\documentclass[reqno]{amsart} \usepackage{epsf} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2001(2001), No. 12, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu \quad ejde.math.unt.edu (login: ftp)} \thanks{\copyright 2001 Southwest Texas State University.} \vspace{1cm}} \begin{document} \title[\hfilneg EJDE--2001/12\hfil Asymptotic behavior of solutions ] {Asymptotic behavior of the solutions of a class of second order differential systems } \author[ Svetoslav Ivanov Nenov \hfil EJDE--2001/12\hfilneg] { Svetoslav Ivanov Nenov } \address{ Svetoslav Ivanov Nenov \hfill\break Department of Mathematics, University of Chemical Technology and Metallurgy; 8, Kliment Ohridsky blvd., Sofia 1756, Bulgaria } \email{ s.nenov@uctm.edu, s\_nenov@hotmail.com } \date{} \thanks{Submitted April 4, 2000; December 28, 2000. Published January 30, 2001.} \subjclass{34D05, 34D10, 34E05} \keywords{asymptotic behaviour, gradient systems, T. Wazewski's theorem} \newtheorem{thm}{Theorem} \newtheorem{lem}{Lemma} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \begin{abstract} In the present paper it is proved that for any solution $x_1(t)$ of the system $M \ddot x + \dot x = f(t,x)$, for which $\lim\limits_{t\to\infty}\|\dot x_1(t)\|=0$, there exists a solution $x_2(t)$ of the system $\dot x = f(t,x)$ such that $\lim\limits_{t\to\infty}\|x_1(t)-x_2(t)\|=0$. Some generalizations of this result are also presented. The case $f(t,x)=-\nabla U(x)$ has been investigated explicitly. \end{abstract} \maketitle \section{Statements and main results} We consider the following two $n$-dimensional systems \begin{gather}\label{eq:1} M \ddot x + \dot x = f(t,x) \intertext{and} \label{eq:2} \dot x = f(t,x), \end{gather} where $M \geq 0$ is a constant; $\cdot$" denotes differentiation with respect to $t$; $f:\mathbb{R}_+ \times \Omega \to \mathbb{R}^n$; $\mathbb{R}_+ \equiv [0,\infty)$; $\Omega$ is a domain in $\mathbb{R}^n$; $\mathbb{R}^n$ is the $n$-dimensional Euclidean space with Euclidean scalar product $\langle\cdot ,\cdot\rangle$ and corresponding norm $\|\cdot\|$. Let $(x_{11}, x_{12})\in\Omega\times\mathbb{R}^n$ and $x_2 \in \Omega$ be fixed initial points and $t_0\in\mathbb{R}_+$ be a fixed initial moment; $x_1 (t; t_0, x_{11}, x_{12})$ and $x_2 (t;t_0,x_2)$ denote the solutions of the systems \eqref{eq:1} and \eqref{eq:2} with initial conditions \begin{gather}\label{ini_for_eq:1} x_1 (t_0;t_0,x_{11},x_{12}) = x_{11}, \quad \dot x_1 (t_0;t_0,x_{11},x_{12}) = x_{12} \intertext{and} \label{ini_for_eq:2} x_2 (t_0;t_0,x_2) = x_2, \end{gather} respectively. We introduce the following hypotheses (H1):\\ \begin{enumerate} \item[(H1.1)] $\Omega$ is a bounded domain in $\mathbb{R}^n$; $f\in {\bf C} (\mathbb{R}_+\times\Omega, \mathbb{R}^n)$. \item[(H1.2)] The function $f$ is Lipschitz with respect to the second argument with Lipschitz constant $L \geq 0$. \item[(H1.3)] For arbitrary initial conditions $(t_0,x_{11}, x_{12})\in\mathbb{R}_+\times\Omega\times\mathbb{R}^n$, $(t_0,x_2) \in \mathbb{R}_+\times\Omega$, the Cauchy problems \eqref{eq:1}, \eqref{ini_for_eq:1} and \eqref{eq:2}, \eqref{ini_for_eq:2} have unique solutions $x_1(t;t_0,x_{11},x_{12})$ and $x_2(t;t_0,x_2)$, respectively, defined on $t$-interval $\mathbb{R}_+$. Moreover $$\{x_1(t;t_0,x_{11},x_{12}) : t \in \mathbb{R}_+\} \subset \Omega,\quad \{x_2(t;t_0,x_2) : t \in \mathbb{R}_+\} \subset \Omega .$$ \end{enumerate} In this article, the following theorem contains one of the basic results. \begin{thm}\label{main-theorem_1} Assume the following conditions hold: \begin{enumerate} \item The hypothesis (H1) holds. \item $(t_0, x_{11},x_{12}) \in \mathbb{R}_+\times\Omega\times\mathbb{R}^n$ is the fixed initial condition for \eqref{eq:1}, \eqref{ini_for_eq:1}. \item $$\label{thm_1:condition-1} \lim\limits_{t\to\infty} \|\dot x_1 (t;t_0,x_{11}, x_{12})\| = 0\,.$$ \end{enumerate} Then there exists at least one initial condition $x_2\in\Omega$ for the system \eqref{eq:2} such that $$\label{thm_1:condition-2a} \lim\limits_{t\to\infty} \|x_1(t;t_0,x_{11}, x_{12}) - x_2(t;t_0,x_2) \| = 0.$$ \end{thm} Theorem \ref{main-theorem_1} is proved in subsection \ref{proof_of_Theorem_1}. \begin{example} \rm Let us consider the differential equations: \begin{gather}\label{ex-1:eq1} \ddot x + \dot x = t, \end{gather} and \begin{gather}\label{ex-1:eq2} \dot x = t. \end{gather} An immediate integration of these equations yields: $$x_1 (t;0,x_{11},x_{12}) = 1+x_{12}+x_{11}-t+\frac{t^2}{2} - e^{-t} (1+x_{12})$$ and $$x_2(t;0,x_2) = x_2 + \frac{t^2}{2}.$$ It is not difficult to check that for any three points $x_{11},x_{12},x_2 \in \mathbb{R}$ we have $$\lim\limits_{t\to\infty}|x_1 (t;0,x_{11},x_{12}) - x_2(t;0,x_2)| \not = 0.$$ Moreover, in this example, for any initial conditions $x_{11},x_{12}\in\mathbb{R}$ of the problem \eqref{ex-1:eq1}, \eqref{ini_for_eq:1} we have $$\lim\limits_{t\to\infty}|\dot x_1 (t;0,x_{11},x_{12})| = \lim\limits_{t\to\infty}|-1+t+e^{-t}(1+x_{12})| = \infty.$$ From the above equality it follows that \eqref{thm_1:condition-1} is not true and, as we have shown, equality \eqref{thm_1:condition-2a} is not valid. Thus \eqref{thm_1:condition-1} is an essential condition. \end{example} The following result is derived similarly to the proof of Theorem \ref{main-theorem_1} (see subsection \ref{proof_of_Theorem_2}). \begin{thm} \label{main-theorem_2} Assume the following conditions are fulfilled: \begin{enumerate} \item The hypothesis (H1) holds. \item $(t_0, x_{11},x_{12}) \in \mathbb{R}_+\times\Omega\times\mathbb{R}^n$ is a fixed initial condition for \eqref{eq:1}, \eqref{ini_for_eq:1}. \item $$\label{thm_2:condition-1} \lim\limits_{t\to\infty} \|\ddot x_1 (t;t_0,x_{11}, x_{12})\| = 0.$$ \end{enumerate} Then there exists at least one initial condition $x_2\in\Omega$ of system \eqref{eq:2} such that the equality \eqref{thm_1:condition-2a} is valid. \end{thm} \section{An application: Second order gradient systems} In the present section we shall discuss some asymptotic properties of the solutions of the following system \begin{gather}\label{II sys} M \ddot x + \dot x = -\nabla U(x), \end{gather} where $M\geq 0$ is a constant; $U\in {\bf C^1} (\Omega, \mathbb{R})$; $\Omega$ is a domain in $\mathbb{R}^n$ such that any solution of \eqref{II sys} starting in $\Omega$ remains in $\Omega$. First, let us write \eqref{II sys} as a first order system in $\mathbb{R}^{2n}$: \begin{gather}\label{II sys-1} \dot x = y,\quad \dot y = M^{-1}\left(-y-\nabla U(x)\right). \end{gather} Setting $$L(x,y) = \frac{M}{2} y^2 + U(x),$$ it is not difficult to see that \begin{gather}\label{for U -1} L^\prime (x,y) = M y \dot y + \nabla U(x) y = (M\dot y + \nabla U(x)) y = - y^2 \leq 0, \end{gather} for any $(x,y)\in\Omega\times\mathbb{R}^{n}$. Therefore, if $U(x) \geq 0$, $x\in\Omega$ then $L(x,y)$ is a Liapunov function (i.e. a continuous non-negative function which satisfies locally a Lipschitz condition) for \eqref{II sys-1}. Let $\mathcal{M}=\{(x,y):L^\prime(x,y) = 0\}=\{(x,0):x\in\Omega\}$ and let $\mathcal{M}_1$ be the union of all points $(x_0,y_0)$ of all orbits $(x(t;x_0,y_0),y(t;x_0,y_0))$ such that $$\{(x(t;x_0,y_0),y(t;x_0,y_0)):t\in\mathbb{R}\}\subset \mathcal{M}.$$ \begin{thm}\label{LaSalle type thm} Suppose that $U(x) \geq 0$ for all $x\in\Omega$, and $\lim\limits_{\|x\|\to\infty} U(x) = \infty$. Then: \begin{enumerate} \item All solutions of \eqref{II sys-1} are bounded. \item Every solution of \eqref{II sys-1} approaches $\mathcal{M}_1$ as $t\to\infty$, i.e. for any $(x_0,y_0)\in\Omega\times\mathbb{R}^n$ we have $$(x(t;x_0,y_0),y(t;x_0,y_0)) \to \mathcal{M}_1,\quad \text{as}\quad t\to\infty.$$ \item For any $(x_0,y_0)\in\Omega\times\mathbb{R}^n$, $$\lim\limits_{t\to\infty} \dot x(t;x_0,y_0)=\lim\limits_{t\to\infty} y(t;x_0,y_0)= 0\,.$$ \end{enumerate} \end{thm} \begin{proof} Obviously, $\lim_{\|x\|^2+\|y\|^2\to\infty}L(x,y)=\infty$. Then the first statement of Theorem \ref{LaSalle type thm} follows from \cite[Theorem 10.1]{Taro}. The second statement follows immediately from results in \cite{LaSalle}, see also \cite[Theorem 14.4, Theorem 14.7]{Taro}. The third statement follows from second one and implication $\mathcal{M}_1 \subset \mathcal{M} = \{ (x,0) : x \in \Omega \}$. \end{proof} The following result follows immediately from Theorem \ref{main-theorem_1} and Theorem \ref{LaSalle type thm}. \begin{thm}\label{main-theorem_3} Let the following conditions hold true: \begin{enumerate} \item $\Omega$ is a domain in $\mathbb{R}^n$; $U\in {\bf C^1} (\Omega, \mathbb{R}^n)$; the hypothesis (H 1.3) is valid, where $f(t,x)=-\nabla U(x)$. \item $(t_0,x_{11},x_{12})\in\mathbb{R}_+\times\Omega\times\mathbb{R}^n$ is a fixed initial condition for the initial-value problem \eqref{II sys}, \eqref{ini_for_eq:1}. \end{enumerate} Then there exists at least one initial condition $x_2\in\Omega$ for the system $\dot x = -\nabla U(x)$ such that $$\label{thm_1:condition-2c} \lim\limits_{t\to\infty} \|x_1(t;t_0,x_{11}, x_{12}) - x_2(t;t_0,x_2) \| = 0.$$ \end{thm} It is not difficult to derive some properties of the solutions of systems \eqref{II sys} or \eqref{II sys-1}: the $\omega$-limit set of a solution of system \eqref{II sys-1} consists of critical points only; if there are two critical points in the $\omega$-limit set then there are infinitely many critical points in the same $\omega$-limit set; there is no non-trivial periodic solutions of \eqref{II sys-1}, etc. The proofs of these facts follow from Theorem \ref{main-theorem_3} and results in \cite[Chapter 1, \S 1]{PalisandMelo}. \section{A Topological Principle} In the present Section we shall deduce the Topological Principle in the theory of autonomous dynamical systems or the so-called T.~Wazewski's Theorem. The Topological Principle is related to the initial value-problem $$\label{eq:topol_principle} \dot x=f(t,x),\quad x(t_0)=x_0,$$ where $f\in {\bf C}({\mathcal{E}},{ R}^n)$; ${\mathcal{E}}$ is an open $(t,x)$-set in $\mathbb{R} \times \mathbb{R}^n$; $(t_0,x_0)\in \mathcal{E}$. Let $\mathcal{E}_0$ be a non-empty open subset in ${\mathcal{E}}$. We recall the following definitions. \begin{defn}\label{defn:egress_points} The point $(t_0,x_0)\in {\mathcal{E}}\cap \partial {\mathcal{E}}_0$ is said to be: \begin{enumerate} \item {\it an egress point} of ${\mathcal{E}}_0$ with respect to the system \eqref{eq:topol_principle} if, for every solution $x(t;t_0,x_0)$ of \eqref{eq:topol_principle} there exists $\theta > 0$ such that $\{(t,x(t;t_0,x_0)):t\in [t_0-\theta,t_0)\}\subset {\mathcal{E}}_0$. \item {\it an strict egress point} of ${\mathcal{E}}_0$ with respect to the system \eqref{eq:topol_principle} if, $(t_0,x_0)$ is an egress point of ${\mathcal{E}}_0$ and $\{(t,x(t;t_0,x_0)):t\in [t_0,t_0+\theta ]\}\subset {\mathcal{E}}\setminus {\mathcal{E}}_0$ for sufficiently small $\theta >0$. See Figure \ref{figure:Topological_principle}. \end{enumerate} \end{defn} In the following, ${\mathcal{E}}_0^e$ (${\mathcal{E}}_0^{se}$) denotes the set of all egress (strict egress) points of ${\mathcal{E}}_0$. It is clear that ${\mathcal{E}}_0^{se} \subset {\mathcal{E}}_0^e$. \begin{defn}\label{defn:UV-set_definition} The open subset ${\mathcal{E}}_0$ in ${\mathcal{E}}$ is said to be an open $[U,V]$-subset in ${\mathcal{E}}$ with respect to the system \eqref{eq:topol_principle} if: \begin{enumerate} \item There exist integers $p,q\geq 1$ and continuous functions $U_j:{\mathcal{E}}\to \mathbb{R}$, $j=1,\dots ,p$ and $V_k:{\mathcal{E}}\to \mathbb{R}$, $k=1,\dots ,q$ such that $${\mathcal{E}}_0=\left\{ (t,x):U_j(t,x)<0\ \text{ and }\ V_k(t,x)<0,\ 1\leq j\leq p,\ 1\leq k\leq q\right\} .$$ \item If for any two indexes $\alpha = 1,\dots , p$ and $\beta = 1,\dots , q$ we denote \begin{eqnarray*} {\mathcal{U}}_\alpha &=& \big\{ (t,x): U_\alpha (t,x)=0,\; U_j(t,x)\leq 0 \text{ and } V_k(t,x) < 0, \\ &&1\leq j\leq p,\; j\neq \alpha,\; 1\leq k\leq q \big\} , \\[3pt] {\mathcal{V}}_\beta &=&\big\{ (t,x): U_j(t,x) < 0,\; V_\beta (t,x)=0 \text{ and } V_k(t,x)\leq 0, \\ &&1\leq j\leq p,\; 1\leq k\leq q,\; k\not=\beta\big\} , \end{eqnarray*} then the trajectory derivatives $$U_\alpha ^{\prime }(t_0,x_0)=\frac{d U_\alpha (t,x(t;t_0,x_0))}{dt}\!\!\mid_{t=t_0},$$ $$V_\beta ^{\prime }(t_0,x_0)=\frac{d V_\beta (t,x(t;t_0,x_0))}{dt}\!\!\mid _{t=t_0}$$ exist, satisfying the inequalities $$U_\alpha ^{\prime }(t_0,x_0)>0,\ \text{ for any point }\ (t_0,x_0)\in {\mathcal{U} }_\alpha ,$$ $$V_\beta ^{\prime }(t_0,x_0)<0,\ \text{ for any point }\ (t_0,x_0)\in {\mathcal{V}}_\beta ,$$ along all solutions of \eqref{eq:topol_principle} through $(t_0,x_0)$. \end{enumerate} \end{defn} \begin{figure}[t] %\includegraphics{nenovg2.eps} \epsfxsize=10cm \epsffile{nenovg2.eps} \caption{Egress points and strict egress points \label{figure:Topological_principle}} \end{figure} The theorem (T. Wazewski's Theorem) is known also as the Topological Principle in the theory of autonomous dynamical systems. \begin{thm}\label{thm:topological_principle} Assume the following conditions: \begin{enumerate} \item ${\mathcal{E}}$ is an open $(t,x)$-set in $\mathbb{R}\times\mathbb{R}^n$; $f\in {\bf C}({\mathcal{E}},R^n)$. \item The initial-value problem \eqref{eq:topol_principle} has a unique solution through every point of ${\mathcal{E}}$, and these solutions depend continuously on initial values. \item ${\mathcal{E}}_0$ is an open subset in ${\mathcal{E}}$. \item All egress points of the set ${\mathcal{E}}_0$ are strict egress points, i.e. ${\mathcal{E}}_0^e={\mathcal{E}}_0^{se}$. \item $\mathcal{W}$ is a non-empty subset in ${\mathcal{E}}_0\cup {\mathcal{E}}_0^e$ such that $\mathcal{W}\cap {\mathcal{E}}_0^e$ is a retract of ${\mathcal{E}}_0^e$, but is not a retract of $\mathcal{W}$. \end{enumerate} Then there exists at least one point $(t_0,x_0)\in \mathcal{W}\cap {\mathcal{E}}_0$ such that $x(t;t_0,x_0)\in {\mathcal{E}}_0$ for any $t$ in the right-maximal interval of existence of $x(t;t_0,x_0)$. \end{thm} An useful tool for checking the validity of condition 4 of Theorem \ref{thm:topological_principle} is the following lemma. \begin{lem}\label{lemma:help_for_topological_principle} Assume the following conditions: \begin{enumerate} \item The conditions 1 and 2 of Theorem \ref{thm:topological_principle} hold. \item ${\mathcal{E}}_0$ is an open $[U,V]$-subset of ${\mathcal{E}}$ with respect to the system \eqref{eq:topol_principle}. \end{enumerate} Then $${\mathcal{E}}_0^e={\mathcal{E}}_0^{se}=\bigcup\limits_{\alpha =1}^p{\mathcal{U}}_\alpha \ \setminus \ \bigcup\limits_{\beta =1}^q{\mathcal{V}}_\beta,$$ where ${\mathcal{U}}_\alpha$ and ${\mathcal{V}}_\beta$ are the sets introduced in Definition \ref{defn:UV-set_definition}. \end{lem} One may find the circumstantial explanation and proofs of all results in this Section in \cite[Chapter X, \S 3]{Hartman}. \section{Proofs} \subsection{Proof of Theorem \ref{main-theorem_1}} \label{proof_of_Theorem_1} Let $(x_{11},x_{12}) \in \Omega\times\mathbb{R}^n$ be a fixed initial condition for the system \eqref{eq:1}. For simplification of notations we suppose that $t_0 = 0$. Further, we shall use the notation $x_1(t) = x_1(t;0,x_{11},x_{12})$. We set $$g:\mathbb{R}_+\times\mathbb{R}_+\to\mathbb{R},\quad g(t,u) = - \frac{6}{5} L u^2 - \frac{6}{5} L M \sqrt[3]{u} \|\dot x_1(t)\|.$$ For any initial condition $u_0\in\mathbb{R}_+$, the differential inequality \begin{gather}\label{proof:ineq-1} 2 u \dot u < g(t,u) \end{gather} has a solution $u(t;u_0)$ for which: \begin{gather}\label{proof:for_ineq-1a} u(0;u_0)=u_0, \end{gather} and \begin{gather}\label{proof:for_ineq-1c} \lim\limits_{t\to\infty} u(t;u_0) = 0. \end{gather} To prove these facts, it is sufficient to see that the initial-value problem \begin{gather}\label{proof:eq-1_for_ineq-1} 2 u \dot u = -2 L u^2 - \frac{6}{5} L M \sqrt[3]{u} \|\dot x_1(t)\|, \quad u(0; u_0) = u_0 \end{gather} has solution for which \eqref{proof:for_ineq-1a} and \eqref{proof:for_ineq-1c} hold true. The mentioned solution is $$u(t,u_0) = e^{-Lt} \left(u_0^{\frac{5}{3}} - LM\int\limits_{0}^{t} e^{\frac{5Ls}{3}}\|\dot x_1(s)\|\,ds\right)^{\frac{3}{5}}.$$ Below, we shall use the notation $u(t) = u(t;u_0)$. We set: $$\begin{array}{ll} U:\mathbb{R}_+\times \Omega\to \mathbb{R} ,&\quad U(t,x)=\|M\dot x_1(t) + x_1(t) - x \|^\frac{6}{5} - u^2(t);\\[2pt] V: \mathbb{R}_+ \to \mathbb{R}_-\equiv (-\infty,0],&\quad V(t)= - t; \end{array}$$ $$\begin{array}{ll} \mathcal{U}=\left\{ (t,x) \in \mathbb{R}_+ \times \Omega : U(t,x)=0\ \text{ and }V(t) < 0\right\}; &\\[2pt] {\mathcal{V}}=\left\{ (t,x) \in \mathbb{R}_+ \times \Omega :U(t,x) < 0\ \text{ and }V(t)=0\right\}; &\\[2pt] {\mathcal{E}}_0=\left\{ (t,x) \in \mathbb{R}_+ \times \Omega :U(t,x)<0\ \text{ and }V(t)<0\right\}; & \quad {\mathcal{E}} = \mathbb{R}_+\times \Omega . \end{array}$$ Our goal is to show that there exists at least one initial condition $\xi_2 \in \Omega$ and initial moment $\tau > 0$ for the problem \eqref{eq:2}, \eqref{ini_for_eq:2} such that \begin{gather}\label{proof:subset_1} \{(t,x_2 (t;\tau,\xi_2)) : t > \tau\} \subset {\mathcal{E}}_0. \end{gather} First, we shall prove if \begin{gather}\label{proof:for_U_prime} (t_*,x_*) \in \mathcal{U}, \text{ then } U^\prime (t_*,x_*)>0, \end{gather} where $^\prime$ denotes the derivative of function $U(t,x)$ along the trajectories of system \eqref{eq:2}, i.e. $U^\prime (t_*,x_*) = \frac{d}{dt} U(t,x_2(t;t_*,x_*))|_{t=t_*}$. Let $(t_*,x_*) \in \mathcal{U}$ be a fixed point. We set $$m: \mathbb{R}_+ \to \mathbb{R}_+ ,\quad m(t) = \|M\dot x_1(t) + x_1(t) - x_2(t;t_*,x_*)\|^2.$$ The definition of the set $\mathcal{U}$ implies \begin{eqnarray*}\label{proof:help_1} 0 = U(t_*,x_*) & =& \|M\dot x_1(t_*) + x_1(t_*) - x_*\|^{\frac{6}{5}} - u^2 (t_*) \\ &=& \|M\dot x_1(t_*) + x_1(t_*) - x_2(t_*;t_*,x_*)\|^{\frac{6}{5}} - u^2 (t_*) \\ &=& m^{\frac{3}{5}}(t_*) - u^2 (t_*) \nonumber \end{eqnarray*} or $m^{\frac{3}{5}}(t_*) = u^2(t_*)$. On the other hand, if $h_0 > 0$ is sufficiently small and if $h \in (-h_0,h_0)$ then \begin{eqnarray*}\label{proof:help_2} m(t_*+h)&=& m(t_*) + 2 h \Big\langle M\dot x_1(t_*) + x_1(t_*) - x_2(t_*;t_*,x_*) ,\\ && M\ddot x_1(t_*)+ \dot x_1(t_*) - \dot x_2(t_*;t_*,x_*)\Big\rangle + \varepsilon_1 (h), \nonumber \end{eqnarray*} where $\varepsilon_1 :(-h_0,h_0)\to\mathbb{R}$ and \begin{gather}\label{proof:limit_of_epsilon} \lim\limits_{h\to 0}\frac{\varepsilon_1 (h)}{h} = 0. \end{gather} The equalities $M \ddot x_1(t) + \dot x_1(t) = f(t,x_1(t))$ and $\dot x_2(t;t_*,x_*) = f(t,x_2(t;t_*,x_*))$ imply \begin{multline}\label{proof:help_2aa} \|M\ddot x_1(t) + \dot x_1(t) - \dot x_2(t;t_*,x_*)\| = \|f(t,x_1(t)) - f(t,x_2(t;t_*,x_*))\| \leq\\ \leq L \|x_1(t) - x_2(t;t_*,x_*)\| = L\|M\dot x_1(t) + x_1(t) - x_2(t;t_*,x_*) - M\dot x_1(t)\|\leq\\ \leq L \sqrt{m(t)} + LM\|\dot x_1(t)\|. \end{multline} From \eqref{proof:help_2} and \eqref{proof:help_2aa} at $t=t_*$ we obtain \begin{gather}\label{proof:help_3} m(t_*+h)\leq m(t_*) + 2 |h| \left(L m(t_*) + LM \sqrt{m(t_*)} \|\dot x_1(t_*)\|\right) + \varepsilon_1 (h). \end{gather} The formula \eqref{proof:help_3} yields \begin{gather}\label{proof:help_4} \begin{array}{lll} \frac{m(t_*+h)-m(t_*)}{h} &\leq 2 L m(t_*) + 2LM \sqrt{m(t_*)} \|\dot x_1(t_*)\| + \frac{\varepsilon_1 (h)}{h}, &\text{ for } h>0,\\[2pt] \frac{m(t_*+h)-m(t_*)}{h} &\geq -2 L m(t_*) - 2LM \sqrt{m(t_*)} \|\dot x_1(t_*)\| + \frac{\varepsilon_1 (h)}{h},&\text{ for } h<0. \end{array} \end{gather} From the definition of the function $m(t)$ it follows that $m(t)$ is ${\bf C^1}$-smooth. Letting $h\to\pm 0$ in the inequalities \eqref{proof:help_4} and using \eqref{proof:limit_of_epsilon} we obtain the following estimates for the derivative of function $m(t)$ at $t=t_*$ \begin{gather}\label{proof:help_5} -2L m(t_*) - 2LM \sqrt{m(t_*)}\|\dot x_1(t_*)\|\leq \dot m(t_*) \leq 2L m(t_*) + 2LM \sqrt{m(t_*)}\|\dot x_1(t_*)\|. \end{gather} Therefore, from definitions of functions $U(t,x)$, $u(t)$, \eqref{proof:help_1} and left hand-side of \eqref{proof:help_5} it follows that $$\begin{array}{rl} U^\prime (t_*,x_*) =& \frac{d}{dt} \left(m^{\frac{3}{5}}(t_*) - u^2(t_*) \right) = \frac{3}{5} m^{-\frac{2}{5}}(t_*) \dot m(t_*) - 2 u \dot u(t_*) \\[2pt] \geq & \frac{3}{5} m^{-\frac{2}{5}}(t_*) \left(-2L m(t_*) - 2LM\sqrt{m(t_*)}\|\dot x_1(t_*)\|\right) -2u(t_*) \dot u(t_*) \\[2pt] =& -\frac{6}{5}L m^{\frac{3}{5}}(t_*) - \frac{6}{5}LM m^{\frac{1}{10}}(t_*) -2u(t_*) \dot u(t_*) \\[2pt] =& -\frac{6}{5}L u^{2}(t_*) - \frac{6}{5}LM u^{\frac{1}{3}}(t_*) -2u(t_*) \dot u(t_*) \\[2pt] =&g(t_*,u(t_*))- 2 u(t_*)\dot u(t_*) > 0. \end{array}$$ The last inequality prove the implication \eqref{proof:for_U_prime}. Immediately, the definition of function $V(t)$ yields \begin{gather}\label{proof:for_V_prime} \text{if }(t_*,x_*) \in {\mathcal{V}}, \text{ then } V^\prime (t_*,x_*) = -1 < 0. \end{gather} From \eqref{proof:for_U_prime} and \eqref{proof:for_V_prime} it follows, ${\mathcal{E}}_0$ is an open $[U,V]$-subset in ${\mathcal{E}}$ with respect to the system \eqref{eq:2}. Therefore, using Lemma \ref{lemma:help_for_topological_principle} we conclude \begin{gather}\label{proof:for_E0-help-1} {\mathcal{E}}_0^e = {\mathcal{E}}_0^{se} = \mathcal{U}\setminus {\mathcal{V}} = \mathcal{U}. \end{gather} Now, from the definitions of the sets $\mathcal{U}$, ${\mathcal{V}}$ and equality \eqref{proof:for_E0-help-1} it is not difficult to conclude (see. Figure \ref{figure:101}) \begin{gather}\label{proof:for_€0-help-2} {\mathcal{E}}_0^e = \left\{ (t,x) \in \mathbb{R} \times \Omega : t>0 \text{ and } \|\phi(t) - x\| = u(t)\right\}, \end{gather} where $\phi(t) = M \dot x_1(t) + x_1(t)$. \begin{figure}[t] %\includegraphics{nenovg1.eps} \epsfxsize=10cm \epsffile{nenovg1.eps} \label{figure:101}\caption{} \end{figure} Let $\tau > 0$ be a fixed number. Setting $${\mathcal{W}} = \left\{ (t,x) \in \mathbb{R}_+ \times \Omega : t = \tau \text{ and } \|\phi(\tau) - x\| \leq u(\tau)\right\} \subset {\mathcal{E}}_0 \cup {\mathcal{E}}_0^e.$$ we obtain that ${\mathcal{W}}$ is a ball in $\mathbb{R}^n$, and \begin{gather}\label{proof:for_W-help_2} {\mathcal{W}} \cap {\mathcal{E}}_0^e = \left\{ (t,x) \in \mathbb{R}_+ \times \Omega : t = \tau \text{ and } \|\phi(\tau) - x\| = u(\tau)\right\}. \end{gather} Obviously, the boundary $\partial {\mathcal{W}}$ of the set ${\mathcal{W}}$ is not a retract of ${\mathcal{W}}$, i.e. the set ${\mathcal{W}}\cap {\mathcal{E}}_0^e$ is not a retract of ${\mathcal{W}}$. We shall show that ${\mathcal{W}} \cap {\mathcal{E}}_0^e$ is a retract of ${\mathcal{E}}_0^e$. For this purpose we introduce the map $$\pi : {\mathcal{E}}_0^e \to \mathbb{R}^{1+n},\quad \pi(t,x) = (\tau,\pi_2 (t,x)),$$ where $$\pi_2(t,x) = \phi(\tau) + (x - \phi(t))\frac{u(\tau)}{u(t)}.$$ Obviously $\pi$ is a continuous map. Moreover, if $(\widetilde t, \widetilde x) \in {\mathcal{E}}_0^e$, then $$\|\phi(\widetilde t) - \widetilde x\| = u(\widetilde t).$$ That is why $$\|\phi(\tau) - \pi_2 (\widetilde t,\widetilde x)\| = \|\phi (\widetilde t) - \widetilde x\| \frac{u(\tau)}{u(\widetilde t)} = u(\tau),$$ or $\pi :{\mathcal{E}}_0^e \to {\mathcal{W}} \cap {\mathcal{E}}_0^e$. For $(\tau, \widetilde x) \in {\mathcal{W}} \cap {\mathcal{E}}_0^e$, we have $$\pi (\tau, \widetilde x) = (\tau, \pi_2(\tau,\widetilde x)) = \left(\tau, \phi(\tau) + (\widetilde x - \phi(\tau))\right) = (\tau,\widetilde x).$$ Therefore, $\pi$ is a retraction. From the Wazewski's Theorem (see Theorem \ref{thm:topological_principle}) it follows that there exists at least one point $(\tau,\xi_2) \in {\mathcal{W}} \cap {\mathcal{E}}_0^e$, such that \eqref{proof:subset_1} holds true. The definition of set ${\mathcal{E}}_0$ yields \begin{gather}\label{proof:the_end} \|M\dot x_1(t) + x_1(t) - x_2(t;\tau,\xi_2)\| < u^{\frac{5}{3}}(t)\text{ for } t>\tau. \end{gather} From \eqref{proof:the_end} and \eqref{proof:for_ineq-1c} we conclude that \begin{gather}\label{proof:the_end-1} \lim\limits_{t\to\infty} \|M\dot x_1(t) + x_1(t) - x_2(t;\tau,\xi_2)\| \leq \lim\limits_{t\to\infty} u^{\frac{5}{3}}(t) = 0. \end{gather} Therefore, \eqref{proof:the_end-1} and \eqref{thm_1:condition-1} imply \begin{multline*} \lim\limits_{t\to\infty} \|x_1(t) - x_2(t;\tau,\xi_2)\| = \lim\limits_{t\to\infty} \|M\dot x_1(t) + x_1(t) - x_2(t;\tau,\xi_2) - M\dot x_1(t)\| \leq\\[2pt] \leq\lim\limits_{t\to\infty} \|M\dot x_1(t) + x_1(t) - x_2(t;\tau,\xi_2)\| + M \lim\limits_{t\to\infty} \|\dot x_1(t)\| = 0. \end{multline*} To complete the proof of Theorem \ref{main-theorem_1} it is enough to set $x_2 = x_2 (0;\tau,\xi_2)$. \subsection{Proof of Theorem \ref{main-theorem_2}} \label{proof_of_Theorem_2} The proof of the Theorem \ref{main-theorem_2} is analogous to the proof of Theorem \ref{main-theorem_1}. We shall present only the appropriate settings: $$g:\mathbb{R}_+\times\mathbb{R} \to \mathbb{R},\quad g(t,u) = - \frac{6}{5}L u^2 - \frac{6}{5} L M u^{\frac{1}{3}}\|\ddot x_1(t)\|$$ and $$U:\mathbb{R}_+\times \Omega\to \mathbb{R},\quad U(t,x)=\|x_1(t) - x \|^{\frac{6}{5}} - u^2(t).$$ \begin{thebibliography}{00} \bibitem{Hartman} Ph. Hartman. \newblock {\em Ordinary Differential Equations}. \newblock John Wiley \& Sons, 1964. \bibitem{LaSalle} J.P. LaSalle. \newblock Asymptotic stability criteria. \newblock {\em Proc. in Symposia in Appl. Math.}, 13:299--307, 1962. \bibitem{PalisandMelo} J.~Palis and W.~De Melo. \newblock {\em Geometric Theory of Dynamical Systems. An Introduction}. \newblock Springer--Verlag, New York, Heidelberg, Berlin, 1982. \bibitem{Taro} T.~Yoshizawa. \newblock {\em Stability Theory by Liapunov's Second Method}. \newblock The Mathematical Society of Japan, 1966. \end{thebibliography} \end{document}