\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 95, pp. 1--9.\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 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/95\hfil Three positive solutions]
{Three positive solutions for p-Laplacian functional dynamic
equations on time scales}

\author[D.-B. Wang\hfil EJDE-2007/95\hfilneg]
{Da-Bin Wang}

\address{Da-Bin Wang \newline
 Department of Applied Mathematics,
 Lanzhou University of Technology,
 Lanzhou, Gansu, 730050, China}
\email{wangdb@lut.cn}

\thanks{Submitted May 17, 2007. Published June 29, 2007.}
\subjclass[2000]{39A10, 34B15}
\keywords{Time scale; $p$-Laplacian functional dynamic equation;
\hfill\break\indent
boundary value problem; positive solution; fixed point}

\begin{abstract}
 In this paper, we establish the existence of three positive solutions
 to the following $p$-Laplacian functional dynamic equation on time scales,
 \begin{gather*}
 [ \Phi _p(u^{\Delta }(t))] ^{\nabla}+a(t)f(u(t),u(\mu (t)))=0,\quad
 t\in (0,T)_{\mathbf{T}}, \\
 u_0(t)=\varphi (t),\quad t\in [-r,0] _{\mathbf{T}},\\
 u(0)-B_0(u^{\Delta }(\eta ))=0,\quad u^{\Delta }(T)=0,.
 \end{gather*}
 using the fixed-point theorem due to Avery and Peterson \cite{a8}.
 An example is given to illustrate the main result.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks


\section{Introduction}

Let $\mathbf{T}$ be a time scale; i.e., $\mathbf{T}$ is a nonempty closed
subset of $R$. Let $0, T$ be points in $\mathbf{T}$,
an interval $(0,T) _{\mathbf{T}}$ denotes time scales interval,
that is, $(0,T) _{\mathbf{T}}:=(0,T) \cap \mathbf{T}$. Other types of
intervals are defined similarly.

The theory of dynamic equations on time scales has been a new important
mathematical branch (see, for example, \cite{a1,a2,b1,b2,k1})
 since it was initiated by Hilger \cite{h4}.
At the same time, boundary value problems (BVPs)
for dynamic equation on time scales have received considerable attention
\cite{a3,a4,a5,a6,c1,e1,h1,h2,h3,k2,s1,s2,s3,w1}.
 However, to the best of our knowledge, few papers can
be found in the literature on bvps of $p$-Laplacian dynamic equations on
time scales \cite{a5,h2,h3,s1,s2,w1},
 especially for $p$-Laplacian functional
dynamic equations on time scales \cite{s1}.

This paper concerns  the existence of positive solutions for the
$p$-Laplacian functional dynamic equation on time scale,
\begin{equation}
\begin{aligned}
\hspace{0.1cm}[\Phi_p(u^{\Delta }(t))]^{\nabla
}+a(t)f(u(t),u(\mu (t)))=0,\quad t\in (0,T) _{\mathbf{T}}, \\
u_0(t)=\varphi (t),\quad t\in [-r,0] _{\mathbf{T}},\\
u(0)-B_0(u^{\Delta }(\eta ))=0,\quad u^{\Delta }(T)=0,
\end{aligned}  \label{e1.1}
\end{equation}
where $\Phi _p(s)$ is $p$-Laplacian operator, i.e.,
$\Phi _p(s)=|s| ^{p-2}s$, $p>1$, $(\Phi _p)^{-1}=\Phi _q$,
 $\frac 1p+\frac 1q=1$, $\eta \in (0,\rho (T))_{\mathbf{T}}$ and
\begin{itemize}
\item[(C1)] $f:(\mathbb{R}^+) ^2\to \mathbb{R}^+$ is continuous;

\item[(C2)] $a:\mathbf{T}\to \mathbb{R}^+$ is left dense continuous
(i.e., $a\in C_{\mathbf{ld}}(\mathbf{T},\mathbb{R}^+)$) and does not vanish
identically on any closed subinterval of $[0,T] $, where
$C_{\mathbf{ld}}(\mathbf{T},\mathbb{R}^+)$ denotes the set of all left dense
continuous functions from $\mathbf{T}$ to $\mathbb{R}^+$;

\item[(C3)] $\varphi :[-r,0] _{\mathbf{T}}\to \mathbb{R}^+$ is
continuous and $r>0$;

\item[(C4)] $\mu :[0,T] _{\mathbf{T}}\to [-r,T]_{\mathbf{T}}$
is continuous, $\mu (t)\leq t$ for all $t$;

\item[(C5)] $B_0:R\to R$ is continuous and there exist constant
$A\geq 1$, $B>0$ such that
\[
Bv\leq B_0(v)\leq Av,\text{ for all }v\geq 0.
\]
\end{itemize}

In \cite{s1}, by using a double-fixed-point theorem due to Avery et
al. \cite{a7} in a cone, Song and Xiao considered the problem
\eqref{e1.1} and obtained the existence of two positive solutions.

In paper \cite{h3}, Hong studied the problem \eqref{e1.1} when
$\varphi (t)=0,t\in [-r,0] _{\mathbf{T}}$ and the nonlinear term
is not involved $u(\mu (t))$. He imposed conditions on $f$ to yield at
least three positive solutions to the problem \eqref{e1.1},
by applying the fixed-point theorem due to
Avery and Peterson \cite{a8}.

Motivated by \cite{h3,s1}, we shall show that the problem \eqref{e1.1},
has at least three positive solutions by means of the fixed point
theorem due to Avery and Peterson.

In the remainder of this section we list the following well known
definitions which can be found in \cite{a2,a6,b1,b2}.

\begin{definition} \label{def1.1} \rm
For  $t<\sup \mathbf{T}$ and $r>\inf\mathbf{T}$, define the forward
jump operator $\sigma $ and the backward jump operator $\rho $,
\[
\sigma (t)=\inf \{\tau \in \mathbf{T}|\tau >t\}\in \mathbf{T},\quad
\rho (r)=\sup \{\tau \in \mathbf{T}|\tau <r\}\in \mathbf{T}
\]
for all $t$, $r\in \mathbf{T}$. If $\sigma (t)>t$, $t$ is said to be right
scattered, and if $\rho (r)<r$, $r$ is sad to be left scattered. If
$\sigma (t)=t$, $t$ is said to be right dense, and if $\rho (r)=r$, $r$
is said to be left dense. If $\mathbf{T}$ has a right scattered minimum $m$,
define $\mathbf{T}_k=\mathbf{T}-\{m\}$; otherwise set
$\mathbf{T}_k=\mathbf{T}$. If $%
\mathbf{T}$ has a left scattered maximum $M$, define $\mathbf{T}^k=\mathbf{T}%
-\{M\}$; otherwise set $\mathbf{T}^k=\mathbf{T}$.
\end{definition}

\begin{definition} \label{def1.2} \rm
For $x:\mathbf{T\to }R$ and $t\in
\mathbf{T}^k$, we define the delta derivative of $x(t)$, $x^{\Delta }(t)$,
to be the number (when it exists), with the property that, for any
$\varepsilon >0$, there is a neighborhood $U$ of $t$ such that
\[
| [x(\sigma (t))-x(s)] -x^{\Delta }(t)[\sigma
(t)-s] | <\varepsilon | \sigma (t)-s| ,
\]
for all $s\in U$. For $x:\mathbf{T\to }R$ and $t\in \mathbf{T}_k$,
 we define the nabla derivative of $x(t)$, $x^\nabla (t)$, to be the
number (when it exists), with the property that, for any $\varepsilon >0$,
there is a neighborhood $V$ of $t$ such that
\[
| [x(\rho (t))-x(s)] -x^\nabla (t)[\rho (t)-s]| <\varepsilon | \rho (t)-s| ,
\]
for all $s\in V$.
If $\mathbf{T}=R$, then $x^{\Delta }(t)=x^\nabla (t)=x^{\prime }(t)$.
If $\mathbf{T}=Z$, then $x^{\Delta }(t)=x(t+1)-x(t)$ is the forward
difference operator while $x^\nabla (t)=x(t)-x(t-1)$ is the backward
difference operator.
\end{definition}

\begin{definition} \label{def1.3}\rm
If $F^{\Delta }(t)=f(t)$, then we define the
delta integral by
\[
\int_a^tf(s)\Delta s=F(t)-F(a).
\]
If $\Phi ^\nabla (t)=f(t)$, then we define the nabla integral by
\[
\int_a^tf(s)\nabla s=\Phi (t)-\Phi (a).
\]
\end{definition}

Throughout this papers, we assume $\mathbf{T}$ is closed subset of
$\mathbb{R}$ with $0\in \mathbf{T}_k$ and $T\in \mathbf{T}^k$.


\begin{lemma}[\cite{a6}] \label{lem1.1}
The following formulas hold:
\begin{itemize}
\item[(i)] $\Big(\int_a^tf(s)\Delta s\Big) ^{\Delta }=f(t)$,

\item[(ii)] $\Big(\int_a^tf(s)\Delta s\Big) ^{\nabla }=f(\rho (t))$,

\item[(iii)] $\Big(\int_a^tf(s)\nabla s\Big) ^{\Delta }=f(\sigma(t))$,

\item[(iv)] $\Big(\int_a^tf(s)\nabla s\Big) ^{\nabla }=f(t)$.
\end{itemize}
\end{lemma}

\section{Preliminaries}

In this section, we provide some background materials from the theory of
cones in Banach spaces and we then state the fixed-point theorem due to
Avery and Peterson.

\begin{definition} \label{def2.1} \rm
 Let $E$ be a real Banach space. A nonempty, closed,
convex set $P\subset E$ is said to be a cone provided the following
conditions are satisfied:
\begin{itemize}
\item[(i)] if $x\in P$ and $\lambda \geq 0$, then $\lambda x\in P$;

\item[(ii)] if $x\in P$ and $-x\in P$, then $x=0$.
\end{itemize}
Every cone $P\subset E$ induces an ordering in $E$ given by
\[
x\leq y \quad\text{if }y-x\in P.
\]
\end{definition}

\begin{definition} \label{def2.2}\rm
 Given a cone $P$ in a real Banach space $E$, the map
$\varsigma :P\to [0,\infty ) $ is called a nonnegative
continuous concave function on cone $P$ provided that $\varsigma $ is
continuous and
\[
\varsigma (tx+(1-t)y)\geq t\varsigma (x)+(1-t)\varsigma (y), \quad
\text{for }x,\;  y\in P\text{ and }0\leq t\leq 1.
\]
Dual to this, we call the map $\tau :P\to [0,\infty ) $
is called a nonnegative continuous convex function on cone $P$ provided that
$\tau $ is continuous and
\[
\tau (tx+(1-t)y)\leq t\tau (x)+(1-t)\tau (y),\quad\text{for }x,\;
y\in P \text{ and }0\leq t\leq 1.
\]
Let $\gamma $ and $\theta $ be nonnegative continuous convex functions on
$P$, $\alpha $ be a nonnegative continuous concave function on $P$ and
$\psi $ be a nonnegative continuous function on $P$.
Then, for positive real numbers $a$, $b$, $c$ and $d$, we define the
following convex sets
\begin{gather*}
P(\gamma ,d) =\{ x\in P:\gamma (x)<d\} , \\
P(\gamma ,\alpha ,b,d) =\left\{ x\in P:b\leq \alpha (x),\; \gamma
(x)\leq d\right\} , \\
P(\gamma ,\theta ,\alpha ,b,c,d) =\{ x\in P:b\leq \alpha (x),\;
\theta (x)\leq c,\; \gamma (x)\leq d\} ,
\end{gather*}
and a closed set
\[
R(\gamma ,\psi ,a,d)=\{ x\in P: a\leq \psi (x),\; \gamma (x)\leq d\} .
\]
\end{definition}

To prove our main results, we need the following fixed-point theorem due to
Avery and Peterson in \cite{a8}.

\begin{theorem} \label{thm2.1}
Let $P$ be a cone in a real Banach space $E$. Let $\gamma $ and
 $\theta $ be nonnegative continuous convex functionals on $P$,
$\alpha $ be a nonnegative continuous concave functionals on $P$ and
$\psi $ be a nonnegative continuous functional on $P$ satisfying
$\psi (\lambda x)\leq \lambda \psi (x)$ for $0\leq \lambda \leq 1$,
such that for some positive numbers $h$ and $d$,
\[
\alpha (x)\leq \psi (x)\quad\text{and}\quad \| x\|  \leq h\gamma
(x),
\]
for all $x\in \overline{P(\gamma ,d)}$. Suppose that
\[
F:\overline{P(\gamma ,d)}\to \overline{P(\gamma ,d)}
\]
is completely continuous and there exist positive numbers $a$, $b$ and $c$
with $a<b$ such that:
\begin{itemize}
\item[(i)] $\{ x\in P(\gamma ,\theta ,\alpha ,b,c,d):\alpha (x)>b\}
 \neq \emptyset $ and $\alpha (Fx)>b$ for
$x$ in the set $P(\gamma ,\theta ,\alpha ,b,c,d)$;

\item[(ii)] $\alpha (Fx)>b$ for $x\in P(\gamma ,\alpha ,b,d)$ with
$\theta (Fx)>c$;

\item[(iv)] $0\notin R(\gamma ,\psi ,a,d)$ and $\psi (Fx)<a$ for
$x\in R(\gamma ,\psi ,a,d)$ with $\psi (x)=a$.

\end{itemize}
Then $F$ has at least three fixed points
$x_1,x_2,x_3\in \overline{P(\gamma ,d)}$ such that
$\gamma (x_i)\leq d$ for $i=1, 2, 3$, and $b<\alpha (x_1)$,
$a<\psi (x_2)$  with $\alpha (x_2)<b$  and $\psi (x_3)<a$.
\end{theorem}

\section{Existence of Three Positive Solutions}

We note that $u(t)$ is a solution of  \eqref{e1.1} if and only if
\[
u(t)=\begin{cases}
B_0(\Phi _q(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r)) \\
+\int_0^t\Phi _q(\int_s^Ta(r)f(u(r),u(\mu (r)))\nabla r)
\Delta s,
& t\in [0,T] _{\mathbf{T}}, \\[5pt]
\varphi (t), & t\in [-r,0] _{\mathbf{T}}.
\end{cases}
\]
Let $E=C_{\mathbf{ld}}^{\Delta }([0,T] _{\mathbf{T}},R)$ be
endowed with the norm
\[
\|u\| =\max \big\{ \max_{t\in [0,T] _{\mathbf{T}}}| u(t)| ,\;
 \max_{t\in [0,T] _{\mathbf{T}^k}}| u^{\Delta }(t)| \big\} ,
\]
 so $E$ is a Banach space. Define cone $P\subset E$ by
\[
P=\big\{ u\in E:u\text{ is concave and nonnegative valued on }
[0,T] _{\mathbf{T}},\text{ and }u^{\Delta }(T)=0\big\} .
\]
For each $u\in E$, extend $u(t)$ to $[-r,T] _{\mathbf{T}}$ with
$u(t)=\varphi (t)$ for $t\in [-r,0] _{\mathbf{T}}$.
Define a completely continuous operator $F$: $P\to E$ by
\begin{align*}
(Fu)(t)= & B_0(\Phi _q(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla
r) ) \\
& +\int_0^t\Phi _q(\int_s^Ta(r)f(u(r),u(\mu (r)))\nabla r)
\Delta s,\quad t\in [0,T] _{\mathbf{T}}.
\end{align*}
We seek for a point, $u_1$, of $F$ in the cone $P$. Define
\[
u(t)=\begin{cases}
u_1(t), & t\in [0,T] _{\mathbf{T}}, \\
\varphi (t), & t\in [-r,0] _{\mathbf{T}}.
\end{cases}
\]
Then $u(t)$ denotes a positive solution of \eqref{e1.1}.

\begin{lemma} \label{lem3.1}
If $u\in P$, then
\begin{itemize}
\item[(i)] $u(t)\geq \frac tT$
$\max_{t\in [0,T] _{\mathbf{T}}}|u(t)| $, $t\in [0,T] _{\mathbf{T}}$.

\item[(ii)] $u(t)$ is increasing on $t\in [0,T] _{\mathbf{T}}$.

\item[(iii)] $u^{\Delta }(t)$ is decreasing on
$t\in [0,T] _{\mathbf{T}^k}$.
\end{itemize}
\end{lemma}

\begin{proof}
Part (i) is  of \cite[Lemma 3.1]{h2}. Parts (ii) and (iii) are easy, so we
omit them here.
\end{proof}

\begin{lemma} \label{lem3.2}
The operator $F$ maps  $P$ into $P$.
\end{lemma}

\begin{proof} For each  $u\in P$, $Fu\in E$ and $(Fu) (t)\geq 0$,
for all $t\in [0,T] _{\mathbf{T}}$. It follows from Lemma \ref{lem1.1} that
\[
(Fu) ^{\Delta }(t)=\Phi _q(\int_t^Ta(r)f(u(r),\varphi
(r))\nabla r) .
\]
Obviously $(Fu) ^{\Delta }(t)$ is a continuous function and
$(Fu) ^{\Delta }(t)\geq 0$, that is $(Fu) (t)$ is
increasing on $[0,T] _{\mathbf{T}}$. Note that $\Phi _q$ is
increasing, we have that $(Fu) ^{\Delta }(t)$ is decreasing.

If $t\in [0,T]_{\mathbf{T}^k\cap \mathbf{T}_k}$, then from
 \cite[Theorem 2.3]{a6}
it follows that $(Fu) ^{\Delta \nabla }(t)\leq 0$;
i.e., $Fu$ is concave on $[0,T]_{\mathbf{T}}$.
This implies that $Fu\in P$ and $F:P\to P$.
\end{proof}

Let $l\in \mathbf{T}$ be fixed such that $0<\eta <l<T$, and set
\[
Y_1=\big\{ t\in [0,T] _{\mathbf{T}}:\mu (t)\leq 0\big\} ;\quad
Y_2=\big\{ t\in [0,T] _{\mathbf{T}}:\mu (t)>0\big\} ; \quad
Y_3=Y_1\cap [l,T] _{\mathbf{T}}.
\]
Throughout this paper, we assume $Y_3\neq \phi $ and
$\int_{Y_3}a(r)\nabla r>0$.

Define the nonnegative continuous concave functionals $\alpha $, the
nonnegative continuous convex functionals $\theta $, $\gamma $, and the
nonnegative continuous functionals $\psi $ on the cone $P$ respectively as
\begin{gather*}
\gamma (u) =\|u\| ,\quad
\theta (u)=\max_{t\in [l,T] _{\mathbf{T}^k}}u^{\Delta }(t), \\
\alpha (u)=\min_{t\in [\eta ,l] _{\mathbf{T}}}u(t),\quad
\psi (u)=\min_{t\in [\eta ,T] _{\mathbf{T}}}u(t).
\end{gather*}
In addition, by Lemma \ref{lem3.1}, we have $\alpha (u)=\psi (u)=u(\eta )$,
$\theta (u)=u^{\Delta }(l)$ for each $u\in P$.
For convenience, we define
\begin{gather*}
\rho =(A+T)\Phi _q(\int_0^Ta(r)\nabla r) ,\quad
\delta =(B+\eta )\Phi_q(\int_{Y_3}a(r)\nabla r) , \\
\lambda =(A+\eta )\Phi _q(\int_0^Ta(r)\nabla r).
\end{gather*}
We now state growth conditions on $f$ so that  \eqref{e1.1} has at least
three positive solutions.


\begin{theorem} \label{thm3.1}
Let $0<\frac T\eta a<b<d$, $\rho b<\delta d$, and
suppose that $f$ satisfies the following conditions:
\begin{itemize}

\item[(H1)] $f(u,\varphi (s))\leq \Phi _p(\frac d\rho ) $, if
$0\leq u\leq d$, uniformly in $s\in [-r,0] _{\mathbf{T}}$;
$f(u_1,u_2)\leq \Phi _p(\frac d\rho )$, if $0\leq u_i\leq d$, $i=1,2$,

\item[(H2)] $f(u,\varphi (s))>\Phi _p(\frac b\delta ) $,
if $b\leq u\leq d$, uniformly in $s\in [-r,0] _{\mathbf{T}}$,

\item[(H3)] $f(u,\varphi (s))<\Phi _p(\frac a\lambda ) $, if
$0\leq u\leq \frac T\eta a$, uniformly in $s\in [-r,0] _{\mathbf{T}}$;
$f(u_1,u_2)<\Phi _p(\frac a\lambda )$, if
$0\leq u_i\leq \frac T\eta a$, $i=1,2$.
\end{itemize}
Then \eqref{e1.1} has at least three positive solutions of the form
\[
u(t)=\begin{cases}
u_i(t), & t\in [0,T] _{\mathbf{T}},\quad i=1,2,3, \\
\varphi (t), & t\in [-r,0] _{\mathbf{T}},
\end{cases}
\]
where $\gamma (u_i)\leq d$ for $i=1$, $2$, $3$, $b<\alpha (u_1)$,
$a<\psi (u_2)$ with $\alpha (u_2)<b$ and $\psi (u_3)<a$.
\end{theorem}

\begin{proof}
We first assert that $F:\overline{P(\gamma ,d)}\to \overline{P(\gamma ,d)}$.
Let $u\in \overline{P(\gamma ,d)}$, then
$\gamma (u)=\|u\| \leq d$, consequently, $0\leq u(t)\leq d$ for
$t\in [0,T] _{\mathbf{T}}$.
From (H1), we have
\begin{align*}
&| (Fu)(t)|\\
&= B_0\Big(\Phi _q\Big(\int_\eta
^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big) \Big)\\
&\quad +\int_0^t\Phi _q\Big(
\int_s^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big) \Delta s \\
&\leq A\Phi _q\Big(\int_0^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big)
+T\Phi _q\Big(\int_0^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big) \\
&=(A+T)\Phi _q\Big[\int_{Y_1}a(r)f(u(r),\varphi (\mu (r)))\nabla r
+\int_{Y_2}a(r)f(u(r),u(\mu (r)))\nabla r\Big] \\
&\leq (A+T)\Phi _q\Big(\int_0^Ta(r)\nabla r\Big) \frac d\rho
=d,
\end{align*}
and
\begin{align*}
| (Fu)^{\Delta }(t)|
&=\Phi _q\Big(\int_t^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big) \\
&\leq \Phi _q\Big(\int_0^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big) \\
&= \Phi _q\Big[\int_{Y_1}a(r)f(u(r),\varphi (\mu (r)))\nabla
r+\int_{Y_2}a(r)f(u(r),u(\mu (r)))\nabla r\Big] \\
&\leq \Phi _q\Big(\int_0^Ta(r)\nabla r\Big) \frac d\rho \\
&= \frac d{(A+T)}
\leq d.
\end{align*}
Therefore $F(u)\in \overline{P(\gamma ,d)}$, i.e.,
$F:\overline{P(\gamma ,d)}\to \overline{P(\gamma ,d)}$.

Secondly, we assert that $\{ u\in P(\gamma ,\theta ,\alpha ,b,c,d):%
\quad \alpha (u)>b\} \neq \phi $ and $\alpha (Fu)>b$ for
$u\in P(\gamma ,\theta ,\alpha ,b,c,d)$.

Let $u(t)=kb$ with $k=\frac \rho \delta >1$, then $u(t)=kb>b$ and
$\theta (u)=0<b$. Furthermore, by $\rho b<\delta d$ we have
$\gamma (u)\leq d$. Let $c=kb$, then
\[
\left\{ u\in P(\gamma ,\theta ,\alpha ,b,c,d):\quad \alpha
(u)>b\right\} \neq \emptyset.
\]
Moreover, for all $u\in P(\gamma ,\theta ,\alpha ,b,kb,d)$, we have
$b\leq u(t)\leq d$, $t\in [\eta ,T] _{\mathbf{T}}$.
From (H2), we see that
\begin{align*}
&\alpha (Fu) \\
&= (Fu)(\eta ) \\
&= B_0(\Phi _q(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r)
) +\int_0^\eta \Phi _q(\int_s^Ta(r)f(u(r),u(\mu (r)))\nabla
r) \Delta s \\
&\geq  B\Phi _q(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r)
+\eta \Phi _q(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r) \\
&\geq  (B+\eta )\Phi _q(\int_l^Ta(r)f(u(r),u(\mu (r)))\nabla r)
\\
&\geq  (B+\eta )\Phi _q(\int_{Y_3}a(r)f(u(r),\varphi (\mu (r)))\nabla
r) \\
&> (B+\eta )\Phi _q(\int_{Y_3}a(r)\nabla r) \frac b\delta
=b,
\end{align*}
as required.

Thirdly, we assert that $\alpha (Fu)>b$ for $u\in P(\gamma ,\alpha
,b,d) $ with $\theta (Fu)>c$. For all $u\in P(\gamma ,\alpha ,b,d) $
with $\theta (Fu)>kb$, from Lemma \ref{lem3.1} we have
\[
\theta (Fu)=(Fu)^{\Delta }(l)=\Phi _q\Big(\int_l^Ta(r)f(u(r),u(\mu
(r)))\nabla r\Big) >kb.
\]
So,
\begin{align*}
&\alpha (Fu) \\
&=(Fu)(\eta ) \\
&=B_0\Big(\Phi _q\Big(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big)\Big)
+\int_0^\eta \Phi _q(\int_s^Ta(r)f(u(r),u(\mu (r)))\nabla
r) \Delta s \\
&\geq  B\Phi _q\Big(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big)
+\eta \Phi _q(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r) \\
&\geq (B+\eta ) \Phi _q\Big(\int_l^Ta(r)f(u(r),u(\mu
(r)))\nabla r\Big) \\
&>(B+\eta ) kb=(B+\eta ) \frac \rho \delta b\\
&\geq (A+T) b>b.
\end{align*}
This implies that $\alpha (Fu)>b$ for $u\in P(\gamma ,\alpha,b,d) $
with $\theta (Fu)>c$.

Finally, we assert that $0\notin R(\gamma ,\psi ,a,d) $ and
$\psi (Fu)<a$ for $u\in R(\gamma ,\psi ,a,d) $ with $\psi (u)=a$ .

Since $\psi (0)=0<a$, we have $0\notin R(\gamma ,\psi ,a,d) $. For
all $u\in R(\gamma ,\psi ,a,d) $ with $\psi (u)=\min_{t\in [\eta ,T]
_{\mathbf{T}}}u(t)=u(\eta )=a$, by Lemma \ref{lem3.1} we have $0\leq
u(t)\leq \frac T\eta a$, for $t\in [0,T] _{\mathbf{T}}$. From (H3),
we have
\begin{align*}
\psi (Fu)
&=(Fu)(\eta ) \\
&=B_0\Big(\Phi _q\Big(\int_\eta ^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big)
\Big) \\
&\quad +\int_0^\eta \Phi _q\Big(\int_s^Ta(r)f(u(r),u(\mu (r)))\nabla
r\Big) \Delta s \\
&\leq A\Phi _q\Big(\int_0^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big)
+\eta \Phi _q\Big(\int_0^Ta(r)f(u(r),u(\mu (r)))\nabla r\Big) \\
&=(A+\eta )\Phi _q\Big[\int_{Y_1}a(r)f(u(r),\varphi (\mu (r)))\nabla
r+\int_{Y_2}a(r)f(u(r),u(\mu (r)))\nabla r\Big] \\
&<(A+\eta )\Phi _q\Big(\int_0^Ta(r)\nabla r\Big) \frac a\lambda
=a,
\end{align*}
which shows that condition (iv) of Theorem \ref{thm2.1} is fulfilled.

Thus, all the conditions of Theorem \ref{thm2.1} are satisfied. Hence, $F$ has at
least three fixed points $u_1$, $u_2$, $u_3$ satisfying
$\gamma (u_i)\leq d $  for $i=1,2,3$,
$b<\alpha (u_1)$, $a<\psi(u_2)$  with $\alpha (u_2)<b$
 and $\psi (u_3)<a$.
Let
\[
u(t)=\begin{cases}
u_i(t), & t\in [0,T] _{\mathbf{T}},\quad i=1,2,3, \\
\varphi (t), & t\in [-r,0] _{\mathbf{T}},
\end{cases}
\]
which are three positive solutions of \eqref{e1.1}.
\end{proof}

\section{Example}

Let $\mathbf{T}=[-\frac 34,-\frac 14] \cup \{ 0,\frac 34\} \cup
\{ (\frac 12) ^{\mathbb{N}_0}\} $,
where $\mathbb{N}_0$ denotes the set of all nonnegative integers.
Consider the following $p$-Laplacian functional dynamic equation on time
scale $\mathbf{T}$,
\begin{equation}
\begin{gathered}
\hspace{0.1cm}[\Phi _p(u^{\Delta }(t))] ^{\nabla
}+\frac{8u^3(t)}{u^3(t)
+u^3(t-\frac 34)+1}=0,\quad t\in (0,1) _{\mathbf{T}}, \\
u_0(t)=\varphi (t)\equiv 0,\quad
t\in [-\frac 34,0] _{\mathbf{T}},\\
 u(0)-B_0(u^{\Delta }(\frac 14))=0, \quad u^{\Delta }(1)=0,
\end{gathered}   \label{e4.1}
\end{equation}
where $T=1$, $p=\frac 32$, $a(t)\equiv 1,B=1$, $A=1$,
$\mu :[0,1] _{\mathbf{T}}\to [-\frac 34,1] _{\mathbf{T}}$
and $\mu (t)=t-\frac 34$, $r=\frac 34$, $\eta =\frac 14$, $l=\frac 12$ and
$f(u,\varphi (s))=\frac{8u^3}{u^3+1}$,
$f(u_1,u_2)=\frac{8u_1^3}{u_1^3+u_2^3+1}$.
We deduce that $Y_1=[0,\frac 34] _{\mathbf{T}}$,
$Y_2=(\frac 34,1] _{\mathbf{T}}$,
$Y_3=[\frac 12,\frac 34] _{\mathbf{T}}$.

Thus it is easy to see by calculating that $\rho =2$, $\delta =\frac 5{64}$,
$\lambda =\frac 54$.
Choose $a=\frac 1{40}$, $b=1$, $d=140$, then we have $0<\frac T\eta a<b\ <d$,
$\rho b<\delta d$, then
\begin{gather*}
f(u,\varphi (s))<8<\Phi _p(\frac d\rho )=\sqrt{\frac{140}2}\approx 8.3666,
\quad 0\leq u\leq 140;
\\
f(u_1,u_2)<8<\Phi _p(\frac d\rho )=\sqrt{\frac{140}2}\approx 8.3666, \quad
0\leq u\leq 140,
\\
f(u,\varphi (s))\geq 4>\Phi _p(\frac b\delta )=\sqrt{\frac{64}5}\approx
3.5777, \quad 1\leq u\leq 140,
\\
f(u,\varphi (s))\leq \frac 8{1001}\approx 0.008<\Phi _p(\frac a\lambda )=
\sqrt{\frac 1{50}}\approx 0.1414, \quad
0\leq u\leq \frac 1{10};
\\
f(u_1,u_2)\leq \frac 8{1002}\approx 0.008<\Phi _p(\frac a\lambda )
=\sqrt{\frac 1{50}}\approx 0.1414, \quad
0\leq u\leq \frac 1{10},
\end{gather*}
Thus by Theorem \ref{thm3.1}, the \eqref{e4.1} has at least three positive
solutions of the form
\[
u(t)=\begin{cases}
u_i(t), & t\in [0,1] _{\mathbf{T}},\quad i=1,2,3, \\
\varphi (t), & t\in [-\frac 34,0] _{\mathbf{T}},
\end{cases}
\]
where $\gamma (u_i)\leq 140$ for $i=1,2,3$, $1<\alpha (u_1)$,
$\frac 1{40}<\psi (u_2)$ with $\alpha (u_2)<1$ and
$\psi (u_3)<\frac 1{40}$.

\subsection*{Acknowledgment}
The author would like to thank the anonymous referees for their
valuable suggestions which led to an improvement of this paper.

\begin{thebibliography}{99}

\bibitem{a1}  R. P. Agarwal and M. Bohner; Quadratic functionals for second
order matrix equations on time scales, \emph{Nonlinear Anal.}, {33}%
(1998), 675--692.

\bibitem{a2}  R. P. Agarwal and M. Bohner; Basic calculus on time scales and
some of its applications, \emph{Results Math.}, 35(1999), 3--22.

\bibitem{a3}  R. P. Agarwal, M. Bohner and P. J. Y. Wong; Sturm-Liouville
eigenvalue problems on time scales, \emph{Appl. Math. Comput.}, {99}
(1999), 153--166.

\bibitem{a4}  D. R. Anderson; Solutions to second-order three-point problems
on time scales, \emph{J. Difference. Equations Appl.}, {8}(2002), 673--688.

\bibitem{a5}  D. R. Anderson, R. Avery and J. Henderson, Existence of
solutions for a one dimensional $p$-Laplacian on time-scales, \emph{J.
Difference. Equations Appl.}, {10}(2004), 889--896.

\bibitem{a6}  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{a7}  R. I. Avery, C. J. Chyan and J. Henderson; Twin solutions of a
boundary value problems for ordinary differential equations and finite
difference equations, \emph{Comput. Math. Appl.}, {42}(2001), 695-704.

\bibitem{a8}  R. I. Avery, A. C. Peterson; Three positive fixed points of
nonlinear operators on ordered Banach spaces, \emph{Comput. Math. Appl.},
{42}(2001), 313-322.

\bibitem{b1}  M. Bohner and A. Peterson; \emph{Dynamic Equations on Time Scales: An
Introduction with Applications}, Birkhauser, Boston, 2001.

\bibitem{b2}  M. Bohner and A. Peterson; \emph{Advances in Dynamic Equations on
Time Scales}, Birkhauser, Boston, 2003.

\bibitem{c1}  C. J. Chyan and J. Henderson; Eigenvalue problems for
nonlinear differential equations on a measure chain, \emph{J. Math. Anal.
Appl}, {245}(2000), 547--559.

\bibitem{e1}  L. Erbe and A. Peterson; Positive solutions for a nonlinear
differential equation on a measure chain, \emph{Math. Comput. Modelling.}, {%
32}(5-6)(2000)571--585.

\bibitem{h1}  J. Henderson; Multiple solutions for 2m$^{th}$-order
Sturm-Liouville boundary value problems on a measure chain, \emph{J.
Difference Equations. Appl}., {6}(2000), 417--429.

\bibitem{h2}  Z. M. He; Double positive solutions of three-point boundary
value problems for $p$-Laplacian dynamic equations on time scales, \emph{J.
Comput. Appl. Math}., {182}(2005), 304--315.

\bibitem{h3}  S. H. Hong; Triple positive solutions of three-point boundary
value problems for $p$-Laplacian dynamic equations on time scales,
\emph{J. Comput. Appl. Math}., 206(2007), 967-976.

\bibitem{h4}  S. Hilger; Analysis on measure chains-a unified approach to
continuous and discrete calculus, \emph{Results Math.}, {18}(1990), 18--56.

\bibitem{k1}  B. Kaymakcalan, V. Lakshmikantham and S. Sivasundaram;
\emph{Dynamical Systems on Measure Chains}, Kluwer Academic
Publishers, Boston, 1996.

\bibitem{k2}  E. R. Kaufmann, Y. N. Raffoul; Positive solutions of nonlinear
functional dynamic equations on a time scales, \emph{Nonlinear Anal.},
62(2005), 1267-1276.

\bibitem{s1}  C. X. Song, C. T. Xiao; Positive solutions for $p$-Laplacian
functional dynamic equations on time scales, \emph{Nonlinear Anal.},
66(2007), 1989-1998.

\bibitem{s2}  H. R. Sun, W. T. Li; Positive solutions for nonlinear $m$%
-point boundary value problems on time scales, \emph{Acta. Math. Sinica}., {%
49}(2006), 369-380 (in Chinese).

\bibitem{s3}  J. P. Sun; A new existence theorem for right focal boundary
value problems on a measure chain, \emph{Appl. Math. Letters}., {18}%
(2005), 41-47.

\bibitem{w1}  D. B. Wang; Existence, multiplicity and infinite solvability
of positive solutions for $p$-Laplacian dynamic equations on time scales,
\emph{Electronic J. Differential equations}., vol. 2006 (2006) no. 96. 1-10.
\end{thebibliography}

\end{document}