\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small {\em 
Electronic Journal of Differential Equations}, 
Vol. 2004(2004), No. 117, pp. 1--6.\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-2004/117\hfil Instability of certain sixth-order DE] 
{On the instability of certain sixth-order nonlinear
differential equations}

\author[Cemil Tun\c{c}\hfil EJDE-2004/117\hfilneg]
{Cemil Tun\c{c}}

\address{Department of Mathematics\\
 Faculty of Arts and Sciences\\
Y\"{u}z\"{u}nc\"{u} Y\i l University\\
65080, VAN -- Turkey} 
\email{cemtunc@yahoo.com}

\date{}
\thanks{Submitted July 13, 2004. Published October 7, 2004.}
\subjclass[2000]{34D05, 34D20} 
\keywords{Nonlinear differential equations of sixth order; instability; \hfill\break\indent
Lyapunov function}

\begin{abstract}
 In this paper, we give an instability criteria for the nonlinear
 sixth-order vector  differential equation
 \begin{gather*}
 X^{(6)} + AX^{(5)} +B(t)\Phi (X,\dot {X},\ddot {X},
 \overset{\dots }{X},X^{(4)},X^{(5)})X^{(4)} +
 C(t)\Psi (\ddot {X})\overset{\dots }{X}\\
 +D(t)\Omega (X,\dot {X},\ddot {X},\overset{\dots }{X},X^{(4)},X^{(5)})
 \ddot {X}+E(t)G(\dot {X})+H(X) = 0\,.
 \end{gather*}
 The result extends and includes earlier results \cite{e5,t3,t7}.
\end{abstract}

\maketitle \numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]

\section{Introduction}

 It is well known from the relevant literature
that there have been intensive studies on the qualitative behavior
of solutions of certain nonlinear ordinary differential equations
in recent years. Meanwhile, many articles have been devoted to the
investigation of the instability properties of solutions for
various third-, fourth-, fifth-, sixth-, seventh and eighth order
certain nonlinear differential equations; see for instance
\cite{b1,e1,e2,e3,e4,e5,l1,l2,s1,s2,s3,t1,t2,t3,t4,t5,t6,t6,t7,t8}
 and the references cited therein. However, to
the best our knowledge for the case $n = 1$, there exist only two
works on the instability properties of solutions of certain
nonlinear differential equations of the sixth order. Namely, in
the case $n = 1$, Ezeilo \cite{e5} and Tiryaki \cite{t3} studied
the instability of the zero solution $x = 0$ of the following
nonlinear differential equations:
\begin{align*}
x^{(6)} + a_1 x^{(5)} + a_2 x^{(4)} + e(x,\dot {x},\ddot
{x},\overset{\dots }{x},x^{(4)},x^{(5)})\overset{\dots }{x}& \\
+ f(\dot{x})\ddot {x} + g(x,\dot {x},\ddot {x},\overset{\dots
}{x}, x^{(4)},x^{(5)})\dot {x} + h(x) &= 0
\end{align*}
and
\begin{align*}
x^{(6)} + a_1 x^{(5)} + f_1 (x,\dot {x},\ddot
{x},\overset{\dots }{x},x^{(4)},x^{(5)})x^{(4)}& \\
+ f_2 (\ddot {x})\overset{\dots }{x} + f_3 (x,\dot {x},\ddot
{x},\overset{\dots }{x},x^{(4)},x^{(5)})\ddot {x} + f_4 (\dot
{x})+ f_5 (x) &= 0
\end{align*}
respectively.

Recently, the author in \cite{t7} investigated the same subject
for the sixth order nonlinear vector differential equations of the
form:
\begin{equation}\label{eq1}
\begin{aligned}
X^{(6)} + AX^{(5)} + \Phi (X,\dot {X},\ddot {X},\overset{\dots
}{X},X^{(4)},X^{(5)})X^{(4)} + \Psi (\ddot
{X})\overset{\dots }{X}& \\
+F(X,\dot {X},\ddot{X},\overset{\dots }{X},X^{(4)},X^{(5)})\ddot
{X}+G(\dot {X})+H(X)& = 0.
\end{aligned}
\end{equation}
In this paper we are concerned with the instability of the trivial
solution $X = 0$ of the nonlinear vector differential equations of
the form:
\begin{equation}\label{eq2}
\begin{aligned}
X^{(6)} + AX^{(5)} + B(t)\Phi (X,\dot {X},\ddot {X},\overset{\dots
}{X},X^{(4)},X^{(5)})X^{(4)} + C(t)\Psi (\ddot
{X})\overset{\dots }{X}&\\
+D(t)\Omega (X,\dot {X},\ddot {X},\overset{\dots
}{X},X^{(4)},X^{(5)})\ddot {X}+E(t)G(\dot {X})+H(X) &= 0
\end{aligned}
\end{equation}
in which $t \in \mathbb{R}^+ $, $\mathbb{R} ^ + = [{0,\infty })$
and $X \in \mathbb{R} ^n$; $A$ is a constant $n\times n$-symmetric
matrix; $B$, $\Phi $, $C$, $\Psi $, $D$, $\Omega $ and $E$ are
continuous $n\times n$ -symmetric matrices depending, in each
case, on the arguments shown; $G:\mathbb{R} ^n \to \mathbb{R} ^n$,
$H:\mathbb{R} ^n \to \mathbb{R} ^n$ and $G(0)=H(0)$=0. It is
supposed that the functions $G$ and $H$ are continuous. Let $J_H
(X)$, $J_G (\dot {X})$ and $J_\Psi (\ddot {X})$ denote the
Jacobian matrices corresponding to the $H(X)$, $G(\dot {X})$ and
$\Psi (\ddot {X})$, respectively, that is,
\[
J_H (X) = \big( {\frac{\partial h_i }{\partial x_j }} \big), \quad
J_G (\dot {X}) = \big( {\frac{\partial g_i }{\partial \dot {x}_j
}} \big), \quad J_\Psi (\ddot {X}) = \big( {\frac{\partial \psi _i
}{\partial \ddot {x}_j }} \big), \quad (i,j = 1,2,\dots ,n),
\]
where $(x_1 ,x_2 ,\dots,x_n )$, $(\dot {x}_1 ,\dot {x}_2 ,\dots
,\dot {x}_n )$, $(\ddot {x}_1 ,\ddot {x}_2 ,\dots ,\ddot {x}_n )$,
$(h_1 ,h_2 ,\dots ,h_n )$, \\ $(g_1 ,g_2 ,\dots ,g_n )$ and $(\psi
_1 ,\psi _2 ,\dots ,\psi _n )$ are the components of $X$, $\dot
{X}$, $\ddot {X}$, $H$, $G$ and $\Psi $, respectively. Other than
these, it is also assumed that the Jacobian matrices $J_H (X)$,
$J_G (\dot {X})$, $J_\Psi (\ddot {X})$ and the derivatives
$\frac{d}{dt}C(t) = \dot {C}(t)$ and $\frac{d}{dt}E(t) = \dot
{E}(t)$ exist and are continuous. Moreover, it is assumed that all
matrices given in the pairs $C$, $\Psi $; $C$,$J_\Psi $; $\dot
{C}$,$\Psi $; $E$, $J_G $; $\dot {E}$, $J_G $; $B$, $\Phi $ and
$D$, $\Omega $ commute with each others.

The symbol $\langle {X,Y}\rangle $ corresponding to any pair$X$,
$Y$ in $\mathbb{R} ^n$ stands for the usual scalar product
$\sum_{i = 1}^n {x_i y_i } $, and $\lambda _i (A)(i = 1,2,\dots
,n)$ are the eigenvalues of the $n\times n$- matrix $A$.

In what follows it will be convenient to use the equivalent
differential system:
\begin{equation}\label{eq3}
\begin{gathered}
\dot {X} = Y, \quad \dot {Y} = Z, \quad \dot {Z} = W, \quad \dot
{W} = U, \quad \dot {U} = V ,\\
\dot {V} = - AV - B(t)\Phi (X,Y,Z,W,U,V)U - C(t)\Psi (Z)W\\
 \qquad - D(t)\Omega (X,Y,Z,W,U,V)Z - E(t)G(Y) - H(X)
\end{gathered}
\end{equation}
which is obtained from \eqref{eq2} by setting $\dot {X} = Y$,
$\ddot {X} = Z$, $\overset{\dots }{X} = W$, $X^{(4)} = U$ and
$X^{(5)} = V$.

\section{Main Result}

The main result is the following theorem. This extends and
includes the results of Ezeilo \cite{e5}, Tiryaki \cite{t3} and
Tun\c{c} \cite{t7}.

\begin{theorem} \label{mainthm}
Suppose that there are constants $a_1 $, $a_2 $ and $a_4 $ with
$a_4 > \frac{1}{4}a_2^2 $ such that
\begin{itemize}
\item [(i)] $A$, $B$, $D$, $J_H (X)$ are symmetric and $\lambda _i
(A) \ge a_1 > 0$, $\lambda _i (B(t)) \ge 1$, $\lambda _i (D(t))
\ge 1$ for all $t \in \mathbb{R} ^ + $, $H(X) \ne 0$ for all $X
\ne 0$, $X \in \mathbb{R} ^n$, and $\lambda _i (J_H (X)) < 0$ for
all $X \in \mathbb{R} ^n$, $(i = 1,2,\dots ,n)$.

\item [(ii)]$\Phi (X,Y,Z,W,U,V)$, $\Omega (X,Y,Z,W,U,V)$ are
symmetric and \\
$\lambda _i (\Phi (X,Y,Z,W,U,V)) \le a_2 $, $\lambda _i (\Omega
(X,Y,Z,W,U,V)) \ge a_4 $ for all $X$, $Y$, $Z$, $W$, $U$, $V \in
\mathbb{R} ^n$, $(i = 1,2,\dots ,n)$.

\item [(iii)] $\dot {E}(t)$, $J_G (Y)$ and $\dot {C}(t)$, $\Psi
(Z)$ are symmetric and have opposite-sign eigenvalues for all $t
\in \mathbb{R} ^ + $and $Y$, $Z \in \mathbb{R} ^n$ .
\end{itemize}
Then the zero solution $X = 0$ of the system (\ref{eq3}) is
unstable.
\end{theorem}

\begin{proof} Our main tool in the proof of the
theorem is the Lyapunov function $\Gamma = \Gamma (t,X,Y,Z,W,U,V)$
given by
\begin{equation} \label{eq4}
\begin{aligned}
\Gamma &=  - \langle {V,Z} \rangle - \langle {Z,AU} \rangle +
\langle {W,U} \rangle + \frac{1}{2}\langle {W,AW} \rangle -
\int_0^1
{\langle {E(t)G(\sigma Y),Y} \rangle d\sigma } \\
&\quad- \int_0^1 {\langle {\sigma C(t)\Psi (\sigma Z)Z,Z} \rangle
d\sigma } -\langle {H(X),Y} \rangle .
\end{aligned}
\end{equation}
It should be noted that this function and its total time
derivative satisfy some fundamental properties: It is clear from
(\ref{eq4}) that $\Gamma (0,0,0,0,0,0,0) = 0$. Obviously, it also
follows from the assumptions of the theorem and (\ref{eq4}) that
\[
\Gamma (0,0,0,0,\varepsilon ,\varepsilon ,0) = \langle
{\varepsilon ,\varepsilon } \rangle + \frac{1}{2}\langle
{\varepsilon ,A\varepsilon } \rangle \ge \| \varepsilon \|^2 +
\frac{1}{2}a_1 \|\varepsilon \|^2 > 0,
\]
for all $\varepsilon \ne 0$ in $\mathbb{R}^n$. Next let
$(X,Y,Z,W,U,V)=(X(t),Y(t),Z(t),W(t),U(t),V(t))$ be an arbitrary
solution of (\ref{eq3}). Differentiating (\ref{eq4}) we obtain
\begin{equation} \label{eq5}
\begin{aligned}
 \dot {\Gamma } &= \frac{d}{dt}\Gamma (t,X,Y,Z,W,U,V)\\
 & =  \langle {U,U} \rangle +\langle {B(t)\Phi (X,Y,Z,W,U,V)U,Z}
\rangle
 +\langle {Z,D(t)\Omega (X,Y,Z,W,U,V)Z} \rangle\\
&\quad -\langle {J_H (X)Y,Y} \rangle +\langle{E(t)G(Y),Z} \rangle
 +\langle {C(t)\Psi (Z)W,Z} \rangle \\
&\quad -\frac{d}{dt}\int_0^1 {\langle {E(t)G(\sigma Y),Y}\rangle
d\sigma } -\frac{d}{dt}\int_0^1 {\langle {\sigma C(t)\Psi (\sigma
Z)Z,Z} \rangle d\sigma }.
\end{aligned}
\end{equation}
Recall that
\begin{equation} \label{eq6}
\begin{aligned}
&\frac{d}{dt}\int_0^1 {\langle {E(t)G(\sigma Y),Y}\rangle d\sigma }\\
&= \int_0^1 {\langle {\dot{E}(t)G(\sigma Y),Y} \rangle d\sigma } +
\int_0^1
 {\sigma \langle {E(t)J_G (\sigma Y)Z,Y} \rangle }d\sigma
 + \int_0^1 {\langle {E(t)G(\sigma Y),Z} \rangle d\sigma } \\
&=\int_0^1 {\langle {\dot {E}(t)G(\sigma Y),Y}
 \rangle d\sigma } +\int_0^1 {\sigma \frac{\partial}
 {\partial \sigma }\langle {E(t)G(\sigma Y),Z} \rangle d\sigma}
 +\int_0^1 {\langle {E(t)G(\sigma Y),Z}\rangle d\sigma} \\
&=\int_0^1 {\langle {\dot {E}(t)G(\sigma Y),Y}
 \rangle d\sigma } +\sigma \langle {E(t)G(\sigma Y),Z}
 \rangle \left| {_0^1 } \right. \\
&=\int_0^1 {\langle {\dot {E}(t)G(\sigma Y),Y} \rangle d\sigma }
+\langle {E(t)G(Y),Z}\rangle.
\end{aligned}
\end{equation}
and
\begin{equation} \label{eq7}
\begin{aligned}
&\frac{d}{dt}\int_0^1 {\langle {\sigma C(t)\Psi
(\sigma Z)Z,Z} \rangle d\sigma } \\
&= \int_0^1 {\langle {\sigma \dot {C}(t)\Psi (\sigma Z)Z,Z}
\rangle d\sigma } + \int_0^1 {\langle {\sigma C(t)\Psi (\sigma
Z)Z,W}
\rangle d\sigma }\\
 &\quad + \int_0^1 {\sigma ^2\langle {C(t)J_\Psi (\sigma
 Z)ZW,Z} \rangle d\sigma } +\int_0^1 {\langle
 {\sigma C(t)\Psi (\sigma Z)W,Z} \rangle d\sigma } \\
&= \int_0^1 {\langle {\sigma \dot {C}(t)\Psi (\sigma
 Z)Z,Z} \rangle d\sigma } +\int_0^1 {\sigma
 \frac{\partial }{\partial \sigma }\langle {\sigma C(t)\Psi
 (\sigma Z)W,Z} \rangle d\sigma } \\
&\quad + \int_0^1 {\langle {\sigma C(t)\Psi (\sigma Z)W,Z}
 \rangle d\sigma } \\
&= \int_0^1 {\langle {\sigma \dot {C}(t)\Psi (\sigma
 Z)Z,Z} \rangle d\sigma } +\sigma ^2\langle {C(t)\Psi
 (\sigma Z)W,Z} \rangle \left| {_0^1 } \right. \\
&= \int_0^1 {\langle {\sigma \dot {C}(t)\Psi (\sigma
 Z)Z,Z} \rangle d\sigma } +\langle {C(t)\Psi (Z)W,Z}\rangle.
\end{aligned}
\end{equation}
On gathering the estimates \eqref{eq6} and \eqref{eq7} into
\eqref{eq5}, we obtain
\begin{equation} \label{eq8}
\begin{aligned}
\dot{\Gamma } &=  \langle {U,U} \rangle + \langle
{B(t)\Phi (X,Y,Z,W,U,V)Z,U} \rangle\\
 &\quad +\langle {Z,D(t)\Omega (X,Y,Z,W,U,V)Z} \rangle
-\langle {J_H (X)Y,Y} \rangle \\
&\quad -\int_0^1 {\langle {\dot {E}(t)G(\sigma Y),Y} \rangle
d\sigma } -\int_0^1 {\langle {\sigma \dot {C}(t)\Psi (\sigma
Z)Z,Z} \rangle d\sigma }.
\end{aligned}
\end{equation}
Since
\[
G(0) = 0, \quad \frac{\partial }{\partial \sigma }G(\sigma Y) =
J_G (\sigma Y)Y , \quad G(Y) = \int_0^1 {J_G (\sigma Y)Yd\sigma },
\]
the assumption (iii) of the theorem shows that
\begin{equation}
\label{eq9} \int_0^1 {\langle \dot {E}(t)G(\sigma Y),Y\rangle
d\sigma } = \int_0^1 {\int_0^1 {\langle \sigma _1 \dot {E}(t)J_G
(\sigma _1 \sigma _2 Y)Y,Y\rangle d\sigma _2 d\sigma _1 \le 0} } .
\end{equation}
Next, by the assumption of (iii) of the theorem it is also clear
that
\begin{equation}\label{eq10}
\int_0^1 {\langle {\sigma \dot {C}(t)\Psi (\sigma Z)Z,Z} \rangle
d\sigma } \le 0.
\end{equation}
Combining the estimates \eqref{eq9} and \eqref{eq10} into
\eqref{eq8} we obtain
\begin{align*}
\dot {\Gamma }
 &\ge \| {U + \frac{1}{2}B(t)\Phi (X,Y,Z,W,U,V)Z}\|^2 +
\langle {Z,D(t)\Omega (X,Y,Z,W,U,V)Z} \rangle\\
&\quad-\langle {J_H (X)Y,Y} \rangle -\frac{1}{4}\langle
{B(t)\Phi (X,Y,Z,W,U,V)Z,B(t)\Phi (X,Y,Z,W,U,V)Z} \rangle \\
&\ge \langle {Z,D(t)\Omega (X,Y,Z,W,U,V)Z} \rangle -
\langle {J_H (X)Y,Y} \rangle\\
&\quad -\frac{1}{4}\langle {B(t)\Phi (X,Y,Z,W,U,V)Z,B(t)\Phi
(X,Y,Z,W,U,V)Z} \rangle
\end{align*}
Hence, it follows from the assumptions (i) and (ii) that
\[
\dot {\Gamma }
 \ge
\langle {Z,a_4 Z} \rangle  - \frac{1}{4}\langle {a_2 Z,a_2 Z}
\rangle =  \big( {a_4 - \frac{1}{4}a_2^2 }\big)\| Z \|^2
> 0.
\]
Thus, the assumptions of the theorem show that $\dot {\Gamma } \ge
0$ for all $t \ge 0$, that is, $\dot {\Gamma }$ is positive
semi-definite. Furthermore, $\dot {\Gamma } = 0(t \ge 0)$
necessarily implies that $Y = 0$ for all $t \ge 0$, and therefore
also that $X = \xi $ (a constant vector), $Z = \dot {Y} = 0$, $W =
\ddot {Y} = 0$, $U = \overset{\dots }{Y} = 0$, $V = Y^{(4)} = 0$
for all $t \ge 0$. The substitution of the estimates
\[
X = \xi , \quad Y = Z = W = U = V = 0
\]
in (\ref{eq3}) leads to the result $H(\xi ) = 0$ which by
assumption (i) of the theorem implies (only) that $\xi = 0$. Hence
$\dot {\Gamma } = 0 \quad (t \ge 0)$ implies that
\[
X = Y = Z = W = U = V = 0\quad \mbox{for all  } t \ge 0.
\]
Therefore, the function $\Gamma $ has the  requisites for
Krasovskii criterion \cite{k1} if the conditions of the theorem
hold. Thus, the basic properties of 
$\Gamma(t,X,Y,Z,W,U,V)$, which are proved just above verify that the zero
solution of the system (\ref{eq3}) is unstable. (See Reissig et al
\cite[Theorem 1.15]{r1} and Krasovskii \cite{k1}). The system of
equations (\ref{eq3}) is equivalent to the differential equation
(\ref{eq2}). Consequently, this completes the proof of the
theorem.
\end{proof}

\subsection*{Acknowledgements}
The author would like to express sincere thanks to the anonymous
referee for his/her valuable comments and suggestions.


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\end{thebibliography}
\end{document}