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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2008(2008), No. 92, pp. 1--14.\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 2008 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2008/92\hfil Existence of positive solutions]
{Existence of positive solutions for $p$-Laplacian three-point
boundary-value problems \\ on time scales}

\author[H.-R. Sun, Y.-H. Wang\hfil EJDE-2008/92\hfilneg]
{Hong-Rui Sun, Ying-Hai Wang}  % in alphabetical order

\address{Hong-Rui Sun \newline
School of Mathematics and Statistics, Lanzhou University\\
Lanzhou, Gansu 730000, China}
\email{hrsun@lzu.edu.cn}

\address{Ying-Hai Wang \newline
School of Physical Science and Technology, Lanzhou University\\
Lanzhou, Gansu 730000, China} 
\email{yhwang@lzu.edu.cn}

\thanks{Submitted January 26, 2007. Published  July 2, 2008.}
\thanks{Supported by grants 10571078 from the NNSF of
China, 10726049 from Tianyuan Youth \hfill\break\indent
Grant of China, Lzu05003 from Fundamental Research Fund for Physics and
Mathematics \hfill\break\indent
of Lanzhou University, and 2005038486 from China Postdoctoral Science
Foundation}
\subjclass[2000]{34B15, 39A10}
\keywords{Time scales; p-Laplacian; positive solution; cone; fixed point}

\begin{abstract}
 This article shows the existence of positive solutions for
 a class of $p$-Laplacian three-point boundary-value problem
 on time scales. By using several fixed point theorems in cones,
 we establish conditions for the existence of at least one,
 two or three positive solutions for the boundary-value problems.
 Our results are new even for the corresponding
 differential ($\mathbb{T}=\mathbb{R})$ and difference equation
 ($\mathbb{T}=\mathbb{Z})$, and for the general time scales setting.
 An example is also given to illustrate our results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
%\allowdisplaybreaks

\section{Introduction}

Dynamic equations on time scales not only unify  differential and
difference equations \cite{Hilger}, but also  exhibit much more
complicated dynamics \cite{ABL, BP1, BP2}. The study of dynamic
equations on time scales has led to important applications in the
study of insect population models, biology, heat transfer, stock
market, wound healing, and epidemic models \cite{Jon, Spedding,
TVY}.

Before introducing the problems of interest for this paper, we
present some basic definitions which can be found in
\cite {a6,BP1, BP2, Hilger}. Another   source on dynamic equations
on time scales is \cite{LSK}.

A time scale $\mathbb{T}$ is a nonempty closed subset of $\mathbb{R}$
with the topology inherited from  $\mathbb{R}$. For notation, we shall
use the convention that, for
each interval $J$ of $\mathbb{R}$,  $J_{\mathbb{T}}=J\cap \mathbb{T}$.

The jump operators $\sigma ,\rho :\mathbb{T}\to \mathbb{T}$ defined
by
\[
\sigma (t)=\inf \{ \tau \in \mathbb{T}:\tau >t\} \quad
\text{and}\quad
\rho (t)=\sup \{ \tau \in \mathbb{T}:\tau <t\}
\]
(supplemented by $\inf \emptyset :=\sup \mathbb{T}$ and 
$\sup \emptyset :=\inf\mathbb{T}$) are well defined. 
The point $t\in \mathbb{T}$ is left-dense,
left-scattered, right-dense, right-scattered if $\rho (t)=t$, $\rho (t)<t$,
$\sigma (t)=t$, $\sigma (t)>t$, respectively. 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 $\mathbb{T}^\kappa =\mathbb{T}$. The forward graininess is
$\mu (t):=\sigma (t)-t$.
Similarly, the backward graininess is $\upsilon (t):=t-\rho (t)$.

For $f:\mathbb{T}\to \mathbb{R}$ and $t\in \mathbb{T}^\kappa $, the
$\Delta$-derivative of $f$ at $t$, denoted by $f^\Delta (t)$, is
the number (provided it exists) with the property that given any
$\epsilon >0$, there is a neighborhood $U\subset \mathbb{T}$ of $t$
such that
\[
|f(\sigma (t))-f(s)-f^\Delta (t)[\sigma (t)-s]|\leq \epsilon
|\sigma (t)-s|,
\]
for all $s\in U$.

For $f:\mathbb{T}\to \mathbb{R}$ and $t\in \mathbb{T}_\kappa $, the
$\nabla$-derivative \cite{a6} of $f$ at $t$, denoted by $f^\nabla
(t)$, is the number (provided it exists) with the property that
given any $\epsilon >0$, there is a neighborhood $U$ of $t$ such
that
\[
|f(\rho (t))-f(s)-f^\nabla (t)[\rho (t)-s]|\leq \epsilon |
\rho (t)-s|,
\]
for all $s\in U$.

A function $f:\mathbb{T\to R}$ is ld-continuous provided it is
continuous at left dense points in $\mathbb{T}$ and its right sided limit
exists (finite) at right dense points in $\mathbb{T}$. If $\mathbb{T}=\mathbb{R}$,
then $f$ is ld-continuous if and only if $f$ is continuous. If $\mathbb{T}=\mathbb{
Z}$, then any function is ld-continuous. It is known \cite{a6} that if $f$
is ld-continuous, then there is a function $F(t)$ such that $F^\nabla
(t)=f(t)$. In this case, we define
\[
\int_a^bf(\tau )\nabla \tau =F(b)-F(a).
\]

For recent results on
positive solutions for second order three point boundary value
problems on time scales the reader is referred to \cite{a5, Anderson2, AT, DDS, SunLi4, k1,
SunLi1}. Our results have been motivated by those of  Anderson, Avery
and Henderson \cite{Anderson3}, and Sun, Tang and Wang
\cite{SunTangWang}.

For convenience, throughout this paper we denote  $\varphi _p(u)=|u|^{p-2}u$ for $p>1$
with $(\varphi _p)^{-1}=\varphi _q$, where $1/p+1/q=1$.

Anderson, Avery and Henderson \cite{Anderson3} considered the problem
\begin{gather*}
(\varphi _p(u^\Delta (t)))^\nabla +c(t)f(u(t))=0,
\quad t\in (a,b)_{\mathbb{T}}, \\
u(a)-B_0(u^\Delta (\nu ))=0,\quad u^\Delta (b)=0,
\end{gather*}
where $\nu \in (a,b)_{\mathbb{T}}$,
$f\in C_{ld}([0,\infty ),[0,\infty
))$, $c\in C_{ld}((a,b)_{\mathbb{T}},[0,\infty ))$ 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.

In \cite{SunTangWang}, the authors considered the eigenvalue
problem for the  $p$-Laplacian three-point boundary value
problem
\begin{gather*}
(\varphi _p(u^\Delta (t)))^\nabla +\lambda h(t)f(u(t))=0,\quad
t\in (0,T)_{\mathbb{T}},\\
u(0)-\beta u^{\Delta }(0)=\gamma u^{\Delta}(\eta ),u^{\Delta}(T)=0.
\end{gather*}
The main tool used in \cite{SunLi3} is Krasnoselskii's fixed point
theorem.

In this paper we study the existence of solutions for the one-dimensional
$p$-Laplacian three-point boundary value problem on time scales
\begin{gather}
(\varphi _p(u^\Delta (t)))^\nabla +h(t)f(t,u(t))=0,\quad t\in
(0,T)_{\mathbb{T}},  \label{1.1} \\
u(0)-\beta u^\Delta (0)=\gamma u^\Delta (\eta ),\quad u^\Delta (T)=0.  \label{1.2}
\end{gather}
We establish sufficient conditions for the existence of at least one,
two or three positive solutions for the boundary value problem. An
example is also given to illustrate the main results.
The results are new  even for the special cases of difference equations
and differential equations.

The rest of the paper is organized as follows. In Section 2, we
first give four lemmas which are needed throughout this paper and
then state several fixed point results: Krasnosel'skii's fixed point
theorem in a cone, a new fixed point theorem due to Avery and
Henderson and the Leggett-Williams fixed point theorem. In Section 3
we use Krasnosel'skii's fixed point theorem to obtain the existence
of at least one positive solutions of problem
\eqref{1.1}-\eqref{1.2}. Section 4 will discuss the existence of
twin positive solutions of problem \eqref{1.1}-\eqref{1.2}. Two new
results and some corollaries will be presented by using a new fixed
point theorem due to Avery and Henderson. In Section 5 we develop
criteria for the existence of  (at least) three positive and
arbitrary odd positive solutions of problem \eqref{1.1} and
\eqref{1.2}. In particular, our results in this section are new when
$\mathbb{T}=\mathbb{R}$ (the continuous case) and
$\mathbb{T}=\mathbb{Z}$ (the discrete case). Finally, in section 6,
we give an example to illustrate our main results.

For the sake of convenience, we have the following hypotheses:
\begin{itemize}
\item[(i)] $\mathbb{T}$ is a time scale, with $0,T\in \mathbb{T}$,
$\beta$, $\gamma $ are nonnegative constants,
$\eta \in (0,\rho (T))_{\mathbb{T}}$.

\item[(ii)] $h\in C_{ld}((0,T)_{\mathbb{T}},[0,\infty ))$ such that
$0<\int_0^Th(s)\nabla s<\infty $, and $f$ is in the space
$C([0,\infty ),(0,\infty))$.
\end{itemize}

\section{Preliminaries}

Let the Banach space $B=C_{ld}([0,T]_{\mathbb{T}})$ (see \cite{a5})
be endowed with the norm
$\| u\| =\sup_{t\in [0,T]_{\mathbb{T}}}|u(t)|$, and choose the cone
$P\subset B$ defined by
\begin{align*}
P=\{&u\in B:u(t)\geq 0\text{ for }t\in [0,T]_{\mathbb{T}}\text{ and} \\
&u^{\Delta \nabla }(t)\leq 0\text{ for }t\in (0,T)_{\mathbb{T}},\;
 u^\Delta (T)=0\}.
\end{align*}
Clearly, $\| u\| =u(T)$ for $u\in P$. Define the operator
$A:P\to B$ by
\begin{equation}
\begin{aligned}
Au(t) &=\int_0^t\varphi _q\Big(\int_s^Th(\tau )f(\tau ,u(\tau ))\nabla
\tau \Big)\Delta s  \label{2.3} \\
&\quad+\beta \varphi _q\big(\int_0^Th(s)f(s,u(s))\nabla s\Big)
+\gamma\varphi _q\Big(\int_\eta ^Th(s)f(s,u(s))\nabla s\Big)
\end{aligned}
\end{equation}
for $ t\in [0, T]_{\mathbb{T}}$.

\begin{lemma}[{\cite[Lemma 2.6]{SunLi3}}]\label{Lma2.1}
Assume $g: \mathbb{R} \to \mathbb{R} $ is continuous, $g:\mathbb{T} \to
\mathbb{R} $ is delta differentiable on $\mathbb{T_{\kappa}}$, and
$f:\mathbb{R} \to \mathbb{R} $
is continuous differentiable. Then there exists $c$ in the  interval
$[\rho(t),t] $ with
\[
(f\circ g)^{\nabla} (t)=f'(g(c))g^\nabla (t).
\]
\end{lemma}

 From the definition of $A$, the monotonicity of $\varphi _q(x)$ and
Lemma 2.1, it is easy to see that for each $u\in P$, $Au\in P$ and
satisfies \eqref{1.2}. In addition, since $(\varphi _p(u^\Delta (t)))
^\nabla =-h(t)f(u(t))<0$, and $u^\Delta (T)=0$, then $Au(T)$ is the maximum
value of $Au(t)$.


\begin{lemma}[{\cite[Lemma 2.2]{SunTangWang}}]\label{lma2.2}
  $A:P\to P$ is completely continuous.
\end{lemma}

\begin{lemma}[{\cite[Lemma 2.3]{SunTangWang}}]\label{lma2.3}
If $u\in P$, then $u(t)\geq \frac tT\| u\| $ for $t\in [0,T]$.
\end{lemma}

From the two lemmas above, we see that each fixed point of the operator
$A$ in $P$ is a positive solution of \eqref{1.1}, \eqref{1.2}.

\begin{lemma}[\cite{2,5}]\label{lma2.4}
Let $P$ be a cone in a Banach space $B$.
Assume $\Omega _1,\Omega _2$ are open subsets of $X$ with
$0\in \Omega _1, \overline{\Omega }_1\subset \Omega _2$.
If $A:P\cap (\overline{\Omega }_2\backslash \Omega _1)\to P$
is a completely continuous operator such that either
\begin{itemize}
\item[(i)]
$\| Ax\| \leq \| x\|$ for all $x\in P\cap \partial
\Omega _1$  and $\| Ax\| \geq \| x\|$ for all
$x\in P\cap \partial \Omega _2$, or

\item[(ii)]
$\| Ax\| \geq \| x\|$ for all
$x\in P\cap \partial \Omega _1$
 and $\| Ax\| \leq \| x\|$ for all $x\in P\cap \partial \Omega _2$.

\end{itemize}
Then $A$ has a fixed point in
$P\cap (\overline{\Omega _2}\backslash \Omega_1)$.
\end{lemma}

In the rest of this section, we provide some background material from the
theory of cones in Banach spaces, and we then state several fixed point
theorems which we needed later.

Let $B$ be a Banach space and $P$ be a cone in $B$.
A map $\psi :P\to [0,+\infty )$ is said to be a nonnegative,
continuous and increasing functional provided $\psi $ is
nonnegative, continuous and satisfies $\psi (x)\leq \psi (y)$ for
all $x,y\in P$ and $x\leq y$.

Given a nonnegative continuous functional $\psi $ on a cone $P$ of
a real Banach space $B$, we define, for each $d>0$, the set
\[
P(\psi ,d)=\{ x\in P:\psi (x)<d\} .
\]

\begin{lemma}[\cite{a7}] \label{lma2.7}
 Let $P$ be a cone in a real Banach space
$E$. Let $\alpha $ and $\psi $ be increasing, nonnegative
continuous functional on $P$, and let $\theta $ be a nonnegative
continuous functional on $P$ with $\theta (0)=0$ such that, for
some $c>0$ and $H>0$,
\[
\psi (x)\leq \theta (x)\leq \alpha (x)\quad{and}\quad
\|x\| \leq H\psi (x)
\]
for all $x\in \overline{P(\psi ,c)}$. Suppose there exist a completely
continuous operator $A:\overline{P(\psi ,c)}\to P$ and $0<a<b<c$
such that
\[
\theta (\lambda x)\leq \lambda \theta (x)\quad \text{ for }
0\leq \lambda \leq 1 { and } x\in \partial P(\theta ,b)
\]
and
\begin{itemize}
\item[(i)] $\psi (Ax)>c$ for all $x\in \partial P(\psi ,c)$;

\item[(ii)] $\theta (Ax)<b$ for all $x\in \partial P(\theta ,b)$;

\item[(iii)] $P(\alpha ,a)\neq \emptyset $ and $\alpha (Ax)>a$
for $x\in \partial P(\alpha ,a)$.
\end{itemize}
Then, $A$ has at least two fixed points, $x_1$ and $x_2$ belonging to
$\overline{P(\psi ,c)}$ satisfying
$a<\alpha (x_1)$  with $\theta (x_1)<b$
and $b<\theta (x_2)$ with $\psi (x_2)<c$.
\end{lemma}

Let $0<a<b$ be given and let $\alpha $ be a nonnegative continuous concave
functional on the cone $P$. Define the convex sets $P_a,P(\alpha ,a,b)$ by
\begin{gather*}
P_a =\{ x\in P:\| x\| <a\} , \\
P(\alpha ,a,b) =\{ x\in P:a\leq \alpha (x),\| x\| \leq b\}.
\end{gather*}
Then we state the Leggett-Williams fixed point theorem \cite{l2}.

\begin{lemma}\label{lma2.9}
Let $P$ be a cone in a real Banach space $B$, $A:
\overline{P}_c\to \overline{P}_c$ be completely continuous and
$\alpha $ be a nonnegative continuous concave functional on $P$ with $\alpha
(x)\leq \| x\| $ for all $x\in \overline{P}_c$. Suppose there
exists $0<d<a<b\leq c$ such that
\begin{itemize}
\item[(i)] $\{ x\in P(\alpha ,a,b):\alpha (x)>a\} \neq \emptyset $ and
$\alpha (Ax)>a$ for $x\in P(\alpha ,a,b)$;

\item[(ii)] $\| Ax\| <d$ for $\| x\| \leq d$;

\item[(iii)] $\alpha (Ax)>a$ for $x\in P(\alpha ,a,c)$ with $\| Ax\| >b$.
\end{itemize}
Then $A$ has at least three fixed points $x_1,x_2,x_3$ satisfying
$\| x_1\| <d$, $a<\alpha (x_2)$, $\| x_3\| >d$,
and $\alpha (x_3)<a$.
\end{lemma}

\section{Existence of One Positive Solution}

For convenience, we define some important constants
\begin{gather}
C_1=(T+\beta +\gamma )^{-1}\varphi _p(\int_0^Th(s)\nabla s),
\label{2.4}\\
C_2=(\eta +\beta +\gamma )^{-1}\varphi _p(\int_\eta ^Th(s)\nabla
s).   \label{2.41}
\end{gather}

\begin{theorem} \label{thm3.1}
Assume there exist positive numbers
$a\neq b$ such that the conditions
\begin{itemize}
\item[(H1)] There is $a>0$ such that $f(t,u)\leq \varphi _p(aC_1)$
for $t\in [0,T]_{\mathbb{T}}$ and $0\leq u \leq a$;

\item[(H2)] There is $b>0$ such that $f(t,u)\geq \varphi _p(bC_2)$
for $t\in [\eta,T]_{\mathbb{T}}$ and $\frac \eta T b\leq u \leq b$.
\end{itemize}
Then  \eqref{1.1}--\eqref{1.2} has at least one positive
solution $u$ such that $\|u\| $
lies between $a$ and $b$.
\end{theorem}

\begin{proof}
Without loss of generality, we may suppose that $0<a<b$.
Define the  bounded open ball centered at the origin by
\[
\Omega _a=\{u\in B:\| u\| \leq a\},\quad
\Omega _b=\{u\in B:\| u\| \leq b\}.
\]
Then $0\in \Omega _a\subset \Omega _b$. For $u\in P\bigcap
\partial \Omega _a $ so that $\Vert u\Vert =a$, by (H1) and
(\ref{2.4}), we have
\begin{align*}
\| Au\|  &= \sup_{t\in [0,T]}\Big[ \int_0^t\varphi _q
\Big(\int_s^Th(\tau )f(\tau ,u(\tau ))\nabla \tau \Big)\Delta s  \\
&\quad  +\beta \varphi _q(\int_0^Th(s)f(s,u(s))\nabla s)
+\gamma \varphi _q\Big(\int_\eta ^Th(s)f(s,u(s))\nabla s\Big)\Big]  \\
&\leq (T+\beta +\gamma )\varphi _q\Big(\int_0^Th(s)f(s,u(s))\nabla
s\Big)\\
&\leq (T+\beta +\gamma )\varphi _q\Big(\int_0^Th(s)\varphi _p(aC_1)\nabla
s\Big)\\
&\leq aC_1(T+\beta +\gamma )\varphi _q\Big(\int_0^Th(s)\nabla s\Big)
\leq a.
\end{align*}
Hence, $\| Au\| \leq \| u\| $ for $u\in P\bigcap
\partial \Omega _a$. Similarly, let $u\in P\bigcap \partial \Omega _b$ so
that $\Vert u\Vert =b$. Then
\[
\min_{t\in [\eta ,T]}u(t)\geq \frac \eta Tb
\]
and
\begin{align*}
\| Au\|  &\geq Au(\eta ) \\
&= \int_0^\eta \varphi _q\Big(\int_s ^Th(\tau )f(\tau ,u(\tau
))\nabla
\tau \Big)\Delta s \\
&\quad +\beta \varphi _q\Big(\int_0^Th(s)f(s,u(s))\nabla s\Big)+\gamma
\varphi _q(\int_\eta ^Th(s)f(s,u(s))\nabla s)\\
&\geq (\eta +\beta +\gamma )\varphi _q(\int_\eta
^Th(s)f(s,u(s))\nabla s)\\
&\geq (\eta +\beta +\gamma )bC_2\varphi _q\big(\int_\eta
^Th(s)\nabla s\Big)\\
&= b=\Vert u\Vert
\end{align*}
by (H2) and (\ref{2.41}). Consequently, $\Vert Au\Vert \geq \Vert u\Vert $
for $u\in P\bigcap \partial \Omega _b$. By Lemma \ref {lma2.4},
$A$ has a fixed point
$u\in P\bigcap (\overline{\Omega }_b\setminus \Omega _a)$, which is
a positive solution of \eqref{1.1}, \eqref{1.2}, such that $a\leq
\Vert u\Vert \leq b$.
\end{proof}

For $t\in [0,T]_{\mathbb{T}}$, we define
\begin{gather}
f_0(t)={\liminf}_{ u\to {0^+}} \frac {f(t,u)} {\varphi_p
(u)},\quad f_\infty(t)={\liminf}_{ u\to {\infty}}
 \frac {f(t,u)} {\varphi_p (u)},  \label{2.51}
\\
f^0(t)={\limsup}_{ u\to {0^+}}  \frac {f(t,u)} {\varphi_p
(u)},\quad f^\infty(t)={\limsup}_{ u\to {\infty}} \frac
{f(t,u)} {\varphi_p (u)}. \label{2.52}
\end{gather}

\begin{corollary}\label{cry2.4}
The boundary-value problem \eqref{1.1}, \eqref{1.2}
has at least one positive solution provided either
\begin{itemize}
\item[(H3)] $f^0(t)<\varphi_p(C_1)$ for $t\in [0,T]_{\mathbb{T}}$ and
$f_{\infty} (t)>\varphi_p\left(\frac {TC_2}{\eta}\right)$ for
$t\in [\eta,T]_{\mathbb{T}} $ or

\item[(H4)] $f_0(t)>\varphi_p\left(\frac {TC_2}{\eta}\right)$ for $t\in [\eta,T]_{\mathbb{T}}$ and
$f^{\infty} (t)<\varphi_p(C_1)$ for $t\in [0,T]_{\mathbb{T}}$,

\end{itemize} where $C_1, C_2, f_0,  f_{\infty}, f^0, f^{\infty}$ are
as in (\ref{2.4}), (\ref{2.41}), (\ref{2.51}), (\ref{2.52}),
respectively. In particular, if $f$ is superlinear in
$\varphi_p(u)$ ($f^0(t)=0$ and $f_\infty(t)=\infty $) or sublinear
in $\varphi_p(u)$ ($f_0(t)=\infty$ and $f^\infty(t)=0) $, then
\eqref{1.1}, \eqref{1.2} has at least one positive solution.
\end{corollary}

\begin{proof} First suppose (H3) holds. Then, there are
sufficiently small $a>0$ and sufficiently large $b>0$ such that
\begin{gather*}
\frac {f(t,u)}{\varphi_p(u)} \leq \varphi_p(C_1) \text{ for }
t\in[0,T]_{\mathbb{T}},  u\in (0,a],
\\
\frac {f(t,u)}{\varphi_p(u)} \geq \varphi_p\big(\frac
{TC_2}{\eta}\big)\quad \text{for } t\in[\eta,T]_{\mathbb{T}},  u\in
[\frac {\eta b} T,+\infty).
\end{gather*}
Then
 \begin{gather*}
 f(t,u)\leq \varphi_p(uC_1) \leq  \varphi_p(aC_1), \quad
t\in[0,T]_{\mathbb{T}}, u\in [0,a],
\\
f(t,u)\geq \varphi_p\big(\frac{TC_2u}{\eta})\geq
\varphi_p(C_2b) , \quad t\in[\eta,T]_{\mathbb{T}},\;  u\in
[\frac {\eta b} T,b].
\end{gather*}
In particular, both (H1) and (H2) hold, so that by Theorem
\ref{thm3.1},  \eqref{1.1}, \eqref{1.2} has at least one positive
solution.

Next assume (H4) holds. Then there exist $0<a<b$ such that
\begin{gather}
\frac {f(t,u)}{\varphi_p(u)} \geq
\varphi_p\big(\frac{TC_2}{\eta})\quad \text{for }
t\in[\eta,T]_{\mathbb{T}},  u\in (0,a], \label{3.66}
\\
\frac {f(t,u)}{\varphi_p(u)} \leq \varphi_p(C_1)\quad \text{for }
t\in[0,T]_{\mathbb{T}},  u\in [b,+\infty). \label{3.67}
\end{gather}
From (\ref{3.66}) we have $f(t,u)\geq \varphi_p\big(\frac {TC_2
u}{\eta}\big)\geq \varphi_p(C_2 a)$ for $t\in [\eta,
T]_\mathbb{T}, u\in [ \frac{\eta a}{T}, a]$ satisfying
(H2) with respect to $a$. Now consider (\ref{3.67}), we wish to
show that (H1) holds. To that end, we consider the two cases: (1)
$f(t,u)$ is bounded or (2) $f(t,u)$ is unbounded.

Case 1: Suppose there exists $C>0$ such that $f(t,u)\leq C$ for
$t\in[0,T]_{\mathbb{T}} $ and $u\in [0,\infty)$. By (\ref{3.67}),
there is $r\geq \max \{b, \frac {\varphi_q(C)}{C_1}\} $ such that
$f(t,u)\leq C\leq \varphi_p(C_1 r)$ for $t\in [0, T]_\mathbb{T}, u\in
[0, r]$. Thus (H1) is satisfied with respect to $r$.

Case 2: If $f$ is unbounded, there exist $t_0\in [0,T]_{\mathbb{T}}$
and $r' \geq b$ such that
\[
f(t,u)\leq f(t_0,r') \leq \varphi_p(C_1 r') \quad \text{for } t\in
[0,T]_{\mathbb{T}} \text { and } u\in [0,r'].
\]
and (H1) is satisfied with respect to $r'$. Thus in both cases
condition (H1) hold and Theorem \ref{thm3.1} yields the
conclusion.
\end{proof}

\section{Twin Solutions}

In this section, we fix $c\in \mathbb{T}$ such that $\eta <c<T$, and denote
\[
C_3=(c+\beta +\gamma )^{-1}\varphi _p\Big(\int_c^Th(s)\nabla s\big).
\]
Define the nonnegative, increasing and continuous functionals $\psi$,
$\theta $, and $\alpha $ on $P$ by
\begin{gather*}
\psi (u)=\min_{t\in [\eta ,c]_{\mathbb{T}}}u(t)=u(\eta ),\quad
 \theta (u)=\max_{t\in [0,\eta ]_{\mathbb{T}}}u(t)=u(\eta ), \\
\alpha (u)=\max_{t\in [0,c]_{\mathbb{T}}}u(t)=u(c).
\end{gather*}
We observe that, for each $u\in P,$
\begin{equation}
\psi (u)=\theta (u)\leq \alpha (u).  \label{3.5}
\end{equation}
In addition, for each $u\in P$,
$\psi (u)=u(\eta )\geq \frac \eta T\|u\| $. Thus
\begin{equation}
\| u\| \leq \frac T\eta \psi (u),\quad u\in P.  \label{3.6}
\end{equation}
Finally, we also note that
\[
\theta (\lambda u)=\lambda \theta (u),\quad 0\leq \lambda \leq
1\text{ and } u\in \partial P(\theta ,b').
\]
We now present the results in this section.

\begin{theorem}\label{thm4.1}
Assume that there are positive numbers $a'<b'<c'$ such that
\[
0<a'<\frac {C_1}{C_3}b'<\frac{\eta C_1}{TC_3}c'.
\]
Assume further that $f(t,u)$ satisfies the following conditions:
\begin{itemize}
\item[(i)] $f(t,u)>\varphi _p(c'C_2)$,
$(t,u) \in [ \eta,T]_{\mathbb{T}} \times [ c',\frac T\eta c'] $,

\item[(ii)] $f(t,u)<\varphi _p(b'C_1)$, $(t,u)\in [
0,T]_{\mathbb{T}} \times [ 0,\frac T\eta b'] $,

\item[(iii)] $f(t,u)>\varphi _p(a'C_3)$, $(t,u)\in [
c,T]_{\mathbb{T}} \times [ a',\frac Tca'] $.

\end{itemize}
 Then \eqref{1.1}-\eqref{1.2} has at least two
positive solutions $u_1$ and $u_2$ such that
\begin{gather*}
a'<\max_{t\in [0,c]_{\mathbb{T}}}u_1(t)\quad \text{with }\max_{t\in
[0,\eta ]_{\mathbb{T}}}u_1(t)<b',\\
b'<\max_{t\in [0,\eta ]_{\mathbb{T}}}u_2(t)\quad \text{with }
\min_{t\in [\eta ,c]_{\mathbb{T}}}u_2(t)<c'.
\end{gather*}
\end{theorem}

\begin{proof}
By the definition of operator $A$ and its properties, it
suffices to show that the conditions of Lemma \ref{lma2.7} hold
with respect to $A$.

We first show that if $u\in \partial P(\psi ,c')$, then $\psi
(Au)>c'$.
Indeed, if $u\in \partial P(\psi ,c')$, then $\psi (u)=\min_{t\in
[\eta ,c]_{\mathbb{T}}}u(t)=u(\eta )=c'$. Since $u\in P$, $\|
u\| \leq \frac T\eta \psi (u)=\frac T\eta c'$, we have
$c'\leq u(t)\leq \frac T\eta c',t\in [\eta ,T]_{\mathbb{T}}$.
As a consequence of (i),
$f(t,u(t))>\varphi _p(c'C_2),\quad t\in [\eta
,T]_{\mathbb{T}}$. Also, $Au\in P$ implies
\begin{align*}
\psi (Au) &= Au(\eta )\\
&= \int_0^\eta \varphi _q\Big(\int_s ^Th(\tau )f(\tau ,u(\tau
))\nabla \tau \Big)\Delta s \\
&\quad +\beta \varphi _q\Big(\int_0^Th(s)f(s,u(s))\nabla s\Big)
 +\gamma \varphi _q\Big(\int_\eta ^Th(s)f(s,u(s))\nabla s\Big)\\
&\geq (\eta +\beta +\gamma )\varphi _q\Big(\int_\eta
^Th(s)f(s,u(s))\nabla s\Big)\\
&> (\eta +\beta +\gamma )\frac{\eta c'C_2}{T}\varphi _q\Big(
\int_\eta ^Th(s)\nabla s\Big)=c'.
\end{align*}

Next, we verify that $\theta (Au)<b'$ for
$u\in \partial P(\theta,b')$.
Let us choose $u\in \partial P(\theta ,b')$, then
$\theta (u)=\max_{t\in [0,\eta ]_{\mathbb{T}}}u(t)=u(\eta )=b'$.
This implies $0\leq u(t)\leq b'$, $t\in [0,\eta ]_{\mathbb{T}}$.
Since $u\in P$, we also have
$b'\leq u(t)\leq \| u\| \leq \frac T\eta u(l)=\frac T\eta b'$
for $t\in [\eta ,T]_{\mathbb{T}}$. So
\[
0\leq u(t)\leq \frac T\eta b',\quad t\in [0,T]_{\mathbb{T}}.
\]
Using (ii), we get
\[
f(t,u(t))<\varphi _p(b'C_1),\quad t\in [0,T]_{\mathbb{T}}.
\]
Also, $Au\in P$ implies that

\begin{align*}
\theta (Au) &= Au(\eta )\leq Au(T) \\
&= \int_0^T\varphi _q(\int_s^Th(\tau )f(\tau ,u(\tau
))\nabla \tau )\Delta s\\
 &\quad +\beta \varphi _q\Big(
\int_0^Th(s)f(s,u(s))\nabla s\Big)
+\gamma \varphi _q(\int_\eta ^Th(s)f(s,u(s))\nabla s)\\
&\leq (T+\beta +\gamma )\varphi _q\Big(\int_0^Th(s)f(s,u(s))\nabla
s\Big)\\
&<(T+\beta +\gamma )b'C_1\varphi _q\Big(\int_0^Th(s)\nabla
s\Big)=b'.
\end{align*}

Finally, we prove that $P(\alpha ,a')\neq \emptyset $ and
$\alpha (Au)>a'$ for all $u\in \partial P(\alpha ,a')$.
In fact, the constant function $\frac{a'}2\in P(\alpha ,a')$.
Moreover, for $u\in \partial P(\alpha ,a')$, we have $\alpha
(u)=\max_{t\in [0,c]_{\mathbb{T}}}u(t)=u(c)=a'$. This implies
$a'\leq u(t)\leq \frac Tca',\quad t\in [c,T]_{\mathbb{T}}$.
Using assumption (iii),
$f(t,u(t))>\varphi _p(a'C_3),\quad t\in [c,T]_{\mathbb{T}}$.
As before $Au\in P$, we obtain
\begin{align*}
\alpha (Au)
&= (Au)(c)\\
&=\int_0^c\varphi _q\big(\int_s ^Th(\tau )f(\tau
,u(\tau ))\nabla \tau \Big)\Delta s \\
&\quad +\beta \varphi _q\Big(\int_0^Th(s)f(s,u(s))\nabla s\Big)
+\gamma \varphi _q\Big(\int_\eta ^Th(s)f(s,u(s))\nabla s\Big)\\
&\geq (c+\beta +\gamma )\varphi _q\Big(\int_c^Th(s)f(s,u(s))\nabla
s\Big)\\
&> (c+\beta +\gamma )\frac{a'}{L}\varphi _q\Big(
\int_c^Th(s)\nabla s\Big)=a'.
\end{align*}
Thus, by Lemma \ref{lma2.7}, there exist at least two fixed points
of $A$ which are positive solutions $u_1$ and $u_2$, belonging to
$\overline{P(\psi ,c')}$, of  \eqref{1.1}-\eqref{1.2}
such that
\[
a'<\alpha (u_1)\quad \text{with }\theta (u_1)<b',\quad
b'<\theta (u_2)\quad \text{with }\psi (u_2)<c'.
\]
\end{proof}

In analogy to Theorem \ref{thm4.1}, we have the following result.

\begin{theorem}\label{thm4.2}
Assume that there are positive numbers $a'<b'<c'$ such that
\[
0<a'<\frac cTb'<\frac{cC_2}{TC_1}c'.
\]
Assume further that $f(t,u)$ satisfies the following conditions:
\begin{itemize}
\item[(i)] $f(t,u)<\varphi _p(c'C_1)$ for $(t,u)\in [0,T]_{\mathbb{T}}\times
[0,\frac T\eta c']$,

\item[(ii)] $f(t,u)>\varphi _p(b'C_2)$ for $(t,u)\in [\eta
,T]_{\mathbb{T}}\times [b',\frac T\eta b']$,

\item[(iii)] $f(t,u)<\varphi _p(a'C_1)$ for $(t,u)\in
[c,T]_{\mathbb{T}}\times [0,\frac Tca']$.

\end{itemize}
Then \eqref{1.1}-\eqref{1.2} has at
least two positive solutions $u_1$ and $u_2$ such that
\begin{gather*}
a'<\max_{t\in [0,c]_{\mathbb{T}}}u_1(t)\quad \text{with }\max_{t\in
[0,\eta ]_{\mathbb{T}}}u_1(t)<b',\\
b'<\max_{t\in [0,\eta ]_{\mathbb{T}}}u_2(t)\quad \text{with }
\max_{t\in [\eta ,c]_{\mathbb{T}}}u_2(t)<c'.
\end{gather*}
\end{theorem}

\begin{corollary} \label{cry4.3}
 Assume that $f$ satisfies conditions
\begin{itemize}
\item[(i)] $f_0(t)>\varphi _p(C_2)$, $t\in [\eta
,T]_{\mathbb{T}}$ and $f_\infty (t)=\liminf_{u\to \infty
}\frac{f(t,u)}{\varphi _p(u)}>\varphi _p(C_3)$, $t\in
[c,T]_\mathbb{T}$;

\item[(ii)] there exists $a'>0$ such that
$f(t,u)<\varphi _p(a'C_1)$ for
$(t,u)\in [0,T]_{\mathbb{T}} \times [0,\frac T \eta a']$.

\end{itemize}
Then \eqref{1.1}-\eqref{1.2} has at least two positive solutions.
\end{corollary}

\begin{corollary}\label{cry4.4}
Suppose that $f$ satisfies conditions
\begin{itemize}
\item[(i)] $f_0(t)<\varphi _p(\frac \eta {T}C_1)$, $t\in [0,T]_{\mathbb{T}}$ and
$f_\infty (t)<\varphi _p(\frac c{T}C_1)$, $t\in
[c,T]_{\mathbb{T}}$;

\item[(ii)] there exists $b'>0$ such that
$f(t,u)>\varphi _p(b'C_2)$, for
$(t,u)\in [ \eta,T]_{\mathbb{T}} \times [ b',\frac T\eta b']$.
\end{itemize}
Then \eqref{1.1}-\eqref{1.2} has at least two positive solutions.
\end{corollary}

By applying Theorems \ref{thm4.1} and \ref{thm4.2}, it is easy to
prove that Corollaries \ref{cry4.3} and \ref{cry4.4} hold,
respectively.

\section{Existence of three solutions}

 Let the nonnegative continuous concave functional $\Psi
:P\to [0,\infty )$ be defined by
\[
\Psi (u)=\min_{t\in [\eta ,T]_\mathbb{T}}u(t)=u(\eta ),\quad u\in
P.
\]
Note that for $u\in P$, $\Psi (u)\leq \| u\| $.

\begin{theorem}\label{thm5.1}
Suppose that there exist constants $0<d'<a'$ such that
\begin{itemize}
\item[(i)] $f(t,u)<\varphi _p(d'C_1)$, $(t,u)\in [0,T]_{\mathbb{T}}\times
[0,d']$;

\item[(ii)] $f(t,u)\geq \varphi _p(a'C_2)$,
$(t,u)\in [\eta ,T]_{\mathbb{T}}\times [a',\frac T\eta a']$;

\item[(iii)] one of the following conditions holds:

(D1) $\limsup_{u\to \infty }\max_{t\in [0,T]_{\mathbb{T}}}\frac{f(t,u)}{\varphi _p(u)
}<\varphi _p(C_1)$;

(D2) there exists a number $c'>\frac T\eta a'$ such that
$f(t,u)<\varphi _p(c'C_1)$ for $(t,u)\in [0,T]_{\mathbb{T}}\times
[0,c']$.

\end{itemize}
Then \eqref{1.1}-\eqref{1.2} has at least three positive solutions.
\end{theorem}

\begin{proof}
By the definition of operator $A$ and its
properties, it suffices to show that the conditions of Lemma
\ref{lma2.9} hold with respect to $A$.

We first show that if (D1) holds, then there exists a number
$l'>\frac T\eta a'$ such that $A:\overline{P}_{l'}\to
P_{l'}$.
Suppose that
\[
\limsup_{u\to \infty }\max_{t\in [0,T]_\mathbb{T}}
\frac{f(t,u)}{\varphi _p(u)}<\varphi _p(C_1)
\]
holds, then there are $\tau >0$ and $\delta <C_1$ such that if
$u>\tau , $ then
\[
\max_{t\in [0,T]_\mathbb{T}}\frac{f(t,u)}{\varphi _p(u)}\leq \varphi
_p(\delta ).
\]
That is to say,
\[
f(t,u)\leq \varphi _p(\delta u),\quad (t,u)\in
[0,T]_\mathbb{T}\times [\tau ,\infty ).
\]
Set $\lambda =\max \{ f(t,u):(t,u)\in [0,T]_\mathbb{T}\times
[0,\tau ]\} $, then
\begin{equation}
f(t,u)\leq \lambda +\varphi _p(\delta u),\quad  (t,u)\in
[0,T]_\mathbb{T}\times [0,\infty ). \label{5.1}
\end{equation}
Take
\begin{equation}
l'>\max \{ \frac T\eta a',\varphi _q(\frac{
\lambda }{\varphi _p(C_1)-\varphi _p(\delta )})\} .
\label{5.2}
\end{equation}
If $u\in \overline{P}_{l'}$, then by (\ref{2.4}), (\ref{5.1})
and (\ref{5.2}), we obtain
\begin{align*}
\| Au\|
&= Au(T) \\
&= \int_0^T\varphi _q(\int_s^Th(\tau )f(\tau ,u(\tau
))\nabla \tau )\Delta s+\beta \varphi _q(
\int_0^Th(s)f(s,u(s))\nabla s)\\
&\quad +\gamma \varphi _q(\int_\eta ^Th(s)f(s,u(s))\nabla s)\\
&\leq (T+\beta +\gamma )\varphi _q(\int_0^Th(s)f(s,u(s))\nabla s)\\
&\leq (T+\beta +\gamma )\varphi _q(
\int_0^Th(s)(\lambda+\varphi _p(\delta u(s))\nabla s)\\
&\leq (T+\beta +\gamma )\varphi_q(\lambda+\varphi_p(\delta
l'))\varphi _q(\int_0^Th(s)\nabla s)\\
&=\varphi_q
(\lambda+\varphi_p(\delta l'))\frac 1 {C_1}<l'.
\end{align*}

Next we verify that if there is a positive number $r'$ such that
if $f(t,u)<\varphi _p(r'/N)$ for $(t,u)\in [0,T]_\mathbb{T}$ $\times
[0,r']$, then $A:\overline{P}_{r'}\to P_{r'}$.

Indeed, if $u\in \overline{P}_{r'}$, then
\begin{align*}
\| Au\|
&= Au(T) \\
&\leq (T+\beta +\gamma )\varphi _q\Big(\int_0^Th(s)f(s,u(s))\nabla s\Big)\\
&<\frac{r'}N(T+\beta +\gamma )\varphi _q\Big(\int_0^Th(s)\nabla s\Big)=r',
\end{align*}
thus, $Au\in P_{r'}$.
Hence, we have shown that either (D$_1$) or (D$_2$) holds, then there exists
a number $c'$ with $c'>\frac T\eta a'$ such that
$A:$ $\overline{P}_{c'}\to $ $P_{c'}$. It is also
note from (i) that $A:$ $\overline{P}_{d'}\to $
$P_{d'}$.

Now, we show that $\{ u\in P(\Psi ,a',\frac T\eta
a'):\Psi (u)>a'\} \neq \emptyset $ and $\Psi (Au)>a'$
for all $u\in P(\Psi ,a',\frac T\eta a')$.
In fact,
\[
u=\frac{(\eta +T)a'}{2\eta }\in \{ u\in P(\Psi
,a',\frac T\eta a'):\Psi (u)>a'\} .
\]
For $u\in P(\Psi ,a',\frac T\eta a')$, we have
\[
a'\leq \min_{t\in [\eta ,T]_\mathbb{T}}u(t)=u(\eta )\leq u(t)\leq
\frac T\eta a',
\]
for all $t\in [\eta ,T]_\mathbb{T}$. Then, in view of (ii), we know
that
\begin{align*}
\Psi (Au)
&= \min_{t\in [\eta ,T]_\mathbb{T}}Au(t)=Au(\eta )\\
&= \int_0^\eta \varphi _q\Big(\int_\tau ^Th(\tau )f(\tau ,u(\tau
))\nabla \tau \Big)\Delta s+\beta \varphi _q(
\int_0^Th(s)f(s,u(s))\nabla s)\\
&\quad +\gamma \varphi _q\Big(\int_\eta ^Th(s)f(s,u(s))\nabla s\Big)
\\
&\geq (\eta +\beta +\gamma )\varphi _q\Big(\int_\eta
^Th(s)f(s,u(s))\nabla s\Big)\\
&\geq (\eta +\beta +\gamma )a'C_2\varphi _q\Big(\int_\eta
^Th(s)\nabla s\Big)=a'.
\end{align*}

Finally, we assert that if $u\in P(\Psi ,a',c')$ and
$\| Au\| >\frac T\eta a'$, then $\Psi (Au)>a'$.
Suppose $u\in P(\Psi ,a',c')$ and $\| Au\|
>\frac T\eta a'$. Then
\begin{align*}
\Psi (Au) &= \min_{t\in [\eta ,T]_\mathbb{T}}Au(t)=Au(\eta )\\
&\geq \frac \eta TAu(T)
= \frac \eta T\| Au\| >a'.
\end{align*}

To sum up, the hypotheses of Lemma \ref{lma2.9} are satisfied, hence
\eqref{1.1}--\eqref{1.2} has at least three positive solutions
$u_1,u_2,u_3$ such that
\[
\| u_1\| <d',a'<\min_{t\in [\eta ,T]_\mathbb{T}}u_2(t)\quad{and}\quad
\| u_3\| >d'\text{ with }\min_{t\in \eta ,T]_\mathbb{T}}u_3(t)<a'.
\]
\end{proof}

From Theorem \ref{thm5.1}, we see that, when assumptions such as
 (i), (ii), (iii) are imposed appropriately on $f$, we can establish
the existence of an arbitrary odd number of positive solutions of
\eqref{1.1}, \eqref{1.2}.

\begin{theorem}\label{thm5.2} If
\[
0<d_1'<a_1'<\frac T\eta a_1'<d_2'<a_2'<\frac T\eta
a_2'<d_3'<\ldots <d_n',\quad n\in \mathbb{N},
\]
\begin{itemize}
\item[(i)] $f(t,u)<\varphi _p(d_i'C_1)$, $(t,u)\in
[0,T]_\mathbb{T}\times [0,d_i']$;

\item[(ii)] $f(t,u)\geq \varphi _p(a_i'C_2)$, $(t,u)\in
[ \eta ,T]_\mathbb{T} \times [a_i',\frac T\eta a_i']$;
\end{itemize}
then \eqref{1.1}-\eqref{1.2} has at least $2n-1$ positive
solutions.
\end{theorem}

\begin{proof}
When $n=1$, it is immediate from condition (i) that $A:
\overline{P}_{d_1'}\to P_{d_1'}\subset \overline{P}_{d_1'}$,
which means that $A$ has at least one fixed
point $u_1\in \overline{P}_{d_1'}$ by the Schauder fixed
point theorem. When $n=2$, it is clear that Theorem 4.1 holds (with
$c_1=d_2')$. Then we can obtain at least three positive solutions
$u_1$, $u_2$, and $u_3$ satisfying
\[
\| u_1\| <d_1',\min_{t\in [\eta
,T]_{\mathbb{T}}}u_2(t)>a_1'\quad{and}\quad
\| u_3\| >d_1' \text{ with }\min_{t\in [\eta ,T]_{\mathbb{T}}}u_3(t)<a_1'.
\]
Following this way, we complete the proof by induction.
\end{proof}

\section{Example}

Let $\mathbb{T}=\{1-(1/2)^{\mathbb{N}_0}\}\cup \{1\}$, where
$\mathbb{N}_0$ denote the set of nonnegative integers.
Take $T=1$, $p=\frac 3 2$, $\beta=\gamma=\frac 1 4$,
$\eta=\frac {15}{16}$, $c=\frac {31}{32}$. If we let $h(s)\equiv 1$,
then by (\ref{2.4}) and (\ref{2.41}) we have
\begin{gather*}
C_1=(T+\beta +\gamma )^{-1}\varphi _p\Big(\int_0^T 1\nabla
s\Big)=\frac 2 3, \\
C_2=(\eta +\beta +\gamma )^{-1}\varphi
_p\Big(\int_\eta ^T 1 \nabla s\Big)=\frac 4 {23} .
\end{gather*}
 Suppose
\[
f(t,u)=f(u):=\frac{6+9u}{20(1+u)}(2+\sin u ) \sqrt u,\quad t\in
[0,1]_{\mathbb{T}},\; u\geq 0,
\]
then $f_0=f^0=\frac 3 5$, $f_\infty=\frac 9 {20}$,
$f^\infty=\frac {27} {20}$.

Firstly, by easy calculation, it is easy to get
\[
\varphi_p(C_1)=\sqrt {\frac 2 3}\approx 1.224, \quad
\varphi_p (\frac T \eta C_2)=\sqrt{\frac
{64}{345}}\approx 0.431,
\]
So  the condition (H3) of Corollary \ref {cry2.4} holds. Thus by
Corollary \ref {cry2.4}, the boundary-value problem
\begin{gather}
\big(|u^{\Delta}(t)|^{-\frac 1 2} u^{\Delta}
(t)\big)^{\nabla}+\frac{6+9u(t)}{20(1+u(t))}(2+\sin u(t))
\sqrt{u(t)}=0,\label{6.1}
\\
u(0)-\frac 1 2u^{\Delta}(0)=\frac 1 2
u^{\Delta}\big(\frac{15}{16}\big), \quad u^\Delta (1)=0,
\label{6.2}
\end{gather}
has at least one positive solution.

Secondly, since
\[
f_0=\frac 3 5<\varphi_p\big(\frac \eta T C_1\big)
=\sqrt{\frac 5 8}\approx 0.791, \quad
f_\infty=\frac 9 {20}<\varphi_p\big(\frac c T
C_1\big)=\sqrt{\frac {11}{48}}\approx 0.479.
\]
If we choose $b'=0.7$, then
\[
\min_{t\in [0,T]_{\mathbb{T}}, u\in [b', \frac {16}
{15}b']} f(t,u)\approx 0.800> \varphi_p(b'C_1)\approx 0.683.
\]
So all the assumptions of Corollary \ref{cry4.4} are satisfied.
Therefore by Corollary \ref{cry4.4} the boundary value problem
\eqref{5.1}--\eqref{5.2} has at lest two solutions $u_1$ and $u_2$
with $0<\|u_1\| \leq 0.7< \|u_2\|$.

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