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

\AtBeginDocument{{\noindent\small {\em 
Electronic Journal of Differential Equations}, 
Vol. 2006(2006), No. 35, pp. 1--10.\newline 
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu
or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/35\hfil Stability and boundedness of solutions] 
{Stability and boundedness of solutions to certain
fourth-order differential equations}
\author[C. Tun\c c\hfil EJDE-2006/35\hfilneg]
{Cemil Tun\c c}

\address{Cemil Tun\c c \newline
Department of Mathematics, Faculty of Arts and Sciences,
Y\"uz\"unc\"u Yil University, 65080, Van, Turkey}
\email{cemtunc@yahoo.com}

\date{}
\thanks{Submitted November 17, 2005. Published March 21,2006.}
\subjclass[2000]{34D20, 34D99} 
\keywords{Fourth order differential equations; stability; boundedness}

\begin{abstract}
 We give criteria for the asymptotic stability and boundedness of
 solutions to the nonlinear fourth-order ordinary differential equation
 $$
 x^{(4)}+\varphi (\ddot{x})\dddot{x}+f(x,\dot{x})
 \ddot{x}+g(\dot{x})+h(x)=p(t,x,\dot{x},\ddot{x},\dddot{x})\,,
 $$
 when $p\equiv 0$ and when $p\neq 0$.
 Our results include and improve some
 well-known results in the literature.
\end{abstract}

\maketitle \numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

Since Lyapunov \cite{l1} proposed his famous theory on the
stability of motion, numerous methods have been proposed for
deriving suitable Lyapunov functions to study the stability and
boundedness of solutions of certain second-, third-, fourth-,
fifth- and sixth order non-linear differential equations. See, for
example, Anderson \cite{a1}, Barbasin \cite{b1}, Cartwright
\cite{c1}, Chin \cite{c2,c3}, Ezeilo \cite{e1,e2}, Harrow
\cite{h1,h2}, Ku and Puri \cite{k1}, Ku et al. \cite{k2}, Ku
\cite{k3,k4}, Krasovskii \cite{k5}, Leighton \cite{l1}, Li
\cite{l2}, Marinosson \cite{m1}, Miyagi and Taniguchi \cite{m2},
Ponzo \cite{p1}, Reissig et al. \cite{r1}, Schwartz and Yan
\cite{s1}, Shi-zong et al. \cite{s2}, Sinha \cite{s3,s4}, Skidmore
\cite{s5}, Szeg\"{o} \cite{s6}, Tiryaki and Tun\c{c} \cite{t1,t2},
Tun\c{c} \cite{t2,t3,t4}, Zubov \cite{z1} and the references
quoted therein. In 1989, Chin \cite{c3} has tried to apply a new
technique (called the intrinsic method) proposed by himself to
construct some new Lyapunov functions to study the stability of
solutions of three fourth order non-linear differential equations
described as follows:
\begin{gather}
x^{(4)}+a_{1}\dddot{x}+a_{2}\ddot{x}+a_{3}\dot{x} +f(x)=0,
\label{e1.1}
\\
x^{(4)}+a_{1}\dddot{x}+\psi (\dot{x})\ddot{x}+a_{3}
\dot{x}+a_{4}x=0,  \label{e1.2}
\\
x^{(4)}+a_{1}\dddot{x}+f(x,\dot{x})\ddot{x}+a_{3}
\dot{x}+a_{4}x=0.  \label{e1.3}
\end{gather}

Later, the authors in \cite{t2} based on the results in \cite{c3}
 have applied the method used in \cite{c3} to construct some new
Lyapunov functions to examine the stability and boundedness of the
solutions of non-linear differential equation described by
\begin{equation}
x^{(4)}+\varphi (\ddot{x})\dddot{x}+f(x,\dot{x})
\ddot{x}+g(\dot{x})+h(x)=p(t,x,\dot{x},\ddot{x}, \dddot{x})
\label{e1.4}
\end{equation}
with $p\equiv 0$ and $p\neq 0$, respectively. In 1998, Wu and
Xiong \cite{w1} proved both that the Lyapunov functions
constructed in Chin \cite{c3} are the same as those obtained by
Cartwright \cite{c1} and Ku \cite{k3}. Chin's results \cite{c1}
are not true for the equations \eqref{e1.1}, \eqref{e1.2},
\eqref{e1.3} in the general cases. Further, the local asymptotic
stability of the zero solution of the equations \eqref{e1.1},
\eqref{e1.2} and \eqref{e1.3} has been investigated in \cite{w1}.
Therefore, in this paper, we will revise our results obtained in
[31] again and extend and improve the results established in
\cite{w1}. Now, we consider the fourth order non-linear
differential equation \eqref{e1.4} or its equivalent system in the
phase variables form
\begin{equation}
\begin{gathered}
\dot{x}=y,\dot{y}=z,\dot{z}=w, \\
\dot{w}=-\varphi (z)w-f(x,y)z-g(y)-h(x)+p(t,x,y,z,w)
\end{gathered}
\label{e1.5}
\end{equation}
in which the functions $\varphi ,f,g,h$ and $p$ depend only on the
arguments displayed and the dots denote differentiation with
respect to $t$. The functions $\varphi ,f,g,h$ and $p$ are assumed
to be continuous on their respective domains. The derivatives
$\frac{dg}{dy}\equiv g'(y)$ and $\frac{dh}{dx}\equiv h'(x)$ exist
and are continuous. Moreover, the existence and the uniqueness of
the solutions of the equation \eqref{e1.4} will be assumed. That
is, the functions $\varphi ,f,g,h$ and $p$  so constructed such
that the uniqueness theorem is valid. It is worth mentioning that
the continuity of the functions $\varphi ,f,g,h$ and $p$
guarantees at the least the existence of \ a solution of the
equation \eqref{e1.4}. Next, the existence and continuity of the
derivatives $\frac{dg}{dy}\equiv g'(y)$ and $ \frac{dh}{dx}\equiv
h'(x)$ in a compact domain ensure that the functions $g$ and $h$
satisfy the locally Lipschitz condition in the closed domain. This
guarantees the uniqueness of the solutions. It should also be
noted that the domain of attraction of the zero solution $x=0$ of
the equation \eqref{e1.4} (for $p\equiv 0$) in the following first
result is not going to be determined here.


\section{Main results}

Before stating the major theorems, we introduce the following
notation:

Set
\begin{gather*}
\varphi _{1}(z)=\begin{cases}
\frac{1}{z}\int_{0}^{z}\varphi (\tau )d\tau , & z\neq 0 \\
\varphi (0),& z=0,
\end{cases}
\\
g_{1}(y)=\begin{cases}
\frac{g(y)}{y}, & y\neq 0 \\
g'(0), & y=0.
\end{cases}
\end{gather*}
In the case $p\equiv 0$, we have the following statement.

\begin{theorem} \label{thm1}
In addition to the basic assumptions on $\varphi ,f,g$ and $h$,
suppose that there are positive constants  $a,b,c,d,\delta
,\varepsilon $ and $\eta $ such that the following conditions are
satisfied:
\begin{itemize}
\item[(i)] $h(0)=g(0)=0$

\item[(ii)] $abc-cg'(y)-ad\varphi (z)\geq \delta >0$
 for all $y$ and $z$

\item[(iii)] $0\leq d-h'(x)\leq \frac{\sqrt{\delta \varepsilon
a}}{4}$  for all $x$ and $h(x)\operatorname{sgn}x\to +\infty $ as
$|x| \to \infty $

\item[(iv)] $0\leq g_{1}(y)-c<\frac{\delta
}{8c}\sqrt{\frac{d}{2ac}}$
 and $g'(y)\geq c$  for all $y$

\item[(v)] $0\leq f(x,y)-b\leq \eta $ for all $x$ and $y$ where
\begin{equation*}
\eta \leq \min \Big[ \frac{c}{8d}\sqrt{\frac{\delta \varepsilon
}{a}},\frac{ a}{8}\sqrt{\frac{\delta \varepsilon }{c}}\Big] ,\quad
\varepsilon \leq \frac{\delta }{2acD},\quad D=ab+\frac{bc}{d}
\end{equation*}

\item[(vi)] $\varphi (z)\geq a,\varphi _{1}(z)-\varphi
(z)<\frac{\delta }{ 2a^{2}c}$ for all $z$.

\end{itemize}
Then the zero solution of the system \eqref{e1.5} is
asymptotically stable.
\end{theorem}

\begin{remark} \label{rmk1} \rm
 Assumptions (ii), (iv) and (vi) imply
\begin{equation*}
\varphi (z)<\frac{bc}{d},\quad g'(y)<ab.
\end{equation*}
\end{remark}

\begin{remark} \label{rmk2} \rm
When $\varphi
(\ddot{x})=a,f(x,\dot{x})=b,g(\dot{x})=c\dot{x},h(x)=dx$, equation
\eqref{e1.4} reduces to a linear constant coefficient differential
equation and conditions (i)-(vi) of Theorem \ref{thm1} reduce to
the corresponding Routh-Hurwitz criterion.
\end{remark}

\begin{remark} \label{rmk3} \rm
Theorem \ref{thm1} revises the first theorem in \cite{t2} and
includes and improves the results of Ezeilo \cite{e1,e2}, Harrow
\cite{h1}, and Wu and Xiong \cite{w1} except the restriction on
$f(x,y)$, that is, $0\leq f(x,y)-b\leq \eta $.
\end{remark}

For the proof of Theorem \ref{thm1} our main tool is the
continuous differentiable function $V=V(x,y,z,w)$ defined by
\begin{equation}
\begin{aligned}
2V= & 2\beta \int_{0}^{x}h(\xi )d\xi +\beta by^{2}-\alpha
dy^{2}+2\int_{0}^{y}g(\rho )d\rho +\alpha
bz^{2}+2\int_{0}^{z}\varphi (\tau )\tau d\tau\\
&\quad  -\beta z^{2}+\alpha w^{2}
+2h(x)y+2\alpha h(x)z+2\alpha g(y)z\\
&\quad +2\beta y\int_{0}^{z}\varphi(\tau )d\tau +2\beta yw+2zw,
\end{aligned} \label{e2.1}
\end{equation}
where
\begin{equation}
\alpha =\varepsilon +\frac{1}{a},\quad \beta =\varepsilon
+\frac{d}{c}.
 \label{e2.2}
\end{equation}

The following lemmas are used for proving that the function
$V(x,y,z,w)$ is a Lyapunov function of the system \eqref{e1.5}.


\begin{lemma} \label{lem1}
Suppose that all the conditions of Theorem \ref{thm1} hold. Then
there are positive constants $D_{i}\equiv
D_{i}(a,b,c,d,\varepsilon ,\delta )$, ($i=1$, $2$, $3$, $4)$, such
that for all $x,y,z, w$,
\begin{equation*}
V\geq D_{1}\int_{0}^{x}h(\xi )d\xi
+D_{2}y^{2}+D_{3}z^{2}+D_{4}w^{2}\,.
\end{equation*}

\end{lemma}

\begin{proof}
We observe that the function $2V$ in \eqref{e2.1} can be
rewrittten as
\begin{align*}
2V= & \frac{1}{c}[ h(x)+cy+\alpha cz] ^{2}+\frac{1}{\varphi
_{1}(z)}[w+\varphi _{1}(z)z+\beta \varphi _{1}(z)y] ^{2}+[
\alpha -\frac{1}{\varphi _{1}(z)}] w^{2} \\
&  \\
& +[\alpha b-\beta -\alpha ^{2}c] z^{2}+[\beta b-\alpha d-\beta
^{2}\varphi _{1}(z)] y^{2}+2\int_{0}^{y}g(\rho )d\rho
-cy^{2} \\
&  \\
& +2\alpha [g_{1}(y)-c] yz+2\beta \int_{0}^{x}h(\xi )d\xi
-(\frac{1}{c})h^{2}(x)+[2\int_{0}^{z}\varphi (\tau )\tau d\tau
-\varphi _{1}(z)z^{2}] .
\end{align*}

 In light of the hypothesis of the theorem, the use of \eqref{e2.2}
and the mean value theorem (both for the derivative and integral),
it can be easily obtained that
\begin{equation*}
2V\geq \varepsilon \int_{0}^{x}h(\xi )d\xi +\big(\frac{\delta d}{
2ac^{2}}\big)y^{2}+\big(\frac{\delta }{4a^{2}c}\big)
z^{2}+\varepsilon w^{2}+2\alpha [g_{1}(y)-c] yz.
\end{equation*}
The remaining of this proof follows the strategy indicated in
\cite{t2}, and
 hence it is omitted. This completes the proof.
\end{proof}

\begin{lemma} \label{lem2}
 Assume that all the conditions of Theorem \ref{thm1}
 hold, Then there exist positive constants
$D_{i}\equiv D_{i}(a,b,c,\varepsilon ,\delta )$, $(i=5,6,7)$, such
that if $(x(t),y(t),z(t),w(t))$ is a solution of the system
\eqref{e1.5},  then
\begin{equation}
\dot{V}\equiv \frac{d}{dt}V(x,y,z,w)\leq
-(D_{5}y^{2}+D_{6}z^{2}+D_{7}w^{2}).  \label{e2.3}
\end{equation}
\end{lemma}

\begin{proof} Along any solution $(x,y,z,w)$ of  system \eqref{e1.5},
it follows from \eqref{e2.1} and \eqref{e1.5} that
\begin{equation}
\begin{aligned}
\dot{V}= & -[f(x,y)-\alpha g'(y)-\beta \varphi
_{1}(z)]z^{2}-[\alpha \varphi (z)-1]w^{2}-[\beta
g_{1}(y)-h'(x)]y^{2}
\\
& -\beta [ f(x,y)-b]yz-\alpha [ f(x,y)-b]zw-\alpha [ d-h'(x)]yz.
\end{aligned} \label{e2.4}
\end{equation}
It is clear from (ii)-(vi) and \eqref{e2.2} that
\begin{equation}
\dot{V}\leq -(\frac{\varepsilon c}{2})y^{2}-(\frac{ \delta
}{8ac})z^{2}-(\frac{3\varepsilon a}{4}) w^{2}-W_{6}-W_{7}-W_{8},
\label{e2.5}
\end{equation}
where
\begin{gather}
W_{6}=(\frac{\varepsilon c}{4})y^{2}+\beta [
f(x,y)-b]yz+(\frac{\delta }{16ac})z^{2},  \label{e2.6}
\\
W_{7}=(\frac{\varepsilon a}{4})w^{2}+\alpha [
f(x,y)-b]zw+(\frac{\delta }{16ac})z^{2},  \label{e2.7}
\\
W_{8}=(\frac{\varepsilon c}{4})y^{2}+\alpha [ d-h'(x)]yz
+(\frac{\delta }{4ac})z^{2}.  \label{e2.8}
\end{gather}
It should be noted that all six coefficients in the expressions
\eqref{e2.6}-\eqref{e2.8} are non-negative. By using the
conditions (ii), (iii),(v), and the inequalities
\begin{gather*}
\beta
^{2}[f(x,y)-b]^{2}<\frac{4d^{2}}{c^{2}}[f(x,y)-b]^{2}<\frac{\delta
\varepsilon }{16a},
\\
\alpha
^{2}[f(x,y)-b]^{2}<\frac{4}{a^{2}}[f(x,y)-b]^{2}<\frac{\delta
\varepsilon }{16c},
\\
\alpha ^{2}[d-h'(x)]^{2}<\frac{4}{a^{2}}[d-h'(x)]^{2}<
\frac{\delta \varepsilon }{4a}
\end{gather*}
respectively, it follows that
\begin{gather}
W_{6}\geq (\frac{\varepsilon c}{4})y^{2}-(\frac{\sqrt{ \delta
\varepsilon }}{4\sqrt{a}})|yz| +( \frac{\delta }{16ac})z^{2}\geq
[\frac{\sqrt{\varepsilon c}}{2} |y| -\frac{1}{4}\sqrt{\frac{\delta
}{ac}}| z| ] ^{2}\geq 0,  \label{e2.9}
\\
W_{7}\geq (\frac{\varepsilon a}{4})w^{2}-(\frac{\sqrt{ \delta
\varepsilon }}{4\sqrt{c}})|zw| +( \frac{\delta
}{16ac})z^{2}=[\frac{\sqrt{\varepsilon a}}{2} |w|
-\frac{1}{4}\sqrt{\frac{\delta }{ac}}| z| ] ^{2}\geq 0,
\label{e2.10}
\\
W_{6}\geq (\frac{\varepsilon c}{4})y^{2}-\frac{1}{2}\sqrt{\frac{
\delta \varepsilon }{a}}|yz| +(\frac{\delta }{4ac}
)z^{2}=[\frac{\sqrt{\varepsilon c}}{2}|y| -
\frac{1}{2}\sqrt{\frac{\delta }{ac}}|z| ] ^{2}\geq 0.
\label{e2.11}
\end{gather}
By collecting the estimates \eqref{e2.9}-\eqref{e2.11} into
 \eqref{e2.5} we obtain
\begin{equation*}
\dot{V}\leq -(\frac{\varepsilon c}{2})y^{2}-(\frac{ \delta
}{8ac})z^{2}-(\frac{3\varepsilon a}{4})w^{2}
\end{equation*}
which proves the lemma.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm1}]
 From Lemma \ref{lem1}, Lemma \ref{lem2} and condition (iii) of
Theorem \ref{thm1}, we see that
\begin{gather*}
V(x,y,z,w)=0\quad \text{if and if only if}\quad
x^{2}+y^{2}+z^{2}+w^{2}=0,
\\
V(x,y,z,w)>0\quad \text{if and if only if}\quad
x^{2}+y^{2}+z^{2}+w^{2}>0,
\\
V(x,y,z,w)\to \infty \quad \text{if and if only if}\quad
x^{2}+y^{2}+z^{2}+w^{2}\to \infty .
\end{gather*}
Let $\gamma $ denote a trajectory $(x(t),y(t),z(t),w(t))$ of
system \eqref{e1.5} with $p(t,x,y,z,w)\equiv 0$ such that $t=0$,
$x=x_{0}$, $y=y_{0}$, $z=z_{0}$, $w=w_{0}$, where
$(x_{0},y_{0},z_{0},w_{0})$ is an arbitrary point in
$x,y,z,w$-space from which motions may originate. Then by Lemma
\ref{lem2} for $t\geq 0$,
\begin{equation*}
V(x,y,z,w)=V(x(t),y(t),z(t),w(t))=V(t)\leq V(0).
\end{equation*}
Moreover, $V(t)$ is nonnegative and non-increasing and therefore
tends to a nonnegative limit, $V(\infty )$ say, as $t\to \infty $.
Suppose $ V(\infty )>0$. Consider the set
\begin{equation*}
S\left\{ (x,y,z,w)\mid V(x,y,z,w)\leq
V(x_{0},y_{0},z_{0},w_{0})\right\} .
\end{equation*}
Because of the properties of the function $V$ we know that $S$ is
bounded, and therefore the set $\gamma \subset S$ is also bounded.
Further, the nonempty set of all limit points of $\gamma $
consists of whole trajectories of the system
\begin{gather*}
\dot{x}=y,\dot{y}=z,\dot{z}=w, \\
\dot{w}=-\varphi (z)w-f(x,y)z-g(y)-h(x)
\end{gather*}
lying on the surface $V(x,y,z,w)=V(\infty )$. Thus if $P$ is a
limit point of $\gamma $, then there exists a half-trajectory, say
$\gamma _{P}$ of the above system, issuing from $P$ and lying on
the surface $V(x,y,z,w)=V(\infty)$. Since for every point
$(x,y,z,w)$ on $\gamma _{P}$ we have $V(x,y,z,w)\geq V(\infty )$,
this implies that $\dot{V}=0$ on $\gamma_{P}$. Also, in view of
the inequality obtained in Lemma \ref{lem2}, that is
\begin{equation*}
\dot{V}\leq -(\frac{\varepsilon c}{2})y^{2}-(\frac{ \delta
}{8ac})z^{2}-(\frac{3\varepsilon a}{4})w^{2},
\end{equation*}
$\dot{V}=0$ implies $y=z=w=0$; and by the above system and
conditions (i) and (iii) of Theorem \ref{thm1}, this means that
$x=0$. Thus, the point $(0,0,0,0) $ lies on the surface
$V(x,y,z,w)=V(\infty )$ and hence $V(\infty)=0$. This completes
the proof of Theorem \ref{thm1}.
\end{proof}

In the case $p\neq 0$ we have

\begin{theorem} \label{thm2}
Suppose the following conditions are satisfied:
\begin{itemize}

\item[(i)] $g(0)=0$

\item[(ii)] the conditions (ii)-(vi) of Theorem \ref{thm1} hold

\item[(iii)] $| p(t,x,y,z,w)| \leq (A+| y| +|z| +| w| )q(t)$,
where $q(t)$ is a non-negative and continuous function of $t$,
and satisfies $\int_{0}^{t}q(s)ds\leq B<\infty $
 for all $t\geq 0$, $A$ and $B$ are positive constants.
\end{itemize}
Then for any given finite constants $x_{0},y_{0},z_{0}$ and
$w_{0}$, there exists a constant $K=K(x_{0},y_{0},z_{0},w_{0})$,
such that any solution $(x(t),y(t),z(t),w(t))$ of the system
\eqref{e1.5} determined by
\begin{equation*}
x(0)=x_{0},\quad y(0)=y_{0}, \quad z(0)=z_{0},\quad w(0)=w_{0}
\end{equation*}
satisfies for all $t\geq 0$,
\begin{equation*}
|x(t)| \leq K,|y(t)| \leq K,|z(t)| \leq K,|w(t)| \leq K.
\end{equation*}
\end{theorem}


\begin{remark} \label{rmk4} \rm
Theorem \ref{thm2} revises the second theorem in \cite{t2},  and
generalizes the results of Ezeilo \cite{e1} and Harrow \cite{h2},
and improves the results of Wu and Xiong \cite{w1} except the
restriction on  $f(x,y)$, that is, $0\leq f(x,y)-b\leq \eta $.
\end{remark}

\begin{proof}[Proof of Theorem \ref{thm2}]
The proof here is based essentially on the method devised by
Antosiewicz \cite{a2}. Let $(x(t),y(t),z(t),w(t))$ be an arbitrary
solution of the system \eqref{e1.5} satisfying the initial
conditions
\begin{equation*}
x(0)=x_{0},\quad y(0)=y_{0},\quad z(0)=z_{0},\quad w(0)=w_{0}.
\end{equation*}
Next, consider the function $V(t)=V(x(t),y(t),z(t),w(t))$, where
$V$ is defined by \eqref{e2.1}. Because $h(0)$ is not necessarily
zero now; we only have the following estimate, in the proof of the
theorem,
\begin{equation}
V\geq D_{1}\int_{0}^{x}h(\xi )d\xi
+D_{2}y^{2}+D_{3}z^{2}+D_{4}w^{2}-(\frac{1}{c})h^{2}(0)
\label{e2.12}
\end{equation}
and since $p\neq 0$, the conclusion of Lemma \ref{lem2} can be
revised as follows
\begin{equation*}
\dot{V}\leq -(D_{5}y^{2}+D_{6}z^{2}+D_{7}w^{2})+(\alpha w+z+\beta
y)p(t,x,y,z,w).
\end{equation*}
Let $D_{8}=\max (\alpha ,1,\beta )$. Then, we have
\begin{equation*}
\dot{V}\leq -D_{8}(|y| +| z| +|w| )(A+| y| +|z| +|w| )q(t).
\end{equation*}
Using the inequalities
\begin{equation*}
|w| \leq 1+w^{2}\text{ and }|2yz| \leq y^{2}+z^{2},
\end{equation*}
we obtain
\begin{equation}
\dot{V}\leq D_{9}[3+4(y^{2}+z^{2}+w^{2})] q(t), \label{e2.13}
\end{equation}
where $D_{9}=D_{8}(A+1)$. It follows from from the result of Lemma
\ref{lem1} that
\begin{equation}
V\geq D_{10}(y^{2}+z^{2}+w^{2})-D_{0},  \label{e2.14}
\end{equation}
$D_{10}=\min (D_{2},D_{3},D_{4})$. Now, from \eqref{e2.13} and
\eqref{e2.14} we have
\begin{equation}
\dot{V}\leq D_{11}q(t)+D_{12}Vq(t)  \label{e2.15}
\end{equation}
where $D_{11}=D_{9}(3+\frac{4D_{0}}{D_{10}})$,
$D_{12}=\frac{4D_{9}}{D_{10}}$. Integrating \eqref{e2.15} from $0$
to $t$, we obtain
\begin{equation*}
V(t)-V(0)\leq
D_{11}\int_{0}^{t}q(s)ds+D_{12}\int_{0}^{t}V(s)q(s)ds.
\end{equation*}
Setting $D_{13}=D_{11}B+V(0)$, and using condition (iii) of
Theorem \ref{thm2} we have
\begin{equation*}
V(t)\leq D_{13}+D_{12}\int_{0}^{t}V(s)q(s)ds.
\end{equation*}
Hence, Gronwall-Bellman inequality yields
\begin{equation*}
V(t)\leq D_{13}\exp (D_{12}\int_{0}^{t}q(s)ds).
\end{equation*}
This completes the proof of Theorem \ref{thm2}.
\end{proof}

Finally, if $p$ is a bounded function, then the constant $K$ above
can be fixed independent of  $x_{0},y_{0},z_{0}$ and $w_{0}$, as
will be seen from our next result.

\begin{theorem} \label{thm3}
Suppose that $g(0)=0$ and conditions (ii)-(vi) of Theorem
\ref{thm1} hold,  and that $p(t,x,y,z,w)$ satisfies
\begin{equation*}
|p(t,x,y,z,w)| \leq \Delta <\infty
\end{equation*}
for all values of $x,y,z$ and $w$, where $\Delta $ is a positive
constant.
\end{theorem}

 Then there exists a constant $K_{1}$ whose magnitude
depends on $a,b,c,d,\delta $ and $\varepsilon $ as well as on the
functions $\varphi ,f,g$ and $h$ such that every solution
$(x(t),y(t),z(t),w(t))$ of the system \eqref{e1.5} ultimately
satisfies
\begin{equation*}
|x(t)| \leq K_{1}, \quad |y(t)| \leq K_{1},\quad |z(t)| \leq
K_{1},\quad |w(t)| \leq K_{1}.
\end{equation*}


\begin{remark} \label{rmk5} \rm
Theorem \ref{thm3} revises \cite[Theorem 3]{t2},
 and improves the results of Wu and Xiong \cite{w1} except the
restriction on $f(x,y)$, that is, $0\leq f(x,y)-b\leq \eta $.
\end{remark}

Now, the actual proof of  Theorem \ref{thm3} will rest mainly on
certain properties of a piecewise continuously differentiable
function $V_{1}=V_{1}(x,y,z,w)$ defined by $V_{1}=V+V_{0}$, where
$V$ is the function \eqref{e2.1} and $V_{0}$ is defined as
follows:
\begin{equation}
V_{0}(x,w)=\begin{cases}
x\operatorname{sgn}w,&|w| \geq |x| \\
w\operatorname{sgn}x,&|w| \leq |x| .
\end{cases}  \label{e2.16}
\end{equation}
The first property of $V_{1}$ is stated as follows.

\begin{lemma} \label{lem3}
Subject to the conditions of Theorem \ref{thm3}, there is a
constant $D_{14}$ such that
\begin{equation}
V_{1}(x,y,z,w)\geq -D_{14}\quad \text{for } x,y,z,w  \label{e2.17}
\end{equation}
and
\begin{equation}
V_{1}(x,y,z,w)\to +\infty \quad \text{as }
x^{2}+y^{2}+z^{2}+w^{2}\to +\infty .  \label{e2.18}
\end{equation}
\end{lemma}

\begin{proof} From \eqref{e2.16} we obtain
$|V_{0}(x,w)| \leq |w| $ for all $x$ and $w$. In view of the last
inequality, it follows that
\begin{equation*}
V_{0}(x,w)\geq -|w| \quad \text{for all } x, w.
\end{equation*}
Using the estimates for $V$ and $V_{0}$ we get the estimate for
$V_{1}$ as follows:
\begin{align*}
2V_{1} & \geq D_{1}\int_{0}^{x}h(\xi )d\xi
+D_{2}y^{2}+D_{3}z^{2}+D_{4}w^{2}-2|w| \\
& =D_{1}\int_{0}^{x}h(\xi )d\xi +D_{2}y^{2}+D_{3}z^{2}+D_{4}(|w|
-D_{4}^{-1})^{2}-D_{4}^{-1}.
\end{align*}
Making use of condition (iii) of Theorem \ref{thm1} we easily
deduce that the integral on the right-hand here is non-negative
and tends to infinity when $x $ does so. Then it is evident that
the expressions \eqref{e2.17} and \eqref{e2.18} are verified,
where $D_{14}=D_{4}^{-1}$ which proves the lemma.
\end{proof}

The next property of the function $V_{1}$ is connected with its
total time derivative and is contained in the following.


\begin{lemma} \label{lem4}
Let $(x,y,z,w)$ be any solution of the differential system
\eqref{e1.5} and the function $v_{1}=v_{1}(t)$ be defined by
$v_{1}(t)=V_{1}(x(t),y(t),z(t),w(t))$. Then the limit
\begin{equation*}
\dot{v}_{1}^{+}(t)=\limsup_{h\to 0^{+}}
\frac{v_{1}(t+h)-v_{1}(t)}{h}
\end{equation*}
exists and there is a constant $D_{15}$ such that
$\dot{v}_{1}^{+}(t)\leq -1$ provided
$$
x^{2}(t)+y^{2}(t)+z^{2}(t)+w^{2}(t)\geq D_{15}.
$$
\end{lemma}

\begin{proof} In accordance with the representation $V_{1}=V+V_{0}$ we
have a representation $v_{1}=v+v_{0}$. The existence of $\dot{v}
_{1}^{+}$ is quite immediate, since $v$ has continuous first
partial derivatives and $v_{0}$ is easily shown to be locally
Lipschitizian in $x$ and $w$ so that the composite function
$v_{1}=v+v_{0}$ is at the least locally Lipschitizian in $x,y,z$
and $w$. Subject to the assumptions of Theorem \ref{thm1} an easy
calculation from \eqref{e2.16} and \eqref{e1.5} shows that
\begin{align*}
\dot{v}_{0}^{+}  &=\begin{cases}
y\operatorname{sgn}w, & \text{if }|w| \geq |x| \\
-h(x)\operatorname{sgn}x-[\varphi
(z)w+f(x,y)z\\
+g(y)-p(t,x,y,z,w)]\operatorname{sgn}x,& \text{if }|w| \leq |x|
\end{cases}
\\
& \leq \begin{cases}
y\operatorname{sgn}w, &\text{if }|w| \geq |x| \\
-h(x)\operatorname{sgn}x+D_{16}[|w| +|z| +|y| +1], &\text{if }|w|
\leq |x| ,
\end{cases}
\end{align*}
where $D_{16}=\max \big\{
\frac{bc}{d},b+\frac{c}{8d}\sqrt{\frac{\delta \varepsilon
}{a}},b+\frac{a}{8}\sqrt{\frac{\delta \varepsilon }{c}},c+\frac{
\delta }{8c}\sqrt{\frac{d}{2ac}},\Delta \big\} $.

In view of the estimates for $\dot{v}$ and $\dot{v}_{0}^{+}$, we
see that
$$
\dot{v}_{1}^{+}=\dot{v}+\dot{v}_{0}^{+}  \leq -( \frac{\varepsilon
c}{2})y^{2}-(\frac{\delta }{8ac}) z^{2}-(\frac{3\varepsilon
a}{4})w^{2}
 +D_{17}(|y| +|z| +|w| )
$$
if $|w| \geq | x|$, or
$$
\dot{v}_{1}^{+}=\dot{v}+\dot{v}_{0}^{+}
 \leq -(\frac{\varepsilon c}{2})y^{2}-(\frac{\delta }{8ac})
z^{2}-(\frac{3\varepsilon a}{4})w^{2}-h(x)\mathop{\rm sgn} x
 +D_{18}(|y| +|z| +|w| ),
$$
 if  $|w| \leq |x|$.
Then by an argument similar to that in the proof of theorem in
\cite{c4},
 one may show that $\dot{v}_{1}^{+}\leq -1$
 provided
$$
x^{2}(t)+y^{2}(t)+z^{2}(t)+w^{2}(t)\geq D_{15}.
$$
The proof of this lemma is now complete.
\end{proof}


\begin{proof}[Proof of Theorem \ref{thm3}]
 We proved through Lemma \ref{lem3} and Lemma \ref{lem4} that the
function $V_{1}=V+V_{0}$ has the following properties:
\begin{gather*}
V_{1}(x,y,z,w)\geq -D_{14}\text{ for all }x,y,z,w,
\\
V_{1}(x,y,z,w)\to \infty \text{ as }x^{2}+y^{2}+z^{2}+w^{2} \to
+\infty ,
\\
\dot{V}_{1}^{+}(t)\leq -1\text{ \ provided
}x^{2}+y^{2}+z^{2}+w^{2} \geq D_{15}.
\end{gather*}
The usual Yoshizawa-type argument,that is Theorem \ref{thm1} in
Chukwu \cite{c4}, applied to the above expressions this implies:
For any solution $(x(t),y(t),z(t),w(t))$ of the system
\eqref{e1.5} we have that
\begin{equation*}
|x(t)| \leq K_{1},|y(t)| \leq K_{1},|z(t)| \leq K_{1},|w(t)| \leq
K_{1}
\end{equation*}
for sufficiently large $t$. Thus the proof of Theorem \ref{thm3}
is complete.
\end{proof}


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\end{document}