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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 45, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/45\hfil Existence of positive solutions]
{Existence of positive solutions for even-order $m$-point
  boundary-value problems on time scales}

\author[\.I. Yaslan \hfil EJDE-2013/45\hfilneg]
{\.Isma\.il Yaslan}  % in alphabetical order

\address{\.Isma\.il Yaslan \newline
Pamukkale University\\
Department of Mathematics\\
20070 Denizli, Turkey}
\email{iyaslan@pau.edu.tr}

\thanks{Submitted October 15, 2012. Published February 8, 2013}
\subjclass[2000]{34B18, 34N05, 39A10}
\keywords{Boundary value problems; cone; fixed point theorems;
\hfill\break\indent positive solutions; time scales}

\begin{abstract}
 In this article, we consider a nonlinear even-order $m$-point
 bound\-ary-value problems on time scales.
 We establish the criteria for the existence of at least one, two and
 three positive solutions for higher order nonlinear $m$-point
 boundary-value problems on time scales by using the four functionals
 fixed point theorem, Avery-Henderson fixed point theorem and the
 five functionals fixed point theorem, respectively.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\allowdisplaybreaks

\section{Introduction}

Higher order multi-point boundary value problems on time scales have
attracted the attention of many researchers in recent years;
see for example  \cite{and04, and06, and08, do10, han08, hu11, hu112, sang07,
su11, su08, wang09, yas12} and the references therein.

In this article, we are concerned with the existence of single and
 multiple positive solutions to the following nonlinear higher
order $m$-point boundary value problem
(BVP) on time scales:
\begin{equation}\label{1.1}
\begin{gathered}
(-1)^{n}y^{\Delta ^{2n}}(t)= f(t,y(t)),\quad t\in [t_1,t_m]\subset \mathbb{T}
,\; n\in \mathbb{N} \\
y^{\Delta ^{2i+1} }(t_m)=0,\quad
\alpha y^{\Delta^{2i}}(t_1)-\beta y^{\Delta^{2i+1}
}(t_1)=\sum_{k=2}^{m-1}y^{\Delta^{2i+1} }(t_{k}),
\end{gathered}
\end{equation}
where $\alpha >0$ and $\beta > 0 $ are given constants,
 $t_1<t_2<\ldots<t_{m-1}<t_m$, $m \geq 3$ and $0\leq i\leq n-1$.
 We assume that $f:[t_1,t_m]\times [0,\infty)\to [0,\infty)$
is continuous.

Throughout this article we assume $\mathbb{T}$ is any time scale and
$[t_1,t_m]$ is a subset of $\mathbb{T}$ such that
$[t_1,t_m]=\{t\in \mathbb{T}: t_1\leq t\leq t_m\}$. Some basic definitions and
theorems on time scales can be found in the books
\cite{boh01,boh03}, which are excellent references for calculus of time scales.

In this article, existence results of at least one
positive solution of \eqref{1.1} are first established as a
result of the four functional fixed point theorem. Second, we apply
the Avery-Henderson
fixed-point theorem to prove the existence of at least two
positive solutions to  \eqref{1.1}. Finally, we use the five functional
fixed-point theorem to show that the existence of at least three
positive solutions to  \eqref{1.1}. The results are even new for the difference
equations and differential equations as well as for dynamic equations
on general time scales.


\section{Preliminaries}

To state the main results of this paper, we will need the following lemmas.

\begin{lemma} \label{lem2.1}
If $\alpha \neq 0$, then Green's function for the boundary value
problem
\begin{gather*}
-y^{\Delta ^{2}}(t)= 0,\quad t\in [t_1,t_m], \\
y^{\Delta}(t_m)=0,\quad \alpha y(t_1)-\beta
y^{\Delta}(t_1)=\sum_{k=2}^{m-1}y^{\Delta}(t_{k}),
\quad m \geq 3
\end{gather*}
is given by
\begin{equation}\label{2.1}
G(t,s)= \begin{cases}
H_1(t,s), &t_1\leq s\leq \sigma(s)\leq t_2, \\
H_2(t,s), &t_2\leq s\leq \sigma(s)\leq t_{3},\\
\dots \\
H_{m-2}(t,s), &t_{m-2}\leq s\leq \sigma(s)\leq t_{m-1},\\
H_{m-1}(t,s), &t_{m-1}\leq s\leq \sigma(s)\leq t_m,
\end{cases}
\end{equation}
where
\[
H_{j}(t,s)=   \begin{cases}
    t+\frac{\beta +j-1}{\alpha}-t_1,& t\leq s,\\
    s+\frac{\beta+j-1}{\alpha}-t_1,& s\leq     t,
  \end{cases}
\]
for all $j=1,2,\dots,m-1$.
\end{lemma}


\begin{proof}
A direct calculation gives that if $h\in C[t_1,t_m]$, then the
boundary-value problem
\begin{gather*}
-y^{\Delta ^{2}}(t)= h(t),\quad t\in [t_1,t_m], \\
y^{\Delta}(t_m)=0,\quad \alpha y(t_1)-\beta
y^{\Delta}(t_1)=\sum_{k=2}^{m-1}y^{\Delta}(t_{k}),
\quad m \geq 3
\end{gather*}
has the unique solution
\begin{align*}
y(t)&= \int_{t_1}^{t_m}(\frac{\beta}{\alpha}+s-t_1)h(s)\Delta
s^+\frac{1}{\alpha}\sum_{k=2}^{m-1}\int_{t_{k}}^{t_m}h(s)\Delta
s + \int_{t}^{t_m}(t-s)h(s)\Delta s\\
&= \int_{t_1}^{t_m}(\frac{\beta}{\alpha}+s-t_1)h(s)\Delta
s-\sum_{j=2}^{m-1}\frac{j-1}{\alpha}\int_{t_{j}}^{t_{j+1}}h(s)\Delta
s + \int_{t}^{t_m}(t-s)h(s)\Delta s.
\end{align*}
Hence, we obtain \eqref{2.1}.
\end{proof}

\begin{lemma}\label{L2.1}
If $\alpha > 0$ and $\beta > 0$, then the Green's function
$G(t,s)$ in \eqref{2.1} satisfies the  inequality
\[
  G(t,s)\geq \frac{t-t_1}{t_m-t_1}G(t_m,s)
\]
for $(t,s)\in [t_1,t_m]\times[t_1,t_m]$.
\end{lemma}

\begin{proof}
(i) Let $ s\in [t_1,t_m]$ and $t\leq s$. Then we obtain
\[
    \frac{G(t,s)}{G(t_m,s)}=\frac{t+\frac{\beta+j-1}{\alpha}-t_1}{t_m+\frac{\beta+j-1}{\alpha}-t_1}> \frac{t-t_1}{t_m-t_1}.
\]
(ii) For $ s\in [t_1,t_m]$ and $s\leq t$, we have
\[
    \frac{G(t,s)}{G(t_m,s)} = 1\geq\frac{t-t_1}{t_m-t_1}.
\]
\end{proof}

\begin{lemma}\label{L2.2}
If $\alpha >0$ and $\beta>0$, then the Green's function
$G(t,s)$ in $\eqref{2.1}$ satisfies
\[
    0<G(t,s)\leq G(s,s)
\]
for $(t,s)\in [t_1,t_m]\times[t_1,t_m]$.
\end{lemma}

\begin{proof}
Since $\alpha >0$ and $\beta>0$, $H_{j}(t,s)>0$ for all
$j=1,2,\dots,m-1$. Then we obtain $G(t,s)>0$ from $\eqref{2.1}$.

Now, we will show that $G(t,s)\leq G(s,s)$.
(i) Let $ s\in [t_1,t_m]$ and $t\leq s$. Since
$G(t,s)$ is nondecreasing in $t$, $G(t,s)\leq G(s,s)$.

(ii) For $s\in [t_1,t_m]$ and $s\leq t$, it is clear
that $G(t,s)= G(s,s)$.
\end{proof}

\begin{lemma}\label{L2.3}
If $\alpha >0$, $\beta >0$ and $s\in [t_1,t_m]$, then
the Green's function $G(t,s)$ in $\eqref{2.1}$ satisfies
\begin{align*}
    \min_{t\in [t_{m-1},t_m]}G(t,s)\geq
    K\|G(.,s)\|,
\end{align*}
where
\begin{equation}\label{2.2}
    K=\frac{\beta+\alpha(t_{m-1}-t_1)}{\beta+m-2+\alpha(t_m-t_1)}
\end{equation}
and  $\|x\|=\max_{t\in [t_1,t_m]}|x(t)|$.
\end{lemma}

\begin{proof}
Since the Green's function $G(t,s)$ in \eqref{2.1} is nondecreasing in $t$,
We have $\min_{t\in [t_{m-1},t_m]}G(t,s)=G(t_{m-1},s)$
In addition, it is obvious that
$\|G(.,s)\|=G(s,s)$ for $s\in [t_1,t_m]$ by Lemma \ref{L2.2}. Then we have
\begin{align*}
    G(t_{m-1},s)\geq K G(s,s)
\end{align*}
from the branches of the Green's function $G(t,s)$.
\end{proof}

If we let $G_1(t,s):=G(t,s)$ for $G$ as in \eqref{2.1}, then we
can recursively define
\begin{align*}
    G_{j}(t,s)=\int_{t_1}^{t_m}G_{j-1}(t,r)G(r,s)\Delta  r
\end{align*}
for $2\leq j\leq n$ and $G_{n}(t,s)$ is Green's function for the
homogeneous problem
\begin{gather*}
(-1)^{n}y^{\Delta ^{2n}}(t)= 0,\quad t\in [t_1,t_m], \\
y^{\Delta ^{2i+1} }(t_m)=0,\quad
\alpha y^{\Delta^{2i}}(t_1)-\beta y^{\Delta^{2i+1} }(t_1)
=\sum_{k=2}^{m-1}y^{\Delta^{2i+1} }(t_{k}),
\end{gather*}
where $m \geq 3$ and $0\leq i\leq n-1$.

\begin{lemma}\label{L2.4}
Let $\alpha >0$, $\beta>0$. The Green's function $G_{n}(t,s)$
satisfies the following inequalities
\begin{gather*}
    0\leq G_{n}(t,s)\leq L^{n-1} \|G(.,s)\|,\quad (t,s)\in
    [t_1,t_m]\times [t_1,t_m], \\
  G_{n}(t,s)\geq K^{n}M^{n-1}\|G(.,s)\|,\quad (t,s)\in
    [t_{m-1},t_m]\times [t_1,t_m]
\end{gather*}
where $K$ is given in \eqref{2.2}, and
\begin{gather}\label{2.3}
    L=\int_{t_1}^{t_m}\|G(.,s)\|\Delta s>0,\\
\label{2.4}
    M=\int_{t_{m-1}}^{t_m}\|G(.,s)\|\Delta s>0.
\end{gather}
\end{lemma}

The proof of the above lemma is done using induction on $n$ and
Lemma \ref{L2.3}.


Let $\mathcal{B}$ denote the Banach space $C[t_1,t_m]$ with the
norm $\|y\|=\max_{t\in [t_1,t_m]}|y(t)|$.
Define the cone $P\subset \mathcal{B}$ by
\begin{equation}\label{2.5}
    P = \{y\in \mathcal{B}: y(t)\geq 0, \min_{t\in
    [t_{m-1},t_m]} y(t) \geq \frac{K^{n} M^{n-1}}{L^{n-1}}
    \|y\|\}
\end{equation}
where $K, L, M$ are given in \eqref{2.2}, \eqref{2.3}, \eqref{2.4},
respectively.

Note that \eqref{1.1} is equivalent to the nonlinear integral equation
\begin{equation}\label{2.6}
    y(t)=\int_{t_1}^{t_m}G_{n}(t,s)f(s,y(s)) \Delta s.
\end{equation}
We can define the operator $A:P \to \mathcal{B}$ by
\begin{equation}\label{2.7}
    Ay(t)=\int_{t_1}^{t_m}G_{n}(t,s)f(s,y(s)) \Delta s,
\end{equation}
where $y\in P$. Therefore, solving \eqref{2.6} in $P$ is
equivalent to finding fixed points of the operator $A$.

It is clear that $AP\subset P$ and $A:P\to P$ is a completely continuous
operator by a standard application of the Arzela-Ascoli theorem.

Now we state the fixed point theorems which will be applied to prove main
theorems.
We are now in a position to present the four functionals fixed point theorem.
Let $\varphi$ and $\Psi$ be nonnegative continuous concave
functionals on the cone $P$, and let $\eta$ and $\theta$ be
nonnegative continuous convex functionals on the cone $P$.
Then for positive numbers $r, \tau, \mu$ and $R$, define the sets
\begin{gather*}
Q(\varphi, \eta, r, R)=\{x\in P: r\leq \varphi(x), \eta (x)\leq R\}, \\
U(\Psi, \tau )= \{x\in Q(\varphi, \eta, r, R): \tau\leq \Psi(x)\}, \\
V(\theta, \mu)= \{x\in Q(\varphi, \eta, r, R): \theta(x)\leq \mu\}.
\end{gather*}
The following theorem can be found in \cite{av08}.

\begin{theorem}[Four Functionals Fixed Point Theorem] \label{T2.2}
 Suppose $P$ is a cone in a real Banach space $E$, $\varphi$ and $\Psi$
are nonnegative continuous concave functionals on $P$, $\eta$ and
$\theta$ are nonnegative continuous convex functionals on $P$, and
there exist nonnegative positive numbers $r, \tau, \mu$ and $R$, such that
$A:Q(\varphi, \eta, r, R)\to P$ is a completely continuous operator,
 and $Q(\varphi, \eta, r, R)$ is a bounded set. If
\begin{itemize}

\item[(i)] $\{x\in U(\Psi, \tau): \eta(x)<R\}\cap\{x\in V(\theta,\mu):
r<\varphi(x)\}\neq \emptyset$

\item[(ii)] $\varphi (Ax)\geq r$, for all $x\in Q(\varphi, \eta, r, R)$,
with $\varphi(x)=r$ and $\mu< \theta(Ax)$,

\item[(iii)] $\varphi(Ax)\geq r$, for all $x\in V(\theta, \mu)$, with
$\varphi(x)=r$,

\item[(iv)] $\eta (Ax)\leq R$, for all $x\in Q(\varphi, \eta, r, R)$, with
$\eta(x)=R$ and $\Psi(Ax)<\tau$,

\item[(v)] $\eta(Ax)\leq R$, for all $x\in U(\Psi, \tau)$, with $\eta(x)=R$,
\end{itemize}
then $A$ has a fixed point $x$ in $Q(\varphi, \eta, r, R)$.
\end{theorem}

\begin{theorem}[Avery-Henderson Fixed Point Theorem \cite{av01}] \label{T2.3}
Let $P$ be a cone in a real Banach space $E$. Set
\begin{align*}
P(\phi ,r)=\{u\in P:\phi (u)<r\}.
\end{align*}
Assume there exist positive numbers $r$ and $M$, nonnegative increasing
continuous functionals $\eta $, $\phi $ on $P$, and a nonnegative continuous
functional $\theta $ on $P$ with $\theta (0)=0$ such that
\[
\phi (u)\leq \theta (u)\leq \eta (u)\quad\text{and}\quad \| u\| \leq
M\phi (u)
\]
for all $u\in \overline{P(\phi ,r)}$. Suppose that there exist
positive numbers $p<q<r$ such that
\[
\theta (\lambda u)\leq \lambda \theta (u),\quad \textmd{for
$0\leq \lambda \leq 1$ and } u\in \partial P(\theta ,q).
\]
If $A:\overline{P(\phi ,r)}\to P$ is a completely
continuous operator satisfying
\begin{itemize}

\item[(i)] $\phi (Au)>r$ for all $u\in \partial P(\phi ,r)$,

\item[(ii)] $\theta (Au)<q$ for all $u\in \partial P(\theta ,q)$,

\item[(iii)] $P(\eta ,p)\neq \emptyset $ and $\eta (Au)>p$ for all
$u\in \partial P(\eta ,p)$,

\end{itemize}
then $A$ has at least two fixed points $u_1$ and $u_2$ such that
\[
p<\eta (u_1) \textmd{ with }\theta (u_1)<q \quad
\textmd{and}\quad  q<\theta (u_2) \textmd{ with } \phi (u_2)<r.
\]
\end{theorem}

Now, we will present the five functionals fixed point theorem.
 Let $\varphi, \eta, \theta$ be nonnegative continuous convex functionals
on the cone $P$, and $\gamma, \Psi$ nonnegative continuous concave
functionals on the cone P. For nonnegative numbers $h, a, b, d$ and $c$,
define the following convex sets:
\begin{equation}\label{2.9}
\begin{gathered}
P(\varphi, c)=\{x\in P: \varphi(x)<c\}, \\
P(\varphi, \gamma, a, c)=\{x\in P: a\leq \gamma (x), \varphi(x)\leq c\}, \\
Q(\varphi, \eta, d, c)=\{x\in P: \eta(x)\leq d, \varphi(x)\leq c\},\\
P(\varphi, \theta, \gamma, a, b, c)=\{x\in P: a\leq \gamma (x), \theta(x)\leq b, \varphi(x)\leq c\}, \\
Q(\varphi, \eta, \Psi, h, d, c)=\{x\in P: h\leq \Psi(x), \eta(x)\leq d, \varphi(x)\leq c\}.
\end{gathered}
\end{equation}

The following theorem can be found in \cite{av99}.

\begin{theorem}[Five Functionals Fixed Point Theorem] \label{T2.4}
Let $P$ be a cone in a real Banach space $E$. Suppose that there exist
nonnegative numbers $c$ and $M$, nonnegative continuous concave functionals
 $\gamma$ and $\Psi$ on $P$, and nonnegative continuous convex functionals
$\varphi, \eta$ and $\theta$ on P, with
$$
\gamma(x)\leq \eta(x), \|x\|\leq M \varphi(x), \forall
x\in \overline{P(\varphi, c)}.
$$
Suppose that $A:\overline{P(\varphi, c)}\to \overline{P(\varphi, c)}$
is a completely continuous and there exist nonnegative numbers
$h, a, k, b$, with $0<a<b$ such that
\begin{itemize}

\item[(i)] $ \{x\in P(\varphi, \theta, \gamma, b, k, c):
\gamma(x)>b\}\neq \emptyset$ and $\gamma (Ax)>b$ \\
for $x\in P(\varphi, \theta, \gamma, b, k, c)$,

\item[(ii)] $ \{x\in Q(\varphi, \eta, \Psi, h, a, c): \eta(x)<a\}\neq \emptyset$
 and $\eta (Ax)<a$ \\
for  $x\in Q(\varphi, \eta, \Psi, h, a, c)$,

\item[(iii)] $\gamma(Ax)>b$, for $x\in P(\varphi, \gamma, b, c)$,
 with $\theta(Ax)>k$,

\item[(iv)] $\eta (Ax)<a$, for $x\in Q(\varphi, \eta, a, c)$,
with $\Psi(Ax)<h$,

\end{itemize}
then $A$ has at least three fixed points
$x_1, x_2, x_{3}\in\overline{P(\varphi, c)}$ such that
$$
\eta(x_1)<a, \quad \gamma (x_2)>b,\quad \eta(x_{3})>a \quad\textmd{with }
\gamma(x_{3})<b.
$$
\end{theorem}

\section{Main Results}

Now, we will give the sufficient conditions to have at least one positive
solution for  \eqref{1.1}. The four functionals fixed point theorem will
be used to prove the next theorem.

\begin{theorem}\label{T3.2}
Let $\alpha >0$ and $\beta > 0$. Suppose that there exist
constants $r, R, \mu, \tau $ with
$0<r<\tau\leq\mu<R, r=\frac{K^{n}M^{n-1}}{L^{n-1}}\mu$ and
$R=\frac{\tau L^{n-1}}{K^{n}M^{n-1}}$.
If the function $f$ satisfies the following conditions:
\begin{itemize}

\item[(i)] $f(t,y)\geq \frac{r}{K^{n}M^{n}}$ for all
$(t,y)\in [t_{m-1},t_m]\times[r,\mu]$,

\item[(ii)] $f(t,y)\leq \frac{R}{L^{n}}$ for all
$(t,y)\in [t_1,t_m]\times[0,R]$,

\end{itemize}
then  \eqref{1.1} has at least one positive solution $y$ such that
$r\leq y(t)\leq R$ for $t\in [t_1,t_m]$.
\end{theorem}

\begin{proof}
Define the maps
\begin{gather*}
\varphi(y)=\Psi(y)=\min_{t\in [t_{m-1},t_m]}y(t),\\
\theta(y)=\max_{t\in [t_{m-1},t_m]}y(t),\\
\eta(y)=\max_{t\in [t_1,t_m]}y(t).
\end{gather*}
Then $\varphi$ and $\Psi$ are nonnegative continuous concave functionals
on $P$, and $\eta$ and $\theta$ are nonnegative continuous convex
functionals on $P$. Since
$$
\|y\|=\max_{t\in [t_1,t_m]}|y(t)|=\eta(y)\leq R
$$
for all $y\in Q(\varphi, \eta, r, R)$, $Q(\varphi, \eta, r, R)$
 is a bounded set. Note that the operator $A:Q(\varphi, \eta, r, R)\to P$
is completely continuous by a standard application of the Arzela-Ascoli theorem.

Now, we verify that the remaining conditions of Theorem \ref{T2.2}.
We obtain
\begin{gather*}
\Psi(\frac{\mu}{2})=\mu\geq\tau , \quad
\eta(\frac{\mu}{2})=\mu<R,\\
\theta(\frac{\mu}{2})=\mu, \quad
\varphi(\frac{\mu}{2})=\mu>r.
\end{gather*}
Then, we have
$\frac{\mu}{2}\in\{y\in U(\Psi,\tau):\eta(y)<R\}\cap\{y\in V(\theta,\mu)
:\varphi(y)>r\}$, which means that $(i)$ in Theorem \ref{T2.2} is fulfilled.

Now, we shall verify that  condition (ii) of Theorem \ref{T2.2} is satisfied.
By Lemma \ref{L2.4}, we obtain
\begin{align*}
\theta (Ay) &=\int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s  \\
&\leq L^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s.
\end{align*}
Since $\theta(Ay)>\mu$, we find that
\begin{equation}\label{3.2}
    \int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s > \frac{\mu}{L^{n-1}}.
\end{equation}
Then, we obtain
\begin{align*}
\varphi (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\
&\geq K^{n}M^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s
>r,
\end{align*}
using Lemma \ref{L2.4} and \eqref{3.2}.

Now, we shall show that  condition (iii) of Theorem \ref{T2.2} holds.
Since $\varphi(y)=r$ and $y\in V(\theta,\mu)$, we find that
$r\leq y(t)\leq \mu$ for $t\in [t_{m-1},t_m]$. By Lemma \ref{L2.4}
and the hypothesis $(i)$, we have
\begin{align*}
\varphi (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\
&\geq K^{n}M^{n-1}\int_{t_{m-1}}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s
\geq r.
\end{align*}

Now, we shall verify that condition (iv) of Theorem \ref{T2.2} is fulfilled.
We get
\begin{align*}
\Psi (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\
&\geq K^{n}M^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s
\end{align*}
using Lemma \ref{L2.4}. Since $\Psi(Ay)<\tau$,
\begin{equation}\label{3.3}
\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s <\frac{\tau}{K^{n}M^{n-1}}.
\end{equation}
Then, by Lemma \ref{L2.4} and \eqref{3.3} we obtain
\begin{align*}
\eta (Ay) &=\int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\
&\leq L^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s  < R.
\end{align*}

Finally, we shall show that  condition $(v)$ of Theorem \ref{T2.2}
is satisfied. Since $\eta(y)=R$, we find $0\leq y(t)\leq R$ for
$t\in [t_1,t_m]$. Using Lemma \ref{L2.4} and the hypothesis $(ii)$, we have
\begin{align*}
\eta (Ay) &=\int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\
&\leq L^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s \leq R.
\end{align*}
Hence, by Theorem \ref{T2.2}, the  \eqref{1.1} has at least one positive
solution $y$ such that
$r\leq y(t)\leq R$ for $t\in [t_1,t_m]$. This completes the proof.
\end{proof}

Now we will use the Avery-Henderson fixed point theorem to prove the next theorem.

\begin{theorem}\label{T4.1}
Assume $\alpha >0$, $\beta>0$. Suppose there exist numbers
$0<p<q<r$ such that the function $f$ satisfies the following
conditions:
\begin{itemize}

\item[(i)] $f(t,y)>\frac{r}{K^{n} M^{n}}$ for $(t,y)\in
[t_{m-1},t_m]\times[r,\frac{rL^{n-1}}{K^{n}M^{n-1}}]$;

\item[(ii)] $f(t,y)<\frac{q}{L^{n}}$ for $(t,y)\in
[t_1,t_m]\times[0,\frac{qL^{n-1}}{K^{n}M^{n-1}}]$;

\item[(iii)] $f(t,y)>\frac{p}{K^{n} M^{n}}$ for $t\in
[t_{m-1},t_m]\times[\frac{K^{n}M^{n-1}}{L^{n-1}}p,p]$,

\end{itemize}
where $K, L, M$, are defined in
\eqref{2.2}, \eqref{2.3}, \eqref{2.4},
respectively. Then  \eqref{1.1} has at least two positive
solutions $y_1$ and $y_2$ such that
\begin{gather*}
p < \max_{t\in [t_1,t_m]}y_1(t)\quad \text{with }\max_{t\in [t_{m-1},t_m]}y_1(t)
  <q \\
q<\max_{t\in [t_{m-1},t_m]}y_2(t) \quad \textmd{with }
\min_{t\in [t_{m-1},t_m]}y_2(t)<r.
\end{gather*}
\end{theorem}

\begin{proof}
Define the cone $P$ as in \eqref{2.5}. From Lemma \ref{L2.4},
$AP\subset P$ and $A$ is completely continuous. Let the
nonnegative increasing continuous functionals $\phi ,\theta $ and
$\eta $ be defined on the cone $P$ by
\begin{gather*}
\phi (y):=\min_{t\in [t_{m-1},t_m]}y(t),\quad
\theta (y):=\max_{t\in [t_{m-1},t_m]}y(t),\quad
\eta (y):=\max_{t\in [t_1,t_m]}y(t).
\end{gather*}
For each $y\in P$, we have
$\phi (y)\leq \theta (y)\leq \eta (y)$,
and from \eqref{2.5}
\[
\|y\| \leq \frac{L^{n-1}}{K^{n}M^{n-1}}\phi (y).
\]
Moreover, $\theta (0)=0$ and for all $y\in P$, $\lambda \in [0,1]$
we obtain $\theta (\lambda y)=\lambda \theta (y)$.

We now verify that the remaining conditions of Theorem \ref{T2.3} hold.

\noindent \textbf{Claim 1:}
If $y\in \partial P(\phi ,r)$, then
$\phi(Ay)>r:$ Since $y\in \partial P(\phi ,r)$ and
$\|y\| \leq \frac{L^{n-1}}{K^{n}M^{n-1}}\phi (y)$, we have
$r\leq y(t)\leq \frac{rL^{n-1}}{K^{n}M^{n-1}}$ for
 $t\in [t_{m-1},t_m]$. Then, by
hypothesis (i) and Lemma \ref{L2.4} we find that
\begin{align*}
\phi(Ay)&= \int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\
&\geq  K^{n} M^{n-1}\int_{t_{m-1}}^{t_m}
\|G(.,s)\|f(s,y(s)) \Delta s
>r.
\end{align*}

\noindent \textbf{Claim 2:}
If $y\in \partial P(\theta ,q)$, then
$\theta(Ay)<q $ : Since $y\in \partial P(\theta ,q)$ and
$\|y\| \leq \frac{L^{n-1}}{K^{n}M^{n-1}}\phi (y)$,
$0\leq y(t)\leq \frac{qL^{n-1}}{K^{n}M^{n-1}}$ for
 $t\in [t_1,t_m]$. Thus, using hypothesis (ii) and Lemma \ref{L2.4},
 we obtain
\begin{align*}
\theta(Ay)&=\int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\
&\leq L^{n-1}\int_{t_1}^{t_m}
\|G(.,s)\|f(s,y(s)) \Delta s < q.
\end{align*}

\noindent \textbf{Claim 3:}
$P(\eta ,p)\neq \emptyset $ and
$\eta(Ay)>p $ for all $y\in \partial P(\eta ,p)$: Since
$\frac{p}{2}\in P$ and $p>0$, $\frac{p}{2}\in P(\eta,p)$.
If $y\in \partial P(\eta ,p)$ and $\eta(y)\geq \frac{K^{n}M^{n-1}}{L^{n-1}}\|y\|$,
 we obtain $\frac{K^{n}M^{n-1}}{L^{n-1}}p\leq y(t)\leq \|y\|=p$ for
$t\in [t_{m-1},t_m]$. Hence, by hypothesis (iii) and
Lemma \ref{L2.4} we have
\begin{align*}
\eta(Ay)&= \int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\
&\geq  K^{n} M^{n-1}\int_{t_{m-1}}^{t_m}
\|G(.,s)\|f(s,y(s)) \Delta s
>p.
\end{align*}
Since the conditions of Theorem \ref{T2.3} are satisfied,  BVP
\eqref{1.1} has at least two positive solutions $y_1$ and
$y_2$ such that
\begin{gather*}
p < \max_{t\in [t_1,t_m]}y_1(t)\quad \textmd{with }
\max_{t\in [t_{m-1},t_m]}y_1(t)<q \\
 q<\max_{t\in [t_{m-1},t_m]}y_2(t) \quad \textmd{with }
\min_{t\in [t_{m-1},t_m]}y_2(t)<r. \qedhere
\end{gather*}
\end{proof}

Now, we will apply the five functionals fixed point theorem to
investigate the existence of at least three positive
solutions for \eqref{1.1}.

\begin{theorem}\label{T4.2}
Let $\alpha >0$ and $\beta >0$. Suppose that there exist constants
$a, b, c$ with $0<a<b<\frac{bL^{n-1}}{K^{n}M^{n-1}}<c$ such that the
function $f$ satisfies the following conditions:
\begin{itemize}

\item[(i)] $f(t,y)\leq\frac{c}{L^{n}}$ for $(t,y)\in[t_1,t_m]\times[0,c]$,

\item[(ii)] $f(t,y)>\frac{b}{K^{n}M^{n}}$ for
 $(t,y)\in[t_{m-1},t_m]\times [b,\frac{bL^{n-1}}{K^{n}M^{n-1}}]$,

\item[(iii)] $f(t,y)<\frac{a}{L^{n}}$ for $(t,y)\in[t_1,t_m]\times[0,a]$,

\end{itemize}
where $K, L, M$ are as defined in \eqref{2.2}, \eqref{2.3}, \eqref{2.4},
respectively. Then  \eqref{1.1} has at least three positive solutions
 $y_1, y_2$ and $y_{3}$ such that
\begin{gather*}
\max_{t\in [t_1,t_m]}y_1(t)<a< \max_{t\in[t_1,t_m]}y_{3}(t),\\
\min_{t\in[t_{m-1},t_m]}y_{3}(t)<b< \min_{t\in[t_{m-1},t_m]}y_2(t).
\end{gather*}
\end{theorem}

\begin{proof}
Define the cone $P$ as in \eqref{2.5} and define these maps
\begin{gather*}
\gamma(y)=\Psi(y)=\min_{t\in [t_{m-1},t_m]}y(t),\\
\theta(y)=\max_{t\in [t_{m-1},t_m]}y(t),\\
\varphi(y)=\eta(y)=\max_{t\in [t_1,t_m]}y(t).
\end{gather*}
Then $\gamma$ and $\Psi$ are nonnegative continuous concave functionals on
$P$, and $\varphi, \eta$ and $\theta$ are nonnegative continuous convex
functionals on $P$. Let $P(\varphi,c)$, $P(\varphi,\gamma,a,c)$,
$Q(\varphi, \eta, d,c)$, $P(\varphi, \theta, \gamma, a,b,c)$ and
$Q(\varphi, \eta, \Psi, h,d,c)$ be defined by \eqref{2.9}. It is clear that
$$
\gamma(y)\leq\eta(y),\quad
\|y\|=\varphi(y),\quad  \forall y\in \overline{P(\varphi,c)}.
$$

If $y\in \overline{P(\varphi,c)}$, then we have $y(t)\in [0,c]$
for all $t\in [t_1,t_m]$. By Lemma \ref{L2.4} and the hypothesis (i),
we obtain
\begin{align*}
\varphi (Ay)
&=\int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s  \\
&\leq L^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s \leq c.
\end{align*}
This proves that $A:\overline{P(\varphi,c)}\to \overline{P(\varphi,c)}$.

Now we verify that the remaining conditions of Theorem \ref{T2.4}.
Let $y_1=\frac{b+\epsilon_1}{2}$ such that
$0<\epsilon_1<(\frac{L^{n-1}}{K^{n}M^{n-1}}-1)b$. Since
\begin{gather*}
\gamma(y_1)=b+\epsilon_1>b,\\
\theta(y_1)=b+\epsilon_1<\frac{bL^{n-1}}{K^{n}M^{n-1}},\\
\varphi(y_1)=b+\epsilon_1<\frac{bL^{n-1}}{K^{n}M^{n-1}}<c,
\end{gather*}
we obtain
\[
 \{y\in P(\varphi, \theta, \gamma, b, \frac{bL^{n-1}}{K^{n}M^{n-1}}, c):
 \gamma(y)>b\}\neq \emptyset.
\]
If $y\in P(\varphi, \theta, \gamma, b, \frac{bL^{n-1}}{K^{n}M^{n-1}}, c)$,
then we have $b\leq y(t)\leq \frac{bL^{n-1}}{K^{n}M^{n-1}}$ for all
$t\in [t_{m-1},t_m]$. By using Lemma \ref{L2.4} and the hypothesis (ii),
we obtain
\begin{align*}
\gamma (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s  \\
&\geq K^{n}M^{n-1}\int_{t_{m-1}}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s > b.
\end{align*}
Thus, the condition (i) of Theorem \ref{T2.4} holds.

Let $y_2=\frac{a-\epsilon_2}{2}$ such that
$0<\epsilon_2<(1-\frac{K^{n}M^{n-1}}{L^{n-1}})a$. Since
\begin{gather*}
\eta(y_2)=a-\epsilon_2<a,\\
\Psi(y_2)=a-\epsilon_2>\frac{K^{n}M^{n-1}}{L^{n-1}}a,\\
\varphi(y_2)=a-\epsilon_2<c,
\end{gather*}
we find that
\[
 \{y\in Q(\varphi, \eta, \Psi, \frac{K^{n}M^{n-1}}{L^{n-1}}a, a, c):
\eta(y)<a\}\neq \emptyset.
\]
If $y\in Q(\varphi, \eta, \Psi, \frac{K^{n}M^{n-1}}{L^{n-1}}a, a, c)$,
then we obtain $0\leq y(t)\leq a$, for $t\in [t_1,t_m]$. Hence,
\begin{align*}
\eta (Ay) &=\int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\
&\leq L^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s <a
\end{align*}
by Lemma \ref{L2.4} and the hypothesis $(iii)$.
It follows that condition (ii) of Theorem \ref{T2.4} is
fulfilled.

Now, we shall show that the condition (iii) of Theorem \ref{T2.4} is satisfied.
We have
\begin{align*}
\theta(Ay)
&=\int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\
&\leq L^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s
\end{align*}
using Lemma \ref{L2.4}.
Since $\theta(Ay)>\frac{bL^{n-1}}{K^{n}M^{n-1}}$, we obtain
\begin{equation}\label{4.1}
\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s >\frac{b}{K^{n}M^{n-1}}.
\end{equation}
Then, by Lemma \ref{L2.4} and \eqref{4.1} we find that
\begin{align*}
\gamma (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\
&\geq  K^{n}M^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s >b.
\end{align*}

Finally, we shall verify that the condition (iv) of Theorem \ref{T2.4} holds.
 By Lemma \ref{L2.4}, we obtain
\begin{align*}
\Psi (Ay) &=\int_{t_1}^{t_m}G_{n}(t_{m-1},s)f(s,y(s)) \Delta s \\
&\geq K^{n}M^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s.
\end{align*}
Since $\Psi (Ay)<\frac{K^{n}M^{n-1}}{L^{n-1}}a$, we have
\begin{equation}\label{4.2}
\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s <\frac{a}{L^{n-1}}.
\end{equation}
Then, we find that
\begin{align*}
\eta (Ay) &= \int_{t_1}^{t_m}G_{n}(t_m,s)f(s,y(s)) \Delta s \\
&\leq L^{n-1}\int_{t_1}^{t_m}\|G(.,s)\|f(s,y(s)) \Delta s < a.
\end{align*}
using Lemma \ref{L2.4} and \eqref{4.2}.

Since the conditions of Theorem \ref{T2.4} are satisfied,  \eqref{1.1}
has at least three positive solutions
$y_1, y_2, y_{3}\in \overline{P(\varphi,c)}$ such that
\begin{gather*}
\max_{t\in [t_1,t_m]}y_1(t)<a< \max_{t\in[t_1,t_m]}y_{3}(t),\\
 \min_{t\in[t_{m-1},t_m]}y_{3}(t)<b< \min_{t\in[t_{m-1},t_m]}y_2(t).
\end{gather*}
This completes the proof.
\end{proof}

\begin{example}\rm
Let $ \mathbb{T}=\{(1/5)^{n}:n\in \mathbb{N} \}\cup \{0\}\cup[3,5]$.
We consider the  boundary value problem
\begin{equation}\label{4.3}
\begin{gathered}
-y^{\Delta ^{4}}(t)= \frac{2013y^{2}}{y^{2}+2013},\quad
 t\in [\frac{1}{5},5]\subset \mathbb{T} \\
y^{\Delta }(5)=0,\quad
\frac{1}{2}y(\frac{1}{5})-2y^{\Delta }(\frac{1}{5})=y^{\Delta}(3)+y^{\Delta}(4), \\
y^{\Delta ^{3} }(5)=0,\quad
\frac{1}{2}y^{\Delta^{2}}(\frac{1}{5})-2 y^{\Delta^{3}
}(\frac{1}{5})=y^{\Delta^{3}}(3)+y^{\Delta^{3}}(4).
\end{gathered}
\end{equation}

If we take $r=84500$, $\tau=187083$, $\mu=633177.9584$ and $R=1401856$, then
all the conditions in Theorem \ref{T3.2} are satisfied.
Thus,  BVP \eqref{4.3} has at least one positive solution $y$ such that
$84500\leq y(t)\leq 1401856$ for $t\in [\frac{1}{5},5]$.


If we take $p = 2\times 10^{-5}$, $q = 26\times10^{-6}$ and
 $r = 24\times10^{-3}$, then
all the conditions in Theorem \ref{T4.1} are fulfilled. Hence,
BVP \eqref{4.3} has at least two positive solutions
$y_1$ and $y_2$ satisfying
\begin{align*}
2\times 10^{-5} < \max_{t\in [\frac{1}{5},5]}y_1(t)\quad \text{with }
 \max_{t\in [4,5]}y_1(t)<26\times 10^{-6}\\
26\times 10^{-6}<\max_{t\in [4,5]}y_2(t) \quad \textmd{with }
 \min_{t\in [4,5]}y_2(t)<24\times 10^{-3}.
\end{align*}

If we take $a=15.10^{-4}$, $b=1$ and $c = 1320000 $, then
all the conditions in Theorem \ref{T4.2} hold. Therefore,
BVP \eqref{4.3} has at least three positive solutions
$y_1, y_2$ and $y_{3}$ such that
\begin{gather*}
\max_{t\in [\frac{1}{5},5]}y_1(t)<15\times 10^{-4}< \max_{t\in[\frac{1}{5},5]}y_{3}(t),
\\
\min_{t\in[4,5]}y_{3}(t)<1< \min_{t\in[4,5]}y_2(t).
\end{gather*}
\end{example}

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\end{document}