\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
\begin{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}
\end{equation}
and the multi-point boundary conditions
\begin{equation}
\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}
\end{equation}
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 
$\rho ^2(x_{1})<x_{2}<\dots <x_{n-1}<\sigma (x_{n})$, while 
$f: \mathbb{X}\times [ 0,\infty )\to [ 0,\infty )$ is continuous, 
$I:[\rho (x_{1}),\sigma (x_{n})]_{\mathbb{X}}\to (0,\infty )$ is left-dense 
continuous and $E>0$ 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
\begin{equation}  \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}
\end{equation}
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
\begin{equation}
(EIW^{\Delta \nabla })^{\nabla \Delta }(x)=y(x),\quad 
x\in [x_{1},x_{n}]_{\mathbb{X}},  \label{b3}
\end{equation}
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
\begin{equation}
(EIW^{\Delta \nabla })^{\nabla \Delta }(x)=0  \label{related}
\end{equation}
satisfying boundary conditions
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
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}
\end{equation}

\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 $0<h<<1$ we have $h\mathbb{Z}=\{0,h,2h,\dots ,1-h,1\}$.
\end{example}

This Green's function satisfies the following properties.

\begin{lemma}[\cite{ah}] \label{signs}
For all $(x,s)\in[\rho^2(x_1),\sigma^2(x_n)]_{\mathbb{X} }
\times[\rho(x_1),\sigma(x_n)]_{\mathbb{X} }$, the Green's function
given by \eqref{greenf} is increasing in $x$ and satisfies
\begin{equation}
0 \le G(x,s) \le G\big(\sigma^2(x_n),s\big).
\end{equation}
\end{lemma}

Now we prove an existence and uniqueness result.

\begin{lemma}\label{lemma22}
Assume the coefficients $a_i$ and $b_i$ in
\eqref{b2} are real non-negative numbers that satisfy the
non-resonant conditions
\begin{equation}
0\leq \sum_{i=2}^{n-1}a_i<1,\quad
0\leq \sum_{i=2}^{n-1}b_i<1.
\label{h2}
\end{equation}
If $y\in C_{rd}[\rho(x_1),\sigma(x_n)]_{\mathbb{X}}$, then the
nonhomogeneous dynamic equation \eqref{b3} with boundary
conditions \eqref{b2} has a unique solution $W$ defined by
\begin{equation}
W(x)=\int_{\rho (x_{1})}^{\sigma (x_{n})}G(x,s)y(s)\Delta
s+A(y)+B(y)\int_{\rho ^2(x_{1})}^{x}\int_{\rho
^2(x_{1})}^{\zeta }\frac{ \sigma (x_{n})-\xi }{EI(\xi )}\nabla
\xi \Delta \zeta ,  \label{form}
\end{equation}
where $G(x,s)$ is the Green's function \eqref{greenf} related with
the boundary value problem \eqref{related}, \eqref{relatedbc} and
the positive constants $A(y)$ and $B(y)$ are given by
\begin{equation}
\begin{aligned}
A(y) &= \Big(1-\sum_{i=2}^{n-1}{a_i}\Big)^{-1}
\sum_{i=2}^{n-1}a_i\Big(\int_{\rho (x_{1})}^{\sigma
(x_{n})}G(x_i,s)y(s)\Delta s   \\
&\quad +B(y)\int_{\rho ^2(x_{1})}^{x_i}\int_{\rho
^2(x_{1})}^{\zeta }\frac{\sigma (x_{n})-\xi }{EI(\xi )}\nabla
\xi \Delta \zeta \Big)
\end{aligned} \label{D}
\end{equation}
and
\begin{equation}
B(y)=\Big(1-\sum_{i=2}^{n-1}{b_i}\Big)^{-1}
\sum_{i=2}^{n-1}b_i\int_{x_i}^{\sigma (x_{n})}y(s)\Delta s.
\label{Bdef}
\end{equation}
\end{lemma}

\begin{proof}
First, we consider equation \eqref{b3} together with conditions
\begin{equation}
\begin{gathered} W(\rho ^2(x_{1}))=A,\quad W^{\Delta }(\rho ^2(x_{1}))=0 \\
(EIW^{\Delta \nabla })(\sigma (x_{n}))=0,\quad (EIW^{\Delta \nabla
})^{\nabla }(\sigma (x_{n}))=B. \end{gathered}  \label{condA-B}
\end{equation}
It is clear that any solution of problem \eqref{b3}, \eqref{condA-B} can be
expressed for some constants $A$ and $B$ as
\[
W(x)=u(x)+Av(x)+Br(x),
\]
where $u$ is the unique solution of problem value problem \eqref{b3}, 
\eqref{relatedbc}, $v$ is the unique solution of \eqref{related} with
boundary conditions
\[
v(\rho ^2(x_{1}))=1,\quad v^{\Delta }(\rho ^2(x_{1}))=(EIv^{\Delta \nabla
})(\sigma (x_{n}))=( EIv^{\Delta \nabla })^{\nabla }(\sigma (x_{n}))=0,
\]
and $r$ is the unique solution of \eqref{related} with boundary conditions
\[
(EIr^{\Delta \nabla })^{\nabla }(\sigma (x_{n}))=1,\quad r(\rho
^2(x_{1}))=r^{\Delta }(\rho ^2(x_{1}))=( EIr^{\Delta \nabla })(\sigma
(x_{n}))=0.
\]
One can verify directly that these functions are
\[
u(x)=\int_{\rho (x_{1})}^{\sigma (x_{n})}G(x,s)y(s)\Delta s,\quad v(x)\equiv
1,\quad r(x)=\int_{\rho ^2(x_{1})}^{x}\int_{\rho ^2(x_{1})}^{\zeta } 
\frac{\xi -\sigma (x_{n})}{EI(\xi )}\nabla \xi \Delta \zeta .
\]
It is clear that $W^{\Delta }(\rho ^2(x_{1}))=0$ and $(EIW^{\Delta \nabla
})(\sigma (x_{n}))=0$. To satisfy the two other boundary conditions in 
\eqref{b2}, we must have at $\sigma (x_{n})$ that
\[
-B=\Big(1-\sum_{i=2}^{n-1}b_i\Big) ^{-1}\sum_{i=2}^{n-1}\int_{x_i}^{\sigma
(x_{n})}y(s)\Delta s,
\]
and at $\rho ^2(x_{1})$ that
\[
A=\sum_{i=2}^{n-1}a_i\Big(\int_{\rho (x_{1})}^{\sigma
(x_{n})}G(x_i,s)y(s)\Delta s+A-B\int_{\rho ^2(x_{1})}^{x_i}\int_{\rho
^2(x_{1})}^{\zeta }\frac{\sigma (x_{n})-\xi }{EI(\xi )}\nabla \xi \Delta
\zeta \Big).
\]
Solving, we arrive at the expression \eqref{form} for $A(y)$ given in 
\eqref{D}.
\end{proof}

For problem \eqref{b3},\eqref{b2} the following maximum principle holds.

\begin{lemma}\label{lemma23}
Assume that \eqref{h2} holds. If
$y\in C_{rd}[\rho(x_1),\sigma(x_n)]_{\mathbb{X} }$ with $y\ge 0$ on
$[\rho(x_1),\sigma(x_n)]_{\mathbb{X}}$, the unique solution $W$
as in \eqref{form} of the problem \eqref{b3}, \eqref{b2} satisfies
$W(x)\ge 0$ for $x\in[\rho^2(x_1),\sigma^2(x_n)]_{\mathbb{X} }$.
\end{lemma}

\begin{proof}
From Lemma \ref{lemma22}, problem \eqref{b3}, \eqref{b2} has a unique
solution $W$ given by \eqref{form} and, by Lemma \ref{signs}, the Green's
function \eqref{greenf} satisfies $G(x,s)\geq 0$ on the set $[\rho
^2(x_{1}),\sigma ^2(x_{n})]_{\mathbb{X}}\times [ \rho (x_{1}),\sigma
(x_{n})]_{\mathbb{X}}$. The result is a direct consequence of assumption 
\eqref{h2} and the fact that $A(y),B(y)\geq 0$.
\end{proof}

\begin{lemma} \label{lemma24}
 Assume that \eqref{h2} holds. If
$y\in C_{rd}[\rho(x_1),\sigma(x_n)]_{ \mathbb{X}}$ with $y\geq 0$ on
$[\rho (x_{1}),\sigma (x_{n})]_{\mathbb{X}}$, then the unique
solution $W$ of the time scale boundary value problem \eqref{b3},
\eqref{b2}, given by \eqref{form}, satisfies
\[
\min_{x\in [ x_{2},x_{3}]_{\mathbb{X}}}W(x)=W(x_{2})\geq \gamma
\|W\|,
\]
where
\begin{equation}
\gamma :=\frac{\int_{\rho ^2(x_{1})}^{x_{2}}\int_{\rho ^2(x_{1})}^{\zeta
}\frac{\sigma (x_{n})-\xi }{EI(\xi )}\nabla \xi \Delta \zeta }{\int_{\rho
^2(x_{1})}^{\sigma (x_{n})}\int_{\rho ^2(x_{1})}^{\zeta }\frac{\sigma
^2(x_{n})-\xi }{EI(\xi )}\nabla \xi \Delta \zeta }\in (0,1),  \label{gamma}
\end{equation}
and
\[
\|W\|:=\max_{x\in [ \rho ^2(x_{1}),\sigma ^2(x_{n})]_{
\mathbb{X}}}W(x)=W(\sigma ^2(x_{n})).
\]
\end{lemma}

\begin{proof}
Using Lemma \ref{signs} and \eqref{form}, we conclude that for all $x\in [
\rho ^2(x_{1}),\sigma ^2(x_{n})]_{\mathbb{X}}$,
\[
W(x)\leq \int_{\rho (x_{1})}^{\sigma (x_{n})}G\big(\sigma ^2(x_{n}),s\big) 
y(s)\Delta s+A(y)+B(y)\int_{\rho ^2(x_{1})}^{\sigma ^2(x_{n})}\int_{\rho
^2(x_{1})}^{\zeta }\frac{\sigma (x_{n})-\xi }{EI(\xi )}\nabla \xi \Delta
\zeta .
\]
For $x\in [ x_{2},x_{3}]_{\mathbb{X}}$, from Lemma \ref{signs} the Green's
function \eqref{greenf} satisfies
\begin{equation}
\frac{G(x,s)}{G(\sigma ^2(x_{n}),s)}\geq \frac{G(x_{2},s)}{G(\sigma
^2(x_{n}),s)}\geq \frac{\int_{\rho ^2(x_{1})}^{x_{2}}\int_{\rho
^2(x_{1})}^{\zeta }\frac{\sigma (x_{n})-\xi }{EI(\xi )}\nabla \xi \Delta
\zeta }{\int_{\rho ^2(x_{1})}^{\sigma (x_{n})}\int_{\rho ^2(x_{1})}^{\zeta }
\frac{\sigma ^2(x_{n})-\xi }{EI(\xi )}\nabla \xi \Delta \zeta }=\gamma
\end{equation}
for $\gamma $ as in \eqref{gamma}, and the constant $A(y)$ in \eqref{D}
satisfies $A(y)\geq \gamma A(y)$ since $\gamma \in (0,1)$ and $A(y)\geq 0$.
Thus for $x\in [ x_{2},x_{3}]_{\mathbb{X}}$, we have
\begin{align*}
W(x) &= \int_{\rho (x_{1})}^{\sigma (x_{n})}
\frac{G(x,s)} { G(\sigma^2(x_{n}),s)}G(\sigma ^2(x_{n}),s)y(s)\Delta s+A(y) \\
&\quad +B(y)\int_{\rho ^2(x_{1})}^{x} \int_{\rho ^2(x_{1})}^{\zeta }
\frac{\sigma (x_{n})-\xi }{ EI(\xi )}\nabla \xi \Delta \zeta \\
&\geq \int_{\rho (x_{1})}^{\sigma (x_{n})}\gamma G(\sigma
^2(x_{n}),s)y(s)\Delta s+\gamma A(y) \\
&\quad +B(y)\int_{\rho ^2(x_{1})}^{x_{2}} \int_{\rho ^2(x_{1})}^{\zeta }
\frac{\sigma (x_{n})-\xi} {EI(\xi )}\nabla \xi \Delta \zeta \\
&= \int_{\rho (x_{1})}^{\sigma (x_{n})}\gamma G(\sigma
^2(x_{n}),s)y(s)\Delta s+\gamma A(y) \\
&\quad +\gamma B(y)\int_{\rho^2(x_{1})}^{\sigma (x_{n})} \int_{\rho
^2(x_{1})}^{\zeta }\frac{\sigma ^2(x_{n})-\xi }{EI(\xi )}\nabla \xi \Delta
\zeta \\
&= \int_{\rho (x_{1})}^{\sigma (x_{n})}\gamma G(\sigma
^2(x_{n}),s)y(s)\Delta s+\gamma A(y) \\
&\quad +\gamma B(y)\int_{\rho^2(x_{1})}^{\sigma ^2(x_{n})} \int_{\rho
^2(x_{1})}^{\zeta }\frac{\sigma (x_{n})-\xi } {EI(\xi )}\nabla \xi \Delta
\zeta \\
&= \gamma W(\sigma ^2(x_{n}))=\gamma \|W\Vert.
\end{align*}
This completes the proof.
\end{proof}

\section{Existence of Positive Solutions}

In this section some criteria are identified whereby the existence of
positive solutions to the multi-point boundary value problem \eqref{b1}, 
\eqref{b2} can be established, where 
$f:\mathbb{X}\times [ 0,\infty )\to [ 0,\infty )$ is continuous such that 
the limits
\[
f_{0}:=\lim_{y\to 0^{+}}\frac{f(x,y)}{y},\quad f_{\infty }:=\lim_{y\to
\infty }\frac{f(x,y)}{y},
\]
exist uniformly for $x\in [ x_{1},x_{n}]_{\mathbb{X}}$.

In the sequel it is assumed that the right-dense continuous mass 
function $m$ satisfies
\begin{equation}
m:[\rho (x_{1}),\sigma (x_{n})]_{\mathbb{X}}\to [ 0,\infty),\quad \exists
\;x_{\ast }\in (x_{2},x_{3})_{\mathbb{X}}: m(x_{\ast })>0.  \label{h3}
\end{equation}

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
\begin{equation}
\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}
\end{equation}
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{equation}
\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}
\end{equation}
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
\[
P_{r}:=\{W\in \mathcal{P}:\|W\|\ <r\},
\]
and, for $\gamma $ in \eqref{gamma}, the set
\[
\Omega _{r}:=\big\{ W\in \mathcal{P}:\min_{x\in [ x_{2},x_{3}]_{\mathbb{X}}}W(x)
<\gamma r\big\} .
\]

\begin{lemma}[\cite{lan}] \label{lemma32}
The set $\Omega _{r}$ has the following properties:
\begin{itemize}
\item[(i)] $\Omega_r$ is open relative to $\mathcal{P}$.

\item[(ii)] $P_{\gamma r}\subset\Omega_r\subset P_r$.

\item[(iii)] $W\in\partial\Omega_r$ if and only if $\min_{x\in[x_2,x_3]_{
\mathbb{X} }}W(x)=\gamma r$.

\item[(iv)] If $W\in\partial\Omega_r$, then $\gamma r\le W(x)\le r$ for $x\in
[x_2,x_3]_{\mathbb{X} }$.
\end{itemize}
\end{lemma}

For $G(x,s)$ in \eqref{greenf} and $A(y)$ in \eqref{D} with $y$ replaced by
the mass function $m$, consider the constant $K$ given by
\begin{equation}  \label{Kdef}
\begin{aligned} K&:=\int_{\rho (x_{1})}^{\sigma (x_{n})}G(\sigma
^2(x_{n}),s)m(s)\Delta s+A(m) \\
&\quad+\Big(1-\sum_{i=2}^{n-1}{b_i}\Big)^{-1}
\Big(\sum_{i=2}^{n-1}b_i\int_{x_i}^{\sigma (x_{n})}m(s)\Delta s\Big)
\int_{\rho ^2(x_{1})}^{\sigma ^2(x_{n})}\int_{\rho ^2(x_{1})}^{\zeta }
\frac{\sigma (x_{n})-\xi }{EI(\xi )}\nabla \xi \Delta \zeta \end{aligned}
\end{equation}
and
\[
f_{\gamma r}^{r}:=\min_{W\in [ \gamma r,r]}\big\{ \min_{x\in [ x_{2},x_{3}]
_{\mathbb{X}}}\frac{f(x,W)}{r}\big\} ,\quad f_{0}^{r}:=\max_{W\in [ 0,r]}
\big\{ \max_{x\in [ \rho (x_{1}),\sigma (x_{n})]_{\mathbb{X}}}\
frac{f(x,W)}{r}\big\} .
\]

The next two lemmas present sufficient conditions on $f$ to evaluate the
index of $\mathcal{L}$.

\begin{lemma}\label{lemma33}
Let $K$ be as in \eqref{Kdef}. If  $f_{0}^{r}<1/K$ holds,
then $i_{P}(\mathcal{L},P_{r})=1$.
\end{lemma}

\begin{proof}
From \eqref{D},
\[
|A(mf(\cdot ,W))|\leq A(m)\|f(\cdot ,W)\|.
\]
For $W\in \partial P_{r}$, by \eqref{Lop} and Lemma \ref{signs},
\begin{align*}
(\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))
 +(1- \sum_{i=2}^{n-1}{b_i})^{-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 \\
&\leq \|f(\cdot ,W)\|\Big[ \int_{\rho (x_{1})}^{\sigma (x_{n})}G(\sigma
^2(x_{n}),s)m(s)\Delta s+A(m)+\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)\Delta s
\Big)\int_{\rho ^2(x_{1})}^{\sigma ^2(x_{n})}\int_{\rho ^2(x_{1})}^{\zeta }
\frac{\sigma (x_{n})-\xi }{ EI(\xi )}\nabla \xi \Delta \zeta \Big] \\
&<(r/K)K=r=\|W\|.
\end{align*}
It follows that for $W\in \partial P_{r}$, $\| \mathcal{L}W\|<\|W\|$. By
Lemma \ref{index} (i), $i_{P}(\mathcal{L},P_{r})=1$.
\end{proof}

\begin{lemma} \label{lemma34}
Let
\begin{equation}  \label{Mdef}
M^{-1}:=\int_{x_2}^{x_3}G(x_2,s)m(s)\Delta s.
\end{equation}
If the inequality $f_{\gamma r}^r> M\gamma$ 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 $0<c_{1}<\gamma
c_{2}$ and $c_{2}<c_{3}$ such that
\begin{itemize}
\item[(H1)]
$f_{0}^{c_{1}},f_{0}^{c_{3}}\leq 1/K$,
$f_{\gamma c_{2}}^{c_{2}}>M\gamma$
\end{itemize}
or there exist constants $c_{1},c_{2},c_{3}\in \mathbb{R}$ with
$0<c_{1}<c_{2}<\gamma c_{3}$ such that
\begin{itemize}
\item[(H2)]
$f_{\gamma c_{1}}^{c_{1}},f_{\gamma c_{3}}^{c_{3}}\geq M\gamma$,
$f_{0}^{c_{2}}<1/K$.
\end{itemize}
Then the multi-point problem \eqref{b1}, \eqref{b2} has two
positive solutions in $\mathcal{P}$, given by (\ref{cone}).
\end{theorem}

\begin{proof}
Assume (H2) holds (the case for (H1) is similar and is omitted). We show
that either $\mathcal{L}$ has a fixed point in $\partial \Omega _{c_{1}}$ or
in $P_{c_{2}}\backslash \overline{\Omega }_{c_{1}}$. From Lemma \ref{lemma34}
, if $W\neq \mathcal{L}W$ for $W\in \partial \Omega _{c_{1}}\cup \partial
\Omega _{c_{3}}$, then $i_{P}(\mathcal{L},\Omega _{c_{1}})=0$ and $i_{P}(
\mathcal{L},\Omega _{c_{3}})=0$. Since $f_{0}^{c_{2}}\leq 1/K$ and $W\neq
\mathcal{L}W$ for $W\in \partial P_{c_{2}}$, Lemma \ref{lemma33} implies
that $i_{P}(\mathcal{L},P_{c_{2}})=1$. By Lemma \ref{lemma32} (ii), $\Omega
_{c_{1}}\subset P_{c_{1}}\subset P_{c_{2}}$. From Lemma \ref{index} (iii), 
$\mathcal{L}$ has a fixed point in $P_{c_{2}}\backslash \overline{\Omega }
_{c_{1}}$. In the same way $P_{c_{2}}\subset P_{\gamma c_{3}}\subset \Omega
_{c_{3}}$ and $\mathcal{L}$ has a fixed point in $\Omega _{c_{3}}\backslash
\overline{P}_{c_{2}}$.
\end{proof}

For $a\in \{0^{+},\infty \}$ define
\[
f_{Wa}:=\liminf_{W\to a}\min_{x\in [ x_{2},x_{3}]_{\mathbb{X}}} 
\frac{f(x,W)}{W}\;,\quad f_{W}^{a}:=\limsup_{W\to a}\max_{x\in [ \rho (t_{1}),\sigma
(x_{n})]_{\mathbb{X}}}\frac{f(x,W)}{W}\,.
\]

\begin{corollary} \label{c-36}
Suppose there exists a positive constant $c$ such that either
one the following to conditions holds:
\begin{itemize}
\item[(H1')]
$0\leq f_{W}^{0}$, $f_{W}^{\infty }<1/K$,
$f_{\gamma c}^{c}>M\gamma$;
\item[(H2')]
$M<f_{W0}$, $f_{W\infty }\leq \infty$, $f_{0}^{c}<1/K$.
\end{itemize}
Then problem \eqref{b1}, \eqref{b2} has two positive
solutions in $ \mathcal{P}$.
\end{corollary}

\begin{proof}
Since (H1') implies (H1) and (H2') implies (H2), the result follows.
\end{proof}

The proofs of the following two results are similar to those given above and
are omitted.

\begin{theorem} \label{thm3.7}
Assume that there exist constants $c_{1},c_{2}\in \mathbb{R}$ with
$ 0<c_{1}<\gamma c_{2}$ such that
\begin{itemize}
\item[(H3)]
$f_{0}^{c_{1}}\leq 1/K$ and $f_{\gamma c_{2}}^{c_{2}}\geq
M\gamma$,
\end{itemize}
or that there exist constants $c_{1},c_{2}\in \mathbb{R}$ with $
0<c_{1}<c_{2} $ such that
\begin{itemize}
\item[(H4)]
$f_{\gamma c_{1}}^{c_{1}}\geq M\gamma $ and
$f_{0}^{c_{2}}\leq 1/K$.
\end{itemize}
Then problem \eqref{b1}, \eqref{b2} has a positive solution.
\end{theorem}

\begin{corollary} \label{c-38} Suppose either one of the following
conditions holds:
\begin{itemize}
\item[(H3')]
$0\leq f_{W}^{0}<1/K$ and $M\gamma <f_{W\infty }\leq \infty$;
\item[(H4')]
$0\leq f_{W}^{\infty }<1/K$ and $M\gamma <f_{W0}\leq \infty$.
\end{itemize}
Then problem \eqref{b1}, \eqref{b2} has a positive
solution.
\end{corollary}

\section{Examples}

In the first example, for $\gamma$, $K$, and $M$ given by \eqref{gamma}, 
\eqref{Kdef}, and \eqref{Mdef}, respectively, assume positive  constants 
$c_{1},c_{2},c_{3}\in \mathbb{R}$ such that $c_{1}<\gamma c_{2}$, 
$c_{2}<c_{3}$ and
\[
\frac{c_{1}}{K}\leq M\gamma c_{2}+\delta \leq \frac{c_{3}}{K},
\]
for some $\delta >0$. Consider a particular case of equation (\ref{b1})
given by
\begin{equation}
(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}
\end{equation}
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
\begin{equation}
W^{(4)}(x)=x\left(\frac{x}{5} +(W(x))^2\right),\quad \text{for } x\in
\mathbb{X},  \label{EqEx}
\end{equation}
with the boundary conditions
\begin{equation}  \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}
\end{equation}
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 $x_3=\frac{2}{3}$.
 Applying the Green's function given in Example \ref{Examp 21}, then 
$K=0.72921$, $\gamma =\frac{14}{27}$ and $M=\frac{2916}{11}$.

For $c_{1}=\frac{1}{2070}$, $c_{2}=1$ and $c_{3}=552$ assumption (H2) holds
and, by Theorem \ref{Thm35}, problem (\ref{EqEx}), (\ref{BCEx}) has two
positive solutions in the cone
\[
\mathcal{P}=\big\{ W\in C([0,1]):W(x)\geq 0\text{ on }[0,1]\text{ and }
W(x)\geq \frac{14}{27}||W||\text{ on }[\frac{1}{3},\frac{2}{3}]
\big\}.
\]

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\end{document}