\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: \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} In this paper we are concerned with the instability of the trivial solution $X = 0$ of the nonlinear vector differential equations of the form: \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} 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: $$\label{eq3} \begin{gathered} \dot {X} = Y, \quad \dot {Y} = Z, \quad \dot {Z} = W, \quad \dot {W} = U, \quad \dot {U} = V ,\\ \dot {V} = - AV - B(t)\Phi (X,Y,Z,W,U,V)U - C(t)\Psi (Z)W\\ \qquad - D(t)\Omega (X,Y,Z,W,U,V)Z - E(t)G(Y) - H(X) \end{gathered}$$ 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 \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} 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 \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} Recall that \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} and \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} On gathering the estimates \eqref{eq6} and \eqref{eq7} into \eqref{eq5}, we obtain \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} 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 $$\label{eq9} \int_0^1 {\langle \dot {E}(t)G(\sigma Y),Y\rangle d\sigma } = \int_0^1 {\int_0^1 {\langle \sigma _1 \dot {E}(t)J_G (\sigma _1 \sigma _2 Y)Y,Y\rangle d\sigma _2 d\sigma _1 \le 0} } .$$ Next, by the assumption of (iii) of the theorem it is also clear that $$\label{eq10} \int_0^1 {\langle {\sigma \dot {C}(t)\Psi (\sigma Z)Z,Z} \rangle d\sigma } \le 0.$$ 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}). 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