\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 152, 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}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/152\hfil Stability properties]
{Stability properties of differential systems
under constantly acting perturbations}

\author[G. Cantarelli, G. Zappal\'a\hfil EJDE-2010/152\hfilneg]
{Giancarlo Cantarelli, Giuseppe Zappal\'a }  % in alphabetical order

\address{Giancarlo Cantarelli \newline
Dipartimento di Matematica dell'Universit\'a di Parma,
Via G.P. Usberti 53/A , 43124  Parma, Italy}
\email{giancarlo.cantarelli@unipr.it}

\address{Giuseppe Zappal\'a \newline
Dipartimento di Matematica e Informatica dell'Universit\'a di Catania,
Viale Doria 6, 95125 Catania, Italy}
\email{zappala@dmi.unict.it}

\dedicatory{In memory of Corrado Risito}

\thanks{Submitted January 17, 2010. Published October 21, 2010.}
\subjclass[2000]{34D20, 34C25, 34A34}
\keywords{Stability; persistent disturbance; two measures; Liapunov functions}

\begin{abstract}
 In this article, we find stability criteria
 for perturbed differential systems, in terms of two measures.
 Our main tool is a  definition of total stability
 based on two classes of perturbations.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}

\section{Introduction}

Let $\mathbb{R}^+$ denote the interval $0 \leq t < \infty$, and
$\mathbb{R}^n$ the $n$ dimensional Euclidean space with
the corresponding norm $\|x\|$ for $x\in \mathbb{R}^n$.
 Let us consider the Cauchy problem
\begin{equation}
\dot x=X(t,x), \quad  x(t_0)=x_0 \label{e1.1}
\end{equation}
and assume that $X(t,0) = 0$ for $t\in \mathbb{R}^+$. Note that
 this differential system has the null solution $x=0$.


In the classical total stability theory, it is required that the null
solution  be stable, not only with respect to
(small) perturbations of the initial conditions but also with
respect to the perturbations of the right-hand side of the
equation. To this end, we associate to the unperturbed
system \eqref{e1.1} a corresponding family of perturbed systems
\begin{equation}
\dot x=X(t,x)+X_p(t,x),\quad  x(t_0)=x_0. \label{e1.2}
\end{equation}
This differential system may not possess null solution, because
we assume only  that the right-hand side of
\eqref{e1.2} be suitably smooth in order to ensure existence,
uniqueness and  continuous dependance of solutions for the
initial value problem.


 For the convenience of the reader, we recall that the null solution
 of  \eqref{e1.1} is said to be \emph{totally
uniformly stable}, according to  Dubosin-Malkin Definition
\cite{d1,r1}, provided that for arbitrary positive $\epsilon$ and
$t_0 \geq 0$ there are $\delta_1 = \delta_1(\epsilon ) > 0$ and
$\delta_2 = \delta_2(\epsilon ) > 0$ such that whenever $\|x_0\| <
\delta_1$ and $\|X_p\| < \delta_2$, the inequality
$\|x(t,t_0,x_0)\| < \epsilon$ is satisfied for all $t \geq t_0$.
Notice that in the classical total stability theory (and in
the present paper) the symbol  $x(t) = x(t,t_0,x_0)$
denotes the solution of  \eqref{e1.2} through a point $(t_0,x_0)$.

We emphasize that our stability criteria in Section 3 generalize
two well-known Malkin theorems. In fact, a  Malkin theorem
\cite{m1,r1}
on the \emph{total uniform stability} is included as a special case
in Theorem \ref{thm3.1}, while Theorem \ref{thm3.5}
 improves another Malkin theorem \cite{m2,r1}.
There under appropriate hypotheses Malkin proves that
\begin{quote}
For arbitrary positive $\epsilon$ and $t_0 \geq 0$, there are
$\delta_1 = \delta_1(\epsilon ) > 0$ and $\delta_2 = \delta_2(\epsilon ) > 0$
and for any $\eta \in ]0,\epsilon[$ there is
$\delta_3(\eta)  \in  ]0,\delta_2]$ such that
whenever $\|x_0\| < \delta_1$ and $\|X_p\| < \delta_3$ there exists a constant
$T_{\eta }>0 $, such that
$\|x(t,t_0,x_0)\| < \epsilon$ is satisfied for all $t \geq t_0 + T_\eta$.
\end{quote}
It is worth noting that this  property  first comes the concept of
the \emph{strong stability under perturbations in generalized
dynamical systems} introduced by Seibert \cite{s2}.

The aim of the present article is to introduce and study a new type
of total stability in terms of two measures,  by splitting
the perturbation terms $X_p$ in two parts.  Namely, by
putting $ X_p = Y + Z$.
In Sections 3,4, 5, we require the usual upper restriction on the
Euclidean norm of vector $Z$,  while we
select vector $Y$ by an appropriate scalar product.
In Section 6, a mechanical example illustrates our
theoretical results.

\section{Preliminaries, notation and  basic ideas}

Let $K := \{a : \mathbb{R}^+\to \mathbb{R}^+:
\text{continuous, strictly increasing}, a(0) = 0 \}$
be the set of functions of class $K$ in the sense of Hahn.
We shall define some concepts in terms of two measures
\cite{l2,l3,m4}.
Namely,  we denote by
$h(t,x)$ and $h_0(t,x)$ two continuous scalar functions satisfying
the conditions:
\begin{itemize}
\item[(i)] $\inf_x h_0(t,x)=0$   for every $t \in \mathbb{R}$;

\item[(ii)]  there exists a positive constant $\lambda$ and a function
$m=m(u) \in K$  such that
$h_0(t,x)< \lambda$ implies $h(t,x) \leq m[h_0(t,x)]<m(\lambda)$.

\end{itemize}

In mathematical language,  condition (ii) means that $h_0$ is
\emph{uniformly finer} than $h$, and it implies that
$\inf_x h(t,x)=0$ for every $t \in \mathbb{R}$.

Putting  $Q(s)=\{ (t,x) \in \mathbb{R}^+\times \mathbb{R}^n:
0<h(t,x)\leq s \}$, we observe that   $0<s'<s$ implies
$Q(s') \subseteq Q(s)$, and moreover the intersection $\cap Q(s)$ for all $s > 0$ is
the empty set.  Hence, the set of the sets
$\{Q(s)\}$ represents a Cartan-Silov direction or, simply, a direction.

 The above theoretical concepts are essential in the following
definition:
For every scalar $V=V(t,x)$ we say that $\lim_{h\to 0}V(t,x)=0$
if and only if for every direction  such that $\lim h(t,x)=0$, we have
$\lim V(t,x)=0$,  see \cite{s3}.


Denote by $U = U(t,x)$ and $G = G(t,x)$ respectively a continuous
scalar function and a continuous $n$-vector function such that
$\|G\| > 0$ on $\mathbb{R}^+\times \mathbb{R}^n$.  For the
unperturbed  differential system
\begin{equation}
\dot x=X(t,x),\quad x(t_0)=x_0 \label{e2.1}
\end{equation}
and a correspondent perturbed differential system
\begin{equation}
\dot x=X(t,x)+Y(t,x)+Z(t,x)\quad x(t_0)=x_0 \label{e2.2}
\end{equation}
without further mention, we will assume
that $Y  G\leq U$, where $Y G$ denotes the scalar product of the
vectors $Y$ and $G$.
Moreover, we assume that the right-hand sides of
\eqref{e2.1} and \eqref{e2.2}, are L-measurable in $t\in
\mathbb{R}^+$, continuous in $x\in \mathbb{R}^n$.
Also we assume that for every compact subset $A\subset \mathbb{R}^n$
there exists a map $\sigma_A =\sigma_A(t)$ locally integrable
such that $\|X(t,x)\|$, $\|Y(t,x)\|$, $\|Z(t,x)\|<\sigma_A(t)$
when $x\in A$.

The previous conditions  (Caratheodory's conditions)
ensure the existence and the general continuity of solutions for
 \eqref{e2.1} and \eqref{e2.2}; see \cite{c1,c2}.
Then, for every $(t_0,x_0)\in \mathbb{R}^+\times \mathbb{R}^n$ we
denote by $x(t)=x(t,t_0,x_0)$ a solution of \eqref{e2.2}, and we
assume that $x(t)$ is defined for $t\ge t_0$.

For every continuous scalar function $V=V(t,x)$ having continuous
partial derivatives we put
\begin{equation}
 V_t={{\partial V}\over
{\partial t}},\quad V_x=\operatorname{grad} V={{\partial V}\over {\partial x}},
\quad \dot{V}_1=V_t+V_x \cdot X=\dot V .\label{e2.3}
\end{equation}
The function $\dot V$ is said to be the derivative of  $V$
computed  along the solutions of the unperturbed system \eqref{e2.1}.
While the related formula given by Malkin  \cite{m1,r1},
\begin{equation}
\dot{V}_2 (t,x)=\dot V (t,x) + V_x (t,x)[Y(t,x)+Z(t,x)] \label{e2.4}
\end{equation} gives
the derivative of $V$  along the solutions of the perturbed system
\eqref{e2.2}.


If $\phi$ and $\theta$ are two scalar functions, it easy to prove
the following results which will be used in the next sections.
\begin{itemize}

\item[(i)] if $V_x=\phi G$ and $YG\leq U$, when $\phi >0$ we deduce
\begin{gather}
\dot{V}_2(t,x)=\dot V(t,x)+\phi (t,x)[GY(t,x)+ GZ(t,x)], \label{e2.4A}\\
\dot{V}_2(t,x)\leq \dot V(t,x)+ \phi (t,x) [U(t,x)+
\|G(t,x)\|\|Z(t,x)\|]; \label{e2.4B}
\end{gather}

\item[(ii)]  if $U(t,x)\leq 0$ and $\phi(t,x)>0,  V_x=\phi G$ we deduce
\begin{equation}
\dot{V}_2(t,x)\leq \dot V(t,x) + \phi (t,x) \|G(t,x)\|
\|Z(t,x)\|; \label{e2.4C}
\end{equation}

\item[(iii)]  if $U(t,x)\ge 0$,   $\phi (t,x)>0$,  $\theta{(t,x)}>0$,
$ V_x(t,x)=\phi G(t,x)$ and $\dot V(t,x)\leq -\theta U(t,x)$, we
deduce
 \begin{equation}
\dot{V}_{2}(t,x) \leq -[\theta (t,x)-\phi
(t,x)]U(t,x) +\phi (t,x)\|G(t,x)\|\|Z(t,x)\|. \label{e2.4D}
\end{equation}
\end{itemize}

We conclude the present section with a list of definitions
concerning the several kinds of the stability in terms of two
measures  and two perturbations.


\begin{definition} \label{def2.1} \rm
 System \eqref{e2.1} is said to be \emph{$(h_0, h)$-stable under two
persistent perturbations}, also called $(h_0,h)$-t.bistable,
if for every $t_0 \in \mathbb{R}^+$ and every $\epsilon >0$,
there exist a number $\delta_1=\delta_1(t_0,\epsilon)$ and a function
$\delta_2= \delta_2(t_0,x,\epsilon)>0$
such that for all $x_0 \in \mathbb{R}^n$ with
$h_0(t_0, x_0)<\delta_1$, all $Z(t,x)$
with $\|Z(t, x)\|<\delta_2$, and  all $Y$ with $YG\leq U$;
we have $h[t, x(t)]<\epsilon$ when
$t\ge t_0$.

If $\delta_1=\delta_1(\epsilon)$ and $\delta_2=\delta_2(\epsilon)$
are independent of $t_0$ and $x$, we have the uniformity.
\end{definition}


\begin{definition} \label{def2.2} \rm
 System \eqref{e2.1} is said to be
\emph{strongly weakly $(h_0,h)$-t.bistable},
if in Definition \ref{def2.1}, $\delta_2(t_0,x,\epsilon)\ge 0$, and
the L-measure of set
\begin{equation}
E_t(\delta_2=0)=\{ x\in \mathbb{R}^n:\delta_2 (t_0,x,\epsilon)=0\}
 \label{e2.5}
\end{equation}
is  zero for $t_0\in \mathbb{R}^+$ and $\epsilon >0$.

In the following we will briefly write $\delta_2 \in GG$ to
indicate this condition.
\end{definition}

\begin{definition} \label{def2.3} \rm
 System \eqref{e2.1} is said to be \emph{weakly  $(h_0,h)-t.bistable$}
  if, for every
$t_0\in \mathbb{R}^+$ and $\epsilon >0$, there exists at the most one
 $x\in \mathbb{R}^n$ such that $\delta_2(t_0,x,\epsilon)=0$.

In the following this condition  will
be briefly denoted as $\delta_2 \in ZZ$.
\end{definition}

\begin{definition} \label{def2.4} \rm
System \eqref{e2.1} is said to be \emph{$(h_0,h)$-eventually
stable under  two  persistent  perturbations}, also called
eventually $(h_0,h)$-t.bistable, if:
For every $\epsilon>0$ there exists $T=T(\epsilon)>0$,
for every $t_0 \ge T$ there exist $\delta_1=\delta_1(t_0,\epsilon)$
and $\delta_2=\delta_2(t_0,x,\epsilon)>0$
such that for every $x_0 \in \mathbb{R}^n$ with
$h_0(t_0, x_0)<\delta_1$, for every $Z$ with $\|Z(t,x)\|<\delta_2$
and  every $Y$ with $YG\leq U$,
we have $h[t, x(t)]<\epsilon$ when $t\ge t_0$.
\end{definition}

\begin{definition} \label{def2.5} \rm
System \eqref{e2.1} is said to be \emph{$(h_0,h)$-semiattractive},
if it is $(h_0,h)$-t.bistable and:
For every $\eta \in ]0,\epsilon[$ there exists a function
${\delta}_3>0$, with ${0< \delta}_3 \leq {\delta}_2$, such that
for every $Z:\|Z\|<{\delta}_3$ and  every $Y$ with $YG\leq U$,
there exists $T_{\eta} >0$ for which $h[t,x(t)]<\eta$
when $t\ge t_0+T_{\eta}$,
where $x(t)=x(t,t_0,x_0)$ is a solution of \eqref{e2.2}.
\end{definition}

\begin{definition} \label{def2.6} \rm
System \eqref{e2.1} is said to be $(h_0,h)$-\emph{stable on
average under two persistent perturbation}, also called
\emph{$(h_0,h)$-t.bistable on average}, if:
For every $t_0\in \mathbb{R}^+$, every $\epsilon>0$ and every $T>0$,
there exist $\delta_1$ and $\delta_2>0$ such that every solution
$x(t)=x(t,t_0,x_0)$ of \eqref{e2.2} with
$h_0(t_0,x_0)<\delta_1$, $YG \leq U$, and
\begin{equation}
\int^{t+T}_t \sup\{\|Z(u,x)\| : x\in \mathbb{R}^n \}
 du<\delta_2 \quad \forall t\ge t_0 \label{e2.6}
\end{equation}
satisfies  $h[t,x(t)]<\epsilon$ for all $t\ge t_0$.
\end{definition}

\section{Theoretical developments}

Suppose that the functions $X,G,U$ are the known start point.
We will use the technique that is known as family of Liapunov
functions introduced by Salvadori \cite{s1}. The basic advantage
of this method is that the single function needs to satisfy less
rigid requirements than in other methods.


\begin{theorem} \label{thm3.1}
Let $U:\mathbb{R}^+\times \mathbb{R}^n \to \mathbb{R}$ be given.
Assume that for every $\epsilon>0$, there exist three scalar functions
$\Theta=\Theta (t,x)$, $\phi=\phi (t,x) \in C$, and $V=V(t,x) \in C^1$,
and exists a constant $l$ such that on the set
$ \mathbb{R}^+\times \mathbb{R}^n$ we have:
\begin{itemize}
\item[(i)] $h(t,x)= \epsilon $ implies $V(t,x) \ge l>0$;

\item[(ii)] ${\lim_{h\to 0}}V(t,x)=0$;


\item[(iii)] $\Theta (t,x) > \phi (t,x)>0$ and $(\Theta-\phi)U>0$;

\item[(iv)] $V_x(t,x)=\phi G(t,x)$;

\item[(v)] $\dot V(t,x)\leq -\Theta U(t,x)$.

\end{itemize}
 Then  system \eqref{e2.1} is $(h_0,h)$-t.bistable.
\end{theorem}



\begin{proof}
Given $t_0, \epsilon, l, \Theta, \phi, V$, by (ii) there exists
$d>0$ such that $h(t_0,x)<d$ implies $V(t_0,x)<l$.
If we select $x_0 \in \mathbb{R}^n$
such that $h_0(t_0,x_0)<\delta_1=\min[\lambda,m^{-1}(d)]$  for the
previous assumptions we recognize that $h(t_0,x_0)< m[h_0(t_0,x_0)]<d$,
hence $V(t_0,x_0)<l$.
 From the Malkin formula \eqref{e2.4}, according to (iv) and (v),
we deduce
\begin{equation}
\dot V_2(t,x)\leq -(\Theta -\phi )U(t,x)+\phi \|G(t,x)\| \|Z(t,x)\|.
 \label{e3.1}
\end{equation}
Then by selecting
$$
\|Z(t,x)\|\leq {{(\Theta -\phi )U(t,x)}\over {\|\phi G(t,x)\|}}=
\delta_2 (t,x,\epsilon)
$$
it follows  that $\dot V_2(t,x)\leq 0$.

Consider a solution $x(t)=x(t,t_0,x_0)$ of the perturbed system and the
correspondent functions $h_1(t)=h[t,x(t)], V_1(t)=V[t,x(t)]$.
If  there exists $t'>t_0$ such that $h_1(t')=\epsilon$ with
$h_1(t)<\epsilon $ for $t\in [t_0,t'[$ then we should deduce that
$V_1(t')\ge l$, which is a contradiction.
\end{proof}


\noindent\textbf{Remark.}
  If $U \ge 0$ the system can be strongly weakly $(h_0,h)$-t.bistable
or weakly $(h_0,h)$-t.bistable.


\begin{corollary} \label{coro3.2}
Suppose that there exist three scalar functions $\Theta=\Theta(t,x),
\phi=\phi(t,x) \in C$ and $V=V(t,x) \in C^1$ such that
on $\mathbb{R}^+\times \mathbb{R}^n$ we have:
\begin{itemize}
\item[(i)] for each $\epsilon >0$ there exists $l>0$ such that
$h(t,x)=\epsilon $ implies $V(t,x) \ge l>0$;

\item[(ii)]  $\lim_{h\to 0} V(t,x)=0$;

\item[(iii)]  $\Theta(t,x)> \phi(t,x)>0$ and $(\Theta-\phi)U >0$;

\item[(iv)] $V_x(t,x)=\phi G(t,x)$;

\item[(v)] $\dot V(t,x)\leq -\Theta U(t,x)$.

\end{itemize}
 Then  \eqref{e2.1} is $(h_0,h)$-t.bistable.
 \end{corollary}


\begin{corollary} \label{coro3.2a}
For a scalar function $U< 0$, suppose that there exist three scalar
 functions $L=L(t,x)$,  $\phi=1$,  $V=V(t,x)$  such that the conditions
 (i)--(iv) in Corollary \ref{coro3.2}
hold, and that $\dot V(t,x)\leq -L(t,x)< 0$. Then  \eqref{e2.1} is
$(h_0,h)-$t.bistable.
\end{corollary}


\begin{proof}
 From \eqref{e2.4C} we have
\begin{equation}
\dot V_2(t,x)\leq -L(t,x)+\|G\|\|Z\| \label{e3.1a}
\end{equation}
hence by choosing  $\|Z\|\leq L/\|G\|$ we have the proof.
\end{proof}


\begin{theorem} \label{thm3.3}
Suppose that for every $\epsilon >0$ there exist two scalar functions
$\phi =\phi (t,x)$, $\Theta =\Theta (t,x) \in C$,
a map $N=N(u)$ L-measurable, the scalar function
$V=V(t,x) \in C^1 $,  and a constant $l$ such that on
$\mathbb{R}^+\times \mathbb{R}^n$ we have:
\begin{itemize}
\item[(i)] $h(t,x)=\epsilon $ implies $V(t,x)\ge l>0$;

\item[(ii)] ${\lim_{h\to 0}} V(t,x)=0$;

\item[(iii)] $\dot V(t,x)\leq -\Theta U(t,x)+N(t)$ with
$0<\int_0^{+\infty}N(u)du <+\infty$ and $U >0 $
(a hypothesis of Hatvani's type);

\item[(iv)] $\Theta (t,x) \ge \phi (t,x)>0$ and $(\Theta-\phi)U >0$;

\item[(v)] $V_x(t,x)=\phi G(t,x)$.
\end{itemize}
 Then \eqref{e2.1} is eventually $(h_0,h)$-t.bistable.
 \end{theorem}


\begin{proof}
 Given $\epsilon>0$ we consider the function
\begin{equation}
W(t,x)=V(t,x)+\int_t^{+\infty}N(u)du \quad (t>0). \label{e3.2}
\end{equation}
Let $T>0$ such that $2\int_{t_0}^{+\infty }N(u)du<l$ for $t_0\ge T$,
and let  $d>0$ such that $h(t_0,x)<d$ implies (by (ii))
$2V(t_0,x)<l$.  If $x_0\in \mathbb{R}^n$ and
$h_0(t_0,x_0)<\delta_1=\min[\lambda,m^{-1}(d)]$,
we deduce that $h(t_0,x_0)\leq m[h_0(t_0,x_0)]<d$ and $2V(t_0,x_0)<l$.
Then it follows that $W(t_0,x_0)<l$.
Consider the derivatives
\begin{gather}
\dot W(t,x)=\dot V(t,x)-N(t)\leq -\Theta U(t,x)< 0, \label{e3.3} \\
\dot W_2(t,x)\leq -(\Theta -\phi )U(t,x)+\phi \|G(t,x)\|\|Z(t,x)\|.
\label{e3.4}
\end{gather}
Provided that
\begin{equation}
\|Z(t,x)\|\leq {{(\Theta -\phi) U(t,x)) }\over {\phi \|G(t,x)\|}}=
\delta_2(t,x,\epsilon) \label{e3.5}
\end{equation}
we obtain $\dot W_2(t,x)\leq 0$. Selecting $Z=Z(t,x)$
 such that $\|Z(t,x)\|\leq \delta_2$, consider $x(t)=x(t,t_0,x_0)$
a solution of the perturbed system \eqref{e2.2}, and put
$H(t)=h[t,x(t)]$, $v(t)=V[t,x(t)]$,
$w(t)=W[t,x(t)]$.  Suppose that there exists $t'>t_0$ such that
$H(t')=\epsilon$ and $H(t)<\epsilon$ for $t_0\leq t<t'$.  So we
have $v(t')>l$ hence $w(t')>l$. This is a contradiction which
completes the proof.
\end{proof}


\begin{lemma} \label{lem3.4}
Suppose that there exist four scalar functions $\phi=\phi(t,x)$,
$U=U(t,x)$, $\Theta=\Theta(t,x) \in C$,  $V=V(t,x) \in C^1$,
a  scalar function $\Psi=\Psi(t,h)$ L-integrable with respect to
$t \in \mathbb{R}^+$, a map
$a=a (u) \in K$ such that on $\mathbb{R}^+\times \mathbb{R}^n$,
we have
\begin{itemize}
\item[(i)] $V(t,x)\ge a [h(t,x)]$;

\item[(ii)] $ {\lim_{h\to 0}}V(t,x)=0$;


\item[(iii)] $V_x(t,x)=\phi G(t,x)$;

\item[(iv)] $\dot V(t,x)\leq -\Theta U(t,x)< 0$;

\item[(v)] $\Theta(t,x)>\phi (t,x)>0$ and $(\Theta-\phi)U >0$;

\item[(vi)]  $(\Theta -\phi)U(t,x)=\Psi [t,h(t,x)]$;

\item[(vii)] $\Psi (t,h) \ge \Psi (t, \mu)$ when $h \ge \mu>0$;

\item[(viii)] $\int_{t'}^{+\infty} \Psi (\tau ,\rho )d\tau =
+\infty $, for all $t'\in I$,  all $\rho >0$.
\end{itemize}
Then  \eqref{e2.1} is $(h_0,h)$-t.bistable.
Also for every $\epsilon >0$,
for every $\eta \in ]0,\epsilon]$, for each  $\gamma>0$
for every  $t_0 \in \mathbb{R}^+$ and
$x_0\in \mathbb{R}^n$ with $h_0(t_0,x_0)<\delta_1$,
for every $Z$  with
\begin{equation}
\|Z(t,x)\|\leq \big({1\over {1+\gamma}}\big)
{{(\Theta -\phi )U(t,x)}\over {\phi \|G(t,x)\|}} = \delta_3;
\label{e3.6}
\end{equation}
there exists  $t_{\eta } \ge t_0$ for which
$h[t_{\eta },x(t_{\eta})]<\eta$ where $x=x(t,t_0,x_0)$  is
solution of \eqref{e2.2}.
\end{lemma}

\begin{proof}
 By contradiction let us assume that that there exist
$\epsilon_1 >0$, $\eta_1\in ]0,\epsilon_1]$,
$(t_1,x_1)\in \mathbb{R}^+\times \mathbb{R}^n$:
$h_0(t_1,x_1)<\delta_1  ,\gamma_1>0$,  $Z_1$ :
$\|Z_1(t,x)\|<\delta_3$ (depending on $\gamma_1$) such that
$h[t,x_1(t)]\ge \eta_1$  when $t\ge t_1$ where
$x_1(t)=x(t,t_1,x_1)$  is obviously a solution of \eqref{e2.2}.

Consider the derivative $\dot{V}_2(t,x)$:  by hypotheses (iii)
and (iv) we have
\begin{equation}
\dot{V}_2(t,x)=\dot{V}+\phi GY(t,x)+\phi GZ(t,x) \le-(\Theta-\phi
)U+ \phi \|G\|\|Z\| \label{e3.7}
\end{equation}
Thus selecting
\begin{equation}
\|Z(t,x)\| \leq {1 \over {1+{\gamma}_1}} {{(\Theta- \phi)U(t,x)} \over
{\phi \|G(t,x)\|}}=\frac {1}{1+\gamma_1}\delta_2=\delta_3 \label{e3.8}
\end{equation}
we obtain
\begin{equation}
\dot{V}_2(t,x)\leq -{{{\gamma}_1} \over
{1+{\gamma}_1}}(\Theta-\phi)U <0. \label{e3.9}
\end{equation}
hence by (vi)
\begin{equation}
{\dot V }_2(t,x) \leq -{{{\gamma}_1}\over {1+{\gamma}_1}} \Psi
[t,h(t,x)]. \label{e3.10}
\end{equation}
On the set $\{ (t,x)\in \mathbb{R}^+\times \mathbb{R}^n: h(t,x)
 \ge \eta_1>0 \}$
we have $ \Psi [t,h(t,x)] \ge \Psi (t, \eta_1)$ and
\begin{equation}
{ \dot V}_2(t,x) \leq -{{{\gamma}_1} \over {1+{{\gamma}_1}}}\Psi (t,{\eta}_1).
\label{e3.11}
\end{equation}
Along the above  solution $x_1(t)$
we obtain, for $t\ge t_1$,
\begin{gather}
\int_{t_1}^{t} \dot{V}_2[u,x_1(u)] du
\leq -{{{\gamma}_1} \over
{1+{\gamma}_1}} \int_{t_1}^{t }\Psi (u,\eta_1) du, \label{e3.12}\\
V[t,x_1(t)]\leq V(t_1,x_1)-{{{\gamma}_1}\over {1+{\gamma}_1}} \int_{t_1}^t
\Psi (u,\eta_1) du \label{e3.13}
\end{gather}
which is a contradiction.
\end{proof}

\begin{theorem} \label{thm3.5}
 Under the hypotheses of Lemma \ref{lem3.4} suppose  that
\begin{itemize}
\item[(ix)] There exist $b=b(u) \in K$ such that
$b[h_0 (t,x)]\leq h(t,x)$ on $\mathbb{R}^+\times \mathbb{R}^n$.
\end{itemize}
Then, for every $\epsilon >0$ and $\sigma \in ]0,\epsilon]$, there
exists a  function  $\delta_3=\delta_3 (t,x,\sigma)\in
]0,\delta_2]$ such that: for every $t_0\in \mathbb{R}^+$,
for every $x_0 \in \mathbb{R}^n$ with $h_0(t_0,x_0)<\delta_1$, and for every $Z:\|Z(t,x)\|<\delta_3$;
there exists $T_{\sigma}>0$ for which $h[t,x(t)]<\sigma$ when $t\ge
t_0+T_{\sigma}$ where $x(t)=x(t,t_0,x_0)$  is a solution of
\eqref{e2.2}.
\end{theorem}


\begin{proof}
 Since the system \eqref{e2.1} is $(h_0,h)$-t.bistable,
given $t_0 \in \mathbb{R}^+$ and  $\epsilon >0$ there exist
$\delta_1 =\delta_1 (t_0,\epsilon)$ and $\delta_2=\delta_2
(t,x,\epsilon)>0$ such that fixed $x_0 \in \mathbb{R}^n$ for which
$h_0(t_0,x_0)<\delta_1$ and select $Z:\|Z(t,x)\|<\delta_2$ we have
$h[t,x(t)]<\epsilon$ for $t\ge t_0$ where $x(t)=x(t,t_0,x_0)$ is a
solution of \eqref{e2.2}.

It is obvious that for every $\sigma \in ]0,\epsilon[$ there exist
$d_1 \in ]0,\delta_1[$ and $d_2 \in ]0,\delta_2[$ such that fixed
$(t_1,x_1) \in \mathbb{R}^+\times \mathbb{R}^n$ for which
$h_0(t_1,x_1)<d_1$, select $Z:\|Z(t,x)\|<d_2$  we have
$h[t,x_1(t)]<\sigma$ for $t\ge t_1$ where $x_1(t)=x_1(t,t_1,x_1)$
is a solution of \eqref{e2.2}.

 From Lemma \ref{lem3.4}, given $\eta \in ]0,\sigma[ \subset ]0,\epsilon]$
there exists $d_3 \in ]0,d_2[$ such that for every
$Z:\|Z(t,x)\|\leq d_3$ there exists $t_{\eta}\ge t_0$ for which
$h[t_{\eta},x(t_{\eta})] <\eta$ where $x(t)=x(t,t_0,x_0)$ is a
solution of \eqref{e2.2}.

If we assume that $\eta =b(d_1)$, we obtain
\begin{equation}
b\{ h_0 [t_{\eta},x(t_{\eta})]\} \leq h[t_{\eta},x(t_{\eta})] <
\eta =b(d_1); \label{e3.14}
\end{equation}
i.e., $h_0[t_{\eta},x(t_{\eta})]<d_1$. Hence when $\|Z(t,x)\| \leq d_3$
we have $h[t,x(t)]\leq \sigma$ for $t\ge t_{\eta}$.  Putting
$T_{\eta}=t_{\eta}-t_0$ we then obtain the semiattractivity.
\end{proof}

\begin{theorem} \label{thm3.6}
Suppose that there exist three functions from $R\times \mathbb{R}^n$
to $R$: $U=U(t,x)$, $ \phi=\phi(t,x) \in C$,  $V=V(t,x) \in C^1$;
three functions $a=a(u)$, $b=b(u)$, $c=c(u)$ belonging to
$K$; and a constant $N>0$; such that on
$\mathbb{R}^+\times \mathbb{R}^n$,  we have:
\begin{itemize}
\item[(i)]  $a[h(t,x)] \leq V(t,x) \leq b[h(t,x)]$;

\item[(ii)] $\phi(t,x) >0$;

\item[(iii)] $V_x(t,x)=\phi G(t,x), \ \|V_x(t,x)\|<N$;

\item[(iv)] $\dot V(t,x)\leq -c[h(t,x)]$;

\item[(v)]  given $r,T,\epsilon >0$, put
$\nu={ r\over T } \epsilon$:  the condition
$\nu <h(t,x)< \epsilon $
implies $ { {\phi U(t,x)}\over
{V(t,x)}}<{ {c(\nu )}\over {2b(\epsilon)}}$.

\end{itemize}
Then \eqref{e2.1} is $(h_0,h)$-t.bistable on average.
\end{theorem}


\begin{proof}
 Given $t_0\in \mathbb{R}^+$, $\epsilon $ and $T>0$,
from (i) $h(t,x)=\epsilon$  implies $V(t,x)\ge a(\epsilon)$.
Select $d \in ]0,\epsilon [$ such that:
\begin{itemize}
\item[(i)] $h(t_0,x)<d$ implies $V(t_0,x)<a(\epsilon)$;

\item[(ii)] $b(d)<\frac 1 2 a(\epsilon)$.
\end{itemize}

 If $x_0\in \mathbb{R}^n: h_0(t_0,x_0)<\delta_1=
\min[\lambda, m^{-1}(d)]$ we have $h(t_0,x_0)\leq m[h_0(t_0,x_0)]\leq d$ \
hence $V(t_0,x_0)<a(\epsilon)$.
 Let $x(t)=x(t,t_0,x_0)$ be a solution of \eqref{e2.2} and
suppose that there exist $t',t''\in \mathbb{R}^+$ with the
following properties:
\begin{itemize}
\item[(iii)] $t_0\leq t'<t''$;

\item[(iv)] $h(t'',x'')=h[t'',x(t'')]=\epsilon$;

\item[(v)] $h(t',x')= h[t',x(t')]= \min h[t,x(t)] $
and $h(t',x')\leq h[t,x(t)] \le h(t'',x'')$ on $t'\leq t\leq t''$.

\end{itemize}

Put $W(t,x)=V(t,x)e^{\beta(t)}$,  where
$\beta =\beta (t):\mathbb{R}^+\to \mathbb{R}$ is a scalar
function that will be defined in \eqref{e3.22},  and consider the
derivatives
\begin{gather}
\dot{W}_1(t,x) =\dot W(t,x) = \dot V(t,x)e^{\beta
(t)}+V(t,x)e^{\beta (t)}\dot {\beta}(t), \label{e3.15}\\
\dot{W}_2(t,x)=\dot W (t,x)+W_x[Y(t,x)+Z(t,x)], \notag\\
\dot{W}_2(t,x)=\dot V (t,x)e^{\beta (t)}+V(t,x)e^{\beta (t)}
\dot {\beta}(t)+ e^{\beta (t)}[V_xY(t,x)+V_xZ(t,x)], \notag\\
\dot{W}_2 (t,x)=W(t,x)\Big[{{\dot V (t,x)}\over V (t,x)}+\dot
{\beta}(t) +{{V_xY(t,x)}\over V(t,x)}+{{V_xZ(t,x)}\over
V(t,x)}\Big] \label{e3.16}
\end{gather}
if we select $Y(t,x)$ such that $V_x Y\leq U$ we obtain
\begin{equation}
\dot{W}_2 (t,x)\leq W(t,x)\Big\{{{\dot V(t,x)}\over V(t,x)}+\dot
{\beta}(t)+ {{\phi U(t,x)}\over V(t,x)}+ {{\|V_x(t,x)\|}\over
{V(t,x)}}\|Z(t,x)\|\Big\}. \label{e3.17}
\end{equation}
On the set
$$
\{ A \}=\{ (t,x)\in \mathbb{R}^+\times \mathbb{R}^n:d=
{r\over T} \epsilon <h(t,x)<\epsilon \}
$$
for suitable $r \in ]0,T[$ we have
\begin{equation}
(s) {{\dot V (t,x)}\over {V(t,x)}}<-{{c(d)}\over
{b(\epsilon)}};\quad \quad (ss) {{\phi U(t,x)}\over
{V(t,x)}}<{{c(d)}\over {2b(\epsilon)}} \label{e3.18}
\end{equation}
hence
\begin{equation}
 \dot{W}_2(t,x)\leq W(t,x)\big\{ \dot {\beta}(t)-{{c(d)}\over {2b(\epsilon)}}+
{N\over {a(d)}}\|Z(t,x)\|\big\}. \label{e3.19}
\end{equation}
Fixed $t\in \mathbb{R}^+$, for every $x\in \mathbb{R}^n$ put
$ \phi (t)= sup \{ \|Z(t,x)\|\} $.  Given $q\in ]0,1[ $ we
construct the function $\Psi=\Psi (t):R \to \mathbb{R}$ such
that the equalities
\begin{equation}
\begin{aligned}
L(T)&=\int^{(\mu +1)T}_{\mu T} \Psi (u)du\\
&=\int^{(\mu +1)T}_{\mu T}
\big\{{{(1-q)}\over 2}{{c(d)}\over {b(\epsilon )}}
-{N\over {a(d)}} \phi (u)\big\} du \\
&={{(1-q)c(d)}\over {2b(\epsilon)}}T-{N\over {a(d)}}
\int^{(\mu +1)T}_{\mu T}\phi (u)du
\end{aligned}\label{e3.20}
\end{equation}
are fulfilled for every non negative integer $\mu$.

If, for every $t\in \mathbb{R}^+$, we select $\|Z(t,x)\|$ such that
\begin{equation}
\int^{(\mu+1)T}_{\mu T}\phi (u)du\le
{{(1-q)c(d)a(d)}\over {2b(\epsilon)N}}T ={{\delta}_2}  \label{e3.21}
\end{equation}
we obtain $L(T)\ge 0$.  On the strength of the previous conditions
we can take it such that $\Psi (t)\ge 0$ for all $t\ge 0$.
We set, for  $t\in \mathbb{R}^+$,
\begin{equation}
{\beta}(t)=\int^t_0\big[-\Psi
(u)+{{(1-q)c(d)}\over{2b(\epsilon)}} -{N\over {a(d)}}\phi
(u)\big]du. \label{e3.22}
\end{equation}
consequently we recognize that $\beta (\mu T)=0$ for every
natural number $\mu$. Also
\begin{gather}
\dot {\beta}(t) =-\Psi (t)+{{(1-q)c(d)}\over
{2b(\epsilon)}}-{N\over {a(d)}} \phi(t), \label{e3.23} \\
\dot{W}_2(t,x)\leq W(t,x)[-\Psi (t)-{{qc(d)}\over
{2b(\epsilon)}}]\leq 0. \label{e3.24}
\end{gather}
Assuming  $\mu T\leq t \leq (\mu +1)T$, put
\begin{gather*}
\Gamma (u)= -\Psi (u) +{{(1-q)c(d)}\over{2b(\epsilon)}}
-{N\over{a(d)}}\phi(u), \\
\Delta (u)=\Psi (u)+{{(1-q)c(d)}\over{2b(\epsilon)}}+ {N\over{a(d)}}\phi (u).
\end{gather*}
Consequently,
\begin{gather}
\beta (t)=\int^t_0 \Gamma (u)du=\int^t_{\mu T}\Gamma(u)du\le
\int^{(\mu+1)T }_{\mu T}\Delta (u)du , \label{e3.25}\\
\begin{aligned}
|\beta(t)|
&\le \big| \int^{(\mu+1)T}_{\mu T}
\big[\Psi (u)+{{(1-q)c(d)}\over{b(\epsilon)}}
+{N\over{a(d)}}\phi (u)\big]du \big|\\
&\leq 3{{(1-q)c(d)}\over{b(\epsilon)}} T=\Theta.
\end{aligned} \label{e3.26}
\end{gather}
Hence we obtain
\begin{gather}
W[t',x']=V[t',x(t')]e^{\beta(t')}\leq b(d)e^{\Theta} <\frac 1 2
a(\epsilon)e^{\Theta}, \label{e3.27} \\
W[t'',x'']=V[t'',x(t'')]e^{\beta (t'') }\ge a(\epsilon)e^{-\Theta}.
\label{e3.28}
\end{gather}
and so according to \eqref{e3.24} we have
\begin{equation}
\frac 1 2 a(\epsilon)e^{\Theta}> b(d)e^{\Theta}\ge
a(\epsilon)e^{-\Theta},\quad {{1}\over {2}}\ge e^{-2\Theta}.
\label{e3.29}
\end{equation}
Since $0<q<1$ is arbitrary we obtain a contradiction.
(Oziraner theorem extension)
\end{proof}

\section{Theoretical developments for inequalities of the second kind }

In this section we assume, as start points, the functions $X, V$, and
select $Y$ from inequalities of the  type (second kind)
$$
F(V_x, W_x, \dot V, \dot W, Y)< 0.
$$
This way we deduce some propositions very useful for applications.


\begin{theorem} \label{thm4.1}
Suppose that there exists a  family of scalar functions $V=V(t,x)\in C^1$
such that on $\mathbb{R}^+\times \mathbb{R}^n$ we
have:
\begin{itemize}
\item[(i)]  for all $\epsilon>0$ there exists $l>0$ such that
$h(t,x)=\epsilon$ implies $V(t,x)>l$;

\item[(ii)] ${\lim_{h\to 0}}V(t,x)=0$;

\item[(iii)] $\dot V(t,x)< 0$;

\item[(iv)] $\|V_x(t,x)\|>0$.

\end{itemize}
 Then  \eqref{e2.1} is  ($h_0,h$)-t.bistable with respect to
the ``aim perturbations'' (friction?)  for which
$V_xY(t,x)\leq 0$.
\end{theorem}


\begin{proof}
The proof is very similar to that of Theorem \ref{thm3.1}.
 We limit ourselves to observe that
\begin{equation}
\dot{V}_2(t,x)\leq \dot V(t,x)+\|V_x(t,x)\| \|Z(t,x)\|
\label{e4.1}
\end{equation}
and thus if
\begin{equation}
\|Z(t,x)\|\leq -{{\dot V(t,x)}\over {\|V_x(t,x)\|}}={\delta}_2
\label{e4.2}
\end{equation}
we have $\dot{V}_2(t,x)\leq 0$.
\end{proof}

\begin{theorem} \label{thm4.2}
Suppose that there exist three functions
$\Phi=\Phi(t,x)\in C$, $V=V(t,x)$ and $W=W(t,x)\in C^1$ such
that on $\mathbb{R}^+\times \mathbb{R}^n$ we have:
\begin{itemize}
\item[(i)] For all $\epsilon >0$ there exists $l>0$ such that
$h(t,x)=\epsilon$ implies $V(t,x)-W(t,x)\ge l$;

\item[(ii)] ${\lim_{h\to 0}}[V(t,x)-W(t,x)]=0$;

\item[(iii)] $\dot V(t,x)< 0$;

\item[(iv)] $ \|V_x(t,x)-W_x(t,x)\|>0$;

\item[(v)] $0<\Phi (t,x)<1$.

\end{itemize}
 Then the system \eqref{e2.1} is  $(h_0,h)$-t.bistable with
respect to the ``aim perturbations'' such that
$(V_x-W_x)Y(t,x)\leq (-\Phi \dot V+\dot W)(t,x)$.
\end{theorem}

\begin{proof}
 Let $T(t,x)=V(t,x)-W(t,x)$ be an auxiliary function. From the
following two conditions
\begin{gather}
\dot{T}_2(t,x)=[\dot V-\dot
W](t,x)+(V_x-W_x)Y(t,x)+(V_x-W_x)Z(t,x), \label{e4.3}\\
\dot{T}_2\leq \dot V-\dot W-\Phi \dot V+\dot W+\|V_x-V_x\|\|Z\|\notag
\end{gather}
if
\begin{equation}
\|Z\|\leq -{{(1-\Phi)\dot V}\over {\|V_x-W_x\|}}=\delta_2 (t,x),
\label{e4.4}
\end{equation}
we obtain $\dot{T}_2(t,x)\leq 0$.
\end{proof}

\begin{theorem} \label{thm4.3}
 Suppose that there exist a constant $a>0$ and two functions
$V=V(t,x)$, $W=W(t,x)\in C^1$ such that on
$\mathbb{R}^+\times \mathbb{R}^n$ we have
\begin{itemize}
\item[(i)] $V(t,x)\ge 0$;

\item[(ii)] $W(t,x)\ge -a$ and  for every $\epsilon >0$ there exist
two constants $r, b>0$ for which $h(t,x)=\epsilon$ with
$V(t,x)<r$ implies   $W(t,x)>b$;

\item[(iii)] ${\lim _{h\to 0}}V(t,x)=
{\lim_{h\to 0}}W(t,x)=0$;

\item[(iv)] $\|V_x(t,x) +\mu W_x(t,x)\|>0$ for every $\mu> 0$.
\end{itemize}
 Then  \eqref{e2.1} is bistable with respect to the
``aim perturbations'' for which
\begin{equation}
\dot V (t,x)+\mu \dot W (t,x)+[V_x(t,x)+\mu W_x(t,x)]Y(t,x)< 0.
\label{e4.5}
\end{equation}
\end{theorem}

\begin{proof}
 Given $\epsilon$ and $r,b>0$, suppose that $0<\mu (a+b)<r$ where
$\mu >0$ is a constant (correspondent to $\epsilon$).
Consider the family of functions
\begin{equation}
 v(t,x)=V(t,x)+\mu W(t,x). \label{e4.6}
\end{equation}
If we assume that $h(t,x)=\epsilon$ and $V(t,x)\ge r$, we obtain
\begin{equation}
v(t,x)-\mu b\ge r-\mu (a+b), v(t,x)\ge \mu b \label{e4.7}
\end{equation}
When $h(t,x)=\epsilon $ implies $V(t,x)<r$,
 we deduce $v(t,x)\ge \mu b$ which  condition (i) of
Theorem \ref{thm3.1}.
Finally, consider the derivative
\begin{equation}
{\dot v}_2(t,x)=\dot V(t,x)+V_xY(t,x)+\mu [\dot
W+W_xY](t,x)+[V_x+\mu W_x] Z(T,x) \label{e4.8}
\end{equation}
if $\dot V+\mu \dot W+[V_x+\mu W_x]Y< 0$, we obtain the proof
by choosing
\begin{equation}
\|Z(t,x)\|\leq -{{\dot V+V_xY+\mu [\dot W+W_xY]}\over {\|V_x+\mu
W_x\|}}. \label{e4.9}
\end{equation}
\end{proof}

\section{Liapunov  functions  in  Salvadori's  sense}

Let us consider a continuous non trivial function
$\phi=\phi(t,x):\mathbb{R}^+\times \mathbb{R}^n \to \mathbb{R}^+$
and a constant $\rho \in ]0,\sup \phi]$.
 We shall set, for  $t\in \mathbb{R}^+$,
\begin{equation}
E_t(\phi \leq \rho)=\{ (t,x)\in \mathbb{R}^+\times \mathbb{R}^n:
\phi (t,x)\leq \rho,  t=\text{constant} \} \label{e5.1}
\end{equation}
The meaning of $E_t(\phi=0)$ and $E_t(\phi \ge \rho)$ is obvious.


\begin{assumption} \label{assum5.1} \rm
 Suppose that there exist two positive numbers $m, m'$  such that
for every $\rho\in ]0,m]$ we have:
\begin{itemize}
\item[(i)] $E_t(\phi =0)\subset E_t(\phi < \rho)$, or for short
$E_t(0)\subset E_t(\rho)$;

\item[(ii)]
$\operatorname{dist}\{ \partial E_t(\phi=0),\partial E_t(\phi
\leq \rho)\} \ge m'$
 for every $t\in \mathbb{R}^+$,
where  dist is the Euclidean distance of sets, and
$\partial E$ is the boundary of $E$.
\end{itemize}
\end{assumption}

\begin{definition} \label{def5.2} \rm
 A function $W=W(t,x):\mathbb{R}^+\times \mathbb{R}^n\to \mathbb{R}$
will be called
\emph{definitively positive, negative, not equal to zero}
on the sets $E_t(\phi=0)$ with respect to $h(t,x)$ if there
exists a constant $m>0$ such that for every $\eta \in ]0,m]$
there exist $\rho, \beta >0$ with the property:
$(t,x)\in \mathbb{R}^+\times \mathbb{R}^n$ with $h(t,x)>\eta$
 and $\phi (t,x)< \rho$ imply respectively
$W(t,x)> \beta$, $W(t,x)< -\beta$, $|W(t,x|> \beta$.
\end{definition}

\begin{theorem} \label{thm5.3}
Suppose that there exist: two functions $V=V(t,x)$ and
$W=W(t,x)\in C^1$ from $\mathbb{R}^+\times \mathbb{R}^n$ to $\mathbb{R}$,
and a constant $a>0$ such that, on $\mathbb{R}^+\times \mathbb{R}^n$,
we have:
\begin{itemize}
\item[(i)] $V(t,x)\ge 0, \sup V(t,x)>0, W(t,x)\ge -a$;

\item[(ii)] for every $\epsilon>0$ two numbers $r, b>0$ exist
such that $h(t,x)=\epsilon$
and $V(t,x)<r$ imply $W(t,x)>b$.

\end{itemize}
Then we can construct a family of functions
that verifies  hypothesis (i) of Theorem \ref{thm3.1}.
\end{theorem}

\begin{proof}
Given $\epsilon>0$ with $r,b$; let $0<\mu \leq r/(a+b)$ and consider
the family of functions
\begin{equation}
v_{\mu}=v_{\mu}(t,x)=V(t,x)+\mu W(t,x). \label{e5.2}
\end{equation}
Suppose $h=h(t,x)=\epsilon$ and $V(t,x)\ge r$ hence
$v_{\mu}(t,x)\ge r-\mu a$ and
\begin{equation}
v_{\mu}(t,x)-\mu b\ge r-\mu (a+b)\ge 0 \label{e5.3}
\end{equation}
hence we have $v_{\mu}(t,x)\ge \mu b>0$.
If $h=h(t,x)=\epsilon$ and $V(t,x)<r$ we have $W(t,x)>b$ and
\begin{equation}
v_{\mu} (t,x)=V(t,x)+\mu W(t,x)\ge \mu b. \label{e5.4}
\end{equation}
\end{proof}

\begin{theorem} \label{thm5.4}
Suppose that there exist three functions  of class $C^1$:
$V=V(t,x)$, $W=W(t,x)$ from $\mathbb{R}^+\times \mathbb{R}^n$ to
$\mathbb{R} $
and $\phi=\phi (t,x) $ from $\mathbb{R}^+\times \mathbb{R}^n$
to $\mathbb{R}^+$, and two constants $M, M'>0$ such that,
 on $\mathbb{R}^+\times \mathbb{R}^n$, we have:
\begin{itemize}
\item[(i)] $\phi(t,x)\ge 0; \phi(t,x)=0$ implies
$\dot V(t,x)\leq 0, \phi(t,x)$ verifies Assumption \ref{assum5.1};

\item[(ii)]  for every $\chi >0$ there exists $\chi'>0$ such
that for every $t\in \mathbb{R}^+$
when $\operatorname{dist}[(t,x),E_t(\phi =0)]>\chi $
we have $\dot V(t,x)<-\chi'$;

\item[(iii)] $|W(t,x)|$ and $\|WX(t,x)\|<M$ on
$\mathbb{R}^+\times \mathbb{R}^n$;

\item[(iv)] $\dot V(t,x)\in GG,  \dot W(t,x)\in ZZ,
\|\dot W(t,x)\|<M'$ on $\mathbb{R}^+\times \mathbb{R}^n$;

\item[(v)] $\dot W(t,x)$ is definitively not equal to zero
with respect to $h$ on the
sets $E_t(\phi =0)$ and $h\in C^1$.
\end{itemize}
Then we can construct a function whose derivative belongs to $ZZ$.
\end{theorem}


\begin{proof}
 Since $\dot W(t,x)$ is  definitively not equal to zero on the sets
$E_t(\phi =0)$ with respect to $h$ there exists $m>0$ such that given
$\eta \in]0,m]$ there exist $\beta , \rho>0$ and three sets:
\begin{gather}
A_1=\{ (t,x)\in \mathbb{R}^+\times \mathbb{R}^n: h(t,x)\ge \eta,
\phi (t,x)\leq \rho, \dot W(t,x) <-\beta \}, \label{e5.5}\\
A_2=\{ (t,x)\in \mathbb{R}^+\times \mathbb{R}^n: h(t,x)\ge \eta,
\phi (t,x)\leq \rho, \dot W(t,x) > \beta \}, \label{e5.6}\\
A_3=\{ (t,x)\in \mathbb{R}^+\times \mathbb{R}^n:h(t,x)
\ge \eta,\phi(t,x) \leq \rho,
(t,x)\notin A_1 \cup A_2 \}=\emptyset \label{e5.7}
\end{gather}
since $W(t,x)\in C^1$ when $A_1,  A_2\neq \emptyset $ we have
$\operatorname{dist}[A_1, A_2]>0 $.

Now, we shall denote, for a fixed $t=t'\in \mathbb{R}^+$, for $i=1, 2$ and for every $r>0$:
\begin{gather}
B_i(t')=\{(t',x)\in A_i:\phi(t',x)=0\};
\operatorname{dist}\{ \partial E_{t'}(0),\partial E_{t'}(\rho)\}
=3\alpha \;(>0) \label{e5.8}\\
S=S(r)=\{ x\in \mathbb{R}^n: \|x\|<r \}. \label{e5.9}
\end{gather}
Consider also the sets:
\begin{gather*}
C_i(t')=S(r)\cap B_i(t'),\\
D_i(t')=\{ (t',x)\in \mathbb{R}\times \mathbb{R}^n:
\operatorname{dist}[(t',x),C_i(t')]<\alpha \},\\
D'_i(t')=\{ (t',x)\in \mathbb{R}\times \mathbb{R}^n:
\operatorname{dist}[(t',x),C_i(t')]<2\alpha \},\\
D''_i(t')=\{ (t',x)\in \mathbb{R}\times \mathbb{R}^n:
\operatorname{dist}[(t',x),C_i(t')]<3\alpha \}\,.
\end{gather*}
Put $\psi_i(t',x)=0$ for $(t',x)\notin D'_i$ and
$\psi_i(t',x)=1$  for $(t',x)\in D'_i$ we consider the functions
\begin{equation}
T_i(t',x)=\int_{ \mathbb{R}^n} \psi_i (t',x)
\Omega_{\alpha}(x-u)du \label{e5.10}
\end{equation}
where $\Omega_{\alpha}$ is the averaging kernel of radius $\alpha$,
$i=1, 2$, and $u\in \mathbb{R}^n$.

Since $h, \phi \in C^1$ we can obtain two functions
$T_i=T_i(t,x):\mathbb{R}^+\times \mathbb{R}^n \to \mathbb{R}^+$
that belong to $C^1$ with respect to the first variable,
and belong to   $C^\infty$ with respect to $x$, and
\begin{itemize}
\item[(i)] $0\leq T_i(t,x) \leq 1$ for $(t,x)\in D''_i$ and
$T_i(t,x)=0$ for $(t,x)\notin D''_i$;

\item[(ii)] $ \big|{{\partial T_i(t,x)} \over {\partial t}}\big|,
\|{{\partial T_i(t,x)} \over {\partial x}}\|\leq N$,
suitable strictly positive constant;

\item[(iii)] $\dot T_i(t,x)={{\partial T_i(t,x)} \over {\partial t}}+
{{\partial T_i(t,x)} \over {\partial x}} X$,
$\|\dot T_i\|\leq N(1+\|X\|)$;

\item[(iv)] $\dot T_i(t,x)=0$ if $(t,x)\notin D''_i$ or
if $(t,x)$ is in the interior of $D_i $.

\end{itemize}

Now, let us consider:

(1) the function $T=T(t,x)$ defined on
$\mathbb{R}^+\times \mathbb{R}^n$
such that $T=T_1(t,x)$ for $(t,x)\in D''_1(t)$, $T=-T_2(t,x)$
for $(t,x)\in D''_2(t)$; $T=0$ when $(t,x)\notin D''_1(t)\cup D''_2(t)$;

(2) the function  $\omega=TW$ defined on the set
\begin{equation}
\Gamma=\{ (t,x)\in \mathbb{R}^+\times \mathbb{R}^n:h(t,x)
\ge \eta , \|x\|\leq r \}.\label{e5.11}
\end{equation}
Since
\begin{equation}
\dot \omega (t,x)=T\dot W(t,x)+\dot T W(t,x),\label{e5.12}
\end{equation}
we have
\begin{gather}
\|\dot TW(t,x)\| = \big| {{\partial T} \over {\partial t}}+
{{\partial T} \over {\partial x}} X \big| |W(t,x)|
\leq  N(|W|+\|WX\|) \leq  2NM, \notag\\
\|T\dot W(t,x)\| \leq \|\dot W(t,x)\| <M', \quad
\|\dot \omega(t,x)\| < 2NM+M'.
\label{e5.13}
\end{gather}

Let us finally consider the following function, defined on $\Gamma$,
\begin{equation}
v_{\nu}=v_{\nu}(t,x)=V(t,x)+\nu \omega(t,x), \nu>0 \label{e5.14}
\end{equation}
and its derivative
\begin{equation}
\dot v_{\nu}=\dot v_{\nu}(t,x)=\dot V(t,x)
+\nu \dot {\omega}(t,x) \label{e5.15}
\end{equation}
It is obvious that for fixed $t\in \mathbb{R}^+, (t,x)\in \Gamma$
with  $(t,x)\notin [D''_1(t) \cup D''_2(t)]$,
we have $v_{\nu}(t,x)=V(t,x), \dot v_{\nu}(t,x)=\dot V(t,x)$.

In this first case since
$\operatorname{dist}[(t,x),E_t(\phi=0)]\ge 3\alpha$ there exists
$\alpha'>0$ such that $\dot V(t,x)<-\alpha'<0$.

If $(t,x)\in D_1(t)\cap \Gamma$ then
  $T=1, \omega (t,x) =W(t,x), v_{\nu} (t,x)=V(t,x)+
\nu W(t,x)$ with $\dot V(t,x)\leq 0, \dot W(t,x)<-\beta$ hence
$\dot v_{\nu} (t,x)<-\nu \beta <0$ for every $\nu>0$.
When $(t,x)\in D''_1(t)\cap \Gamma$ with $(t,x)\notin D_1(t)$, we have
$\alpha \leq \operatorname{dist}[(t,x),E_t(\phi=0)]\leq 3\alpha$
then there exists $\alpha''>0$ such that $\dot V(t,x)<-\alpha''$;
 therefore,
$\dot v_{\nu} (t,x)<-\alpha''+\nu (2MN+M')$.
If  $0<2\nu [2MN+M']<\alpha''$ we obtain
\begin{equation}
2\dot v_{\nu} (t,x)<-\alpha''.\label{e5.16}
\end{equation}
The cases $(t,x)\in D_2(t)\cap \Gamma$ and
$(t,x)\in D''_2(t)\cap \Gamma$ with
$(t,x)\notin D_2(t)$ are trivial as $A_1=\emptyset$ or
$A_2=\emptyset$.
\end{proof}

\section{Application to the motion of rigid bodies}

In this section we present an illustrative mechanical example.
 Putting
$$
CD=\{ (p,q,r,\gamma_1,\gamma_2)\in \mathbb{R}^5: \gamma_1^2+
\gamma_2^2 \leq 1\}
$$
on the set $\mathbb{R}^+\times CD$ let us consider the
system of equations
\begin{equation}
\begin{gathered}
\dot Ap+2A\dot p+2(C-A)qr=2Pz\gamma_2 \gamma_3-2f_1p-2f_4r, \\
\dot Aq+2A\dot q+2(A-C)pr=-2Pz\gamma_1 \gamma_3-2f_2q-2f_5r,\\
\dot Cr+2C\dot r=2f_4p+2f_5q-2f_3r,\\
\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, \quad
\gamma_1^2+\gamma_2^2+\gamma_3^2=1\,.
\end{gathered} \label{e6.1}
\end{equation}
This system, with the usual designation and when $Pz=0$,
constitutes the basic dynamical system for the motion of a
symmetrical rigid body about a fixed point and
variable mass \cite{o1}; if $P=0$ the body is non heavy,
if $z=0$ the center of gravity is a fixed point.

\begin{assumption} \label{assum6.1} \rm
 Assume that  the given functions
 $A(t), C(t)\in C^1(\mathbb{R}^+\to \mathbb{R}^+)$,
$P(t)\in C(\mathbb{R}^+\to \mathbb{R}^+)$,
$z(t)\in C(\mathbb{R}^+\to ]0,\infty[)$  and
$G(t,p,q,r,\gamma_1,  \gamma_2),U(t,\dots \gamma_2)$
satisfy the following properties:
\begin{itemize}
\item[(i)] $\inf  \{ A(t),C(t),P(t),-z(t)\} >0$, $-Pz=\text{const} >0$
and $A'=\inf A(t)\leq \sup A(t)=A''$;

\item[(ii)] $0<f'=\inf \{ f_i(t,p,q,r,\gamma_1,\gamma_2)\}
\leq \sup \{ f_i(..)\}=f''$
for $i=1,2,3$;

\item[(iii)]  for every $t_0,p_0,q_0,r_0,\gamma_1^0,\gamma_2^0$
there exists only one solution, defined for $t\ge t_0 $;

\item[(iv)] $G=\{ Ap,Aq,Cr,-Pz\gamma_1,-Pz\gamma_2 \}$,
 $U=A^2 p^2+A^2 q^2+C^2 r^2$;

\item[(v)]   as measures of stability we select the following
functions $h$ and $h_0$:
\begin{equation}
4h = A(p^2+q^2)+Cr^2-Pz(\gamma_1^2+\gamma_2^2) = h_0; \label{e6.2}
\end{equation}

\item[(vi)]  as auxiliary Liapunov's functions we  select
the following functions of Matrosov's type
\begin{gather}
V = \frac 1 2 [A(p^2+q^2)+Cr^2]-\frac 1 2 Pz(\gamma_1^2+\gamma_2^2)
= 2h, \label{e6.3}\\
W = A(p\gamma_2-q\gamma_1) .\label{e6.4}
\end{gather}

\item[(vii)] $f_1 = 2A^2$, $f_2 = 2A^2$, $f_3 = 2C^2$.
\end{itemize}
\end{assumption}

\begin{theorem} \label{thm6.2}
Under Assumption \ref{assum6.1}, we deduce that:
\begin{itemize}
\item[(i)]
 The measure $h_0$ is uniformly finer than $h$, $h=0$ is equivalent
to $p=q=r=\gamma_1=\gamma_2=0$.

\item[(ii)] $V=2h$ and ${\lim_{h\to 0}}V=0$ hence condition
{\rm (3.2)}(ii) hold, and we obtain ${\lim_{h\to 0}}W=0$.

\item[(iii)] $G=\operatorname{grad} V$ hence, for $\phi =1$, condition
{\rm (3.2)}(iv) is  verified.


\item[(iv)] $\dot V=-f_1p^2-f_2q^2-f_3r^2=-2U<0$, so conditions
{\rm (3.2)}(iii) and {\rm (3.2)}(v) hold for $\theta =2>\phi =1$,
$-f''(p^2+q^2+r^2)\leq \dot V \leq -f'(p^2+q^2+r^2)$.

\item[(v)] $\dot V=0$ if and only if $p=q=r=0$; therefore,
the L-measure on  $\mathbb{R}^5$ of the set
$$
E_1=E(\dot V=0)=\{  p=q=r=0,(\gamma_1,\gamma_2)\in \mathbb{R}_2
:\gamma_1^2+\gamma_2^2 \leq 1 \}
$$
is equal to zero, hence the system \eqref{e6.1} is strongly weakly
$(h_0,h)$-t.bistable with respect to perturbation
\begin{equation}
\sigma\{ Ap, Aq, Cr, 0, 0 \} \label{e6.5}
\end{equation}
where $\sigma=\sigma(t,p,q,r,\gamma_1,\gamma_2)>0$ belongs to $C^1$.

\item[(vi)] Since
\begin{equation}
\begin{aligned}
\dot W&
=2(A-C)qr\gamma_2 -A\dot p\gamma_2 +2Pz\gamma_2^2 -2f_1
p\gamma_2 -2f_4 r\gamma_2+Ap\dot{\gamma_2}\\
&\quad -2(C-A)pr\gamma_1+A\dot q\gamma_1+2Pz\gamma_1^2
 +2f_2q\gamma_1+ 2f_5r\gamma_1+Aq\dot {\gamma_1},
\end{aligned}\label{e6.6}
\end{equation}
hence on the set $E_1$, we obtain
$\dot W=2Pz(\gamma_1^2+\gamma_2^2)\leq 0$.


\item[(vii)] If $0<\eta <1$ and
$\gamma_1^2+\gamma_2^2=\gamma \ge \eta$ we deduce
$2Pz\gamma\leq 2Pz\eta <0$ i.e.
on the set $E_2=E(\dot V=0,\eta \leq \gamma \leq 1)$
\ we have $\dot W\leq 2Pz\eta <0$ and $4h=-Pz\gamma\ge -Pz\eta $.
Since $W\in C^1$  there exists $b>0$ such that on the set
\begin{equation}
(CD)_1=\{ (p,q,r)\in \mathbb{R}^3:p^2+q^2+r^2\leq 9b^2 \}
\times \{ (\gamma_1,\gamma_2)
\in \mathbb{R}^2:\gamma \ge \eta \} \label{e6.7}
\end{equation}
we have $\dot W \leq Pz\eta<0$, i.e. the function $\dot W$
is definitely negative on the set  $E(\dot V=0)$ with respect
to the measure $h$ when $\gamma \ge \eta$.
According to Theorem \ref{thm5.4}  we have $A_2=0$ and
\begin{equation}
\begin{aligned}
A_1&=\{ (p,q,r,\gamma_1,\gamma_2):h\ge -\frac 1 2 Pz\gamma
 \ge -\frac 1 2 Pz\eta; \phi=p^2+q^2+r^2\leq 9b^2 \} \\
&=\{ (p,q,r)\in \mathbb{R}^3:\phi=p^2+q^2+r^2 \leq 9b^2 \}
 \times \{ (\gamma_1,\gamma_2) \in \mathbb{R}^2:\gamma \ge \eta \}.
\end{aligned}\label{e6.8}
\end{equation}

\item[(viii)]  Consider the  function $\psi =\psi(p,q,r)$ from
$\mathbb{R}^3$ to $\mathbb{R}^+$ such that
$\psi=0$ when $4b^2\leq \phi$, $\psi=1$ when $0\leq \psi<4b^2 $
and their regularized function, defined on $\mathbb{R}^3$:
\begin{equation}
T(x)=\int_{\mathbb{R}^3}\psi_i(x)\Omega_{b}(x-u)du \label{e6.9}
\end{equation}
where $x=(p,q,r)$ and $\Omega_b$ is the averaging kernel of
radius $b$.
It is obvious that $0\leq T \leq 1,   T\in C^{\infty}$ and:
$T =0$ when $\phi\ge 9b^2$,
$0<T \leq 1$ when $b^2\leq \phi<9b^2$,$T=1$ when $\phi<b^2$,
$|\dot T|<M''(>0)$
being $M''$ a suitable constant.
On the set $(CD)_1$ we obtain
\begin{equation}
|TW|=|T\|W|\leq |W|\leq A''[|p\gamma_2|+|q\gamma_1|]\leq 6A''b.
\label{e6.10}
\end{equation}

\item[(ix)]  Successively consider the family of functions
$w_{\mu}=V+\mu TW$  defined on the set
$$
\mathbb{R}^+\times (CD)_2=\{ t\in \mathbb{R}^+\}
\times \{ (p,q,r)\in \mathbb{R}^3;
(\gamma_1,\gamma_2)\in \mathbb{R}^2:\eta \leq \gamma<1\}
$$
when $\mu >0$ and suppose that:
\begin{itemize}
\item[(1)] $\phi \ge 9b^2$ in this case $T=0$, $w_{\mu}=V$ hence for $h=s$ we obtain $w_{\mu}=2s$;
\item[(2)] $\phi<9b^2$ now $|TW|\leq 6A''b$ therefore $h=s$ implies
\begin{equation}
 [w_{\mu}]_{h=s}=[V+\mu TW]_{h=s}\ge 2s-6\mu A''b>s \leftrightarrow
\mu<\frac {s}{6A''b}. \label{e6.11}
\end{equation}
\end{itemize}
\item[(x)]  Consider the derivatives
\begin{equation}
{\dot w}_{\mu}=\dot V+\mu \dot T W+\mu T\dot W \label{e6.12}
\end{equation}
and suppose that:
\begin{itemize}
\item[(1)] $\phi \ge 9b^2$ then $T=0$ i.e.
${\dot w}_{\mu}=\dot V=-f_1p^2-f_2q^2-fr^3r^2\leq -f'\phi \le
-9f'b^2$.
\item[(2)] $\phi \leq b^2$ hence $T=1, {\dot w}_{\mu}
=\dot V+\mu \dot W, {\dot w}_{\mu}<\mu Pz\eta$.

\item[(3)] $b^2<\phi <9b^2$ then $\dot W \leq 0, {\dot w}_{\mu}
 \leq \dot V+\mu \dot T W$  but \ $|\dot T W|\leq M''|W|\leq 6M''A''b$
 and $\dot V \leq -f'b^2$ therefore we obtain
${\dot w}_{\mu}\leq -f'b^2+6\mu M''A''b\leq -3\mu M''A''2b$
 if and only if  $\mu \leq {{f'b}\over {9M''A''}}$.
\end{itemize}
\end{itemize}
 When $\mu \leq  \mu'=\min [{{s}\over{6A''b}} ,  {{f'b}\over{9M''A''}}]$
all the conditions of Theorem \ref{thm3.1} are verified, hence
 \eqref{e6.1} is weakly $(h_0,h)$-t.bistable with respect
to the perturbations
\begin{equation}
Y=\sigma \{ -(w_{\mu})_p , -(w_{\mu})_q ,
-(w_{\mu})_r , 0 , 0 \}  \label{e6.13}
\end{equation}
where $\mu \leq \mu'$ and $\sigma =\sigma (p,q,r,\gamma_1,\gamma_2)$
is an arbitrary continuous function.
\end{theorem}

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\end{document}