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\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 
\begin{equation} \label{fuera}
\limsup_{t \to \infty }(|y(t)|+|y'(t)|)>0\,, 
\end{equation}
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
\begin{equation} 
\label{ec7} |f(t,x)|\leq \gamma(t)|x|^m\,,
\end{equation}
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 
\begin{equation}\label{30}
|\Phi(t)\Phi^{-1}(s)|\leq K\frac{h(t)}{h(s)}\,,\quad s\geq t\geq t_0\,,
\end{equation}
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
\begin{equation}
\label{ec6} \limsup_{t\to \infty} |y(t)|0\,,
\end{equation}
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,
\begin{equation} \label{beta}
0<\limsup_{t\to \infty} |y(t)|  \leq \infty\,,
\end{equation}
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 
\begin{equation}\label{40}
 0<\limsup_{t\to \infty}|\frac{y(t)}{h(t)}| < \infty,
\end{equation}
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 $0 < \varepsilon <1$, 
\begin{equation}
\label{ec12}|h(t)^{-1}y(t)|\leq 1-\varepsilon ,\quad  \forall t\geq 0\,.
\end{equation}
Since $\gamma \in L^1_{h^{1-m}}$, there exists a positive $t_0$, such that 
\begin{equation}
\label{ec13} K\int^\infty_t h(s)^{m-1}\gamma (s)\, ds < \varepsilon \,,\quad
\forall t\geq t_0, 
\end{equation}
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  
\begin{equation}
\label{ec14}{\cal U}(x)(t) = y(t) - \int^\infty_t\Phi (t)\Phi^{-1}(s)f(s,x(s))\,ds.
\end{equation}
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  
\begin{equation}
\label{ec15}K\int^\infty_th(s)^{-1}\gamma (s)\,ds \leq \frac{\varepsilon}{4}\,.
\end{equation}
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  
\begin{equation}
\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]. 
\end{equation} 
For $t\in [t_0,T]$ we write  
\begin{equation}
\label{ec17}|h(t)^{-1}\left[ {\cal U}(x_n)(t)-{\cal U}(x)(t)\right] |\leq I_1+I_2+I_3,
\end{equation}
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  
\begin{equation}
\label{ec18}\lim_{t \to \infty }|h(t)^{-1}[x(t)-y(t)]|=0\,.
\end{equation} 
>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 
\begin{equation}\label{44}
 0<\limsup_{t\to \infty}|h(t)^{-1}y(t)| \leq \infty\,,
\end{equation}
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
\begin{equation}
\label{ec21}x(t) = \Phi (t)c + \Phi (t)\int^t_0 \Phi^{-1}(t) b(s)\,ds\,.
\end{equation}
Let $y(t)=\Phi (t)c$ be a solution that satisfies (\ref{44}).
If 
\begin{equation}
\label{ec22}\limsup_{t\to \infty}|h(t)^{-1}\Phi (t)\int^t_0h(s)^{-1}\Phi^{-1}(s)
b(s)ds|=0, 
\end{equation}
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  
\begin{equation}
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\,,
\end{equation}
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}