\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 69, pp. 1--6.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2005/69\hfil Interval oscillation criteria] {Interval oscillation criteria for second order forced nonlinear matrix differential equations} \author[W.-T. Li, R.-K. Zhuang\hfil EJDE-2005/69\hfilneg] {Wan-Tong Li, Rong-Kun Zhuang} % in alphabetical order \address{Wan-Tong Li \hfill\break Department of Mathematics, Lanzhou University\\ Lanzhou, Gansu 730000, China} \email{wtli@lzu.edu.cn} \address{Rong-Kun Zhuang \hfill\break Department of Mathematics, Huizhou University\\ Huizhou, Guangdong 516015, China} \email{rkzhuang@163.com} \date{} \thanks{Submitted April 20, 2004. Published June 28, 2005.} \subjclass[2000]{34C10} \keywords{Interval oscillation; nonlinear matrix differential equation; \hfill\break\indent forcing term} \begin{abstract} New oscillation criteria are established for the nonlinear matrix differential equations with a forced term \[ [ r(t)Y'(t)] '+p(t)Y'(t)+Q(t)G( Y'(t)) F( Y(t)) =e(t)I_n. \] Our results extend and improve the recent results of Li and Agarwal for scalar cases. Furthermore, one example that dwell upon the importance of our results is included. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \section{Introduction} In this paper, we consider the oscillatory behavior of solutions of the forced second order nonlinear matrix differential equation \begin{equation} [ r(t)Y'(t)] '+p(t)Y'(t)+Q(t)G(Y'(t)) F( Y(t)) =e(t)I_n, \label{1.1} \end{equation} where $t\geq t_0$, $r(t)\in C^1([t_0,\infty )$, $(0,\infty ))$, $p(t)\in C([t_0,\infty ),(-\infty ,\infty ))$, $Q(t),G(Y'(t))$ are positive semi-definite matrices, $Q(t)$ is continuous, $F\in C^1(R^{n^2},R^{n^2})$, and the inverse of the matrix $F(Y(t))$ exists for all $Y(t)\neq 0$ and is denoted by $[F(Y(t))]^{-1}$. Moreover, $[F(Y(t))]^{-1}$ is positive definite and satisfies \cite{19} \begin{equation} ( [ F(Y(t))]^{-1}) ^T( Y'(t))^T=Y'(t)[F(Y(t))]^{-1} \label{1.2} \end{equation} for every solution $Y(t)$ of \eqref{1.1}, where $A^T$ is the transpose of $A$. We call a matrix function $Y(t)\in C^2([t_0,\infty ),R^{n^2})$ a prepared nontrivial solution of \eqref{1.1} if $\det Y(t)\neq 0$ for at least one $t\in [t_0,\infty ),$ $r(t)Y'(t)\in C^1([t_0,\infty ),R^{n^2})$ and $Y(t)$ satisfies \eqref{1.2}. A prepared solution $Y(t)$ of \eqref{1.1} is called oscillatory if $\det Y(t)$ has arbitrary large zeros. \eqref{1.1} is called oscillatory if every nontrivial prepared solution of \eqref{1.1} is oscillatory. Otherwise it is called non-oscillatory. For $n=1$, $p(t)=0$ and $G=1$, \eqref{1.1} has been studied by many authors, for example, Jaros, Kusano and Yoshida \cite{20} and their references. On the one hand, many authors assume that $Q(t)$ is nonnegative; see Skidmore and Leighton \cite{11} and Tenfel \cite{12}. In this case, one can usually establish oscillation criteria for more general nonlinear differential equation by employing a technique introduced by Kartsators \cite{4} where it is additionally assumed that $e(t)$ is the second derivative of an oscillatory function $h(t)$. On the other hand, most oscillation results involve the integral of $Q(t)$ and hence require the information of $r$ and $Q$ on the entire half-line $[t_0,\infty )$, see Li and Yan \cite{35} and their references. For $n>1$, Erbe, Kong and Ruan \cite{15}, Meng, Wang and Zheng \cite{17} and Etgen and Pawlowski \cite{16} obtain some generalized Kamenev type oscillation criterion for the linear matrix differential equation \begin{equation} ( R(t)Y'(t)) '+Q(t)Y(t)=0. \label{1.4} \end{equation} In 1999, Kong \cite{21} employed the technique from Philos \cite{40} for the second-order linear differential equations, and presented several interval oscillation criteria for \eqref{1.4} with $n=1$ (see Theorems 2.1 and 2.2 and their corollaries 2.1-2.4 in \cite{21}) involving the Kamenev's type condition. These results have been generalized by Li \cite{26}, Li and Agarwal \cite{29, 13} and Li and Cheng \cite{31}. Recently, Zhuang \cite{18} and Yang \cite{19} extended the results of \cite{21} to the matrix differential equation (\ref{1.4}) and to the nonlinear matrix differential equation \[ [ r(t)Y'(t)] '+p(t)Y'(t)+Q(t)G(Y'(t)) F( Y(t)) =0,t\geq t_0, \] respectively. However, the above results cannot be applied to the non-homogeneous nonlinear matrix differential equation \eqref{1.1}. Motivated by the ideas from Li and Agarwal \cite{29, 13}, in this paper we obtain, by using a matrix Riccati type transformation, some results of \cite{29, 13} are generalized to the nonlinear matrix differential equation \eqref{1.1}. For convenience of the reader, we introduce the following notation. Let $M$ be the linear space of $n\times n$ real matrices, $M_0\subset M$ be the subspace of symmetric matrices. For any real symmetric matrices $A, B, C\in M_0$, we write $A\geq B$ to mean that $A-B\geq 0$, that is, $A-B$ is positive semi-definite and $A>B$ to mean that $A-B>0$, that is $A-B$ is positive definite. We will use some properties of this ordering, viz., $A\geq B$ implies that $CAC\geq CBC$ and $A\geq B$ and $B\geq 0$ implies $A\geq 0$. Moreover, $A\geq B$ implies $\int_a^bAds\geq\int_a^bBds$ \section{Main Results} In the sequel we say that a function $H=H(t)$ belongs to a function class $D(a,b)=\{H\in C^1[a,b]:H(t)\not{\equiv }0,H(a)=H(b)=0\}$, denoted by $H\in D(a,b)$. \begin{lemma} \label{lemA} If $Y(t)$ is a nontrivial prepared solution of \eqref{1.1} and $\det Y(t)>0$ for $t\geq t_0$, then, for any $\rho (t)\in C^1([t_0,\infty ),(0,\infty ))$, the matrix \begin{equation} W(t)=\rho (t)r(t)Y'(t)[F(Y(t))]^{-1} \label{2.1} \end{equation} satisfies the equation \begin{equation} \label{2.2} \begin{aligned} W'(t) &=\big[ \frac{\rho '(t)}{\rho (t)}-\frac{p(t)}{r(t) }\big] W(t)-\rho (t)Q(t)G( Y'(t)) \\ &-\frac{W(t)F'( Y(t)) W(t)}{\rho (t)r(t)}+\rho (t)e(t)[F(Y(t))]^{-1} . \end{aligned} \end{equation} \end{lemma} \begin{proof} From \eqref{1.1}, we obtain \begin{align*} W'(t) &=\frac{\rho '(t)}{\rho (t)}W(t)+\rho (t)( r(t)Y'(t)) '[F(Y(t))]^{-1} +\rho (t)r(t)Y'(t)[ [F(Y(t))]^{-1} ] ' \\ &=\frac{\rho '(t)}{\rho (t)}W(t)+\rho (t)[ e(t)I_n-p(t)Y'(t)-Q(t)G( Y'(t)) F( Y(t)) ] [F(Y(t))]^{-1} \\ &\quad-\rho (t)r(t)Y'(t)[F(Y(t))]^{-1} F'( Y(t))Y'(t)[F(Y(t))]^{-1} \\ &=[ \frac{\rho '(t)}{\rho (t)}-\frac{p(t)}{r(t)}] W(t)-\rho (t)Q(t)G( Y'(t)) \\ &\quad -\frac{W(t)F'( Y(t)) W(t)}{\rho (t)r(t)}+\rho (t)e(t)[F(Y(t))]^{-1}. \end{align*} The proof is complete. \end{proof} \begin{theorem} \label{thm2.1} Suppose that for any $T\geq t_0$, there exist $T\leq aa\geq T_0$ such that $e(t)<0$ on the interval $I=[a,b]$. From \eqref{2.5} we see that $W(t)$ satisfies \begin{equation} \rho (t)Q(t)G( Y'(t)) <-W'(t)+\big[ \frac{\rho '(t)}{\rho (t)}-\frac{p(t)}{r(t)}\big] W(t)-\frac{W(t)F'( Y(t)) W(t)}{\rho (t)r(t)}. \label{2.6} \end{equation} Let $H\in D(a,b)$ be given as in hypothesis. Multiplying $H^2$ through (\ref {2.6}) and integrating over $I=[a,b]$, we have \begin{equation} \begin{aligned} &\int_a^bH^2(t)\rho (t)Q(t)G( Y'(t)) dt\\ &<-\int_a^bH^2(t)W'(t)dt +\int_a^bH^2(t)\big[ \frac{\rho '(t)}{\rho (t)}-\frac{p(t)}{r(t)} \big] W(t)dt\\ &\quad -\int_a^bH^2(t)\frac{W(t)F'( Y(t)) W(t)}{\rho (t)r(t)}dt. \label{2.7} \end{aligned} \end{equation} Integrating (\ref{2.7}) by parts and using that $H(a)=H(b)=0$, we have \begin{align*} &\int_a^bH^2(t)\rho (t)Q(t)G( Y'(t)) dt \\ &<-\int_a^b\Big\{ \frac{H^2(t)W(t)F'( Y(t)) W(t)}{\rho (t)r(t)} \\ &\quad -2H(t)H'(t)W(t)-H^2(t)[ \frac{\rho '(t)}{ \rho (t)}-\frac{p(t)}{r(t)}] W(t)\Big\} dt \\ &=-\int_a^b\Big\{ \frac{H^2(t)W(t)F'( Y(t)) W(t)}{\rho (t)r(t)} \\ &\quad -\Big[ 2H'(t)+( \frac{\rho '(t)}{\rho (t)}- \frac{p(t)}{r(t)}) H(t)\Big] H(t)W(t)\Big\} dt \\ &=-\int_a^b\Big\{ \frac{H(t)W(t)}{\sqrt{\rho (t)r(t)}} \\ &\quad -\frac{\sqrt{\rho (t)r(t)}\big[ 2H'(t)+( \frac{\rho '(t)}{\rho (t)}-\frac{p(t)}{r(t)}) H(t)\big] [ F'( Y(t)) ] ^{-1}}2\Big\} F'(Y(t)) \\ &\quad\times \Big\{ \frac{H(t)W(t)}{\sqrt{\rho (t)r(t)}}-\frac{\sqrt{\rho (t)r(t)}[ 2H'(t)+( \frac{\rho '(t)}{\rho (t)}- \frac{p(t)}{r(t)}) H(t)] [ F'( Y(t))] ^{-1}}2\Big\} dt \\ &\quad+\frac 14\int_a^b\rho (t)r(t)\Big[ 2H'(t)+( \frac{\rho '(t)}{\rho (t)}-\frac{p(t)}{r(t)}) H(t)\Big] ^2[ F'( Y(t)) ] ^{-1}dt \\ &\leq \frac 14\int_a^b\rho (t)r(t)\Big[ 2H'(t)+( \frac{\rho '(t)}{\rho (t)} -\frac{p(t)}{r(t)}) H(t)\Big] ^2[F'( Y(t)) ] ^{-1}dt, \end{align*} which contradicts the condition \eqref{2.3}. Hence every solution of \eqref{1.1} is oscillatory. The proof is complete. \end{proof} From Theorem \ref{thm2.1}, it is easy to see that the following important corollary is true. \begin{corollary} \label{coro2.1} Under the assumptions in Theorem \ref{thm2.1}, assume that $F'(Y)\geq A>0$ and $G(Y)\geq B>0$, where $A,B\in M_0$ are constant positive definite matrices such that \begin{equation} \int_a^bH^2(t)\rho (t)Q(t)Bdt\geq \frac 14\int_a^b\rho (t)r(t) \Big[ 2H'(t)+( \frac{\rho '(t)}{\rho (t)}-\frac{p(t)}{r(t)} ) H(t)\Big] ^2A^{-1}dt\,. \label{2.8} \end{equation} Then every solution of \eqref{1.1} is oscillatory. \end{corollary} We remark that if $n=1$, then Corollary \ref{coro2.1} reduces to the main result of Li and Agarwal \cite{13}. % Remark 2.1 \subsection*{Example} Consider the linear $n\times n$ matrix differential equation \begin{equation} ( \sqrt{t}Y'(t)) '-2Y'(t)+\frac 5{4% \sqrt{t}}Y(t)=\frac 1{\sqrt{t}}( \sin \sqrt{t}-\cos \sqrt{t}) I_n, \label{2.9} \end{equation} where $r(t)=\sqrt{t}$, $p(t)=-2$, $Q(t)=\frac{5}{4\sqrt{t}}$, $G(Y')=I_n$, $F(Y)=Y(t)$, and $F'(Y)=I_n$. Clearly, the zeros of the forcing term $\frac 1{\sqrt{t}}( \sin \sqrt{t}-\cos \sqrt{t}) I_n$ are $[ k\pi +\frac \pi 4] ^2$. Let \[ H(t)=\sin ( \sqrt{t}-\frac \pi 4) . \] For any $T>1$, choose $k$ sufficient large so that $((2k+1)\pi +\frac \pi 4)>T$ and set \[ a=[ (2k+1)\pi +\frac \pi 4] ^2,\quad b=[ 2(k+1)\pi +\frac \pi 4] ^2, \] then $e(t)\leq 0$ for $t\in [a,b]$. Pick up $\rho (t)\equiv 1$. It is easy to verify that \begin{align*} \int_a^bH^2(t)Q(t)Bdt &=\int_a^b\sin ^2( \sqrt{t}-\frac \pi 4) \frac{5}{4\sqrt{t}}I_ndt \\ &=\int_{(2k+1)\pi +\frac \pi 4}^{2(k+1)\pi +\frac \pi 4}\sin ^2( s-\frac \pi 4) \frac 5{4s}2sI_nds \\ &=\int_{(2k+1)\pi +\frac \pi 4}^{2(k+1)\pi +\frac \pi 4}\frac 52\sin ^2( s-\frac \pi 4) I_nds=\frac{5\pi }4I_n, \end{align*} and \begin{align*} &\frac 14\int_a^b\rho (t)r(t)\Big[ 2H'(t)+\big( \frac{\rho '(t)}{\rho (t)}-\frac{p(t)}{r(t)}\big) H(t)\Big] ^2A^{-1}dt \\ &=\frac 14\int_a^b\sqrt{t}\Big[ 2\frac{\cos ( \sqrt{t}-\frac \pi 4) }{2\sqrt{t}}+\frac 2{\sqrt{t}}\sin ( \sqrt{t}-\frac \pi 4) \Big] ^2I_ndt \\ &=\frac 14\int_{(2k+1)\pi +\frac \pi 4}^{2(k+1)\pi +\frac \pi 4}s\Big[ \frac{\cos ( s-\frac \pi 4) }s+\frac 2s\sin ( s-\frac \pi 4) \Big] ^2 2sI_nds \\ &=\frac 12\int_{(2k+1)\pi +\frac \pi 4}^{2(k+1)\pi +\frac \pi 4}\Big[ \cos ^2( s-\frac \pi 4) +2\sin ( 2s-\frac \pi 2) +4\sin ^2( s-\frac \pi 4) \Big] I_nds \\ &=\frac{5\pi } 4I_n, \end{align*} which implies that \eqref{2.8} holds. 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