\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 $$x^{(4)}+\varphi (\ddot{x})\dddot{x}+f(x,\dot{x}) \ddot{x}+g(\dot{x})+h(x)=p(t,x,\dot{x},\ddot{x}, \dddot{x}) \label{e1.4}$$ 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 $$\begin{gathered} \dot{x}=y,\dot{y}=z,\dot{z}=w, \\ \dot{w}=-\varphi (z)w-f(x,y)z-g(y)-h(x)+p(t,x,y,z,w) \end{gathered} \label{e1.5}$$ 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)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, $$V\geq D_{1}\int_{0}^{x}h(\xi )d\xi +D_{2}y^{2}+D_{3}z^{2}+D_{4}w^{2}-(\frac{1}{c})h^{2}(0) \label{e2.12}$$ 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 $$\dot{V}\leq D_{9}[3+4(y^{2}+z^{2}+w^{2})] q(t), \label{e2.13}$$ where $D_{9}=D_{8}(A+1)$. It follows from from the result of Lemma \ref{lem1} that $$V\geq D_{10}(y^{2}+z^{2}+w^{2})-D_{0}, \label{e2.14}$$ $D_{10}=\min (D_{2},D_{3},D_{4})$. Now, from \eqref{e2.13} and \eqref{e2.14} we have $$\dot{V}\leq D_{11}q(t)+D_{12}Vq(t) \label{e2.15}$$ 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: $$V_{0}(x,w)=\begin{cases} x\operatorname{sgn}w,&|w| \geq |x| \\ w\operatorname{sgn}x,&|w| \leq |x| . \end{cases} \label{e2.16}$$ 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 $$V_{1}(x,y,z,w)\geq -D_{14}\quad \text{for } x,y,z,w \label{e2.17}$$ and $$V_{1}(x,y,z,w)\to +\infty \quad \text{as } x^{2}+y^{2}+z^{2}+w^{2}\to +\infty . \label{e2.18}$$ \end{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 $t$. 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