\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2006(2006), No. 96, 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  (login: ftp)}
\thanks{\copyright 2006 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2006/96\hfil Existence, multiplicity and infinite solvability]
{Existence, multiplicity and infinite solvability of positive
solutions for $p$-Laplacian dynamic equations on time scales}

\author[D.-B. Wang\hfil EJDE-2006/96\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}

\date{}
\thanks{Submitted April 14, 2006. Published August 22, 2006.}
\subjclass[2000]{34B10, 34B18, 39A10}
\keywords{Time scales; $p$-Laplacian; boundary value problem;
\hfill\break\indent
 positive solution; existence; multiplicity; infinite solvability}

\begin{abstract}
 In this paper, by using Guo-Krasnosel'skii fixed point theorem in cones,
 we study the existence, multiplicity and infinite solvability of positive
 solutions for the following three-point boundary value problems for
 $p$-Laplacian dynamic equations on time scales
 \begin{gather*}
 [ \Phi _p(u^{\triangle }(t))] ^{\triangledown}+a(t)f(t,u(t))
 =0,\quad t\in [0,T]_{\mathbf{T}}, \\
 u(0)-B_0(u^{\triangle }(\eta )) = 0,\quad u^{\triangle }(T)=0.
 \end{gather*}
 By multiplicity we mean the existence of arbitrary number of solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{example}[theorem]{Example}

\section{Introduction}

Let $\mathbb{T}$ be a closed nonempty subset of $\mathbb{R}$,
  and let $\mathbb{T}$  have the subspace topology
inherited from the Euclidean topology on $\mathbb{R}$.  In some of
the current literature, $\mathbb{T}$  is called a time scale (or
measure chain). For notation, we shall use the convention that, for each
interval $J$ of $\mathbb{R}$,
\[
J_{\mathbb{T}}=J\cap \mathbb{T}.
\]
The theory of dynamic equations on time scales has become a new important
mathematical branch (see, for example \cite{a1,b1,k2}) since it was initiated by
Hilger \cite{h2}. At the same time, boundary-value problems (BVPs) for scalar
dynamic equations on time scales has received considerable attention
\cite{a2,a3,a4,a5,a6,c1,e1,e2}. The purpose of this paper is to investigate the existence,
multiplicity and infinite solvability of positive solutions for
$p$-Laplacian dynamic equations on time scales
\begin{equation} \label{e1.1}
[ \Phi _p(u^{\triangle }(t))]^{\triangledown }+a(t)f(t,u(t))=0,
\quad t\in [0,T]_{\mathbb{T}},
\end{equation}
satisfying the boundary conditions
\begin{equation} \label{e1.2}
u(0)-B_0(u^{\triangle }(\eta ))=0,\quad u^{\triangle }(T)=0,
\end{equation}
or
\begin{equation} \label{e1.3}
u^{\triangle }(0)=0,\quad u(T)+B_1(u^{\triangle }(\eta ))=0,
\end{equation}
where $\Phi _p(s)$ is $p$-Laplacian operator, i.e.,
 $\Phi _p(s)=|s| ^{p-2}s$, $p>1,(\Phi _p)^{-1}=\Phi _{q}$,
$\frac 1p+\frac 1q=1$, $\eta \in (0,\rho (T))_{\mathbb{T}}$.
Here, by multiplicity we mean the existence of $m$ solutions, where $m$
is an arbitrary natural number.

In this paper we  assume the following hypotheses:
\begin{itemize}
\item[(H1)] $f:[0,T]_{\mathbb{T}}\times \mathbb{R}^{+}\to\mathbb{R}^{+}$
is continuous ($\mathbb{R}^{+} $ denotes the nonnegative real numbers)

\item[(H2)] $a:\mathbb{T}\to\mathbb{R}^{+} $ is left dense continuous
(i.e., $a\in C_{\rm ld}(\mathbb{T},\mathbb{R}^{+} )$) and does not
vanish identically on any closed subinterval of $[0,T]_{\mathbb{T}}$, where
$C_{\rm ld}(\mathbb{T},\mathbb{R}^{+} )$ denotes the set of all left
dense continuous functions from $\mathbb{T}$ to $\mathbb{R}^{+} $.

\item[(H3)] $B_0(v)$ and $B_1(v)$ are both continuous odd functions defined on
$R $ and satisfy that there exist $C,D>0$ such that
\[
Dv\leq B_j(v)\leq Cv,\quad \text{for all }v\geq 0,\quad j=0,1.
\]
\end{itemize}
We remark that by a solution $u$ of \eqref{e1.1}, \eqref{e1.2}
(respectively \eqref{e1.1},\eqref{e1.3}), we mean $u:\mathbb
{T}\to\mathbb {R}$ is delta differentiable, $u^{\triangle
}:\mathbb{T}^\kappa \to \mathbb {R}$ is nabla differentiable on
$\mathbb{T}^\kappa \cap \mathbb{T} _\kappa $ and $u^{\triangle
\nabla }:\mathbb{T}^\kappa \cap \mathbb{T} _\kappa \to\mathbb {R}$
is continuous, and $u$ satisfies boundary conditions \eqref{e1.2}
(respectively \eqref{e1.3}). If $u^{\triangle \nabla }(t)\leq 0$ on
$[ 0,T]_{\mathbb{T}^\kappa \cap \mathbb{T}_\kappa }$, then we say
$u$ is concave on $[0,T]_{\mathbb{T}}$.

Anderson, Avery and Henderson \cite{a5} considered the  problem
\begin{gather*}
[ \Phi _p(u^{\triangle }(t))]^{\triangledown }+c(t)f(u(t)) = 0,
\quad t\in (a,b)_{\mathbb{T}}, \\
u(a)-B_0(u^{\triangle }(v)) = 0,\quad u^{\triangle }(b)=0,
\end{gather*}
where $v\in (a,b)_{\mathbb{T}}$, $f\in C(
\mathbb{R}^{+},\mathbb{R}^{+})$, $c\in C_{\rm
ld}((a,b)_{\mathbb{T}}, \mathbb{R}^{+})$ and $K_mx\leq B_0(x)\leq
K_Mx$ for some positive constants $K_m$, $K_M$. They established the
existence result of at least one positive solution by a fixed point
theorem of cone expansion and compression of functional type.

Very recently, in the case $f(t,l)=f(l)$, the existence of two positive
solutions for the problem \eqref{e1.1}, \eqref{e1.2} and \eqref{e1.1}, \eqref{e1.3} has been established
by He \cite{h1} by using a new double fixed-point theorem due to Avery,
et al \cite{a7} in a cone.

In this paper, we shall apply the method arising in papers
\cite{y1,y2} to problem \eqref{e1.1}, \eqref{e1.2}.
The main ingredient is the Guo-Krasnosel'skii fixed
point theorem in cone. By considering the properties of $f$ on a bounded set
of $[ 0,T]_{\mathbb{T}}\times \mathbb{R}^{+} $, we shall
establish a basic existence criterion, that is theorem \ref{thm3.1}. Then, we shall
prove the existence of $m$ positive solutions (in section 4) and the
existence of infinitely many positive solutions (in section 5).

In the remainder of this section we will provide without proof several
foundational definitions and results from the calculus on time scales so
that the paper is self-contained. For more details, one can see
\cite{a1,a6,b1,h2,k2}.

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

\begin{definition} \label{def1.2}\rm
For $x:\mathbb{T\to}\mathbb{R}$
and $t\in \mathbb{T}^\kappa$, we define the delta derivative of $x(t)$,
$x^{\triangle }(t)$, to be the number (when it exists), with the property
that, for any $\varepsilon >0$, there is a neighborhood $U$ of $t$ such that
\[
| [ x(\sigma (t))-x(s)]-x^{\triangle }(t)[ \sigma
(t)-s]| <\varepsilon | \sigma (t)-s| ,
\]
for all $s\in U$. For $x:\mathbb{T\to}\mathbb{R}$ and
 $t\in \mathbb{T}_\kappa$, 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 $s\in V$.
\end{definition}

If $\mathbb{T}=\mathbb{R}$, then $x^{\triangle }(t)=x^\nabla (t)=x'(t)$.
If $\mathbb{T}=\mathbb {Z}$, then $x^{\triangle }(t)=x(t+1)-x(t)$
is the forward difference operator while $x^\nabla (t)=x(t)-x(t-1)$
is the backward difference operator.

\begin{definition} \label{def1.3}\rm
If $F^{\triangle }(t)=f(t)$, then we define the
delta integral by
\[
\int_a^tf(s)\triangle s=F(t)-F(a).
\]
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 paper, we assume $\mathbb{T}$ is a nonempty closed subset of
$\mathbb{R}$ with $0\in \mathbb{T}_\kappa$, $T\in \mathbb{T}^\kappa$.

\section{Preliminaries}

Consider the Banach space
$E=C_{\rm ld}([ 0,T]_{\mathbb{T}},\mathbb{R})$ with norm
$\|u\|=\sup_{t\in [ 0,T]_{\mathbb{T}}}| u(t)|$.
Then define the cone by
\[
K=\left\{ u\in E|\quad u\text{ is concave and nonnegative valued on }
[ 0,T]_{\mathbb{T}},\text{ and }u^{\triangle }(T)=0\right\}.
\]
 From \cite{h1}, we know that if $u\in K$, then
\[
\inf_{t\in [ \eta ,T]_{\mathbb{T} }}u(t)\geq (\eta /T)\|u\|.
\]
We define a operator $F:K\to E$ by
\[
(Fu)(t)=B_0(\Phi _q(\int_\eta ^Ta(r)f(r,u(r))\nabla r))+\int_0^t\Phi
_q(\int_s^Ta(r)f(r,u(r))\nabla r)\triangle s,
\]
and from \cite{h1}, we also know $F:K\to K$ is completely continuous.
We denote the constants
\[
A=\Big[ C\Phi _q(\int_\eta ^Ta(r)\nabla r)+\int_0^T\Phi
_q(\int_s^Ta(r)\nabla r)\triangle s\Big]^{-1}, \quad
B=\Big[ D\Phi _q(\int_\eta ^Ta(r)\nabla r)\Big]^{-1}.
\]
Clearly, $0<A<B$. The following symbols are used in this paper:
\begin{gather*}
\alpha (l) =\max \{f(t,c):(t,c)\in [ 0,T]_{\mathbb{T}}\times [ 0,l]\}, \\
\beta (l) =\min \{f(t,c):(t,c)\in [ \eta ,T]_{\mathbb{T}
}\times [ (\eta /T)l,l]\}, \\
\underline{\alpha }_0 =\liminf _{l\to0}\alpha (l)/l^{p-1},\quad
\underline{\alpha }_\infty =\liminf _{l\to+\infty }\alpha (l)/l^{p-1}, \\
\overline{\beta }_0 = \limsup _{l\to0}\beta (l)/l^{p-1}, \quad
\overline{\beta }_\infty =\limsup _{l\to+\infty }\beta (l)/l^{p-1}; \\
\max \overline{f}_0 =\limsup _{l\to0}\max_{t\in [0,T]_{\mathbb{T}}}
f(t,l)/l^{p-1},\quad
\max \overline{f}_\infty = \limsup _{l\to+\infty }\max_{t\in [ 0,T]_{
\mathbb{T}}}f(t,l)/l^{p-1}, \\
\min \underline{f}_0 =\liminf _{l\to0}\min_{t\in [
\eta ,T]_{\mathbb{T}}}f(t,l)/l^{p-1},\quad
 \min \underline{f} _\infty =\liminf _{l\to+\infty }\min_{t\in [ \eta
,T]_{\mathbb{T}}}f(t,l)/l^{p-1}, \\
\max f_0 =\lim_{l\to0}\max_{t\in [ 0,T]_{\mathbb{T} }}f(t,l)/l^{p-1}, \quad
\max f_\infty =\lim_{l\to+\infty }\max_{t\in [ 0,T]_{\mathbb{T}}}
f(t,l)/l^{p-1}, \\
\min f_0 =\lim_{l\to0}\min_{t\in [ \eta ,T]_{\mathbb{T}}}f(t,l)/l^{p-1},\quad
\min f_\infty =\lim_{l\to+\infty}\min_{t\in [ \eta ,T]_{\mathbb{T}}}
f(t,l)/l^{p-1}.
\end{gather*}

\begin{lemma} \label{lem2.1}
\begin{enumerate}
\item If $\max \overline{f}_0<A^{p-1}$, then $\underline{\alpha }_0<A^{p-1}$.

\item If $\max \overline{f}_\infty <A^{p-1}$, then $\underline{\alpha }_\infty
<A^{p-1}$.

\item If $\min \underline{f}_0>(TB^{p-1})/\eta $, then $\overline{
\beta }_0>B^{p-1}$.

\item If $\min \underline{f}_\infty >(TB^{p-1})/\eta $, then
$\overline{\beta }_\infty >B^{p-1}$.
\end{enumerate}
\end{lemma}

\begin{proof}
It is easy to show that the following inequalities hold:
\begin{gather*}
\limsup _{l\to0}\max_{t\in [ 0,T]_{\mathbb{T}}}f(t,l)/l^{p-1}
\geq \liminf _{l\to0}\max \{f(t,c):(t,c)\in [ 0,T]_{\mathbb{T}}\times [ 0,l]
\}/l^{p-1},
\\
\limsup _{l\to+\infty }\max_{t\in [ 0,T]_{
\mathbb{T}}}f(t,l)/l^{p-1}
\geq \liminf _{l\to+\infty }\max \{f(t,c):(t,c)\in [ 0,T]_{\mathbb{T}}
\times [ 0,l]\}/l^{p-1},
\\
\begin{aligned}
&\liminf _{l\to0}\min_{t\in [ \eta ,T]_{\mathbb{T}}}f(t,l)/l^{p-1}\\
&\leq \limsup _{l\to0}\min
\{f(t,c):(t,c)\in [ \eta ,T]_{\mathbb{T}}\times [ (
\eta /T)l,l]\}/((l^{p-1}\eta )/T),
\end{aligned}
\\
\begin{aligned}
&\liminf _{l\to+\infty }\min_{t\in [ \eta ,T]_{
\mathbb{T}}}f(t,l)/l^{p-1}\\
&\leq \limsup _{l\to+\infty }\min
\{f(t,c):(t,c)\in [ \eta ,T]_{\mathbb{T}}\times [ (
\eta /T)l,l]\}/((l^{p-1}\eta )/T).
\end{aligned}
\end{gather*}
The statements of the lemma follow from these inequalities.
\end{proof}

The following Lemma is  crucial in our argument, which is the well-known
Guo-Krasnosel'skii fixed point theorem in cone.

\begin{lemma}[\cite{g1,k1}] \label{lem2.2}
 Let $X$ be a Banach space and $K\subset E$
be a cone in $X$. Assume $\Omega _1$, $\Omega _2$ are bounded open subsets
of $K$ with $0\in \Omega _1\subset \overline{\Omega }_1\subset \Omega _2$
and $T:K\to K$ is a completely continuous operator such that either:
\begin{enumerate}
\item  $\|Tw\|\leq \|w\|$, $w\in \partial \Omega _1$,
and $\|Tw\|\geq \|w\|$, $w\in \partial \Omega _2$;
or
\item  $\|Tw\|\geq \|w\|$, $w\in \partial \Omega _1$,
and $\|Tw\|\leq \|w\|$, $w\in \partial \Omega _2$.
\end{enumerate}
Then $T$ has a fixed point in $\overline{\Omega }_2\backslash \Omega _1$.
\end{lemma}

\section{Existence results}

\begin{theorem} \label{thm3.1}
 Assume that there exist two positive numbers $a$, $b$
such that $\alpha (a)\leq (aA)^{p-1}$, $\beta (b)\geq (bB)^{p-1}$. Then
problem \eqref{e1.1}, \eqref{e1.2} has at least one positive solution $u^{*}\in K$
satisfying
\[
\min \{a,b\}\leq u^{*}\leq \max \{a,b\}.
\]
\end{theorem}

\begin{proof}
First of all, we claim $a\neq b$. If not, $a=b$. Noticing
that $A<B$, then
\begin{align*}
&\max \{f(t,l) :(t,l)\in [ 0,T]_{\mathbb{T}}\times [0,a]\}\\
&=\alpha (a)\leq (aA)^{p-1}\\
&<(aB)^{p-1}\leq \beta (a) \\
&=\min \{f(t,c):(t,c)\in [ \eta ,T]_{\mathbb{T}}\times [
(\eta /T)a,a]\}.
\end{align*}
This is impossible.

Without loss of generality, we may assume $a<b$.
We denote $\Omega _c=\{u\in K:\|u\|<c\}$, $\partial \Omega
_c=\{u:\|u\|=c\}$.
If $u\in \partial \Omega _a$, then $0\leq u\leq a$, $t\in [ 0,T]
_{\mathbb{T}}$. So,
\[
f(t,u(t))\leq \alpha (a)\leq (aA)^{p-1},\quad t\in [ 0,T]_{
\mathbb{T}}.
\]
It follows that
\begin{align*}
\|Fu\|&= B_0(\Phi _q(\int_\eta ^Ta(r)f(r,u(r))\nabla
r))+\int_0^T\Phi _q(\int_s^Ta(r)f(r,u(r))\nabla r)\triangle s \\
&\leq aAC\Phi _q(\int_\eta ^Ta(r)\nabla r)+aA\int_0^T\Phi
_q(\int_s^Ta(r)\nabla r)\triangle s \\
&= aA[ C\Phi _q(\int_\eta ^Ta(r)\nabla r)+\int_0^T\Phi
_q(\int_s^Ta(r)\nabla r)\triangle s]\\
&= a=\|u\|.
\end{align*}
If $u\in \partial \Omega _b$, then
$(\eta /T)b=(\eta/T)\|u\|\leq \min_{t\in [ \eta ,T]_{
\mathbb{T}}}u(t)\leq u(t)\leq b$, $t\in [ \eta ,T]_{\mathbb{T}}$.
So
\[
f(t,u(t))\geq \beta (b)\geq (bB)^{p-1},\quad t\in [ \eta ,T]_{
\mathbb{T}}.
\]
It follows that
\begin{align*}
\|Fu\|&= B_0(\Phi _q(\int_\eta ^Ta(r)f(r,u(r))\nabla
r))+\int_0^T\Phi _q(\int_s^Ta(r)f(r,u(r))\nabla r)\triangle s \\
&\geq  B_0(\Phi _q(\int_\eta ^Ta(r)f(r,u(r))\nabla r)) \\
&\geq  DbB\Phi _q(\int_\eta ^Ta(r)\nabla r) \\
&= b=\|u\|.
\end{align*}
By Lemma \ref{lem2.2}, $F$ has a fixed point $u^{*}\in \overline{\Omega }
_b\backslash \Omega _a$.
\end{proof}

\begin{corollary} \label{coro3.2}
Assume  $\inf_{l>0}\alpha (l)/l^{p-1}<A^{p-1}$
and $\sup_{l>0}\beta (l)/l^{p-1}>B^{p-1}$. Then problem \eqref{e1.1}, \eqref{e1.2} has at
least one positive solution.
\end{corollary}

By Theorem \ref{thm3.1} and Lemma \ref{lem2.1} we have the following
results.

\begin{corollary} \label{coro3.3}
Assume that one of the following conditions holds:
\begin{enumerate}
\item $\underline{\alpha }_0<A^{p-1}$ and $\overline{\beta }_\infty >B^{p-1}$
(in particular, $\underline{\alpha }_0=0$ and $\overline{\beta }_\infty
=+\infty $,

\item $\overline{\beta }_0>B^{p-1}$ and $\underline{\alpha }_\infty <A^{p-1}$
(in particular, $\overline{\beta }_0=+\infty $ and
$\underline{\alpha }_\infty =0$).
\end{enumerate}
Then problem \eqref{e1.1}, \eqref{e1.2} has at least one positive solution.
\end{corollary}

\begin{corollary} \label{coro3.4}
Assume that one of the following conditions holds:
\begin{enumerate}
\item $\max \overline{f}_0<A^{p-1}$ and
$\min \underline{f}_\infty >(TB^{p-1})/\eta $ (in particular,
$\max f_0=0$ and $\min f_\infty =+\infty $)

\item $\min \underline{f}_0>(TB^{p-1})/\eta $ and
$\max \overline{f}_\infty <A^{p-1}$(in particular,
$\min f_0=+\infty $ and $\max f_\infty =0$).
\end{enumerate}
Then problem \eqref{e1.1}, \eqref{e1.2} has at least one positive solution.
\end{corollary}

The special case of Corollary \ref{coro3.4} is a useful result for superlinear and
sublinear problems.

\begin{corollary} \label{coro3.5} Assume that
\begin{enumerate}
\item $\inf_{l>0}\alpha (l)/l^{p-1}<A^{p-1}$ (in particular, there exists $a>0$
such that $\alpha (a)<(aA)^{p-1})$,

\item $\max \{\min \underline{f}_0,\min \underline{f}_\infty \}
>(TB^{p-1})/\eta $ (in particular when  $\min f_0=+\infty $ or
$\min f_\infty =+\infty$).
\end{enumerate}
Then problem \eqref{e1.1}, \eqref{e1.2} has at least one positive solution.
\end{corollary}

\begin{corollary} \label{coro3.6}
Assume that:
\begin{enumerate}
\item $\sup_{l>0}\beta (l)/l^{p-1}>B^{p-1}$ (in particular, there exists $b>0$
such that $\beta (b)>(bB)^{p-1})$,

\item $\min \{\max \overline{f}_0,\max \overline{f}_\infty \}<A^{p-1}$ (in
particular, $\max f_0=0$ or $\max f_\infty =0)$.
\end{enumerate}
Then problem \eqref{e1.1}, \eqref{e1.2} has at least one positive solution.
\end{corollary}

\section{Multiplicity}

Let $[c]$ be the integer part of $c$.

\begin{theorem} \label{thm4.1}
Let $0<a_1<a_2<\dots <a_{m+1}$. If one of
the following conditions holds:
\begin{enumerate}
\item $\alpha (a_{2k-1})<(a_{2k-1}A)^{p-1}$, $k=1,\dots $, $[ \frac{m+2}
2]$, $\beta (a_{2k})>(a_{2k}B)^{p-1}$, $k=1,\dots $, $[ \frac{
m+1}2]$

\item $\beta (a_{2k-1})>(a_{2k-1}B)^{p-1}$, $k=1,\dots $, $[ \frac{m+2}
2]$, $\alpha (a_{2k})<(a_{2k}A)^{p-1}$, $k=1,\dots $, $[ \frac{
m+1}2]$.
\end{enumerate}
Then problem \eqref{e1.1}, \eqref{e1.2} has at least $m$ positive solutions $u_1^{*}$,
$u_2^{*}$, $\dots $, $u_m^{*}$ satisfying
\[
a_k<\|u_k^{*}\|<a_{k+1},\quad k=1, 2, \dots , m.
\]
\end{theorem}

\begin{proof}
We prove only Case (2). The proof of Case (1) is similar. By the
continuity of $\alpha $ and $\beta $, there exist
\[
0<b_1<a_1<c_1<b_2<a_2<c_2<\dots <c_m<b_{m+1}<a_{m+1}<+\infty
\]
such that
\begin{gather*}
\beta (b_{2k-1}) \geq (b_{2k-1}B)^{p-1},\quad \beta (c_{2k-1})\geq
(c_{2k-1}B)^{p-1},\quad k=1,\dots ,[ \frac{m+2}2], \\
\alpha (b_{2k}) \leq (b_{2k}A)^{p-1},\quad \alpha (c_{2k})\leq
(c_{2k}A)^{p-1},\quad k=1,\dots ,[ \frac{m+1}2].
\end{gather*}
Applying Theorem \ref{thm3.1} for each pair of numbers $\{c_k,b_{k+1}\}$,
$k=1, 2, \dots , m$, we complete the proof.
\end{proof}

By Theorem \ref{thm4.1} and Lemma \ref{lem2.2} we have the following existence results of two
or three positive solutions.

\begin{corollary} \label{coro4.2} Assume that
\begin{enumerate}
\item $\inf_{l>0}\alpha (l)/l^{p-1}<A^{p-1}$ (in particular, there exists $a>0$
such that $\alpha (a)<(aA)^{p-1}$)

\item $\min \{\min \underline{f}_0$, $\min \underline{f}_\infty \}>(
TB^{p-1})/\eta $ (in particular, $\min f_0=\min f_\infty =+\infty $).
\end{enumerate}
Then problem \eqref{e1.1}, \eqref{e1.2} has at least two positive solutions.
\end{corollary}

\begin{corollary} \label{coro4.3} Assume that:
\begin{enumerate}
\item $\sup_{l>0}\beta (l)/l^{p-1}>B^{p-1}$ (in particular, there exists $b>0$
such that $\beta (b)>(bB)^{p-1}$)

\item $\max \{\max \overline{f}_0$, $\max \overline{f}_\infty \}<A^{p-1}$ (in
particular, $\max f_0=\max f_\infty =0$).
\end{enumerate}
Then problem \eqref{e1.1}, \eqref{e1.2} has at least two positive solutions.
\end{corollary}

\begin{corollary} \label{coro4.4} Let $0<a_1<a_2<+\infty $. If
\begin{enumerate}
\item $\min \underline{f}_0>(TB^{p-1})/\eta $ and $\max \overline{
f}_\infty <A^{p-1}$ (in particular, $\min f_0=+\infty $ and $\max f_\infty
=0 $)

\item $\alpha (a_1)<(a_1A)^{p-1}$ and $\beta (a_2)>(a_2B)^{p-1}$.
\end{enumerate}
Then problem \eqref{e1.1}, \eqref{e1.2} has at least three positive solutions.
\end{corollary}

\begin{corollary} \label{coro4.5} Let $0<a_1<a_2<+\infty $. If
\begin{enumerate}
\item $\max \overline{f}_0<A^{p-1}$ and $\min $\underline{$f$}$_\infty >(
TB^{p-1})/\eta $ (in particular, $\max f_0=0$ and $\min f_\infty
=+\infty $)

\item $\beta (a_1)>(a_1B)^{p-1}$ and $\alpha (a_2)<(a_2A)^{p-1}$.
\end{enumerate}
Then problem \eqref{e1.1}, \eqref{e1.2} has at least three positive
solutions.
\end{corollary}

Obviously, analogous results still hold for arbitrary number $m$. Also
we have the following result.

\begin{theorem} \label{thm4.6}
Let $0<a_1<a_2<\dots <a_{2m}<+\infty $. If
one of the following conditions holds:
\begin{enumerate}
\item $\alpha (a_{2k-1})\leq (a_{2k-1}A)^{p-1}$, $\beta (a_{2k})\geq
(a_{2k}B)^{p-1}$, $k=1,\dots $, $m$

\item $\beta (a_{2k-1})\geq (a_{2k-1}B)^{p-1}$, $\alpha (a_{2k})\leq
(a_{2k}A)^{p-1}$, $k=1,\dots $, $m;$
\end{enumerate}
then problem \eqref{e1.1}, \eqref{e1.2} has at least $m$ positive solutions
$u_1^{*}, u_2^{*}, \dots , u_m^{*}$ satisfying
\[
a_1\leq \|u_1^{*}\|<\|u_2^{*}\|<\dots <\|
u_m^{*}\|\leq a_{2m}.
\]
\end{theorem}

\begin{proof} Applying Theorem \ref{thm3.1} for each pair of numbers
$\{a_{2k-1},a_{2k}\}$, $k=1,\dots $, $m$, the proof is completed.
\end{proof}

\section{Infinite Solvability}

\begin{theorem} \label{thm5.1}
Assume that $\underline{\alpha }_0<A^{p-1}$ and
$\overline{\beta }_0>B^{p-1}$ (in particular, $\underline{\alpha }_0=0$ and
$\overline{\beta }_0=+\infty $).
Then problem \eqref{e1.1}, \eqref{e1.2} has a sequence of positive solutions
$\{u_k^{*}\} _{k=1}^\infty $ satisfying $\|u_k^{*}\| \to 0$
as $k\to\infty$.
\end{theorem}

\begin{proof}  Since $\liminf _{l\to0}\alpha (l)/l^{p-1}<A^{p-1}$ and
$\limsup _{l\to0}\beta (l)/l^{p-1}>B^{p-1}$, there exist two sequences
of positive numbers $a_k\to 0$ and $b_k\to 0$ such that
\[
\alpha (a_k)\leq (a_kA)^{p-1},\quad \beta (b_k)\geq (b_kB)^{p-1},\quad
k=1,\quad 2,\dots
\]
Without loss of generality, we may assume
\[
a_1>b_1>a_2>b_2>\dots >a_k>b_k>\dots .
\]
Now applying Theorem \ref{thm3.1} to each pair of numbers $\{ b_k,a_k\} $,
$k=1,2,\dots $, problem \eqref{e1.1}, \eqref{e1.2} has a sequence of positive
solutions $\{ u_k^{*}t\} _{k=1}^\infty $ satisfying $b_k\leq
\|u_k^{*}\|\leq a_k$. The proof is complete.
\end{proof}

\begin{corollary} \label{coro5.2}
 Assume that there exists an $l_0$ such that
\[
\inf_{0<l\leq l_0}\min_{t\in [ \eta ,T]_{\mathbb{T}
}}f(t,l)/\alpha (l)\geq k_1>0,\quad
\sup_{0<l\leq l_0}\max_{t\in [0,T]_{\mathbb{T}}}f(t,(l\eta )/T)/\beta (l)\leq
k_2<+\infty .
\]
If $\max \overline{f}_0>(k_2TB^{p-1})/\eta $ and
$\min \underline{f}_0<k_1A^{p-1}$ (in particular,
$\max \overline{f}_0=+\infty $
and $\min \underline{f}_0=0$), then problem \eqref{e1.1}, \eqref{e1.2}
has a sequence of positive solutions $\{ u_k^{*}\} _{k=1}^\infty $ satisfying
$\|u_k^{*}\|\to 0$  as $k\to\infty$.
\end{corollary}

\begin{proof}
 Clearly, $\alpha (l)\leq \min_{t\in [ \eta ,T]_{\mathbb{T}}}f(t,l)/k_1$,
$\beta (l)\geq \max_{t\in [ 0,T]_{\mathbb{T}}}f(t,(l\eta )/T)/k_2$, for
$l\in (0,l_0]$. Then
\begin{align*}
\underline{\alpha }_0
&=\liminf _{l\to0}\alpha (l)/l^{p-1}
  \leq (1/k_1)\liminf _{l\to 0}\min_{t\in [ \eta ,T]_{\mathbb{T}}}
  f(t,l)/l^{p-1} \\
&=(1/k_1)\min \underline{f}_0<(1/k_1)(k_1A^{p-1})=A^{p-1},
\end{align*}
\begin{align*}
\overline{\beta }_0
&=\limsup _{l\to0}\beta
(l)/l^{p-1}\geq (1/k_2)\limsup _{l\to
0}\max_{t\in [ 0,T]_{\mathbb{T}}}f(t,(l\eta )/T)/l^{p-1} \\
&=\eta /(Tk_2)\limsup _{l\to0}\max_{t\in
[ 0,T]_{\mathbb{T}}}f(t,(l\eta )/T)/(l\eta /T)\\
&=\eta /(Tk_2)\max \overline{f}_0>\eta /(Tk_2)
(k_2TB^{p-1}/\eta )=B^{p-1}.
\end{align*}
Now the conclusion follows from Theorem \ref{thm5.1}.
\end{proof}

Similarly, we have the following statement.

\begin{theorem} \label{thm5.3}
Assume that $\underline{\alpha }_\infty <A^{p-1}$ and
$\overline{\beta }_\infty >B^{p-1}$ (in particular, $\underline{\alpha }
_\infty =0$ and $\overline{\beta }_\infty =+\infty $).
Then problem \eqref{e1.1}, \eqref{e1.2} has a sequence of positive
solutions $\{u_k^{*}\} _{k=1}^\infty $ satisfying
$\|u_k^{*}\|\to+\infty $ as $k\to\infty $.
\end{theorem}

\begin{corollary} \label{coro5.4} Assume that
\begin{gather*}
\inf_{0<l\leq +\infty }\min_{t\in [ \eta ,T]_{\mathbb{T}
}}f(t,l)/\alpha (l)\geq k_1>0,\\
 \sup_{0<l\leq +\infty }\max_{t\in
[ 0,T]_{\mathbb{T}}}f(t,l\eta /T)/\beta (l)\leq k_2<+\infty .
\end{gather*}
If $\max \overline{f}_\infty >(k_2TB^{p-1})/\eta $ and
$\min \underline{f}_\infty <k_1A^{p-1}$ (in particular,
$\max \overline{f}_\infty =+\infty $ and $\min \underline{f}_\infty =0$),
then problem \eqref{e1.1}, \eqref{e1.2} has a sequence of positive
solutions $\{ u_k^{*}\} _{k=1}^\infty $
satisfying $\|u_k^{*}\|\to+\infty $  as $k\to\infty$.
\end{corollary}

\section{Examples}

\begin{example} \label{exa6.1}\rm
 Let $\mathbb{T}=\{ 1-(\frac 12)^{\mathbb{N}_{0}}\} \cup \{ 1\} $,
where $\mathbb{N}_{0}$ denotes the set of all non-negative integers.
Consider the $p$-Laplacian dynamic equation
\begin{equation} \label{e6.1}
[ \Phi _p(u^{\triangle }(t))]^{\triangledown }+f(u(t))=0,t\in
[0,1]_{\mathbb{T}},
\end{equation}
satisfying the boundary conditions
\begin{equation} \label{e6.2}
u(0)-2u^{\triangle }(1/2)=0,\quad u^{\triangle }(1)=0,  %\label{5.2}
\end{equation}
where $p=3/2$, $q=3$, $a(t)\equiv 1$, $C=D=2$, $T=1$ and
$$
f(u)=\begin{cases}
1, & 0\leq u\leq 1, \\
4u-3 & 1\leq u\leq \frac 32, \\
3 & \frac 32\leq u\leq 10, \\
2u-17 & 10\leq u\leq 12, \\
7 & 12\leq u\leq 24.
\end{cases}
$$
Then problem \eqref{e6.1}, \eqref{e6.2} has at least two positive solutions.
\end{example}

To proof the statement of the above example, choose $a_1=1$, $a_2=3$,
$a_3=10$, $a_4=24$. It is easy to see that $A=6/5$, $B=2$, and
\begin{gather*}
\alpha (a_1)=\max \{ f(u):u\in [0,1]\} =1<\sqrt{\frac 65}=(a_1A)^{p-1},\\
\alpha (a_3)=\max \{ f(u):u\in [0,10]\} =3<\sqrt{\frac{60}5}=(a_3A)^{p-1},\\
\beta (a_2)=\min \{ f(u):u\in [ \frac 32,3]\} =3> \sqrt{6}=(a_2B)^{p-1},\\
\beta (a_4)=\min \{ f(u):u\in [ 12,24]\} =7>\sqrt{48}=(a_4B)^{p-1}.
\end{gather*}
Therefore, by Theorem \ref{thm4.6},  problem \eqref{e6.1},
\eqref{e6.2} has at least two positive solutions $u_1^{*}$,
$u_2^{*}$ satisfying $1\leq \|u_1^{*}\| < \|u_2^{*}\|\leq 24$.

The methods in this paper can also be used for studying problem \eqref{e1.1},
\eqref{e1.3}.

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\end{document}