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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 176, 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 2010 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2010/176\hfil Multiple positive solutions]
{Multiple positive solutions for a nonlinear 3n-th order
three-point boundary-value problem}

\author[K. L. S. Devi, K. R. Prasad, \hfil EJDE-2010/176\hfilneg]
{K. L. Saraswathi Devi, Kapula R. Prasad}  % in alphabetical order

\address{K. L. Saraswathi Devi \newline
Department of Applied Mathematics, Andhra University,
Visakhapatnam, 530 003, India \newline
Department of Mathematics,
Ch. S. D. St. Theresa's Degree College for Women, 
Eluru, 534 003, India}
\email{saraswathikatneni@gmail.com}

\address{Kapula Rajendra Prasad \newline
Department of Applied Mathematics, Andhra University,
Visakhapatnam, 530 003, India}
 \email{rajendra92@rediffmail.com}

\thanks{Submitted June 23, 2010. Published December 17, 2010.}
\subjclass[2000]{34B18, 34A10}
\keywords{Boundary value problem; multiple positive solutions;
 fixed point; cone}

\begin{abstract}
 In this article we establish the existence
 of at least three positive solutions for 3n-th order
 three-point boundary value problem by using five functional
 fixed point theorem.
 We also establish the existence of at least $2m-1$ positive
 solutions of the problem for arbitrary positive integer $m$.
\end{abstract}

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

\section{Introduction}

The general theory of differential
equations has emerged as an important area of investigation due to
its powerful and versatile applications to almost all areas of
science, engineering and technology. Much interest has been
developed since last decade  regarding the  study of existence of
positive solutions to the boundary value problems as they are
arising in the branches of applied mathematics, physics and
technological problems.

 In this article, we prove the existence of multiple
positive solutions of  $3n^{th}$  order  ordinary differential
equation
\begin{equation}\label{e11}
{(-1)}^ny^{(3n)}= f(y(t), y^{(3)}(t),
y^{(6)}(t),\dots,y^{(3(n-1))}(t)),\quad t\in[t_1,t_3],
\end{equation}
  satisfying the general three point boundary conditions
\begin{equation}\label{e12}
\begin{gathered}
\alpha_{3i-2,1}y^{(3i-3)}(t_1)+\alpha_{3i-2,2}y^{(3i-2)}(t_1)
+\alpha_{3i-2,3}y^{(3i-1)}(t_1)=0,\\
\alpha_{3i-1,1}y^{(3i-3)}(t_2)+
\alpha_{3i-1,2}y^{(3i-2)}(t_2)+\alpha_{3i-1,3}y^{(3i-1)}(t_2)=0,\\
\alpha_{3i,1}y^{(3i-3)}(t_3)+\alpha_{3i,2}y^{(3i-2)}(t_3)
+\alpha_{3i,3}y^{(3i-1)}(t_3)=0,
\end{gathered}
\end{equation}
where the  coefficients
$\alpha_{3i-j,1},\alpha_{3i-j,2},\dots,\alpha_{3i-j,3}$  for
$j=0,1,2$ and $i=1,\dots, n-1$, are  real constants.

Boundary-value problems of the form \eqref{e11}-\eqref{e12}
constitute a natural extension
of third order  three-point boundary-value problems studied in many
papers with simple boundary conditions. Here we refer to
Graef, Yang \cite{gra1},  Eloe and Henderson \cite{pweaj}, Yang
\cite{by},   Anderson\cite{dra}, Anderson and Davis \cite{dra1},
Guo, Sun and Zhao \cite{li} and references there in.
 Recently  Prasad and Murali \cite{prasad}  studied  the multiple
positive solutions for nonlinear third order general three-point
boundary-value problem.

For convenience we adopt the notation Let
\begin{gather*}
\beta_{ij}=\alpha_{3i-3+j,1}t_j+\alpha_{3i-3+j,2}, \quad
\gamma_{ij}=\alpha_{3i-3+j,1}t_j^2+2\alpha_{3i-3+j,2}t_j+2\alpha_{3i-3+j,3},\\
l_{ij}=\alpha_{3i-3+j,1}s^2-2\beta_{ij}s+\gamma_{ij},
\end{gather*}
and define
\[
m_{i_{kj}}=\frac{\alpha_{3i-2+k,1}\gamma_{ij}-\alpha_{3i-2+j,1}
\gamma_{ik}}{2(\alpha_{3i-2+k,1}\beta_{ij}-\alpha_{3i-2+j,1}\beta_{ik})},\quad
M_{i_{kj}}=\frac{\beta_{3i-2+k,1}
\gamma_{ij}-\beta_{ij}\gamma_{ik}}{(\alpha_{3i-2+k,1}\beta_{ij}
-\alpha_{3i-2+j,1}\beta_{ik})}
\]
for $k=1,2,3$, $j=1,2,3$. Also let $m=\max{
\{m_{i_{12}},m_{i_{13}},m_{i_{23}}}\}$,
$$
M_i=\min\Big\{m_{i_{23}}+\sqrt{m_{i_{23}}^2-M_{i_{23}}},\, m_{i_{13}}
+\sqrt{m_{i_{13}}^2-M_{i_{13}}}\Big\}
$$
and
$$
d_i=[\alpha_{3i-2,1}(\beta _{i2}\gamma_{i3}-\beta
_{i3}\gamma_{i2)}-\beta_{i1}(\alpha_{3i-1,1}\gamma_{i3}-\alpha_{3i,1}\gamma_{i2})
+\gamma_{i1}(\alpha_{3i-1,1}\beta_{i3}-\alpha_{3i,1}\beta_{i2})].
$$
We assume the
following conditions throughout this paper:
\begin{itemize}
\item[(A1)]
$f:\mathbb{R}^n\to \mathbb{R}^+$ is continuous;

\item[(A2)]
$\alpha_{3i-2,1}>0$, $\alpha_{3i-1,1}>0$ and 
$\alpha_{3i,1} >0$ for $1\leq i\leq n$ are real constants,
$\frac{\alpha_{3i-2,2}}{\alpha_{3i-2,1}}
<\frac{\alpha_{3i-1,2}}{\alpha_{3i-1,1}}<\frac{\alpha_{3i,2}}{\alpha_{3i,1}}$.
\item[(A3)]
$m_i\leq t_1\leq t_2\leq t_3\leq M_i$,
$2\alpha_{3i-1,3}\alpha_{3i-1,1}>
 \alpha_{3i-1,2}^2$,\\
  $2\alpha_{3i-2,3}\alpha_{3i-2,1} < \alpha_{3i-2,2}^2$, 
 $2\alpha_{3i,3} \alpha_{3i,3} >
  \alpha_{3i,2}^2$.

\item[(A4)] $ m_{i_{23}}^2  >M_{i_{23}}$, $m_{i_{12}}^2  < M_{i_{12}}$, 
$  m_{i_{13}}^2  >  M_{i_{13}} $ and $ d_i>0$.
\end{itemize}

The rest of the paper is
organized as follows. In Section 2, we construct the Green's
function for the homogeneous boundary value problem corresponding to
\eqref{e11}-\eqref{e12} and estimate the bounds for the Green's
function. In Section 3, we  establish the existence of at least
three positive solutions for
\eqref{e11}-\eqref{e12}, using five functional fixed point theorem.
We also establish  the existence of at least $2m-1$ positive
solutions of  \eqref{e11}-\eqref{e12}, for
arbitrary positive integer $m$.

\section{The Green's Function and Bounds}

In this section, we construct the Green's function for the
homogeneous boundary value
problem corresponding to \eqref{e11}-\eqref{e12} and estimate the
bounds of the Greens function. We prove certain lemmas which are
needed to establish our main results.

  Let $G_i(t,s)$ be the Green's function for the homogeneous
problem
\begin{equation}\label{e21}
-y'''=0,\quad t\in [t_1,t_3]
\end{equation}
satisfying the general three point boundary conditions \eqref{e12}.
 First we  establish results on
the related third order homogeneous boundary-value problem
\eqref{e21} and \eqref{e12}.

\begin{lemma}\label{l21}
The homogeneous boundary-value problem \eqref{e21} and \eqref{e12}
has only the trivial solution if and only if
$d_i=[\alpha_{3i-2,1}(\beta _2\gamma_3-\beta
_3\gamma_2)-\beta_1(\alpha_{3i-1,1}\gamma_{3}-\alpha_{3i,1}\gamma_{2})
+\gamma _1(\alpha_{3i-1,1}\beta_3-\alpha_{3i,1}\beta_2)] \neq0$ for
$1\leq i\leq n$.
\end{lemma}

\begin{proof}
On application of boundary conditions  \eqref{e12} to
the general solution of \eqref{e21}, it can be established.
\end{proof}

\begin{lemma}\label{l22}
For $1 \leq i\leq n$, the Green's function  for the
homogeneous boundary value problem \eqref{e21} and \eqref{e12} is
\begin{equation}\label{e22}
G_i(t,s)= \begin{cases}
G_{i_1}(t,s), &  t_1< s<t\leq t_2<t_3\\
G_{i_2}(t,s), &  t_1\leq t<s< t_2<t_3\\
G_{i_3}(t,s), &  t_1\leq t< t_2<s<t_3\\
G_{i_4}(t,s), &  t_1<t_2< s<t\leq t_3\\
G_{i_5}(t,s), & t_1<t_2\leq t<s< t_3\\
G_{i_6}(t,s), &  t_1\leq s< t_2<t<t_3
\end{cases}
\end{equation}
where
\begin{gather*}
\begin{aligned}
 G_{i_1}(t,s)&=\frac{1}{2d_i}[-(\beta _{i2}\gamma_{i3}-\beta_{i3}\gamma_{i2})
+t(\alpha_{3i-1,1}\gamma_{i3}-\alpha_{3i,1}\gamma_{i2})\\
&\quad -t^2 (\alpha_{3i-1,1}\beta _{i3}-\alpha_{3i,1}\beta
_{i2})]\times l_{i1},
\end{aligned}\\
\begin{aligned}
G_{i_2}(t,s)&=\frac{1}{2d_i}[-(\beta
_{i1}\gamma_{i3}-\beta_{i3}\gamma_{i1})
+t(\alpha_{3i-2,1}\gamma_{i3}-\alpha_{3i,1}\gamma_{i1})\\
&\quad -t^2 (\alpha_{3i-2,1}\beta _{i3}-\alpha_{3i,1}\beta
_{i1})]\times l_{i2}\\
&\quad+\frac{1}{2d_i}[(\beta
_{i1}\gamma_{i2}-\beta_{i2}\gamma_{i1})-t(\alpha_{3i-2,1}\gamma_{i2}-\alpha_{3i-1,1}\gamma_{i1})\\
&\quad +t^2 (\alpha_{3i-2,1}\beta
_{i2}-\alpha_{3i-1,1}\beta_{i1}]\times l_{i3} ,
\end{aligned}\\
\begin{aligned}
G_{i_3}(t,s)&=\frac{1}{2d_i}[(\beta
_{i1}\gamma_{i2}-\beta_{i2}\gamma_{i1})
-t(\alpha_{3i-2,1}\gamma_{i2}-\alpha_{3i-1,1}\gamma_{i1})\\
&\quad +t^2 (\alpha_{3i-2,1}\beta
_{i2}-\alpha_{3i-1,1}\beta_{i1})]\times l_{i3},
\end{aligned}\\
\begin{aligned}
G_{i_4}(t,s)&=\frac{1}{2d_i}[-(\beta
_{i2}\gamma_{i3}-\beta_{i3}\gamma_{i2})+t(\alpha_{3i-1,2,1}\gamma_{i3}-\alpha_{3i,1}\gamma_{i2})\\
&\quad -t^2 (\alpha_{3i-1,1}\beta _{i3}-\alpha_{3i,1}\beta
_{i2})]\times l_{i1}\\
&\quad +\frac{1}{2d_i}[(\beta
_{i1}\gamma_{i3}-\beta_{i3}\gamma_{i1})-t(\alpha_{3i-2,1}\gamma_{i3}-\alpha_{3i,1}\gamma_{i1})\\
&\quad +t^2 (\alpha_{3i-2,1}\beta _{i3}-\alpha_{3i,1}\beta
_{i1}]\times l_{i2} ,
\end{aligned}\\
\begin{aligned}
G_{i_5}(t,s)&=\frac{1}{2d_i}[(\beta
_{i1}\gamma_{i2}-\beta_{i2}\gamma_{i1})
-t(\alpha_{3i-2,1}\gamma_{i2}-\alpha_{3i-1,1}\gamma_{i1})\\
&\quad +t^2 (\alpha_{3i-2,1}\beta _{i2}-\alpha_{3i-1,2,1}\beta
_{i1})]\times l_{i3} ,
\end{aligned}\\
\begin{aligned}
G_{i_6}(t,s)&=\frac{1}{2d_i}[-(\beta
_{i2}\gamma_{i3}-\beta_{i3}\gamma_{i2})+t(\alpha_{3i-1,1}\gamma_{i3}-\alpha_{3i,1}\gamma_{i2})\\
&\quad -t^2 (\alpha_{3i-1,1}\beta _{i3}-\alpha_{3i,1}\beta_
{i2})]\times l_{i1}.
\end{aligned}
\end{gather*}
\end{lemma}

\begin{proof}
$G_i(t, s)$ is constructed by using standard methods \cite{rao}.
\end{proof}

\begin{lemma}\label{l23}
Assume the conditions {\rm (A1)--(A4)} are satisfied. Then, for
$1\leq i\leq n$, the Green's function $G_{i}(t,s)$ of the
 boundary-value problem \eqref{e21} and \eqref{e12} satisfies
$G_{i}(t,s)>0$, for $(t,s)\in[t_1,t_3]\times[t_1,t_3]$.
\end{lemma}

\begin{proof} For $(t,s)\in[t_1,t_3]\times[t_1,t_3]$, $G_{i}(t,s)$
as stated in \eqref{e22}, if we consider sequentially,
from {\rm (A2)--(A4)}, we obtain
\begin{equation}\label{e23}
G_{i}(t,s)>0, \quad\text{for }(t,s)\in[t_1,t_3]\times[t_1,t_3].
\end{equation}
\end{proof}

\begin{lemma}\label{l24}
Assume the conditions {\rm (A1)--(A4)} are satisfied. Then,
for $1\leq i\leq n$, the Green's function $G_{i}(t,s)$ given
by \eqref{e22} satisfies
$$
G_{i}(t,s)\leq\max\big\{G_{i}(t_1,s),
G_{i}(s,s),G_{i}(t_3,s)\big\}.
$$
\end{lemma}

\begin{proof}
This can be proved by proceeding sequentially with the branches of
$G_{i}(t,s)$ in \eqref{e22}.

\textbf{Case 1.} For $t_{1}<s<t<t_{2}<t_{3}$.
\begin{align*}
G_{i}(t,s)=G_{i_1}(t,s)
&=\frac{1}{2d_i}[-(\beta_{i2}\gamma_{i3}-\beta_{i3}\gamma_{i2})
+t(\alpha_{3i-1,1}\gamma_{i3}-\alpha_{3i,1}\gamma_{i2})\\
&\quad -t^2 (\alpha_{3i-1,1}\beta _{i3}-\alpha_{3i,1}\beta
_{i2}]\times l_i{1}
\end{align*}
which is decreasing in $t$, by (A2)-(A4). Therefore,
 $G_{i_{1}}(t,s)\leq
G_{i_{1}}(s,s)\leq G_{i_{1}}(t_1,s)$.
 Hence $G_i(t,s)\leq G_i(t_1,s)$.

\textbf{Case 2.} For $t_{1}\leq t<t_{2}<s<t_{3}$.
\begin{align*}
G_{i}(t,s)=G_{i_{3}}(t,s) &=\frac{1}{2d_i}[(\beta
_{i1}\gamma_{i2}-\beta_{i2}\gamma_{i1})
-t(\alpha_{3i-2,1}\gamma_{i2}-\alpha_{3i-1,1}\gamma_{i1})\\
&\quad +t^2 (\alpha_{3i-2,1}\beta
_{i2}-\alpha_{3i-1,1}\beta_{i1})]\times l_3
\end{align*}
 which is increasing in $t$ from
(A2)-(A4). Therefore, $G_{i_{3}}(t,s)\leq G_{i_{3}}(s,s)\leq
G_{i_{3}}(t_3,s)$.  Hence $G_{i}(t,s)\leq G_{i}(t_3,s)$.

\textbf{Case 3.} For $t_{1}\leq t<s<t_{2}<t_{3}$.
\begin{align*}
G_{i}(t,s)=G_{i_{2}}(t,s) &=\frac{1}{2d_i}[-(\beta
_{i1}\gamma_{i3}-\beta_{i3}\gamma_{i1})
+t(\alpha_{3i-2,1}\gamma_{i3}-\alpha_{3i,1}\gamma_{i1})\\
&\quad -t^2 (\alpha_{3i-2,1}\beta _{i3}-\alpha_{3i,1}\beta
_{i1})]\times l_{i2}\\
&\quad +\frac{1}{2d_i}[(\beta
_{i1}\gamma_{i2}-\beta_{i2}\gamma_{i1})-t(\alpha_{3i-2,1}
  \gamma_{i2}-\alpha_{3i-1,1}\gamma_{i1})\\
&\quad
+t^2 (\alpha_{3i-2,1}\beta_{i2}-\alpha_{3i-1,1}\beta_{i1}]\times
l_{i3}
\end{align*}
which is increasing in $t$ by  (A2)-(A4) and  case 2. Therefore,
$G_{i_{2}}(t,s)\leq G_{i_2}(s,s)$. Hence $G_{i}(t,s)\leq
G_{i}(s,s)$.

\textbf{Case 4.} For $t_{1}<t<t_{2}<s<t<t_{3}$.
\begin{align*}
G_{i}(t,s) =G_{i_4}(t,s)
&=\frac{1}{2d_i}[-(\beta_{i2}\gamma_{i3}-\beta_{i3}\gamma_{i2})
 +t(\alpha_{3i-1,1}\gamma_{i3}-\alpha_{3i,1}\gamma_{i2})\\
&\quad -t^2 (\alpha_{3i-1,1}\beta _{i3}-\alpha_{3i,1}\beta_{i2})]\times l_{i1}\\
&\quad +\frac{1}{2d_i}[(\beta
_{i1}\gamma_{i3}-\beta_{i3}\gamma_{i1})
 -t(\alpha_{3i-2,1}\gamma_{i3}-\alpha_{3i,1}\gamma_{i1})\\
&\quad +t^2 (\alpha_{3i-2,1}\beta _{i3}-\alpha_{3i,1}\beta
_{i1})]\times l_{i2}
\end{align*}
which is decreasing in $t$ from case 1 and case 2. Therefore,
$G_{i_4}(t,s)\leq G_{i_4}(s,s)$. Hence $G_{i}(t,s)\leq G_{i}(s,s)$.

Similarly  we  can establish the inequality when the Green's
function $G_{i}(t,s) =  G_{i_5}(t,s)$  and
 $G_{i}(t,s) = G_{i_6}(t,s)$ as in  case 2  and case 1 respectively,
where $ G_{i_5}(t,s)$, $G_{i_6}(t,s)$
are given as in  \eqref{e22}. From all above cases
$$
G_{i}(t,s)\leq \max\{G_{i}(t_1,s),G_{i}(s,s),G_{i}(t_3,s)\}.
$$
\end{proof}

\begin{lemma}\label{l25}
Assume that the conditions {\rm (A1)--(A4)} hold.
For $1\leq i\leq n$,
and  fixed $s\in[t_1,t_3]$,   the Green's function $G_{i}(t,s)$ in
\eqref{e22} satisfies
$$
\min_{t\in [t_2, t_3]} G_{i}(t,s)\geq
m_i\| G_i(.,s)\|,
$$
where
$$
m_i=\min\Big\{\frac{G_{i_1}(t_3,s)}{G_{i_1}(t_2,s)},
\frac{G_{i_4}(t_3,s)}{G_{i_4}(t_2,s)},
\frac{G_{i_5}(t_2,s)}{G_{i_5}(t_3,s)}\Big\}
$$
and $ \|\cdot\|$ is defined by $\|x\|=\max\{x(t):t\in[t_1,t_3]\}$.
\end{lemma}

\begin{proof}
For $s\in[t_1,t_2]$,  $G_{i}(t,s)=G_{i_1}(t,s)$ which is
decreasing in $t$ by (A2)-(A4). Therefore,
$$
\frac{G_{i}(t,s)}{G_{i}(s,s)}=\frac{G_{i_1}(t,s)}{G_{i_1}(s,s)}
\geq\frac{G_{i_1}(t_3,s)}{G_{i_1}(t_2,s)}.
$$
 For $s\in[t_2,t_3]$ and $t_{1}<t_{2}\leq t<s<t_{3}$.
$G_{i}(t,s)=G_{i_5}(t,s)$ which is increasing in $t$ on $[t_1,t_3]$ by
(A2)-(A4). Therefore,
$$
\frac{G_{i}(t,s)}{G_{i}(s,s)}
=\frac{G_{i_5}(t,s)}{G_{i_5}(s,s)}
\leq\frac{G_{i_5}(t_2,s)}{G_{i_5}(t_3,s)}.
$$
 For $s\in[t_2,t_3]$ and $t_{1}<t_{2}<s<t<t_{3}$.
$G_{i}(t,s)=G_{i_4}(t,s)$ which is decreasing in $t$ on $[t_1,t_3]$ by
$(A2)$-$(A4)$. Therefore
$$
\frac{G_{i}(t,s)}{G_{i}(s,s)}
=\frac{G_{i_4}(t,s)}{G_{i_4}(s,s)}
\geq\frac{G_{i_4}(t,s)}{G_{i_4}(t_2,s)}
\geq\frac{G_{i_4}(t_3,s)}{G_{i_4}(t_2,s)}.
$$
Therefore, from Lemma
\ref{l24} and by all the above cases we have $$\min_{t\in [t_2,
t_3]} G_{i}(t,s)\geq m_i\| G(.,s)\|,
$$
where
$$
m_i=\min\Big\{\frac{G_{i_1}(t_3,s)}{G_{i_1}(t_2,s)},
\frac{G_{i_4}(t_3,s)}{G_{i_4}(t_2,s)},
\frac{G_{i_5}(t_2,s)}{G_{i_5}(t_3,s)}\Big\}.
$$
\end{proof}

\begin{lemma}\label{l26}
Assume the conditions {\rm (A1)-(A4)} are satisfied and $G_i(t,s)$
as in \eqref{e22}. Let us define $H_1(t,s)=G_1(t,s)$ and recursively
define
$$
H_j(t,s)=\int_{t_1}^{t_3} H_{j-1}(t,r) G_j(r,s)dr
$$
for $2\leq j\leq n$, then $H_n(t,s)$ is the Green's function for the
homogeneous problem corresponding  to \eqref{e11}-\eqref{e12}.
\end{lemma}

\begin{lemma} \label{l27}
Assume the conditions {\rm (A1)-(A4)} holds. If we
define
$$
K=\prod_{j=1}^{n-1} K_j,\quad
L=\prod_{j=1}^{n-1}m_jL_j,
$$
then the Green's function $H_n(t,s)$ in Lemma \ref{l26} satisfies
\begin{gather}\label{e24}
0\leq H_n(t,s)\leq K \| G_n(s,s)\|, \quad
(t,s)\in[t_1,t_3]\times[t_1,t_3],\\
\label{e25}
H_n(t,s)\geq m_n L
\| G_n(s,s)\|,\quad (t,s)\in[t_2,t_3]\times[t_1,t_3]
\end{gather}
where $m_n$ is given as in Lemma \ref{l25},
\begin{gather*}
K_j=\int_{t_1}^{t_3}\| G_j(s,s)\| ds>0,\quad\text{for } 1\leq j \leq n,
\\
L_j=\int_{t_2}^{t_3}\| G_j(s,s)\| ds>0, \quad\text{for } 1\leq j\leq n.
\end{gather*}
\end{lemma}

 Using Lemma \ref{l25} and  induction on $n$, we can
easily establish the proof of the above lemma.

 Let
$C=\{v\mid v:[t_1,t_3]\to \mathbb{R} \text{ is continuous function } \}$.
For each $1\leq j\leq n-1$, define the operator $T_j:C\to C$
by
$$
(T_jv)(t)=\int_{t_1}^{t_3}H_j(t,s)v(s)ds,\quad t\in [t_1,t_3].
$$
By the construction of $T_j$, and the properties of $H_j(t,s)$,
it is clear that
\begin{gather*}
(-1)^j(T_jv)^{(3j)}(t)=v(t),\quad t\in [t_1,t_3],\\
\alpha_{3i-2,1}{T_jv}^{(3i-3)}(t_1)
+\alpha_{3i-2,2}{T_jv}^{(3i-2)}(t_1)
+\alpha_{3i-2,3}{T_jv}^{(3i-1)}(t_1)=0,\\
\alpha_{3i-1,1}{T_jv}^{(3i-3)}(t_2)+
\alpha_{3i-1,2}{T_jv}^{(3i-2)}(t_2)
+\alpha_{3i-1,3}{T_jv}^{(3i-1)}(t_2)=0,\\
\alpha_{3i,1}{T_jv}^{(3i-3)}(t_3)+\alpha_{3i,2}{T_jv}^{(3i-2)}(t_3)
+\alpha_{3i,3}{T_jv}^{(3i-1)}(t_3)=0,
\end{gather*}
for $i=1,2,\dots, j-1$.  Hence, we see that
\eqref{e11}-\eqref{e12} has a solution if and only if the following
boundary-value problem has a solution
\begin{gather}\label{e26}
v^{(3)}(t)+f(T_{n-1}v(t),T_{n-2}v(t),\dots,T_1v(t),v(t))=0,
\quad t\in[t_1,t_3], \\
\label{e27}
\begin{gathered}
\alpha_{3i-2,1}{v}^{(3i-3)}(t_1)
+\alpha_{3i-2,2}{v}^{(3i-2)}(t_1)
+\alpha_{3i-2,3}{v}^{(3i-1)}(t_1)=0,\\
\alpha_{3i-1,1}{v}^{(3i-3)}(t_2)+
\alpha_{3i-1,2}{v}^{(3i-2)}(t_2)+\alpha_{3i-1,3}{v}^{(3i-1)}(t_2)=0,\\
\alpha_{3i,1}{v}^{(3i-3)}(t_3)+\alpha_{3i,2}{v}^{(3i-2)}(t_3)
+\alpha_{3i,3}{v}^{(3i-1)}(t_3)=0.
\end{gathered}
\end{gather}
for $i=1,2,\dots, j-1$. Indeed, if $y$ is a solution of
 \eqref{e11}-\eqref{e12}, then $v(t)=y^{3 (n-1)}(t)$ is
a solution of  \eqref{e26}-\eqref{e27}.
Conversely, if $v$ is a solution of
\eqref{e26}-\eqref{e27}, then $y(t)=T_{n-1}v(t)$ is a solution of
 \eqref{e11}-\eqref{e12}.
 In fact, $y(t)$ is represented as
$$
y(t)=\int_{t_1}^{t_3}H_n(t,s)v(s)ds,
$$
where
$$
v(s)=\int_{t_1}^{t_3}G_1(s,\tau)f(T_{n-1}v(\tau),T_{n-2}v(\tau),
\dots,T_1v(\tau),v(\tau))d \tau.
$$
is a solution of  \eqref{e11}-\eqref{e12}.

\section{Existence of multiple positive solutions}

In this section, we establish the existence of multiple positive
solutions for  \eqref{e11}-\eqref{e12},
by using five functional fixed point theorem  which is Avery
generalization of the Leggett-Williams fixed point theorem. And
then, we establish $2m-1$ positive solutions for an arbitrary
positive integer $m$.

 Let $B$ be a real Banach space with cone $P$. A map
$\alpha:P\to [0,\infty)$ is said to be a nonnegative
continuous concave functional on $P$ if $\alpha$ is continuous and
$$
\alpha(\lambda x+(1-\lambda )y)\geq \lambda \alpha(x)
+(1-\lambda )\alpha(y),
$$
for all $x, y\in P$ and $\lambda \in [0,1]$. Similarly, we say that
a map $\beta:P\to [0, \infty)$ is said to be a nonnegative
continuous convex functional on $P$ if $\beta$ is continuous and
$$
\beta(\lambda x+(1-\lambda )y)\leq \lambda \beta(x)
+(1-\lambda )\beta(y),
$$
for all $x,y\in P$ and $\lambda \in [0,1]$. Let
$\gamma, \beta, \theta$ be nonnegative continuous convex functional
on $P$ and $\alpha, \psi$ be nonnegative continuous concave
functionals on $P$, then for nonnegative numbers $h', a', b', d'$
and $c'$, we define the following convex sets
\begin{gather*}
P(\gamma,c')=\{y\in P|\gamma(y)<c'\},\\
P(\gamma,\alpha,a',c')=\{y\in P|a'\leq\alpha(y), \gamma(y)\leq c'\},\\
Q(\gamma,\beta,d',c')=\{y\in P|\beta(y)\leq d', \gamma(y)\leq c'\},\\
P(\gamma,\theta,\alpha,a',b',c')=\{y\in P|a'\leq\alpha(y),
\theta(y)\leq b', \gamma(y)\leq c'\},\\
Q(\gamma,\beta,\psi,h',d',c')=\{y\in P|h'\leq \psi(y), \beta(y)\leq
d', \gamma(y)\leq c'\}.
\end{gather*}
In obtaining multiple positive
solutions of   \eqref{e11}-\eqref{e12},
the following so called Five Functionals Fixed Point Theorem will be
fundamental.

\begin{theorem}\label{t31}
Let $P$ be a cone in a real Banach space $B$.  Suppose $\alpha$ and
$\psi $ are nonnegative continuous concave functionals on $P$ and
$\gamma,\beta$ $\theta$ are nonnegative continuous convex
functionals on $P$ such that, for some positive numbers $c'$ and
$k$,
$$
\alpha(y)\leq \beta(y)\quad\text{and}\quad\| y\|\leq
k\gamma(y)\quad \text{for all }y\in\overline{P(\gamma,c')}.
$$
Suppose further that $T:\overline{P(\gamma,c')}\to
\overline{P(\gamma,c')} $ is completely continuous and there exist
constants $h', d', a', b'\geq 0$ with $ 0<d'<a'$ such that each of
the following is satisfied.
\begin{itemize}
\item[(B1)] $\{y\in P(\gamma, \theta,\alpha,a',b',c')|\alpha(y)>a'\}\neq
\emptyset$ and $\alpha(Ty)>a'$ \\
for $y\in P(\gamma,\theta,\alpha,a',b',c')$;

\item[(B2)] $\{y\in Q(\gamma,\beta,\psi,h',d',c')|\beta(y)<d'\}\neq
 \emptyset$
and $\beta(Ty)<d'$ \\
for $y\in Q(\gamma,\beta,\psi,h',d',c')$;

\item[(B3)] $\alpha(Ty)>a'$ provided $y\in P(\gamma,\alpha,a',c') $
with $\theta(Ty)>b'$;

\item[(B4)] $\beta(Ty)<d'$ provided $y\in Q(\gamma,\beta,d',c')$ with
$\psi(Ty)<h'$.

\end{itemize}
Then $T$ has at least three fixed points $y_1,y_2,y_3\in
\overline{P(\gamma,c')}$ such that
 $$
\beta(y_1)<d',  a<\alpha(y_2)  and  d'<\beta(y_3)
\quad\text{with }\alpha(y_3)<a'.
$$
\end{theorem}

 Let $B=\{v|v:C[t_1,t_3]\to \mathbb{R}\}$ be the Banach
space equipped with the norm
$$
\| v\|= \max_{t\in[t_1,t_3]}|v(t)|.
$$
Define the cone $P\subset B$ by
$$
P=\Big\{v \in B:v(t)\geq 0\text{ on }[t_1,t_3]\text{ and }
 \min_{t\in [t_2,t_3]}v(t)\geq M \| v\| \Big\}.
$$
where $M =m_jL/K$ and $m_j,L,K$ are defined as in
Lemma \ref{l27}. Now, let
$$
[t_2', t_3']\subset [t_2, t_3],
$$
and define the nonnegative
continuous concave functionals $\alpha, \psi$ and the nonnegative
continuous convex functionals $\beta, \theta, \gamma$ on $P$ by
\begin{gather*}
\gamma(v)=\max_{t\in[t_1,t_3]} |v(t)|, \quad
\psi(v)=\min_{t\in[t_2',t_3'] }|v(t)|, \quad
\beta(v)=\max_{t\in[t_2',t_3']}|v(t)|,\\
\alpha(v)=\min_{t\in [t_2,t_3]}|v(t)|, \quad
\theta(v)=\max_{t\in[t_2,t_3]}|v(t)|.
\end{gather*}
We observe that for any $v\in P$,
\begin{gather}\label{e31}
\alpha(v)=\min_{t\in [t_2,t_3]}|v(t)|\leq \max_{t\in[t_2',t_3']
}|v(t)|=\beta(v),\\
\label{e32}
\| v \| \leq \frac{1}{M}
\min_{t\in [t_2,t_3]}v(t)\leq \frac{1}{M}\max_{t\in[t_1,t_3]}
|v(t)|=\frac{1}{M}\gamma(v).
\end{gather}
 We are now ready to present the main result of this section.
Let
\begin{gather*}
\overline L=\min\Big\{\int_{t_2}^{t_3}G_1(s,s)ds,
\int_{t_2}^{t_3}G_2(s,s)ds,\dots, \int_{t_2}^{t_3}G_n(s,s)ds\Big\},\\
\overline L'=\min\Big\{\int_{t_2'}^{t_3'} G_1(s,s)ds, \int_{t_2'}^{t_3'} G_2(s,s)ds,\dots,
\int_{t_2'}^{t_3'} G_n(s,s)ds\Big\},\\
\overline{K}=\max\Big\{\int_{t_1}^{t_3}G_1(s,s)ds,
\int_{t_1}^{t_3}G_2(s,s)ds,\dots, \int_{t_1}^{t_3}G_n(s,s)ds\Big\}.
\end{gather*}

\begin{theorem}\label{t32}
Suppose there exist $0<a'<b'<b'/M\leq c'$ such that $f$
satisfies the following conditions:
\begin{itemize}
\item[(D1)] $f(u_{n-1},u_{n-2},\dots,u_1,u_0) <a'/\overline L'$
for all $(|u_{n-1}|,|u_{n-2}|,\dots,|u_1|,|u_0|)$ in  $\prod_{j=n-1}^1
[\frac{a'LL'm_n}{M},\frac{c'KK_j}{M}] \times[Ma',a']$;

\item[(D2)] $f(u_{n-1},u_{n-2},\dots,u_1,u_0)>b'/M\overline K$
for all $(|u_{n-1}|,|u_{n-2}|,\dots,|u_1|,|u_0|)$ in
$\prod_{j=n-1}^1[\frac{b'm_n\overline L'L}{M},\frac{c'KK_j}{M}]\times
[b',\frac{b'}{M}]$;

\item[(D3)] $f(u_{n-1},u_{n-2},\dots,u_1,u_0)<c'/\phi$
for all $(|u_{n-1}|,|u_{n-2}|,\dots,|u_1|,|u_0|)$ in \break
$\prod_{j=n-1}^1 [0,\frac{c'KK_j}{M}]\times[0,c']$.

\end{itemize}
Then  \eqref{e11}-\eqref{e12} has at least
three positive solutions.
\end{theorem}

\begin{proof}
Define the completely continuous operator $T:P\to B$ by
\begin{equation}\label{e33}
Tv(t)=\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds.
\end{equation}
 It is obvious that a fixed point of $T$ is a solution of
 \eqref{e26}-\eqref{e27}.  We seek three fixed
points $v_1,v_2,v_3\in P$ of $T$. First, we show that
$T:P\to P$. Let $v\in P$. Clearly, $Tv(t)\geq 0$ for
$t\in[t_1,t_3]$. Also, noting that $Tv$ satisfies the boundary
conditions \eqref{e12},  we have
\begin{align*}
\min_{t\in [t_2,t_3]}Tv(t)
&=\min_{t\in[t_2,t_3] }\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),
 T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&\geq M\int_{t_1}^{t_3}G_1(s,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,
T_1v(s),v(s))ds\\
&=M\| Tv\|.
\end{align*}
Thus, $T:P\to P$.  Next, for all $v\in P$, by
\eqref{e31}-\eqref{e32}, we have $\alpha(v)\leq \beta(v)$ and
$\| v\|\leq\frac{c'}{M}$.  To show that $T:\overline{P(\gamma,c')}\to
\overline{P(\gamma,c')}$, let $v\in \overline{P(\gamma,c')}$. This
implies $\| v\|\leq\frac{c'}{M}$. For $1\leq j\leq
n-1$ and $t\in [a,b]$,
\begin{align*}
T_jv(t)
&=\int_{t_1}^{t_3}H_j(t,s)v(s)ds \\
&\leq\frac{c'}{M}\int_{t_1}^{t_3}H_j(t,s)ds\\
&\leq\frac{c'}{M}K\int_{t_1}^{t_3}G_j(s,s) ds
\\&=\frac{c'}{M}KK_j.
\end{align*}
We may now use condition $(D3)$ to obtain
\begin{align*}
\gamma(Tv)
&=\max_{t\in[t_1, t_3]}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&\leq \frac{c'}{\overline{K}}\int_{t_1}^{t_3}G_1(s,s) ds\\&\leq c'.
\end{align*}
Therefore,
$T:\overline{P(\gamma,c')}\to\overline{P(\gamma,c')}$.

We first verify that conditions (B1), (B2) of Theorem \ref{t31}
are satisfied. It is obvious that
\begin{gather*}
\{v\in P(\gamma,\theta,\alpha,b', \frac{b'}{M},c'):\alpha(v)>b'\}
\neq \emptyset,\\
\{v\in Q(\gamma,\beta,\psi, Ma',a',c'):\beta(v)<a'\}\neq \emptyset.
\end{gather*}
Next, let $v\in P(\gamma,\theta,\alpha,b',\frac{b'}{M},c')$ or $v\in
Q(\gamma,\beta,\psi, Ma',a',c')$.  Then, for $1\leq j\leq n-1,$
\begin{align*}
T_jv(t)
&=\int_{t_1}^{t_3} H_j(t, \tau)v(\tau)d\tau\\
&\leq \frac{c'}{M}\int_{t_1}^{t_3} H_j(t,\tau)d\tau,\\
&\leq  \frac{c'}{M}K\int_{t_1}^{t_3} \| G_j(\tau,\tau)\| d\tau,\\
&\leq\frac{c'KK_j}{M}
\end{align*}
and for $v\in P(\gamma,\theta,\alpha,b',\frac{b'}{M},c')$,
\begin{align*}
T_jv(t)&=\int_{t_1}^{t_3} H_j(t, \tau)v(\tau)d\tau\\
&\geq\min_{t\in[t_2, t_3]}\int_{t_2}^{t_3} H_j(t, \tau)v(\tau)d\tau\\
&\geq\frac{m_nLb'}{M}\int_{t_2}^{t_3} \| G_j(\tau, \tau)\| d\tau\\
&\geq\frac{m_nLb' \overline{L}}{M}.
\end{align*}
Also for $v\in Q(\gamma,\beta,\psi,Ma',a',c')$,
\begin{align*}
T_jv(t)&=\int_{t_1}^{t_3} H_j(t, \tau)v(\tau)d\tau\\
&\geq\max_{t\in[t_2', t_3']}\int_{t_2'}^{t_3'} H_j(t,
\tau)v(\tau)d\tau\\
&\geq\frac{m_nLa'}{M}\int_{t_2'}^{t_3'} \| G_j(\tau, \tau)\| d\tau\\
&\geq\frac{m_nLa' \overline{L'}}{M}.
\end{align*}
Now, we may apply condition (D2) to obtain
\begin{align*}
\alpha(Tv)
&=\min_{t\in [t_2, t_3]}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&\geq M\int_{t_1}^{t_3}G_1(s,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds
\geq b'.
\end{align*}
Clearly, by condition (D1),  we obtain
\begin{align*}
\beta(Tv)
&=\max_{t\in [t_2', t_3']}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&\leq\frac{a'}{\overline{L'}}\max_{t\in [t_2', t_3']}\int_{t_1}^{t_3}G_1(s,s)ds\\
&\leq\frac{a'}{\overline{L'}}\overline{L'}=a'.
\end{align*}
To see that (B3) is satisfied, let $v\in P(\gamma,\alpha,b',c')$
with $\theta(Tv)>b'/M$, we obtain
\begin{align*}
\alpha(Tv)&=\min_{t\in [t_2,
t_3]}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))d s \\
&\geq M\int_{t_1}^{t_3}G_1(s,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&\geq M\max_{t\in[t_1,
t_3]}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&\geq M\max_{t\in [t_2,
t_3]}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&=M\theta(Tv)>b'.
\end{align*}
Finally, we show that (B4) holds.  Let $v\in
Q(\gamma,\beta,a',c')$ with $\psi(Tv)<Ma'$,   we have
\begin{align*}
\beta(Tv)&=\max_{t\in [t_2', t_3']}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds \\
&\leq\max_{t\in [t_1,
t_3]}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&\leq\int_{t_1}^{t_3}G_1(s,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&=\frac{1}{M}\int_{t_1}^{t_3}G_1(s,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&\leq\frac{1}{M}\min_{t\in [t_2, t_3]
}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&\leq\frac{1}{M}\min_{t\in [t_2', t_3']
}\int_{t_1}^{t_3}G_1(t,s)f(T_{n-1}v(s),T_{n-2}v(s),\dots,T_1v(s),v(s))ds\\
&\leq\frac{1}{M} \psi(Tv)<a'.
\end{align*}
We have proved that all the conditions of Theorem \ref{t31} are
satisfied and so there exist at least three positive solutions
$v_1,v_2,v_3\in \overline{P(\gamma,c')}$ for
\eqref{e26}-\eqref{e27}.  Therefore,
\eqref{e11}-\eqref{e12} has at least three  positive
solutions $y_1,y_2,y_3$ of the form
$$
y_i(t)=T_{n-1}v_i(t)=\int_a^{b}H_{n-1}(t,s)v_i(s)ds, \quad i=1,2,3.
$$
This completes the proof.
\end{proof}

Now we prove the existence of $2m-1$ positive solutions for
 \eqref{e11}-\eqref{e12} by using induction on $m$.

\begin{theorem}\label{t3}
Let $m$ be an arbitrary positive integer. Assume that there exist
numbers $a_i(1\leq i\leq m)$ and $b_j(1\leq j\leq m-1)$ with
$0<a_1<b_1<\frac{b_1}{M}<a_2<b_2<\frac{b_2}{M}
<\dots<a_{m-1}<b_{m-1}<\frac{b_{m-1}}{M}<a_{m}$
such that
\begin{equation} \label{e41}
f(u_{n-1},u_{n-2},\dots,u_1,u_0)
<\frac{a_i}{\overline{L'}}
\end{equation}
for all $(|u_{n-1}|,|u_{n-2}|,\dots,|u_1|,|u_0|)$ in
$\prod_{j=n-1}^1
[\frac{a_iLL'm_n}{M},\frac{a_mKK_j}{M}]\times[Ma_i,a_i]$,
$1\leq i \leq m$,
and
\begin{equation}\label{e42}
f(u_{n-1},u_{n-2},\dots,u_1,u_0)>\frac{b_l}{K
\overline M}
\end{equation} for all
$(|u_{n-1}|,|u_{n-2}|,\dots,|u_1|,|u_0|)$
in $\prod_{j=n-1}^1[\frac{b_lm_n\overline
L'L}{M},\frac{b_{m-1}KK_j}{M}]\times [b_l,\frac{b_l}{M}]$,
$1\leq l \leq m-1$.
Then  \eqref{e11}-\eqref{e12} has at
least $2m-1$ positive solutions in $\overline{P}_{a_{m}}$.
\end{theorem}

\begin{proof}
We use induction on $m$.  First, for $m=1 $, we know from
\eqref{e41} that $T:\overline{P}_{a_{1}}\to P_{a_{1}}$,
then, it follows from Schauder fixed point theorem that
 \eqref{e11}-\eqref{e12} has at least one
positive solution in $\overline{P}_{a_{1}}$. Next, we assume that
this conclusion holds for $m=k$. In order to prove that this
conclusion holds for $m=k+1$, we suppose that there exist numbers
$a_i(1\leq i\leq k+1)$ and $b_j(1\leq j\leq k)$ with
$0<a_1<b_1<\frac{b_1}{M}<a_2<b_2<\frac{b_2}{M}<\dots<a_{k}<b_k<\frac{b_k}{M}<a_{k+1}$
such that
\begin{equation}\label{e43}
f(u_{n-1},u_{n-2},\dots,u_1,u_0)
<\frac{a_i}{\overline{L'}}
\end{equation}
for all $(|u_{n-1}|,|u_{n-2}|,\dots,|u_1|,|u_0|)$
in $\prod_{j=n-1}^1 [\frac{a_iLL'm_n}{M},\frac{a_mKK_j}{M}]
\times[Ma_i,a_i]$, $1\leq i \leq k+1$;
\begin{equation}\label{e44}
 f(u_{n-1},u_{n-2},\dots,u_1,u_0)>\frac{b_l}{K
\overline M}
\end{equation}
for all $(|u_{n-1}|,|u_{n-2}|,\dots,|u_1|,|u_0|)$
in $\prod_{j=n-1}^1[\frac{b_lm_n\overline
L'L}{M},\frac{b_{m-1}KK_j}{M}]\times [b_l,\frac{b_l}{M}]$,
$1\leq l \leq k$.

By assumption,  Problem \eqref{e11}-\eqref{e12}
has at least $2k-1$ positive solutions $u_i$ $(i=1,2,\dots,2k-1)$ in
$\overline{P}_{a_{k}}$. At the same time, it follows from
Theorem \ref{t32}, and \eqref{e43} and \eqref{e44} that
 \eqref{e11}-\eqref{e12} has at least three  positive
solutions $u,v$ and $w$ in $\overline{P}_{a_{k+1}}$ such that,
$\| u\|<a_k,b_k<\min_{t\in [t_2, t_3]}v(t),\| w
\|>a_k,\min_{t \in [t_2, t_3]}w(t)<b_k$. Obviously, $v$ and $w$
 are different from $u_i$ $(i=1,2,\dots,2k-1)$. Therefore,
\eqref{e11}-\eqref{e12} has at least $2k+1$  positive solutions in
$\overline{P}_{a_{k+1}}$ which shows that this conclusion also holds
for $m=k+1$.
\end{proof}

\subsection*{Acknowledgements}
K. L. Saraswathi Devi is grateful to the U. G. C. (India) and to the
management of Ch. S. D. St. Theresa's College for Women, Eluru, for
being  accepted in the Faculty Development Programme. The authors
thank the referees for their valuable suggestions.


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