\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2011 (2011), No. 130, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2011 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2011/130\hfil Fourth-order periodic boundary value problems] {Periodic boundary-value problems for fourth-order differential equations with delay} \author[S. A. Iyase\hfil EJDE-2011/130\hfilneg] {Samuel A. Iyase} \address{Samuel A. Iyase \newline Department of Mathematics, Computer Science and Information Technology, Igbinedion University, Okada, P.M.B. 0006, Benin City, Edo State, Nigeria} \email{driyase2011@yahoo.com, iyasesam@gmail.com} \thanks{Submitted June 3, 2011. Published October 11, 2011.} \subjclass[2000]{34B15} \keywords{Periodic solution; uniqueness, uniqueness; \hfill\break\indent Carathoeodory conditions; fourth order ODE; delay} \begin{abstract} We study the periodic boundary-value problem \begin{gather*} x^{(iv)}(t)+f(\ddot{x})\dddot{x}(t)+b\ddot{x}(t) +g(t,\dot{x}(t-\tau))+dx=p(t)\\ x(0)=x(2\pi),\quad \dot{x}(0)=\dot{x}(2\pi),\quad \ddot{x}(0)=\ddot{x}(2\pi),\quad \dddot{x}(0)=\dddot{x}(2\pi), \end{gather*} Under some resonant conditions on the asymptotic behaviour of the ratio $g(t,y)/(by)$ for $|y|\to\infty$. Uniqueness of periodic solutions is also examined. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \section{Introduction} In this article we study the periodic boundary-value problem $$\label{1.1} \begin{gathered} x^{(iv)}(t)+f(\ddot{x})\dddot{x}(t)+b\ddot{x} +g(t,\dot{x}(t-\tau))+dx=p(t)\\ x(0)=x(2\pi),\quad \dot{x}(0)=\dot{x}(2\pi),\quad \ddot{x}(0)=\ddot{x}(2\pi),\quad \dddot{x}(0)=\dddot{x}(2\pi), \end{gathered}$$ with fixed delay $\tau\in[0,2\pi)$, $f:\mathbb{R}\to\mathbb{R}$ is a continuous function, $P:[0,2\pi]\to\mathbb{R}$ and $g:[0,2\pi]\times\mathbb{R}\to\mathbb{R}$ are $2\pi-$periodic in $t$ and $g$ satisfies Caratheodory conditions with $b$ and $d$ real constants. The unknown function $x:[0,2\pi]\to\mathbb{R}$ is defined for $00$ and \label{2.2} 0<\frac{c}{b}0$and$\Gamma(t)=b^{-1}c(t)\in L^2_{2\pi}$. Suppose that $$\label{2.4} 0 < \Gamma(t)< 1\,.$$ Then \eqref{2.3} has no non-trivial periodic solution for every fixed$\tau\in[0,2\pi)$. \end{theorem} \begin{proof} Let$x(t)be any solution of \eqref{2.3}. Then \begin{align*} 0&=\frac{1}{2\pi}\int^{2\pi}_0\ddot{\tilde{x}}(t) \Big[\frac{b^{-1}}{2\pi}\Big\{x^{(iv)}+a\dddot{x}+dx+\{\ddot{x} +\Gamma(t)\dot{x}(t-\tau )\}\Big\}\Big]dt\\ &=-\frac{b^{-1}}{2\pi}\int^{2\pi}_0\ddot{\tilde{x}}^2(t)dt -\frac{db^{-1}}{2\pi}\int^{2\pi}_0\dot{\tilde{x}}^2(t)dt +\frac{1}{2\pi}\int^{2\pi}_0\ddot{\tilde{x}}(t)[\dddot{x}(t) +\Gamma(t)\dot{x}(t-\tau)]dt\\&\geq \frac{1}{2\pi} \int^{2\pi}_0\ddot{\tilde{x}}(t)[\ddot{x}(t) +\Gamma(t)\dot{x}(t-\tau)]dt\\ &=\int^{2\pi}_0[\ddot{\tilde{x}}^2(t) +\Gamma(t)\ddot{\tilde{x}}(t)\dot{x}(t-\tau)]dt\\ &=\frac{1}{2\pi}\int^{2\pi}_0\Big[\ddot{\tilde{x}}^2(t) -\frac{\Gamma(t)}{2}\ddot{\tilde{x}}^2(t) -\frac{\Gamma(t)}{2}\dot{\tilde{x}}^2(t-\tau)\Big]dt\\ &\quad +\frac{1}{2\pi}\int^{2\pi}_0\frac{\Gamma(t)}{2} \Big[\ddot{\tilde{x}}(t)+\dot{x}(t-\tau)\Big]^2dt. \end{align*} In the above expression we used the equality $$ab=\big(\frac{a+b}{2}\big)^2-\frac{a^2}{2}-\frac{b^2}{2}.$$ From the periodicity of\dot{x}(t), it follows that $$\frac{1}{2\pi}\int^{2\pi}_0\ddot{\tilde{x}}^2(t)dt =\frac{1}{2\pi}\int^{2\pi}_0\ddot{\tilde{x}}^2(t-\tau)dt.$$ Hence, \begin{align*} 0&\geq\frac{1}{2}\Big[\frac{1}{2\pi}\int^{2\pi}_0[\ddot{\tilde{x}}^2(t) -\Gamma(t)\ddot{\tilde{x}}^2(t)]dt\Big]\\ &= \frac{1}{2}\Big[\frac{1}{2\pi}\int^{2\pi}_0[\ddot{\tilde{x}}^2 (t-\tau)-\Gamma(t)\dot{\tilde{x}}^2(t-\tau)]dt\\ &\geq\delta|\dot{\tilde{x}}|^2_{H^{1}_{2\pi}} =\delta|\dot{x}|_{H^{1}_{2\pi}}. \end{align*} By \cite[Lemma 1]{a4} where\delta>0$is a constant. This implies that$x$is constant a.e. But since$d\neq 0$we must have$x =0$, a. e. \end{proof} \section{The non-linear problem} We shall consider here a preliminary Lemma which will enable us obtain a priori estimates required for our results. \begin{lemma}\label{lem:3.1} Let all the conditions of Lemma \ref{lem:2.1} hold and let$\delta$be related to$\Gamma(t)$by Theorem \ref{thm:2.1}. Suppose that$v\in L^2_{2\pi}and $$00. Then$$ \frac{1}{2\pi}\int^{2\pi}_0\ddot{\tilde{x}}(t)\Big[b^{-1}\{x^{(iv)} +a\dddot{x}+dx\}+\ddot{x}+\Gamma(t)\dot{x}(t-\tau)\Big]dt \geq(\delta-\epsilon)|\dot{x}|^2_{H^{1}_{2\pi}}. \end{lemma} \begin{proof} From the proof of Theorem \ref{thm:2.1}, we have \begin{align*} &\frac{1}{2\pi}\int^{2\pi}_0\ddot{\tilde{x}}(t) \Big[b^{-1}\{x^{(iv)}+a\dddot{x}+dx\}+\ddot{x}+v(t)\dot{x}(t-\tau) \Big]dt\\ &\geq\frac{1}{2}\Big[\frac{1}{2\pi}\int^{2\pi}_0[\ddot{\tilde{x}}^2(t) -\Gamma(t)\ddot{\tilde{x}}^2(t)]dt\Big] +\frac{1}{2}\Big[\frac{1}{2\pi}\int^{2\pi}_0[\ddot{\tilde{x}}^2 (t-\tau)-\Gamma(t)\dot{\tilde{x}}^2(t-\tau)]dt\Big]\\ &\quad -\epsilon\frac{1}{2\pi}\int^{2\pi}_0(\dot{\tilde{x}}^2 (t-\tau)+\ddot{\tilde{x}}^2(t))dt\\ & \geq\frac{1}{2}\Big[\frac{1}{2\pi}\int^{2\pi}_0[\ddot{\tilde{x}}^2 (t-\tau)-\Gamma(t)\dot{\tilde{x}}^2 (t-\tau)]dt\Big] -\frac{\epsilon}{2\pi}\int^{2\pi}_0\dot{x}^2 (t-\tau)\\ &\quad -\frac{\epsilon}{2\pi}\int^{2\pi}_0\ddot{\tilde{x}}^2(t-\tau)dt\\ &\geq\delta|\dot{\tilde{x}}|^2_{H^{1}_{2\pi}} -\epsilon|\ddot{\tilde{x}}|^2_{H^{1}_{2\pi}}\\ &\geq(\delta-\epsilon)|\dot{\tilde{x}}|^2_{H^{1}_{2\pi}}. \end{align*} \end{proof} We shall consider the non-linear delay equation $$\label{3.1} x^{(iv)}+f(\ddot{x})\dddot{x}+b\ddot{x}+g(t,\dot{x}(t-\tau))+dx=p(t)$$ where f:\mathbb{R}\to\mathbb{R} is a continuous function and g: [0 ,2\pi]\times \mathbb{R}\to\mathbb{R} are 2\pi periodic in t and g satisfies Caratheodory condition; that is, g(\cdot, x) is measurable on [0, 2\pi] for each x\in\mathbb{R} and g(t,\cdot) is continuous on \mathbb{R} for almost each t\in [0, 2\pi]. We assume moreover that for r > 0 there exists Y_{r}\in L^2_{2\pi} such that |g(t,y)|\leq Y_{r}(t) for a.e. t\in[0, 2\pi] and x\in [-r, r]. \begin{theorem}\label{thm:3.1} Let b<0 and d>0. Suppose that g is Caratheodory function satisfying the inequality \begin{gather}\label{3.2} g(t,y)\geq 0,\quad |y|\leq r \\ \label{3.3} \lim_{|y|\to\infty}\sup\frac{g(t,y)}{by} \leq\Gamma (t) \end{gather} uniformly a.e., t\in[0,2\pi] where r>0 is a constant and \Gamma(t)\in L^2_{2\pi} is such that $$\label{3.4} 0<\Gamma(t)<1$$ Then for arbitrary continuous function f, the boundary-value problem \eqref{3.1} has at least one 2\pi-periodic solution. \end{theorem} \begin{proof} Let \delta>0 be associated to the function \Gamma by Theorem \ref{thm:2.1}. Then by \eqref{3.2}, \eqref{3.3} there exists a constant R_1 > 0 such that $$\label{3.5} 0\leq\frac{g(t,y)}{by}<\Gamma(t)+\frac{\delta}{2}$$ if |y|\geq R_1 for a. e., t\in[0,2\pi] and all y\in\mathbb{R}. Define \bar{Y}:[0,2\pi]\times\mathbb{R}\to\mathbb{R} by \label{3.6} \overline{Y}=\begin{cases} y^{-1}g(t,y), &|y|\geq R_1\\ R^{-1}g(t,R), & 00. Hence, $$\label{3.11} |\dot{x}|_{H^{1}_{2\pi}}\leq\frac{2\beta}{\delta}=c_1,$$ with c_1>0. This implies \begin{gather} \label{3.12} |\ddot{x}|_2\leq c_2 \\ \label{3.13} |\ddot{x}|_{\infty}\leq c_3 \end{gather} where c_2>0 and c_3>0. Using Wirtinger's inequality in \eqref{3.12}, we obtain $$\label{3.14} |\dot{x}|_2\leq c_4$$ with c_4>0. Multiplying \eqref{3.9} by -\ddot{x}(t) and integrating over [0,2\pi], we obtain |\dddot{x}|^2_2\leq|\ddot{x}|^2_2|1+\frac{\delta}{2}|\ddot{x}|_2 +|\alpha|_2+d|\dot{x}|_2+|p|_2|\ddot{x}|_2 $$Applying Wirtingers inequality we obtain $$\label{3.15} |\dddot{x}|^2_2\leq c_5$$ with c_5>0 and hence$$ |\ddot{x}|_{\infty}\leq c_6 with c_6>0. We multiply \eqref{3.9} by x^{(iv)}(t) and integrate over [0,2\pi] to get \begin{align*} -b^{-1}|x^{(iv)}|^2_2 &\leq |f(\ddot{x})|_{\infty}|\ddot{x}|_2|x^{(iv)}|_2|b^{-1}| +|\ddot{x}|_2|x^{(iv)}|_2+|1+\frac{\delta}{2}||\dot{x}|_2|x^{i(iv)}|_2\\ &\quad +|b^{-1}||d||\ddot{x}|_2+|\alpha|_2|x^{(iv)}|_2 +|p|_2|x^{i(iv)}|_2\\ &\leq|f(\ddot{x})|_{\infty}|\ddot{x}|_2|x^{(iv)}|_2|b^{-1}| +|\ddot{x}|_2|x^{(iv)}|_2\\ &\quad +|1+\frac{\delta}{2}||\dot{x}|_2|x^{(iv)}|_2|b^{-1}|d|x^{(iv)}|_2 +|\alpha|_2|x^{(iv)}|_2+|p|_2|x^{i(iv)}|_2|b^{-1}|, \end{align*} where we used the Wirtinger's inequality. Thus $$\label{3.16} |x^{(iv)}|_2\leq c_7$$ with c_7>0. Finally multiplying \eqref{3.9} by x(t) and integrating over [0,2\pi] we obtain $$\label{3.17} |x|_2\leq c_8$$ with c_8>0. Hence, |x|_{W^{4,2}_{2\pi}}=|x|_2+|\dot{x}|_2+|\ddot{x}|_2 +|\dddot{x}|_2+|x^{(iv)}|_2\leq c_8+c_4+c_2+c_5+c_7=C_{9} $$Taking R>C_{9}>0, the required a priori bound in W^{4,2}_{2\pi} is obtained independently of x and \lambda. \end{proof} \section{Uniqueness Result} For f(x) = a, a constant, in \eqref{1.1}, we have the following uniqueness result. \begin{theorem}\label{thm:4.1} Let a, b, d be constants with b <0 and d >0. Suppose g is a Caratheodory function satisfying $$\label{4.1} 0<\frac{g(t,\dot{x}_1)-g(t,\dot{x}_2)}{b(\dot{x}_1-\dot{x}_2)} <\Gamma(t)$$ for all x_1,x_2\in\mathbb{R} with x_1\neq x_2 where \Gamma(t)\in L^2_{2\pi} is such that 0<\Gamma(t)<1. Then for all arbitrary constant a and every \tau\in[0,2\pi) the boundary-value problem $$\label{4.2} \begin{gathered} x^{(iv)}(t)+a\dddot{x}+b\ddot{x}+g(t,\dot{x}(t-\tau))+dx=p(t)\\ x(0)=x(2\pi),\dot{x}(0)=\dot{x}(2\pi),\ddot{x}(0)=\ddot{x}(2\pi), \dddot{x}(0)=\dddot{x}(2\pi), \end{gathered}$$ has at most one solution. \end{theorem} \begin{proof} Let x_1,x_2 be any two solutions of \eqref{4.2}. Set x =x_1-x_2. Then x satisfies the boundary value problem \begin{gather*} b^{-1}x^{(iv)}(t)+a\dddot{x}+\Gamma(t)\dot{x}(t-\tau)+b^{-1}dx=0\\ x(0)=x(2\pi),\quad \dot{x}(0)=\dot{x}(2\pi),\quad \ddot{x}(0)=\ddot{x}(2\pi),\quad \dddot{x}(0)=\dddot{x}(2\pi), \end{gather*} where the function \Gamma(t)\in L^2_{2\pi} is defined by$$ \Gamma(t)=\begin{cases} \frac{g(t,\dot{x}_1(t-\tau))-g(t,\dot{x}_2(t-\tau))}{\dot{x}(t)} &\text{if }\ddot{x}(t)\neq 0\\ \frac{1}{2} &\text{if }\ddot{x}(t)=0 \end{cases}$if$\dot{x}(t)$on every subset of$[0, 2\pi]$of positive measure, then$x$is constant Since$d\neq 0$we must have$x =0$and hence$x_1 = x_2$a.e. Suppose on the other hand that$\dot{x}(t)\neq 0$on a certain subset of$[0, 2\pi]$of positive measure, then using the arguments of Theorem \ref{thm:2.1} we obtain that$x =0$and hence$x_1 = x_2\$ a .e. \end{proof} \begin{thebibliography}{99} \bibitem{a1} R. Gaines and J. Mawhim; Coincidence degree and non-linear differential equations. \emph{Lecture Notes in Math} No. 568, Springer Verlag Berlin, (1977). \bibitem{a2} S. A. Iyase; Non-resonant oscillations for some fourth-order differential equations with delay. \emph{Mathematical Proceedings of the Royal Irish Academy,} Vol. 99A, No.1, (1999), 113- 121. \bibitem{a3} S. A. Iyase and P. O. K. Aiyelo; Resonant oscillation of certain fourth order non linear differential equations with delay, \emph{International Journal of Mathematics and Computation}, Vol.3, No. Jo9, June (2009), 67 - 75. \bibitem{a4} J. Mawhin and J. R. Ward; Periodic solution of some forced Lienard differential equations at resonance. \emph{Arch. Math.}, vol. 41, 337 -351 (1983). \bibitem{a5} Oguztoreli and Stein; An analysis of oscillation in neuromuscular systems. \emph{Journal of Mathematical Biology 2.} (1975), 87 -105. \bibitem{a6} E. De Pascale and R. Iannaci; Periodic solution of a Generalised Lienard equation with delay. Proceedings of the International Conference (Equadiff), Wurzburg (1982) Lecture Note Math No. 1017. Springer Verlag Berlin, (1983). \end{thebibliography} \end{document}