\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2004(2004), No. 143, pp. 1--8.\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-2004/143\hfil Twin positive solutions] {Twin positive solutions for fourth-order two-point boundary-value problems with sign changing nonlinearities} \author[Y. Tian, W. Ge\hfil EJDE-2004/143\hfilneg] {Yu Tian, Weigao Ge} % in alphabetical order \address{Yu Tian\hfill\break Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China} \email{tianyu2992@163.com} \address{Weigao Ge \hfill\break Department of Applied Mathematics, Beijing Institute of Technology, Beijing 100081, China} \email{gew@bit.edu.cn} \date{} \thanks{Submitted August 28, 2004. Published December 3, 2004.} \thanks{Supported by grant 10371006 from the National Natural Sciences Foundation of China} \subjclass[2000]{34B10, 34B15} \keywords{Fourth-order two-point boundary-value problem; fixed point theorem; \hfill\break\indent double cones; positive solution} \begin{abstract} A new fixed point theorem on double cones is applied to obtain the existence of at least two positive solutions to \begin{gather*} (\Phi_p(y''(t))''-a(t)f(t,y(t),y''(t))=0,\quad 01$. When$p=2$Problem \eqref{e1.1}-\eqref{e1.2} describes the deformations of an elastic beam. The boundary conditions are given according to the control at the ends of the beam. A great deal of research has been devoted to the existence of solutions for the fourth-order two point boundary value problem. We refer the reader to \cite{a1,d1,g2,g3,m1,y1} and their references. Aftabizadeh \cite{a1}, \cite{y1}, del Pino and Manasevich \cite{d1}, Gupta \cite{g2,g3}, Ma and Wang \cite{m1}, Liu \cite{l1} have studied the existence problem of positive solutions of the following fourth-order two-point boundary-value problem \begin{gather*} y^{(4)}(t)-f(t,y(t),y''(t))=0,\quad 00$, let $$K_{r}=\{x\in K:\|x\|a>0 such that \begin{itemize} \item[(C1)] \|Tx\|b for x\in \partial K^{'}(b) \item[(C3)] Tx=T'x, for x\in K'_{a}(b)\cap\{u:T'u=u\} \end{itemize} then T has at least two fixed points y_1 and y_2 in K such that$$ 0\le \|y_1\| 0, t\in[0,t_2), (Ay_1)(0)<0$, which contradicts (1.2). If$t_1>0$. So$y_1(t)=0, (Ay_1)(t_1)=0, (Ay_1)(t)<0$for$t\in(t_1,t_2). (Ay_1)'(t_1)\le 0. \begin{align*} (Ay_1)''(t)&=-\Phi_q(\int_{0}^{1}G(t,s)a(s)f(s,y(s),y''(s))ds)\\ &=-\Phi_q(\int_{0}^{1}G(t,s)a(s)f(s,0,0)ds)\le 0. \end{align*} So(Ay_1)'(t)$is decreasing for$t\in(t_1,t_2)$. So$t_2=1, (Ay_1)(1)<0$, which contradicts boundary condition (1.2). So$y_1$is a solution of \eqref{e1.1}-\eqref{e1.2}. We now show that (C2) of Theorem \ref{thm1.1} is satisfied. For$x\in\partial K^{'}_{a}$, i.e.,$\|x\|=a$, then$0 \Phi_q\Big(\min\big\{f(t,u,v):t\in[0,1],u\in[0,b],v\in[-b,-\delta^{q-1} b]\big\} \\ &\quad \times \min_{t\in[\delta,1-\delta]} \int_{\delta}^{1-\delta}G(t,s)a(s)ds\Big)\\ &= b>\delta^{q-1} b. \end{align*} Finally we show that $(C_3)$ of Theorem \ref{thm1.1} is also satisfied. Let $x\in K^{'}_{a}(\delta^{q-1} b)\cap\{u: T'u=u\}$, then $\|x\|<\frac{1}{\delta^{q-1}}\alpha(x)$. From $\alpha(x)\le\|x\|\le \frac{1}{\delta^{q-1}}\alpha(x)$, we have $$\min_{t\in[\delta, 1-\delta]}\{-x''(t)\}=\alpha(x)\ge \delta^{q-1}\|x\|>\delta^{q-1} a>d\,.$$ So $-x''\in[d, b]$. From (H3), we have $f(t,u,v)=f^+(t,u,v)$, which implies $Ty=T'y$. Therefore, there exist two positive solutions $y_1,y_2$ satisfying \eqref{e2.2}. \end{proof} \noindent{\bf Remark.} When $p=2, a(t)\equiv 1, f(t,u,v)>0, \delta=1/4$, Theorem \ref{thm2.4} reduces to \cite[Theorem 3.1]{l1}. But our result shows at least two positive solutions, whereas there is at least one positive solution in B. Liu \cite[Theorem 3.1]{l1}. \begin{theorem} \label{thm2.5} Suppose (H1),(H2) hold. Also assume that \begin{itemize} \item[(H6)] $\delta_i\in(0,1/2)$, $i=1,2,\dots, n$, $0<\int_{\delta_i}^{1-\delta_i}a(s)ds<\infty$ \item[(H7)] There exists constants $a_i, b_i, d_i>0$, $i=1,2,\dots, n$, where $00$, $d$, $i=1,2,\dots, n$, where $0\sqrt{\frac{1}{24}}\,. $$For (t, u, v)\in[0, 1]\times[0, \pi/2]\times[-\pi/2, -\pi/36], we have f(t, u,v)=\frac{u+\pi/6}{6}(\sqrt{3}\cos(v+\frac{5}{12}\pi))^{19}+\frac{t}{10}>0. So (H3) holds. For (t, u, v)\in[0, 1]\times[0, \pi/12]\times[-\pi/12, 0], f(t, u,v)=\frac{u+\pi/6}{6}\big(\sqrt{3}\cos(v+\frac{5}{12}\pi)\big)^{19} +\frac{t}{10}<\frac{\pi}{24}\times(\frac{\sqrt{3}}{2})^{19}+\frac{1}{10} <0.6<(\frac{\pi}{12}\times 3^{5/4})^{2}=\Phi_{3}(a/Q). So (H4) holds. For (t, u, v)\in[0, 1]\times[0, \pi/2]\times[-\pi/2, -\pi/4], f(t, u, v)=\frac{u+\pi/6}{6}\big(\sqrt{3}\cos(v+\frac{5}{12}\pi)\big)^{19} +\frac{t}{10}>(\pi\sqrt{6})^2>\Phi_{3}(b/m). So (H5) holds. Thus by Theorem \ref{thm2.4}, this boundary-value problem has at least two positive solutions y_1, y_2 such that$$ 0<\|y_1\|<\frac{\pi}{12}<\|y_2\|,\quad \alpha(y_1)<\frac{\pi}{16},\quad \|y_1\|_{\infty}<\frac{\pi}{12},\quad \|y_2\|_{\infty}<\frac{\pi}{2}.$\$ \begin{thebibliography}{00} \bibitem{a1} A. R. Aftabizadeh; \emph{Existence and uniqueness theorems for fourth-order boundary problems}, J. Math. Anal. Appl. 116(1986), 415-426. \bibitem{y1} Yisong Yang; \emph{Fourth-order two-point boundary value problem}, Proc. 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