\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 19, pp. 1--5.\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 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/19\hfil An oscillation theorem] {An oscillation theorem for a second order nonlinear differential equations with variable potential} \author[J. Tyagi\hfil EJDE-2009/19\hfilneg] {Jagmohan Tyagi} \address{Jagmohan Tyagi \newline Department of Mathematics and Statistics, Indian Institute of Technology Kanpur, Kanpur - 208016, India} \email{jagmohan.iitk@gmail.com} \thanks{Submitted September 28, 2007. Published January 20, 2009.} \subjclass[2000]{34C10, 34C15} \keywords{Nonlinear; ordinary differential equations; oscillation} \begin{abstract} We obtain a new oscillation theorem for the nonlinear second-order differential equation \begin{equation*} (a(t)x'(t))' + p(t)f(t, x(t), x'(t))+ q(t)g(x(t))=0,\quad t\in [0,\infty), \end{equation*} via the generalization of Leighton's variational theorem. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \section{Introduction} The purpose of this study is to establish a new oscillation criteria for the nonlinear differential equation \begin{equation} \label{e1.1} (a(t)x'(t))' + p(t)f(t, x(t), x'(t))+ q(t)g(x(t))=0, \end{equation} where $a, p, q \in C(\mathbb{R}^+, \mathbb{R})$, $f \in C(\mathbb{R}^+\times \mathbb{R}^2, \mathbb{R})$, $g\in C(\mathbb{R}, \mathbb{R})$, $a(t)> 0$ and $p(t)\geq 0$. Komkov \cite{Komko} generalized a well-known variational theorem of Leighton \cite{leig}. In this note, we establish a new oscillation theorem for \eqref{e1.1} via Komkov's result. Also, we do not impose restriction on the sign of the potential $q$. Here, we consider only solution of \eqref{e1.1} which are defined for all large $t$. A solution of \eqref{e1.1} is called \textit{oscillatory} if it has arbitrarily large zeros, otherwise it is called \textit{nonoscillatory}. Oscillation criteria for the special cases of \eqref{e1.1} \begin{gather} x''(t) + q(t)g(x(t))=0, \label{e1.2} \\ x''(t) + q(t)x(t)=0, \label{e1.3} \end{gather} have been extensively investigated; (see, e.g., \cite{coles, grace, Hartman, Kamenev, Leighton}, \cite{wli}--\cite{jswon} for an excellent bibliography). The most important simple oscillation criterion for linear differential equations is the well-known Leighton's theorem \cite{Leighton}, which states that if $q(t)\geq 0$ and satisfies \begin{equation} \label{e1.4} \lim_{t\to \infty}\int_{0}^{t} q(s)ds = \infty, \end{equation} then \eqref{e1.3} is oscillatory. Wintner \cite{Wintner} modified the Leighton's criteria and proved a stronger result which required a weaker condition \begin{equation} \label{e1.5} \lim_{t\to \infty} \frac{1}{t}\int_{0}^{t}\int_{0}^{s}q(\tau)d\tau ds = \infty. \end{equation} Also, Wintner did not impose any condition on the sign of $q(t)$. Wintner's result was further improved by Hartman \cite{Hartman} who proved that \eqref{e1.5} can be substituted by the weaker condition \begin{equation} \label{e1.6} -\infty< \liminf_{t\to \infty }\frac{1}{t} \int_{0}^{t}\int_{0}^{s}q(\tau)d\tau ds < \limsup_{t\to \infty }\frac{1}{t} \int_{0}^{t}\int_{0}^{s}q(\tau)d\tau ds \leq \infty. \end{equation} Later in 1978, Kamenev \cite{Kamenev} showed that if for some positive integer $n>2$, \begin{equation} \label{e1.7} \limsup_{t\to \infty }\frac{1}{t^{n-1}} \int_{0}^{t}(t-s)^{n-1}q(s)ds = \infty, \end{equation} then \eqref{e1.3} is oscillatory. Also, there is a good amount of literature on oscillation of \eqref{e1.2} (see \cite{coles, grace, wli, patricia, wtli, wongi, jswon} and the literature cited therein). In 1992, James S. W. Wong \cite{wongi} proved the following extension of Cole's result \cite{coles} to the more general equation \eqref{e1.2}. \begin{theorem} \label{thm0} Let $g(x)$ satisfy the superlinearity condition \begin{equation*} 0 < \int_{x}^{\infty}\frac{du}{g(u)} < \infty , \quad 0 < \int_{-x}^{-\infty}\frac{du}{g(u)} < \infty, \quad \forall 0< x \in \mathbb{R}. \end{equation*} Also, let $A(t)= \int_{t}^{\infty} q(s)ds $ exists for each $t\geq 0 $ and satisfy \begin{equation*} \lim_{T\to \infty}\int_{0}^{T} A(t)dt = \infty. \end{equation*} Then \eqref{e1.2} is oscillatory. \end{theorem} The above cited results do not include a damping term. The main result is stated and proved in section 2 which includes a nonlinear damping term. \section{Main Result} In this section, we state and prove the main theorem of the paper. \begin{theorem} \label{thm2.1} Let there exist two divergent sequences $\{\tau_{n}\},\, \{\eta_{n}\} \subset \mathbb{R}^+ $ such that $0<\tau_{n}< \eta_{n}\leq \tau_{n+1}<\eta_{n+1}\leq \dots$, for all $n\in \mathbb{N}$. Let there exist a $C^1$ function $y$ defined on $[\tau_{n}, \eta_{n}]$ such that $y(\tau_{n})= 0 = y(\eta_{n})$, for all $n\in \mathbb{N}$. Let $g'(u)$ exist and there exist $\mu > 0$ such that $g'(u)\geq \mu^2>0$, $u g(u)> 0$, for all $0\neq u\in \mathbb{R}$ and $x f(t, x, u)\geq 0$, for all $(t, x, u)\in \mathbb{R}^+\times \mathbb{R}^2$, $x\neq 0$. Assume that there exist a $C^1$ function $F$ defined on $\mathbb{R}$ and a continuous function $h$ on $\mathbb{R}$ such that $F(0)=0$, $F(y(t))$ is not constant on $[\tau_{n}, \eta_{n}]$, for all $n\in \mathbb{N}$, $F'(y)=\mu h(y)$ with $[h(y(t))]^2\leq 4 F(y(t))$ and \begin{equation} \label{e2.1} \int_{\tau_{n}}^{\eta_{n}}[a(t)(y'(t))^2 - q(t) F(y(t))] dt < 0, \,\forall\, t\in [\tau_{n}, \eta_{n}], \quad \forall\,n\in \mathbb{N}. \end{equation} Then every solution of \eqref{e1.1} will vanish on $[\tau_{n}, \eta_{n}]$, for all $n\in \mathbb{N}$, and hence \eqref{e1.1} is oscillatory. \end{theorem} \begin{proof} Suppose on the contrary, there exist a solution $x$ of \eqref{e1.1} such that $x(t)\neq 0$, for all $t\in [\tau_{p}, \eta_{p}]$ for some $p\in \mathbb{N}$. Now there are two cases. \noindent\textbf{Case 1.} $x(t)> 0$, for all $t\in [\tau_{p}, \eta_{p}]$. We observe that the following is valid on $[\tau_{p}, \eta_{p}]$: \begin{align*} &a(t)(y'(t))^2 - q(t)F(y(t))+ \frac{F(y(t))}{g(x(t))}[(a(t)x'(t))' + p(t)f(t, x(t), x'(t))+ q(t)g(x(t))]\\ &= a(t)(x(t))^2\big[ \big(\frac{y(t)}{x(t)}\big)' \big]^2 +\Big( \frac{a(t)x'(t)F(y(t))}{g(x(t))}\Big)' -\Big(\frac{a(t)x'(t)F'(y(t))y'(t)}{g(x(t))}\Big)\\ &\quad -\Big(\frac{a(t)(x'(t))^2(y(t))^2}{(x(t))^2}\Big) +\Big(\frac{a(t)(x'(t))^2 g'(x(t)) F(y(t))}{(g(x(t)))^2}\Big) + \Big(\frac{2 a(t)y'(t)y(t)x'(t)}{(x(t))}\Big)\\ &\quad + \frac{F(y(t))}{g(x(t))} p(t)f(t, x(t), x'(t))\\ &\geq a(t)(x(t))^2\big[ \big(\frac{y(t)}{x(t)}\big)' \big]^2 + \Big( \frac{a(t)x'(t)F(y(t))}{g(x(t))}\Big)' - \Big(\frac{a(t)x'(t)\mu h(y(t))y'(t)}{g(x(t))}\Big)\\ &\quad -\Big(\frac{a(t)(x'(t))^2 (y(t))^2}{(x(t))^2}\Big) +\Big(\frac{a(t)(x'(t))^2 \mu^2 (h(y(t)))^2}{4 (g(x(t)))^2}\Big) + \Big(\frac{2 a(t)y'(t)y(t)x'(t)}{(x(t))}\Big)\\ &\quad + \frac{F(y(t))}{g(x(t))} p(t)f(t, x(t), x'(t))\\ &\geq \Big( \frac{a(t)x'(t)F(y(t))}{g(x(t))}\Big)' + a(t)\big[y'(t)- \frac{x'(t)\mu h(y(t))}{2 g(x(t))}\big]^2\\ &\quad +\frac{F(y(t))}{g(x(t))} p(t)f(t, x(t), x'(t)). \end{align*} Since $x$ is a solution of \eqref{e1.1}, so, we have \begin{equation} \label{e2.2} \begin{aligned} &a(t)(y'(t))^2 - q(t)F(y(t)) \\ &\geq \Big(\frac{a(t)x'(t)F(y(t))}{g(x(t))}\Big)' + a(t)\big[y'(t)- \frac{x'(t) \mu h(y(t))}{2 g(x(t))}\big]^2 \\ &\quad+\frac{F(y(t))}{g(x(t))} p(t)f(t, x(t), x'(t)). \end{aligned} \end{equation} An integration of \eqref{e2.2} on $[\tau_{p}, \eta_{p}]$ yields \begin{equation} \label{e2.3} \begin{aligned} &\int_{\tau_{p}}^{\eta_{p}} [a(t)(y'(t))^2 - q(t)F(y(t))]dt \\ &\geq \Big(\frac{a(t)x'(t)F(y(t))}{g(x(t))}\Big)_{\tau_{p}}^{\eta_{p}} + \int_{\tau_{p}}^{\eta_{p}} a(t) \big[y'(t)- \frac{x'(t)\mu h(y(t))}{2 g(x(t))}\big]^2 dt \\ &\quad +\int_{\tau_{p}}^{\eta_{p}}\frac{F(y(t))}{g(x(t))} p(t)f(t, x(t), x'(t))dt. \end{aligned} \end{equation} From this inequality, it follows that \begin{equation*} \int_{\tau_{p}}^{\eta_{p}} [a(t)(y'(t))^2 - q(t)F(y(t))]dt \geq 0, \end{equation*} which contradicts (2.1). \noindent\textbf{Case 2.} $x(t)< 0$ for all $t\in [\tau_{p}, \eta_{p}]$. The proof of case 2 is similar to that of case 1 and is omitted for the sake of brevity. This completes the proof. \end{proof} \begin{remark} \label{rmk2.2} \rm Consider the differential equation \begin{equation} \label{e2.4} (a(t)x'(t))' + p(t)f(t, x(t), x'(t))x'(t) + q(t)g(x(t))=0, \end{equation} where $a, p, q \in C(\mathbb{R}^+, \mathbb{R})$, $f \in C(\mathbb{R}^+\times \mathbb{R}^2, \mathbb{R})$, $g\in C(\mathbb{R}, \mathbb{R})$, $a(t)> 0$ and $p(t)\geq 0$. With the hypotheses of Theorem \ref{thm2.1}, if we replace the condition $xf(t, x, u)\geq 0$ for all $(t, x, u)\in \mathbb{R}^+\times \mathbb{R}^2$, $x\neq 0$ in Theorem \ref{thm2.1} by $x u f(t, x, u)\geq 0$ for all $(t, x, u)\in \mathbb{R}^+\times \mathbb{R}^2$, $x\neq 0$, then \eqref{e2.4} is oscillatory. \end{remark} \section{Examples} In this section, we construct some examples for illustration. \begin{example} \label{exa3.1} \rm Consider the differential equation \begin{equation} \label{e3.1} (a(t)x'(t))' + p(t)f(t, x(t), x'(t))+ q(t)g(x(t))=0, \end{equation} where $a(t)\equiv 1$, $p(t)\equiv 1$, $f(t, x, y)= x^3 e^y$, $q(t)= t^2 \sin t$ and $ g(x)= x+ x^{2n+1}, n\in \mathbb{N}$. With the choice of $y(t)= \sin t$, $\tau_{n}=(2n-1)\pi$, $\eta_{n}=(2n+1)\pi$, $F(y)= y^2$, $\mu=1$, it is easy to see that the hypotheses of Theorem \ref{thm2.1} are satisfied. Also, it is easy to verify \begin{equation*} \int_{(2n-1)\pi}^{(2n+1)\pi}[\cos^2 t - t^2 \sin t\, \sin^2 t ] dt < 0, \quad \forall\,n\in \mathbb{N}. \end{equation*} An application of Theorem \ref{thm2.1} implies that \eqref{e3.1} is oscillatory. \end{example} \begin{remark} \label{rmk3.2} \rm Let $a(t)\equiv 1$, $p(t)\equiv 0$, $q(t)= t^2 \sin t$ and $ g(x)= x$ in \eqref{e3.1}. Then none of the known criteria (see, \cite{Hartman, Leighton, Wintner}, \cite[Thms.\,3.3,\,3.5]{patricia}, \cite[Thm.\,3.1]{wtli}) can be applied to \eqref{e3.1}. \end{remark} \begin{remark} \label{rmk3.3} \rm Let $a(t)\equiv 1$, $p(t)\equiv 0$, $g(x)= x+x^3 $ in \eqref{e3.1}. Then \cite[Thm.\,3]{grace} cannot be applied to \eqref{e3.1}. \end{remark} \begin{example} \label{exa3.4} \rm Let $a, b\in \mathbb{R} $ and $a > 4$. Consider the damped Mathieu's equation \begin{equation} \label{e3.2} x''(t) + e^t x(t)(x'(t))^2 + (a+ b\cos 2t )x(t)=0. \end{equation} This equation can be viewed as \eqref{e3.1} with $a(t)\equiv 1$, $p(t)=e^t$, $f(t, x, y)= x y^2, q(t)=a+ b\cos 2t $ and $g(x)= x$. With the selection of $y(t)= \sin 2t$, $\tau_{n}= \frac{(n-1)\pi}{2}$, $\eta_{n}=\frac{(n+1)\pi}{2}$, $F(y)= y^2$, $\mu=1$, it is easy to verify the hypotheses of Theorem \ref{thm2.1}. Also, the condition \begin{equation*} \int_{\frac{(n-1)\pi}{2}}^{\frac{(n+1)\pi}{2}}[4\cos^2 2t - (a+ b\cos 2t) \sin^2 2t ] dt < 0, \forall\,a> 4, \quad \forall\,n\in \mathbb{N} \end{equation*} holds. Thus, from Theorem \ref{thm2.1}, \eqref{e3.2} is oscillatory. \end{example} \begin{example} \label{exa3.5} \rm Consider the equation \begin{equation} \label{e3.3} x''(t) + \cos t\, x'(t) + \sin t\, x(t) =0. \end{equation} This equation is oscillatory; see \cite[Cor.\,3]{jswon}. Here, we give another alternative which is simple. \eqref{e3.3} can be converted into \begin{equation} \label{e3.4} u''(t) + \Big(\frac {3\sin t}{2}- \frac{\cos^2 t}{4}\Big) u(t)=0, \end{equation} where $u(t)= x(t)e^{(\sin t)/2}$. \eqref{e3.4} can be viewed as \eqref{e3.1} with $a(t)\equiv 1$, $p(t)=0$, $q(t)=\big(\frac {3\sin t}{2}-\frac{\cos^2 t}{4}\big)$ and $ g(x)= x$. After setting $y(t)=\sin t$, $\tau_{n}=2n\pi$, $\eta_{n}=(2n+1)\pi$, $F(y)= y^2$, $\mu=1$, it is not difficult to satisfy the hypotheses of Theorem \ref{thm2.1} with \begin{equation*} \int_{2n\pi}^{(2n+1)\pi}\big[\cos^2 t -\big(\frac {3\sin t}{2}- \frac{\cos^2 t}{4}\big) \sin^2 t \big] dt < 0, \quad \forall\,n\in \mathbb{N}. \end{equation*} It follows from Theorem \ref{thm2.1} that \eqref{e3.4} is oscillatory. Since $u(t)= x(t)e^{(\sin t)/2}$ is an oscillation preserving substitution, so, \eqref{e3.3} is oscillatory. \end{example} \begin{remark}\rm The results of Li and Agarwal \cite{wli} cannot be applied to \eqref{e3.3}. \end{remark} Finally, it remains an open question if the result of this note can be modified for \eqref{e1.1} with linear damping and variable potential. \subsection*{Acknowledgements} The author thank the anonymous referee for his/her remarks concerning the style of the paper. \begin{thebibliography}{0} \bibitem{coles} W. J. Coles; \emph{Oscilllation criteria for nonlinear second order equations}, Ann. Mat. Pura Appl. \textbf{82} (1969), 123--134. \bibitem{grace} S. R. Grace, B. S. Lalli and C. C. Yeh; \emph{Oscillation theorems for nonlinear second order differential equations with a nonlinear damping term}, SIAM J. Math. Anal. \textbf{15} (1984), 1082--1093. \bibitem{Hartman} P. Hartman; \emph{On nonoscillatory linear differential equations of second order}, Amer. J. 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