\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 87, pp. 1--16.\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 2006 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2006/87\hfil Restricted total stability]
{Restricted total stability and total attractivity}
\author[G. Zappal\'a\hfil EJDE-2006/87\hfilneg]
{Giuseppe Zappal\'a}
\address{Giuseppe Zappala' \newline
Dipartimento di Matematica e Informatica dell'Universit\`a di
Catania, Viale Doria 6, 95125 Catania, Italy}
\email{zappala@dmi.unict.it}
\date{}
\thanks{Submitted February 13, 2006. Published August 2, 2006.}
\thanks{Supported by G.N.F.M. of INdAM and by P.R.A.}
\subjclass[2000]{34D20, 34C25, 34A34}
\keywords{Stability; attractivity; persistent disturbance}
\begin{abstract}
In this paper the new concepts of restricted total stability and total
attractivity is formulated. For this purpose the classical theory of
Malkin with suitable changes and the theory of limiting equations,
introduced by Sell developed by Artstein and Andreev, are used.
Significant examples are presented.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\section{Introduction}
Consider the differential system
\begin{equation}
\dot{x}=f(t,x),\quad x(t_{0})=x_{0} \label{a11}
\end{equation}
where $f\in C[ \mathbb{R}^{+}\times {R}^{n},{R}^{n}] $.
Assume that
\begin{equation*}
f(t,0)\equiv 0.
\end{equation*}
Denote the perturbation of (\ref{a11}) by
\begin{equation}
\dot{x}=f(t,x)+F(t,x),\quad x(t_{0})=x_{0} \label{a22}
\end{equation}
where $F\in C[\mathbb{R}^{+}\times {R}^{n},{R}^{n}]$ and
denote by $x(t,t_{0},x_{0})$ a solution of \eqref{a22}
through $(t_{0},x_{0})$.
Now, the zero solution, $x\equiv 0$, of (\ref{a11}) is called
\textbf{uniformly totally stable} [13] if, for every $\epsilon >0$ and for
each $t_{0}\geq 0$ there exist $\delta _{1}(\epsilon )$ and
$\delta_{2}(\epsilon )>0$ so that for all $x_{0}\in {R}^{n}$ with
$\Vert x_{0}\Vert <\delta _{1}$ and for all $F$ with
$\Vert F\Vert <\delta _{2}$ we obtain
$\Vert x(t,t_{0},x_{0})\Vert <\epsilon $ for $t\geq t_{0}$ where
$x(t,t_{0},x_{0})$ is a solution of \eqref{a22}.
Malkin [13] deduced, under appropriate hypotheses,
the following two properties.
\begin{enumerate}
\item The solution $x=0$ of $(A_{1})$ is uniformly totally stable (1944).
\item For all $\epsilon >0$ there exist $\delta _{1}, \delta _{2}>0$ and
for any $\eta \in ]0,\epsilon ]$ there exists
$\delta _{3}\in ]0,\delta _{2}]$ with the property that for all
$t_{0}\geq 0$, for all $x_{0}$ satisfying
$\Vert x_{0}\Vert <\delta _{1}$ and for all $F$ with
$\Vert F\Vert <\delta_{3}$ there exists $T>0$ such that
$\Vert x(t,t_{0},x_{0})\Vert <\eta $ for $t\geq T+t_{0}$ where
$x(t,t_{0},x_{0})$ is a solution of the
perturbed system (1952).
\end{enumerate}
The second property brings to mind the strong stability under perturbations
in generalized dynamical systems introduced by Seibert [15].
In the classical total stability theory of Malkin, the norm
$\Vert F(t,x)\Vert $ of the perturbing function must either be
bounded by a constant [12,13] or by a suitable function depending upon
$\epsilon $ [18]. Under either of these restrictions a solution $x(
t,t_{0},x_{0})$ through $(t_{0},x_{0})$ of the perturbed
differential equation remains in some arbitrarily small neighborhood of the
solution $x\equiv 0$ for all time $t\geq 0$ and $x_{0}$ small. Note that the
limit of the solution $\lim_{t\to \infty }x(t,t_{0},x_{0})$ is
uncertain.
In this paper we replace the bound for $\Vert F\Vert $ with a different
criteria on $F$. Under this new criteria we can define and motivate a new
concept called the restricted total stability of \eqref{a11}.
We then use this concept to motivate the total attractivity of \eqref{a11}.
The Liapunov submethods employed in this work are the families of Liapunov
functions used by Salvadori [14] and the theory of limiting equations by
Sell [16], Artstein [5,6,7] and Andreev [1,2,3,4]. The topological limit
theory by Cartan-Silov [17] is used to formulate several theorems. Finally,
the theoretical results are supported by significant examples.
\section{Preliminaries and basic concepts}
Let $I=\mathbb{R}^{+}=[0,+\infty [$ and $I'=]0,+\infty [$. Let
$\mathbb{R}^n$ be a real $n$-dimensional vector space.
Let $\|x\|$ be the Euclidean
norm of $x$ for any $x\in \mathbb{R}^n$. Following the convention
of Hahn [13], let
$K$ denote the class of all strictly increasing functions $c:I\to
I\in C$ which satisfy $c(0)=0$. Throughout this paper let
$b=b(u)$, $c=c(u)$, $m=m(u)\in K$,
let $\chi=\chi (t,x)$, $U=U(t,x)$, $V=V(t,x)$,
$W=W(t,x)\in C[R\times \mathbb{R}^n \to R]$ and $G=G(t,x)$,
$\Gamma =\Gamma(t,x)$ be continuous vector valued functions mapping
$I\times \mathbb{R}^n$ to $\mathbb{R}^n$ .
With $F\cdot G$ denote the scalar product of the vectors $F$
and $G$ in $\mathbb{R}^n$.
Also, let $h=h(t,x):I\times \mathbb{R}^n\to I$ and
$h_{0}=h_{0}(t,x):I\times \mathbb{R}^n\to I$ be continuous functions
having the property that
\begin{equation*}
\inf_{x\in \mathbb{R}^n}h(t,x)=\inf_{x\in \mathbb{R}^n}h_{0}(t,x)
=0 \quad \text{for every }t\in I.
\end{equation*}
We shall select the functions $h$ and $h_0$ as measures of stability.
Also, assume that the measure $h_{0}$ is uniformly finer than $h$. This
implies that there exists a constant $\lambda >0$ and a function $m\in K$ so
that when $h_{0}(t,x)<\lambda $ we have
$h(t,x)\leq m[h_{0}(t,x)]0$. Consider only the
perturbed systems
\begin{equation}
\dot{x}=\Gamma (t,x)=f(t,x)+F(t,x),\quad x(t_{0})=x_{0} \label{e2}
\end{equation}
such that $F\cdot G$$\leq 0$ where $f$ and $F$ are measurable functions for
$t\in I$ and continuous for $x\in \mathbb{R}^n$. Further assume that for every
closed and bounded subset $B\subseteq \mathbb{R}^n$ there exists a locally
integrable function $\sigma _{B}=\sigma _{B}(t)$ so that
\begin{equation}
\Vert f(t,x)\Vert \leq \sigma _{B}(t)\text{ and }
\Vert F(t,x)\Vert \leq \sigma _{B}(t)\quad\text{when }x\in B.
\label{care}
\end{equation}
These conditions of Caratheodory [8] ensure the existence and the general
continuity of the solutions for \eqref{e1} and \eqref{e2}.
Finally, let $[G]=\{ F(t,x):F\cdot G\leq 0\} $ be
the set of selected perturbations. And, as before, let
$x(t)=x(t,t_{0},x_{0}) $ be a solution of \eqref{e2} through
$(t_{0},x_{0})$.
We introduce the following definitions of restricted total stability and
attractivity.
\begin{definition}\label{rst} \rm
System \eqref{e1} is said to be \textbf{restrictedly totally
$(h_{0},h)$-stable} or $t$-stable, if, for every $t_{0}\in I$ and
$\epsilon >0$ there exists $\delta =\delta (t_{0},\epsilon )>0$
such that for all $x_{0}\in \mathbb{R}^n$ with $h_{0}(t_{0},x_{0})<\delta $
we have $h(t,x(t))<\epsilon $ for all $t\geq t_{0}$ where
$x(t)=x(t,t_{0},x_{0})$. If $\delta =\delta (\epsilon )$
we have the uniformity, in short $t,u$-stability.
\end{definition}
\begin{definition} \label{ta} \rm
System \eqref{e1} is called \textbf{totally $(h_{0},h)$-attractive}
or $t$-attractive, if there exists a function set $[G_{1}]\subseteq [
G]$ with the property that for all $t_{0}\in I$ there exists $\delta
_{1}=\delta _{1}(t_{0})>0$ so that for all $x_{0}\in {R}^{n}$
satisfying $h_{0}(t_{0},x_{0})<\delta _{1}$ and for all $\eta >0$ and $F\in
[G_{1}]$ there exists $T>0$ so that $h(t,x(t))<\eta $
for $t\geq t_{0}+T$. If $\delta $ is a constant and $T=T(\eta )$ the system
\eqref{e1} is $t,u$-attractive. If $\delta =+\infty $ we have global
$t$-attractivity.
\end{definition}
\begin{definition} \label{rtas} \rm
System \eqref{e1} is said to be \textbf{restrictedly totally
asymptotically stable} or $t$-asymptotically stable if it is $t$-stable and
$t$-attractive.
\end{definition}
\begin{definition} \label{rtusa} \rm
The system \eqref{e1} is said to be \textbf{restrictedly totally
uniformly asymptotically stable} or $t,u$-asymptotically stable if it is
$t,u$-stable and $t,u$-attractive.
\end{definition}
\subsection{The derivatives}
Let $q=q(u)$ be a function from $R$ to $R$.
Denote by $q_{u}$ the derivative of $q$ with respect to $u$. For a function
$V=V(t,x)$ we shall denote by $\dot{V}$ or $\dot{V_{1}}$ the derivative of $V$
computed along the solutions of the differential system \eqref{e1} and
also denote $V_{t}={\frac{{\partial V}}{{\partial t}}}$ and
$\mathop{\rm grad}V={\frac{{\partial V}}{{\partial x}}}$.
If we assume that $V=V(t,x)\in C^{1}$ is continuous with continuous
derivative, then Malkin [13] shows that
\begin{equation}
\dot{V_{2}}(t,x)=\dot{V_{1}}(t,x)+F(t,x)\cdot \mathop{\rm grad}V(t,x)
\label{e3}
\end{equation}
where $\dot{V_{1}}=V_{t}+f\mathop{\rm grad}V$ and
$\dot{V_{2}}$ is the derivative over \eqref{e2}.
Observe that if $\mathop{\rm grad}V=\chi G$ where $\chi =\chi (t,x)$$\geq 0$
and if $F\cdot G\leq 0$ we deduce that
$\dot{V_{2}}=\dot{V_{1}}+\chi F\cdot G\leq \dot{V_{1}}$.
Throughout this paper, assume that
$\lim (f_{1},\dots ,f_{m})=(a_{1},\dots ,a_{m})$
if and only if $\lim f_{j}=a_{j}$ for every $j=1,\dots,m$.
\section{The restricted total stability}
We use the Liapunov families of functions [14] in this section. The basic
advantage of this method is that the single function needs to satisfy less
rigid requirements.
\begin{theorem}\label{t31}
Suppose that for every $s>0$ there exists two functions
$V=V(t,x)\in C^{1}$ and $\chi =\chi (t,x)\in C$ and a constant
$l$ so that:
\begin{itemize}
\item[(i)] $h(t,x)=s$ implies $V(t,x)\geq l>0$
\item[(ii)] ${\lim_{h\to 0}V(t,x)}=0$
\item[(iii)] $\dot{V}(t,x)\leq 0$ on $Q(s)$
\item[(iv)] $\mathop{\rm grad}V(t,x)=(\chi G)(t,x)$ on $Q(s)$.
\end{itemize}
Then the system \eqref{e1} is $t$-stable.
\end{theorem}
\begin{proof}
Given $t_{0}\in I$, $\epsilon >0$ and $V, \chi , l$, by (ii) there
exists
$d>0$ so that $h(t_{0},x)t_{0}$ so that
$h(t',x(t'))=\epsilon $ with $h(t,x(t))<\epsilon $ for
$t\in [t_{0},t'[$ then we have $V(t',x(t'))\geq l$,
hence there exists $\tau \in ]t_{0},t'[$ where
$\dot{V_{2}}(\tau ,x(\tau ))>0$ which is a contradiction.
\end{proof}
The following proposition ensures the $t,u$-stability.
\begin{theorem} \label{t32}
Suppose that for every $s>0$ there exists two functions
$V=V(t,x)\in C^{1}$, $\chi =\chi (t,x)\in C$ and two constants
$l,L\in \mathbb{R}$ so that:
\begin{itemize}
\item[(i)] $\dot{V}(t,x)\leq 0$ on $Q(s)$
\item[(ii)] $h(t,x)=s$ implies $V(t,x)\geq l$
\item[(iii)] $00$ with $V,\chi ,l,L$; from (v) we can select
$\epsilon_{1} \in ]0, \epsilon[$ with
$V_{1},\chi _{1},l_{1},L_{1}$ so that $L_{1}0$ we obtain all the hypotheses of Theorem \ref{t31}. In
the following formulation, fix $\epsilon >0$. If $c[h(t,x)]**\lambda $ so that the sets
where $h_{0}(t,x)<\lambda '$ and $h(t,x)<\lambda '$ are
bounded in $x$. Further, assume that there exists a compact set
$D\subset \mathbb{R}^n$ so that $I\times D$ contains the set
$\{(t,x):h(t,x)\leq m(\lambda )\}$.
We combine the limiting equations theory formulated by Sell,
the convergence of limiting equations by Artstein
and the Liapunov's second method formulated by Andreev to study the
asymptotic aspect of $t$-stability.
\subsection{Precompactness}
Let $X=X(t,x):I\times \mathbb{R}^n\to \mathbb{R}^n$
be a vector valued function which is continuous in $x$ and measurable
in $t$. Then the precompactness conditions for $X(t,x)$ are [5]:
For every compact set $A\subset \mathbb{R}^n$ there exist two
locally $L_{1}$ functions $M_{A}(t)$ and $N_{A}(t)$ so that
if $x,y\in A$ and $t\in I$ then
\begin{itemize}
\item[(i)] $\Vert X(t,x)\Vert \leq M_{A}(t)$
\item[(ii)] $\Vert X(t,x)-X(t,y)\Vert \leq N_{A}(t)\Vert x-y\Vert $
\end{itemize}
where the functions $M_{A}(t)$ and $N_{A}(t)$ satisfy the following two
criteria:
\begin{itemize}
\item[(a)] For every $t,\epsilon >0$ there exists a
$\mu_{A}=\mu _{A}(\epsilon )>0$ so that if $E$ is a measurable
set in $I$, with measure less than $\mu _{A}$, and $E$ is contained in the
interval $[t, t+1]$, then $\int_{E}M_{A}(\tau )d{\tau }\leq \epsilon$.
\item[(b)] There exists a number $L_{A}>0$ so that
$\int_{t}^{t+1}N_{A}(\tau )d{\tau }\leq L_{A}$ for all $t\in I$.
\end{itemize}
We assume that the functions $f$ and $F$ satisfy the precompactness
conditions. Observe that: (i) we can select the compact sets $A,B,D$ so that
$A=B=D$; (ii) the above precompactness condition (ii) implies the uniqueness
of solutions of \eqref{e1}. We denote with $\Im $ the family of functions
from $I\times \mathbb{R}^n$ to $\mathbb{R}^n$ which satisfy the
precompactness conditions.
\subsection{The translates of the first kind}
Given $T>0, x\in D$, a sequence $t_{m}$ diverging to $+\infty $
and a solution $x=x(t)$ of \eqref{e2},
the functions $\Gamma _{m}(t,x)=\Gamma (t+t_{m},x)$, $x_{m}(t)=x(t+t_{m})$,
defined for $t\in [0,T]$ and $x\in D$, are called the translates of
$\Gamma (t,x)$ and $x(t)$; obviously ${\dot{x}}_{m}(t)=\Gamma
_{m}[t,x_{m}(t)]$ for $m\in \{1,2,\dots \}$ [16].
\subsection{The convergence and the limiting equations}
In this subsection, the mathematical problem, which we face,
is basically the convergence of the
translates previously considered and, particularly, to embed the functions
$\Gamma _{m}$ in a compact metric space [5]. According to Artstein, the
convergence in $\Im $ can be induced by a suitable metric defined on
equivalence classes of $\Im $. Therefore we do not distinguish between two
elements that differ only on some $t$-set of measure zero [6].
Hence, under the previous hypotheses of
precompactness, there exists, with respect to a suitable metric, a
subsequence $\{\Gamma _{r}(t,x)\}$ of $\{\Gamma_{m}(t,x)\}$
convergent to $g(t,x)$. In other words, the sequence
$\int_{0}^{t}\Gamma _{r}(u,x)du$ converges in $\mathbb{R}^n$ to
$\int_{0}^{t}g(u,x)du $ when $t\in [0,T]$ and $x\in D$ [5]. This
convergence concept is fairly weak and covers a wide family of functions.
More clearly; let $x\in A$, $t\in [0,T],m=1,2,\dots $ and
$G_{m}(t,x)=\int_{0}^{t}\Gamma _{m}(u,x)du$. Then, from the precompactness
conditions, one deduces the following:
\begin{itemize}
\item[(i)] $\|G_{m}(t,x)\|\leq \int_{0}^{t}\|\Gamma _{m}(u,x)\|\leq
2\int_{0}^{T}M_{A}(u)du<\rho (A)$
\item[(ii)] $\|G_{m}(t',x')-G_{m}(t'',x'')\|\leq
2\|x'-x''\|\int_{0}^{T}N_{A}(u)du+2\|\int_{t'}^{t''}M_{A}(u)du\|$
\end{itemize}
where $\rho (A)>0$ is a suitable constant. Hence we obtain, in order, the
equiboundedness and the equicontinuity for $G_{m}(t,x)$.
From the Ascoli-Arzela' theorem we deduce the uniform convergence for a
suitable subsequence $G_{r}$ of $G_{m}$. Since, on every closed interval
$[0,T]$, the functions $x_{m}(t)=x(t_{m})+\int_{0}^{t}\Gamma_{m}[u,x_{m}(u)]du$
are equibounded and equicontinuous, there exists a
subsequence $x_{r}(t)$ which converges uniformly to $y=y(t)$ where
$\dot{y}(t)=g(t,y(t))$. The functions $g(t,x)$ and $y(t)$ are called
limiting or limit functions with respect to $t_{m}$ for $\Gamma (t,x)$ and
$x(t)$; we similarly define\ the limiting equation of \eqref{e2} the
the following equation $\dot{x}=g(t,x)$.
[6,16]. Hence the limiting equations of the nonautonomous ordinary
differential equations are limit points of the translated equations. The
limit is taken in a prespecified and suitable space. The general motivation
for introducing the limiting equations is that there is a strong connection
between the asymptotic behavior of the solutions of the original equation
and the solutions of the limiting equations [6].
We suppose also that the perturbation $F(t,x)$ is \textbf{integrally
convergent to zero}. Then
\begin{equation}
{\lim_{m\to +\infty }\int_{\alpha }^{\beta }F}({\tau +t_{m},
{\phi }_{m}(\tau )}){d{\tau }}=0 \label{e4}
\end{equation}
whenever $\{{\phi }_{m}\}$ is a uniformly convergent sequence
of functions from $[\alpha ,\beta ]\subset I$ to $\mathbb{R}^n$ and
$\{t_{m}\}$ is a sequence diverging to $+\infty $ [7]. Consequently
systems \eqref{e1} and \eqref{e2} have identical limiting equations [4].
This result implies that we can develop the theory of total attractvity
according to the restricted total stability.
Consider the limit set of Andreev. This set is very important and is denoted
by $V_{\infty }^{-1}(t,c)$ [2]; Now, $x\in V_{\infty }^{-1}(t,c)$ if there
exists a sequence $\{t_{r}\}$ which diverges to $+\infty $ and
a sequence $\{x_{r}\}$ which converges to $x$ so that for
$t\geq 0$ and $c\in \mathbb{R}$ we have
\[
{\lim_{r\to +\infty }V(t+t_{r},x_{r})=c}.
\]
Let $\Im _{1}$ denote the metric space of the scalar precompact functions $U$
mapping $I\times \mathbb{R}^n$ to $I$. Then $U\in \Im _{1}$.
\section{The restricted total asymptotic stability and applications}
In this section we consider theorems and examples on restricted total
asymptotic stability in bounded domains. According to Artstein [7], the
Liapunov and La Salle [10] theoretical constructions are a very powerful
tool to establish the asymptotic and the uniform asymptotic stability of
ordinary differential equations. These two methods are direct methods. A
major role in our technique is played by the limiting equations. This
abstract characterization becomes practical in those cases where the
structure of the limiting equations is relatively easy.
\begin{theorem} \label{t51}
Under the first hypothises of Theorem \ref{t33} and under the
hypotheses and results of section 4, suppose that there exists
$U=U(t,x)\in \Im _{1}$ so that:
\begin{itemize}
\item[(i)] $\dot{V}(t,x)\leq -U(t,x)\leq 0$ on $I\times D$
\item[(ii)] ${\lim_{V\to 0}h(t,x)=0}$
\item[(iii)] for every limit pair $(g,W)$ of $(f,U)$ the set $\{(W=0)\cap
V_{\infty }^{-1}(t,c)\}$
contains no solutions of $\dot{x}=g(t,x)$, the limiting equation of
\eqref{e2}, for $c>0$.
\end{itemize}
Then the system \eqref{e1} is $t$-asymptotically stable.
\end{theorem}
\begin{proof}
Suppose that there exists $t_{0}\in I$ with the property that for all
$\delta >0$ there exists $x_{0}\in D$ satisfying $h_{0}(t_{0},x_{0})<\delta $
and there exists $\eta >0$ and a sequence $\{t_{m}\}$ diverging
to $+\infty $ so that we have $h(t_{m}',x(t_{m}'))\geq \eta $ where
$t_{m}'=t_{0}+t_{m}$ and $x(t)=x(t,t_{0},x_{0})$. Let $v(t)=V(t,x(t))$.
Then by (i), we deduce
\begin{equation*}
{\lim_{t\to +\infty }v(t)=c}\geq 0
\end{equation*}
If $c=0$ we have the proof; otherwise, when $c>0$, consider for every $T>0$
the following three sequences of translates, of the first kind, and an
obvious equality
\begin{gather}
\Gamma _{m}(t,x)=f(t+t_{m}',x)+F(t+t_{m}',x),\quad
U_{m}(t,x)=U(t+t_{m}',x) \label{e5}
\\
x_{m}(t)=x(t+t_{m}'),\quad \dot{x}_{m}(t)=\Gamma _{m}(
t,x_{m}(t))\label{e6}
\end{gather}
where $t\in [0,T]$ and $x\in D$. According to the convergence of
limiting equations, we can select from $\Gamma _{m}(t,x)$ and $U_{m}(t,x)$
two weakly converging subsequences $\Gamma _{r}(t,x)$ and $U_{r}(t,x)$ which
converge respectively to $g(t,x)$\ and $W(t,x)$. We can also select a
subsequence $x_{r}(t)$ which converges uniformly to $y(t)$ with
$\dot{y}(t)=g(t,y(t))$.
From $\dot{v}(t)\leq -U(t,x(t))\leq 0$ and for $r$ diverging to
$+\infty $ deduce successively:
\begin{equation}
v(t+t_{r}')-v(t_{r}')\leq -\int_{0}^{t}U_{r}(
u,x_{r}(u))du\leq 0 \label{e7}
\end{equation}
and
\begin{equation}
0=c-c\leq -\int_{0}^{t}W(u,y(u))du\leq 0. \label{e8}
\end{equation}
Hence $y(t)\in \{(W=0)\cap V_{\infty }^{-1}(t,c)\}$ for $c>0$ which is an
absurdity. Therefore
\begin{equation*}
{\lim_{t\to +\infty }h}({t,x(t)})={0}.
\end{equation*}
\end{proof}
The set $[G_{1}]$, relative to the definition of attractivity given in
section 2, is the set of functions belonging to $[G]$ which are measurable,
precompact and integrally convergent to zero.
\begin{corollary} \label{c52}
If $V(t,x)$ and $h(t,x)$ satisfy the
hypotheses of Theorem \ref{t51} and if there exists a function $c\in K$ so
that $V\leq c(h)$ then the system \eqref{e1} is $t,u$-stable and $t$-attractive.
\end{corollary}
We obtain the following theorem if we suppose also that the measure $h(t,x)$
is precompact.
\begin{theorem} \label{t53}
Under the assumptions and the results of section 4, suppose that
the system \eqref{e1} is $t,u$-stable and $t$-attractive. Let
$h\in \Im _{1}$ and assume that for every limit pair $(g,l)$ of $(f,h)$ the
set $h_{\infty }^{-1}(t,0)$ contains only the solutions $y=y(t)$ of the
limiting system $\dot{x}=g(t,x)$ so that $l(0,y(0))=0$ . Then the system
\eqref{e1} is $t,u$-asymptotically stable.
\end{theorem}
\begin{proof}
It is sufficient to show that the system \eqref{e1} is
uniformly attractive. Assume the contrary. Then for all $\delta >0$ there
exists $\eta >0,$ there exists $t_{m}\in I$ and there exists $x_{m}\in D$
satisfying $h_{0}(t_{m},x_{m})<\delta $, so that for some divergent sequence
$\{t_{m}'\}$ we have $h(t_{m}'',x_{m}(t_{m}''))\geq \eta $ where
$t_{m}''=t_{m}'+t_{m}$ and $x_{m}(t)=x(t,t_{m},x_{m})$. Consider
the sequences of translates
\begin{equation}
h_{m}(t,x)=h(t+t_{m}'',x),\quad \Gamma _{m}(t,x)=\Gamma
(t+t_{m}'',x),\quad z_{m}(t)=x_{m}(t+t_{m}'')
\label{e9}
\end{equation}
where $t\in [0,T(>0)]$, $x\in D$. We obtain
\begin{equation}
\dot{z}_{m}=\Gamma _{m}(t,z_{m}(t)),\quad
z_{m}(0)=x_{m}(t_{m}''),\quad h_{m}(0,z_{m}(0))
\geq \eta. \label{e10}
\end{equation}
Since the functions $z_{m}(t)$ are equibounded and equicontinuous we can
select, under the conditions of precompactness, three converging
subsequences $\Gamma _{r}(t,x)$ converging to $g(t,x)$, $h_{r}(t,x)$
converging to $l(t,x)$, $z_{r}(t)$ converging uniformly to $y(t)$ where
$t\in [0,T]$ and $x\in D$. Obviously
\begin{equation}
\dot{y}(t)=g(t,y(t)),\quad l(0,y(0))\geq \eta >0.
\label{e11}
\end{equation}
Since the attractivity implies that
\begin{equation}
{\lim_{r\to +\infty }}h(t+t_{r}'',x(t+t_{r}''))=0\label{e12}
\end{equation}
we deduce $y(t)\in h_{\infty }^{-1}(t,0)$ which is a contradiction. See
Theorem 3.3 of [19].
\end{proof}
\begin{remark} \label{r54} \rm
In this paper $t\in I$ and $x\in \mathbb{R}^n$, the restriction of the
present theory on a suitable subset of $\mathbb{R}^n$ is obvious.
\end{remark}
We conclude this section by studying the behavior of two differential
systems. We establish the more general classes of perturbations
corresponding to these two different types of stability.
\subsection{Example}
Consider the differential system from $I\times \mathbb{R}^{3}$
to $\mathbb{R}^{3}$
\begin{equation}
\begin{gathered}
\dot{x}=a^{2}(x^{2}-by^{2}+z^{2})x^{3}y^{4}+a^{2}(x^{2}
+by^{2}+z^{2})xy^{2}+a^{3}z=f_{1} \\
\dot{y}
=-a(x^{2}+by^{2}+z^{2})x^{2}y+a(x^{2}-by^{2}+z^{2})x^{4}y^{3}-a^{3}z=f_{2}
\\
\dot{z}=-a^{2}x+3a^{3}y=f_{3}
\end{gathered} \label{e13}
\end{equation}
where $a=a(t):I\to I$ and $b=b(t):I\to I$ are locally
integrable functions.
\begin{theorem}\label{t551}
Under the previous hypotheses suppose that
\begin{itemize}
\item[(i)] $0\leq a(t)\leq 2$
\item[(ii)] $h=x^{2}+3ay^{2}+az^{2}$ and $h_{0}=x^{2}+6y^{2}+2z^{2}$
\item[(iii)] $2x^{2}y^{2}\leq 1$
\item[(iv)] $\dot{a}(t)\leq 0$.
\end{itemize}
Then the system \eqref{e13} is $t,u$-stable with respect to
the perturbations
\[
F[-2xM_{1},-6ayM_{2},-2azM_{3}]
\]
where
$M_{\iota}=M_{\iota }(t,x,y,z)\geq 0$ $(\iota =1,2,3)$ are arbitrary
functions that satisfy, locally, the Caratheodory conditions
given in \eqref{care}.
\end{theorem}
\begin{proof}
Let $V=h$ and $G=\mathop{\rm grad}V=(2x,6ay,2az)$. Note that we obtain the
conditions of Theorem \ref{t33}. In fact we have
%\begin{eqnarray}
\begin{itemize}
\item[(i)] $V\geq h$
\item[(ii)] for every direction $\lim h=\lim V$
\item[(iii)] on the set $I\times \mathbb{R}^{3}$ the derivative $\dot{V}$ satisfies
\end{itemize}
\[
=8a^{2}(x^{2}-by^{2}+z^{2})x^{4}y^{4}-4a^{2}(x^{2}+by^{2}+z^{2})x^{2}y^{2}
+\dot{a}(3y^{2}+z^{2}).
\]
Also, on the set $\{(x, y, z):\,2x^{2}y^{2}\leq 1\}$, we obtain
\begin{equation}
\dot{V}\leq \dot{a}(3y^{2}+z^{2})=-U\leq 0. \label{e14}
\end{equation}
Observing that $F\cdot \mathop{\rm grad}V\leq 0$ we have the proof.
\end{proof}
\begin{theorem} \label{t552}
Under the hypotheses of Theorem \ref{t551}, suppose that
\begin{itemize}
\item[(i)] $a(t)$ and $b(t)$ are bounded
\item[(ii)] $M_{\iota }$ is integrally convergent to zero
\item[(iii)] ${\lim_{t\to +\infty }}\{a, \dot{a}, b\}=\{\alpha
>0, -\alpha _{1}$$<0, \beta >0\}$
\end{itemize}
Then the system \eqref{e13} is $t,u$-asymptotically stable.
\end{theorem}
\begin{proof}
Since, for j=1,2,3, the derivatives of $f_{j}$ with respect to $x,y,z$ are
bounded on every compact set $D\subset \mathbb{R}^{3}$, we deduce that system \eqref{e13} is precompact. From (ii) and (iii) the limiting system of \eqref{e13}
and the limiting system of a perturbed system are identical and unique. Let
$W=\lim $ $U=\alpha _{1}(3y^{2}+z^{2})$ and $l(x,y,z)=\lim $ $
h(t,x,y,z)=x^{2}+3\alpha y^{2}+\beta z^{2}$. Then on the set
$\{W=0\}=\{y=z=0\}$ the limiting system of \eqref{e13} assumes the form $\dot{x
}=0$, $0=0$ and $0=x$. Therefore, only the solution $x=y=z=0$ of the
limiting system belongs to $\{W=0\}$. Hence, according to Theorems \ref{t51}
and \ref{t53}, the result follows by observing that the set $\{(W=0)\cap
V_{\infty }^{-1}(t,c)\}$ contains no solutions of limiting equations for
$c>0 $ and $h(t,0,0,0)=l(0,0,0)=0$.
\end{proof}
We next consider the following application.
\subsection{Application}
Suppose that the adimensionate motion equations of
a particular rigid body, with a fixed point and variable mass, are given by
\begin{equation}
\begin{gathered}
\dot{A}p+2A\dot{p}+2(C-A)qr=2Pz\gamma _{2}-2f_{1}p-2f_{4}qr \\
\dot{A}q+2A\dot{q}+2(A-C)pr=-2Pz\gamma _{1}-2f_{2}q-2f_{5}pr \\
\dot{C}r+2C{\dot{r}}=2(f_{4}+f_{5})pq-2f_{3}r \\
\dot{\gamma}_{1}=r\gamma _{2}-q\gamma _{3},\quad \dot{\gamma}_{2}=p\gamma
_{3}-r\gamma _{1},\quad \dot{\gamma}_{3}=q\gamma _{1}-p\gamma _{2},~~\gamma
^{2}=1-\gamma _{3}.
\end{gathered} \label{e15}
\end{equation}
where
\begin{itemize}
\item[(i)] $A=A(t), C=C(t)$ from $I$ to $I'$ are continuous functions
\item[(ii)] $\dot{A}=\dot{A}(t)$, $\dot{C}=\dot{C}(t)$, $P=P(t),z=z(t)$
from $I$ to $R$ are locally integrable functions;
\item[(iii)] for $j=1,2,\dots ,5$ the functions
$f_{j}=f_{j}(t,p,q,r,\gamma_{1},\gamma _{2})$ from $I\times \mathbb{R}^{5}$ to $R$
satisfy the conditions of Caratheodory given in \eqref{care}
\item[(iv)] $p,q,r,\gamma _{1},\gamma _{2},\gamma _{3}$ are the unknown
variables.
\end{itemize}
We assume also that
\begin{itemize}
\item[(i)] $Pz<0$
\item[(ii)] There exists three constants $A_{0}, C_{0}, P_{0}$ so that
$A\geq A_{0}>0$, $C\geq C_{0}>0$, and $P\geq P_{0}>0$.
\end{itemize}
\begin{lemma}\label{l562}
If we select the auxiliary function of Matrosov's type [11]
\begin{equation}
V=\frac{1}{2}[A(p^{2}+q^{2})+C{r^{2}}]-\frac{1}{2}Pz(\gamma _{1}^{2}+\gamma
_{2}^{2}+\gamma^{4}) \label{e16}
\end{equation}
we deduce $\dot{V}=-(f_{1}p^{2}+f_{2}q^{2}+f_{3}r^{2})=-U$.
\end{lemma}
\begin{lemma}\label{l563}
If $f_{1}$, $f_{2}$, $f_{3}>0$ and $\min (A,C)>0$ then $V$ is
positive definite with $\dot{V}=-U\leq 0$.
\end{lemma}
During the rest of this section, we will assume that $4h=2V=h_{0}$. Then let
$G=[Ap,Aq,Cr,-Pz\gamma _{1},-Pz\gamma _{2},Pz(1-\gamma _{3})]$ and consider
only the perturbations
\begin{equation*}
F=[-ApM_{1},-AqM_{2},-CrM_{3},pz\gamma _{1}M_{4},Pz\gamma
_{2}M_{5},-Pz(1-\gamma _{3})M_{6}]
\end{equation*}
where $M_{\iota }=M_{\iota }(t,p,\dots,\gamma _{2}):I\times \mathbb{R}^{5}\to
I $ for $\iota =1,2,\dots ,6$ are arbitrary functions that satisfy
Caratheodory's conditions.
\begin{theorem}\label{t566}
Under the hyptheses of Lemma \ref{l563} and the above
assumptions and the definition of the perturbation $F$, the system \eqref{e15}
is $t,u$-stable. In fact we have
\begin{itemize}
\item[(i)] $V\geq h$
\item[(ii)] for every direction $\lim 2h=\lim V$
\item[(iii)] $\dot{V}\leq 0$
\item[(iv)] $\mathop{\rm grad}V=G$
\item[(v)] $F\cdot \mathop{\rm grad}V\leq 0$.
\end{itemize}
\end{theorem}
\begin{theorem}\label{t567} Under the previous hypotheses suppose that
\begin{itemize}
\item[(i)] ${\lim_{t\to +\infty }}\{A,C,\dot{A},\dot{C}, P, f_{3}\}
=\{A_{1}, C_{1}, A_{2}, C_{2}, P_{1}, \Lambda >0\}=d$
for some constant $d$ and where $A_{2}=2A_{1}\alpha >0$
\item[(ii)] ${\lim_{t\to +\infty }}\{f_{(j=1,2,4,5)}, z\}=\{0, 0\}$
\item[(iii)] $A, C, \dot{A}, \dot{C}, Pz$ are bounded
\item[(iv)] $M_{\iota }$ are integrally convergent to zero
\item[(v)] the functions $f_{j}$ and the derivatives of $f_{j}$ with
respect to $p, q, r, \gamma _{1}, \gamma _{2}$
are bounded on every compact set $D\subset \mathbb{R}^{5}$.
\end{itemize}
Then the system \eqref{e15} is $t,u$-asymptotically stable.
\end{theorem}
\begin{proof}
From the previous hypotheses and assumptions we have:
(i) the system \eqref{e15} is precompact;
(ii) the limit of \eqref{e15} and the limit of a
perturbed system are identical and unique. The mutual limiting system is
\begin{equation}
\begin{gathered}
A_{2}p+2A_{1}\dot{p}+2(C_{1}-A_{1})qr=0 \\
A_{2}q+2A_{1}\dot{q}+2(A_{1}-C_{1})pr=0 \\
C_{2}r+2C_{1}\dot{r}=-2\Lambda r
\end{gathered} \label{17}
\end{equation}
and the fourth line of \eqref{e15} is unchanged;
(iii) the limit of $U$ is $W=\Lambda r^{2}$ hence $W=0$ implies $r=0$.
Also, the limiting system on $\{W=0\}$ assumes the form
\begin{equation}
A_{2}p+2A_{1}\dot{p}=0,\quad A_{2}q+2A_{1}\dot{q}=0\,,\quad 0=0. \label{e18}
\end{equation}
From this we deduce $p=q=Ze^{-\alpha t}$ where $Z$ is some arbitrary
constant.
We complete the proof with the following considerations.
Since over the solutions of system \eqref{e18} we have
\begin{equation*}
{\lim_{t\to +\infty }}V=0
\end{equation*}
then the set $\{(W=0)\cap V_{\infty }^{-1}(t,c)\}$ contains no solutions of
limiting equations for $c>0$. According to Theorem \ref{t53}, since the
limit of $h$ is $l={\frac{{1}}{{4}}}[A_{1}(p^{2}+q^{2})+C_{1}r^{2}]$, the
unique solution of \eqref{e18} which belongs to $h^{-1}(t,0)$ and satisfies
$l=0$, is $p=q=r=0$. Therefore the system \eqref{e15} is $t,u$
-asymptotically stable with respect to the select measures.
\end{proof}
\subsection{Partial stability of motion}
In system \eqref{e15} suppose that:
\begin{itemize}
\item[(i)] $f_{j}=f_{j}(t,p,q,\gamma _{1},\gamma _{2})$ for $j=1, 2$
\item[(ii)] $f_{3}=f_{3}(r)$
\item[(iii)] $f_{4}=f_{5}=0$
\item[(iv)] the third equation $\dot{C}r+2C\dot{r}=-2f_{3}r$ determines,
as a solution for every $(t_{0},r_{0})$, a continuous function
$r=r(t,t_{0},r_{0})$ defined for $t\geq t_{0}$.
\end{itemize}
\begin{theorem}\label{t569}
Under the previous hypotheses, the system
\begin{equation}
\begin{gathered}
\dot{A}p+2A\dot{p}+2(C-A)qr=2Pz\gamma _{2}-2f_{1}p \\
\dot{A}q+2A\dot{q}+2(A-C)pr=-2Pz\gamma _{1}-2f_{2}q \\
\dot{\gamma}_{1}=r\gamma _{2}-q\gamma _{3}, \quad
\dot{\gamma}_{2}=p\gamma _{3}-r\gamma _{1}, \quad
\dot{\gamma}_{3}=q\gamma _{1}-p\gamma _{2}, \quad
\gamma^{2}=1-\gamma _{3}
\end{gathered} \label{e19}
\end{equation}
is $t,u$-stable with respect to the measures
$h={\frac{{1}}{{4}}}[A(p^{2}+q^{2})-Pz(\gamma _{1}^{2}
+\gamma _{2}^{2}+\gamma ^{4})]$ and $h_{0}=4h$.
\end{theorem}
\begin{proof}
Consider the new positive definite auxiliary function
\begin{equation*}
V={\frac{{1}}{{2}}}[A(p^{2}+q^{2})-Pz(\gamma _{1}^{2}+\gamma _{2}^{2}+\gamma
^{4})]
\end{equation*}
with the derivative
$\dot{V}=-(f_{1}p^{2}+f_{2}q^{2})=-U_{1}\leq 0$.
Select
\begin{gather*}
G=[Ap, Aq, -Pz\gamma _{1}, -Pz\gamma _{2}, Pz(1-\gamma _{3})],\\
F=[-ApM_{1},-AqM_{2},pz\gamma _{1}M_{3},pz\gamma _{2}M_{4},-pz(1-\gamma
_{3})M_{5}]
\end{gather*}
where $M_{\iota }$ satisfies the Caratheodory conditions. We deduce the
proof using the previous theory.
\end{proof}
Observe that the Matrosov motion [11] $p=q=\gamma_{1}=\gamma _{2}=\gamma =0$
and $r=r(t)$ describing an irregular rotation of
the body around the vertical axis of symmetry, is $t,u$-stable with respect
to the variables $p, q, \gamma _{1}, \gamma _{2}, \gamma $ and to the
select measures. Under the corresponding hypotheses of Theorem \ref{t567}
we also obtain the partial $t,u$ -attractivity with respect to
$p, q, \gamma _{1}, \gamma _{2}, \gamma $ of the system \eqref{e19}.
\section{Assumptions for the attractivity in $x$-unbounded domains}
\subsection*{Definitions and theorems}
In this section we consider the case where
\begin{equation*}
{\lim_{\|x\|\to +\infty }h(t,x)=0.}
\end{equation*}
Hence we can consider only the solutions $x=x(t)$ of \eqref{e2} so that
$h(t,x(t))\leq m(\lambda )$ for $t\geq t_{0}$. Since the
hypotheses of precompactness are not sufficient to obtain the previous
convergences discussed in section 4, we assume more restrictive conditions
that ensure the uniform convergence for suitable subsequences of translates.
In this section we use the known translates of the first kind and the
translates of the second kind which are defined later. We shall assume also
that the perturbation $F$ is integrally convergent to zero with respect to
two different divergent sequences.
Throughout this section, assume that the set $\{x\in
\mathbb{R}^n:h_{0}(t,x)<\lambda $ for some $t\in I\}$ is bounded and
that the system
\eqref{e1} is $t$-stable. Let $S$ be the set $S=\{x\in
\mathbb{R}^n:h(t,x)0$ are constants and where $F=F(t,x)$ and $U=U(t,x)$ satisfy the
above two criteria as well. Observe that criteria (ii) implies the
uniqueness of solutions. Let $t_{m}\in I$ and $z_{m}\in \mathbb{R}^n$
denote two sequences so that $\lim_{m\to \infty }t_{m}=\infty $ and
$\lim_{m\to \infty }\Vert z_{m}\Vert =\infty $.
\begin{definition} \label{d65} \rm
We say that $f_{2m}(t,x)=f(t+t_{m},x+z_{m})$,
$F_{2m}(t,x)=F(t+t_{m},x+z_{m})$, $U_{2m}(t,x)=U(t+t_{m},x+z_{m})$ are the
translates of second kind [3], respectively for $f,F,U$ when $t\in I$ and
$x\in \mathbb{R}^n$.
\end{definition}
Assume that the perturbation $F(t,x)$ is \textbf{integrally bi-convergent to
zero} with respect to the sequences $t_{m}$ and $z_{m}$ i.e: for every
sequence of uniformly convergent functions
$\phi _{m}(u):[\alpha ,\beta ]\subset I\to \mathbb{R}^n$ we have
\begin{equation}
\lim_{m\to +\infty }\int_{\alpha }^{\beta }F(t_{m}+s,\phi_{m}(s)+z_{m})ds=0.
\label{e20}
\end{equation}
Hence, as in section 4, the limit of the original system \eqref{e1} is the
same as the limit of the perturbed system \eqref{e2}.
\begin{theorem} \label{t67}
Given a differential system \eqref{e2}, a constant $T>0$ and a
sequence $\{t_{m}\}$ suppose there exists a solution $x=x(t)$
so that
\begin{equation*}
{\lim_{t\to +\infty }}\Vert x(t)\Vert =+\infty.
\end{equation*}
Let $z_{m}=x(t_{m})$, $z_{m}(t)=x(t+t_{m})$ and $z_{2m}(t)=z_{m}(t)-z_{m}$.
Then the translates of second kind, $z_{2m}(t),$ are equibounded and
equicontinuous for $t\in [0,T]$.
\end{theorem}
\begin{proof}
Since $x(t)=x(0)+{\int_{0}^{t}\Gamma }({u,x(u)}){du}$ for all
$t\in [0,T]$ we have
\begin{equation}
z_{2m}(t)=\int_{0}^{t}\Gamma _{2m}(u,z_{2m}(u))du=z_{m}(t)-z_{m}
\label{e21}
\end{equation}
and therefore from criteria (i) given in equation \eqref{criteria} we deduce
$\|z_{2m}(t)\|<2MT$ and $\|z_{2m}(t')-z_{2m}(t'')\|<2M|t'-t''|$.
\end{proof}
We now give a corollary to Theorem \ref{t67}.
\begin{corollary}\label{c671}
For all $T>0$, there exists a compact set $E\subset \mathbb{R}^n$ so
that $z_{2m}(t)\in E$ for $0\leq t\leq T$ and there exists a subsequence
$z_{2r}(t)$, defined on $[0,T]$ which converges uniformly to $\phi (t)$ [or
$y_{2}(t)$]. This subsequence converges to the limit of the
second kind of the solution $x(t)$.
\end{corollary}
\begin{theorem} \label{t68}
Suppose that $x(t)$ and $E$ are respectively the solution and the
compact set defined in Corollary \ref{c671}. Then the sequence of functions
$G_{2m}(t,x)=\int_{0}^{t}{\Gamma _{2m}(u,x)}du$, where $0\leq t\leq T, x\in
E, m=1,2,\dots $ is equibounded and equicontinuous.
\end{theorem}
\begin{proof}
From the two criteria given in equation \eqref{criteria} we deduce
that $\|G_{2m}(t,x)\|<2MT$, and $\|G_{2m}(t',x')-G_{2m}(t'',x'')\|<\epsilon $
when $|t'-t''|<{\frac{{\epsilon }}{{4M}}}$ and $\|x'-x''\|<{\frac{{\epsilon
}}{{4LT}}}$. Obviously the sequence of functions
$W_{2m}(t,x)=\int_{0}^{t}U_{2m}(u,x)du$ are also equibounded
and equicontinuous.
Consequently there exists two subsequences $G_{2r}(t,x)$ and $W_{2r}(t,x)$
which are uniformly convergent respectively to $\int_{0}^{t}{g_{2}(u,x)}du$
and to $\int_{0}^{t}W_{2}(u,x)du$. This implies that the functions $g_{2}$
and $W_{2}$ are limiting functions of second kind for $\Gamma$ and $U$.
Observe that from $\dot{x}(t)=f(t,x(t))+F(t,x(t))$
for $0\leq t\leq T$ we deduce
\begin{equation}
\dot{z}_{2m}(t)=f_{2m}(t,z_{2m}(t))+F_{2m}(
t,z_{2m}(t))\label{e22}
\end{equation}
and hence $\dot{\phi}(t)=g_{2}(t,\phi (t))$ for $0\leq t\leq T$.
\end{proof}
Theorems \ref{t67} and \ref{t68} are, respectively, the analog of
Lemmas 1 and 2 of [3]. We denote by $g_{1}(t,x)$, $y_{1}(t)$ and $W_{1}(t,x)$
respectively the limits of the first kind for $f(t,x)$, $x(t)$ and $U(t,x)$
with regard to $t_{m}$.
\begin{definition} \label{d69} \rm
Given $V=V(t,x)$, $t\geq 0$ and a constant $c\in \mathbb{R}$ we
define and denote by $V_{\infty }^{-2}(t,c)$ the set $\{x\in \mathbb{R}^n\}$
such that there exists two sequences $\{t_{m}\}$, $\{z_{m}\}$ for which
\begin{equation*}
{\lim_{m\to +\infty }V(t+t_{m},x+z_{m})}={\lim_{m\to +\infty
}V_{2m}(t,x)=c.}
\end{equation*}
\end{definition}
The previous definition recalls the definition of $N_{2}(t,c)$ given in [3].
\begin{theorem}\label{t610}
Under the first hypotheses of Theorem \ref{t33}, suppose that
\begin{itemize}
\item[(i)] $\dot{V}(t,x)\leq -U(t,x)\leq 0$ on $S$
\item[(ii)] ${\lim_{V\to 0}h=0}$
\item[(iii)] $j=1, 2$
\item[(iv)] for every pair $(g_{j},W_{j})$ limit of $(f,U)$ the set
$\{(W_{j}=0)\cap V_{\infty }^{-j}(t,c)\}$
contains no solutions of the limiting system $\dot{x}=g_{j}(t,x)$ for
every $c>0$.
\end{itemize}
Then the system \eqref{e1} is $t$-asymptotically stable.
\end{theorem}
\begin{proof}
The system \eqref{e1} is $t$-stable so suppose that
$t_{0}\in I$ exists so that for all $\delta >0$ there exists an $\eta >0$ and
there exists $x_{0}\in \mathbb{R}^{n}$ satisfying $h_{0}(t_{0},x_{0})<\delta
$ and there exists a sequence $t_{m}$ so that we have $h(t_{m}',x(t_{m}'))\geq \eta $ where $t_{m}'=t_{0}+t_{m}$
and $x(t)=x(t,t_{0},x_{0})$. Let $v(t)=V(t,x(t))$. Then, since
$\dot{V}\leq 0$, we have
\begin{equation*}
{\lim_{t\to +\infty }v(t)}=c\geq 0
\end{equation*}
and if $c=0$ we have the proof. If $c>0$ with $\Vert x(t)\Vert $ bounded we
proceed as in Theorem \ref{t51}. If $c>0$ and $\sup \{x^{2}(t)\}=+\infty $
then consider the sequences $x(t_{m}')=z_{m}$, $x(t+t_{m}')=z_{m}(t)$, $z_{2m}(t)=z_{m}(t)-z_{m}$ and the corresponding translates of
the second kind $\Gamma _{2m}(t,x)$, $U_{2m}(t,x)$ where $t\in [0,T]$
and $x\in E$. The set E is defined in Corollary \ref{c671}. We have
\begin{equation}
\dot{z}_{2m}(t)=\Gamma _{2m}(t,z_{2m}(t)). \label{e23}
\end{equation}
From the previous theorem and their assumptions there exist three subsequences
$\Gamma _{2r}(t,x)$ converging to $g_{2}(t,x)$, $U_{2r}(t,x)$ converging to
$W_{2}(t,x)$, and $z_{2r}(t)$ converging uniformly to $\phi (t)$ with $\dot{
\phi}(t)=g_{2}(t,\phi (t))$ on $[0,T]$. From (i) we deduce
\begin{equation}
v(t+t_{m}')-v(t_{m}')\leq -\int_{t_{m}'}^{t_{m}'+t}U(u,x(u))du\leq 0.
\label{e24}
\end{equation}
With the known procedure we obtain $W_{2}(t,\phi (t))=0$
$\forall t\in [0,T]$ and also $\phi (t)\in V_{\infty }^{-2}(t,c)$ with
$c>0$, which is a contradiction.
\end{proof}
\begin{theorem}\label{t611}
Under the hypotheses of Theorem \ref{t610}, suppose that:
\begin{itemize}
\item[(i)] $h$ satisfies the assumptions of Section 6
\item[(ii)] the system \eqref{e1} is $t,u$-stable
\item[(iii)] for every limiting pair $(g_{j},l_{j})$ of $(f,h)$ the set
$\{h_{\infty }^{-j}(t,0)\}$
contains only the solutions $y_{j}=y_{j}(t)$ of the limiting
system $\dot{x}=g_{i}(t,x)$ so that $l_{j}(0,y_{j}(0))=0$
when $j=1$ and $2$.
\end{itemize}
Then the system \eqref{e1} is $t,u$-asymptotically stable.
\end{theorem}
This theorem brings to mind Theorem \ref{t53}. In fact, given $T>0$ and
$t\in [0,T]$, if we let
\begin{gather*}
x_{m}(t)=x(t,t_{m},x_{m}),\quad
z_{m}(t)=x_{m}(t+t_{m}''),\\
z_{m}=z_{m}(0)=x_{m}(t_{m}''),\quad
z_{2m}(t)=z_{m}(t)-z_{m}
\end{gather*}
we find that the functions $z_{2m}(t)$ are equibounded and equicontinuous in
$[0,T]$, and so it is sufficient to establish the proof.
We proceed with an example.
\subsection{Example}
Consider the differential system from $I\times \mathbb{R}^{3}$
to $\mathbb{R}^{3}$
\begin{equation}
\begin{gathered}
\dot{u}={-qq_{u}+[(1-\mu )q-\frac{r}{p}]v} \\
\dot{v}={[1-(1-\mu )\frac{pq}{r}]qq_{u}+[\frac{1}{2}
(\frac{\dot{p}}{p}-\frac{\dot{r}}{r})-\frac{r}{p}]v-qrz}
\\
\dot{z}={\frac{r^{2}q}{p}v-\nu ^{2}z}
\end{gathered} \label{e25}
\end{equation}
and the vector ${G=[qq_{u},\frac{r}{p}v,z]}$. In system
\eqref{e25} $u,v,z$ are the unknown variables. The function
$q=q(u):\mathbb{R}\to \mathbb{R}$
belongs to $C^{2}$; $\mu =\mu (t,u,v,z):I\times \mathbb{R}^{3}\to I$
satisfies the Caratheodory conditions given in equation \eqref{care};
$p=p(t)$, $r=r(t):I\to R$ are continuous with locally integrable
derivatives $\dot{p}(t), \dot{r}(t)$; also $\nu =\nu (t):I\to \mathbb{R}$
is locally integrable. Relatively to $p$ and $r$ assume
$p_{1}\geq p(t)\geq p_{0}>0, r_{1}\geq r(t)\geq r_{0}>0$ with
$p_{0}, p_{1}$ and $r_{0}, r_{1} $ as constants.
\begin{theorem}\label{t6121}
Suppose that
\begin{itemize}
\item[(i)] $q(u)q_{u}(u)\neq 0$ for $u\in \mathbb{R}$
\item[(ii)] ${0<\frac{r}{p}\leq 2}$ for $t\geq 0$
\item[(iii)] ${h=q^{2}+\frac{r}{p}v^{2}+z^{2}}$, $h_{0}=q^{2}+2v^{2}+z^{2}$.
\end{itemize}
Then the system \eqref{e25} is $t$-stable with respect to the
following perturbations:
\begin{equation}
F=[-qq_{u}M_{1},\quad -\frac{r}{p}vM_{2},\quad -zM_{3}]
\label{e26}
\end{equation}
where $M_{\iota }=M_{\iota }(t,u,v,z):I\times \mathbb{R}^{3}\to I$ are
arbitrary functions for $\iota =1,2,3$ that satisfy the conditions given in
equation \eqref{care}. If the ratio of $r$ to $p$ is constant then the
system \eqref{e25} is $(t,u)$-stable.
\end{theorem}
\begin{proof}
If we select the auxiliary function
\begin{equation*}
{V=q^{2}+\frac{r}{p}v^{2}+z^{2}=h}
\end{equation*}
we obtain
\begin{equation}
\dot{V}=-2[(qq_{u})^{2}+(\frac{r}{p})^{2}v^{2}+\nu
^{2}z^{2}]\leq 0;\quad \mathop{\rm grad}V=2G. \label{e27}
\end{equation}
Now the proof follows by Theorem \ref{t33}, since ${\lim_{h\to
0}V=0}$ and vice versa and since $F\cdot \mathop{\rm grad}V\leq 0$.
\end{proof}
Observe that the sets where $h$ and $\|f\|<\sigma $, for every $\sigma >0$,
are bounded in $v,z$ and unbounded in $u$.
\begin{theorem}\label{t6122}
Under the hypothesis of Theorem \ref{t6121} suppose that
\begin{itemize}
\item[(i)] ${\lim_{u\to +\infty }}q(u)=0$ and
\[
{\lim_{t\to +\infty }}\{(1-\mu ), {\frac{r}{p}}, \nu , {
(\frac{\dot{p}}{p}-\frac{\dot{r}}{r})}\}=\{
0, r_{2}\neq 0, \nu _{2}\neq 0, 0\}
\]
\item[(ii)] the functions $q, q_{u}$ and $\mu $ are bounded with their
derivatives
with respect to $u, v, z$ on every compact set of $\mathbb{R}^{3}$
\item[(iii)] $M_{\iota }$ is integrally bi-convergent to zero.
\end{itemize}
Then the system \eqref{e25} is $t$-asymptotically stable.
\end{theorem}
\begin{proof}
System \eqref{e25} is precompact. The limit of system \eqref{e25} and
the limit of a perturbed system are identical and unique. Letting
\begin{equation*}
{U=(qq_{u})^{2}+(\frac{r}{p}v)^{2}+(\nu z)^{2}}
\end{equation*}
we obtain
\begin{equation*}
W_{1}=(qq_{u})^{2}+(r_{2}v)^{2}+(\nu _{2}z)^{2}
\end{equation*}
and so $\{W_{1}=0\}=\emptyset $. Since
\begin{equation*}
W_{2}=(r_{2}v)^{2}+(\nu _{2}z)^{2}
\quad\text{and}\quad
\{W_{2}=0\}=\{v=z=0\}
\end{equation*}
we consider the limiting system of second kind given by
\begin{equation}
\dot{u}=-r_{2}v,\quad \dot{v}=-r_{2}v,\quad \dot{z}=-{\nu _{2}}^{2}z.
\label{e28}
\end{equation}
On the set $\{W_{2}=0\}$ we have the system $\dot{u}=0$, $0=0$, $0=0$ with
the solutions $\{u=c,v=z=0\}$ for some constant $c$. Also $V=q^{2}(u)$.
Hence
\begin{equation*}
{\lim_{m\to \infty }V(t+t_{m},u+x_{m})}={\lim_{m\to \infty
}q^{2}(u+x_{m})=0.}
\end{equation*}
Therefore, the set $\{(W_{j}=0)\cap V_{\infty }^{-j}(t,0)\}$ contains no
solutions of first and second type for $c>0$ and $j=1,2$.
\end{proof}
\subsection*{Conclusion}
The total stability is a property of some differential equations that has
the power to influence the behavior of the solutions of some other
differential equations. Since, in literature, there is not an analogous work
relative to the attractivity, this paper is written to fill this gap.
\subsection*{Acknowledgements}
The author wants to thank A. S. Andreev for
the conversations on limiting equations theory and P. Anderson
for her help in translating and improving the look of this paper.
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\end{document}
**