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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 37, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/37\hfil Existence of positive solutions]
{Existence of positive solutions for $p$-Laplacian an
$m$-point boundary value problem  involving the derivative on time scales}

\author[A. Dogan \hfil EJDE-2014/37\hfilneg]
{Abdulkadir Dogan}  % in alphabetical order

\address{Abdulkadir Dogan  \newline
Department of  Applied Mathematics,
Faculty of Computer Sciences,
Abdullah Gul University, Kayseri, 38039 Turkey}
\email{abdulkadir.dogan@agu.edu.tr
Tel: +90 352 224 88 00  Fax: +90 352 338 88 28}

\thanks{Submitted December 3, 2013. Published January 30, 2014.}
\subjclass[2000]{34B15, 34B16, 34B18, 39A10}
\keywords{Time scales; boundary value problem;
  $p$-Laplacian; \hfill\break\indent positive solution; fixed point theorem}

\begin{abstract}
 We  are interested  in the existence  of positive solutions
 for the $p$-Laplacian  dynamic equation  on time scales,
 $$
 (\phi_p(u^\Delta(t)))^\nabla+a(t)f(t,u(t),u^\Delta(t))=0,\quad
 t\in(0,T)_{\mathbb{T}},
 $$
 subject to the  multipoint boundary condition,
 $$
 u(0)=\sum_{i=1}^{m-2}\alpha_i u(\xi_i), \quad  u^\Delta(T)=0,
 $$
 where $\phi_p(s)=|s|^{p-2} s$, $p>1$, $\xi_i\in [0,T]_{\mathbb{T}}$,
 $ 0<\xi_1<\xi_2<\dots<\xi_{m-2}<\rho(T)$.
 By using fixed point theorems, we prove the existence of at least three
 non-negatvie solutions, two of them positive, to the above boundary value
 problem.  The interesting point is  the nonlinear term $f$ is involved with
 the first order derivative explicitly.
 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}

The theory of  time scales, which has recently received a lot of attention,
was introduced and developed  by Aulbach and Hilger \cite{Hilger} in 1988.
It has been created in order to unify continuous and  discrete analysis,
and it allows a simultaneous treatment of differential and difference equations,
extending those theories to so-called dynamic equations.
 Further, the study of time scales has led to several important applications,
e.g., in the study of insect population models, heat transfer, neural
networks, phytoremediation of metals, wound healing, and epidemic models,
see \cite{Jones,Sped,Thomas}.


Recently,  much  attention has been paid to the existence of multipoint
 positive solutions of  boundary value problems (BVPs)
 on time scales. When the nonlinear term $f$ does not depend on the
first order derivative, many researchers have studied multipoint
boundary conditions on time scales; see
\cite{Anderson,Bohner3,Dogan,Dogan2,Dogan3,Han,Luo,Sang1,Sang2,Sun1,Zhu}.
However, little work has done on the existence of positive solutions
for multipoint BVP on time scales when the nonlinear term is involved in
first order derivative explicitly;  see \cite{Dogan4,Su1,Sun2}.

There is recent work in fixed point applications using convex and concave
functionals in which there is nonlinear dependence on higher order derivatives;
see \cite{Anderson2,Mavr}.


Motivated by all the above works, we are interested in the existence  of
at least three non-negative solutions, two of them  positive, 
for $p$-Laplacian dynamic equation on time scales,
\begin{equation}
(\phi_p(u^\Delta(t)))^\nabla+a(t)f(t,u(t),u^\Delta(t))=0,\quad
t\in(0,T)_{\mathbb{T}}, \label{e1.1}
\end{equation}
subject to  boundary condition
\begin{equation}
u(0)=\sum_{i=1}^{m-2}\alpha_i u(\xi_i), \quad u^\Delta(T)=0,
\label{e1.2}
\end{equation}
where $\phi_p(u)$ is $p$-Laplacian operator; i.e.,
$\phi_p(s)=|s|^{p-2} s$, for  $p>1$, with $(\phi_p)^{-1}=\phi_q$
and $\frac{1}{p}+ \frac{1}{q}=1$.
 The usual notation and terminology for time scales as can be found
in \cite{Bohner,Bohner2}, will be used here. The  interesting point
is that the nonlinear term $f$ is involved with the first order derivative
explicitly. Our main results will depend on an application of a generalization
of the Leggett-Williams fixed point theorem due to Bai and Ge.
An example is also given to illustrate the main results. The results are
even new for the special cases of difference equations and differential
equations, as well as in the general time scale setting. We shall prove
that the BVP \eqref{e1.1} and \eqref{e1.2} has at least three non-negative
solutions.

Throughout the paper, we will suppose that the following conditions are satisfied:
\begin{enumerate}
\item[(H1)]  $0,T \in \mathbb{T}$,  $0<\xi_1<\xi_2<\dots<\xi_{m-2}<\rho(T)$,
$\xi_i \in \mathbb{T}$, $\alpha_i \in[0,\infty) $ for
$i=1,\dots,m-2$, and $1-\sum_{i=1}^{m-2}\alpha_i>0$;
\item[(H2)]  $\eta=\min\{t\in \mathbb{T} : \frac{T}{2} \leq t <T \}$ exists;
\item[(H3)]  $a(t)\in C_{ld}([0,T]_{\mathbb{T}},[0,\infty))$ with
 $0<\int_{\eta}^{T} a(t) \nabla t<\infty$;
\item[(H4)]  $f:(0,T)_{\mathbb{T}}\times[0,\infty)\times
\mathbb{R} \to[0,\infty)$   is continuous;
\item[(H5)]  $a(t)f(t,0,0)\not\equiv  0$, $f(t,0,0)\geq 0$ on $[0,T]_{\mathbb{T}}$.
\end{enumerate}

The rest of this article is arranged as follows.
We state some definitions, notation,  lemmas and prove several preliminary
results in Section 2.
The main theorem on the existence of at least three non-negative solutions
and its proof are presented in Section 3.
In  section 4, we give an example to demonstrate our results.


\section{Preliminaries}

In this section, we provide some background materials from theory of cones
in Banach spaces. The following
definitions can be found in the book by Deimling \cite{Deim}
as well as in the book by Guo and Lakshmikantham \cite{Guo}.

\begin{definition} \label{def2.1} \rm
  Let $E$  be a real Banach space. A nonempty, closed, convex set
 $P \subset E$ is a cone if it satisfies the following two conditions:
\begin{enumerate}
\item[(i)]  $x \in P$, $\lambda \geq 0$ imply   $\lambda x\in P$;
\item[(ii)] $x \in P$, $-x\in P$ imply   $ x=0$.
\end{enumerate}
Every cone $P \subset E$ induces an ordering in $E$ given by
$x\leq y$ if and only if $y-x \in P$.
\end{definition}

\begin{definition} \label{def2.2} \rm
A map $\psi$  is said to be a nonnegative continuous concave functional
on a cone $P$ of a real Banach space $E$ if $\psi:P \to [0,\infty)$
is continuous and
$$
\psi(tx+(1-t)y)\geq t\psi(x)+(1-t)\psi(y)
$$
 for all $x,y\in P$ and $t\in[0,1]$.

Similarly, we say the map $\alpha$ is a nonnegative continuous convex
functional on a cone $P$ of a real Banach space $E$ if
$\alpha: P \to [0,\infty)$ is continuous and
$$
\alpha (tx+(1-t)y) \leq t\alpha(x)+(1-t)\alpha(y)
$$
for all $x,y \in P$ and $t\in [0,1]$.
\end{definition}

Let $\psi$ be a nonnegative continuous concave functional on $P$, and
$\alpha$ and  $\beta$ be nonnegative continuous convex functionals on  $P$.
For nonnegative real numbers  $r, a$  and $l$, we define the following
convex sets.
\begin{gather*}
P (\alpha,r;\beta,l)=\{u\in P : \alpha(u)<r, \beta(u)<l \},\\
\bar{P}  (\alpha,r;\beta,l)=\{u\in P : \alpha(u)\leq r, \beta(u)\leq l \}, \\
P (\alpha,r;\beta,l;\psi,a)=\{u\in P : \alpha(u)<r, \beta(u)<l, \psi(u)>a \},\\
\bar{P}  (\alpha,r;\beta,l;\psi,a)=\{u\in P : \alpha(u)\leq r, \beta(u)\leq l,
\psi(u)\geq a \}.
\end{gather*}
To prove our main results,  we need the following fixed point theorem,
which comes from Bai and Ge in  \cite{Bai}.

\begin{lemma}[\cite{Bai}] \label{lem2.1}
Let $P$ be a cone in a real Banach space $E$. Assume that constants
$r_1,b,d,r_2,l_1$ and  $l_2$  satisfy $0<r_1<b<d\leq r_2$ and $0<l_1\leq l_2$.
If there exist two nonnegative continuous convex functionals  $\alpha$
 and  $\beta$  on $P$ and a nonnegative continuous concave functional
$\psi$ on $P$ such that
\begin{itemize}
\item [(A1)]  there exists $M>0$ such that
 $\|u \| \leq M \max\{\alpha(u),\beta(u) \}$ for all $u\in P$;
\item[(A2)]  $P(\alpha,r;\beta,l)\ne \emptyset$ for any $r>0$ and $l>0$;
\item[(A3)]  $\psi(u)\leq \alpha(u)$ for all $u\in \bar{P}(\alpha, r_2;
\beta,l_2)$;
\end{itemize}
and if
$F: \bar{P}(\alpha, r_2; \beta,l_2)\to(\alpha, r_2; \beta,l_2)$
is completely continuous operator, which satisfies
\begin{itemize}
\item[(B1)]  $\{u \in \bar{P}(\alpha, d;\beta,l_2;\psi,b)
 :\psi(u)>b\}\ne \emptyset$, $\psi(Fu)>b$  for \\
 $u \in \bar{P}(\alpha, d;\beta,l_2;\psi,b)$;

\item[(B2)]  $\alpha(Fu)<r_1$,    $\beta(Fu)<l_1$ for
$ u\in \bar{P}(\alpha,r_1;\beta,l_1)$;
\item[(B3)]  $\psi(Fu)>b $  for  $u \in \bar{P}(\alpha, r_2;\beta,l_2;\psi,b) $
with  $\alpha(Fu)>d$.
\end{itemize}
Then  $F$  has at least three different fixed points  $u_1,u_2$ and
$u_3$ in $\bar{P}( \alpha,r_2;\beta,l_2)$  with
\begin{gather*}
u_1\in P(\alpha,r_1;\beta,l_1),\quad
u_2 \in\{\bar{P}(\alpha,r_2;\beta,l_2;\psi,b):\psi(u)>b\},\\
u_3\in \bar{P}(\alpha,r_2;\beta,l_2) \setminus
\big ( \bar{P}(\alpha,r_2;\beta,l_2;\psi,b)
\cup \bar{P}(\alpha,r_1;\beta,l_1)\big).
\end{gather*}
\end{lemma}

Let the Banach space
\[
E= \{ u :  [0,T]_{\mathbb{T}}\to\mathbb{R}  : 
\text{$u$ is $\Delta$-differentiable and $u^{\Delta}$ is ld-continuous on }
[0,T]_{\mathbb{T}} \}
\]
be endowed with norm
$$
\| u\|= \max \Big \{ \sup_{t\in [0,T]_{\mathbb{T}}} | u(t)|,
\sup_{t\in [0,T]_{\mathbb{T}}}  | u^{\Delta}(t)| \Big \}.
$$
Define
$$
P=\{u\in E : u(t)\geq 0, u^{\Delta}(t)\geq 0, \text{ and $u(t)$ is concave on }
[0,T]_{\mathbb{T}}\}.
$$
Clearly, $P$ is a cone.

\begin{lemma} \label{lem2.2}
If  $\sum_{i=1}^{m-2}\alpha_i \ne 1$,
then for  $h\in C_{ld}[0,T]_{\mathbb{T}}$   and  $h \geq 0$,
\begin{gather}
(\phi_p(u^\Delta(t)))^\nabla+h(t)=0,\quad  t\in(0,T)_{\mathbb{T}}, \label{e2.1} \\
u(0)=\sum_{i=1}^{m-2}\alpha_i u(\xi_i), \quad u^\Delta(T)=0 \label{e2.2}
\end{gather}
has the unique solution
\begin{equation}
u(t)=\int_0^t\phi_q \Big( \int_s^T h(\tau) \nabla \tau \Big) \Delta s
   +\frac{ 1} {1-\sum_{i=1}^{m-2} \alpha_i} \sum_{i=1}^{m-2}
 \alpha_i \int_0^{\xi_i} \phi_q \Big(\int_s^T h(\tau) \nabla \tau \Big)\Delta s.
   \label{e2.3}
\end{equation}
Moreover, if $ h(t) \geq 0 $ on $ [0,T]_{\mathbb{T}} $ and {\rm (H1)} is
satisfied, then $ u(t) \geq 0 $ on $ [0,T]_{\mathbb{T}}$.
\end{lemma}


\begin{proof}
 Let  $u$  be as in \eqref{e2.3},  taking the delta derivative of \eqref{e2.3},
we have
$$
u^\Delta(t)=\phi_q \Big(\int_t^T h(\tau) \nabla \tau \Big),
$$
moreover, we obtain
$$
\phi _p (u^\Delta(t))= \int_t^T h(\tau) \nabla \tau,
$$
taking the nabla derivative of this expression yields
$(\phi_p(u^\Delta(t)))^\nabla=-h(t)$.
Routine calculations verify that $u$ satisfies the boundary value
conditions in \eqref{e2.2}, so that
$u$ given in \eqref{e2.3} is a solution of \eqref{e2.1} and \eqref{e2.2}.
It is easy to see that BVP
$(\phi_p(u^\Delta))^\nabla =0 $,
$u(0)=\sum_{i=1}^{m-2}\alpha_i u(\xi_i)$, $u^\Delta(T)=0$
 has only the trivial solution. Thus $u$
in  \eqref{e2.3} is the unique solution of \eqref{e2.1} and \eqref{e2.2}.
The proof is complete.
\end{proof}

\begin{lemma} \label{lem2.3}  The solution of BVP \eqref{e2.1}  and
\eqref{e2.2} satisfies $u(t)\geq 0$, for $t\in[0,T]_{\mathbb{T}}$.
\end{lemma}

\begin{proof}  Let
 $$
\varphi(s)=\phi_q \Big( \int_s^T h(\tau) \nabla \tau \Big).
$$
Since
$\int_s^T h(\tau) \nabla \tau \geq 0$, it follows that
$\varphi(s)\geq 0$. According to Lemma \ref{lem2.2},  we obtain
\begin{gather*}
u(0)=\frac {1}{1-\sum_{i=1}^{m-2} \alpha_i}
\sum_{i=1}^{m-2} \alpha_i
 \int_0^{\xi_i} \varphi(s)\Delta s \geq 0 , \\
u(T)=\int_0^T \varphi(s) \Delta s+
\frac {1 }{1-\sum_{i=1}^{m-2} \alpha_i} \sum_{i=1}^{m-2} \alpha_i
 \int_0^{\xi_i} \varphi(s)\Delta s \geq 0.
\end{gather*}
If  $t\in (0,T)$, we have
$$
u(t)=\int_0^t \varphi(s) \Delta s+
\frac {1 }{1-\sum_{i=1}^{m-2} \alpha_i} \sum_{i=1}^{m-2} \alpha_i
 \int_0^{\xi_i} \varphi(s)\Delta s \geq 0.
$$
So  $ u(t)\geq 0$ for $t\in [0,T]$.
\end{proof}

\begin{lemma} \label{lem2.4}
The solution of  \eqref{e1.1}  and  \eqref{e1.2}  satisfies
$$
\inf_{t\in [0,T]_{\mathbb{T}}} u(t)\geq \gamma \|u\|
$$
where
$$
\gamma=\frac{\sum_{i=1}^{m-2} \alpha_i \xi_i} {T-\sum_{i=1}^{m-2}
\alpha_i(T- \xi_i)}.
$$
\end{lemma}

\begin{proof}
Clearly $u^{\Delta}(t)=\varphi(t)\geq0$. This implies that
$$
\min_{t\in[0,T]}u(t)=u(0),\quad  \|u \|=u(T).
$$
It is easy to see that $u^{\Delta}(t_2)\leq u^{\Delta}(t_1)$
for any $t_1,t_2\in [0,T]$ with $t_1\leq t_2$.
Hence $u^{\Delta}(t)$ is a decreasing function on $[0,T]$.
This means that graph of $u(t)$ is concave down on $(0,T)$.
For each $i\in \{1,2,\dots,m-2\}$, we have
$$
\frac{u(T)-u(0)}{T-0}\geq \frac{u(T)-u(\xi_i)}{T-\xi_i},
$$
i.e.,
$$
Tu(\xi_i)-\xi_i u(T) \geq (T-\xi_i) u(0),
$$
so that
$$
T\sum_{i=1}^{m-2} \alpha_i u(\xi_i)- \sum_{i=1}^{m-2} \alpha_i \xi_i u(T)
\geq \sum_{i=1}^{m-2} \alpha_i (T-\xi_i)u(0).
$$
With the boundary condition $u(0)=\sum_{i=1}^{m-2}\alpha_i u(\xi_i)$, we have
$$
u(0) \geq \frac{\sum_{i=1}^{m-2} \alpha_i \xi_i} {T-\sum_{i=1}^{m-2}
\alpha_i (T-\xi_i)}u(T).
$$
This completes the proof.
\end{proof}

Define the operator  $F:P\to E$  by
\begin{align*}
(Fu)(t)
& = \int_0^t \phi_q \Big(\int_s^T a(\tau) f( \tau,u(\tau),u^{\Delta}(\tau) )
 \nabla \tau \Big)\Delta s\\
  &\quad  +\frac{ 1} {1-\sum_{i=1}^{m-2} \alpha_i}
\sum_{i=1}^{m-2} \alpha_i   \int_0^{\xi_i} \phi_q
\Big(\int_s^T a(\tau) f(\tau,u(\tau),u^{\Delta}(\tau) )
\nabla \tau \Big)\Delta s
\end{align*}
for  $t\in [0,T]_{\mathbb{T}}$.
  By the definition of  $F$, the monotonicity of  $\phi_q(u)$  and
assumption of (H1)-(H5), it is   easy to see that for each
$u\in P$, $Fu \in P$ and $Fu(T)$ is the maximum value of $ Fu(t)$.
Moreover, by direct calculation, we obtain that each fixed point of the
 operator  $F$  in  $P$  is a positive solution of
 \eqref{e1.1}  and  \eqref{e1.2}.
 It is easy to see that  $F:P\to P$  is completely continuous.


\section{Existence of positive solutions}

For  $u\in P$  we define
\begin{gather*}
\alpha(u)=\max_{t\in [0,T]_{\mathbb{T}}} | u(t)|=u(T),\quad
\beta(u)=\sup_{t\in [0,T]_{\mathbb{T}}} | u^{\Delta}(t)|=u^{\Delta}(0),\\
\psi(u)=\min_{t\in [\eta,T]_{\mathbb{T}}}  u(t)=u(\eta).
\end{gather*}
It is easy to see that $\alpha,\beta:P\to [0,\infty)$ are
nonnegative continuous convex functionals with
$\|u\|=\max\{\alpha(u),\beta(u)\}; \psi:P\to [0,\infty)$
is nonnegative concave functional.
We have $\psi(u)\leq \alpha(u)$ for $u\in P$ and assumptions
 (A1), (A2) and (A3) in Lemma \ref{lem2.1} hold.

For notational convenience, we denote
\begin{gather*}
 M  =  \int_0^\eta \phi _q \Big( \int_{\eta}^T a(\tau) \nabla \tau \Big)
 \Delta s, \\
 N  =  \int_0^T \phi_q \Big( \int_s^T a(\tau) \nabla \tau \Big) \Delta s
  +\frac { 1} {1-\sum_{i=1}^{m-2} \alpha_i}\sum_{i=1}^{m-2} \alpha_i
  \int_0^{\xi_i} \phi _q \Big( \int_s^T a(\tau) \nabla \tau \Big) \Delta s, \\
 L  =  \phi_q \Big( \int_0^T a(\tau) \nabla \tau \Big).
\end{gather*}


\begin{theorem} \label{thm3.1}
Assume that {\rm (H1)--(H5)} hold, and there exists
$0<r_1 < b < 2b \leq r_2$,  $0 <l_1\leq l_2$ such that
$\frac{b}{M} \leq \min \{r_2/N, l_2/L \}$.
If  $f$ satisfies the following three conditions:
\begin{itemize}
\item[(i)] $ f(t,w,v) \leq \min \{\phi_p(r_2/N),\phi_p(l_2/L) \}$ for
$(t,w,v) \in [0,T]_{\mathbb{T}}\times[ 0,r_2]\times [-l_2,l_2]$;

\item[(ii)] $ f(t,w,v) > \phi_p(b/M) $ for
$(t,w,v) \in [\eta,T]_{\mathbb{T}}\times[ b,2b]\times [-l_2,l_2]$;

\item[(iii)] $f(t,w,v) <\min \{\phi_p(r_1/N),\phi_p(l_1/L) \}$  for
$(t,w,v) \in [0,T]_{\mathbb{T}}\times[ 0,r_1]\times [-l_1,l_1]$;
\end{itemize}
then BVP \eqref{e1.1} and \eqref{e1.2} has at least three 
three non-negative solutions, two of them positive,
$u_1,u_2,u_3$, which satisfy
\begin{gather*}
\max_{t\in [0,T]_{\mathbb{T}} } \{ u_1(t) \} <r_1,  \quad
 \sup_{t\in [0,T]_{\mathbb{T}} } | u_1^{\Delta}(t) | < l_1;  \\
b <\min_{t\in [\eta,T]_{\mathbb{T}} } \{ u_2(t) \}
 \leq  \max_{t\in [0,T]_{\mathbb{T}} } \{ u_2(t) \} \leq r_2, \quad
\sup_{t\in [0,T]_{\mathbb{T}} } | u_2^{\Delta}(t) | \leq l_2;  \\
\min_{t\in [\eta,T]_{\mathbb{T}} } \{ u_3(t) \} < b, \quad
r_1< \max_{t\in [0,T]_{\mathbb{T}} } \{ u_3(t) \} < 2b, \quad
 l_1 < \sup_{t\in [0,T]_{\mathbb{T}} } | u_3^{\Delta}(t) | \leq l_2.
\end{gather*}
\end{theorem}

\begin{proof}  To show Lemma \ref{lem2.1} holds, it  is sufficient to show that
conditions in Lemma \ref{lem2.1} are satisfied with respect to operator $F$.
We first prove that if the assumption (i) is satisfied,
then $F:\bar{P}(\alpha,r_2;\beta,l_2)\to \bar{P}(\alpha,r_2;\beta,l_2)$.
If $u \in \bar{P}(\alpha,r_2;\beta,l_2)$, then
$$
\alpha(u)=\max_{t\in [0,T]_{\mathbb{T}} } | u(t) | \leq r_2,\quad
\beta(u)=\sup_{t\in [0,T]_{\mathbb{T}} } | u^{\Delta}(t) | \leq l_2
$$
and assumption (i) implies
$$
f(t,u(t),u^{\Delta}(t)) \leq \min \Big\{\phi_p(r_2/N),\phi_p(l_2/L) \Big \},
 t \in [0,T]_{\mathbb{T}}.
$$
For $u\in P$, there is $Fu\in P$, therefore
\begin{align*}
 \alpha(Fu)
&=  \max_{t\in [0,T]_{\mathbb{T}} } | (Fu)(t) | \\
&=  \max_{t\in [0,T]_{\mathbb{T}} } \Big| \int_0^t \phi_q \Big(
\int_s^T a(\tau) f(\tau,u(\tau ),u^{\Delta}(\tau ) ) \nabla \tau \Big)\Delta s\\
& \quad +\frac{1} {1-\sum_{i=1}^{m-2} \alpha_i}
 \sum_{i=1}^{m-2} \alpha_i   \int_0^{\xi_i} \phi_q \Big(
\int_s^T a(\tau) f(\tau,u(\tau ) ,u^{\Delta}(\tau ) ) \nabla \tau \Big)\Delta s
 \Big|\\
&=  \int_0^T \phi_q \Big(
\int_s^T a(\tau) f(\tau,u(\tau ),u^{\Delta}(\tau ) ) \nabla \tau \Big)\Delta s\\
& \quad +\frac{1} {1-\sum_{i=1}^{m-2} \alpha_i}
  \sum_{i=1}^{m-2} \alpha_i   \int_0^{\xi_i} \phi_q \Big(
\int_s^T a(\tau) f(\tau,u(\tau ) ,u^{\Delta}(\tau ) ) \nabla \tau \Big)\Delta s  \\
& <   \int_0^T \phi_q \Big(
\int_s^T a(\tau) \phi_p(r_2/N) \nabla \tau \Big)\Delta s\\
& \quad +\frac{1} {1-\sum_{i=1}^{m-2} \alpha_i}
    \sum_{i=1}^{m-2} \alpha_i \int_0^{\xi_i} \phi_q \Big(
\int_s^T a(\tau)  \phi_p(r_2/N)    \nabla \tau \Big)\Delta s  \\
& = \frac{r_2}{N} \Big[  \int_0^T \phi_q \Big(
\int_s^T a(\tau)  \nabla \tau \Big)\Delta s\\
&\quad +\frac{1} {1-\sum_{i=1}^{m-2} \alpha_i}
   \sum_{i=1}^{m-2} \alpha_i  \int_0^{\xi_i} \phi_q \Big(
\int_s^T a(\tau)    \nabla \tau \Big)\Delta s \Big]=r_2
\end{align*}
and
\begin{align*}
\beta(Fu) & = \sup_{t\in [0,T]_{\mathbb{T}} } | (Fu)^{\Delta}(t) |   \\
&=  \sup_{t\in [0,T]_{\mathbb{T}} } \Big|  \phi_q \Big(
\int_t^T a(\tau) f(\tau,u(\tau ),u^{\Delta}(\tau ) ) \nabla \tau \Big) \Big|\\
&=    \phi_q \Big(
\int_0^T a(\tau) f(\tau,u(\tau ),u^{\Delta}(\tau ) ) \nabla \tau \Big) \\
&\leq   \phi_q \Big( \int_0^T a(\tau) \phi_p(l_2/L) \nabla \tau \Big) \\
&=  \frac{l_2}{L}  \phi_q \Big( \int_0^T a(\tau)  \nabla \tau \Big)=l_2.
\end{align*}
Therefore $F:\bar{P}(\alpha,r_2;\beta,l_2)\to \bar{P}(\alpha,r_2;\beta,l_2)$.
Similarly, if  $u\in \bar{P}(\alpha,r_1;\beta,l_1)$, then the assumption (iii)
implies
$$
f (t,u(t),u^{\Delta}(t))< \min\{\phi_p(r_1/N),\phi_p(l_1/L) \} \quad
  \text{for }  t\in[0,T]_{\mathbb{T}}.
$$
We can get that $F:\bar{P}(\alpha,r_1;\beta,l_1)\to P(\alpha,r_1;\beta,l_1)$.
So condition (B2) of Lemma \ref{lem2.1} is satisfied.

To prove that condition (B1) of Lemma \ref{lem2.1} holds. We choose
$u(t)=2b$ for $t\in [0,T]_{\mathbb{T}}$. It is obvious that
$u(t)=2b\in \bar{P}(\alpha,2b;\beta,l_2;\psi,b)$ and
$\psi(u)=2b >b$,
and consequently
$$
\{ u\in \bar{P}(\alpha,2b;\beta,l_2;\psi,b):\psi(u) >b \}\neq \emptyset.
$$
So, for $u\in \bar{P}(\alpha,2b;\beta,l_2;\psi,b)$, there are
$b\leq u(t)\leq 2b$ and  $| u^\Delta(t)|\leq l_2$ for
$ t\in[\eta,T]_{\mathbb{T}}$.
Thus from the assumption (ii) we have
$$
f(t,u(t),u^\Delta(t)) > \phi_p(b/M) \quad \text{for }t\in[\eta,T]_{\mathbb{T}}.
$$
From the definition of the functional $\psi$ we see that
\begin{align*}
\psi(Fu)
&= \min_{t\in[\eta,T]_{\mathbb{T}}} Fu (t)=Fu (\eta)\\
&=   \int_0^{\eta} \phi_q \Big(\int_s^T a(\tau)
f(\tau,u(\tau ),u^{\Delta}(\tau ) ) \nabla \tau \Big)\Delta s\\
& \quad +\frac{1} {1-\sum_{i=1}^{m-2} \alpha_i}
 \sum_{i=1}^{m-2} \alpha_i \int_0^{\xi_i} \phi_q \Big(
\int_s^T a(\tau) f(\tau,u(\tau ) ,u^{\Delta}(\tau ) ) \nabla \tau \Big)\Delta s  \\
& \geq   \int_0^{\eta} \phi_q \Big(\int_ {\eta}^T a(\tau)
f(\tau,u(\tau ),u^{\Delta}(\tau ) ) \nabla \tau \Big)\Delta s\\
& >   \int_0^{\eta} \phi_q \Big(\int_ {\eta}^T a(\tau) \phi_p(b/M)
\nabla \tau \Big)\Delta s\\
& = \frac{b}{M}  \int_0^{\eta} \phi_q \Big(\int_ {\eta}^T a(\tau)
 \nabla \tau \Big)\Delta s=b.
\end{align*}
So, we obtain $\psi(Fu)>b$ for $u\in \bar{P}(\alpha,2b;\beta,l_2;\psi,b)$,
and condition (B1) of Lemma \ref{lem2.1} holds.

Finally, we prove that condition (B3) of Lemma \ref{lem2.1} holds.
If  $u\in \bar{P}(\alpha,r_2;\beta,l_2;\psi,b)$ and $\alpha(Fu)> 2b$, we have
$$
\psi(Fu)=\min_{t\in[\eta,T]_{\mathbb{T}}} Fu(t)
=Fu(\eta)\geq \frac{\eta}{T}\max_{t\in [0,T]_{\mathbb{T}}}
 Fu(t)\geq \frac{1}{2}\alpha(Fu)>b.
$$
Hence, condition (B3) of Lemma \ref{lem2.1} is satisfied.
Then using Lemma \ref{lem2.1} and the  assumption that
$f(t,0,0)\not\equiv 0$   on $[0,T]_{\mathbb{T}}$, we find that there
exist at least three non-negative  solutions
of \eqref{e1.1} and \eqref{e1.2} such that
\begin{gather*}
u_1\in P (\alpha,r_1;\beta,l_1), \quad
u_2\in \{ P(\alpha,r_2;\beta,l_2;\psi,b) | \psi(u) >b \}, \\
u_3\in \bar{P} (\alpha,r_2;\beta,l_2)\setminus
\Big( \bar{P}(\alpha,r_2;\beta,l_2;\psi,b)
\cup\bar{P}(\alpha,r_1;\beta,l_1)\Big ).
\end{gather*}
Otherwise, as $u_3$ satisfies $\alpha(u_3) \leq 2\psi(u_3)$, we have
$\max_{ t \in[0,T]_{\mathbb{T}}} u_3(t) <2b$.
\end{proof}

In  the following section, we now give  an example to illustrate our results.

\section{An example}

Let $\mathbb{T}=\{1-(\frac{1}{2})^{\mathbb{N}_0}  \}\cup[1,2]$, and let
$\mathbb{N}_0$   denote the set of  nonnegative integers.
Take $\alpha_1=1/2$,  $\alpha_2=1/6$,
     $\xi_1=1/4$,  $\xi_2=3/4$,
  $T=2$, $p=q=2$,   and   $a(t)\equiv 1$ for $t\in[0,T]_{\mathbb{T}}$.
Consider the BVP
\begin{gather}
\Big( u^{\Delta}(t) \Big )^{\nabla }+ f(t,u(t),u^{\Delta}(t))=0, \quad
 t\in[0,2]_{\mathbb{T}}, \label{e4.1}
\\
u(0)=\frac{1}{2}u\Big(\frac{1}{4} \Big)+\frac{1}{6}u \Big(\frac{3}{4}\Big),\quad
u^{\Delta}(2)=0, \quad \label{e4.2}
\end{gather}
where
\[
 f(t,w,v)=  \begin{cases}
\frac{t}{1000}+ \frac{2w^3}{3}+( \frac{v}{100})^3, &    w \leq 3,\\
\frac{t}{1000}+18+(\frac{v}{100})^3,    &  w>3.
  \end{cases}
\]
Clearly, assumptions (H1)--(H5) hold and
$f(t,0,0)\not\equiv 0$ on $[0,2]_{\mathbb{T}}$.
We choose  $r_1=1/2$,  $r_2=140$,  $b=2$, and $l_1=1/4$,  $l_2=80$.
So $0 < r_1 < b < 2b < r_2$ and $0 <l_1 < l_2$.
By doing some calculations, we obtain
\begin{gather*}
 M  =  \int_0^\eta \phi _q \Big( \int_{\eta}^T a(\tau) \nabla \tau \Big)
 \Delta s=1, \\
 L  =  \phi_q \Big( \int_0^T a(\tau) \nabla \tau \Big)=2,
\end{gather*}
and
\begin{align*}
 N & =  \int_0^T \phi_q \Big( \int_s^T a(\tau) \nabla \tau \Big) \Delta s
  +\frac { 1} {1-\sum_{i=1}^{m-2} \alpha_i}
  \sum_{i=1}^{m-2} \alpha_i
  \int_0^{\xi_i} \phi _q \Big( \int_s^T a(\tau) \nabla \tau \Big) \Delta s \\
& <  \tilde{N}\\
&=\int_0^T \phi_q \Big( \int_0^T a(\tau) \nabla \tau \Big) \Delta s
  +\frac { 1} {1-\sum_{i=1}^{m-2} \alpha_i}
  \sum_{i=1}^{m-2} \alpha_i
  \int_0^{\xi_i} \phi _q \Big( \int_0^T a(\tau) \nabla \tau \Big) \Delta s\\
&=\frac{19}{4}.
\end{align*}
As a result,  $f(t,w,v)$  satisfies
$$
f(t,w,v)\leq \min \big\{ \phi_p \Big(\frac{r_2}{\tilde{N}}\Big),\phi_p
\Big(\frac{l_2}{L} \Big )\big\}
\approx 29.4736 <  \min \big \{ \phi_p \Big(\frac{r_2}{N}\Big),
\phi_p \Big(\frac{l_2}{L} \Big )\big \},
$$
for   $0\leq t \leq 2$, $0\leq w \leq 140$, $| v| \leq 80$;
$$
f(t,w,v) > \phi_p \big (\frac{b}{M}\big )=2,
$$
for  $1\leq t \leq 2$, $2\leq w \leq 4$, $| v| \leq 80$;
$$
f(t,w,v) < \min \big \{ \phi_p \Big(\frac{r_1}{\tilde{N}}\Big),\phi_p
\Big(\frac{l_1}{L} \Big )\big \}
\approx 0.1053 <  \min \big \{ \phi_p \Big(\frac{r_1}{N}\Big),
\phi_p \Big(\frac{l_1}{L} \Big )\big \},
$$
for  $0\leq t \leq 2$, $0\leq w \leq \frac{1}{2}$, $| v| \leq 1/4$.
Hence, by Theorem \ref{thm3.1}, BVP   \eqref{e4.1} and \eqref{e4.2}
 has at least three non-negative solutions, two of them positive, 
$u_1, u_2,  u_3$ such that
\begin{gather*}
\max_{t\in [0,2]_{\mathbb{T}} } \{ u_1(t) \} < \frac{1}{2},  \quad
 \sup_{t\in [0,2]_{\mathbb{T}} } | u_1^{\Delta}(t) | < \frac{1}{4};  \\
2 <\min_{t\in [1,2]_{\mathbb{T}} } \{ u_2(t) \}
 \leq  \max_{t\in [0,2]_{\mathbb{T}} } \{ u_2(t) \} \leq 140, \quad
\sup_{t\in [0,2]_{\mathbb{T}} } | u_2^{\Delta}(t) | \leq 80;   \\
 \min_{t\in [1,2]_{\mathbb{T}} } \{ u_3(t) \} < 2, \quad
\frac{1}{2}< \max_{t\in [0,2]_{\mathbb{T}} } \{ u_3(t) \} < 4, \quad
  \frac{1}{4} < \sup_{t\in [0,2]_{\mathbb{T}} } | u_3^{\Delta}(t) | \leq 80.
\end{gather*}

\subsection*{Acknowledgments}
The author would like to thank the anonymous referees and the editor for
their helpful comments and suggestions. The project is supported by
Abdullah Gul University Foundation of Turkey (Project No. 5).

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\end{document}