\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 109, 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-2006/109\hfil Instability results] {Instability results for certain third order nonlinear vector differential equations} \author[C. Tun\c c, E.Tun\c c\hfil EJDE-2006/109\hfilneg] {Cemil Tun\c c, Ercan Tun\c c} % in alphabetical order \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} \address{Ercan Tun\c c \newline Department of Mathematics, Faculty of Arts and Sciences, Gaziosmanpa\c sa University, 60250, Tokat, Turkey} \email{ercantunc72@yahoo.com} \date{} \thanks{Submitted July 5, 2006. Published September 11, 2006.} \subjclass[2000]{34D05, 34D20} \keywords{Instability, Lyapunov's second (or direct) method; \hfill\break\indent nonlinear differential equations of third order} \begin{abstract} Our goal in this paper is to obtain sufficient conditions for instability of the zero solution to the non-linear vector differential equation \begin{equation*} \dddot{X}+F(X,\dot{X})\ddot{X}+G(\dot{X})+H(X)=0. \end{equation*} An example illustrates the results obtained. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} It is well-known that, since Lyapunov \cite{l7} proposed his famous second (or direct) method on the stability of motion, the problems related to the investigation of instability of solutions of certain second-, third-, fourth-, fifth-, sixth-, seventh and eighth-order linear and nonlinear differential equations have been given great attention in the past decade due to the importance of the subject. During this period, instability of solutions for various higher order linear and nonlinear differential equations have been extensively studied and many results have been obtained in the literature (see, e.g., Bereketo\u{g}lu \cite{b2}, Bereketo\u{g}lu and Kart \cite{b3}, Ezeilo \cite{e1,e2,e3,e4,e5}, Kipnis \cite{k1}, Krasovskii \cite{k2}, Liao and Lu \cite{l1}, Li and Yu \cite{l2}, Li and Duan \cite{l3}, Losprime \cite{l4}, Lu and Liao \cite{l5}, Lu \cite{l6}, Reissig et al \cite{r1}, Sadek \cite{s1,s2}, Skrapek \cite{s3,s4}, Tejumola \cite{t1}, Tiryaki \cite{t2,t3,t4}, C. Tun\c{c} \cite{t5,t6,t7,t8,t9,t10,t11,t12,t13}, C.Tun\c{c} and E. Tun\c{c} \cite{t14,t15,t16}, C. Tun\c{c} and \c{S}evli \cite{t17}, E. Tun\c{c} \cite{t18} and the references cited in that works). Among which, the results performed on instability properties of linear and nonlinear scalar and vector differential equations of third order can briefly be summarized as follows: First, in 1966, Losprime \cite{l4} took into consideration the third-order scalar linear differential equation with periodic coefficients \begin{equation*} \dddot{x}+\ddot{x}+S(t)\dot{x}+T(t)x=0. \end{equation*} Losprime \cite{l4} found the regions of stability and instability of this differential equation by means of some expansions and the Lyapunov's second (or direct) method (see, Lyapunov \cite{l7}). Then, in 1974, Kipnis \cite{k1} discussed the instability of the scalar linear differential equation \begin{equation*} \dddot{x}+p(t)x=0. \end{equation*} The author presented that if the function $p$ is continuous, $\omega$-periodic, non-positive, and satisfies an inequality involving $\omega$, then the above equation is unstable. Later, in 1980, by using Lyapunov's second (or direct) method, Skrapek \cite{s4} established sufficient conditions which guarantee the instability of the trivial solution of the scalar non-linear differential equation as \begin{equation*} \dddot{x}+f_{1}(\ddot{x})+f_{2}(\dot{x})+f_{3}(x) +f_{4}(x,\dot{x},\ddot{x})=0. \end{equation*} In 1995, Lu \cite{l6} discussed a similar problem for the third order nonlinear scalar differential equation \begin{equation*} \dddot{x}+f(x,\dot{x})\ddot{x}+g(x)=0. \end{equation*} In a similar manner, in 1996, Bereketo\u{g}lu and Kart \cite{b3} also studied instability of the trivial solution of scalar differential equation \begin{equation*} \dddot{x}+f(\dot{x})\ddot{x}+g(x)\dot{x}+h(x, \dot{x},\ddot{x})=0. \end{equation*} Together the above works, by using Lyapunov function approach, more recently the authors in \cite{t13,t18} also established some instability results for the zero solution of the non-linear vector differential equations of third order \begin{equation*} \dddot{X}+F(\dot{X})\ddot{X}+G(\dot{X})+H(X)=0, \end{equation*} and \begin{equation*} \dddot{X}+F(\dot{X})\ddot{X}+G(X)\dot{X}+H(X,\dot{X},\ddot{X})=0 \end{equation*} respectively. Furthermore, to the best of our knowledge in the relevant literature, no author except that mentioned above has investigated the instability of solutions of third order nonlinear vector differential equations of the form \begin{equation*} \dddot{X}+A_{1}\ddot{X}+A_{2}\dot{X}+A_{3}X=0 \end{equation*} in which $X\in \mathbb{R}^{n},A_{1},A_{2}$ and $A_{3}$ are not necessarily $n\times n$-constant matrices. In the present paper, we concern with the instability of the trivial solution $X=0$ of nonlinear vector differential equation $$\dddot{X}+F(X,\dot{X})\ddot{X}+G(\dot{X})+H(X)=0, \label{e1.1}$$ in which $X\in \mathbb{R}^{n}$; $F$ is a continuous $n\times n$ -symmetric matrix; $G:\mathbb{R}^{n}\to \mathbb{R}^{n},H:\mathbb{R}^{n}\to \mathbb{R}^{n}$ and $G(0)=H(0)=0$. It will be supposed that the functions $G$ and $H$ are continuous. Throughout this paper, we use the following differential system $$\begin{gathered} \dot{X}=Y,\dot{Y}=Z, \\ \dot{Z}=-F(X,Y)Z-G(Y)-H(X), \end{gathered} \label{e1.2}$$ which was obtained as usual by setting $\dot{X}=Y,\ddot{X}=Z$ in \eqref{e1.1}. Let $J\left( F(X,Y)Y\mid X\right) ,J\left( F(X,Y)Y\mid Y\right) ,J_{G}(Y)$ and $J_{H}(X)$ denote the Jacobian matrices corresponding to $F(X,\dot {X}),G(Y)$ and $H(X)$, respectively: \begin{gather*} J( F(X,Y)Y\mid X) =\Big( \frac{\partial }{\partial x_{j}} \sum_{k=1}^{n}f_{ik}y_{k}\Big) =\Big( \sum_{k=1}^{n}\frac{ \partial f_{ik}}{\partial x_{j}}y_{k}\Big) , \\ J( F(X,Y)Y\mid Y) =\Big( \frac{\partial }{\partial y_{j}} \sum_{k=1}^{n}f_{ik}y_{k}\Big) =F(X,Y)+\Big( \sum_{k=1}^{n} \frac{\partial f_{ik}}{\partial y_{j}}y_{k}\Big) , \\ J_{G}(Y)=\Big( \frac{\partial g_{i}}{\partial y_{j}}\Big) ,\quad J_{H}(X)=\Big( \frac{\partial h_{i}}{\partial x_{j}}\Big) \quad (i,j=1,2,\dots ,n), \end{gather*} where $(x_{1},x_{2},\dots ,x_{n})$, $(y_{1},y_{2},\dots ,y_{n})$, $(z_{1},z_{2},\dots ,z_{n})$, $(f_{ik})$, $(g_{1},g_{2},\dots ,g_{n})$ and $(h_{1},h_{2},\dots ,h_{n})$ are the components of $X,Y,Z,F,G$ and $H$, respectively. It will also be assumed as basic throughout the paper that the Jacobian matrices, $J( F(X,Y)Y\mid X),J( F(X,Y)Y\mid Y),J_{G}(Y)$ and $J_{H}(X)$ exist, and are symmetric and continuous. The symbol $\langle X,Y\rangle$ will be used to denote the usual scalar product in $\mathbb{R}^{n}$ for given any $X,Y$ in $\mathbb{R}^{n}$, that is, $\langle X,Y\rangle =\sum_{i=1}^{n}x_{i}y_{i}$; thus $\langle X,X\rangle =\Vert X\Vert^{2}$. It is well-known that the real symmetric matrix $A=(a_{ij})$, $(i,j=1,2,\dots ,n)$ is said to be positive definite if and only if the quadratic form $X^{T}AX$ is positive definite, where $X\in \mathbb{R}^{n}$ and $X^{T}$ denotes the transpose of $X$. The reason for investigation equation \eqref{e1.1} has been inspired basically by the papers mentioned above. It is worth mentioning that the papers performed on the instability of solutions of third order nonlinear differential equation (see, e.g., \cite{k1,l4,l6,s4,t13,t18}) have been published without an example. But, this paper includes an explanatory example on the subject. It should be noted that the Lyapunov's second (or direct) method is used to verify the results established here. \section{main results} Now, above all, we state the following algebraic results, lemmas, which are needed in the proofs of the main results. \begin{lemma} \label{lem1} Let $A$ be a real symmetric $n\times n$-matrix and \begin{equation*} a'\geq \lambda _{i}(A)\geq a>0\quad (i=1,2,\dots ,n), \end{equation*} where $a',a$ are constants. Then \begin{gather*} a'\langle X,X\rangle \geq \langle AX,X\rangle \geq a\langle X,X\rangle, \\ {a'}^2 \langle X,X\rangle \geq \langle AX,AX\rangle \geq a^{2}\langle X,X\rangle . \end{gather*} \end{lemma} For a proof of the above lemma, see Bellman \cite{b1}. \begin{lemma} \label{lem2} Let $Q,D$ be any two real $n\times n$ commuting symmetric matrices. Then \noindent(i) The eigenvalues $\lambda _{i}(QD)$, $(i=1,2,\dots ,n)$, of the product matrix $QD$ are real and satisfy \begin{equation*} \max_{1\leq j,k\leq n}\lambda _{j}(Q)\lambda _{k}(D)\geq \lambda _{i}(QD)\geq \min_{1 \leq j,k\leq n}\lambda _{j}(Q)\lambda _{k}(D). \end{equation*} \noindent(ii) The eigenvalues $\lambda _{i}(Q+D)$, $(i=1,2,\dots ,n)$, of the sum of matrices $Q$ and $D$ are real and satisfy \begin{equation*} \big\{ \max_{1\leq j\leq n}\lambda_{j}(Q) +\max_{1\leq k\leq n}\lambda _{k}(D)\big\} \geq \lambda _{i}(Q+D)\geq \big\{ \min_{1\leq j\leq n}\lambda _{j}(Q) +\min_{1\leq k\leq n}\lambda _{k}(D)\big\} , \end{equation*} where $\lambda _{j}(Q)$ and $\lambda _{k}(D)$ are, respectively, the eigenvalues of $Q$ and $D$. \end{lemma} For a proof of the above lemma, see Bellman \cite{b1}. We can now state our first main result. \begin{theorem} \label{thm1} In addition to the fundamental assumptions imposed on $F,G$ and $H$ appeared in \eqref{e1.2}, suppose that there are constants $a_{1},a_{2},\overline{a}_{2}$ and $a_{3}$ such that one of the following conditions is satisfied: \begin{itemize} \item[(i)] $\lambda _{i}(F(X,Y))\leq a_{1}$, $-\overline{a}_{2}\leq \lambda_{i}(J_{G}(Y))\leq -a_{2}<0$ and $\lambda _{i}(J_{H}(X))\geq a_{3}>0$, $(i=1,2,\dots ,n)$, for all $X,Y\in \mathbb{R}^{n}$, and $J(F(X,Y)Y\mid X)$ is positive-definite for all $X,Y\in \mathbb{R}^{n}$ \item[(i')] $\lambda _{i}(F(X,Y))\leq a_{1}$, $-\overline{a}_{2}\leq \lambda _{i}(J_{G}(Y))\leq -a_{2}<0$ and $\lambda _{i}(J_{H}(X))\leq -a_{3}<0$, $(i=1,2,\dots ,n)$, for all $X,Y\in \mathbb{R}^{n}$, and $J( F(X,Y)Y\mid X)$ is positive-definite for all $X,Y\in \mathbb{R}^{n}$. \end{itemize} Then the trivial solution $X=0$ of the system \eqref{e1.2} is unstable. \end{theorem} \begin{proof} In order to prove the theorem it will suffice (see Krasovskii \cite{k2}) to show that there exists a continuous function $V_{0}=V_{0}(X,Y,Z)$ which has the following Krasovskii properties: \begin{itemize} \item[(K1)] In every neighborhood of $(0,0,0)$ there exists a point $(\xi ,\eta ,\zeta )$ such that $V_{0}(\xi ,\eta ,\zeta )>0$. \item[(K2)] The time derivative $\dot{V}_{0}=\frac{d}{dt}V_{0}(X,Y,Z)$ along solution paths of the system \eqref{e1.2} is positive-semi definite. \item[(K3)] The only solution $(X,Y,Z)=(X(t),Y(t),Z(t))$ of the system \eqref{e1.2} which satisfies $\dot{V}_{0}=0$ $(t\geq 0)$ is the trivial solution $(0,0,0)$. \end{itemize} We claim that the function $V_{0}=V_{0}(X,Y,Z)$ defined by \begin{aligned} 2V_{0}&= 2\alpha \int_{0}^{1}\langle H(\sigma X),X\rangle d\sigma +2\alpha \langle Y,Z\rangle +\alpha \int_{0}^{1}\sigma \langle F(X,\sigma Y)Y,Y\rangle d\sigma \\ &\quad +\langle Y,Y\rangle -2\langle X,Z\rangle , \end{aligned} \label{e2.1} has all the three properties, where $\alpha$ is a positive constant. Indeed, it is clear from \eqref{e2.1} that $V_{0}(0,0,0)=0$. Since \begin{equation*} H(0)=0,\frac{\partial }{\partial \sigma }H(\sigma X)=J_{H}(\sigma X)X, \end{equation*} then $$H(X)=\int_{0}^{1}J_{H}(\sigma X)Xd\sigma . \label{e2.2}$$ Hence, in view of assumption (i) of Theorem \ref{thm1} and \eqref{e2.2}, we obtain \begin{aligned} \int_{0}^{1}\langle H(\sigma X),X\rangle d\sigma & =\int_{0}^{1}\int_{0}^{1}\langle \sigma _{1}J_{H}(\sigma_{1}\sigma _{2}X)X,X\rangle d\sigma _{2}d\sigma _{1} \\ & \geq \int_{0}^{1}\int_{0}^{1}\langle \sigma_{1}a_{3}X,X\rangle d\sigma _{2}d\sigma _{1}\\ &=\frac{a_{3}}{2}\langle X,X\rangle =\frac{a_{3}}{2}\| X\| ^{2}. \end{aligned} \label{e2.3} Obviously, it follows from assumption (i) of Theorem \ref{thm1}, \eqref{e2.1} and \eqref{e2.3} that \begin{equation*} V_{0}(\varepsilon ,0,0)\geq \frac{a_{3}}{2}\langle \varepsilon ,\varepsilon \rangle =\frac{a_{3}}{2}\left\Vert \varepsilon \right\Vert ^{2}>0 \end{equation*} for all arbitrary $\varepsilon \in \mathbb{R}^{n}$ , $\varepsilon \neq 0$. Thus, in every neighborhood of $(0,0,0)$ there exists a point $(\xi ,\eta ,\zeta )$ such that $V_{0}(\xi ,\eta ,\zeta )>0$ for all $\xi ,\eta$ and $\zeta$ in $\mathbb{R}^{n}$. Next, let $(X,Y,Z)=(X(t),Y(t),Z(t))$ be an arbitrary solution of the system \eqref{e1.2}. Then, the total derivative of the function $V_{0}$ with respect to $t$ along this solution path is \begin{aligned} \dot{V}_{0} &=\frac{d}{dt}V_{0}(X,Y,Z)\\ &= \alpha \langle Z,Z\rangle -\alpha \langle Y,G(Y)\rangle+\langle X,H(X)\rangle +\langle X,F(X,Y)Z\rangle \\ &\quad -\alpha \langle F(X,Y)Z,Y\rangle +\langle X,G(Y)\rangle -\alpha \langle H(X),Y\rangle \\ &\quad +\alpha \frac{d}{dt}\int_{0}^{1}\langle H(\sigma X),X\rangle d\sigma +\alpha \frac{d}{dt}\int_{0}^{1}\sigma \langle F(X,\sigma Y)Y,Y\rangle d\sigma . \end{aligned} \label{e2.4} Check that \begin{aligned} \frac{d}{dt}\int_{0}^{1}\langle H(\sigma X),X\rangle d\sigma & =\int_{0}^{1}\sigma \langle J_{H}(\sigma X)Y,X\rangle d\sigma +\int_{0}^{1}\langle H(\sigma X),Y\rangle d\sigma \\ & =\int_{0}^{1}\sigma \frac{\partial }{\partial \sigma }\langle H(\sigma X),Y\rangle d\sigma +\int_{0}^{1}\langle H(\sigma X),Y\rangle d\sigma \\ & =\sigma \langle H(\sigma X),Y\rangle \big|_0^1 =\langle H(X),Y\rangle \end{aligned} \label{e2.5} and \begin{aligned} &\frac{d}{dt}\int_{0}^{1}\sigma \langle F(X,\sigma Y)Y,Y\rangle d\sigma \\ & =\int_{0}^{1}\langle \sigma F(X,\sigma Y)Z,Y\rangle d\sigma +\int_{0}^{1}\langle \sigma F(X,\sigma Y)Y,Z\rangle d\sigma \\ &\quad +\int_{0}^{1}\langle \sigma J(F(X,\sigma Y)Y\mid X)Y,Y\rangle d\sigma +\int_{0}^{1}\langle \sigma ^{2}J(F(X,\sigma Y)Z\mid Y)Y,Y\rangle d\sigma \\ & =\int_{0}^{1}\langle \sigma F(X,\sigma Y)Z,Y\rangle d\sigma +\int_{0}^{1}\langle \sigma J(F(X,\sigma Y)Y\mid X)Y,Y\rangle d\sigma \\ &\quad +\int_{0}^{1}\sigma \frac{\partial }{\partial \sigma }\langle \sigma F(X,\sigma Y)Z,Y\rangle d\sigma \\ & =\sigma ^{2}\langle F(X,Y)Z,Y\rangle \big|_0^1 +\int_{0}^{1}\langle \sigma J(F(X,\sigma Y)Y\mid X)Y,Y\rangle d\sigma \\ & =\langle F(X,Y)Z,Y\rangle +\int_{0}^{1}\langle \sigma J(F(X,\sigma Y)Y\mid X)Y,Y\rangle d\sigma . \end{aligned} \label{e2.6} Combining the estimates \eqref{e2.5} and \eqref{e2.6} with \eqref{e2.4}, we obtain \begin{aligned} \dot{V}_{0}&= \alpha \langle Z,Z\rangle -\alpha \langle Y,G(Y)\rangle +\langle X,H(X)\rangle \\ &\quad +\langle X,F(X,Y)Z\rangle +\langle X,G(Y)\rangle +\alpha \int_{0}^{1}\langle \sigma J(F(X,\sigma Y)Y\mid X)Y,Y\rangle d\sigma . \end{aligned} \label{e2.7} Since \begin{equation*} G(0)=0,\quad \frac{\partial }{\partial \sigma }G(\sigma Y)=J_{G}(\sigma Y)Y, \end{equation*} it follows that \begin{equation*} G(Y)=\int_{0}^{1}J_{G}(\sigma Y)Yd\sigma . \end{equation*} Thus, assumption (i) of Theorem \ref{thm1} shows that \begin{aligned} \alpha \langle Y,G(Y)\rangle &= \alpha \int_{0}^{1}\langle Y,J_{G}(\sigma Y)Y\rangle d\sigma \\ &\leq -\alpha a_{2}\int_{0}^{1}\langle Y,Y\rangle d\sigma \\ & =-\alpha a_{2}\langle Y,Y\rangle =-\alpha a_{2}\|Y\| ^{2}. \end{aligned} \label{e2.8} By noting assumption (i) of Theorem \ref{thm1} and then combining the estimate \eqref{e2.8} with \eqref{e2.7} we can easily find that $$\dot{V}_{0}\geq \alpha \|Z\| ^{2}+\alpha a_{2}\|Y\| ^{2}+a_{3}\| X\| ^{2} +\langle X,F(X,Y)Z\rangle +\langle X,G(Y)\rangle . \label{e2.9}$$ Now, for some constants $k_{1}$ and $k_{2}$ conveniently chosen later, we have \begin{aligned} \langle X,G(Y)\rangle & =\frac{1}{2}\left\Vert k_{1}X+k_{1}^{-1}G(Y)\right\Vert ^{2}-\frac{1}{2}k_{1}^{2}\langle X,X\rangle -\frac{1}{2}k_{1}^{-2}\langle G(Y),G(Y)\rangle\\ & \geq -\frac{1}{2}k_{1}^{2}\langle X,X\rangle -\frac{1}{ 2k_{1}^{2}}\overline{a}_{2}^{2}\langle Y,Y\rangle \\ & =-\frac{1}{2}k_{1}^{2}\| X\| ^{2}-\frac{1}{2k_{1}^{2}} \overline{a}_{2}^{2}\|Y\| ^{2} \end{aligned} \label{e2.10} and \begin{aligned} &\langle X,F(X,Y)Z\rangle \\ & =\frac{1}{2}\left\Vert k_{2}X+k_{2}^{-1}F(X,Y)Z\right\Vert ^{2}-\frac{1}{2}k_{2}^{2}\langle X,X\rangle -\frac{1}{2k_{2}^{2}}\langle F(X,Y)Z,F(X,Y)Z\rangle \\ & \geq -\frac{1}{2}k_{2}^{2}\langle X,X\rangle -\frac{1}{ 2k_{2}^{2}}\langle F(X,Y)Z,F(X,Y)Z\rangle \\ & \geq -\frac{1}{2}k_{2}^{2}\| X\| ^{2} -\frac{1}{2k_{2}^{2}} a_{1}^{2}\|Z\| ^{2}. \end{aligned} \label{e2.11} From the estimates \eqref{e2.9}-\eqref{e2.11}, we deduce that \begin{equation*} \dot{V}_{0}\geq [ a_{3}-\frac{1}{2}k_{1}^{2}-\frac{1}{2} k_{2}^{2}] \Vert X\Vert ^{2}+[ \alpha a_{2}-\frac{1}{ 2k_{1}^{2}}\overline{a}_{2}^{2}] \Vert Y\Vert ^{2} +[ \alpha -\frac{1}{2k_{2}^{2}}a_{1}^{2}]\Vert Z\Vert ^{2}. \end{equation*} Let \begin{equation*} k_{1}^{2}=\min \big\{ \frac{a_{3}}{2},\frac{\overline{a}_{2}^{2}}{ a_{2}\alpha }\big\} ,k_{2}^{2}=\min \big\{ \frac{a_{3}}{2} ,a_{1}^{2}\alpha ^{-1}\big\} . \end{equation*} Then \begin{align*} \dot{V}_{0} & \geq \left( \frac{a_{3}}{2}\right) \| X\| ^{2}+\left( \frac{\alpha a_{2}}{2}\right) \|Y\| ^{2}+\left( \frac{3\alpha }{4}\right) \|Z\| ^{2} \\ & \geq k\left( \| X\| ^{2}+\|Y\| ^{2}+\|Z\| ^{2}\right) >0, \end{align*} where \begin{equation*} k=\min \big\{ \frac{a_{3}}{2},\frac{\alpha a_{2}}{2},\frac{\alpha }{2} \big\} . \end{equation*} Thus, assumption (i) of Theorem \ref{thm1} shows that $\dot{V}_{0}(t)\geq 0$ for all $t\geq 0$, that is, $\dot{V}_{0}$ is positive semi-definite. Furthermore, the equality $\dot{V}_{0}=0$ $(t\geq 0)$ necessarily implies that $Y=0$ for all $t\geq 0$. Hence, we obtain that $X=\xi$ (a constant vector), $Z=\dot{Y}=0$ for all $t\geq 0$. Substituting the estimates \begin{equation*} X=\xi ,\quad Y=Z=0 \end{equation*} in the system \eqref{e1.2} it follows that $H(\xi )=0$ which necessarily implies that $\xi =0$ because of $H(0)=0$. So \begin{equation*} X=Y=Z=0\quad \text{for all }t\geq 0. \end{equation*} Therefore, the function $V_{0}$ has the entire requisite Krasovskii's criteria \cite{k2} if assumption (i) in Theorem \ref{thm1} holds. This proves part (i) of Theorem \ref{thm1}. Similarly, for the proof of part (i') of Theorem \ref{thm1}, we consider the Lyapunov function $V_{1}=V_{1}(X,Y,Z)$ defined by: \begin{aligned} 2V_{1}&= -2\overline{\alpha }\int_{0}^{1}\langle H(\sigma X),X\rangle d\sigma +2\overline{\alpha }\langle Y,Z\rangle + \overline{\alpha }\int_{0}^{1}\sigma \langle F(X,\sigma Y)Y,Y\rangle d\sigma \\ &\quad -\langle Y,Y\rangle +2\langle X,Z\rangle , \end{aligned} \label{e2.12} where $\overline{\alpha }$ is a positive constant. When we follow the lines indicated in the proof of part (i) of Theorem \ref{thm1}, we can easily obtain \begin{equation*} V_{1}(0,0,0)=0,V_{1}(\overline{\varepsilon },0,0)\geq \frac{a_{3}}{2} \left\Vert \overline{\varepsilon }\right\Vert ^{2}>0 \end{equation*} for all arbitrary $\overline{\varepsilon }\neq 0,\overline{\varepsilon }\in \mathbb{R}^{n}$ and \begin{equation*} \dot{V}_{1}\geq \overline{k}\left( \| X\| ^{2}+\|Y\| ^{2}+\|Z\| ^{2}\right) >0, \end{equation*} where $\overline{k}$ is a certain positive constant. This proves the proof of the part (i') of Theorem \ref{thm1}. The basic properties of $V_{0}(X,Y,Z)$ and $V_{1}(X,Y,Z)$, which we have proved just above, justify that the zero solution of the system \eqref{e1.2} is unstable. See \cite[theorem 1.15]{r1}, see also \cite{k2}. The system \eqref{e1.2} is equivalent to the differential equation \eqref{e1.1}. It follows thus the original statement of the theorem. \end{proof} \noindent\textbf{Example:} As a special case of the system \eqref{e1.2}, let us choose, for the case $n=3,F,G$ and $H$ that appeared in \eqref{e1.2} as follows: \begin{gather*} F(X,Y)=\begin{bmatrix} 1 & -5x_{1}^{2}+\frac{1}{1+y_{1}^{2}} & x_{3}+2y_{3} \\ 0 & \frac{1}{1+x_{2}^{2}+y_{2}^{2}} & 0 \\ 0 & 0 & \frac{1}{2+x_{3}^{4}+y_{3}^{4}} \end{bmatrix}, \\ G(Y)=\begin{bmatrix} -y_{1}-y_{1}^{3} \\ -y_{2}-y_{2}^{3} \\ -y_{3}-y_{3}^{3} \end{bmatrix}, \quad H(X)=\begin{bmatrix} x_{1}+x_{1}^{3} \\ x_{2}+x_{2}^{3} \\ x_{3}+x_{3}^{3} \end{bmatrix} \end{gather*} Then, clearly, the eigenvalues of the matrix $F(X,Y)$ are \begin{gather*} \lambda _{1}(F(X,Y))=1,\quad \lambda _{2}(F(X,Y))=\frac{1}{1+x_{2}^{2}+y_{2}^{2}} \leq 1,\\ \lambda _{3}(F(X,Y))=\frac{1}{2+x_{3}^{4}+y_{3}^{4}}\leq 1. \end{gather*} Next, observe that \begin{equation*} J_{G}(Y)=\begin{bmatrix} -1-3y_{1}^{2} & 0 & 0 \\ 0 & -1-3y_{2}^{2} & 0 \\ 0 & 0 & -1-3y_{3}^{2} \end{bmatrix}, \end{equation*} and hence $\lambda _{1}(J_{G}(Y))=-1-3y_{1}^{2}$, $\lambda_{2}(J_{G}(Y))=-1-3y_{2}^{2}$, $\lambda_{3}(J_{G}(Y))=-1-3y_{3}^{2}$. Clearly, $-1\leq \lambda _{1}(J_{G}(Y))\leq -\frac{1}{2}$, $-1\leq\lambda_{2}(J_{G}(Y))\leq -\frac{1}{2}$ and $-1\leq \lambda _{3}(J_{G}(Y))\leq -\frac{1}{2}$. Finally, we have that \begin{equation*} J_{H}(X)=\begin{bmatrix} 1+3x_{1}^{2} & 0 & 0 \\ 0 & 1+3x_{2}^{2} & 0 \\ 0 & 0 & 1+3x_{3}^{2} \end{bmatrix}, \end{equation*} and $\lambda _{1}(J_{H}(X))=1+3x_{1}^{2}\geq 1>0$, $\lambda_{2}(J_{H}(X))=1+3x_{2}^{2}\geq 1>0$, $\lambda_{3}(J_{H}(X))=1+3x_{3}^{2}\geq 1>0$. Thus all the conditions of part (i) of Theorem \ref{thm1} are satisfied. The next theorem is our second main result. \begin{theorem} \label{thm2} Further to the basic assumptions imposed on $F,G$ and $H$ appeared in \eqref{e1.2}, suppose that there are constants $a_{1},a_{2}$ and $a_{3}$ such that one of the following conditions is satisfied: \begin{itemize} \item[(i)] $\lambda _{i}(F(X,Y))\leq -a_{1}<0$, $\lambda _{i}(J_{G}(Y))\leq a_{2}$ and $\lambda _{i}(J_{H}(X))\geq a_{3}>0$, $(i=1,2,\dots ,n)$, for all $X,Y\in \mathbb{R}^{n}$. \item[(i')] $\lambda _{i}(F(X,Y))\geq a_{1}>0$, $\lambda_{i}(J_{G}(Y))\leq a_{2}$ and $\lambda _{i}(J_{H}(X))\leq -a_{3}<0$, $(i=1,2,\dots ,n)$, for all $X,Y\in \mathbb{R}^{n}$. \end{itemize} Then the zero solution $X=0$ of the system \eqref{e1.2} is unstable. \end{theorem} \begin{proof} Consider the function $V_{2}=V_{2}(X,Y,Z)$ defined by $$2V_{2}= \beta \langle Z,Z\rangle +2\beta \langle Y,H(X)\rangle +2\beta \int_{0}^{1}\langle G(\sigma Y),Y\rangle d\sigma +\langle Y,Y\rangle -2\langle X,Z\rangle , \label{e2.13}$$ where $\beta$ is a positive constant. Observe that $V_{2}(0,0,0)=0$. It is also clear from assumption (i) of Theorem \ref{thm2} that \begin{equation*} V_{2}(0,0,\varepsilon )\geq \beta \langle \varepsilon ,\varepsilon \rangle =\beta \Vert \varepsilon \Vert ^{2}>0 \end{equation*} for all arbitrary $\varepsilon \in \mathbb{R}^{n}$, $\varepsilon \neq 0$. So that in every neighborhood of $(0,0,0)$ there exists a point $(\xi ,\eta ,\zeta )$ such that $V_{2}(\xi ,\eta ,\zeta )>0$ for all $\xi ,\eta$ and $\zeta$ in $\mathbb{R}^{n}$. Next, let $(X,Y,Z)=(X(t),Y(t),Z(t))$ be an arbitrary solution of the system \eqref{e1.2}. An easy calculation from \eqref{e2.13} and \eqref{e1.2} yields that \begin{align*} \dot{V}_{2} &=\frac{d}{dt}V_{2}(X,Y,Z)\\ &= -\beta \langle Z,F(X,Y)Z\rangle +\beta \langle Y,J_{H}(X)Y\rangle +\langle X,H(X)\rangle \\ &\quad +\langle X,F(X,Y)Z\rangle +\langle X,G(Y)\rangle -\beta \langle G(Y),Z\rangle +\beta \frac{d}{dt}\int_{0}^{1}\langle G(\sigma Y),Y\rangle d\sigma . \end{align*} But \begin{aligned} \frac{d}{dt}\int_{0}^{1}\langle G(\sigma Y),Y\rangle d\sigma & =\int_{0}^{1}\sigma \langle J_{G}(\sigma Y)Z,Y\rangle d\sigma +\int_{0}^{1}\langle G(\sigma Y),Z\rangle d\sigma \\ & =\int_{0}^{1}\sigma \frac{\partial }{\partial \sigma }\langle G(\sigma Y),Z\rangle d\sigma +\int_{0}^{1}\langle G(\sigma Y),Z\rangle d\sigma \\ & =\sigma \langle G(\sigma Y),Z\rangle \big|_0^1 =\langle G(Y),Z\rangle . \end{aligned} \label{e2.14} Therefore, by using \eqref{e2.14} and assumption (i) of Theorem \ref{thm2}, we get \begin{aligned} \dot{V}_{2} & =-\beta \langle Z,F(X,Y)Z\rangle +\beta \langle Y,J_{H}(X)Y\rangle +\langle X,H(X)\rangle \\ &\quad +\langle X,F(X,Y)Z\rangle +\langle X,G(Y)\rangle \\ & \geq \beta a_{1}\|Z\| ^{2}+\beta a_{3}\|Y\| ^{2}+a_{3}\| X\|^{2}+\langle X,F(X,Y)Z\rangle +\langle X,G(Y)\rangle . \end{aligned} \label{e2.15} Similarly, as shown just above for some constants $\overline{k}_{1}$ and $\overline{k}_{2}$ conveniently chosen later, we can easily obtain from \eqref{e2.15} that \begin{equation*} \dot{V}_{2}\geq \big( a_{3}-\frac{1}{2}\overline{k}_{1}^{2}-\frac{1}{ 2}\overline{k}_{2}^{2}\big) \| X\| ^{2} +\big(\beta a_{3}-\frac{1}{2}\overline{k}_{1}^{-2}a_{2}^{2}\big) \| Y\| ^{2}+\big( \beta a_{1}-\frac{1}{2}\overline{k} _{2}^{-2}a_{1}^{2}\big) \| Z\| ^{2}. \end{equation*} Let \begin{equation*} \overline{k}_{1}^{2}=\min \big\{\frac{a_{3}}{2},\frac{a_{2}^{2}}{\beta a_{3}}\big\} ,\overline{k}_{2}^{2}=\min \big\{ \frac{a_{3}}{2},\frac{a_{1} }{\beta }\big\} . \end{equation*} Hence \begin{align*} \dot{V}_{2} & \geq \left( \frac{a_{3}}{2}\right) \| X\| ^{2} +\left( \frac{\beta a_{3}}{2}\right) \|Y\| ^{2} +\left( \frac{\beta a_{1}}{2}\right) \|Z\| ^{2} \\ & \geq \overline{k}\left( \| X\| ^{2}+\|Y\| ^{2}+\|Z\| ^{2}\right) >0, \end{align*} where \begin{equation*} \overline{k}=\min \big\{ \frac{a_{3}}{2},\frac{\beta a_{3}}{2},\frac{\beta a_{1}}{2}\big\} . \end{equation*} The rest of the proof of part (i) of Theorem \ref{thm2} is the same as the proof of part (i) of Theorem \ref{thm1} just proved above and hence it is omitted the details. \end{proof} Finally, for the proof of part (i') of Theorem \ref{thm2}, we consider the Lyapunov function \begin{equation*} V_{3}(X,Y,Z)=V_{2}(X,Y,Z)-2\beta \int_{0}^{1}\langle H(\sigma X),X\rangle d\sigma , \end{equation*} where $V_{2}(X,Y,Z)$ is defined as the same the function in \eqref{e2.13}. 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