\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 85, pp. 1--20.\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 2004 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2004/85\hfil Asymptotic stability]
{Asymptotic stability for second-order differential equations
with complex coefficients}

\author[G. R. Hovhannisyan\hfil EJDE-2004/85\hfilneg]
{Gro R. Hovhannisyan}

\address{Gro R. Hovhannisyan \hfill\break
Kent State University, Stark Campus\\
6000 Frank Ave. NW\\
Canton, OH 44720-7599, USA}
\email{ghovhannisyan@stark.kent.edu}

\date{}
\thanks{Submitted April 19, 2004. Published June 18, 2004.}
\subjclass[2000]{34D20, 34E05}
\keywords{Asymptotic stability; asymptotic representation;
  WKB solution; \hfill\break\indent  second order differential equation}

\begin{abstract}
 We prove asymptotical stability and instability results for a general
 second-order differential equations with complex-valued functions as
 coefficients. To prove asymptotic stability of linear second-order 
 differential equations,  we use the technique of asymptotic representations 
 of solutions and error estimates.  For nonlinear second-order 
 differential equations, we extend the asymptotic  stability theorem 
 of Pucci and Serrin to the case of complex-valued coefficients.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}{Example}[section]
\allowdisplaybreaks

\section{Main Results}

Consider the linear second-order differential equation
\begin{equation}
L[x(t)]=x''(t)+2f(t)x'(t)+g(t)x(t)=0,\quad t>T>0,\label{e1.1}
\end{equation}
where the coefficients $2f(t)$ and $g(t)$ are complex-valued
continuous functions of time $t$. The rest state
$x(t)=x'(t)=0$ of (\ref{e1.1}) is called asymptotically stable if
\begin{equation}
\lim_{t\to\infty}x(t)=\lim_{t\to \infty}x'(t)=0\label{e1.2}
\end{equation}
for every solution of (\ref{e1.1}).

The asymptotic stability for the classical equation \ref{e1.1} has been
widely studied \cite{a1,b1,c1,h1,i1,l1,m1,y1}.
However, most of the studies consider real-valued coefficients and
are based on  Lyapunov stability theorems. In
this paper, we prove asymptotical stability and instability
theorems for a general linear second-order equation (\ref{e1.1})
with complex-valued coefficients. For a linear case, we use the
technique of asymptotic representations of solutions and error
estimates \cite{l2,h2}. For a nonlinear second-order
equations (\ref{e1.25}) with complex-valued variable
coefficients, we generalize the asymptotic stability theorem of
Pucci and Serrin (Theorem \ref{thm1.8}).

Denote
\begin{gather}
g_0(t)\equiv g(t)-f^2(t)-f'(t),\quad k_0(t)\equiv{g_0'(t)\over
4g_0^{3/2}(t)},\label{e1.3} \\
G_0(t)\equiv-k_0'(t)-k_0^2(t)\sqrt{g_0(t)}={5g_0'(t)^2\over
16g_0^{5/2}(t)}-{g_0''(t)\over
4g_0^{3/2}(t)},\label{e1.4} \\
\mu_{1,2}(t)=-f(t)-{g_0'(t)\over
4g_0(t)}\pm\sqrt{f^2(t)+f'(t)-g(t)}.\label{e1.5}
\end{gather}
Denote by $L^1(T,\infty)$ the class of Lebesgue integrable in
$(T,\infty)$ functions and by $C^1(T,\infty)$ the class of
differentiable functions on $(T,\infty)$.

\begin{theorem}\label{thm1.1}
Let $f\in C^3(T,\infty),g\in C^2(T,\infty)$ be the complex-valued functions,
and assume that there exists positive number $N$ such that
\begin{equation}
\int^{\infty}_T\left|k_0^2(t)\sqrt{g_0(t)}+k_0'(t)\right|e^{\pm
2\int^t_T\Re\sqrt{-g_0(s)}ds}dt\leq N. \label{e1.6}
\end{equation}
Then the rest state of (\ref{e1.1}) is asymptotically stable if
and only if
\begin{equation}
\lim_{t\to \infty}\int^t_T\Re[\mu_j]dt=-\infty,  \quad\lim_{t\to
\infty}\int^t_T\Re\big[\mu_j+{\mu'_j\over\mu_j}\big]dt=-\infty,\quad
j=1,2.\label{e1.7}
\end{equation}
\end{theorem}

\noindent{\bf Remark.} When f(t) and g(t) are constant, $k_0(t)\equiv 0$,
conditions (\ref{e1.6}) are satisfied, and conditions (\ref{e1.7}) becomes the
Routh-Hurwitz criterion of asymptotical stability:
$$ \Re\big(-f\mp\sqrt{f^2-g}\big)<0.
$$

\noindent{\bf Remark.} Theorem \ref{thm1.1} shows that asymptotic
stability of (\ref{e1.1}) depends on the behavior of
$\Re(f)$ and $g_0(t)$ as $t\to \infty$.

\begin{example} \rm
Let $f(t)=t^{\alpha}(\ln t)^{\beta},g(t)=1$.
 From Theorem \ref{thm1.1}, equation (\ref{e1.1}) is asymptotically stable if
 $-1<\alpha<0$ or  $\alpha=-1,\beta>-1$ (see section 3).
\end{example}

\begin{example} Let $f(t)=t^{\alpha}+it^{\beta},g(t)=1$.
 From Theorem \ref{thm1.1}, equation (\ref{e1.1}) is asymptotically
stable if $\quad-1<\alpha<-\beta-1, \alpha<0$ (see section 3).
\end{example}

\noindent{\bf Remark.} For the small damping case: $g(t)=1$,
$\lim_{t\to \infty}f(t)= 0$, we have $\Re\sqrt{-g_0(t)}=0$ and
conditions (\ref{e1.6}) are not restrictive. For the large damping
case: $\lim_{t\to \infty}f(t)=\infty $, we have
$ \lim_{t\to \infty}\Re\sqrt{-g_0(t)}=\infty$. and conditions (\ref{e1.6}) are
very restrictive.

\noindent{\bf Remark.}
If $g_0(t)=g(t)-f^2(t)-f'(t)\geq 0$ then $ \Re\sqrt{-g_0(t)}\equiv 0$ and
(\ref{e1.6}) becomes
\begin{equation}
\int^{\infty}_T\big|k_0^2(t)\sqrt{g_0(t)}+k_0'(t)\big|dt\leq N.
\label{e1.8}
\end{equation}

\noindent{\bf Remark.}
Condition (\ref{e1.8}) is close to the main assumption of asymptotic stability
theorems in Pucci and Serrin, that $k_0(t)$ is the function of bounded
variation ($\int^{\infty}_T|k_0'(t)|dt<\infty$).
See \cite{p1,p2} or Theorem \ref{thm1.8} in this paper.

\begin{theorem}\label{thm1.2}
Assume there exist the complex-valued functions $\varphi_{1,2}\in C^2(T,\infty)$
that satisfy the conditions
\begin{gather}
\lim_{t\to \infty}\exp \int_T^t\Re\big({\varphi_j'\over
\varphi_j}\big)ds =0,\quad j=1,2,\label{e1.9} \\
\varphi_2(t)L\varphi_1(t)=\varphi_1(t)L\varphi_2(t),\label{e1.10} \\
\Re\big({\varphi_j'(t)\over\varphi_j(t)}\big)\leq 0,\quad
j=1,2,\; t\geq b\, \hbox{ for some } b\geq T,
\label{e1.11} \\
\int^{\infty}_T|B_{21}(s)|ds<\infty,\label{e1.12} \\
B_{21}(t)\equiv{\varphi_2(t)L\varphi_1(t)\over
W[\varphi_1(t),\varphi_2(t)]},\quad
W[\varphi_1(t),\varphi_2(t)]\equiv\varphi_1(t)\varphi'_2(t)
-\varphi'_1(t)\varphi_2(t)\label{e1.13}
\end{gather}
then every solution of (\ref{e1.1}) satisfies
$\lim_{t\to \infty} x(t)=0$.
\end{theorem}

\begin{theorem}\label{thm1.3}
Assume there exist complex-valued functions $ \varphi_{1,2}\in C^2(T,\infty)$
that satisfy conditions (\ref{e1.9})-(\ref{e1.12}) and
\begin{gather}
\lim_{t\to \infty}\int^t_T\Re\big[{\varphi_j''(s)\over\varphi_j'(s)}\big]ds
=-\infty,\quad j=1,2,\label{e1.14} \\
\big|{\varphi_j'(t)\over\varphi_j(t)}\big|\leq
c\Big(\int^t_b|B_{21}(s)|ds\Big)^{-\delta},\quad j=1,2,\;
t\geq b,\; 0<\delta< 1,\label{e1.15}
\end{gather}
for some positive constants c and $\delta$. Then the rest state of
(\ref{e1.1}) is asymptotically stable and for $k=0,1$ we have
\begin{equation}
|x^{(k)}(t)|\leq
\sum_{j=1}^2\big|C_j\varphi_j^{(k)}(t)\big|
+C\Big(\int^t_b|B_{21}(s)|ds\Big)^{-k\delta}
\Big(-1+\exp\big(\int^t_b|2B(s)|ds\big)\Big).\label{e1.16}
\end{equation}
\end{theorem}

\begin{theorem}\label{thm1.4}
 Let conditions (\ref{e1.8}) and
\begin{gather}
\lim_{t\to \infty}\int^t_T\Re\big[f(t)+{g_0'(t)\over
4g_0(t)}\pm\sqrt{f^2(t)+f'(t)-g(t)}\big]dt=\infty,\label{e1.17} \\
\Re\big[f(t)+{g_0'(t)\over
4g_0(t)}\pm\sqrt{f^2(t)+f'(t)-g(t)}\big]\geq 0,\quad t\geq b\,
\label{e1.18}
\end{gather}
be satisfied for some number $b\ge T$. Then every solution $x(t)$
 of (\ref{e1.1}) satisfies
$ \lim_{t\to \infty} x(t)=0$.
\end{theorem}

\begin{example}\rm
$f(t)=t^{\alpha}$, $g(t)=1$.
From Theorem \ref{thm1.4} follows that if
$-1\leq\alpha\leq1$ then all solutions of (\ref{e1.1}) approach to zero
as $t\to \infty$.
It is known that condition $-1\leq\alpha\leq1$ is necessary and sufficient
condition of asymptotic stability in this case.
\end{example}

\noindent{\bf Remark.}
Example 1.3 shows that Theorem \ref{thm1.4} covers small and large damping
cases although example 1.1 shows that Theorem \ref{thm1.1} covers only the small damping case ($\alpha<0$).

\begin{example} \rm
$f(t)=t^{\alpha}+it^{\beta}$, $g(t)=1$.
It can be checked that (\ref{e1.8}) is satisfied (see section 3).
 From Theorem \ref{thm1.4} follows that if
 $-1\leq\alpha< 1$, $\beta\leq(\alpha+1)/2$ then all solutions of
(\ref{e1.1}) approach to zero as $t\to \infty $.
\end{example}

\begin{theorem}\label{thm1.5}
Let conditions (\ref{e1.8}),(\ref{e1.17}), (\ref{e1.18}) and
\begin{gather}
\lim_{t\to \infty}\int^t_T\Re\big[\mu_j+{\mu'_j\over\mu_j}\big]dt
=-\infty,\quad j=1,2.\label{e1.19} \\
|\mu_j(t)|\leq c\Big(\int^t_T\big|k_0^2(t)\sqrt{g_0(t)}+k_0'(t)
\big|dt\Big)^{-\delta},\quad 0<\delta<1,\quad j=1,2 \label{e1.20}
\end{gather}
be satisfied for some positive numbers $c,\delta$. Then the rest state of
(\ref{e1.1}) is asymptotically stable.
\end{theorem}

\begin{theorem}\label{thm1.6}
Let the complex-valued functions $\varphi_{1,2}\in C^2(T,\infty)$ satisfy
conditions  (\ref{e1.10}), (\ref{e1.12}) and
\begin{gather}
 |\varphi_{1,2}(t)|\quad \hbox{be decreasing},\label{e1.21} \\
|\varphi_1(\infty)|=\gamma>0  \label{e1.22}
\end{gather}
 then
the rest state of  (\ref{e1.1}) is not asymptotically stable.
\end{theorem}

\begin{theorem}\label{thm1.7}
Let $f\in C^3(T,\infty)$, $g\in C^2(T,\infty)$ satisfy conditions (\ref{e1.8})
and
\begin{gather}
\Re[f(t)+{g_0'(t)\over 4g_0(t)}\pm\sqrt{f^2(t)+f'(t)-g(t)}]dt\geq
0,\quad t\geq T, \label{e1.23} \\
\int^{\infty}_T\Re[f(t)+{g_0'(t)\over
4g_0(t)}-\sqrt{f^2(t)+f'(t)-g(t)}]dt<\infty \,.\label{e1.24}
\end{gather}
Then the rest state of equation (\ref{e1.1}) is
not asymptotically stable.
\end{theorem}

Consider a nonlinear second order differential equation
\begin{equation}
x''(t)+h(t,x(t),x'(t))x'(t)+j(t,x(t))=0,\quad t\in
J=[T,\infty).\label{e1.25}
\end{equation}
 The following theorem is
a generalization of the asymptotic stability theorem of Pucci and
Serrin \cite[Theorem 3.1]{p1}, to the case of complex-valued
coefficients.

\begin{theorem}\label{thm1.8}
If there exist a non-negative continuous function $k(t)$ of bounded variation
on $(T,\infty)$, non-negative measurable functions
$\sigma(t), \delta(t), \psi\in L^1(J)$ and positive numbers
$\beta,\chi,M,c,m$ such that
\begin{gather}
 0\leq \sigma\leq Re[h(t,x,x')],\quad t\in J, \label{e1.26} \\
 |h(t,x,x')|\leq\delta(t), \quad t\in J,\label{e1.27} \\
 |h(t,x,x')|\leq \gamma Re[h(t,x,x')], \quad t\in J,\; \gamma\geq 1,\label{e1.28}\\
 0\leq k(t)\leq\beta\sigma(t), \quad t\in J,\label{e1.29} \\
\lim_{t\to \infty}\int^t_Tk(s)ds=\infty, \label{e1.30} \\
\int^t_T\delta(s)k(s)e^{\int_t^sk(z)dz}ds\leq M, \quad t\in J,\label{e1.31} \\
\bar{x}j(t,x,x')+x\bar{j}(t,x,x')\geq\chi>0,\quad
\hbox{for }|x|>0,\; t\in J, \label{e1.32} \\
F(t,x)=\int j(t,x,x')d\bar{x}=\int\bar{j}(t,x,x')dx>0,\quad
\hbox{for }|x|>0,\label{e1.33} \\
F(t,0)=0,\quad F(t,x)\geq c|x|^m,\quad
\partial_tF(t,x)\leq\psi(t),\quad t\in J.\label{e1.34}
\end{gather}
 Then the rest state of  (\ref{e1.25}) is
asymptotically stable.
\end{theorem}

\begin{example} \rm
 Let $j(t,x)=l(t)x|x|^{2q}$, $h(t,x,x')=t^{\alpha}+it^{\beta}$, $q>0$
then from Theorem  \ref{thm1.8} it follows that the rest state of  (\ref{e1.1})
 is asymptotically stable if
\begin{equation}
0\leq l_0\leq l(t)\leq l_1<\infty,\quad
\int_T^{\infty}|l_1(t)|dt<\infty,\quad -1\leq\alpha<0,\quad
\beta\le\alpha.\label{e1.35}
\end{equation}
\end{example}

\section{Auxiliary theorems}

Consider the system of ordinary differential equations
\begin{equation}
a'(t)=A(t)a(t),\quad t>T,\label{e36}
\end{equation}
where $a(t)$ is a $n$-vector function and $A(t)$ is a continuous on $(T,\infty)$
$n\times n$ matrix-function. Suppose we can find the exact
solutions of the system
\begin{equation}
\psi'(t)=A_1(t)\psi(t),\quad t>T,  \label{e37}
\end{equation}
 with the matrix-function $A_1$ close to the matrix-function A. Let
$\Psi(t)$ is the $n\times n$ fundamental matrix of the auxiliary
system  (\ref{e37}). Then the solutions of (\ref{e36}) can be
represented in the form
\begin{equation}
a(t)=\Psi(t)(C+\varepsilon(t)), \label{e38}
\end{equation}
where $a(t),\varepsilon(t),C$ are the $n$-vector columns:
$a(t)=\mathop{\rm colomn}(a_1(t),\dots,a_n(t))$,
$\varepsilon(t)=\mathop{\rm colomn}(\varepsilon_1(t),\dots,\varepsilon_n(t))$,
$C=\mathop{\rm colomn}(C_1,\dots,C_n), C_k$ are an arbitrary constants. We can
consider (\ref{e38}) as definition of the error vector-function
$\varepsilon(t)$.

\begin{theorem}[\cite{h2}]\label{thm2.1}
Assume there exist an invertible matrix function $\Psi(t)\in C^1[T,\infty)$
such that
\begin{equation}
H(t)\equiv \Psi^{-1}(t)(A(t)\Psi(t)-\Psi'(t))=\Psi^{-1}(t)(A(t)-A_1(t))\Psi(t)\in
L^1(T,\infty). \label{e39}
\end{equation}
Then every solution of (\ref{e36}) can be represented in form (\ref{e38}) and
the error vector-function  $\varepsilon(t)$ can be estimated as
\begin{equation}
\|\varepsilon(t)\|\le\|C\|\Big(
-1+\exp\Big[\int_T^t\|\Psi^{-1}(s)(A\Psi(s)-\Psi'(s))\|ds\Big]\Big),\label{e40}
\end{equation}
 where $\|\cdot\|$ is the Euclidean vector (or matrix)
norm:  $\|C\|=\sqrt{C_1^2+\dots+C_n^2}$.
\end{theorem}

\noindent {\bf Remark.} From (\ref{e40}) the error $\varepsilon(t)$  is small when
$\int_T^t\|\Psi^{-1}(A-A_1)\Psi\|ds$ is small.

\begin{proof}[Proof of Theorem \ref{thm2.1}]
Let $a(t)$ be a solution of (\ref{e36}). The substitution $a(t)=\Psi(t)u(t)$
transforms (\ref{e36}) into
\begin{equation}
u'(t)=H(t)u(t),\quad t>T,\label{e41}
\end{equation}
 where $H$ is defined by (\ref{e39}). By integration we obtain
\begin{equation}
u(t)=C+\int_T^tH(s)u(s)ds,\quad t>T, \label{e42}
\end{equation}
where the constant vector $C$ is chosen as in (\ref{e38}). Estimating $u(t)$,
\begin{equation}
\|u(t)\|\le\|C\|+\int_T^t\|H(s)\|\cdot\|u(s)\|ds,
\label{e43}
\end{equation}
 and by Gronwall's lemma we have
\begin{equation}
\|u(t)\|\le\|C\|\exp\Big(\int_T^t\|H(s)\|ds\Big).\label{e44}
\end{equation}
 From representation (\ref{e38}) and expression
(\ref{e42}), we have
$$\varepsilon(t)=\Psi^{-1}a-C=u-C=\int_T^tH(s)u(s)ds.
 $$
Then using (\ref{e44}) we obtain the estimate (\ref{e40}):
\begin{align*}
\|\varepsilon(t)\|&\le  \int_T^t\|Hu\|ds\\
&\le\|C\|\int_T^t\|H(s)\| \exp\Big(\int_T^s\|H\|dy\Big)ds\\
&=\|C\|\Big(-1+\exp\big(\int_T^t\|H\|ds\big)\Big).
\end{align*}
\end{proof}

\begin{theorem}\label{thm2.2}
Let $\varphi_{1,2}(t)\in C^2(T,\infty)$ be complex-valued functions
such that
\begin{equation}
\int_T^{\infty}|B_{kj}(t)|dt<\infty,\quad k,j=1,2,
\label{e45}
\end{equation}
 where
\begin{equation}
B_{kj}(t)\equiv{\varphi_{k}(t)L\varphi_{j}(t)\over
W(\varphi_1,\varphi_2)}, \quad L\equiv {d^2\over
dt^2}+2f(t){d\over dt}+g(t),\quad j=1,2. \label{e46}
\end{equation}
Then for arbitrary constants $C_1$,$C_2$ there exist solution of
(\ref{e1.1}) that can be written in the form
\begin{gather}
x(t)=\left[C_1+\varepsilon_1(t)\right]\varphi_1(t)+\left[C_2+\varepsilon_2(t)\right]
\varphi_2(t), \label{e47} \\
x'(t)=\left[C_1+\varepsilon_1(t)\right]\varphi_1'(t)+\left[C_2
+\varepsilon_2(t)\right]\varphi_2'(t), \label{e48}
\end{gather}
 where the error function is estimated as
\begin{equation}
\|\varepsilon(t)\|\le \|C\|\big(-1 +\exp\int_T^t\|B(s)\|ds\big), \label{e49}
\end{equation}
the matrix $B$ has entries  $B_{kj}$ and has norm $\|B\|$.
\end{theorem}

\begin{proof} Equation (\ref{e1.1}) we can rewrite in the form
\begin{equation}
v'(t)=A(t)v(t), \label{e50}
\end{equation}
 where $$v(t)=\begin{pmatrix}x \\   x'(t)\end{pmatrix},\quad
A(t)=\begin{pmatrix} 0  &   1     \\
              -g(t)& -2f(t)   \end{pmatrix}.
$$
By substitution
\begin{equation}
v(t)=\Psi w(t),\quad \Psi =\begin{pmatrix}  \varphi_1(t) & \varphi_2(t)\\
\varphi_1'(t) & \varphi_2'(t)\end{pmatrix}. \label{e51}
\end{equation}
in (\ref{e50}), we get
\begin{equation}
w'(t)=H(t)w(t),\quad H(t)=\begin{pmatrix} B_{21}(t) & B_{22}(t)\\
-B_{11}(t) & -B_{12}(t)\end{pmatrix}.\label{e52}
\end{equation}
To apply  Theorem \ref{thm2.1} to the system (\ref{e52}), we choose $A(t)=H(t)$
in (\ref{e36}) and $A_1(t)\equiv 0$ in (\ref{e37}). Then the identity matrix
$\Psi=I$ is the fundamental solution of (\ref{e37}) with $A_1(t)\equiv 0$.
By direct calculations we get $\|H\|=\|\Psi^{-1}(A\Psi-\Psi')\|=\|B\|$,
so condition (\ref{e39}) of Theorem \ref{thm2.1} follows from (\ref{e45}).
From Theorem \ref{thm2.1} we have
\begin{equation}
w(t)=(C+\varepsilon(t)),\quad \hbox{or}\quad
v(t)=\Psi(t)w(t)=\Psi(t)(C+\varepsilon(t)).\label{e53}
\end{equation}
Representations (\ref{e47}), (\ref{e48}) and estimates (\ref{e49})
follow from Theorem \ref{thm2.1}.
\end{proof}

Denote
\begin{equation}
x_j(t)=\exp\big(\int_T^t\mu_j(s)ds),\quad j=1,2,\quad
\mu_{1,2}=-f(t)-{g_0'(t)\over 4g_0(t)}\pm i\sqrt{g_0(t)}.
\label{e54}
\end{equation}


\begin{theorem}\label{thm2.3}
Let $g\in C^2(T,\infty)$, $f\in C^3(T,\infty)$ and
\begin{equation}
\int^{\infty}_T\left|G_0(t)\right|e^{\pm
2\int^t_T\Im\sqrt{g_0(s)}ds}dt=\int_T^{\infty}\big|G_0(t)e^{\pm
2\int_T^t\Re[\sqrt{-g_0(s)}]ds}\big|dt<\infty,\label{e55}
\end{equation}
where $G_0(t)$ is defined by (\ref{e1.4}). Then for any constants
$C_1,C_2$ there exist solution of (\ref{e1.1}) that can be written
in the form
\begin{gather}
x(t)=\left[C_1+\varepsilon_1(t)\right]x_1(t)+\left[C_2+\varepsilon_2(t)\right]x_2(t),
\label{e56} \\
x'(t)=\left[C_1+\varepsilon_1(t)\right]x_1'(t)+\left[C_2+\varepsilon_2(t)\right]x_2'(t),
\label{e57}
\end{gather}
 and for the error vector-function
$\varepsilon(t)=\begin{pmatrix}
\varepsilon_1(t)   \\    \varepsilon_2(t) \end{pmatrix}$
we have the estimate
\begin{gather}
\|\varepsilon(t)\|\le \|C\|\big(-1
+\exp\int_T^t|G(t)|dt\big),\label{e58} \\
G(t)\equiv \max\Big(\big|G_0(s)e^{2\int_T^t\Im\sqrt{g_0}dz}\big|,
\big|G_0(s)e^{-2\int_T^t\Im\sqrt{g_0}dz}\big|\Big).
\label{e59}
\end{gather}
\end{theorem}

\begin{proof} We apply Theorem \ref{thm2.2} with
$\varphi_j(t)=x_j(t)$. By direct calculations, we have
\begin{equation}
\begin{gathered}
{x_1(t)Lx_1(t)\over W[x_1,x_2]}={iG_0(t)\over 2}e^{2i\int_T^t\sqrt{g_0(s)}ds},\\
{x_2(t)Lx_2(t)\over W[x_1,x_2]}={G_0(t)\over 2i}e^{-2i\int_T^t\sqrt{g_0(s)}ds},\\
{x_1(t)Lx_2(t)\over W[x_1,x_2]}={x_2(t)Lx_1(t)\over
W[x_1,x_2]}={G_0\over 2i}. \end{gathered}\label{e60}
\end{equation}
 From
(\ref{e55}) and Cauchy-Schwarz inequality follows
$\int_T^{\infty}|G_0|dt<\infty$. So conditions (\ref{e45}) of
Theorem \ref{thm2.2} follow from (\ref{e55}). Theorem \ref{thm2.3}
follows from Theorem \ref{thm2.2}.
\end{proof}

\begin{theorem}\label{thm2.4}
Let $\varphi_{1,2}\in C^2(T,\infty)$ satisfied (\ref{e1.10})-(\ref{e1.12}).
Then for any constants $C_1,C_2$ there exist solution $x(t)$ of (\ref{e1.1}) that can be written in the form (\ref{e47}),(\ref{e48})
and the error functions $\varepsilon_j(t)$ are estimated as
\begin{equation}
|\varepsilon_j(t)|\le {C\big(-1
+\exp\int_b^t|B_{21}|ds\big)\over |\varphi_j(t)|},\quad
j=1,2\label{e61}
\end{equation}
 with some positive constant $C$ not depending on $b$.
\end{theorem}

\noindent{\bf Remark.} For the given functions $\varphi_1(t)$, $W(t)$
we can construct
$$\varphi_2(t)=\varphi_1(t)\int_T^t{W(s)ds\over \varphi_1^2(s)} $$
such that (\ref{e1.10}) and (\ref{e1.13}) are satisfied.

\begin{proof}[Proof of Theorem \ref{thm2.4}]
 From (\ref{e1.11}) we have
$$
{d\over dt}|\varphi_j(t)|=|\varphi_j(b)|{d\over dt}
\Big|\exp{\int_b^t{\varphi_j'\over\varphi_j}ds}\Big|
=|\varphi_j(t)|Re\big({\varphi_j'(t)\over\varphi_j(t)}\big)\leq 0,
 \quad j=1,2, \; t\geq b,
$$
which means that the functions $|\varphi_j(t)|$  are decreasing.
When (\ref{e1.10}) is satisfied then the functions  $\varphi_{1,2}(t)$ are
solutions of the homogeneous equation
$$
u''(t)+2f(t)u'(t)+\big(g(t)-{L\varphi_1\over \varphi_1}\big)u(t)=0
$$
and any solution of (also of (\ref{e1.1}))
$$
x''(t)+2f(t)x'(t)+\big(g(t)-{L\varphi_1\over \varphi_1}\big)x(t)
=-{L\varphi_1\over \varphi_1}x(t)
$$
can be written in the form:
\begin{gather}
x(t)=\varphi_1(t)C_1+\varphi_2(t)C_2+\varphi_1(t)\int_b^t{x(s)L\varphi_2ds\over
W[\varphi_1,\varphi_2]}-\varphi_2(t)\int_b^t{x(s)L\varphi_1ds\over
W[\varphi_1,\varphi_2]},\label{e62} \\
x(t)=\varphi_1(t)C_1+\varphi_2(t)C_2+\int_b^t\big({\varphi_1(t)\over\varphi_1(s)}-{\varphi_2(t)
\over\varphi_2(s)}\big){\varphi_1(s)L\varphi_2(s)\over
W[\varphi_1(s),\varphi_2(s)]}x(s)ds,\label{e63} \\
x'(t)=\varphi_1'(t)C_1+\varphi_2'(t)C_2+\int_b^t\big({\varphi_1'(t)\over\varphi_1(s)}
-{\varphi_2'(t)\over\varphi_2(s)}\big){\varphi_1(s)L\varphi_2(s)\over
W[\varphi_1(s),\varphi_2(s)]}x(s)ds,\label{e64}
\end{gather}
 or
(\ref{e47}), (\ref{e48}) where
\begin{equation}
\varepsilon_1(t)=\int_b^t{x(s)L\varphi_2(s)ds\over
W[\varphi_1(s),\varphi_2(s)]},\quad
\varepsilon_2(t)=-\int_b^t{x(s)L\varphi_1(s)ds\over
W[\varphi_1(s),\varphi_2(s)]}.\label{e65}
\end{equation}
Here $C_1,C_2 $ and $b$ are arbitrary constants and $C_1,C_2 $ do not
depend on b. Because the functions $|\varphi_j(t)|$ are decreasing
they are bounded:
\begin{equation}
|\varphi_j(t)|\leq N_j(T), \quad j=1,2,\quad t\geq T.
\label{e66}
\end{equation}
 From representation (\ref{e63}) we have
the estimates:
\begin{gather*}
|x(t)|\le\left|\varphi_1(t)C_1|+|\varphi_2(t)C_2\right|+2\int_b^t
\big|{x(s)\varphi_1(s)L\varphi_2(s)\over W(s)}\big|ds,\\
|x(t)|\leq|N_1C_1|+|N_2C_2|+2\int_b^t|B_{21}x(s)|ds.
\end{gather*}
Applying Gronwall's lemma we have
\begin{equation}
|x(t)|\leq C\exp\Big(\int_b^t2|B_{21}(s)|ds\Big),\quad
C=|N_1C_1|+|N_2C_2|. \label{e67}
\end{equation}
 From (\ref{e65})
and (\ref{e67}), because $\varphi_{1,2}(t)$ are decreasing, we
obtain estimates (\ref{e61}):
$$\left|\varphi_j\varepsilon_j(t)\right|\leq
C\int_b^t|B_{21}(s)|e^{\int_b^s|B_{21}dz}ds
=C\Big(-1+\exp {\int_b^t|2B_{21}|dz}\Big), \quad j=1,2.
$$
\end{proof}


\section{Proofs of the main statements}

\begin{proof}[Proof of Theorem \ref{thm1.1}]
Let us choose $x_j(t)$  as in (\ref{e54}) and apply Theorem \ref{thm2.3}.
 From conditions (\ref{e55}) (which coincide with conditions (\ref{e1.6}))
of Theorem \ref{thm1.1}, by Theorem \ref{thm2.3} we have representations
(\ref{e56}), (\ref{e57}) and estimates (\ref{e58}). From
\begin{equation}
\begin{gathered}
|x_j(t)|= \exp\int_T^tRe(\mu_j)ds,\\
|x_j'(t)|=|\mu_j(T)|exp\int_T^t \Re\big(\mu_j(s)+{\mu_j'(s)\over\mu_j(s)}\big)ds,
\quad j=1,2,
\end{gathered} \label{e3.1} %\label{e68}
\end{equation}
 and (\ref{e1.7}) we have
$$
\lim_{t\to \infty}|x_j(t)|=\lim_{t\to \infty}|x_j'(t)|=0,\quad j=1,2.
$$
 From (\ref{e56})-(\ref{e58}) and (\ref{e1.6}) we have
 $|\varepsilon_j(t)|\le\mbox{const}$, $t>T$, $j=1,2$
and the asymptotic stability.

Now prove that if one of the conditions in (\ref{e1.7}) is not satisfied,
then there exist an asymptotically unstable solution $x(t)$.
By contradiction assume that (\ref{e1.2}) is satisfied,  and, for
example, the first condition of (\ref{e1.7}) is not satisfied. Then
there exist the sequence  $t_n \to \infty $ such that
\begin{equation}
\lim_{t_n\to \infty}|x_1(t_n)|=\lambda_1\neq 0.\label{e69}
\end{equation}
 There exist the subsequence
$t_{n_j}\equiv t_m$ of the sequence $t_n$   such that
\begin{equation}
\lim_{t_m\to \infty}|x_2(t_m)|=\lambda_2.\label{e70}
\end{equation}
From Theorem \ref{thm2.3} for any constants $C_1,C_2$ there exists
the solution $x(t)$ of (\ref{e1.1}) that can be represented in
form (\ref{e56}), or
\begin{equation}
x(t_m)=\left[C_1+\varepsilon_1(t_m)\right]x_1(t_m)+\left[C_2+\varepsilon_2(t_m)\right]x_2(t_m),
\label{e71}
\end{equation}
 where $|\varepsilon_j(t)|\le const,\quad
t>T,\quad j=1,2$.
 From representation (\ref{e69}),(\ref{e71})
$\lambda_{1,2}$  must be finite numbers, otherwise left side of
(\ref{e71}) vanished and right side approaches to infinity when
$t_m\to \infty$  by appropriate choice of $C_j$. Let choose
$C_1=1,\quad C_2=0$ and denote
$N=\exp\big(\int_T^{\infty}|G|ds\big)$ then from (\ref{e58}) we
get
\begin{equation}
|\varepsilon_j(t)|\le\|\varepsilon\|\le
e^N-1.\label{e72}
\end{equation}
 There exist the subsequence $t_k$
of sequence $t_m$ such that exist $\lim_{t_k\to
\infty}\varepsilon_j(t_k).$ So from (\ref{e71}) we obtain
\begin{gather*}
0=\lambda_1+\lambda_1\lim_{t_k\to \infty}\varepsilon_1(t_k)
+\lambda_2\lim_{t_k\to \infty}\varepsilon_2(t_k),\\
-1=\lim_{t_k\to \infty}\varepsilon_1(t_k)+{\lambda_2\over\lambda_1}
\lim_{t_k\to \infty}\varepsilon_2(t_k),
\end{gather*}
which is impossible because the right side can be made small in view of
estimate (\ref{e72}) by choosing T big, which makes $N$ and $\varepsilon_j$  small.
\end{proof}

To prove the statement of Example 1.1 let us show that if
$-1<\alpha<0$,  or $\alpha=-1$, $\beta>-1$,
then conditions (\ref{e1.6}), (\ref{e1.7}) of Theorem \ref{thm1.1} are satisfied.
 From the estimates
\begin{gather*}
f(t)=o(1),\quad g_0\equiv 1-f^2(t)-f'(t)=1+o(1),\quad t\to\infty,\\
g_0\ge 0.5,\quad  \Im\sqrt{g_0}=0,\quad t>T,\\
|g_0'(t)|\le C|f'(t)|,\quad |g_0''(t)|\le C|f'(t)|\in L_1(T,\infty),
\end{gather*}
conditions (\ref{e1.6}) follows:
\begin{align*}
\int_T^{\infty}\big|G_0(s)e^{\pm 2\int_T^s\Im\sqrt{g_0(y)}dy}ds\big|
&\le \int_T^{\infty}\left(|g_0'(s)|^2+|g_0''(s)|\right)ds\\
&\le C\int_T^{\infty}|f'(s)|ds<\infty.
\end{align*}
Further from the estimates
\begin{gather*}
\mu_{1,2}=-f-{g_0'\over 4g_0}\pm i\sqrt{g_0}=-f+O(f'f)\pm i\sqrt{g_0}=\pm i+o(1),\\
|\mu_j'|=\Big|-f'(t)\pm {ig_0'(t)\over 2g_0^{3/2}}-\big({g_0'\over
4g_0}\big)'\Big|
\le|f'|+c_1|g_0'|+c_2|g_0'|^2+c_3|g_0''|\le C|f'|,\\
Re({\mu_j'\over \mu_j})\le|{\mu_j'\over
\mu_j}|\le c_4|\mu_j'(t)|\le C |f'|,\quad f'\in L_1(T,\infty),\\
\int_T^{\infty}fdt=\int_T^{\infty}t^{\alpha}\ln^{\beta}tdt=\infty,\quad
 \alpha>-1,\hbox{ or } \alpha=-1, \beta>-1,\\
\int_T^{\infty}\Re(\mu_j)dt=\int_T^{\infty}\left(-f+O(f'f)dt\right)=-\infty
\end{gather*}
conditions (\ref{e1.7}) follows.

To prove the statement of example 1.2 let us show that if
$-1<\alpha<-1-\beta$, $\alpha<0$
then conditions (\ref{e1.6}), (\ref{e1.7}) of Theorem \ref{thm1.1} are satisfied.
 From the estimates
\begin{gather*}
f(t)=o(1),\quad g_0=1-f^2(t)-f'(t)=1+o(1),\quad t\to\infty, \\
|f'(t)|\le {\sqrt{\alpha^2 t^{2\alpha} +\beta^2
  t^{2\beta}}\over t}\in L_1(0,\infty),\\
|g_0|\ge 0.5,\quad |g_0'(t)|\le C|f'(t)|,\quad |g_0''(t)|\le C|f'(t)|,\\
P\equiv\Re(-g_0)=-1+t^{2\alpha}-t^{2\beta}+\alpha t^{\alpha-1},\quad
Q\equiv\Im(-g_0)=2t^{\alpha+\beta}+\beta t^{\beta-1},\\
P=-1+o(1),\quad Q=2t^{\alpha+\beta}(1+o(1)),\quad R\equiv\sqrt{P^2+Q^2}=1+o(1),
\quad t\to\infty,\\
\Re\sqrt{-g_0}=\sqrt{{P+R\over 2}}=\sqrt{{R^2-P^2\over 2(R-P)}}={|Q|\over
2(1+o(1))},\\
\int_T^t\Re\sqrt{-g_0}dt=\int_T^t {|Q|\over 2(1+o(1))}
\le C\int_T^t s^{\alpha+\beta}ds <\mathrm{const},\quad
t\to\infty
\end{gather*}
conditions (\ref{e1.6}) follow:
\begin{align*}
\int_T^{\infty}\Big|G_0(s)e^{\pm\int_T^t\Re\sqrt{-g_0}dy}ds\Big|
&\le C\int_T^{\infty}|G_0(s)|ds\le C\int_T^{\infty}(|g'_0|^2+|g_0''|)ds\\
&\le C\int_T^{\infty}|f'(t)|dt <\infty.
\end{align*}
 Further from the estimates
\begin{gather*}
\Re(\mu_j)=Re\big(-f-{g_0'\over 4g_0}\pm i\sqrt{g_0}\big)
=-Re(f)+O(g_0')\pm O(|Q|/2),\;\; j=1,2,\; t\to\infty,\\
\Re(\mu_j)=-t^{\alpha}+O(g_0')\pm O(t^{\alpha+\beta}),\quad  j=1,2,\;
t\to\infty,\\
\int_T^{t}\Re(\mu_j)dt\to -\infty,\quad \alpha>-1,\; t\to\infty,\\
\mu_{j}=-f-{g_0'\over 4g_0}\pm i\sqrt{g_0}=\pm i+o(1),\quad  j=1,2,\quad
t\to\infty,\\
|\mu_j'|=\big|f'+({g_0'\over 4g_0})'\pm {ig_0'\over g_0^{3/2}}\big|
\le C|f'(t)|\quad f'\in L_1(T,\infty),\\
{\mu_j'\over\mu_j}\in L_1(T,\infty),\quad j=1,2
\end{gather*}
 conditions (\ref{e1.7}) follow.
%\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.2}]
 From representation (\ref{e63}) of the solutions of (\ref{e1.1}) we have
the estimates:
\begin{equation}
|x(t)|\leq\sum_{j=1}^2|C_j\varphi_j(t)|
+\int_b^t\big|{\varphi_1(t)\over\varphi_1(s)}-{\varphi_2(t)
\over\varphi_2(s)}\big| \big|{x(s)\varphi_2(s)L\varphi_1(s)\over
W[\varphi_1(s),\varphi_2(s)]}\big|ds\label{e73}
\end{equation}
 or because the functions $|\varphi_j(t)|$ are decreasing:
\begin{equation}
|x(t)|\leq\sum_{j=1}^2|C_j\varphi_j(t)|+2\int_b^t
\big|{x(s)\varphi_2(s)L\varphi_1(s)\over
W[\varphi_1(s),\varphi_2(s)]}\big|ds.\label{e74}
\end{equation}
 From (\ref{e66}),
\begin{equation}
|x(t)|\leq C+2\int_b^t|x(s)B_{21}(s)|ds,\label{e75}
\end{equation}
where $C=|N_1C_1|+|N_2C_2|$ depends on T and does not depend on b.
 From (\ref{e1.10}) we have $B_{21}=B_{12}.$
Applying Gronwall's lemma we have
\begin{equation}
|x(t)|\leq C\exp\int_b^t2|B_{21}(s)|ds.\label{e76}
\end{equation}
 From (\ref{e74}), (\ref{e76}) we have
\begin{equation}
|x(t)|\leq\sum_{j=1}^2|C_j\varphi_j(t)|+C\int_b^t|B_{21}|
\exp\big(2\int_b^s|B_{21}(y)|dy\big)ds. \label{e77}
\end{equation}
 From (\ref{e1.9}),
$$
\lim_{t\to \infty}|\varphi_j(t)|=|\varphi_j(T)|\lim_{t\to \infty}\exp
{\int_T^t\Re({\varphi_j'\over \varphi_j})ds}=0,\quad j=1,2.
$$
In view of (\ref{e1.12})
$$
|x(t)|\leq\sum_{j=1}^2|C_j\varphi_j(t)|
+C\Big(-1+\exp\int_b^t|2B_{21}(s)|ds\Big)\to 0,
$$
when $ t\to \infty$ and $ b\to\infty$, because $C_j,C$ do not depend
on $b$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.3}]
 From
\begin{equation}
|\varphi_j^{(k-1)}(t)|=|\varphi_j^{(k-1)}(T)|
\exp\int_T^tRe\Big({\varphi_j^{(k)}(s)\over\varphi_j^{(k-1)}(s)}\Big)ds,
\quad k=1,2\label{e78}
\end{equation}
 and conditions
(\ref{e1.9}), (\ref{e1.14}) we have
\begin{equation}
\lim_{t\to\infty}|\varphi_j^{(k-1)}(t)|=0,\quad j,k=1,2.
\label{e79}
\end{equation}
 From (\ref{e79}) we have
\begin{equation}
|\varphi_j^{(k-1)}(t)\le S<\infty,\quad j,k=1,2,\; t\ge T,\label{e80}
\end{equation}
 where $S$ depend on $T$ and does not
depend on $b$. From representations (\ref{e63}),(\ref{e64}) we have the estimates:
\begin{equation}
|x^{(k)}(t)|\le
\sum_{j=1}^2|\varphi_j^{(k)}(t)C_j|+\int_b^t
\Big|{\varphi_1^{(k)}(t)\over\varphi_1(s)}
-{\varphi_2^{(k)}(t)\over\varphi_2(s)}\Big|\cdot|B_{21}(s)x(s)|ds,\quad
k=0,1\label{e81}
\end{equation}
 or because the functions
$|\varphi_j(t)|$ are decreasing we get for $k=0,1$
\begin{equation}
|x^{(k)}(t)|\le
\sum_{j=1}^2|\varphi_j^{(k)}(t)C_j|+\Big(\Big|{\varphi_1^{(k)}(t)\over\varphi_1(t)}\Big|
+\Big|{\varphi_2^{(k)}(t)\over\varphi_2(t)}\Big|\Big)\int_b^t|B_{21}(s)x(s)|ds.\label{e82}
\end{equation}
In view of (\ref{e1.15}) and (\ref{e76}) we obtain for $k=0,1$ the estimates
$$
|x^{(k)}(t)|\le \sum_{j=1}^2|\varphi_j^{(k)}(t)C_j|
+cC\Big(\int_b^t|B_{21}(s)|ds\Big)^{-k\delta}
\int_b^t|B_{21}(s)|e^{\int_b^s|B_{21}|dz}ds,
$$
from which follow estimates (\ref{e1.16}):
$$
|x^{(k)}(t)|\leq \sum_{j=1}^2\big|C_j\varphi_j^{(k)}(t)\big|
+cC\Big(\int^t_b|B(s)|ds\Big)^{-k\delta}\Big(-1+e^{\int^t_b|2B(s)|ds}\Big)\to
0,
$$
when $t\to \infty$ and $b\to \infty,$ and $C_1,C_2,c,C$ do
not depend on $b$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.4}]
Let us choose $\varphi_j(t)=x_j(t)$ as in (\ref{e54}).
From (\ref{e1.17}), (\ref{e1.18}), (\ref{e60}) and (\ref{e1.8}),
conditions (\ref{e1.9})-(\ref{e1.12}) of Theorem \ref{thm1.2} follow.
Theorem \ref{thm1.4} follows from Theorem \ref{thm1.2}.
\end{proof}

Note that Theorem \ref{thm1.5} follows from Theorem \ref{thm1.3}
by choosing $\varphi_j(t)=x_j(t)$ as in (\ref{e54}).

The statement in the example 1.3. follows
from example 1.4 when $\beta\to -\infty$.
To prove the statement in example 1.4 let us show that if
$$
-1\le\alpha<1,\quad \beta\le{\alpha+1\over 2}
$$
then conditions (\ref{e1.8}),(\ref{e1.17}), (\ref{e1.18}) of Theorem \ref{thm1.4}
are satisfied. From conditions (\ref{e1.17}) or
\begin{equation}
\exp\Big(\int_T^t\Re\big(f+{g_0'\over 4g_0}\pm
\sqrt{-g_0}\big)dt\Big)=C|g_0|^{1/4}
\exp\Big(\int^t_T \Re\left(f\pm\sqrt{-g_0}\right)dt\Big)\to\infty
\label{e83}
\end{equation}
it  follows that for the product of these
expressions we have
$$
|g_0|^{1/2}\exp\Big(\int^t_T
Re(f)dt\Big)=|g_0|^{1/2}\exp\Big(\int^t_Tt^{\alpha}dt\Big)
\to\infty
$$
from which, because $g_0$ has polynomial growth or
decay, we get the necessary condition $\alpha\ge-1$. Denote
\begin{gather*}
U(t)\equiv Re(f-\sqrt{-g_0})=t^{\alpha}-\sqrt{(R+P)/2}, \\
P\equiv\Re(-g_0),\quad Q\equiv\Im(-g_0),\quad R\equiv\sqrt{P^2+Q^2}.
\end{gather*}
Then one of conditions (\ref{e83}) turns
$$
R^{1/4}\exp(\int_T^tU(s)ds) \to\infty.
$$
Because $R(t)$  has polynomial growth or decay in most of the cases
to prove (\ref{e1.17}) it is sufficient to prove that
$$
\int_T^tU(s)ds=O(t^{\lambda})\to\infty,\quad
t\to\infty,\quad \lambda>0.
$$
By direct calculations
\begin{align*}
 U&={t^{2\alpha}-(P+R)/2\over t^{\alpha}+\sqrt{(P+R)/2}}
={2t^{2\alpha}-P-R\over 2t^{\alpha}+\sqrt{2P+2R}}\\
&={K\over (2t^{\alpha}+\sqrt{2P+2R})(2t^{2\alpha}+R-P)},
\end{align*}
$K=4t^{2\alpha}-4\alpha t^{3\alpha-1}-4\beta t^{\alpha+2\beta-1}-\beta^2
t^{2\beta-2}$.
Dividing the plane $(\alpha,\beta)$ on 6 regions
$\{\alpha\ge\beta, \alpha>0\}$, $\{\alpha\le\beta, \beta>0\}$,
$\{\alpha< 0,\beta< 0\}$, $\{\alpha<0,\beta=0\},\{\alpha= 0,\beta< 0\}$ and
$\{\alpha= 0,\beta= 0\}$ we check conditions of Theorem 1.4 in each region
separately.

\noindent{\bf Case 1:} $\alpha\ge\beta,\;  \alpha>0$.\quad From
\begin{gather*}
g_0=O(t^{2\alpha}),\quad g_0'(t)=O(t^{2\alpha-1}),\quad t\to\infty, \\
|G_0(t)|\le C\left(|g_0'(t)|^2/|g_0(t)|^{5/2}+|g_0''(t)|/|g_0(t)|^{3/2}\right)
\le Ct^{-\alpha-2}
\end{gather*}
condition (\ref{e1.8}) follows.

To prove (\ref{e1.17}) note that
\begin{gather*}
P=-1+t^{2\alpha}-t^{2\beta}+\alpha t^{\alpha-1}=t^{2\alpha}(1+o(1)),
\quad  t\to\infty,\\
Q=2t^{\alpha+\beta}(1+o(1)),\quad R=\sqrt{P^2+Q^2}=t^{2\alpha}(1+o(1)),\quad
t\to\infty,\\
(2t^{\alpha}+\sqrt{2P+2R})(2t^{2\alpha}+2R-P)=12t^{3\alpha}(1+o(1)),\quad
t\to\infty,\\
U(t)={4t^{2\alpha}-4\alpha t^{3\alpha-1}-4\beta
t^{\alpha+2\beta-1}-\beta^2 t^{2\beta-2}
\over 12t^{3\alpha}(1+o(1))}
\end{gather*}
When $\alpha<1$,
$$
\int_T^tU(s)ds={t^{1-\alpha}(1+o(1))\over 3}\to\infty,\quad
t\to\infty.
$$
Conditions (\ref{e1.17}),(\ref{e1.18}) are satisfied.

\noindent{\bf Case 2:} $-1<\alpha\le\beta,\;  \beta>0$.\quad
From
\begin{gather*}
g_0=O(t^{2\beta}),\quad g_0'(t)=O(t^{2\beta-1}),\quad t\to\infty, \\
|G_0(t)|\le C\left(|g_0'(t)|^2/|g_0(t)|^{5/2}+|g_0''(t)|/|g_0(t)|^{3/2}\right)
\le Ct^{-\beta-2}
\end{gather*}
condition (\ref{e1.8}) follows. Further
\begin{gather*}
P=t^{2\beta}(-1+o(1)),\quad Q=O(t^{\alpha+\beta}),\quad  t\to\infty,\\
R=\sqrt{P^2+Q^2}=t^{2\beta}(1+o(1))\quad  R-P=2t^{2\beta}(1+o(1)),\quad
t\to\infty,\\
\begin{aligned}
(2t^{\alpha}+\sqrt{2P+2R})(2t^{2\alpha}+2R-P)
&=(2t^{\alpha}+{|Q|\sqrt{2}\over\sqrt{R-P}})(2t^{2\alpha}+2R-P)\\
&=6t^{\alpha+2\beta}(1+o(1)),\quad t\to\infty,
\end{aligned} \\
U(t)={4t^{2\alpha}-4\alpha t^{3\alpha-1}-4\beta
t^{\alpha+2\beta-1}-\beta^2 t^{2\beta-2}
\over 6t^{\alpha+2\beta}(1+o(1))}.
\end{gather*}
If $\beta<{\alpha +1\over 2}$ then (\ref{e1.17}) is satisfied:
$$
\int_T^tU(s)ds={4t^{1+\alpha-2\beta}\over
\alpha-2\beta+1}(1+o(1))\to\infty, \quad t\to\infty.
$$
If
$\beta={\alpha +1\over 2}<1$, we have
\begin{gather*}
U(t)={2\over 3}(1-\beta)t^{-1}-{2\over 3}(2\beta-1)t^{2\beta-3}
-{\beta^2 \over 6t^{2\beta+1}},\\
\int_T^tU(s)ds={2\over 3}(1-\beta)\ln t-{4\beta-2\over 3t^{2(1-\beta)}}
-{t^{-2\beta}\over 6} \to \infty\\
R^{1/4}\exp(\int_T^tU(s)ds) \to\infty,\quad t\to\infty.
\end{gather*}
Then conditions (\ref{e1.17}),(\ref{e1.18}) are satisfied.

\noindent {\bf Case 3:} $\beta< 0,\;  -1<\alpha<0$. \quad From
\begin{gather*}
g_0=O(1),\quad |g_0'(t)|\le {C\over t},\quad |g_0''(t)|\le {C\over t^2},
\quad t\to\infty, \\
|G_0(t)|\le C|g_0'(t)|^2/|g_0(t)|^{5/2}+|g_0''(t)|/|g_0(t)|^{3/2}\le Ct^{-2}
\end{gather*}
condition (\ref{e1.8}) follows.

Then
\begin{gather*}
P=-1+o(1),\quad  t\to\infty,\\
Q=2t^{\alpha+\beta}(1+o(1)),\quad R=\sqrt{P^2+Q^2}=1+o(1),\quad
t\to\infty,\\
(2t^{\alpha}+\sqrt{2P+2R})(2t^{2\alpha}+2R-P)=6t^{\alpha}(1+o(1)),\quad
t\to\infty,\\
U(t)={4t^{2\alpha}-4\alpha t^{3\alpha-1}-4\beta
t^{\alpha+2\beta-1}-\beta^2 t^{2\beta-2}
\over 6t^{\alpha}(1+o(1))},\\
\int_T^tU(s)ds={2t^{1+\alpha}\over 3(1+\alpha)}(1+o(1))\to\infty,\quad
t\to\infty.
\end{gather*}
Conditions (\ref{e1.17}), (\ref{e1.18}) are
satisfied.

\noindent{\bf Case 4:} $\beta=0,\;  -1<\alpha<0$.
\quad From
$$
g_0=O(2),\quad |g_0'(t)|\le {C\over t},\quad |g_0''(t)|\le {C\over t^2},\quad
t\to\infty, $$
condition (\ref{e1.8}) follows. Then
\begin{gather*}
P=-2+o(1),\quad  t\to\infty, \\
Q=2t^{\alpha}(1+o(1)),\quad R=2+o(1),\quad t\to\infty,\\
(2t^{\alpha}+\sqrt{2P+2R})(2t^{2\alpha}+2R-P)=
\Big(2t^{\alpha}+{|Q|\sqrt{2}\over\sqrt{R-P}}\Big)
(2t^{2\alpha}+2R-P)=O(t^{\alpha}),\\
U(t)=O\Big({4t^{2\alpha}-4\alpha t^{3\alpha-1}\over t^{\alpha}}\Big),\\
\int_T^tU(s)ds=O\left(t^{1+\alpha}\right)\to\infty,\quad
t\to\infty.
\end{gather*}
Conditions (\ref{e1.17}), (\ref{e1.18}) are satisfied.

\noindent{\bf Case 5:} $\beta<0,\;  \alpha=0$.
\quad From
$$
g_0=O(t^{2\beta}-2it^{\beta})=O(-2it^{\beta}),\quad |g_0'(t)|\le {C\over t},
\quad |g_0''(t)|\le {C\over t^2},\quad t\to\infty, $$
 condition (\ref{e1.8}) follows. Then
\begin{gather*}
P=-t^{2\beta},\quad  Q=O(2t^{\beta}),\quad R=O(2t^{\beta}),\quad t\to\infty\\
(2t^{\alpha}+\sqrt{2P+2R})(2t^{2\alpha}+2R-P)=(2+\sqrt{4t^{\beta}-2t^{2\beta}})
(2+4t^{\beta}+t^{2\beta})=O(4), \\
U(t)=O(1-\beta t^{2\beta-1}-{\beta^2\over 4} t^{2\beta-2}),\quad t\to\infty, \\
R^{1/4}\int_T^tU(s)ds=O\Big(t^{\beta/4}\exp(4t-2t^{2\beta}-{\beta^2\over 2\beta-1}
t^{2\beta-1})\Big)\to\infty,\quad t\to\infty.
\end{gather*}
Conditions (\ref{e1.17}), (\ref{e1.18}) are satisfied.

\noindent{\bf Case 6:} $\beta=0,\;  \alpha=0$. \quad From
$$P=-1,\quad Q=2,\quad R=\sqrt{5}, \quad U=O(1) $$
conditions (\ref{e1.8}), (\ref{e1.17}), (\ref{e1.18}) follow.
%\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.6}] From representation
(\ref{e62}) we have the estimate:
\begin{equation}
|x(t)|\le\sum_{j=1}^2|\varphi_j(t)C_j|+|\varphi_1(t)|\int_b^t
\Big|{x(s)L\varphi_2(s)\over
W[\varphi_1,\varphi_2]}\Big|ds+|\varphi_2(t)|\int_b^t
\Big|{x(s)L\varphi_1(s)\over W[\varphi_1,\varphi_2]}\Big|ds\label{e84}
\end{equation}
 or
because the functions $|\varphi_j|$ are decreasing and bounded:
\begin{equation}
|x(t)|\le\sum_{j=1}^2|\varphi_j(t)C_j|
+2\int_b^t\Big|{x(s)\varphi_1L\varphi_2(s)\over
W[\varphi_1,\varphi_2]}\Big|ds
\le C+2\int_b^t|x(s)B_{21}(s)|ds.\label{e85}
\end{equation}
 Applying Gronwall's lemma we have
\begin{equation}
|x(t)|\le C\exp\int_b^t|2B_{21}(s)|ds\le
C\exp\int_T^\infty|2B_{21}|ds\equiv C_0. \label{e86}
\end{equation}
By choosing $C_2=0$ from representation (\ref{e62}) we have the estimates:
$$
|x(t)|\ge|\varphi_1(t)C_1|-|\varphi_1(t)|\int_b^t
\Big|{x(s)L\varphi_2(s)\over W[\varphi_1,\varphi_2]}\Big|ds
-|\varphi_2(t)|\int_b^t\Big|{x(s)L\varphi_1(s)\over W[\varphi_1,\varphi_2]}\Big|ds
$$
or
\begin{equation}
|x(t)|\ge|\varphi_1(t)C_1|-2\int_b^t\Big|{x\varphi_2L\varphi_1\over
W[\varphi_1,\varphi_2]}\Big|ds.\label{e87}
\end{equation}
 From (\ref{e1.12})
\begin{equation}
\alpha(b)\equiv\int_b^{\infty}\Big|{\varphi_2L\varphi_1\over
W[\varphi_1,\varphi_2]}\Big|ds\to 0\label{e88}
\end{equation}
when $b\to\infty$. Because positive constants $|C_1|,C_0,\gamma$
do not depend on $b$  by choosing $b$ big enough we can make
$$ \alpha(b)<{|C_1|\gamma\over 2 C_0}. $$
Thus from (\ref{e87}) and $|\varphi_1(t)|\ge |\varphi_1(\infty)|=\gamma>0$
for $t>b$ we have
$$ |x(t)|\ge|C_1|\gamma-2\alpha(b) C_0>0 $$
and Theorem \ref{thm1.6} is proved.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.7}]
 By choosing $\varphi_j(t)=x_j(t)$ as in (\ref{e54}) from (\ref{e60}),
(\ref{e1.23}), (\ref{e1.8}) it follows (\ref{e1.10}), (\ref{e1.12}) and that
the functions $|\varphi_j|$ are decreasing.
 From (\ref{e1.24}) follows $|\varphi_1(\infty)|=\gamma>0$.
So Theorem \ref{thm1.7} follows from Theorem \ref{thm1.6}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1.8}]
 We prove this theorem by the method of Pucci and Serrin. First we prove the
theorem in the case the function of bounded variation $k(t)$ is of class $C^1(J)$.
Multiplying equation (\ref{e1.25}) by $\bar{x}'(t)$ we get
\begin{equation}
 \bar{x}'(t)x''(t)+h(t,x,x')|x'(t)|^2+\bar{x}'(t)j(t,x)=0.\label{e89}
\end{equation}
Adding the conjugate equation
$$
\bar{x}''(t)x'(t)+\bar{h}(t,x,x')|x'(t)|^2+x'(t)\bar{j}(t,x)=0
$$
we get
\begin{gather}
{d\over dt}(|x'(t)|^2+F(t,x))+2Re[h(t,x,x')]|x'(t)|^2=F_t(t,x),\label{e90} \\
|x'(t)|^2+F(t,x)+2\int_T^tRe[h(s,x,x')]|x'(s)|^2ds=C+\int_T^tF_s(s,x)ds.\label{e91}
\end{gather}
 From $F(t,x)\geq 0$ and $\int_T^{\infty}F_t(t,x)dt<\infty$  we
have
\begin{equation}
\int_T^{\infty}|x'(t)|^2Re[h(t,x,x')]dt<\infty.\label{e92}
\end{equation}
Indeed otherwise the right side of (\ref{e91}) is finite when
$t\to\infty$  and the left side approaches to positive infinity
and we get contradiction. So when $t\to\infty$  from (\ref{e91})
we get
\begin{equation}
|x'(t)|^2+F(t,x)=l+\varepsilon(t),\quad l\geq 0,\quad
\lim_{t\to\infty}\varepsilon(t)=0.\label{e93}
\end{equation}
 From this expression and (\ref{e1.34}) we see that $x(t)$  and $x'(t)$
are bounded:
\begin{equation}
|x(t)|\le L,\quad |x'(t)|\le C,\quad \hbox{ for } t\in
J.\label{e94}
\end{equation}
 To prove that $l=0$ assume
for contradiction $l>0$. Multiplying (\ref{e90}) by the positive non
decreasing function
\begin{equation}
\omega(t)=\exp{\int_T^tk(s)ds}\label{e95}
\end{equation}
 we get
\begin{align*}
&{d\over dt}(\omega|x'(t)|^2+\omega F(t,x))\\
&=\omega F_t(t,x)+\omega'(t)(|x'(t)|^2+ F(t,x)) -2\omega Re[h(t,x,x')]|x'(t)|^2,
\end{align*}
or
\begin{equation}
\begin{aligned}
&{d\over dt}\left(\omega|x'|^2+\omega F+\alpha\omega'\bar{x}'x
+\alpha\omega'\bar{x}x'\right)\\
&=\alpha\omega'' (\bar{x}'x+x'\bar{x})-\alpha\omega'\bar{x}(x'h+j)
-\alpha\omega'x(\bar{x}'\bar{h}+\bar{j})\\
&\quad +2\alpha\omega'|x'|^2+\omega F_t+\omega'(|x'|^2+ F)-2\omega
Re[h)]|x'|^2,
\end{aligned}\label{e96}
\end{equation}
 where $\alpha$ is a positive number. Denote
\begin{equation}
R(t)\equiv {d\over dt}\left(\omega|x'|^2+\omega F+\alpha
k\omega(\bar{x}'x+x'\bar{x})\right),\label{e97}
\end{equation}
 then from $\omega'=k\omega$, $\omega''=(k'+k^2)\omega$ and
\begin{equation}
\begin{aligned}
{R\over\omega}&=F_t+k\left(|x'|^2+F-\alpha\bar{x}j-\alpha x\bar{j}\right)
-2Re(h)|x'|^2\\
&\quad +\alpha(k'+k^2)(x\bar{x})'+2k\alpha|x'|^2-\alpha
k(x'\bar{x}h+x\bar{x'}\bar{h}).
\end{aligned}\label{e98}
\end{equation}
 We take $T_1$  large so that
\begin{equation}
|x'|^2+F\geq {3l\over 4}  \hbox{ on } J_1=(T_1,\infty)\quad
\hbox{ and } \int_{T_1}^{\infty}|\psi(t)|dt\leq {l \over
4}.\label{e99}
\end{equation}
 Let us estimate $R$ when $t\in J_1$ and
$\alpha$ is suitably small. Suppose that $k=k'\equiv 0$ on $ t\in
J\setminus I,\quad J=[T,\infty)$. Then from
(\ref{e1.26}),(\ref{e1.34}), (\ref{e98}):
\begin{equation}
 {R\over\omega}\leq \psi(t),\quad t\in J_1\setminus I. \label{e100}
\end{equation}
On the remaining set $I'=I\bigcap J_1$, we partition $R$ in the form
\begin{equation}
 {R\over\omega}=F_t+k(|x'|^2+F)+\sum_{k=1}^5R_k, \label{e101}
\end{equation}
where
\begin{gather*}
R_1=-{2Re(h)\over 5}|x'|^2-k\alpha(\bar{x}j+x\bar{j}),\\
R_2=-{2Re(h)\over 5}|x'|^2+2k\alpha|x'|^2,\\
R_3=-{2Re(h)\over 5}|x'|^2+k^2\alpha(x\bar{x}'+\bar{x}x'),\\
R_4=-{2Re(h)\over 5}|x'|^2+k'\alpha(x\bar{x}'+\bar{x}x'),\\
R_5=-{2Re(h)\over 5}|x'|^2-k\alpha(\bar{x}x'h+\bar{x'}x\bar{h}).
\end{gather*}
To prove the estimate
\begin{equation}
R_1\le -k\alpha\chi,\quad \hbox{ for } t\in I' \hbox{ and small
}\alpha\label{e102}
\end{equation}
 let us fix $p_1=\sqrt{l/4}$ so that
$|x'|^2=|p|^2\leq {l\over 4}$when $|x|\le L$ and $|p|\le|p_1|$.
 From (\ref{e99})
$$ F(t,x)\ge {l\over 2}\quad \hbox{ on } I_1=\{t\in I':|x'(t)|\le p_1\}.
$$
On other hand
$$
F(t,x)=F(T_1,x)+\int_{T_1}^tF_s(s,x)ds\le F(T_1,x)+\int_{T_1}^t\psi(s)ds
\le F(T_1,x)+{l\over 4}
$$
for $t\in J_1$. Thus
$F(t,x)\ge l/4$  in $I_1$.
Since $F(T,0)=0$ it follows that there exist a number $u_0>0$ such that
$|x(t)|>u_0$ for $t\in I_1$. From  (\ref{e1.26}), (\ref{e1.32}) we get
(\ref{e102}) for $t\in I_1$.
In the remaining set $I'\setminus I_1$ we have $|x'(t)>p_1$ and if
\begin{equation}
\alpha\le {2p_1^2\over 5\beta\chi}.\label{e103}
\end{equation}
 then
\begin{gather*}
2Re(h)|x'(t)|^2\ge 2\sigma p_1^2\ge{2kp_1^2\over \beta}\ge5\alpha k\chi.\\
R_1=-{2\over 5}Re(h) |x'(t)|^2-\alpha k(\bar{x}j+x\bar{j})\le-\alpha k\chi
\end{gather*}
and (\ref{e102}) is valid for all $t\in I'$.
We claim that
\begin{equation}
R_2\le{\alpha k\chi\over 8},\quad \hbox{ for } t\in I' \hbox{ and
small }\alpha. \label{e104}
\end{equation}
 Indeed
\begin{gather*}
|x'(t)|^2\le {\chi\over 16} \quad \hbox{ if }\quad |x'(t)|
\le p_2\equiv{\chi\over 16}, \\
R_2=\big(2\alpha k-{2Re(h)\over 5}\big)|x'(t)|^2\le 2\alpha k |x'(t)|^2
\le {\alpha k\chi\over 8}.
\end{gather*}
Otherwise, if $|x'(t)|>p_2$, then from
\begin{equation}
\alpha\le{1\over 5\beta} \label{e105}
\end{equation}
 we have
$$
2\alpha k\le 2\alpha\beta\sigma\le {2\sigma\over 5}\le {2Re(h)\over 5}
\quad \hbox{ and } R_2\le 0.
$$
Let us prove that
\begin{equation}
R_3\le{\alpha k\chi\over 8},\quad \hbox{ for } t\in I'  \hbox{ and
small }  \alpha . \label{e106}
\end{equation}
 Indeed for
$p_3\equiv{\chi\over 16L\sup(k)}$, we have
$$
2\alpha k^2|x'(t)\bar{x}|\le 2\alpha k^2Lp_3
\le {\alpha k\chi\over 8}\quad \hbox{ if }\quad |x'(t)|\le p_3.
$$
Otherwise if $|x'(t)|>p_3$ then from $|x'(t)|\le C$ and
\begin{equation}
\alpha \le {p_3\over 5\beta L\sup(k)}\label{e107}
\end{equation}
 we have
$$
2\alpha k^2|x'\bar{x}|\le {2\alpha k^2L\over p_3}|x'|^2
\le {2k\over 5\beta}|x'|^2\le {2\sigma\over 5}|x'|^2
\le {2Re(h)\over 5}|x'|^2, \quad R_3\le 0 .
$$
So (\ref{e106}) is proved. From (\ref{e94}) we have
\begin{equation}
R_4=-{2\over 5} Re(h)|x'|^2+\alpha k'(\bar{x}'x+\bar{x}(t)x')\le 2 \alpha|k'|CL,\quad\hbox{ for } \quad t\in I'.\label{e108}
\end{equation}
To prove the estimate
\begin{equation}
R_5\le 10\alpha^2 L^2\gamma\delta
k\sup(k)\label{e109}
\end{equation}
 define the set
$$
I_5=\{t\in I':|x'(t)|\ge\alpha\Lambda,\quad\Lambda=5L\gamma \sup(k)\}.
$$
In this set
$$
-\alpha k(\bar{x}x'h+\bar{x}'x\bar{h})\le 2\alpha k|x'xh|
\le 2\alpha kL{|x'|^2\gamma \Re(h)\over 5\alpha L\gamma \sup(k)}
\le {2Re(h)\over 5}|x'|^2
$$
and we have $R_5\le 0$.

In $I'\setminus I_5$ we have $|x'(t)|\le \alpha L$ and estimate (\ref{e109}):
$$
R_5\le2\alpha k|\bar{x}x'(t)h|\le 10\alpha^2 L^2\gamma\delta k\sup(k)
=2\alpha^2\delta kL\Lambda .
$$
Thus we have the estimates
$$
{R\over\omega}\le \psi+k\big(|x'(t)|^2+F-\alpha\chi+{2\alpha\chi\over 8}\big)
+2\alpha CL|k'(t)|+10\alpha^2 L^2\gamma\delta k\sup(k),
$$
\begin{equation}
R\le \omega\left(\psi+2\alpha CL|k'|+10\alpha^2 L^2\gamma\delta
k\sup(k)\right)+\omega'\big(l+\varepsilon-\alpha\chi+{2\alpha\chi\over
8}\big),\label{e110}
\end{equation}
 where $\delta k\equiv 0$ on
$J\setminus I.$ Let us fix $\alpha$ so small that
(\ref{e103}), (\ref{e105}), (\ref{e107}) and
\begin{equation}
\alpha \le{\chi\over 80ML^2\gamma
\sup(k)}\label{e111}
\end{equation}
 are satisfied. Moreover in view
of (\ref{e93}) and $k\in BV(J),k'\in L_1(J)$ we can take $T_2>T_1$
such that
\begin{gather}
|\varepsilon(t)|\le {\alpha\chi\over 8}\quad \hbox{ for }
t>T_2,\label{e112} \\
\int_{T_2}^\infty\psi(s)ds\le {\alpha\chi\over 8},\quad
\int_{T_2}^\infty|k'(s)ds\le{\chi\over
16CL}.\label{e113}
\end{gather}
 Then from (\ref{e97}) and (\ref{e110}),
\begin{align*}
&{d\over dt}\left(\omega|x'|^2+\omega F+\alpha k\omega(\bar{x}'x+x'\bar{x})\right)\\
&\equiv R\le
\omega\left(\psi+2\alpha CL|k'|+10\alpha^2L^2\gamma\delta k\sup(k)\right)
+\omega'(l+\alpha\chi/8-3\alpha\chi/4)
\end{align*}
which integrating yields
\begin{align*}
&\omega(|x'(t)|^2+F+\alpha k(\bar{x}'x+x'\bar{x}))\\
&\le \int_{T_2}^\infty\omega\psi ds+2\alpha CL\int_{T_2}^t\omega|k'|ds
+10\alpha^2L^2\gamma \sup(k)\int_{T_2}^t\omega\delta k\,ds
+\omega(l-5\alpha\chi/8)+c.
\end{align*}
So the function
\begin{equation}
\begin{aligned}
\Psi&=\omega\left(|x'(t)|^2+F+ \alpha k(\bar{x}'x+x'\bar{x})-l+5\alpha\chi/8 \right)\\
&\quad -\int_{T_2}^t\omega\psi ds-2\alpha
CL\int_{T_2}^t\omega|k'|ds-10\alpha^2L^2\gamma
\sup(k)\int_{T_2}^t\omega\delta k\,ds
\end{aligned} \label{e114}
\end{equation}
 is decreasing.

Now we claim that there exist a sequence $t_n$ such that $t_n\uparrow\infty$
and
\begin{equation}
k(t_n)|x'(t_n)|^2\to 0,\quad n\to\infty. \label{e115}
\end{equation}
 Otherwise, because of boundedness of
$k(t),|x'(t)|$ there exist numbers $k_0>0$, $p_0>0$, $\bar{t}$ such that
$$
k(t)\ge k_0>0 \quad\hbox{and}\quad |x'(t)|\ge p_0>0\quad\hbox{for } t>\bar{t}.
$$
In turn, since $k(t)\equiv 0$ on $J\setminus I$, we must have
$I\supset[\bar{t},\infty)$,
$$
\sigma(t)\ge {k_0\over\beta}\quad\hbox{and}\quad |x'(t)|\ge p_0>0\quad\hbox{for }
t>\bar{t}.
$$
So $\mathop{\rm Re}(h)|x'(t)|^2>0$ for $ t>\bar{t}$, which contradicts (\ref{e92}).

From (\ref{e111})-(\ref{e114}) we have
\begin{align*}
{\Psi(t_n)\over\omega(t_n)}
&\ge \varepsilon-2\alpha Lk|x'|+5\alpha\chi/8-3\alpha\chi/8\\
&\ge -\alpha\chi/8-2\alpha Lk|x'|+2\alpha\chi/8\\
&\ge \alpha\chi/8-2\alpha Lk|x'|.
\end{align*}
 From (\ref{e115}), $Lk(t_n)|x'(t_n)|\le\chi/32$ for $n>n_0$  and
$$ {\Psi(t_n)\over\omega(t_n)}\ge \alpha\chi/16.
$$
Hence $\Psi(t_n)\to\infty$ as $\to\infty$. This contradicts the
fact that $\Psi(t)$ is decreasing. So $l=0$ or
$\lim_{t\to\infty}(|x'|^2+F)=l=0$ from which follows (\ref{e1.2}).

The proof of the general case $k\in BV(J)$ follows from the lemma below.
\end{proof}

\begin{lemma}[\cite{p1}]
Let $k(t)$ be a non-negative continuous function of bounded variation on $J$
  $(k\in BV(J))$.
Then for every constant $\theta>1$ there exists a function $\bar{k}\in C^1(J)$
and an open set $E\subset J$ such that
\begin{itemize}
\item[(i)] $\theta k\ge \bar{k}\ge \begin{cases} k, & \mbox{in } J\setminus E\\
0,  &\mbox{in } E
\end{cases}$
\item[(ii)]  $\mathop{\rm Var}(\bar{k})\le \theta \mathop{\rm Var}(k)$
\item[(iii)] $\int_E kdt\le 1$.
\end{itemize}
\end{lemma}

\subsection*{Acknowledgment}
The author wants  to thank the anonymous referee for his/her
comments that helped improving the original manuscript.

\begin{thebibliography}{00}

\bibitem{a1} Z. Arstein and E. F. Infante;
\emph{On asymptotic stability of oscillators with unbounded damping},
Quart. Appl. Mech. 34 (1976), 195-198.

\bibitem{b1}  R. J. Ballieu and K. Peiffer;
\emph{Asymptotic stability of the origin for the equation,
$x''(t)+f(t,x,x'(t))|x'(t)|^{\alpha}+g(x)=0$}
J.Math Anal. Appl 34 (1978) 321-332

\bibitem{c1}  L. Cesary
\emph{Asymptotic behavior and stability problems in ordinary differential},
 3rd ed., Springer Verlag, Berlin, 1970.

\bibitem{h1} L. Hatvani
\emph{Integral conditions on asymptotic stability for the damped linear oscillator with small damping},
Procedings of the American Mathematical Society. 1996, Vol. 124 No.2, p.415-422.

\bibitem{h2} G. R. Hovhannisyan
\emph{Estimates for error functions of asymptotic solutions of ordinary linear differential equations},
Izv. NAN Armenii, Matematika, [English translation: Journal of Contemporary Math. Analysis -- (Armenian Academy of Sciences) 1996, Vol.31, no.1, p.9-28.

\bibitem{i1} A. O. Ignatyev
\emph{Stability of a linear oscillator with variable parameters},
Electronic Journal of Differential Equations Vol.1997(1997), No. 17, p.1-6.

\bibitem{l1} J. J. Levin and J. A. Nobel
\emph{Global asymptotic stability of nonlinear systems of differential equations to reactor dynamics},
Arch. Rational Mech. Anal. 5(1960), 104-211.

\bibitem{l2} N. Levinson
\emph{The asymptotic nature of solutions of linear systems of differential equations},
Duke Math. J. 15,111-126, (1948).

\bibitem{m1} N. N. Moiseev
\emph{Asymptotic Methods of Nonlinear Mechanics}
Nauka, 1969. (Russian)

\bibitem{p1}  P. Pucci and J. Serrin
\emph{Precise damping conditions for global asymptotic stability for nonlinear second order systems},
Acta Math. 170(1993), 275-307.

\bibitem{p2} P. Pucci and J. Serrin
\emph{Asymptotic stability for ordinary differential systems with time dependent restoring potentials},
Archive Rat. Math. Anal. 113(1995), 1-32.

\bibitem{s1} V. M. Starzhinsky
\emph{Sufficient conditions of stability of the mechanical system with one degree of freedom},
J. Appl. Math. And Mech. 16(1952), 369-374. (Russian).

\bibitem{y1} V. A. Yakubovich and V. M. Starzhinsky
\emph{ Linear differential equations with periodic coefficients and their applications},
Nauka, Moscow,1972. (Russian).


\end{thebibliography}

\end{document}