\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2009(2009), No. 98, 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} \thanks{\copyright 2009 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2009/98\hfil Existence of positive solutions] {Existence of positive solutions for a fourth-order multi-point beam problem on measure chains} \author[D. R. Anderson, F. Minh\'{o}s \hfil EJDE-2009/98\hfilneg] {Douglas R. Anderson, Feliz Minh\'{o}s} % in alphabetical order \address{Douglas R. Anderson \newline Department of Mathematics, Concordia College, Moorhead, MN 56562 USA} \email{andersod@cord.edu} \address{Feliz Minh\'{o}s \newline Department of Mathematics, University of \'{E}vora, Portugal} \email{fminhos@uevora.pt} \thanks{Submitted February 6, 2009. Published August 11, 2009.} \subjclass[2000]{34B15, 39A10} \keywords{Measure chains; boundary value problems; Green's function; \hfill\break\indent fixed point; fourth order; cantilever beam} \begin{abstract} This article concerns the fourth-order multi-point beam problem \begin{gather*} (EIW^{\Delta \nabla }) ^{\nabla \Delta }(x)=m(x)f(x,W(x)),\quad x\in [x_{1},x_{n}]_{\mathbb{X}} \\ W(\rho ^2(x_{1}))=\sum_{i=2}^{n-1}a_iW(x_i),\quad W^{\Delta}(\rho ^2(x_{1}))=0, \\ (EIW^{\Delta \nabla }) (\sigma (x_{n}))=0,\quad (EIW^{\Delta \nabla })^{\nabla }(\sigma(x_{n})) =\sum_{i=2}^{n-1}b_i(EIW^{\Delta \nabla })^{\nabla}(x_i). \end{gather*} Under various assumptions on the functions $f$ and $m$ and the coefficients $a_i$ and $b_i$ we establish the existence of one or two positive solutions for this measure chain boundary value problem using the Green's function approach. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \section{Introduction} The aim of this work is to obtain sufficient conditions for the existence of positive solutions of the measure chain fourth-order multi-point boundary value problem composed by the equation $$(EIW^{\Delta \nabla })^{\nabla \Delta }(x)=m(x)f(x,W(x))\quad \text{for all } x\in [ x_{1},x_{n}]_{\mathbb{X}} \label{b1}$$ and the multi-point boundary conditions $$\begin{gathered} W(\rho ^2(x_{1}))=\sum_{i=2}^{n-1}a_iW(x_i),\quad W^{\Delta }(\rho ^2(x_{1}))=0, \\ (EIW^{\Delta \nabla })(\sigma (x_{n}))=0,\quad (EIW^{\Delta\nabla})^{\nabla }(\sigma (x_{n})) =\sum_{i=2}^{n-1}b_i(EIW^{\Delta \nabla })^{\nabla }(x_i), \end{gathered} \label{b2}$$ on a measure chain $\mathbb{X}$, $n\geq 4$. The boundary points satisfy $x_{1}\in \mathbb{X}_{\kappa ^2}$ and $x_{n}\in \mathbb{X}^{\kappa ^2}$ with $\rho ^2(x_{1})0$ is constant. The mass function $m:[\rho (x_{1}),\sigma (x_{n})]_{\mathbb{X}}\to [ 0,\infty )$ is right-dense continuous, not identically zero on $[x_{2},x_{3}]_{\mathbb{X}}$ and the non-negative coefficients $a_i$ and $b_i$ satisfy the non-resonant conditions $\sum_{i=2}^{n-1}a_i<1$ and $\sum_{i=2}^{n-1}b_i<1$. Physically, the motivation for this fourth-order problem is a nonuniform cantilever beam of length $L$ in transverse vibration such that the left end is clamped and the right end is free with vanishing bending moment and shearing force. Let $E$ be the modulus of elasticity, $I(x)$ the area moment of inertia about the neutral axis and $m(x)$ the mass per unit length of the beam. After separation of variables, the space-variable problem is formulated as $$\label{original} \begin{gathered} (EI(x)W''(x))'' =m(x)W(x),\quad \mbox{for all }x\in [ 0,L], \\ W(0) = W'(0)=(EIW'')(L)=( EIW'')'(L)=0\,; \end{gathered}$$ see Meirovitch \cite{meir1, meir2}. Throughout this work we assume a working knowledge of measure chains (time scales) and measure chain notation, where any arbitrary nonempty closed subset of $\mathbb{R}$ can serve as a measure chain $\mathbb{X}$. See Hilger \cite{hilger} for an introduction to measure chains; other excellent sources on delta dynamic equations include \cite{albohner1,albohner2}, and for nabla dynamic equations, see \cite{atici}. For more on beam and other fourth-order continuous problems we refer to the recent papers \cite{anderson, ek, pang, yang, yao}, and for functional boundary value problems see \cite{ACFM, ACRPFM}. Related to fourth-order dynamic equations, see \cite{agh, ah, iyk, wang}. However, as far as we know, this is the first time where multi-point boundary conditions as in (\ref{b2})\ are considered in fourth order nonlinear problems on time scales. The second section contains some preliminary lemmas needed to evaluate explicitly the unique solution $W$ of a related fourth-order equation, by a Green's function approach, and to prove some properties of $W$. Section three provides some sufficient conditions on the nonlinearity to obtain the existence and the multiplicity of positive solutions, via index theory in cones. Two examples are referred in the last section, to illustrate the existence of multiple positive solutions. \section{Foundational lemmas} For the related fourth-order multi-point boundary value problem composed by the equation $$(EIW^{\Delta \nabla })^{\nabla \Delta }(x)=y(x),\quad x\in [x_{1},x_{n}]_{\mathbb{X}}, \label{b3}$$ with $y:[x_{1},x_{n}]_{\mathbb{X}}\to \mathbb{R}$ right-dense continuous, and boundary conditions \eqref{b2}, it is referred \cite[Theorem 7.1]{agh}, where the Green's function $G(x,s)$ for the corresponding homogeneous equation $$(EIW^{\Delta \nabla })^{\nabla \Delta }(x)=0 \label{related}$$ satisfying boundary conditions $$\begin{gathered} W(\rho ^2(x_{1}))=W^{\Delta }(\rho ^2(x_{1}))=0,\\ (EIW^{\Delta \nabla })(\sigma (x_{n}))=(EIW^{\Delta \nabla })^{\nabla }(\sigma (x_{n}))=0, \end{gathered} \label{relatedbc}$$ is given, for $(x,s)\in [ \rho ^2(x_{1}),\sigma ^2(x_{n})]_{ \mathbb{X}} \times [ \rho (x_{1}),\sigma (x_{n})]_{\mathbb{X}}$, by $$G(x,s)= \begin{cases} \int_{\rho ^2(x_{1})}^{s}\Big(\int_{\rho ^2(x_{1})}^{\zeta } \frac{x-\xi } {EI(\xi )}\nabla \xi \Big) \Delta \zeta & s\in [ \rho (x_{1}),x] _{\mathbb{X}},\; x\leq \sigma ^2(x_{n}), \$3pt] \int_{\rho ^2(x_{1})}^{x}\Big(\int_{\rho ^2(x_{1})}^{\zeta } \frac{s-\xi }{EI(\xi )}\nabla \xi \Big) \Delta \zeta & s\in [ x,\sigma (x_{n})]_{\mathbb{X}},\; x\geq \rho ^2(x_{1}). \end{cases} \label{greenf}$$ \begin{example} \label{Examp 21} \rm Consider the Green's function \eqref{greenf} for \rho ^2(x_{1})=0 and \sigma ^2(x_{n})=1, with EI(x)\equiv 1. Then we have the following continuous and discrete illustrations: \begin{gather*} \mathbb{X}=\mathbb{R}:\quad G(x,s)= \begin{cases} {\frac{s^2(3x-s)}{6}} & s\in [ 0,x],\; x\in [0,1], \\[3pt] {\frac{x^2(3s-x)}{6}} & s\in [ x,1],\; x\in [ 0,1], \end{cases} \\ \mathbb{X}=h\mathbb{Z}:\quad G(x,s)= \begin{cases} {\frac{s(s-h)(3x-s-h)}{6}} & s\in [ h,x]_{h\mathbb{Z} },\;x\leq 1, \\[3pt] {\frac{x(x-h)(3s-x-h)}{6}} & s\in [ x,1-h]_{h\mathbb{Z} },\;x\geq 0, \end{cases} \end{gather*} where for 00. \label{h3} Let \mathcal{B} denote the Banach space C[\rho^2(x_1),\sigma^2(x_n)]_{\mathbb{X}} with the norm \[ \|W\|=\sup_{x\in [ \rho ^2(x_{1}),\sigma ^2(x_{n})]_{ \mathbb{X}}}|W(x)|.$ Define the cone $\mathcal{P}\subset \mathcal{B}$ by $$\mathcal{P}=\big\{W\in \mathcal{B}:W(x)\geq 0\text{ on } [\rho^2(x_{1}), \sigma ^2(x_{n})]_{\mathbb{X}},\; W(x)\geq \gamma \|W\|\text{ on } [x_{2},x_{3}]_{\mathbb{X}}\big\}, \label{cone}$$ where $\gamma$ is given in \eqref{gamma}. Since $W$ is a solution of \eqref{b1}, \eqref{b2} if and only if it satisfies equation \eqref{form} replacing in this case $y(s)$ by $m(s)f(s,W(s))$, define for $W\in \mathcal{P }$ the operator $\mathcal{L}:\mathcal{P}\to \mathcal{B}$ by \begin{aligned} \mathcal{L}W(x) &= \int_{\rho (x_{1})}^{\sigma (x_{n})}G(x,s)m(s)f(s,W(s))\Delta s+A(mf(\cdot ,W)) +\Big(1-\sum_{i=2}^{n-1}{b_i}\Big)^{-1} \\ &\quad\times \Big(\sum_{i=2}^{n-1}b_i\int_{x_i}^{\sigma (x_{n})}m(s)f(s,W(s))\Delta s\Big)\int_{\rho ^2(x_{1})}^{x}\int_{\rho ^2(x_{1})}^{\zeta }\frac{\sigma (x_{n})-\xi }{EI(\xi )}\nabla \xi \Delta \zeta . \end{aligned} \label{Lop} By Lemmas \ref{lemma23} and \ref{lemma24}, $\mathcal{L}:\mathcal{P} \to \mathcal{P}$. Moreover, $\mathcal{L}$ is completely continuous by a typical application of the Ascoli-Arzela Theorem. \begin{lemma}[\cite{gu,lan}]\label{index} Let $P$ be a cone in a Banach space $S$ and $B$ an open, bounded subset of $S$ with $B_{P}:=B\cap P\neq\emptyset$ and $\overline{B}_{P}\neq P$. Assume that $L:\overline{B}_{P}\to P$ is a compact map such that $y\neq Ly$ for $y\in \partial B_{P}$, and the following results hold: \begin{itemize} \item[(i)] If $\|Ly\|\le\|y\|$ for $y\in\partial B_P$, then $i_P(L,B_P)=1$. \item[(ii)] If there exists an $\eta\in P\backslash\{0\}$ such that $y\neq Ly+\lambda\eta$ for all $y\in\partial B_P$ and all $\lambda>0$, then $i_P(L,B_P)=0$. \item[(iii)] Let $U$ be open in $P$ such that $\overline{U}_P\subset B_P$. If $i_P(L,B_P)=1$ and $i_P(L,U_P)=0$, then $L$ has a fixed point in $B_P\backslash\overline{U}_P$; the same is true if $i_P(L,B_P)=0$ and $i_P(L,U_P)=1$. \end{itemize} \end{lemma} For the cone $\mathcal{P}$ given in \eqref{cone} and any positive real number $r$, define the convex set P_{r}:=\{W\in \mathcal{P}:\|W\|\ M\gamma is satisfied, then i_P(\mathcal{ L},\Omega_r)=0. \end{lemma} \begin{proof} Let \eta (x)\equiv 1 for x\in [ \rho ^2(x_{1}),\sigma ^2(x_{n})]_{\mathbb{ X}}, so that \eta \in \partial P_{1}. Suppose there exist W_{\ast }\in \partial \Omega _{r} and \lambda _{\ast }\geq 0 such that W_{\ast }= \mathcal{L}W_{\ast }+\lambda _{\ast }\eta . Then for x\in [ x_{2},x_{3}]_{ \mathbb{X}}, \begin{align*} W_{\ast }(x) &= (\mathcal{L}W_{\ast })(x)+\lambda _{\ast }\eta (x) \\ &\geq \int_{x_{2}}^{x_{3}}G(x,s)m(s)f(s,W_{\ast }(s))\Delta s+\lambda_{\ast } \\ &> M\gamma r\int_{x_{2}}^{x_{3}}G(x_{2},s)m(s)\Delta s+\lambda _{\ast }=\gamma r+\lambda _{\ast }, \end{align*} with \gamma  given in \eqref{gamma}, and, by Lemma \ref{lemma32} (iv), this contradiction is obtained: \gamma r>\gamma r+\lambda _{\ast }. Consequently, W_{\ast }\neq \mathcal{L}W_{\ast }+\lambda _{\ast }\eta  for W_{\ast }\in \partial \Omega _{r} and \lambda _{\ast }\geq 0, so, by Lemma \ref{index} (ii), i_{P}(\mathcal{L},\Omega _{r})=0. \end{proof} \begin{theorem}\label{Thm35} Let \gamma , K and M be as given in \eqref{gamma}, \eqref{Kdef} and \eqref{Mdef}, respectively. Assume that one of the following assumptions holds:\newline there exist constants c_{1},c_{2},c_{3}\in \mathbb{R} with 0M\gamma \end{itemize} or there exist constants c_{1},c_{2},c_{3}\in \mathbb{R} with 0M\gamma; \item[(H2')] M0. Consider a particular case of equation (\ref{b1}) given by $$(EIW^{\Delta \nabla })^{\nabla \Delta }(x)=m(x)f(W)\quad \text{for all}\quad x\in [ x_{1},x_{n}]_{\mathbb{X}}, \label{EqEx1}$$ where \[ f(W)=% \begin{cases} \frac{1}{K}W & \text{if } W\in [ 0,c_{1}], \\[3pt] \frac{M\gamma c_{2}+\delta -\frac{c_{1}}{K}}{\gamma c_{2}-c_{1}}(W-c_{1})+ \frac{c_{1}}{K} & \text{if } W\in [ c_{1},\gamma c_{2}], \\[3pt] \frac{\frac{c_{3}}{K}-M\gamma c_{2}-\delta }{c_{3}-\gamma c_{2}}(W-c_{3})+ \frac{c_{3}}{K} & \text{if } W\geq \gamma c_{2}.% \end{cases} As $f$ satisfies assumption (H1), by Theorem \ref{Thm35}, problem \eqref{EqEx1}, \eqref{b2} has two positive solutions. \smallskip For the second example consider, on the time scale $\mathbb{X} =[0,1]$, the boundary value problem composed by the equation $$W^{(4)}(x)=x\left(\frac{x}{5} +(W(x))^2\right),\quad \text{for } x\in \mathbb{X}, \label{EqEx}$$ with the boundary conditions $$\label{BCEx} \begin{gathered} W(0) = 0.2 W\Big(\frac{1}{3}\Big)+0.5 W\Big(\frac{2}{3}\Big), \\ W'(0) = 0,\quad W'' (1)=0, \\ W'''(1) = 0.1 W'''\Big(\frac{1}{3}\Big)+0.3 W'''\Big(\frac{2}{3}\Big). \end{gathered}$$ In fact this is a particular case of the initial problem (\ref{b1}), \eqref{b2}, with $EI(x)\equiv 1$, $m(x)=x$, $f(x,W(x))= \frac{x}{5}+(W(x))^2$, $n=4$, $\rho (x)=x$, $\sigma(x)=x$, $x_2=\frac{1}{3}$ and $x_3=\frac{2}{3}$. 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