\documentstyle[twoside]{article} %\input amssymb.def % used for R in Real numbers \pagestyle{myheadings} \markboth{\hfil Asymptotic instability of nonlinear differential equations \hfil EJDE--1997/16}% {EJDE--1997/16\hfil Rafael Avis \& Ra\'ul Naulin \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol.\ {\bf 1997}(1997), No.\ 16, pp. 1--7. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp (login: ftp) 147.26.103.110 or 129.120.3.113} \vspace{\bigskipamount} \\ Asymptotic instability of nonlinear differential equations \thanks{ {\em 1991 Mathematics Subject Classifications:} 39A11, 39A10.\hfil\break\indent {\em Key words and phrases:} Liapunov instability, $h$-stability. \hfil\break\indent \copyright 1997 Southwest Texas State University and University of North Texas. \hfil\break\indent Submitted July 9, 1997. Published October 15, 1997.} } \date{} \author{Rafael Avis \& Ra\'ul Naulin} \maketitle \newtheorem{theo}{Theorem} \newtheorem{defi}{Definition} \begin{abstract} This article shows that the zero solution to the system $$x'=A(t)x+f(t,x),\quad f(t,0)=0$$ is unstable. To show instability, we impose conditions on the nonlinear part $f(t,x)$ and on the fundamental matrix of the linear system $y'=A(t)y$. Our results generalize the instability results obtained by J. M. Bownds, Hatvani-Pint\'er, and K. L. Chiou. \end{abstract} \section{Introduction} Bownds \cite{bow} studied stability properties of the second order differential equations \begin{eqnarray} \label{ec1} &y''+a(t)y=0, & t\geq 0 \\ \label{ec2} &x''+a(t)x=f(t,x,x'), & t \geq 0\,, \end{eqnarray} where $a(t)$ is a continuous real-valued function. It is proved in \cite{bow} that, if (\ref{ec1}) has a stable zero solution and has another solution with the property $$\label{fuera} \limsup_{t \to \infty }(|y(t)|+|y'(t)|)>0\,,$$ then, under suitable conditions on $f$, there exists a solution $x$ to (\ref{ec2}) which satisfies (\ref{fuera}). Bownds \cite{bow} conjectured that this result is true without the stability assumption for (1), conjecture that was later proven in \cite{hat}. This result and some other ideas from \cite{hat} have opened interesting possibilities in the study of asymptotic instability, as shown in \cite{nau}. This article concerns the generalization of the results given in \cite{chi} for the systems \begin{eqnarray} \label{ec3} y'(t)& =& A(t) y(t), \quad t \geq 0 \\ \label{ec4} x'(t)& =& A(t)x(t)+f(t,x(t)), \quad f(t,0)=0,\; t \geq 0 \\ \label{ec5} x'(t)& =& A(t) x(t) + b(t), \quad t \geq 0\,, \end{eqnarray} where $A(t)$, $f(t,x)$ and $b(t)$ are continuous functions, and $f(t,x)$ satisfies $$\label{ec7} |f(t,x)|\leq \gamma(t)|x|^m\,,$$ where $m$ is a positive constant and $\gamma$ is an integrable function. The following two theorems are proven in \cite{chi}. {\flushleft Theorem A \cite{chi}} {\it Assume that the fundamental matrix, $\Phi$, of System (\ref{ec3}) satisfies $$\label{30} |\Phi(t)\Phi^{-1}(s)|\leq K\frac{h(t)}{h(s)}\,,\quad s\geq t\geq t_0\,,$$ for some constant $K$. If (\ref{ec7}) is fulfilled with $\gamma \in L^1[0,\infty)$, and there exists a solution $y(t)$ of (\ref{ec3}) such that $$\label{ec6} \limsup_{t\to \infty} |y(t)|0\,,$$ then there exists a nontrivial solution $x(t)$ to (\ref{ec4}) satisfying (\ref{ec6}). } {\flushleft Theorem B \cite{chi}} {\it If the linear system (\ref{ec3}) has a solution $y$ such that, $$\label{beta} 0<\limsup_{t\to \infty} |y(t)| \leq \infty\,,$$ then there exists a solution $x(t)$ of (\ref{ec5}) satisfying (\ref{beta}). } Our goal is to extend Theorems A and B for functions $f(t,x)$ for which (\ref{ec7}) holds more general functions $\gamma$. This generalization is obtained by using the notion of $h$-asymptotic instability. % PRELIMINARIES \section{Preliminaries } Let $V^n$ denote one of the spaces ${\bf R}^n$ or ${\bf C}^n$. In this space $|x|$ denotes a fixed norm of a vector $x$, and $|A|$ denotes the corresponding matrix-norm of matrix $A$. Throughout this article, the function $h$ is assumed to be positive and continuous, the interval $[0,+\infty)$ is denoted by $J$, and we use the following notation: \begin{itemize} \item $\displaystyle |x|_h =\sup_{t \geq 0} \left|\frac{x(t)}{h(t)}\right|$ \item $C_h = \{x :J \to V^n: \;x\mbox{ is continuous and }|x|_h<\infty\}$, \item $B_h[0,1] :=\{x \in C_h:\; |x|_h \leq 1 \; \}$, \item $L^1_h=\{ x :J\to V^n: \;\displaystyle \int^\infty_0 \frac{|x(t)|}{h(t)} ds < \infty \}$. \end{itemize} The following definitions are taken from \cite{pon}. \begin{defi} We say that the null solution to (\ref{ec4}) is: \paragraph{ $h$-Unstable } on $J$ iff there exist an $\varepsilon >0$ and $t_0 \in J$, such that for each $\delta >0$, there exist an initial condition $\xi_\delta$ and a $t_\delta>0$, such that $$|h(t_0)^{-1}\xi_\delta| < \delta,\; \mbox{ and } |h(t_\delta)^{-1}x(t_\delta,t_0, \xi_\delta)|\geq \varepsilon.$$ \paragraph{Asymptotically $h$-unstable} on $J$ iff $x=0$ is $h$-unstable or there exists a $t_0 \in J$, such that for any $\delta >0$, there exists $\xi \in V^n$ such that $$|h(t_0)^{-1}\xi_\delta| < \delta,\; \mbox{ and } \limsup_{t \to \infty }|h(t)^{-1} x(t,t_0,\xi_\delta)| >0.$$ \end{defi} % $h$-ASYMPTOTIC INSTABILITY \section{$h$-Asymptotic instability} \begin{theo} Assume that the fundamental matrix of system (\ref{ec3}) satisfies (\ref{30}), and the function $f(t,x)$ in (\ref{ec4}) satisfies (\ref{ec7}) with $\gamma \in L^1_{h^{1-m}}$. If there exists a solution of (\ref{ec3}), such that $$\label{40} 0<\limsup_{t\to \infty}|\frac{y(t)}{h(t)}| < \infty,$$ then there exists a nontrivial solution $x$ of (\ref{ec4}) with Property (\ref{40}). \end{theo} \paragraph{ Proof.} From (\ref{40}), we may assume that for a fixed $\varepsilon$, with $0 < \varepsilon <1$, $$\label{ec12}|h(t)^{-1}y(t)|\leq 1-\varepsilon ,\quad \forall t\geq 0\,.$$ Since $\gamma \in L^1_{h^{1-m}}$, there exists a positive $t_0$, such that $$\label{ec13} K\int^\infty_t h(s)^{m-1}\gamma (s)\, ds < \varepsilon \,,\quad \forall t\geq t_0,$$ where $K$ is the same constant that appears in (\ref{30}). We find a solution to (\ref{ec4}) by finding a solution to the integral equation $$x(t) =y(t)-\Phi (t)\int^\infty_t \Phi^{-1}(s)f(s,x(s))\,ds,\quad t\geq t_0\,,$$ on the set $B_h[0,1]$. For $x \in B_h[0,1]$, define $$\label{ec14}{\cal U}(x)(t) = y(t) - \int^\infty_t\Phi (t)\Phi^{-1}(s)f(s,x(s))\,ds.$$ Using (\ref{ec7}), (\ref{30}), and (\ref{ec12}), we obtain $$|h(t)^{-1}{\cal U}(x)(t)| \leq 1-\varepsilon + \int^\infty_t|h(t)^{-1}\Phi(t)\Phi^{-1}(s)|\gamma (s)|x(s)|^m\,ds\,.$$ For $t \geq t_0$ we obtain \begin{eqnarray*} |h(t)^{-1}{\cal U}(x)(t)| & \leq & 1-\varepsilon +\int^\infty_t|h(t)^{-1}\Phi (t)\Phi^{-1}(s)|\gamma (s)|h(s)h^{-1}(s)x(s)|^m\,ds\\ &\leq &1-\varepsilon +K\int^\infty_th(s)^{m-1}\gamma (s)\,ds \\ &\leq& 1-\varepsilon +\varepsilon = 1\,. \end{eqnarray*} Hence ${\cal U} : B_h[0,1] \to B_h[0,1]$. Now, we prove that ${\cal U}$ is continuous in the following sense: Suppose that a sequence $\left\{x_n\right\}$ in $C_h$ converges uniformly to $x$ on each compact subinterval of $J$, then ${\cal U}(x_n)$ converges uniformly to ${\cal U}(x)$ on each compact subinterval of $J$. For a fixed $T>t_0$, we will show the uniform convergence of $\left\{{\cal U}(x_n)\right\}$ on $[t_0,T]$. Choose $t_1>T$, such that $t>t_1$ implies $$\label{ec15}K\int^\infty_th(s)^{-1}\gamma (s)\,ds \leq \frac{\varepsilon}{4}\,.$$ By the uniform convergence of $\left\{x_n\right\}$ on the interval $[t_0,t_1]$, there exists a positive integer $N=N(\varepsilon,t_1)$, such that $n\geq N$ implies $$\label{ec16}|f(s,x_n(s))-f(s,x(s))|\leq \varepsilon \left[ 2Kt_1\sup_{[t_0,t_1]}|h(t)^{-1}|\right]^{-1},\;\forall s\in[t_0,t_1].$$ For $t\in [t_0,T]$ we write $$\label{ec17}|h(t)^{-1}\left[ {\cal U}(x_n)(t)-{\cal U}(x)(t)\right] |\leq I_1+I_2+I_3,$$ where \begin{eqnarray*} I_1&=&\int^{t_1}_t |h(t)^{-1}\Phi (t)\Phi^{-1}(s)||f(s,x_n(s))-f(s,x(s))|\,ds\\ I_2&=&\int^\infty_{t_1}|h(t)^{-1}\Phi (t)\Phi^{-1}(s)||f(s,x_n(s))|\,ds\\ I_3&=&\int^\infty_{t_1}|h(t)^{-1}\Phi (t)\Phi^{-1}(s)||f(s,x(s))|\,ds\,. \end{eqnarray*} From (\ref{ec7}) and (\ref{ec13}) we obtain $I_2 \leq \frac{\varepsilon}{4}$ and $I_3\leq \frac{\varepsilon}{4}$. From (\ref{ec15}) we have $I_1 \leq \frac{\varepsilon}{2}$. These estimates and (\ref{ec17}) yield $$|h(t)^{-1}[{\cal U}(x_n)(t) - {\cal U}(x)(t)]| \leq \varepsilon ,\;\forall t\in [t_0,T],$$ which proves the uniform convergence of ${\cal U}(x_n)$ to ${\cal U}(x)$ on $[t_0,T]$. Now, we prove that the set of functions ${\cal U}(B_h[0,1])$ is equicontinuous at each point $t\in [t_0, \infty)$. For each $x\in B_h[0,1]$, the function $z(t)={\cal U}(x)(t)$ is a solution of the non-homogeneous linear system $$z'(t)=A(t)z(t) + f(t,x(t)).$$ Since $$|h(t)^{-1}z(t)|=|h(t)^{-1}{\cal U}(x)(t)|\leq 1$$ and $|f(t,x(t))|$ is uniformly bounded on any finite $t$-interval, the set of all functions $z(t)={\cal U}(x)(t)$, with $x\in B_h[0,1]$, is equicontinuous at each point of $[t_0, \infty)$. In this manner all the hypotheses of the Schauder-Tychonoff theorem \cite{cop} are satisfied. Consequently, there exists $x\in B_h[0,1]$ such that $x(t)={\cal U}(x)(t)$, i.e. $x$ satisfies the integral equation $$x(t)=y(t) - \Phi (t)\int^\infty_t\Phi^{-1}(s)f(s,x(s))\,ds\,.$$ >From (\ref{40}) and $$\displaystyle\lim_{t \to \infty}\int^\infty_t \Phi(t) \Phi ^{-1}(s)f(s,x(s))\,ds=0\,,$$ we obtain $$\label{ec18}\lim_{t \to \infty }|h(t)^{-1}[x(t)-y(t)]|=0\,.$$ >From (\ref{40}) and (\ref{ec18}) we conclude that (\ref{40}) is satisfied with $y$ replaced by $x$, and this proof is complete. \hfil$\diamondsuit$ \paragraph{Remarks} Note that Theorem~A follows from Theorem 1, by putting $h(t)=1$. Also note that under the conditions of Theorem 1, if we assume that $$\limsup_{t \to \infty} h(t)=\infty\,,$$ then the trivial solution of (\ref{ec4}) is unstable in the sense of Liapounov. Let us consider Equation (\ref{ec4}) with $$A(t)=\left( \begin{array}{cr} -1 & 0\\ 0 & \frac{1}{t} \end{array}\right).$$ In this case the fundamental matrix $\Phi$ for system (\ref{ec3}) satisfies $$|\Phi(t)\Phi^{-1}(s)|\leq t/s ,\quad s\geq t.$$ Assume that $f(t,x)$ satisfies (\ref{ec7}) with $t^{m-1}\gamma \in L^1$. Then, according to Theorem~1, Equation (\ref{ec4}) yields a solution $x$ satisfying $$\limsup_{t\to \infty}|t^{-1}x(t)|>0.$$ This property implies instability in the sense of Liapounov. Note that this result cannot be obtained from Perron's theorem \cite{lev}, from Coppel's instability theorem \cite{co}, or from Theorem A. Our next goal is to generalize Theorem B. \begin{theo} If there exists a solution $y$ of (\ref{ec3}) satisfying $$\label{44} 0<\limsup_{t\to \infty}|h(t)^{-1}y(t)| \leq \infty\,,$$ then there exists a solution $x$ of (\ref{ec5}) with the same property. \end{theo} \paragraph{Proof.} Note that every solution $x(t)$ of (\ref{ec5}) has the form $$\label{ec21}x(t) = \Phi (t)c + \Phi (t)\int^t_0 \Phi^{-1}(t) b(s)\,ds\,.$$ Let $y(t)=\Phi (t)c$ be a solution that satisfies (\ref{44}). If $$\label{ec22}\limsup_{t\to \infty}|h(t)^{-1}\Phi (t)\int^t_0h(s)^{-1}\Phi^{-1}(s) b(s)ds|=0,$$ we multiply (\ref{ec21}) by $h(t)^{-1}$ to obtain \begin {eqnarray*} \lefteqn{\limsup_{t\to\infty}|h(t)^{-1}x(t)|} & & \\ &>&\limsup_{t\to\infty}|h(t)^{-1}y(t)| - \limsup_{t\to\infty}|h(t)^{-1}\Phi(t)\int^t_0h(s)^{-1}\Phi^{-1}(s)b(s)\,ds|\,. \end{eqnarray*} >From (\ref{44}) and (\ref{ec22}) it follows that $\limsup_{t\to\infty}|h(t)^{-1}x(t)|$ belongs to $(0, \infty]$. Therefore, (\ref{44}) is satisfied with $y$ replaced by $x$. On the other hand, if $$0<\label{ec23}\limsup_{t\to \infty}|h(t)^{-1}\Phi (t)\int^t_0h(s)^{-1}\Phi^{-1}(s)b(s)ds| \leq \infty\,,$$ the assertion of this theorem follows independently of (\ref{44}). \hfil$\diamondsuit$ \paragraph{Acknowledgments.} The authors express their gratitude to Consejo de Investigaci\'on of Universidad de Oriente for the financial support of Proyecto CI-5-025-00730/95. \begin{thebibliography}{99} \bibitem{bow} Bownds J. M. , Stability implications on the asymptotic between of second order differential equations, {\it Proc. Amer. Math. Soc}., 39, 169-172 (1973). \bibitem{chi} Chiou K. L. , Stability implications on asymptotic behavior of non linear systems, {\it Internat. J. Math and Math Sci}. , Vo. 5, No 1, 105-112 (1982). \bibitem{lev} Coddington E. A., and Levinson N., {\it Theory of Ordinary Differential Equation}, New York, McGraw-Hill (1955). \bibitem{co} Coppel W.A., On the stability of ordinary differential equations, {\it J. London Math. Soc.}, 39, 255-260 (1964). \bibitem{cop} Coppel W. A. , {\it Stability and Asymptotic Behavior of Differential Equations}, D. C. Heath and Company, Boston (1965). \bibitem{hat} Hatvani L., Pint\'er L., On perturbation of unstable second order linear differential equations, {\it Proc. Amer. Soc.}, 61, 36-38 (1976). \bibitem{nau} Naulin R., Instability of nonautonomous differential systems, to appear in {\it Differential Equations and Dynamical Systems} (1997). \bibitem{pon} Pinto M., Asymptotic integration of a system resulting from the perturbation of an h-system, {\it J. Math. Anal. and App.,} 131, 194-216 (1988). \end{thebibliography} {\sc Rafael Avis \& Ra\'ul Naulin \newline Departamento de Matem\'aticas, Universidad de Oriente \newline Cuman\'a 6101 A-285. Venezuela}\newline E-mail address: rnaulin@cumana.sucre.udo.edu.ve \end{document}