\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 06, 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 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/06\hfil Boundedness of solutions] {Boundedness of solutions to fourth order differential equations with oscillatory restoring and forcing terms} \author[B. S. Ogundare\hfil EJDE-2006/06\hfilneg] {Babatunde S. Ogundare} \address{Babatunde S. Ogundare \hfill\break Department of Mathematics\\ Obafemi Awolowo University\\ Ile-Ife, Nigeria} \email{bogunda@oauife.edu.ng \quad ogundareb@yahoo.com} \date{} \thanks{Submitted September 7, 2005. Published January 11, 2006.} \subjclass[2000]{34C10, 34C11} \keywords{Fourth order differential equation; bounded solution; \hfill\break\indent oscillatory solution; restoring and forcing terms} \begin{abstract} This article concerns the fourth order differential equation $$x^{(iv)}+ax'''+bx''+g(x')+h(x)=p(t).$$ Using the Cauchy formula for the particular solution of non-homogeneous linear differential equations with constant coefficients, we prove that the solution and its derivatives up to order three are bounded. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} In this article, we study the boundedness of solutions to the fourth-order nonlinear differential equation $$x^{(iv)}+ax'''+bx''+g(x')+h(x)=p(t)\label{e1.1}$$ where $a>0$ and $b>0$ are positive constants with $a^{2}>4b$, $g, h,$ and $p$ and their first derivatives are continuous functions depending on the arguments shown. In addition, $h$ and $p$ are oscillatory. Several authors have investigated the boundedness of solutions of certain differential equations of the fourth order. We can mention in this direction, the works of Afuwape and Adesina \cite{a1} where the frequency-domain approach was used. Other articles in this connection include Tiryaki and Tunc \cite{t1}, Tunc \cite{t2,t3,t4}, Tunc and Tiryaki \cite{t5} where the second Lyapunov method was used. All these results generalize in one way or another some results on third order nonlinear differential equations see for instance \cite{a2,a3,a4,a5,c1,r1,s1,s2}. The present work was motivated by a relatively recent paper of Andres \cite{a2}, where the existence of a bounded solution for a third order non-linear differential equation with oscillatory restoring and forcing terms was proved. We shall use the Cauchy formula for the particular solution of non homogeneous linear part of the equation \eqref{e1.1}, to prove that the solution $x(t)$ and its derivatives $x'(t)$, $x''(t)$ and $x'''(t)$ are bounded. \section{Assumptions and Main Result} The basic assumptions on the functions which appear in \eqref{e1.1} are the following: \begin{itemize} \item[(i)] $h$ and $p$ are oscillatory in the following sense: For each argument there exist such numbers $\beta _{1}>\alpha _{1}>u>\beta _{-1}>\alpha _{-1}$ that for $f(\alpha _{1})<0, f(\beta _{1})>0$, $f(\alpha _{-1})<0, f(\beta _{-1})>0$ where $f$ is either $h(x)$ or $p(t)$, $u$ is either $x$ or $t$ and all the roots of the restoring term $h(x)$ are isolated. \item[(ii)] (a) $|h(x)|\leq H$,\quad (b) $|h'(x)| \leq H'$; \item[(iii)] (a) $|g(x')|0$. \end{itemize} The main result of this paper is as follows. \begin{theorem} \label{thm2.1} Assume there exist positive constants $H, H', G, G', P, P', P_{0}, R$ such that for $|x|>R$ and $t>0$ the conditions (ii) and (iii) hold. If in addition, $$\min[d(\overline{x}_{k},\overline{x}_{k+1}), d(\overline{x}_{k}, \overline{ x}_{k-1})] > 2 \frac{G+H+P}{b} \big( \frac{2}{a}+\frac{a}{b}\big) +\frac{P_{0}}{b},$$ where $\overline{x}_{k}$ are roots of $h(x)$, $h'(x_{k})>0$ and $\overline{x}_{k-1}, \overline{x}_{k+1}$ denote the couple adjacent roots of $\overline{x}_{k}(k=0,\pm 2,\pm 4,\dots )$; then all solutions $x(t)$ of equation \eqref{e1.1} are bounded and for each of them there exists a root $\overline{x}$ of $h(x)$ such that $(x(t)-\overline{x})$ oscillates. \end{theorem} \section{Preliminary Results} To prove our main result, we shall need the following result. \begin{lemma} \label{lem3.1} If there exist positive constants, $H, G, P$ such that for all $x\in \Re ^{1}$ and $t\geq 0$ the assumptions (ii)(a), (iii)(a) and (iv)(a) hold, then each solution $x(t)$ of \eqref{e1.1} satisfies the inequalities \begin{gather} {\limsup_{t\to \infty }}|x'(t)|\leq \frac{G}{c}:=D',\label{e3.1a} \\ {\limsup_{t\to \infty }}|x''(t)|\leq \frac{ H+G+P}{b}:=D'',\label{e3.1b} \\ {\limsup_{t\to \infty }}|x'''(t)| \leq \frac{2(H+G+P)}{a}:=D'''. \label{e3.1c} \end{gather} \end{lemma} \begin{proof} Let $z=x''$ then the equation \eqref{e1.1} reduces to $$z''+az'+bz=P(t)-g(x'(t))-h(x).\label{e3.2}$$ Equation \eqref{e3.2} can also be rewritten as $$z''+az'+bz=B,\label{e3.3}$$ where $B=P(t)-g(x'(t))-h(x)$. Thus the general solution of the equation \eqref{e3.3} satisfies $$|x''(t)|=|z(t)|=C_{1}e^{a_{1}t}+C_{2}e^{a_{2}t} + \int_{0}^{t}{\frac{e^{a_{1}\tau}-e^{a_{2}\tau }} {a_{1}-a_{2}}(B)}d\tau,$$ where $a_{1,2}=\big(-a\pm \sqrt{a^{2}-4b}\big)/2$ and constants $C_{1}$ and $C_{2}$ are arbitrary. Hence by the virtue of assumptions (i)-(iv) for $t\geq 0$, we have not only $$\big|\int_{0}^{t}{\frac{e^{a_{1}\tau }-e^{a_{2}\tau }}{a_{1}-a_{2}}[ P(t)-g(x'(t))-h(x)]}d\tau \big| \leq \frac{H+G+P}{b}(1+\frac{a_{2}e^{a_{1}t}-a_{1}e^{a_{2}t}}{a_{1}-a_{2}}),$$ but also $$\limsup_{t\to \infty }|x''(t)|\leq \frac{H+G+P}{b}=:D''.$$ Furthermore on substituting $w=z'$ in \eqref{e3.2}, we have $$w'+aw=P(t)-bx''(t)-g(x'(t))-h(x).$$ Following the same argument used in obtaining the general solution for the equation \eqref{e3.3}, we have $$|x'''(t)|=|w(t)|= Ce^{-at}+\int_{0}^{t}{e^{-a\tau } [P(t)-bx'' (t)-g(x'(t))-h(x)]}d\tau,$$ and by assumptions (i)-(iii) for $t\geq T_{x}$, we have not only \begin{align*} |\int_{0}^{t}{e^{-a\tau }[P(t)-bx''(t)-g(x'(t))-h(x)]}d\tau | &\leq 2\frac{H+G+P}{a}\int_{0}^{t}{e^{-a\tau }} d\tau \\ &\leq 2\frac{H+G+P}{a}(1-e^{-a(t-T_{x})}), \end{align*} but also $$\limsup_{t\to \infty }|x'''(t)|\leq 2\frac{H+G+P}{a}=:D'''.$$ To establish the inequality \eqref{e3.1a}, we use the assumption (iii)(a); i.e., given that $|g(x')|< cx'\leq G$, we have $$|cx'(t)|\leq c|x'(t)|\leq G;$$ i.e., $|x'(t)|\leq G/c$. Hence $$\limsup_{t\to\infty} |x'(t)|\leq \frac{G}{c}:=D'.$$ This completes the proof of the lemma \ref{lem3.1}. \end{proof} \begin{lemma} \label{lem3.2} Under the assumptions of Lemma \ref{lem3.1}. If (ii)(b) and (v)(d) hold for $x\in \Re ^{1}$, then every solution $x(t)$ of \eqref{e1.1} either satisfies the relation $$\lim_{t\to \infty }x(t)=\overline{x}, \quad \lim_{t\to \infty} x'(t) = \lim_{t\to \infty }x''(t)=\lim_{t\to \infty }x'''(t)=0 \quad (h(\overline{x} )=0,)\label{e*}$$ or there exists a root $\overline{x}$ of $h(x)$ such that $(x(t)-\overline{x})$ oscillates. \end{lemma} \begin{proof} Substituting a fixed bounded solution $x(t)$ of \eqref{e1.1} into itself and integrating the result from $T_{x}$ to $t$, we have \begin{aligned} &\int_{T_{x}}^{t}{h(x(\tau )}d\tau \\ &=-\{ b[x'(t)-x'(T_{x})]+a[x''(t)-x''(T_{x})] +\int_{T_{x}}^{t}{g(x'(\tau ))}d\tau \} +\int_{T_{x}}^{t}{P(\tau )}d\tau. \end{aligned}\label{e3.4} By condition (iii)(a), we have that \begin{aligned} &\int_{T_{x}}^{t}{h(x(\tau )}d\tau \\ &<-\{ b[x'(t)-x'(T_{x})]+a[x''(t)-x''(T_{x})]+c[x(t)-x(T_{x})]\} +\int_{T_{x}}^{t}{P(\tau )}d\tau. \end{aligned}\label{e3.5} Define $I(t)\equiv{\int_{T_{x}}^{t}}{P(\tau ))}d\tau$. By the virtue of the above condition, the assertion of the Lemma \ref{lem3.1} and the boundedness of $x(t)$, there exists a constant $M_{x}$ such that for $t\geq T_{x}$, $$|I(t)|\leq M_{x};$$ i.e., $$\big|\int_{T_{x}}^{t}{h(x(\tau ))}d\tau \big|\leq|I(t)| \leq M_{x}.\label{e3.6}$$ Now let us assume that $x(t)$ does not converge to any root $\overline{x}$ of $h(x)$; i.e., $$\limsup_{t\to \infty }|x(t)-\overline{x}|>0 \label{e3.7}$$ and simultaneously, for $t\geq T_{x}$, $$h(x(t))\geq 0 \quad\text{or}\quad h(t)\leq 0.\label{e3.8}$$ Then $$H(t):\equiv \int_{T_{x}}^{t}{h(x(\tau )})d\tau$$ for $t\geq T_{x}$ which is a composed monotone function with a finite or infinite limit for $t\to \infty$. Since \eqref{e3.6} implies that divergent case can be disregarded, it follows from \eqref{e3.7} that not only $$\lim_{t\to \infty }\int_{T_{x}}^{t}{|h(x(\tau ))|}d\tau =\lim_{t\to \infty }|\int_{T_{x}}^{t}{|h(x(\tau ))|}d\tau |\leq M_{x}, \label{e3.9}$$ but also $$\lim_{t\to \infty }|x(t)-\overline{x}| =0\,.\label{e3.10}$$ Otherwise (i.e., if ${ \limsup_{t\to \infty} }|x(t) -\overline{x}|>0$) the inequality \eqref{e3.7} together with the fact that the roots of $h(x)$ are isolated yields $$\liminf_{t\to \infty} |h(x(t))| ={\liminf_{t\to \infty} }|h(x(t))-h(\overline{x})|>0$$ which is a contradiction to \eqref{e3.9}. Thus \eqref{e3.6} and \eqref{e3.8} imply $${\limsup_{t\to \infty }}|h(x(t))| ={\limsup_{t\to \infty }}|h(x(t))-h(\overline{x})|>0 ={\liminf_{t\to \infty}}|h(x(t))|.$$ In what follows, $d(x,y)$ denotes the distance between $x$ and $y$. Consequently, there exists such a sequence ${t_{i}}\geq T_{x}$ and a constant $\widetilde{H}>0$, such that $$\liminf_{t\to \infty \Rightarrow t_{i}\to \infty } d(t_{i},t_{i-1})>0, \quad |h(x(t_{i})|\geq \widetilde{H},$$ and such that $$M_{x}\geq \lim_{t\to \infty }\int_{t_{i-1}}^{t_{i}}{|h(x(\tau ))|}d\tau =\sum_{i=2}^{\infty }{\int_{t_{i-1}}^{t_{i}}{| h(x(\tau ))|}d\tau }$$ implies $$\lim_{t\to \infty \Rightarrow t_{i}\to \infty }{\int_{t_{i-1}}^{t_{i}}{|h(x(\tau ))|} d\tau }=0,$$ or $$H'\limsup_{t\to \infty }|x'(t)|\geq \lim_{t\to \infty }|\frac{dh(x(t))}{dx(t)}x'(t)| =\limsup_{t\to \infty }|\frac{dh(t)}{dt}|=\infty.$$ According to the assertion of the Lemma \ref{lem3.1}, this is impossible and that is why $(x(t)-\overline{x})$ necessarily oscillates. The remaining part of the lemma follows from the assertion $$x(t)\in C^{n}[0, \infty ), \quad \lim_{t\to \infty} |x^{n}(t)|< \infty, \label{e3.11}$$ ${\lim_{t\to \infty }}|x(t)|< \infty$ implies $\lim_{t\to \infty }x^{k}(t)=0$, where $n\geq 2$ is a natural numbers and $k=1,\dots (n-1)$. This completes the proof. \end{proof} \begin{lemma} \label{lem3.3} Under the assumptions of the Lemma \ref{lem3.2}, suppose that (iv)(b) holds for all $t\geq 0$, and ${\limsup_{t\to \infty}}|p(t)|>0$ hold, where $P'$ is a suitable constant, then for every bounded solution $x(t)$ of \eqref{e1.1} there exists a root $\overline{x}$ of $h(x)$ such that $(x(t)-\overline{x})$ oscillates. \end{lemma} \begin{proof} If Lemma \ref{lem3.3} does not hold, then according to Lemma \ref{lem3.2}, equations \eqref{e*} hold and the fifth derivative of $x(t)$ satisfies $$x^{v}(t)=p'(t)-ax^{^{iv}}(t)-bx'''(t)-g'(x'(t))x''(t)-h'(x(t))x'(t).$$ But by the ultimate boundedness of $x'(t), x''(t), x'''(t)$ and $x^{iv}(t)$, there exists a constant $D_{5}$ such that $${\limsup_{t\to \infty}} |x^{v}(t)|\leq D_{5}$$ which according to \eqref{e3.11} gives the relations $${\lim_{t\to \infty}}x(t)=\overline{x}\Rightarrow \lim_{t\to \infty }h(x(t))=h(\overline{x})=0, {\lim_{t\to \infty }}x^{j}(t)=0,$$ $j=1,2,3$, or $${\limsup_{t\to \infty} }|p(t) |=\big| x^{iv}(t)+ax''' (t)+bx''(t)+g(x'(t))+h(x(t))\big|=0,$$ which is a contradiction to $\limsup_{t\to \infty}|p(t)|>0$. \end{proof} \begin{proof}[Proof of Theorem \ref{thm2.1}] Let us assume on the contrary, that $x(t)$ is an unbounded solution of \eqref{e1.1}; i.e., $\limsup_{t\to \infty } x(t)= \infty$. It will follow from Lemma \ref{lem3.1} that there exists a number $T_{0}\geq 0$ large enough such that for $t\geq T_{0}$, \begin{gather*} |x'(t)|\leq D'+\epsilon _{1},\\ |x''(t)|\leq D''+\epsilon _{2}\\ |x'''(t)|\leq D'''+\epsilon_{3} \end{gather*} with $\epsilon _{i}>0$, ($i=1,2,3$) small enough constants. Let $T_{1}\geq T_{0}$ be the last point with $x(T_{1})=\overline{x}_{k}$, ($k$ even) and $T_{2}>T_{1}$ be the first point with $x(T_{2})=\overline{x}_{k+1}$. Integrating \eqref{e1.1} from $T_{1}$ to $t$, $T_{1}\leq t\leq T_{2}$, we have \begin{aligned} &[ x'''(t)-x'''(T_{1})]+a[x''(t)-x''(T_{1})]+b[x'(t)-x' (T_{1})]\\ &+ {\int_{T_{1}}^{t}}{g(x'(\tau ))}d\tau +\int_{T_{1}}^{t}{h(\tau)}d\tau \\ &= \int_{T_{1}}^{t}{P(\tau )}d\tau. \end{aligned} \label{e3.12} However, on replacing ${\int_{T_{1}}^{t}}{g(x'(\tau ))}d\tau$ with $c[x(t)-x(T_{1})]$, for $T_{1}\leq t\leq T_{2}$, we have $h(x(t))sgn x(t)\geq 0$. Multiplying \eqref{e3.12} by $\mathop{\rm sgn}x$, we obtain $$|x(t)|\leq|x(T_{1})|+\frac{1}{c}[D'''+aD''+bD' + P_{0}] + \epsilon,$$ where $c>0$ and $\epsilon >0$ is arbitrary small constant, a contradiction to $x(T_{2})=\overline{x}_{k+1}$ with respect to condition (ii) of Theorem \ref{thm2.1}. The remaining part of the proof follows from the Lemma \ref{lem3.3}. \end{proof} \subsection*{Acknowledgments} The Author wishes to express his profound gratitude to the anonymous referees for his/her valuable suggestions and contribution that saw this work to its present form. \begin{thebibliography}{99} \bibitem{a1} A .U. Afuwape and O. A. Adesina: \textit{Frequency-Domain Approach to Stability and Periodic Solutions of Certain Fourth-Order Nonlinear Differential Equations\/}, Nonlinear Studies Vol. 12 (2005), No. 3, 259-269. \bibitem{a2} J. Andres: \textit{Boundedness of Solutions of the Third Order Differential Equation with Oscillatory Restoring and Forcing Terms\/}, Czech. Math. 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