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\markboth{\hfil Instability of discrete systems \hfil EJDE--1998/33}
{EJDE--1998/33\hfil Ra\'ul Naulin \& Carmen J. Vanegas \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc Electronic Journal of Differential Equations},
Vol. {\bf 1998}(1998), No.~33, pp. 1--11. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp 147.26.103.110 or 129.120.3.113 (login: ftp)}
\vspace{\bigskipamount} \\
Instability of discrete systems
\thanks{ {\em 1991 Mathematics Subject Classifications:} 39A11, 39A10.
\hfil\break\indent
{\em Key words and phrases:} Instability, Perron's Theorem, discrete dichotomies.
\hfil\break\indent
\copyright 1998 Southwest Texas State University and University of
North Texas. \hfil\break\indent
Submitted September 17, 1998. Published December 8, 1998. \hfil\break\indent
Partially supported by Proyecto CI-5-025-00730/95} }
\date{}
\author{Ra\'ul Naulin \& Carmen J. Vanegas}
\maketitle
\begin{abstract}
In this paper, we give criteria for instability and asymptotic instability
for the null solution to the non-autonomous system of difference equations
$$ y(t+1)=A(t)y(t) + f(t,y(t)),\quad f(t,0)=0\,, $$
when the system $x(t+1)=A(t)x(t)$ is unstable.
In particular for $A$ constant, we study instability from a new point of view.
Our results are obtained using the method
of discrete dichotomies, and cover a class of difference systems for
which instability properties cannot be deduced from the classical results
by Perron and Coppel.
\end{abstract}
\newtheorem{theo}{Theorem}
\section{Introduction}
A classical result on Liapounov instability for the difference equation
\begin{equation}\label{1}
y(t+1)=Ay(t)+f(t,y(t)),\quad f(t,0)=0,\quad
t=0,1,2,\dots
\end{equation}
states that the null solution is unstable if the matrix $A$ has
an eigenvalue $\lambda $ satisfying $|\lambda | >1$, and
the nonlinear term $f(t,y)$ satisfies
$$
\lim_{|y|\to 0}\frac{f(t,y)}{|y|}=0\,.
$$
This result is known as Perron's Theorem on instability \cite{li,per}, and
has played an important role in the study of difference systems \cite{la}.
We are interested in the study of two questions related to Perron's
Theorem. First, when the matrix $A$ depends on $t$, and
second, when above limit is replaced by condition (F) below.
For the first question consider the non-autonomous difference system
\begin{equation} \label{2}
y(t+1)=A(t)y(t) + f(t,y(t)), \quad f(t,0)=0,
\end{equation}
where $f(t,y)$ is continuous in $y$ and
$A(t)$ is invertible at $t=0, 1, 2, \dots $. We remark that instability of
this system cannot be obtained through Perron's Theorem. Coppel \cite{cp2}
studied this problem for ordinary differential equations,
and Agarwal \cite{ag}
studied the difference equation case. Here we reproduce the result obtained
in \cite{ag}, which requires the concept of fundamental matrix.
For the nonautonomous system of difference equations
\begin{equation} \label{3}
x(t+1) = A(t)x(t), \quad t=0,1,2,3, \dots \,,
\end{equation}
the fundamental matrix is defined as
$$
\Phi(t)= \prod_{s=0}^{t-1}A(s)=A(t-1) \cdots A(1)A(0)\,,
$$
where $I$ denotes the identity matrix, and
$\prod_{s=0}^{-1}A(s)= I$.
\begin{theo}[\cite{ag}] \label{A} Assume that $f(t,y)$ is
continuous in the variable $y$, and that for some constant $\gamma$ and
$t=0,1,2,\dots$,
$$
|f(t,y)| \leq \gamma |y|\,.
$$
Also assume that there is a projection matrix $P \neq I$, and a constant
$K$ such that
$$
\sum^{t-1}_{s=t_0}|\Phi(t)P\Phi^{-1}(s+1)|+
\sum^\infty_{s=t}|\Phi(t)(I-P)\Phi^{-1}(s+1)| \leq K.
$$
Then the null solution to \mbox{\rm (\ref{2})} is unstable if $K\gamma<1$.
\end{theo}
This theorem is important because of its applications.
For example, Perron's Theorem can be proven easily form Theorem~\ref{A}.
However, instability of a large class of difference systems
cannot be obtained
using Theorem~\ref{A}.
The aim of this paper is to provide a method for
investigating the instability of (\ref{2}), relying on the dichotomy
properties of the non-autonomous system (\ref{3}).
According to Coppel \cite{cp2}, System (\ref{2}) must inherit
some kind of instability of (\ref{3}) under certain conditions on $f(t,y)$.
This idea was also proposed in \cite{rn} for ordinary differential
systems, and in \cite{lr2,lr3} for difference equations.
For the second question about Perron's Theorem, we assume that
$f(t,y)$ satisfies
\paragraph{Condition (F)} There exists a sequence of positive numbers
$\{\gamma (t)\}$, and $\alpha \geq 0$ such that
$$
|f(t,y)| \leq \gamma(t) |y|^\alpha\,, \forall t\,.
$$
Assuming that the matrix $A$ has an eigenvalue satisfying
$|\lambda| > 1$ we formulate the question: Under what conditions on
$\{\gamma(t) \}$ is the system (\ref{1}) unstable?
Notice that the hypothesis in Perron's Theorem is implied by Condition (F)
with $\alpha>1$ and $\gamma$ bounded. Also notice that the hypothesis of
Theorem~\ref{A} is implied by Condition (F) with $\alpha=1$ and
$\gamma$ bounded.
In this article we show instability under the assumption that $A$ has an
eigenvalue with magnitude larger than 1, and $f$ satisfies conditions
weaker than those of Perron's Theorem. See the remark in \S 5.
To the best of our knowledge, this is the first publication in
response to the question above.
\section{Notation and preliminaries}
The summation (discrete integral) $\sum_{s=m}^na_s $
is assumed to be equal to zero if $m>n$.
The set of non-negative integers is denoted by $\mbox{\bf N}$, i.e.,
$\mbox{\bf N}=\{0,1,2,\dots\}$.
Functions $h(t)$ and $k(t)$ denote sequences of positive numbers.
For $t_0 \in \mbox{\bf N}$, we put
$\mbox{\bf N}_{t_0}= \{t \in \mbox{\bf N}: t \geq t_{0}\}$. For
$m \leq n$, we define $\overline{m,n}=
\{ s: s \in \mbox{\bf N}, m \leq s \leq n\}$.
The sequences $\{y(t,t_0,\xi)\}$ and $\{x(t,t_0,\xi)\}$, respectively,
stand for the solutions to systems (\ref{2}) and (\ref{3}) with initial
condition $\xi$ at time $t_0$.
The spaces ${\mathbb R}^r$ and ${\mathbb C}^r$
with the norm $|\cdot |$ are denoted by $V$.
The term \lq\lq sequential space" means the space of sequences
with range in $V$.
For a sequence $x :\mbox{\bf N} \to V$, we define
$$
|x|_\infty =\sup \{|x(t)|: \; t \in \mbox{\bf N}\}\,, \quad
|x|_k= |k(\cdot)^{-1}x(\cdot)|_\infty\,.
$$
The space of sequences such that $|x|_\infty <\infty $
is denoted by $\ell^\infty$, and the space of sequences
such that $|x|_k<\infty$ by $\ell^\infty_k$.
In the space $\ell^\infty_k $, the closed ball
with center 0 with radius $\rho$ is denoted by
$B_k[0,\rho]= \{x \in \ell_k^\infty: |x|_k \leq \rho \} $.
On the set of initial conditions, we define
$$\displaylines{
V_k =\{\xi \in V: \{ k(t)^{-1}x(t,t_0,\xi)\} \in \ell^\infty \}\,,\cr
V_{k,0} =\{\xi \in V_k: \lim_{t \to \infty} k(t)^{-1}x(t,t_0,\xi)=0 \}. }
$$
Based on \cite{rn}, solutions to (\ref{2}) on an interval
$\mbox{\bf N}_{t_0}$ are classified as follows:
\paragraph{$h$-stable:}
If for each positive $\varepsilon $ there exists a
positive $\delta$ such that for any initial condition $y_0$ satisfying
$|h(t_0)^{-1}y_0| < \delta$, the solution
$y(t, t_0, y_0)$ satisfies
$|y(\cdot ,t_0,y_0)|_h <\varepsilon$ on $\mbox{\bf N}_{t_0}$.
\paragraph{$h$-unstable: }
If the null solution is not $h$-stable.
\paragraph{Asymptotically $h$-stable: }
If for each positive $\varepsilon $ there exists a positive $\delta$
such that any initial condition $y_0$ satisfying
$|h(t_0)^{-1}y_0| < \delta$, the solution
$y(t, t_0, y_0)$ satisfies
$|y(\cdot ,t_0,y_0)|_h <\varepsilon$
on $\mbox{\bf N}_{t_0}$, and
\begin{equation} \label{4}
\lim_{t \to \infty }
h(t)^{-1}y(t,t_0,y_0)=0\,. \end{equation}
\paragraph{Asymptotically $h$-unstable: } If the null solution is not
asymptotically $h$-stable.
We will assume that System (\ref{3}) has a certain dichotomy
behavior, but the analysis of instability would be restricted if we
limited our attention to the dichotomy properties described
by ordinary and exponential dichotomies only, \cite{ag}.
Therefore, we use $(h,k)$-dichotomies \cite{np4, m1} to study
system (\ref{3}).
\paragraph{Definition}
System (\ref{3}) has an $(h,k)$-dichotomy on {\bf N}, if
there exist a constant $K$ and a projection matrix $P$ such that
\begin{equation} \label{5}
\begin{array}{c}
|\Phi(t)P\Phi^{-1}(s)| \leq Kh(t)h(s)^{-1}, \quad 0\leq s \leq t , \\[6pt]
|\Phi(t)(I-P)\Phi^{-1}(s)| \leq K k(t)k(s)^{-1}, \quad 0\leq t \leq s.
\end{array}
\end{equation}
For short notation, $(h,h)$-dichotomies are called $h$-dichotomies.
An important class of $(h,k)$-dichotomies is given by those having
the following property.
\paragraph{Definition} An ordered pair $(h,k)$ is uniformly
compensated \cite{np4} if there exists a
positive constant $C$ such that
$$
h(t)h(s)^{-1} \leq C k(t)k(s)^{-1}, \quad t \geq s.
$$
\paragraph{Remark} If System (\ref{3}) has an $(h,k)$-dichotomy
(with projection $P$ and constant $K$),
and the pair $(h,k)$ is uniformly compensated (with constant $C$),
then the system has both an $h$ and a $k$-dichotomies with projection $P$
and constant $CK$.
Uniformly compensated dichotomies have the following property \cite{np}.
\begin{theo} \label{B} Assume that (\ref{3}) has an $(h,k)$-dichotomy,
and that the pair $(h,k)$ is compensated.
Then (\ref{3}) has an
$(h,k)$-dichotomy with projection $Q$ if and only if
$$
V_{h,0}\subset V_{k,0} \subset Q[V] \subset V_{h} \subset V_{k}.
$$
\end{theo}
We need the following version of the
Schauder fixed point theorem \cite{lloyd} in a later proof.
\begin{theo} \label{C}
Let $E$ be a Banach space with norm $|\cdot|$, and let ${\cal T}$
be an operator, ${\cal T}: \Omega \to \Omega$, where
$\Omega$ is a bounded, closed and convex subset of $E$.
If ${\cal T}(\Omega)$ is precompact, and
${\cal T}$ is continuous,
then there exists $x \in \Omega$, such that $ {\cal T}(x)=
x$. \end{theo}
In discrete analysis, the application of the Schauder theorem frequently
is accompanied by the following criterion for compactness.
\paragraph{Definition} A subset $\Omega$ of the sequential space
is equiconvergent to $0$, if for every
$\epsilon>0$ there exists $N \in \mbox{\bf N} $
such that for all $x \in \Omega$ and all $ n \geq N$,
$ |x(n)| <\varepsilon$.
\begin{theo} \label{D} If $\Omega\subset S$ is bounded, closed and
equiconvergent to $0$, then $\Omega $ is compact.
\end{theo}
For a future use, we also define the operator
$$
{\cal U}(y)(t) = \sum_{s=t_0}^{t-1}\Phi (t)
P\Phi^{-1}(s+1)f(s,y(s)) -
\sum_{s=t}^\infty \Phi (t)(I-P)\Phi^{-1}(s+1)f(s,y(s)).
$$
\section{Instability under contraction conditions }
In this section we assume that the nonlinear term of (\ref{2}) satisfies
\paragraph{Condition (L)} Assume that
for some positive $\rho_0$, and all $\rho \in (0, \rho_0)$
there exists a sequence $\gamma(t,\rho)$, such that
$$
|f(t,h(t)y)-f(t,h(t)z)|
\leq \gamma(t, \rho )|y-z|,\quad |z|,|y|\leq \rho\,.
$$
\begin{theo} \label{teo-1}
Assume that (\ref{3}) has an $h$-dichotomy and
(\ref{2}) satisfies Condition (L), with
\begin{equation} \label{6}
K \sum_{s=t_0}^\infty h(s+1)^{-1} \gamma(s,\rho) < 1.
\end{equation}
If $V_h \neq V$,
then the null solution of (\ref{2}) is not $h$-stable.
\end{theo}
\paragraph{Proof.} Assume that the null solution is
$h$-stable. Then for $\varepsilon = \rho \in (0, \rho_0)$
there exists a $\delta$ such that
$|h(t_0)^{-1}y_0|<\delta$
implies $|y(\cdot ,t_0,y_0)|_h< \rho $. We will show a contradiction to
this statement.
The estimate
\begin{equation} \label{7}
\begin{array}{rcl}
|h(t)^{-1}{\cal U}(y)(t)| & \leq & K \displaystyle
\sum_{s=t_0}^\infty h(s+1)^{-1}
|f(s,y(s))| \\[6pt] & \leq &
K \displaystyle \sum_{s=t_0}^\infty
h(s+1)^{-1} \gamma(s,\rho)\rho
\end{array}
\end{equation}
implies that ${\cal U}: B_h[0, \rho] \to B_h[0, \rho]$. Moreover, we
have the estimate
\begin{equation} \label{8}
|h(t)^{-1}({\cal U}(y)(t)-{\cal U}(z)(t))| \leq
K \displaystyle \sum_{s=t_0}^\infty
h(s+1)^{-1} \gamma(s,\rho) |y-z|_h.
\end{equation}
Let us consider the sequence
$$
x(t)=y(t,t_0, y_0) -
{\cal U}(y(\cdot ,t_0,y_0))(t),\;|h(t_0)^{-1}y_0|<\delta.
$$
It is easy to see
that $x$ is an $h$-bounded solution of (\ref{3}).
Hence $x(t_0) \in \Phi (t_0) [V_h]$. From
Theorem~\ref{B} we may assume that
$x(t_0) \in \Phi (t_0) P[V]$.
Let $y_0$ be chosen with the properties
\begin{equation} \label{9} y_0 \in \Phi (t_0)(I-P)[V],
\quad y_0 \neq 0, \quad
|h(t_0)^{-1}y_0|<\delta\,.
\end{equation}
From the definition of the sequence $x$ we obtain
$$
x(t_0)=y_0 -\Phi(t_0)(I-P)
\sum_{s=t_0}^\infty\Phi^{-1} (s+1)f(s,y(s,t_0,y_0))
$$
that belongs to $\Phi(t_0)(I-P)[V]$, which implies $x(t_0)=0$.
In this case $y(\cdot ,t_0,y_0)$ satisfies the integral equation
$$
y(\cdot ,t_0, y_0 )= {\cal U}(y(\cdot ,t_0, y_0)).
$$
Thus, any solution $y(\cdot , t_0, y_0)$, where $y_0$ satisfies
(\ref{9}), is a fixed point of the
dichotomy operator ${\cal U}$. But from
(\ref{7}) and (\ref{8}) we see that operator ${\cal U}$ is a contraction
acting from $B_h[0, \rho]$ to $B_h[0, \rho ]$. Moreover ${\cal
U}(0)=0$, therefore
$y(\cdot ,t_0,y_0)=0$
giving $y_0=y(t_0,t_0,y_0)=0$ which is a contradiction.
\begin{theo}\label{teo-2}
Under the hypotheses of Theorem \ref{teo-1},
if $V_{h,0} \neq V_h$, then
the null solution of (\ref{2}) is not asymptotically
$h$-stable on the interval $\mbox{\bf N}_{t_0}$.
\end{theo}
\paragraph{Proof.}
By contradiction assume that the null solution is
asymptotically $h$-stable. Then for $\varepsilon=1$ there
exists a positive
$\delta $ such that $|h(t_0)^{-1}y_0|< \delta $ implies
(\ref{4}). Let $0<\rho < \mbox{min}\{\rho_0, \delta\}$
and $\sigma $ be a small number such that
\begin{equation}\label{10}
\sigma + K \rho
\sum_{s=t_0}^\infty h(s+1)^{-1}\gamma(s,\rho) \leq \rho.
\end{equation}
Fixing a vector
$x_0 \in V_h\setminus V_{h,0}$ with the property
$|x(\cdot ,t_0,x_0)|_h < \sigma $, we introduce the
operator
$$
{\cal T}(y)(t)=x(t,t_0, x_0)+{\cal U}(y)(t).
$$
From the property (\ref{7}) and (\ref{10}) we obtain that
${\cal T}: B_h[0,\rho] \to B_h[0,\rho]$.
From condition (L), it follows that
$$
|{\cal T}(y)-{\cal T}(z) |_h \leq
K \sum_{s=t_0}^\infty h(s+1)^{-1} \gamma(s,\rho)|y-z|_h .
$$
Thus, condition (\ref{6}) implies that the
operator ${\cal T}$ is a contraction from the ball $B_h[0,\rho]$
into itself and
therefore has a unique fixed point $y(\cdot )$. This fixed point
is a solution of (\ref{2}). From Theorem~\ref{B} we
may assume that projection $P$ defining
the $h$-dichotomy satisfies the condition
\begin{equation} \label{11}
\lim_{t \to \infty }h(t)^{-1}\Phi (t)P=0 .
\end{equation}
From this property it follows the asymptotic formula
\begin{equation} \label{12} y(t)=x(t,t_0,x_0)+o(h(t)),
\end{equation}
where \lq\lq small o" is the standard Landau symbol.
Inasmuch as the initial
condition of the solution $y(\cdot )$ satisfies
$$
|h(t_0)^{-1}y(t_0)|\leq \rho < \delta,
$$
then $\lim_{t \rightarrow \infty} h(t)^{-1}y(t)=0$. But
$\lim_{t \rightarrow \infty} h(t)^{-1}x(t,t_0,x_0)\neq 0$
which contradicts (\ref{12}).
\section{General conditions for instability }
The contraction property of ${\cal U}$ is implied by the stringent Condition
(L), and it plays a very important role in proof of Theorem~\ref{teo-1}.
A more general situation can be considered by a small modification to the
monotone conditions imposed by Brauer and Wong in \cite{bw}.
In this section we will assume that $f(\cdot ,y)$ satisfies
\paragraph{Condition M} There exists a scalar-valued function $\psi(t,r)$
defined for $t\in \mbox{\bf N}$, $r\geq 0$, which is continuous, and
nondecreasing in $r$ for each fixed $t$,
such that
$$
|f(t,y)|\leq \psi (t,|y|).
$$
\begin{theo}\label{teo-3}
Assume that (\ref{3}) has an $(h,k)$-dichotomy, with
$(h,k)$ a compensated pair, and $f(\cdot ,y)$ satisfying
Condition (M). Also assume that there exists $\rho_0$ such that
for $0<\rho <\rho_0$,
\begin{equation} \label{13}
KC \sum_{s=t_0}^\infty k(s+1)^{-1} \psi(s,k(s)\rho) < \rho\,.
\end{equation}
Then, if $V_h \neq V_k$, the null solution to (\ref{2}) is $h$-unstable.
\end{theo}
\paragraph{Proof.}
By contradiction, assume that the null solution to (\ref{2}) is $h$-stable.
Then for $\varepsilon >0$, there exists a $\delta >0$
such that $|y(\cdot ,t_0, y_0)|_h < \varepsilon$ if
$|h(t_0)^{-1}y_0| < \delta$. Let
\begin{equation}
\label{14}\rho < \frac{h(t_0)}{k(t_0)} \delta \,.
\end{equation}
Choose a positive $\sigma $ satisfying
$$
\sigma + KC \sum_{s=t_0}^\infty
k(s+1)^{-1} \psi(s,k(s)\rho) \leq \rho,
$$
and fix an initial value $x_0 \in \Phi(t_0)[V_{k}]\setminus
\Phi(t_0)[V_{h}]$ such
that $|x(\cdot , t_0, x_0)|_{k} \leq \sigma $.
Let us consider the integral equation
$$
y={\cal T}(y),
$$
where
$$
{\cal T}(y)(t)=x(t, t_0, x_0)+{\cal U}(y)(t).
$$
{\flushleft \bf Step 1: } Show that ${\cal T}: B_{k}[0, \rho]
\to B_{k}[0, \rho]$.
From (\ref{5}) and (\ref{13}), we obtain
\begin{eqnarray*}
|k(t)^{-1}{\cal T}(y)(t)| & \leq & |k(t)^{-1}x(t,t_0,x_0)|+
k(t)^{-1}|{\cal U}(y)(t)| \\
& \leq & |k(t)^{-1}x(t,t_0,x_0)|+
KC \sum_{s=t_0}^\infty k(s+1)^{-1} \psi(s,k(s)\rho) \leq \rho\,.
\end{eqnarray*}
{\flushleft \bf Step 2: }
Prove that the operator ${\cal U}$ is continuous in the
$\ell^\infty_k$ metric.
Let $\mu > 0$, choose $T$ large enough such that
$$
KC \sum_{s=T}^\infty k(s+1)^{-1} \psi(s,k(s)\rho) \leq \mu /3\,.
$$
Therefore, for all $n=0,1,\ldots$, and all $t\geq T$ we have
$$
|k(t)^{-1} \sum_{s=T}^\infty \Phi(t)(I-P)\Phi^{-1}(s+1)f(s,y_n(s))|
\leq \mu /3\,.
$$
From this estimate we obtain
\begin{eqnarray}
({\cal U})(y_n)(t) & =& \sum_{s=t_0}^{t-1}
\Phi(t)P\Phi^{-1}(s+1)f(s,y_n(s)) \label{15}\\
&& - \sum_{s=t}^T \Phi(t)(I-P)\Phi^{-1}(s+1)f(s,y_n(s)) +
k(t) O(\mu /3). \nonumber
\end{eqnarray}
From this asymptotic formula, we observe that the uniform convergence
of $\{y_n\}$ to $y_\infty $
on the interval $\overline{0,T}$ implies the
convergence of $\{{\cal
U}(y_n)\}$ to ${\cal U}(y_\infty )$ in the
metric of the space
$\ell^\infty_k$.
{\flushleft \bf Step 3:} Prove that if $\{y_n\}$ is contained in
$B_{k}[0, \rho] $, then $\{k(t)^{-1 }{\cal U}(y_n)(t)\}$
is equiconvergent to zero. Notice that given a
positive number $\mu$, then there
exists a $T \in \mbox{\bf N}$ such that
(\ref{15}) is valid. From Theorem~\ref{B}, we may assume that
$$
\lim_{t \to \infty } k(t)^{-1}\Phi(t)P=0\,.
$$
From this limit and (\ref{15}), it follows that
$\{k(t)^{-1 }{\cal U}(y_n)(t)\}$ is equiconvergent to zero.
Because of steps 1--3, the conditions of Theorem~\ref{C} are fulfilled, and
therefore the operator ${\cal T}$ has a fixed point $y(\cdot )$
in the ball $B_{k}[0, \rho]$.
Since $|k(t_0)^{-1}y(t_0)|< \rho$, from (\ref{14}) we obtain
$|h(t_0)^{-1}y(t_0)|< \delta $, implying
that $y(\cdot )$ is an $h$-bounded
function. But condition (\ref{13}) and the compensation of the
$(h,k)$-dichotomy
imply the $h$-boundedness of the sequence ${\cal
U}(y)$. Since
$$
y(t)=x(t,t_0,x_0)+ {\cal U}(y)(t),
$$
the sequence $x(\cdot ,t_0,x_0)$ is $h$-bounded.
But this contradicts the choice of $x_0$.
\begin{theo}\label{teo-4}
Assume that (\ref{3}) has an $h$-dichotomy and $f(\cdot ,y)$ satisfies
Condition (M). Also assume that there exists a $\rho_0>0$ such that for
$0<\rho <\rho_0$,
$$
K \sum_{s=t_0}^\infty h(s+1)^{-1} \psi(s,h(s)\rho) < \rho\,.
$$
Then, if $V_{h,0} \neq V_h$, the null solution of (\ref{2}) is
asymptotically $h$-unstable.
\end{theo}
\paragraph{Proof.}
By contradiction, assume that the null solution to (\ref{2}) is
asymptotically $h$-stable. Then, for $\varepsilon=1$ there
exists a positive
$\delta $ such that $|h(t_0)^{-1}y_0|<\delta$ implies
$\lim_{t \to \infty } h(t)^{-1}y(t,t_0,y_0)=0$.
Let $0<\rho < \mbox{min}\{\rho_0, \delta\}$, and choose $\sigma $ positive
such that
$$
\sigma + K \sum_{s=t_0}^\infty h(s+1)^{-1}
\psi(s,h(s)\rho) \leq \rho.
$$
For an initial condition
$x_0\in \Phi(t_0)[V_h]\setminus \Phi(t_0)[V_{h,0}]$
such that
$|x(\cdot , t_0, x_0 )|_h<\sigma , $
we consider the
operator
$$
{\cal T}(y)(t)=x(t,t_0,x_0)+{\cal U}(y)(t).
$$
For any $y \in B_h[0,\rho] $ we have the estimate
$$
|{\cal T(y)}|_h\leq \sigma +
K \sum_{s=t_0}^\infty h(s+1)^{-1}\psi(s, h(s)\rho)\leq \rho\,,
$$
which implies that ${\cal T}: B_h[0,\rho] \to B_h[0,\rho]$.
By repeating the arguments given in the proof of Theorem~\ref{teo-3},
we conclude that this operator satisfies the conditions of
Theorem~\ref{C}. Therefore, there is a fixed point $y(\cdot )$
in the ball $B_h[0,\rho ]$; hence
$$
y=x( \cdot ,t_0, x_0)+{\cal U}(y).
$$
Because of Theorem~\ref{B}, we assume that projection $P$ defining
the $h$-dichotomy satisfies the condition (\ref{11}).
Therefore,
$$
y(t)=x(t,t_0, x_0)+o(h(t))
$$
which contradicts $y(\cdot )$ satisfying (\ref{4}) with
$x_0 \in V_h\setminus V_{h,0}$.
\section{A Perron like result}
In this section we assume that the matrix $A$ is constant and has
an eigenvalue with magnitude greater than 1. We also assume that
Conditions (F) is satisfied under two possible cases.
\paragraph{Case $0 \leq \alpha < 1$:}
Then there exists a real number $r$ in $(0,1)$, such that none of the
eigenvalues has magnitude 1, and at least one
eigenvalue $\lambda$ of matrix $rA$ satisfies $|\lambda | >1$.
The change of variable $y(t)=r^{-t}z(t)$ in (\ref{1}) yields
\begin{equation} \label{16}
z(t+1)=rAz(t) + r^{t+1}f(t, r^{-t}z(t)), \quad f(t,0)=0\,.
\end{equation}
Let
$$
R_1 = \min \{ |\lambda|: |\lambda|>1, \lambda \mbox{
is an eigenvalue of } rA \},
$$
and $\Phi_r (t)$ be the fundamental matrix of the
equation
$$
x(t+1)=r A x(t)\,.
$$
Let $R$ be a positive number satisfying $R_1^\alpha < R < R_1$. Then is is easy
to prove the existence of a projection matrix $P$ and a
constant $K\geq 1$, such that
\begin{eqnarray*}
& |\Phi_r(t) P \Phi^{-1}_r(s)| \leq K R^{t-s}, \quad 0 \leq s \leq t,& \\
& |\Phi_r(t) (I-P) \Phi^{-1}_r(s)| \leq K R^{t-s}_1, \quad 0 \leq t \leq s\,. &
\end{eqnarray*}
These estimates imply that the difference system $x(t+1)=rAx(t)$
has an $(R^{t},R_1^t)$-dichotomy (This is not an exponential dichotomy).
Since the condition $V_h \neq V_k$ is satisfied, then
we aim to apply Theorem~\ref{teo-3}. If condition (F)
is satisfied then the monotone condition (M) is valid with
$$
\psi(t,s)=\gamma(t)r^{(1-\alpha)t+1} s^\alpha.
$$
To satisfy (\ref{13}) we need
\begin{equation} \label{17}
KrR_1^{-1}\rho^\alpha \sum_{s=t_0}^\infty
\left(\frac{R_1}{r}\right)^{(\alpha-1)s} \gamma
(s) < \rho\,.
\end{equation}
Because $(R_1/r)^{(\alpha-1)} < 1$, the series in the above inequality
converges even for a $\gamma(t)$ of exponential growth, and (\ref{17}) is
satisfied for all $\rho$ sufficiently small.
Then by Theorem~\ref{teo-3} the null solution to (\ref{16})
is $R^t$-unstable. This implies the instability of the null solution
to (\ref{1}).
\paragraph{Remark} Instability of (\ref{1}) has been obtained under
conditions weaker than those in Theorem~\ref{A}. In Condition (F)
$\gamma$ is unbounded ($\gamma(t)= R^t$ with $R>1$), as opposed to
$\gamma$ being bounded in Theorem~\ref{A}.
\paragraph{Case $1 \leq \alpha $:}
This case can be reduced to the previous one, because stability of the
null solution to (\ref{1}) is equivalent to stability of the null
solution to
$$
y(t+1)=Ay(t) + F(t,y(t)), \quad F(t,0)=0,
$$
where $F(t,y)$ is defined by
$$
F(t,y)= \left\{
\begin{array}{rl}
f(t,y), & |y| < 2^{-1}, \\[5pt]
f(t, \frac{y}{2|y|}), & |y| \geq 2^{-1}.
\end{array}
\right.
$$
Notice that $F(t,y)$ satisfies Condition (F) with
$$
|F(t,y)| \leq \gamma(t) |y|^\beta, \quad \forall \beta \in [0,1)\,.
$$
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{\sc Ra\'ul Naulin}\\
Departamento de Matem\'aticas,
Universidad de Oriente \\
Cuman\'a 6101 Apartado 285, Venezuela \\
Email address: rnaulin@sucre.udo.edu.ve \bigskip
{\sc Carmen J. Vanegas} \\
Departamento de Matem\'aticas, Universidad Sim\'on Bol\'{\i}var \\
Caracas, Apartado 89000, Venezuela \\
Email address: cvanegas@usb.ve
\end{document}