\documentclass[reqno]{amsart} 

\AtBeginDocument{{\noindent\small 
{\em Electronic Journal of Differential Equations},
Vol. 2004(2004), No. 47, pp. 1--12.\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-2004/47\hfil Triple-point problems on time scales]
{Nonlinear triple-point problems on time scales} 

\author[Douglas R. Anderson\hfil EJDE-2004/47\hfilneg]
{Douglas R. Anderson}  

\address{Douglas R. Anderson \hfill\break
Department of Mathematics and Computer Science,
Concordia College, Moorhead, Minnesota 56562 USA}
\email{andersod@cord.edu}

\date{}
\thanks{Submitted March 7, 2004. Published April 6, 2004.}
\subjclass[2000]{34B10, 34B15, 39A10}
\keywords{Fixed-point theorems, time scales, dynamic equations, cone}


\begin{abstract}
 We establish the existence of multiple positive solutions to the
  nonlinear second-order triple-point boundary-value problem on
  time scales,
 \begin{gather*}
 u^{\Delta\nabla}(t)+h(t)f(t,u(t))=0, \\
 u(a)=\alpha u(b)+\delta u^\Delta(a),\quad  
 \beta u(c)+\gamma u^\Delta(c)=0
 \end{gather*}
 for $t\in[a,c]\subset\mathbb{T}$, where $\mathbb{T}$ is a time scale,
 $\beta, \gamma, \delta\ge 0$ with $\beta+\gamma>0$,
 $0<\alpha<\frac{c-a}{c-b}$ and $b\in(a,c)\subset\mathbb{T}$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{cor}[theorem]{Corollary}
\newtheorem{defi}[theorem]{Definition}


\section{Introduction to the boundary-value problem}

We are concerned with proving the existence of multiple positive
solutions to the second-order triple-point nonlinear
boundary-value problem on a time scale $\mathbb{T}$ given by the
time-scale dynamic equation
\begin{equation}\label{bvp}
  u^{\Delta\nabla}(t)+h(t)f(t,u(t))=0, \quad t\in(a,c)\subset\mathbb{T}
\end{equation}
with boundary conditions
\begin{equation}\label{bc}
 u(a)=\alpha u(b)+\delta u^\Delta(a),\quad  \beta u(c)+\gamma u^\Delta(c)=0,
\end{equation}
where $\beta, \gamma, \delta\ge 0$ with $\beta+\gamma>0$, $0<\alpha<\frac{c-a}{c-b}$ and $b\in(a,c)\subset\mathbb{T}$
for $a\in\mathbb{T}_\kappa$, $c\in\mathbb{T}^\kappa$.  The function $h\in C_{ld}[a,c]$ is nonnegative with $h(t_0)>0$ for at
least one $t_0\in(a,b]$, and the nonlinearity $f:[a,c]\times[0,\infty)\to[0,\infty)$ is continuous such
that $f(t,\cdot)>0$ on any subset of $\mathbb{T}$ containing $t_0$.  This problem is related to that first studied in
the case $\mathbb{T}=\mathbb{R}$ on the unit interval by He and Ge \cite{he} and Ma \cite{ma1,ma2,ma3},
\begin{equation}\label{maprob}
  u''+f(t,u)=0,\quad t\in (0,1) \quad u(0)=0,\quad\alpha u(\eta)=u(1)
\end{equation}
where $0<\eta<1$ and $0<\alpha<1/\eta$. The boundary-value problem
\eqref{maprob} has since been extended
to general time scales in Anderson \cite{and} and Kaufmann \cite{kauf} as
$$ u^{\Delta\nabla}(t)+f(t,u(t))=0,\quad u(0)=0,\; \alpha u(\eta)=u(T), $$
and in a slightly different way by Sun and Li \cite{li}
$$ u^{\Delta\nabla}(t)+a(t)f(t,u(t))=0, \quad u^\Delta(0)=0,\; \alpha u(\eta)=u(T). $$
In this paper there is a nexus between the boundary conditions at $a$ and $b$ instead of at $b=\eta$ and $c=T$,
and we add the $\delta\ge 0$ term as well. Consequently these results are new for differential and difference
equations as well as for dynamic equations on general time scales.

For more on time scales and positive solutions, see the books by Bohner and
Peterson \cite{bp,bp2} and the following
articles: \cite{akin,and2,ah,atici,aa,chyan,erbepete1,erbepete2,hend,hilger}.


\section{preliminary results}

To prove the main existence results we will employ several straightforward
lemmas.  These lemmas are
based on the linear boundary-value problem
\begin{gather}\label{line3}
 u^{\Delta\nabla}(t)+y(t)=0 \quad t\in(a,c)\subset\mathbb{T} \\
\label{line4}
  u(a)=\alpha u(b)+\delta u^\Delta(a),\quad  \beta u(c)+\gamma u^\Delta(c)=0.
\end{gather}


\begin{lemma}\label{lemma1}
Let
\begin{equation}\label{dee}
 d:=\gamma(1-\alpha)+\beta\left[(c-a)-\alpha(c-b)+\delta\right].
\end{equation}
If $d\neq 0$, then for $y\in C_{ld}[a,c]$ the boundary-value problem \eqref{line3}, \eqref{line4}
has the unique solution
\begin{equation}\label{unique}
\begin{aligned}
 u(t)=& \frac{1}{d}\left(\gamma+\beta(c-t)\right)
 \Big[ \int_a^c (s-a+\delta)y(s)\nabla s \\
      &  - \alpha\int_b^c (s-b)y(s)\nabla s \Big] - \int_t^c (s-t)y(s)\nabla s.
\end{aligned}
\end{equation}
\end{lemma}

\begin{proof}
Let $u$ be as in \eqref{unique}.  Then the delta derivative of $u$ is given by
$$ u^{\Delta}(t) = -\frac{\beta}{d} \left[\int_a^c (s-a+\delta)y(s)\nabla s - \alpha\int_b^c (s-b)y(s)\nabla s \right]
       + \int_t^c y(s)\nabla s $$
and
$$ u^{\Delta\nabla}(t)=-y(t), $$
so that $u$ given in \eqref{unique} is a solution of \eqref{line3}.  It is routine to check that the boundary conditions
\eqref{line4} are met by $u$ in \eqref{unique} as well.
\end{proof}


\begin{lemma}\label{lemma2}
If $d>0$ and $y\in C_{ld}[a,c]$ with $y\geq 0$, the unique solution $u$ of \eqref{line3}, \eqref{line4} given in
\eqref{unique} satisfies
$$ u(t)\geq 0, \quad t\in[a,c]\subset\mathbb{T}. $$
\end{lemma}

\begin{proof}
  From the fact that $u^{\Delta\nabla}(t)=-y(t)\le 0$, we know that if $u(a)\ge 0$ and $u(c)\ge 0$,
then $u(t)\ge 0$ for $t\in[a,c]$.  For $0 < \alpha \le 1$,
\begin{align*}
 u(c) & =  \frac{\gamma}{d}\left[\int_a^c (s-a+\delta)y(s)\nabla s - \alpha\int_b^c (s-b)y(s)\nabla s\right] \\
      &\ge \frac{\gamma}{d}\left[(1-\alpha)\int_b^c (s-b)y(s)\nabla s + \delta\int_a^c y(s)\nabla s\right]  \\
      &\ge 0.
\end{align*}
For $1 < \alpha < \frac{c-a}{c-b}$,
\begin{align*}
 u(c) & =  \frac{\gamma}{d}\left[\int_a^b (s-a+\delta)y(s)\nabla s+\int_b^c (s-a-\alpha(s-b)+\delta)y(s)\nabla s\right]\\
      &\ge \frac{\gamma}{d}\left[\int_a^b (s-a+\delta)y(s)\nabla s+\int_b^c (c-a-\alpha(c-b)+\delta)y(s)\nabla s\right]\\
      &\ge 0
\end{align*}
since $\alpha<(c-a)/(c-b)$.  Finally,
\begin{align*}
 u(a) & =  \frac{1}{d}\left(\gamma + \beta(c-a)\right)
       \Big[\int_a^c(s-a+\delta)y(s)\nabla s-\alpha\int_b^c(s-b)y(s)\nabla s\Big]
       -\frac{d}{d}\int_a^c(s-a)y(s)\nabla s\\
      & =  \frac{\alpha}{d}(\gamma+\beta(c-b))\int_a^b(s-a)y(s)\nabla s+\frac{\alpha}{d}(b-a)\int_b^c(\gamma+\beta(c-s))y(s)\nabla s  \\
      & \quad +\frac{\delta}{d}\int_a^c (\gamma+\beta(c-s))y(s)\nabla s \\
      &\ge 0.
\end{align*}
\end{proof}

\begin{lemma}\label{lemma3}
Let $d>0$.  If $y\in C_{ld}[a,c]$ with $y$ nonnegative but nontrivial, then the
unique solution $u$ as in
\eqref{unique} of \eqref{line3}, \eqref{line4} satisfies
\begin{equation}
 \inf_{t\in[a,b]}u(t)\ge k\|u\|, \quad \|u\|:=\sup_{t\in[a,c]}|u(t)|,
\end{equation}
where
\begin{equation}\label{kdef}
   k:=\min\Big\{\frac{\alpha(c-b)}{c-a}, \frac{c-b}{c-a},
   \frac{\alpha(b-a)}{c-a-\alpha(c-b)} \Big\}\in(0,1).
\end{equation}
\end{lemma}

\begin{proof}
Note that $u^{\Delta\nabla}(t)=-y(t)\le 0$ for all $t\in(a,c)$, so that
$$ \min_{t\in[a,b]}u(t)=\min\{u(a), u(b)\}. $$
Then for any $\tau\in[a,c)$,
$$ \eta(t):=u(t)-\big(\frac{c-t}{c-\tau}\big)u(\tau)
$$
satisfies $\eta(\tau)=0$, $\eta(c)=u(c)\ge 0$, and
$\eta^{\Delta\nabla}(t)=u^{\Delta\nabla}(t)\le 0 $ on $[\tau,c)$.
In particular,
$$ \frac{u(t)}{c-t} \ge \frac{u(\tau)}{c-\tau}
$$
for all $t\in[\tau,c)$.  Now fix $\tau\in[a,c)$ such that $u(\tau)=\|u\|$.
If $a\le \tau \le b$ and $u(a)\le u(b)$, then
$$ \frac{\alpha u(b)}{c-b}\ge \frac{\alpha u(\tau)}{c-\tau}
\ge \frac{\alpha u(\tau)}{c-a}, $$
rewritten with the boundary condition at $b$ as
$$ u(a)-\delta u^\Delta(a)\ge\big(\frac{c-b}{c-a}\big)\alpha \|u\|. $$
Therefore,
$$ \min_{t\in[a,b]}u(t)=u(a)\ge \big(\frac{c-b}{c-a}\big)\alpha\|u\|,
$$
since $u^\Delta(a)\ge 0$ in this case.  If $a\le \tau \le b$ and $u(b)\le u(a)$,
then
$$ u(b)\ge \big(\frac{c-b}{c-\tau}\big)u(\tau) \ge
\big(\frac{c-b}{c-a}\big)u(\tau), $$
so that
$$ \min_{t\in[a,b]}u(t)=u(b)\ge \big(\frac{c-b}{c-a}\big)\|u\|.
$$
If $b < \tau < c$, then $u(a)=\min_{t\in[a,b]}u(t)$, and by the concavity of $u$
 and a secant line we have
\begin{align*}
 u(\tau) &\le u(a)+\Big(\frac{u(b)-u(a)}{b-a}\Big)(c-a) \\
         & =  \frac{\left[c-a-\alpha(c-b)\right]u(a)-\delta (c-a)u^\Delta(a)}{\alpha(b-a)} \\
         &\le \frac{\left[c-a-\alpha(c-b)\right]u(a)}{\alpha(b-a)},
\end{align*}
since again $u^\Delta(a)\ge 0$.
Consequently,
$$ \min_{t\in[a,b]}u(t)=u(a)\ge\frac{\alpha(b-a)}{c-a-\alpha(c-b)}\|u\|.
$$
\end{proof}

\section{Existence of at Least Two Positive Solutions}

We apply the Avery-Henderson Fixed Point Theorem \cite{AvHe} to prove the existence of at least two positive
solutions to the nonlinear boundary-value problem \eqref{bvp}, \eqref{bc}, where $h\in C_{ld}[a,c]$ is
nonnegative with $h(t_0)>0$ for at least one $t_0\in(a,b]$,and the nonlinearity
$f:[a,c]\times[0,\infty)\to[0,\infty)$ is continuous such that $f(t,\cdot)>0$ on any subset of $\mathbb{T}$
containing $t_0$.  The solutions are the fixed points of the operator $A$ defined by
\begin{align*}
   A u(t) &= \frac{1}{d}\left(\gamma+\beta(c-t)\right)
   \Big[ \int_a^c (s-a+\delta)h(s)f(s,u(s))\nabla s \\
   &\quad - \alpha\int_b^c (s-b)h(s)f(s,u(s))\nabla s \Big]
    - \int_t^c (s-t)h(s)f(s,u(s))\nabla s
\end{align*}
by Lemma \ref{lemma1}.  Notationally, the cone $P$ has subsets of the form
$P(\phi,r):=\{u\in P:\phi(u)<r\}$ for a given functional $\phi$.

% Theorem 21   (AvHe)

\begin{theorem}\cite{AvHe}\label{averyhen}
Let $P$ be a cone in a real Banach space $\mathcal{S}$. If $\eta$ and $\phi$ are increasing,
nonnegative continuous functionals on $P$, let $\theta$ be a nonnegative continuous functional
on $P$ with $\theta(0)=0$ such that, for some positive constants $r$ and $M$,
$$ \phi(u)\le \theta(u)\le \eta(u) \quad \text{and}\quad ||u||\le M\phi(u)
 $$
for all $u\in \overline{P(\phi,r)}$.
Suppose that there exist positive numbers $p<q<r$ such that
$$ \theta(\lambda u)\le \lambda\theta(u),
\quad\text{ for all }0\le\lambda\le 1\text{ and } u\in\partial P(\theta,q). $$
If $A:\overline{P(\phi,r)}\to P$ is a completely continuous operator satisfying
\begin{itemize}
  \item[(i)] $\phi(A u)>r$ for all $u\in \partial P(\phi,r)$,
  \item[(ii)] $\theta(A u)<q$ for all $u\in\partial P(\theta,q)$,
  \item[(iii)] $P(\eta,p)\not=\emptyset$ and $\eta(A u)>p$ for all $u\in\partial P(\eta,p)$,
\end{itemize}
then $A$ has at least two fixed points $u_1$ and $u_2$ such that
$$ p < \eta(u_1) \text{ with } \theta(u_1) < q \quad\text{ and }\quad q < \theta(u_2) \text{ with }\phi(u_2) < r. $$
\end{theorem}

Let $\mathcal{S}$ denote the Banach space $C[\rho(a),\sigma(c)]$
with the supremum norm. Define the cone $P\subset\mathcal{S}$ by
\begin{equation}\label{P}
  P=\{u\in\mathcal{S}: u(t)\ge 0, u \mbox{ is concave, and } \min_{t\in[a,b]}u(t) \ge k\|u\|\},
\end{equation}
where $k$ is given in \eqref{kdef}.
Finally, let the nonnegative, increasing, continuous functionals
$\phi$, $\theta$, and $\eta$ be defined on the cone $P$ by
$$ \phi(u):=\min_{t\in[a,b]} u(t),
\quad \theta(u):=\max_{t\in[b,c]} u(t), \quad \eta(u):=\max_{t\in[a,c]} u(t). 
$$ 
Observe that, for each $u\in P$,
the concavity of $u$ implies
\begin{gather}\label{gpd}
  \phi(u)\le\theta(u)\le\eta(u), \\
\label{normgam}
  \|u\| = \le \frac{1}{k}\min_{t\in[a,b]}u(t) = \frac{1}{k}\phi(u)\le\frac{1}{k}\theta(u)\le\frac{1}{k}\eta(u).
\end{gather}

\begin{theorem}\label{averyhen2}
Let $d>0$ and $k$ be as in \eqref{kdef}.  Suppose there exist positive numbers $0 < p < q < r$
such that the function $f$ satisfies the following conditions:
\begin{enumerate}
  \item[$(i)$]   $f(s,u) > pM$ for $s\in[a,b]$ and $u\in[kp,p]$,
  \item[$(ii)$]  $f(s,u) < qm$ for $s\in[a,c]$ and $u\in[0,q/k]$,
  \item[$(iii)$] $f(s,u) > rM$ for $s\in[a,b]$ and $u\in[r,r/k]$
\end{enumerate}
for some positive constants $m$ and $M$.  Then the second-order boundary-value problem \eqref{bvp}, \eqref{bc},
has at least two positive solutions $u_1$ and $u_2$ such that
\begin{gather*}
 p < \max_{t\in[a,c]}u_1(t) \quad \text{with } \max_{t\in[b,c]}u_1(t) < q, \\
 q < \max_{t\in[b,c]}u_2(t) \quad \text{ with } \min_{t\in[a,b]}u_2(t) < r.
\end{gather*}
\end{theorem}

\begin{proof}
For $u\in P$, $u(t)\ge k\|u\|$ for all $t\in[a,b]$. By Lemma \ref{lemma3},
$A(P)\subset P$.  Standard arguments
show that $A:P\to P$ is completely continuous.
For any $u\in P$, \eqref{gpd} and \eqref{normgam} imply
$$ \phi(u)\le\theta(u)\le\eta(u), \quad \|u\|\le\frac{1}{k}\phi(u).
$$
It is clear that $\theta(0)=0$, and for all $u\in P$, $\lambda\in[0,1]$ we have
$$ \theta(\lambda u) = \max_{t\in[b,c]}(\lambda u)(t)
= \lambda\max_{t\in[b,c]} u(t) = \lambda\theta(u).
$$
Since $0\in P$ and $p>0$, $P(\eta,p)\ne\emptyset$.

In the following claims, we verify the remaining conditions of
Theorem \ref{averyhen}. \smallskip

\noindent {\bf Claim 1:} If $u\in\partial P(\eta,p)$, then $\eta(A u)>p$:
Since $u\in\partial P(\eta,p)$, $kp\le u(t)\le\|u\|=p$ for $t\in[a,b]$. Define
\begin{equation}\label{bigm}
  M:=d\Big(\min\{\alpha,1\}(\gamma+\beta(c-b))\int_a^b(s-a+\delta)h(s)\nabla s
\Big)^{-1}.
\end{equation}
Thus
\begin{align*}
\eta(A u)
  & =  \max_{t\in[a,c]} A u(t) \\
  &\ge A u(b) \\
  & =  \frac{1}{d}\left(\gamma+\beta(c-b)\right)
  \Big[ \int_a^c (s-a+\delta)h(s)f(s,u(s))\nabla s\\
  &\quad - \alpha\int_b^c (s-b)h(s)f(s,u(s))\nabla s \Big]
   - \int_b^c (s-b)h(s)f(s,u(s))\nabla s \\
  & =  \frac{\gamma+\beta(c-b)}{d}\int_a^b (s-a+\delta)h(s)f(s,u(s))\nabla s\\
  & \quad+\frac{b-a+\delta}{d}\int_b^c (\gamma+\beta(c-s))h(s)f(s,u(s))\nabla s \\
  &\ge \frac{\gamma+\beta(c-b)}{d}\int_a^b (s-a+\delta)h(s)f(s,u(s))\nabla s \\
  & >  pM\frac{\gamma+\beta(c-b)}{d}\int_a^b (s-a+\delta)h(s)\nabla s\\
  &\ge p
\end{align*}
by hypothesis $(i)$ and \eqref{bigm}. \smallskip

\noindent {\bf Claim 2:} If $u\in\partial P(\theta,q)$, then $\theta(A u)<q$:
Note that $u\in\partial P(\theta,q)$ and \eqref{normgam} imply that
$0\le u(t)\le \|u\|\le q/k$
for $t\in[a,c]$.  Define
\begin{equation}\label{littlem}
  m:=d\Big((\gamma+\beta(c-b))\int_a^c(s-a+\delta)h(s)\nabla s \Big)^{-1}.
\end{equation}
Then
\begin{align*}
\theta(A u)
  & =  \max_{t\in[b,c]} A u(t) \\
  &\le \max_{t\in[b,c]} \frac{1}{d}\left(\gamma+\beta(c-t)\right)
  \Big[ \int_a^c (s-a+\delta)h(s)f(s,u(s))\nabla s\\
  &\quad - \alpha\int_b^c (s-b)h(s)f(s,u(s))\nabla s \Big] \\
  &\le \frac{1}{d}\left(\gamma+\beta(c-b)\right)\int_a^c (s-a+\delta)h(s)f(s,u(s))\nabla s \\
  & <  \frac{qm}{d}\left(\gamma+\beta(c-b)\right)\int_a^c (s-a+\delta)h(s)\nabla s \\
  & =  q
\end{align*}
using \eqref{unique}, hypothesis $(ii)$, and \eqref{littlem}. \noindent

\noindent {\bf Claim 3:} If $u\in\partial P(\phi,r)$, then $\phi(A u)>r$:
Since $u\in\partial P(\phi,r)$, from \eqref{normgam} we have that
$\displaystyle\min_{t\in[a,b]}u(t)=r$ and $r\le \|u\|\le \frac{r}{k}$.
Since $\delta\ge 0$, as in the proof of Claim 1 we have
\begin{align*}
A u(b)&=\frac{\gamma+\beta(c-b)}{d}\int_a^b (s-a+\delta)h(s)f(s,u(s))\nabla s\\
      &\quad +\frac{b-a+\delta}{d}\int_b^c (\gamma+\beta(c-s))h(s)f(s,u(s))\nabla s.
\end{align*}
In a similar fashion from the proof of Lemma \ref{lemma2} we have
\begin{align*}
A u(a)
  &=\alpha\Big[\frac{\gamma+\beta(c-b)}{d}\int_a^b (s-a)h(s)f(s,u(s))\nabla s\\
  &\quad+\frac{b-a}{d}\int_b^c (\gamma+\beta(c-s))h(s)f(s,u(s))\nabla s\Big] \\
  & \quad +\frac{\delta}{d}\int_a^c(\gamma+\beta(c-s))h(s)f(s,u(s))\nabla s.
\end{align*}
By concavity,
$$ \min_{t\in[a,b]} A u(t) = \min\{A u(a), A u(b)\}. $$
If $\alpha\ge 1$, then $A u(a)\ge A u(b)$.  If $0<\alpha<1$, then $A u(a) \ge \alpha A u(b)$.
It follows that
\begin{align*}
 \phi (A u)
   & =  \min_{t\in[a,b]} A u(t) \\
   &\ge \min\{\alpha,1\}A u(b) \\
   &\ge \min\{\alpha,1\}\big(\frac{\gamma+\beta(c-b)}{d}\big)
   \int_a^b (s-a+\delta)h(s)f(s,u(s))\nabla s \\
   & > \min\{\alpha,1\}rM\big(\frac{\gamma+\beta(c-b)}{d}\big)
   \int_a^b (s-a+\delta)h(s)\nabla s \\
   &\ge r
\end{align*}
by hypothesis $(iii)$ and \eqref{bigm}.
Therefore the hypotheses of Theorem \ref{averyhen} are satisfied and there exist at least two positive fixed points
$u_1$ and $u_2$ of $A$ in $\overline{P(\phi,r)}$. Thus, the second-order triple-point boundary-value problem
\eqref{bvp}, \eqref{bc}, has at least two positive solutions $u_1$ and $u_2$
such that
\begin{gather*}
 p < \eta(u_1) \quad \mbox{with }  \theta(u_1) < q, \\
 q < \theta(u_2) \quad \mbox{with }  \phi(u_2) < r.
\end{gather*}
\end{proof}

\begin{cor}\label{ahcor}
Let $d>0$.  If there exists $q>0$ such that the function $f$ satisfies the
following conditions:
\begin{enumerate}
  \item[$(i)$]   $\displaystyle\liminf_{u\to 0^+}\frac{f(s,u)}{u} > M/k$
  for $s\in[a,b]$,
  \item[$(ii)$]  $f(s,u) < qm$ for $s\in[a,c]$ and $u\in[0,q/k]$,
  \item[$(iii)$] $\displaystyle\liminf_{u\to\infty}\frac{f(s,u)}{u} > M$
  for $s\in[a,b]$,
\end{enumerate}
then the second-order boundary-value problem \eqref{bvp}, \eqref{bc},
has at least two positive solutions $u_1$ and $u_2$ such that
\begin{gather*}
 p < \max_{t\in[a,c]}u_1(t) \quad \text{with } \max_{t\in[b,c]}u_1(t) < q, \\
 q < \max_{t\in[b,c]}u_2(t) \quad \text{with } \min_{t\in[a,b]}u_2(t) < r.
\end{gather*}
\end{cor}

\begin{proof}
  From assumption $(i)$ of the corollary we know there exists $p\in(0,q)$ such
that
$$ \frac{f(s,u)}{u} > \frac{M}{k}, \quad u\in(0,p], \; s\in[a,b]. $$
In particular,
$$ f(s,u) > uM/k \ge pk(M/k) = pM, \quad u\in[kp,p], \; s\in[a,b],
$$
and $(i)$ of Theorem \ref{averyhen2} holds.  Now let
$$ f_\infty:=\liminf_{u\to\infty}\Big(\min_{s\in[a,b]}\frac{f(s,u)}{u}\Big),
$$
and $\eta\in(M,f_\infty)$.  Then there exists $r'>q$ such that
$\min_{s\in[a,b]}f(s,u)\ge \eta u$, $u\in[r',\infty)$.
Set
$$ \varpi:=\min\big\{\min_{s\in[a,b]}f(s,u):u\in[0,r']\big\} $$
and take
$$ r>\max\big\{r', \frac{\varpi}{\eta-M}\big\}. $$
Then
$$
\min_{s\in[a,b]}f(s,u)\ge \eta u-\varpi \ge \eta r-\varpi > rM,
\quad u\in[r,\infty), $$
so that $(iii)$ of Theorem \ref{averyhen2} holds.  The conclusion follows.
\end{proof}

Similar to Corollary \ref{ahcor}, one can prove the following statement.


\begin{cor} \label{coro7}
Let $d>0$.  If there exists $q>0$ such that the function $f$ satisfies the following conditions:
\begin{enumerate}
  \item[$(i)$]   $\displaystyle\limsup_{u\to 0^+}\frac{f(s,u)}{u} < m$ for $s\in[a,c]$,
  \item[$(ii)$]  $f(s,u) > qM$ for $s\in[a,b]$ and $u\in[kq,q]$,
  \item[$(iii)$] $\displaystyle\limsup_{u\to\infty}\frac{f(s,u)}{u} < mk$ for $s\in[a,c]$,
\end{enumerate}
then the second-order boundary-value problem \eqref{bvp}, \eqref{bc},
has at least two positive solutions.
\end{cor}


\section{Existence of at Least Three Positive Solutions}

To prove the existence of at least three positive solutions
to \eqref{bvp}, \eqref{bc} we will use the Leggett-Williams
fixed point theorem \cite{gu,le}:

\begin{theorem}\label{LWTHM}
Let $P$ be a cone in the real Banach space $\mathcal{S}$, $A:\overline{P_r}
\to \overline{P_r}$ be completely continuous and $\phi $ be a
nonnegative continuous concave functional on $P$ with $\phi(u)\le\|u\|$
for all $u\in \overline{P_r}$. Suppose there exists $0<p<q<\ell\le r$ such
that the following conditions hold:
\begin{enumerate}
 \item[$(i)$] $\{u\in P(\phi,q,\ell): \phi(u) > q\}\neq\emptyset$ and $\phi(A u)>q$
 for all $u\in P(\phi,q,\ell)$;
 \item[$(ii)$] $\|A u\| < p$ for $\|u\|\le p$;
 \item[$(iii)$] $\phi(A u) > q$ for $u\in P(\phi,q,r)$ with $\|A u\| > \ell$.
\end{enumerate}
Then $A$ has at least three fixed points $u_1$, $u_2$, and $u_3$ in $\overline{P_r}$
satisfying:
$$ \|u_1\| < p, \quad \phi(u_2) > q, \quad p < \|u_3\|\;\;
\mbox{with}\;\; \phi(u_3) < q.$$
\end{theorem}

Again define the continuous concave functional $\phi:P\to[0,\infty)$ to be
$ \phi(u):=\min_{t\in[a,b]} u(t)$,
the cone $P$ as in \eqref{P}, $M$ as in \eqref{bigm}, and
$$   m:=d\Big((\gamma+\beta(c-a))\int_a^c(s-a+\delta)h(s)\nabla s \Big)^{-1}. $$
Moreover, we take
$$ P_r:= \{u\in P: \|u\| \ < r \},
\quad P(\phi,p,q) := \{u\in P : p\le\phi(u),\; \|u\| \le q \}. $$

\begin{theorem}\label{3sols}
Let $d>0$.  Suppose that there exist constants $0<p<q<q/k<r$ such that
\begin{enumerate}
 \item[(D1)] $f(s,u)\le rm$ for $s\in[a,c]$, $u\in[0,r]$;
 \item[(D2)] $f(s,u)\ge qM$ for $s\in[a,b]$, $u\in[q,q/k]$;
 \item[(D3)] $f(s,u)  < pm$ for $s\in[a,c]$, $u\in[0,p]$,
\end{enumerate}
where $k$ is given in \eqref{kdef}.  Then the boundary-value problem \eqref{bvp}, \eqref{bc}
has at least three positive solutions $u_1$, $u_2$, $u_3$ satisfying
$$ \|u_1\| < p, \quad q < \min_{t\in[a,b]}u_2(t), \quad \|u_3\| > p \mbox{ with }
\min_{t\in[a,b]}u_3(t) < q. $$
\end{theorem}

\begin{proof}
Again the solutions are the fixed points of the operator $A$ defined by
\begin{align*}
   A u(t) &= \frac{1}{d}\left(\gamma+\beta(c-t)\right)
   \Big[ \int_a^c (s-a+\delta)h(s)f(s,u(s))\nabla s\\
   &\quad - \alpha\int_b^c (s-b)h(s)f(s,u(s))\nabla s \Big]
 - \int_t^c (s-t)h(s)f(s,u(s))\nabla s.
\end{align*}
The conditions of Theorem \ref{LWTHM} will now be shown to be satisfied.  For all $u\in P$
we have $\phi(u)\le\|u\|$.  If $u\in\overline{P_r}$, then $\|u\|\le r$ and assumption
(D1) implies $f(s,u(s))\le rm$ for $s\in[a,c]$.  Consequently,
\begin{align*}
 \|A u\|
  & =  \max_{t\in[a,c]} A u(t) \\
  &\le \max_{t\in[a,c]} \frac{1}{d}\left(\gamma+\beta(c-t)\right)
  \Big[ \int_a^c (s-a+\delta)h(s)f(s,u(s))\nabla s\\
  &\quad - \alpha\int_b^c (s-b)h(s)f(s,u(s))\nabla s \Big] \\
  &\le \frac{1}{d}\left(\gamma+\beta(c-a)\right)
  \int_a^c (s-a+\delta)h(s)f(s,u(s))\nabla s \\
  & <  \frac{rm}{d}\left(\gamma+\beta(c-a)\right)
  \int_a^c (s-a+\delta)h(s)\nabla s \\
  & =  r.
\end{align*}
This proves that $A: \overline{P_r}\to\overline{P_r}$.  Similarly, if $u\in P_p$, then
assumption (D3) yields $f(s,u(s))<pm$ for $s\in[a,c]$.  Just as above, we have
$A: \overline{P_p}\to P_p$.  It follows that condition $(ii)$ of Theorem \ref{LWTHM} is satisfied.

We now consider condition $(i)$ of Theorem \ref{LWTHM}; pick $u_P(t)\equiv q/k$ for $t\in[a,c]$,
for $k$ given in \eqref{kdef}.  Then $u_P\in P(\phi,q,q/k)$ and $\phi(u_P)=\phi(q/k) > q$,
so that $\{u\in P(\phi,q,q/k): \phi(u)>q\}\neq\emptyset$.  Consequently, if $u\in P(\phi,q,q/k)$,
then $q\le u(s)\le q/k$ when $s\in[a,b]$.  From assumption (D2) we have that
$$ f(s,u(s))\ge qM$$ for $s\in[a,b]$.
As in Claim 3 of the proof of Theorem \ref{averyhen2},
$$ \min_{t\in[a,b]} A u(t) = \min\{A u(a), A u(b)\}. $$
If $\alpha\ge 1$, then $A u(a)\ge A u(b)$; if $0<\alpha<1$,
then $A u(a) \ge \alpha A u(b)$.
It follows that
\begin{align*}
 \phi (A u)
   & =  \min_{t\in[a,b]} A u(t) \\
   &\ge \min\{\alpha,1\}A u(b) \\
   &\ge \min\{\alpha,1\}\big(\frac{\gamma+\beta(c-b)}{d}\big)
   \int_a^b (s-a+\delta)h(s)f(s,u(s))\nabla s \\
   & > \min\{\alpha,1\}qM\big(\frac{\gamma+\beta(c-b)}{d}\big)
   \int_a^b (s-a+\delta)h(s)\nabla s \\
   &\ge q
\end{align*}
by hypothesis (D2) and \eqref{bigm}.  Therefore,
$$ \phi(A u) > q, \quad u\in P(\phi,q,q/k), $$
so that condition $(i)$ of Theorem \ref{LWTHM} holds.
To check on Theorem \ref{LWTHM} $(iii)$, we suppose that
$u\in P(\phi,q,r)$ with $\|A u\|>q/k$.  Then, Lemma \ref{lemma3} and the
definition of $\phi$ yield
$$ \phi(A u) = \min_{t\in[a,b]}A u(t) \ge k\|A u\| > kq/k = q. $$
\end{proof}

Using the ideas in the proof of the above theorem, we can establish the existence
of an arbitrary odd number of positive solutions of \eqref{bvp}, \eqref{bc},
assuming the right conditions on the nonlinearity $f$.

\begin{theorem} \label{thm10}
Let $d>0$.  Suppose that there exist constants
$$ 0 < p_1 < q_1 < q_1/k < p_2 < q_2 < q_2/k < p_3 < \dots < p_n, \quad n\in\{2,3,4,\cdots\}, $$
such that
\begin{enumerate}
 \item[(D1)] $f(s,u)\ge q_iM$ for $s\in[a,b]$, $u\in[q_i,q_i/k]$;
 \item[(D2)] $f(s,u)  < p_im$ for $s\in[a,c]$, $u\in[0,p_i]$,
\end{enumerate}
where $k$ is given in \eqref{kdef}.  Then the boundary-value problem
\eqref{bvp}, \eqref{bc}
has at least $2n-1$ positive solutions.
\end{theorem}

\section{Example}

Let $\mathbb{T}= \{1-(1/2)^{\mathbb{N}_0}\}\cup\{1\}$. Taking $a=0$, $b=31/32$, $c=\beta=1$, 
$\alpha=20$, $\gamma=1/19$, and
$\delta=3/4$, we have $d=1/8$ and $k=1/32$.  If we let $h(s)\equiv 1$, then $m=57/680$ and $M=77824/71145$.  Suppose
$$ f(t,u)=f(u):=\frac{2000e^u}{99e^{700}+e^u}, \quad t\in[0,1], \quad u\ge 0. 
$$
Clearly $f$ is always increasing.  If we take $p=701$, $q=705$, and $r=24000$, then
$$ 0 < p < q < q/k < r. $$
We check that (D1), (D2), and (D3) of Theorem \ref{3sols} are satisfied.
Since $f(715)\approx 1999.94$ and $\lim f(u)=2000$,
$$ f(u)\le 24000m\approx 2011.76, \quad u\in[0,r], 
$$
so that (D1) is met.  To verify (D2), note that $f(705)\approx 1199.72$, so that
$$ f(u)\ge 705M\approx 771.18, \quad u\in[q,32q]. $$
Lastly, as $f(701)\approx 53.45$,
$$ f(u) < 701m\approx 58.76, \quad u\in[0,p] $$
and (D3) holds.  Therefore by Theorem \ref{3sols}, the boundary-value problem
$$ u^{\Delta\nabla}(t)+f(u(t))=0, \quad u(0)=20u(31/32)+\frac{3}{4}u^\Delta(0), \quad u(1)+\frac{1}{19}u^\Delta(1)=0 $$
has at least three positive solutions $u_1$, $u_2$, $u_3$ satisfying
$$ \|u_1\| < 701, \quad 705 < \min_{t\in[0,31/32]}u_2(t), \quad \|u_3\| > 701
\quad \mbox{with }   \min_{t\in[0,31/32]}u_3(t) < 705. $$


\begin{thebibliography}{99}

\bibitem{akin}
  E. Akin,
  Boundary value problems for a differential equation on a measure chain,
  \emph{Panamerican Mathematical Journal}, 10:3 (2000) 17--30.

\bibitem{and}
  D. R. Anderson,
  Solutions to second-order three-point problems on time scales,
  \emph{J. Difference Eq. Appl.}, 8 (2002) 673--688.

\bibitem{and2}
  D. R. Anderson,
  Extension of a second-order multi-point problem to time scales,
  \emph{Dynamic Systems and Applications}, 12:3-4 (2003) 393--404.

\bibitem{ah}
  D. R. Anderson and J. Hoffacker,
  Green's Function for an Even Order Mixed Derivative Problem on Time Scales,
  \emph{Dynamic Systems and Applications}, 12:1-2 (2003) 9--22.

\bibitem{atici}
  F. M. Atici and G. Sh. Guseinov,
  On Green's functions and positive solutions for boundary value problems on time scales,
  \emph{J. Comput. Appl. Math.}, 141 (2002) 75--99.

\bibitem{aa}
  R. I. Avery and D. R. Anderson,
  Existence of three positive solutions to a second-order boundary value problem on a measure chain,
  \emph{J. Comput. Appl. Math.}, 141 (2002) 65--73.

\bibitem{AvHe}
  R.I. Avery and J. Henderson,
  Two positive fixed points of nonlinear operators on ordered Banach spaces,
  \emph{Comm. Appl. Nonlinear Anal.} 8 (2001) 27--36.

\bibitem{bp}
  M. Bohner and A. C. Peterson,
  \emph{Dynamic Equations on Time Scales: An Introduction with Applications},
  Birkh\"auser, Boston, 2001.

\bibitem{bp2}
  M. Bohner and A. C. Peterson, editors,
  \emph{Advances in Dynamic Equations on Time Scales},
  Birkh\"auser, Boston, 2003.

\bibitem{chyan}
  C. J. Chyan and J. Henderson,
  Twin solutions of boundary value problems for differential equations on measure chains,
  \emph{Journal of Computational and Applied Mathematics}, 141 (2002) 123--131.

\bibitem{erbepete1}
  L. H. Erbe and A. C. Peterson,
  Positive Solutions for a Nonlinear Differential Equation on a Measure Chain,
  {\em Mathematical and Computer Modelling} 32 (2000) 571--585.

\bibitem{erbepete2}
  L. H. Erbe and A. C. Peterson,
  Green's functions and comparison theorems for differential equations on measure chains,
  {\em Dynam. Continuous, Discrete \& Impulsive Systems}, 6 (1999) 121--137.

\bibitem{gu}
  D. Guo and V. Lakshmikantham,
  {\em Nonlinear Problems in Abstract Cones},
  Academic Press, San Diego, 1988.

\bibitem{he}
  X. He and W. Ge,
  Triple Solutions for Second-Order Three-Point Boundary Value Problems,
  {\em Journal of Mathematical Analysis and Applications}, 268 (2002) 256--265.

\bibitem{hend}
  J. Henderson,
  Multiple solutions for 2mth-order Sturm-Liouville boundary value problems on a measure chain,
  {\em J. Difference Eq. Appl.}, 6 (2000) 417--429.

\bibitem{hilger}
  S. Hilger,
  Analysis on measure chains--a unified approach to continuous and discrete calculus,
  \emph{Results Math.}, 18 (1990) 18--56.

\bibitem{kauf}
  E. R. Kaufmann,
  Positive solutions of a three-point boundary value problem on a time scale,
  {\em Electronic Journal of Differential Equations}, 2003:82 (2003) 1--11.

\bibitem{le}
  R. W. Leggett and L. R. Williams,
  {\em Multiple Positive Fixed Points of Nonlinear Operators on Ordered Banach Spaces},
  Indiana University Mathematics Journal, 28 (1979) 673--688.

\bibitem{ma1}
  R. Ma,
  Positive solutions of a nonlinear three-point boundary-value problem,
  {\em Electronic Journal of Differential Equations}, 1999:34 (1999) 1--8.

\bibitem{ma2}
  R. Ma,
  Multiplicity of positive solutions for second-order three-point boundary-value problems,
  {\em Computers \& Mathematics with Applications}, 40 (2000) 193--204.

\bibitem{ma3}
  R. Ma,
  Positive solutions for second-order three-point boundary-value problems,
  {\em Applied Mathematics Letters}, 14 (2001) 1--5.

\bibitem{li}
  H. R. Sun and W. T. Li,
  Positive solutions of nonlinear three-point boundary value problems on time scales,
  {\em Nonlinear Analysis TMA}, (2004) to appear.

\end{thebibliography}


\end{document}