mathbb{R}$ is a right-dense continuous function satisfying $q\ge 0$ on $[t_0,\infty)_\mathbb{T}$ with $q$ nonzero and not eventually zero; note that $p$ and $q$ are delta integrable. Moreover, we assume that $f,g:\mathbb{R}\to\mathbb{R}$ are nondecreasing continuous functions that satisfy $zf(z), zg(z)>0$ for $z\ne 0$, and that there exist positive real numbers $F$ and $G$ such that $f(y)/y\ge F$ and $g(x)/x\ge G$. A solution $(x,y)$ of \eqref{system} is oscillatory if both component functions $x$ and $y$ are oscillatory, that is to say neither eventually positive nor eventually negative; otherwise, the solution is nonoscillatory. The nonlinear dynamic system \eqref{system} is oscillatory if all its solutions are oscillatory. \begin{lemma}\label{lem21} The component functions $x$ and $y$ of a nonoscillatory solution $(x,y)$ of \eqref{system} are themselves nonoscillatory. \end{lemma} \begin{proof} Assume to the contrary that $x$ oscillates but $y$ is eventually positive. Then $x^\Delta=pf(y)>0$ eventually, so that $x(t)>0$ or $x(t)<0$ for all large $t\in\mathbb{T}$, a contradiction. The case where $y$ is eventually negative is similar. Likewise, assuming that $y$ oscillates while $x$ is eventually positive or eventually negative leads to comparable contradictions. \end{proof} \begin{lemma}\label{lem22} If \begin{equation}\label{2infty} \int_{t_0}^{\infty}p(r)\Delta r=\infty \quad\text{and}\quad \int_{t_0}^{\infty}q(s)\Delta s=\infty, \end{equation} then each solution of nonlinear system \eqref{system} is oscillatory. \end{lemma} \begin{proof} Let $(x,y)$ be a nonoscillatory solution of \eqref{system}. First assume that $x>0$; then $y^\Delta=-qg(x)\le 0$, and in view of Lemma \ref{lem21}, $y$ must be of constant sign eventually. If $y(t_1)<0$ for some $t_1\in[t_0,\infty)_\mathbb{T}$, then $y<0$ on $[t_1,\infty)_\mathbb{T}$ and $x^\Delta=pf(y)<0$ on $[t_1,\infty)_\mathbb{T}$; after delta integrating from $t_1$ to $t$, we have \begin{equation}\label{lem22eq} x(t)=x(t_1)+\int_{t_1}^{t} p(r)f\big(y(r)\big)\Delta r. \end{equation} Since $y$ is negative and nonincreasing, and $yf(y)>0$ with $f$ nondecreasing, we know $f(y)<0$, and by the first assumption in \eqref{2infty} the right-hand side of \eqref{lem22eq} tends to $-\infty$, a contradiction of $x>0$. Consequently, $y>0$ with $y^\Delta\le 0$ on $[t_0,\infty)_\mathbb{T}$, and $x^\Delta>0$ on $[t_0,\infty)_\mathbb{T}$ by the first equation of \eqref{system}. Thus there exists a constant $c>0$ and $t_1\in[t_0,\infty)_\mathbb{T}$ such that $x(t)\ge c$ for $t\in[t_1,\infty)_\mathbb{T}$. Delta integrating the second equation of \eqref{system}, we obtain \[ g(c) \int_{t_1}^{\infty}q(s)\Delta s \le y(t_1) < \infty, \] and this contradicts the second assumption in \eqref{2infty}. Similar contradictions are reached for $x<0$. \end{proof} \begin{lemma}\label{lem23} If \begin{equation}\label{2finite} \int_{t_0}^{\infty}p(r)\Delta r<\infty \quad\text{and}\quad \int_{t_0}^{\infty}q(s)\Delta s<\infty, \end{equation} then nonlinear system \eqref{system} is nonoscillatory. \end{lemma} \begin{proof} Suppose that \eqref{2finite} holds. Then there exists $t_1\in[t_0,\infty)_\mathbb{T}$ such that \begin{equation} \label{less1} \int_{t_1}^{\infty}p(r)f\Big(1+g(2)\int_{r}^{\infty}q(s)\Delta s\Big) \Delta r<1. \end{equation} Let $\mathcal{B}=\mathop{\rm C_{rd}}(\mathbb{T})$ be the Banach space of right-dense continuous functions on $\mathbb{T}$, with norm $\|x\|=\sup_{t\ge t_1, t\in\mathbb{T}}|x(t)|$ and the usual pointwise ordering $\le$. Define a subset $\mathcal{S}$ of $\mathcal{B}$ as follows: \[ \mathcal{S}=\{x\in\mathcal{B}:1\le x(t)\le 2, \; t\in[t_1,\infty)_\mathbb{T}\}. \] For any subset $\mathcal{Q}$ of $\mathcal{S}$, we have that $\inf\mathcal{Q}\in\mathcal{S}$ and $\sup\mathcal{Q}\in\mathcal{S}$. Let $L:\mathcal{S}\to\mathcal{B}$ be the functional given via \[ (Lx)(t)=1+\int_{t_1}^{t}p(r)f\Big(1+\int_{r}^{\infty}q(s)g\big(x(s)\big) \Delta s\Big)\Delta r, \quad t\in[t_1,\infty)_\mathbb{T}. \] By the assumptions on $x\in\mathcal{S}$ and $p$ and $q$ and the fact that $f$ and $g$ are nondecreasing, $(Lx)(t)\ge 1$ for all $t\in[t_1,\infty)_\mathbb{T}$, and \[ (Lx)(t) \le 1+\int_{t_1}^{t}p(r)f \Big(1+\int_{r}^{\infty}q(s)g(2)\Delta s\Big)\Delta r \le 2 \] by \eqref{less1}. Moreover, \begin{equation}\label{Ldel} (Lx)^\Delta(t) = p(t)f\Big(1+\int_{t}^{\infty}q(s)g\big(x(s)\big)\Delta s\Big) > 0, \end{equation} ensuring that $L:\mathcal{S}\to\mathcal{S}$ is increasing. By Knaster's fixed-point theorem \cite{knast}, we can conclude that there exists an $x\in\mathcal{S}$ such that $x=Lx$. If we let \[ y(t)= 1+\int_{t}^{\infty}q(s)g\big(x(s)\big)\Delta s, \quad t\in[t_1,\infty)_\mathbb{T} \] using the fixed point $x\in\mathcal{S}$, then we have \[ x^\Delta(t)=(Lx)^\Delta(t)=p(t)f\big(y(t)\big) \quad\text{and}\quad y^\Delta(t)=-q(t)g\big(x(t)\big) \] for $t\in[t_1,\infty)_\mathbb{T}$ by using \eqref{Ldel}. Thus $(x,y)$ is a nonoscillatory solution of \eqref{system}. \end{proof} In view of Lemmas \ref{lem22} and \ref{lem23}, respectively, we could assume that either \begin{align} &\int_{t_0}^{\infty}p(r)\Delta r=\infty \quad\text{and}\quad \int_{t_0}^{\infty}q(s)\Delta s<\infty, \quad\text{or} \label{qfinite} \\ &\int_{t_0}^{\infty}p(r)\Delta r<\infty \quad\text{and}\quad \int_{t_0}^{\infty}q(s)\Delta s=\infty; \label{pfinite} \end{align} in fact, we will focus on \eqref{qfinite}. Moreover, in preparation for what follows, we introduce the following notation. Let \begin{equation}\label{Pdef} P(t):=\int_{t_0}^tp(r)\Delta r. \end{equation} \begin{lemma}\label{lem24} Assume that \eqref{qfinite} holds, $P$ is given by \eqref{Pdef}, and $\lambda\in[0,1)$ is a real number. If \begin{equation}\label{limpies} \lim_{t\to\infty}\frac{\mu(t)p(t)}{P(t)}=0, \quad \Big(\text{equivalently}, \; \lim_{t\to\infty}\frac{P^\sigma(t)}{P(t)}=1\Big) \end{equation} then given $\epsilon>0$ there exists a $t_1\equiv t_1(\epsilon)\in(t_0,\infty)_\mathbb{T}$ such that for any $t\in[t_1,\infty)_\mathbb{T}$, \begin{gather} \int_{t}^{\infty}\frac{\big[\big(P^\lambda\big)^\Delta(r)\big]^2} {p(r)P^\lambda(r)}\Delta r \le \frac{\lambda^2}{1-\lambda} \left(1+\epsilon\right)^{2-\lambda} P^{\lambda-1}(t), \quad\text{and} \label{e208} \\ \int_{t}^{\infty}\frac{p(r)}{P^{2-\lambda}(r)}\Delta r \le \frac{(1+\epsilon)^{2-\lambda}}{1-\lambda}P^{\lambda-1}(t). \label{e209} \end{gather} \end{lemma} \begin{proof} For $r\in(t_0,\infty)_\mathbb{T}$, by the chain rule \cite[Theorem 1.90]{bp1} we have \[ \big(P^\lambda\big)^\Delta(r) = \begin{cases} \dfrac{P^\lambda(\sigma(r))-P^\lambda(r)}{\mu(r)} & :\mu(r)>0, \\[3pt] \lambda p(r)P^{\lambda-1}(r) & :\mu(r)=0.\end{cases} \] By \cite[Theorem 1.16 (iv)]{bp1}, $\mu P^\Delta=P^\sigma-P$, so that $\mu p=P^\sigma-P$ on $\mathbb{T}$. If $r\in(t_0,\infty)_\mathbb{T}$ is a right-scattered point, then $\mu(r)>0$ and, suppressing the $r$, \begin{align*} \frac{\big[\big(P^\lambda\big)^\Delta\big]^2}{pP^\lambda} &= \frac{p}{\mu^2p^2P^\lambda}\left(\big(P^\sigma\big)^\lambda-P^\lambda \right)^2 \\ &= \frac{p}{P^\lambda}\Big(\frac{\big(P^\sigma\big)^\lambda -P^\lambda}{P^\sigma-P}\Big)^2 \\ &\overset{\text{MVT}}{=} \frac{p}{P^\lambda} \left(\lambda\xi^{\lambda-1}\right)^2, \quad \xi\in\big(P(r), P^\sigma(r)\big)_\mathbb{R} \\ &\le \frac{p\lambda^2}{P^\lambda}P^{2\lambda-2}, \quad \lambda-1<0 \\ &= \lambda^2 pP^{\lambda-2}. \end{align*} If $r\in(t_0,\infty)_\mathbb{T}$ is a right-dense point, then $\mu(r)=0$ and \[ \frac{\big[\big(P^\lambda\big)^\Delta\big]^2}{pP^\lambda} = \frac{\big[\lambda pP^{\lambda-1}\big]^2}{pP^\lambda} = \lambda^2 pP^{\lambda-2}. \] It follows that in either case, \begin{equation}\label{delP1} \frac{\left[\big(P^\lambda\big)^\Delta(r)\right]^2}{p(r)P^\lambda(r)} \le \lambda^2 p(r)P^{\lambda-2}(r), \quad r\in(t_0,\infty)_\mathbb{T}. \end{equation} Similarly, if $r\in(t_0,\infty)_\mathbb{T}$ is a right-scattered point, then once again $\mu(r)>0$ and, suppressing the $r$, \begin{align*} -\big(P^{\lambda-1}\big)^\Delta &= \frac{-p}{\mu p}\left(\big(P^\sigma\big)^{\lambda-1}-P^{\lambda-1} \right) \\ &= -p\Big(\frac{\big(P^\sigma\big)^{\lambda-1} -P^{\lambda-1}}{P^\sigma-P}\Big) \\ &\overset{\text{MVT}}{=} p(1-\lambda)\eta^{\lambda-2}, \quad \eta\in\big(P(r),P^\sigma(r)\big)_\mathbb{R} \\ &\geq p(1-\lambda)\big(P^\sigma\big)^{\lambda-2}. \end{align*} If $r$ is a right-dense point, then $P^\sigma=P$, $\mu(r)=0$, and $p(1-\lambda)P^{\lambda-2}=-\big(P^{\lambda-1}\big)^\Delta$. Summarizing, in either case we have \begin{equation} \label{delP2} -\big(P^{\lambda-1}\big)^\Delta \ge p(1-\lambda)\big(P^\sigma\big)^{\lambda-2}, \quad r\in(t_0,\infty)_\mathbb{T}. \end{equation} Combining \eqref{delP1} and \eqref{delP2}, we see that \[ \frac{\big[\big(P^\lambda\big)^\Delta(r)\big]^2}{p(r)P^\lambda(r)} \le \frac{\lambda^2}{1-\lambda}\Big(\frac{P(r)}{P^\sigma(r)}\Big)^{\lambda-2} \big[-\big(P^{\lambda-1}\big)^\Delta(r)\big]. \] By \eqref{limpies}, given $\epsilon>0$ there exists a $t_1\in[t_0,\infty)_\mathbb{T}$ such that $P^\sigma/P\le (1+\epsilon)$ on $[t_1,\infty)_\mathbb{T}$. Consequently, for any $t\in[t_1,\infty)_\mathbb{T}$, \begin{align*} \int_{t}^{\infty}\frac{\big[\big(P^\lambda\big)^\Delta(r)\big]^2} {p(r)P^\lambda(r)}\Delta r &\leq \frac{\lambda^2}{1-\lambda}(1+\epsilon)^{2-\lambda} \int_{t}^{\infty}\big[-\big(P^{\lambda-1}\big)^\Delta(r)\big]\Delta r \\ &\overset{\eqref{qfinite}, \eqref{Pdef}}{=} \frac{\lambda^2}{1-\lambda}(1+\epsilon)^{2-\lambda} P^{\lambda-1}(t), \end{align*} which is \eqref{e208}. Moreover, again for any $r\in[t_1,\infty)_\mathbb{T}$, \begin{equation}\label{e212} \begin{aligned} \frac{p(r)}{P^{2-\lambda}(r)} & = \frac{p(r)}{P^{2-\lambda}(\sigma(r))} \frac{P^{2-\lambda}(\sigma(r))}{P^{2-\lambda}(r)} \le (1+\epsilon)^{2-\lambda} \frac{p(r)}{P^{2-\lambda}(\sigma(r))} \\ &\overset{\eqref{delP2}}{\le} \frac{(1+\epsilon)^{2-\lambda}}{\lambda-1} \big(P^{\lambda-1}\big)^\Delta(r). \end{aligned} \end{equation} Delta integrating \eqref{e212} from $t$ to infinity, we obtain \[ \int_{t}^{\infty}\frac{p(r)}{P^{2-\lambda}(r)}\Delta r \le \frac{(1+\epsilon)^{2-\lambda}}{\lambda-1} \int_{t}^{\infty}\big(P^{\lambda-1}\big)^\Delta(r)\Delta r \overset{\eqref{qfinite}, \eqref{Pdef}}{=}\frac{(1+\epsilon)^{2-\lambda}}{1-\lambda}P^{\lambda-1}(t), \] which is \eqref{e209}. \end{proof} Note that if $\mathbb{T} =\mathbb{R}$, then \eqref{limpies} is automatically satisfied, as $\mu(t)\equiv 0$. \begin{lemma}\label{lem25} Assume that \eqref{qfinite} holds, that $P$ is given by \eqref{Pdef}, and that \eqref{limpies} holds. If for some real number $\lambda<1$ we have \begin{equation}\label{Plamq} \int_{t_1}^{\infty} q(r)P^\lambda(r)\Delta r = \infty \quad\text{for}\quad t_1 \ge \sigma(t_0), \end{equation} then nonlinear system \eqref{system} is oscillatory. \end{lemma} \begin{proof} By Lemma \ref{lem23}, we can focus on $\lambda\in(0,1)$. Assume that $(x,y)$ is a nonoscillatory solution of nonlinear system \eqref{system}, and assume that $x>0$ on $[t_0,\infty)_\mathbb{T}$; the case where $x<0$ on $[t_0,\infty)_\mathbb{T}$ is similar and consequently omitted. As in the proof of Lemma \ref{lem22}, $y>0$ with $y^\Delta\le 0$ and $x^\Delta>0$ on $[t_0,\infty)_\mathbb{T}$. Let $w:=y/x$. Then $w>0$, and suppressing the argument, we have by the delta quotient rule and \eqref{system} that on $[t_0,\infty)_\mathbb{T}$, \begin{equation}\label{wdelorig} w^\Delta = \frac{x^\sigma y^\Delta-y^\sigma x^\Delta}{xx^\sigma} = -q\frac{g(x)}{x}-pww^\sigma \frac{f(y)}{y} \le -qG -pww^\sigma F < 0. \end{equation} In fact this gives us \begin{equation}\label{wdel} w^\Delta \le -qG-p(w^\sigma)^2F, \end{equation} and from the previous line we obtain on $[t_0,\infty)_\mathbb{T}$ that \[ \big(\frac{1}{w}\big)^\Delta=\frac{-w^\Delta}{ww^\sigma} \ge \frac{qG+pww^\sigma F}{ww^\sigma} \ge pF; \] delta integrating from $t_0$ to $t$ we see that \begin{equation}\label{wPbd} 1 > 1-\frac{w(t)}{w(t_0)} \ge Fw(t)\int_{t_0}^{t}p(r)\Delta r=Fw(t)P(t)\ge 0, \quad t\in[t_0,\infty)_\mathbb{T}. \end{equation} Again by the mean value theorem, $\big(P^{\lambda}\big)^\Delta \le\lambda pP^{\lambda-1}$ for $\lambda\in(0,1)$. Multiplying \eqref{wdel} by $P^\lambda$ and delta integrating from $t_1 \ge \sigma(t_0)$ to $t$ we obtain \begin{equation} \begin{aligned} G\int_{t_1}^{t}q(r)P^\lambda(r)\Delta r &\leq -\int_{t_1}^{t}P^\lambda(r)w^\Delta(r)\Delta r - F\int_{t_1}^{t}p(r)P^\lambda(r)(w^\sigma)^2(r)\Delta r \\ &\overset{\text{parts}}{=} -P^\lambda(t)w(t) + P^\lambda(t_1)w(t_1) + \int_{t_1}^{t}\big(P^\lambda\big)^\Delta(r)w^\sigma(r)\Delta r \\ & \quad - F\int_{t_1}^{t}p(r)P^\lambda(r)(w^\sigma)^2(r)\Delta r \\ &\leq -P^\lambda(t)w(t) + P^\lambda(t_1)w(t_1) + \int_{t_1}^{t}\lambda p(r)P^{\lambda-1}(r)w^\sigma(r)\Delta r \\ &\quad - F\int_{t_1}^{t}p(r)P^\lambda(r)(w^\sigma)^2(r)\Delta r \\ &= -P^\lambda(t)w(t) + P^\lambda(t_1)w(t_1) \\ &\quad + \int_{t_1}^{t} p(r)P^{\lambda-2}(r)\big[P(r)w^\sigma(r)\Big(\lambda - F P(r)w^\sigma(r)\Big)\big]\Delta r. \end{aligned}\label{qsigP} \end{equation} Since by \eqref{wPbd} we have \begin{equation}\label{wPineq} 0 \le FP(t)w^\sigma(t) \le FP(t)w(t) < 1, \quad t\in[t_0,\infty)_\mathbb{T}, \end{equation} there exists a positive real number $k$ such that \[ \left|P(r)w^\sigma(r)\big(\lambda - FP(r)w^\sigma(r)\big)\right|1. \end{cases} \] In either case, take \[ g_*(\lambda):=\liminf_{t\to\infty}g(t,\lambda) \quad\text{and}\quad g^*(\lambda):=\limsup_{t\to\infty}g(t,\lambda). \] \begin{lemma}\label{lem26} Assume that \eqref{finitecond} holds, that $P$ is given by \eqref{Pdef}, and that \eqref{limpies} holds. If $(x,y)$ is a nonoscillatory solution of nonlinear system \eqref{system}, then \begin{gather} \liminf_{t\to\infty} w(t)P(t)\ge\frac{1}{2F} \Big(1-\sqrt{1-4Fg_*(0)}\Big), \label{e220} \\ \limsup_{t\to\infty} w(t)P(t)\le\frac{1}{2F} \Big(1+\sqrt{1-4Fg_*(2)}\Big), \label{e221} \end{gather} where again $w:=y/x$. \end{lemma} \begin{proof} By \eqref{wPbd}, we can introduce the constants \begin{equation}\label{rRdef} r:=\liminf_{t\to\infty} w(t)P(t), \quad R:=\limsup_{t\to\infty} w(t)P(t), \end{equation} and by \eqref{finitecond}, we must have \begin{equation}\label{wdies} \lim_{t\to\infty}w(t)=0. \end{equation} From \eqref{wdelorig} we have $w^\Delta \le -qG-pww^\sigma F$; delta integrate this from $t$ to $\infty$, use \eqref{wdies}, and multiply by $P$ to see that \begin{equation}\label{e224} w(t)P(t) \ge GP(t)\int_{t}^{\infty} q(\tau)\Delta \tau + FP(t)\int_{t}^{\infty} p(\tau)w(\tau)w^\sigma(\tau)\Delta \tau \end{equation} holds for $t\in[t_1,\infty)_\mathbb{T}$. From \eqref{rRdef} this yields \begin{equation}\label{e225} r \ge g_*(0). \end{equation} This time multiply \eqref{wdel} by $P^2$ and delta integrate from $t_1$ to $t$ to get \begin{align*} G\int_{t_1}^{t} q(\tau)P^2(\tau)\Delta \tau &\leq -\int_{t_1}^{t}P^2(\tau)w^\Delta(\tau)\Delta \tau - F\int_{t_1}^{t} p(\tau)P^2(\tau)\big(w^\sigma\big)^2(\tau)\Delta \tau \\ &= -P^2(t)w(t)+P^2(t_1)w(t_1)+\int_{t_1}^{t}(P^2)^\Delta(\tau)w^\sigma(\tau)\Delta \tau \\ &\quad -F\int_{t_1}^{t}p(\tau)P^2(\tau)\big(w^\sigma\big)^2(\tau)\Delta \tau \\ &= -P^2(t)w(t) + P^2(t_1)w(t_1) + \int_{t_1}^{t}\mu(\tau)p^2(\tau)w^\sigma(\tau)\Delta \tau \\ &\quad + \int_{t_1}^{t}p(\tau)P(\tau)w^\sigma(\tau) [2-FP(\tau)w^\sigma(\tau)]\Delta \tau \end{align*} for $t\in[t_1,\infty)_\mathbb{T}$, which leads to \begin{equation} \begin{aligned} &w(t)P(t) \\ &\leq -GP^{-1}(t)\int_{t_1}^{t} q(\tau)P^2(\tau)\Delta \tau + P^{-1}(t)\int_{t_1}^{t}\mu(\tau)p^2(\tau)w^\sigma(\tau)\Delta \tau \\ &\quad + P^{-1}(t)P^2(t_1)w(t_1) + P^{-1}(t) \int_{t_1}^{t}p(\tau)P(\tau)w^\sigma(\tau) [2-FP(\tau)w^\sigma(\tau)]\Delta \tau. \end{aligned} \label{bdthis} \end{equation} Using \eqref{wPineq}, $0<(1-FPw^\sigma)^2$, leading to $FPw^\sigma[2-FPw^\sigma]<1$. Thus for large $t\in\mathbb{T}$, \[ P^{-1}(t)\int_{t_1}^{t}p(\tau)P(\tau)w^\sigma(\tau)\left[2-P(\tau)w^\sigma(\tau)\right]\Delta \tau\le 1/F. \] Applying L'H\^{o}pital's rule \cite[Theorem 1.120]{bp1}, \eqref{wPineq} again, and \eqref{limpies} we have \[ 0\le\lim_{t\to\infty}\frac{\int_{t_1}^{t}\mu(\tau)p^2(\tau)w^\sigma(\tau)\Delta \tau}{P(t)} = \lim_{t\to\infty}\mu(t)p(t)w^\sigma(t) \le \lim_{t\to\infty}\frac{\mu(t)p(t)}{P(t)}=0. \] Altogether then, inequality \eqref{bdthis} implies that \begin{equation}\label{e227} R\le 1/F - g_*(2). \end{equation} If $g_*(0)=0=g_*(2)$, then estimates \eqref{e220} and \eqref{e221} follow directly from \eqref{e225} and \eqref{e227}, respectively. Thus we pick a real number $\epsilon\in\big(0,\min\{g_*(0),g_*(2)\}\big)$ and $t_2\in[t_1,\infty)_\mathbb{T}$ such that for $t\in[t_2,\infty)_\mathbb{T}$, \begin{gather*} r-\epsilon < w(t)P(t) < R+\epsilon, \quad w(t)P(t) \ge GP(t)\int_{t}^{\infty} q(\tau)\Delta \tau>g_*(0)-\epsilon, \\ GP^{-1}(t)\int_{t_0}^{t} q(\tau)P^2(\tau)\Delta \tau>g_*(2)-\epsilon. \end{gather*} From \eqref{e224} and L'H\^{o}pital's rule we have for $t\in[t_2,\infty)_\mathbb{T}$ that \[ w(t)P(t) \ge g_*(0)-\epsilon + F(r-\epsilon)^2. \] Multiply \eqref{wdelorig} by $P^2$ and delta integrate from $t_1$ to $t$ to see that this leads to \begin{equation} \begin{aligned} w(t)P(t) &\leq -GP^{-1}(t)\int_{t_1}^{t} q(\tau)P^2(\tau)\Delta \tau + P^{-1}(t)\int_{t_1}^{t}\mu(\tau)p^2(\tau)w^\sigma(\tau)\Delta \tau \\ &\quad + P^{-1}(t)P^2(t_1)w(t_1) + P^{-1}(t)\int_{t_1}^{t}p(\tau)P(\tau)w^\sigma(\tau) [2-Fw(\tau)P(\tau)]\Delta \tau. \end{aligned} \label{bdthis2} \end{equation} From \eqref{bdthis2} we have for $t\in[t_2,\infty)_\mathbb{T}$ that \[ w(t)P(t) \le \frac{P^2(t_1)w(t_1) + \int_{t_1}^{t}\mu(\tau)p^2(\tau)w^\sigma(\tau)\Delta \tau}{P(t)} -g_*(2)+\epsilon +(R+\epsilon)(2-F(R+\epsilon)), \] since $Fw^\sigma P\le FwP<1$. These two inequalities lead to \begin{equation}\label{rRineq} r \ge g_*(0) + Fr^2, \quad R \le R(2-FR)-g_*(2). \end{equation} Consequently, \[ r \ge \frac{1}{2F}\left(1-\sqrt{1-4Fg_*(0)}\right), \quad R \le \frac{1}{2F}\left(1+\sqrt{1-4Fg_*(2)}\right), \] and the lemma is proven. \end{proof} \section{main oscillation results} We use the lemmas obtained previously to prove our main results. \begin{theorem}\label{thm31} Assume that \eqref{finitecond} holds, that $P$ is given by \eqref{Pdef}, and that \eqref{limpies} holds. If \begin{gather} g_*(0)=\liminf_{t\to\infty} P(t)\int_{t}^{\infty} q(\tau)\Delta \tau > \frac{1}{4F}, \quad\text{or} \label{e301} \\ g_*(2)=\liminf_{t\to\infty} \frac{1}{P(t)}\int_{t_0}^{t} q(\tau)P^2(\tau)\Delta \tau > \frac{1}{4F}, \label{e302} \end{gather} then every solution of nonlinear system \eqref{system} is oscillatory. \end{theorem} \begin{proof} Suppose to the contrary that $(x,y)$ is a nonoscillatory solution of \eqref{system} with $x(t)>0$ for $t\in[t_0,\infty)_\mathbb{T}$. Let \[ r:=\liminf_{t\to\infty} w(t)P(t),\quad R:=\limsup_{t\to\infty} w(t)P(t), \] where $w=y/x$. By Lemma \ref{lem26} and its proof (in particular \eqref{rRineq}) and simple calculus, we have \[ g_*(0)\le r-Fr^2\le\frac{1}{4F} \quad\text{and}\quad g_*(2) \le R-FR^2\le \frac{1}{4F}, \] a contradiction of both \eqref{e301} and \eqref{e302}. The case with $x(t)<0$ for $t\in[t_0,\infty)_\mathbb{T}$ is similar. \end{proof} \begin{theorem} \label{thm32} Assume that \eqref{finitecond} holds, that $P$ is given by \eqref{Pdef}, and that \eqref{limpies} holds. Let $g_*(2)\le1/(4F)$, and assume there exists a real number $\lambda\in[0,1)$ such that \begin{equation} \label{e303} g^*(\lambda)>\frac{\lambda^2}{4F(1-\lambda)}+\frac{1}{2F} \left(1+\sqrt{1-4Fg_*(2)}\right). \end{equation} Then every solution of nonlinear system \eqref{system} is oscillatory. \end{theorem} \begin{proof} Suppose to the contrary that $(x,y)$ is a nonoscillatory solution of % \eqref{system} with $x(t)>0$ for $t\in[t_0,\infty)_\mathbb{T}$. By \eqref{wdel} we have \[ Gq(t) \le -w^\Delta(t) - Fp(t)(w^\sigma)^2(t), \quad t\in[t_0,\infty)_\mathbb{T}, \] where $w=y/x$; multiply this by $P^\lambda$ and delta integrate from $t$ to infinity to get \begin{align*} G\int_{t}^{\infty} q(\tau)P^\lambda(\tau)\Delta \tau &\leq -\int_{t}^{\infty} w^\Delta(\tau)P^\lambda(\tau)\Delta \tau - F\int_{t}^{\infty} p(\tau)(w^\sigma)^2(\tau)P^\lambda(\tau)\Delta \tau \\ &= P^\lambda(t)w(t) + \int_{t}^{\infty}\big(P^\lambda\big)^\Delta(\tau)w^\sigma(\tau)\Delta \tau \\ &\quad - F\int_{t}^{\infty} p(\tau)P^\lambda(\tau)(w^\sigma)^2(\tau)\Delta \tau \\ &= P^\lambda(t)w(t) + \frac{1}{4F}\int_{t}^{\infty}\frac{\left((P^\lambda)^\Delta\right)^2(\tau)}{p(\tau)P^\lambda(\tau)}\Delta \tau \\ &\quad - \int_{t}^{\infty} \Big(\sqrt{Fp(\tau)}P^{\lambda/2}(\tau) w^\sigma(\tau) - \frac{\big(P^\lambda\big)^\Delta(\tau)}{2\sqrt{Fp(\tau)} P^{\lambda/2}(\tau)}\Big)^2\Delta \tau \\ &\leq P^\lambda(t)w(t) + \frac{1}{4F}\int_{t}^{\infty} \frac{\big((P^\lambda)^\Delta\big)^2(\tau)}{p(\tau)P^\lambda(\tau)} \Delta \tau. \end{align*} It follows that \begin{equation}\label{e304} P^{1-\lambda}(t)G\int_{t}^{\infty} q(\tau)P^{\lambda}(\tau)\Delta \tau < P(t)w(t) + \frac{P^{1-\lambda}(t)}{4F} \int_{t}^{\infty}\frac{\left((P^\lambda)^\Delta\right)^2(\tau)} {p(\tau)P^\lambda(\tau)}\Delta \tau. \end{equation} By \eqref{e208}, \eqref{e221}, and \eqref{e304}, \[ g^*(\lambda) \le \frac{1}{2F} \left(1+\sqrt{1-4Fg_*(2)}\right) + \frac{\lambda^2}{4F(1-\lambda)}, \] a contradiction of \eqref{e303}. Similarly if $x(t)<0$ for $t\in[t_0,\infty)_\mathbb{T}$. \end{proof} \begin{corollary} Assume that \eqref{finitecond} holds, that $P$ is given by \eqref{Pdef}, and that \eqref{limpies} holds. If $g_*(2)\le1/(4F)$ and $g^*(0) > \frac{1}{2F}\left(1+\sqrt{1-4Fg_*(2)}\right)$, then every solution of nonlinear system \eqref{system} is oscillatory. \end{corollary} \begin{theorem} \label{thm34} Assume that \eqref{finitecond} holds, that $P$ is given by \eqref{Pdef}, and that \eqref{limpies} holds. Let $g_*(0), g_*(2)\le1/(4F)$, and assume there exists a real number $\lambda\in[0,1)$ such that \begin{gather} g_*(0) > \frac{\lambda(2-\lambda)}{4F}, \quad\text{and} \label{e305} \\ g^*(\lambda) > \frac{g_*(0)}{1-\lambda}+\frac{1}{2F}\left(\sqrt{1-4Fg_*(0)} + \sqrt{1-4Fg_*(2)}\right). \label{e306} \end{gather} Then every solution of nonlinear system \eqref{system} is oscillatory. \end{theorem} \begin{proof} Suppose to the contrary that $(x,y)$ is a nonoscillatory solution of \eqref{system} with $x(t)>0$ for $t\in[t_0,\infty)_\mathbb{T}$; the case with $x(t)<0$ for $t\in[t_0,\infty)_\mathbb{T}$ is omitted. Let $r=\liminf_{t\to\infty} w(t)P(t)$ and $R=\limsup_{t\to\infty} w(t)P(t)$, where $w=y/x$. By \eqref{e220} and \eqref{e221}, \begin{equation}\label{e307} r \ge m:=\frac{1}{2F} \left(1-\sqrt{1-4Fg_*(0)}\right), \quad R\le M:=\frac{1}{2F} \left(1+\sqrt{1-4Fg_*(2)}\right). \end{equation} Using \eqref{e305} and \eqref{e307} we find that $m>\lambda/(2F)$, whence given $\epsilon\in\left(0,m-\frac{\lambda}{2F}\right)$, there exists a $t_1\in[t_0,\infty)_\mathbb{T}$ such that \begin{equation}\label{e308} m-\epsilon < w(t)P(t) < M+\epsilon, \quad t\in[t_1,\infty)_\mathbb{T}. \end{equation} Similar to what we did in \eqref{qsigP}, multiply \eqref{wdel} by $P^\lambda$ and delta integrate from $t$ to infinity to get \begin{align*} &G\int_{t}^{\infty} q(\tau)P^\lambda(\tau)\Delta \tau\\ &\le w(t)P^\lambda(t) + \int_{t}^{\infty} p(\tau)P^{\lambda-2}(\tau) \big[\lambda w^\sigma(\tau)P(\tau) - F\big(P(\tau)w^\sigma(\tau)\big)^2\big]\Delta \tau; \end{align*} this leads to \begin{equation} \begin{aligned} P^{1-\lambda}(t)G\int_{t}^{\infty} q(\tau)P^\lambda(\tau)\Delta \tau &\leq w(t)P(t) + P^{1-\lambda}(t)\int_{t}^{\infty} p(\tau)P^{\lambda-2}(\tau) \\ &\quad \times\big[\lambda w^\sigma(\tau)P(\tau) - F\big(P(\tau)w^\sigma(\tau)\big)^2\big]\Delta \tau. \end{aligned} \label{e309} \end{equation} Since the function $\gamma(z):=\lambda z-Fz^2$ is decreasing over the real interval $[\frac{\lambda}{2F},\infty)$, it follows from \eqref{e308}, \eqref{e309}, and Lemma \ref{lem24} that \begin{align*} &P^{1-\lambda}(t)G\int_{t}^{\infty} q(\tau)P^\lambda(\tau)\Delta \tau\\ &< M+\epsilon + (m-\epsilon)(\lambda-F(m-\epsilon)) P^{1-\lambda}(t)\int_{t}^{\infty} p(\tau)P^{\lambda-2}(\tau)\Delta \tau \\ &< M+\epsilon + \frac{(m-\epsilon)(\lambda-F(m-\epsilon))(1+\epsilon)^{2-\lambda}}{1-\lambda}. \end{align*} This in tandem with \eqref{e307} yields \[ g^*(\lambda)\le M+\frac{m(\lambda-Fm)}{1-\lambda} = \frac{g_*(0)}{1-\lambda} + \frac{1}{2F}\left(\sqrt{1-4Fg_*(0)} + \sqrt{1-4Fg_*(2)}\right), \] a contradiction of \eqref{e306}. \end{proof} \begin{corollary} \label{coro3.5} Assume that \eqref{finitecond} holds, that $P$ is given by \eqref{Pdef}, and that \eqref{limpies} holds. Let $0 g_*(0) + \frac{1}{2F}\left(\sqrt{1-4Fg_*(0)} + \sqrt{1-4Fg_*(2)}\right), \] then every solution of nonlinear system \eqref{system} is oscillatory. \end{corollary} \section{example} We illustrate Theorem \ref{thm31} with the following example. \begin{example}\label{ex41} \rm Let $\mathbb{T}$ be an arbitrary time scale unbounded above, and let $p$ and $F$ be positive constants. Then the linear system \begin{equation}\label{exampeq2} x^\Delta(t)=pFy(t), \quad y^\Delta(t)=\frac{-1}{t\sigma(t)}x(t), \quad t\in[t_0,\infty)_\mathbb{T} \end{equation} for $t_0>0$, is nonoscillatory for $01/(4F)$. In other words, the inequality in \eqref{e301} is sharp on all time scales. \end{example} \begin{proof} Note that $p(t)\equiv p$, $f(y)=Fy$, $q(t)=\frac{1}{t\sigma(t)}$, and $g(x)=x$. Thus we have $P(t)=p(t-t_0)$, $f(y)/y=F$, and $G\equiv 1$, so that $$ g_*(0)=\liminf_{t\to\infty} GP(t)\int_{t}^{\infty} q(r)\Delta r =\liminf_{t\to\infty} \frac{p(t-t_0)}{t} =p. $$ By Theorem \ref{thm31} and \eqref{e301}, any solution $(x,y)$ of \eqref{exampeq2} oscillates if $p>1/(4F)$. Converting \eqref{exampeq2} to a second-order dynamic equation for $x$, we arrive at a Cauchy-Euler equation \cite[Section 2.3]{bp2} of the form $$ t\sigma(t)x^{\Delta\Delta}(t)+pFx(t)=0, $$ with general solution \begin{equation}\label{gensol} x(t)=Ae_{\frac{1+\sqrt{1-4Fp}}{2t}}(t,t_0) +Be_{\frac{1-\sqrt{1-4Fp}}{2t}}(t,t_0), \end{equation} where we have used a linear combination involving the time-scale exponential function \cite[Section 2.2]{bp1}. From elementary analysis and Euler's formula we know that $x$ is nonoscillatory for $p\le1/(4F)$ and oscillatory for $p>1/(4F)$, showing in particular that the $1/(4F)$ in \eqref{e301} is sharp for all time scales $\mathbb{T}$. \end{proof} \begin{remark} \rm In Example \ref{ex41} we can identify the exponential functions that occur in \eqref{gensol} for specific time scales \cite[Example 2.19]{bp2}. Letting $\lambda=\frac{1+\sqrt{1-4Fp}}{2}$, we get that \begin{gather*} \mathbb{T}=\mathbb{R}: e_{\frac{1+\sqrt{1-4Fp}}{2t}}(t,t_0) = \Big(\frac{t}{t_0} \Big)^\lambda, \\ \mathbb{T}=q^\mathbb{Z}: e_{\frac{1+\sqrt{1-4Fp}}{2t}}(t,t_0) = \Big(\frac{t}{t_0} \Big)^{\log_q[1+(q-1)\lambda]}, \\ \mathbb{T}=\mathbb{Z}: e_{\frac{1+\sqrt{1-4Fp}}{2t}}(t,t_0) = \frac{\Gamma(t+\lambda)\Gamma(t_0)}{\Gamma(t)\Gamma(t_0+\lambda)}, \end{gather*} where $\Gamma$ is the gamma function. \end{remark} \begin{thebibliography}{0} \bibitem{abbs} E. Akin-Bohner, M. Bohner and S. H. Saker, Oscillation criteria for a certain class of second order Emden-Fowler dynamic equations, \emph{Electron. Trans. Numer. Anal.} 27 (2007), 1---12. \bibitem{ah} D. R. Anderson and W. R. Hall, Oscillation criteria for two-dimensional systems of first-order linear dynamic equations on time scales, \emph{Involve} (2009), to appear. \bibitem{bep} M. Bohner, L. Erbe, and A. Peterson, Oscillation for nonlinear second order dynamic equations on a time scale, \emph{J. Math. Anal. Appl.} 301 (2005), no. 2, 491---507. \bibitem{bp1} M. Bohner and A. Peterson, \emph{Dynamic Equations on Time Scales, An Introduction with Applications}, Birkh\"auser, Boston, 2001. \bibitem{bp2} M. Bohner and A. Peterson, editors, \emph{Advances in Dynamic Equations on Time Scales}, Birkh\"auser, Boston, 2003. \bibitem{bs1} M. Bohner and S. H. Saker, Oscillation of second order nonlinear dynamic equations on time scales, \emph{Rocky Mountain J. Math.} 34 (2004), no. 4, 1239---1254. \bibitem{bs2} M. Bohner and S. H. Saker, Oscillation of second order half-linear dynamic equations on discrete time scales, \emph{Int. J. Difference Equ.} 1 (2006), no. 2, 205---218. \bibitem{bt} M. Bohner and C. C. Tisdell, Oscillation and nonoscillation of forced second order dynamic equations, \emph{Pacific J. Math.} 230 (2007), no. 1, 59---71. \bibitem{ep} L. Erbe and A. Peterson, Boundedness and oscillation for nonlinear dynamic equations on a time scale, \emph{Proc. Amer. Math. Soc.} 132 (2004) 735--–744. \bibitem{eps1} L. Erbe, A. Peterson, and S. H. Saker, Oscillation criteria for second-order nonlinear dynamic equations on time scales, \emph{J. London Math. Soc.} 67 (2003) 701--–714. \bibitem{eps2} L. Erbe, A. Peterson, and S. H. Saker, Oscillation criteria for second-order nonlinear delay dynamic equations, \emph{J. Math. Anal. Appl.} 333:1 (2007) 505---522. \bibitem{eps3} L. Erbe, A. Peterson, and S. H. Saker, Oscillation criteria for a forced second-order nonlinear dynamic equation, \emph{J. Difference Eq. Appl.} 14:10 (2008) 997---1009. \bibitem{hilger} S. Hilger, Analysis on measure chains---a unified approach to continuous and discrete calculus, \emph{Results Math.} 18 (1990) 18--56. \bibitem{jt} J. C. Jiang and X. H. Tang, Oscillation criteria for two-dimensional difference systems of first order linear difference equations, \emph{Comput. Math. Appl.} 54 (2007) 808---818. \bibitem{knast} B. Knaster, Un th\'{e}or\`{e}me sur les fonctions d'ensembles, \emph{Ann. Soc. Polon. Math.} 6 (1928) 133---134. \bibitem{li} W. T. Li, Classification schemes for nonoscillatory solutions of two-dimensional nonlinear difference systems, \emph{Comput. Math. Appl.} 42 (2001) 341---355. \bibitem{lp} A. Lomtatidze and N. Partsvania, Oscillation and nonoscillation criteria for two-dimensional systems of first order linear ordinary differential equations, \emph{Georgian Math. J.} 6 (1999) 285---298. \bibitem{s} S. H. Saker, Oscillation of nonlinear dynamic equations on time scales, \emph{Appl. Math. Comput.} 148 (2004) 81--–91. \bibitem{xu} Y. J. Xu and Z. T. Xu, Oscillation criteria for two-dimensional dynamic systems on time scales, \emph{J. Computational Appl. Math.} 225:1 (2009) 9---19. \end{thebibliography} \end{document}