\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 173, pp. 1--11.\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-2014/173\hfil Existence of solutions]
{Existence of solutions for three-point BVPs arising in bridge design}

\author[A. K. Verma, M. Singh \hfil EJDE-2014/173\hfilneg]
{Amit K. Verma, Mandeep Singh}  % in alphabetical order

\address{Amit Kumar Verma \newline
Department of Mathematics, BITS Pilani, Pilani-333031, Rajasthan, India}
\email{amitkverma02@yahoo.co.in}

\address{Mandeep Singh \newline
Department of Mathematics, BITS Pilani, Pilani-333031, Rajasthan, India}
\email{mandeep04may@yahoo.in}

\thanks{Submitted February 25, 2014. Published August 12, 2014.}
\subjclass[2000]{34L30, 34B27, 34B15}
\keywords{Monotone iterative technique; Lipschitz continuous;
\hfill\break\indent 
 reversed ordered upper and lower solutions; three point BVP; nonlinear ODE;
 Green's function}

\begin{abstract}
 This article deals with a class of three-point nonlinear boundary-value problems
 (BVPs) with Neumann type boundary conditions which arises in bridge design.
 The source term (nonlinear term) depends on the derivative of the solution,
 it is also Lipschitz continuous. We use monotone iterative technique in the
 presence of upper and lower solutions for both well order and reverse order case.
 Under some sufficient conditions we prove existence results. We also construct
 two examples to validate our results. These result can be used to generate a
 user friendly package in Mathematica or MATLAB so that solutions of nonlinear
 boundary-value problems can be computed.
\end{abstract}

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

\section{Introduction}

In the past few years, there has been much attention focused on
 multipoint BVPs for nonlinear ordinary differential equations,  see
\cite{aftabizadeh,gregus,henderson,ma,webb,wei}.
 Multipoint BVPs have lots of applications in modern science and engineering.

It is observed \cite{lazer} that a linear model is insufficient
to explain the large oscillatory behavior in suspension bridges also 
suspension bridges have other nonlinear behaviors such as traveling waves. 
If the roadbed of a suspension bridge is treated as a one-dimensional 
vibrating beam the following equation is derived 
(see \cite[Section 3]{lazer})
\begin{gather}
\label{sb} u_{tt}+EIu_{xxxx}+\delta u_t=-k u^{+}+W(x)+\varepsilon f(x,t),\\
\label{sb-bc} u(0,t)=u(L,t)=u_{xx}(0,t)=u_{xx}(L,t)=0.
\end{gather}
Thus the suspension bridge is seen as a beam of length $L$, with hinged ends, 
whose downward deflection is measured by $u(x, t)$, with a small viscous
 damping term, subject to three separate forces; the stays, holding it up as 
nonlinear springs with spring constant $k$, the weight per unit length of the 
bridge $W(x)$ pushing it down, and the external forcing term $\varepsilon f (x, t)$. 
The loading $W(x)$ would usually be constant.

If $W$ is replaced by the term $W(x) = W_0 \sin{(x/L)}$, an error of magnitude 
around $10\%$ is introduced in the loading and little less in the steady-state 
deflection. Second, if the forcing term is given by $f (x, t) = f (t) \sin{(x /L)}$ 
and general solutions of \eqref{sb}-\eqref{sb-bc}, is of the form 
$u(x, t) = y(t)\sin{(x/L)}$. These no-nodal solutions were the most commonly 
observed type for low velocities on the Tacoma Narrows Bridge. When this 
$u(x, t)$ is substituted into \eqref{sb}, this results into the 
differential equation
\begin{equation}
\label{ode-sb} -y''(t)=f(t,y,y')
\end{equation}
were $f(t,y,y')=\delta y'+EI(\pi/L)^4y+ky^{+}-W_0-\varepsilon f(t)$,
where $y^+$ denotes $y$ if $y$ is non-negative, and zero if
$y$ is negative.

Large size bridges are sometimes contrived with multi-point supports,
 which gives rise to multi-point boundary condition.

In this paper we focus on monotone iterative technique related to upper 
and lower solutions. Lot of aspects of this technique has been explored
 for two point BVPs. When upper $\beta_0$ and lower $\alpha_0$  solution 
are well ordered; i.e., $\beta_0\geq\alpha_0$ lots of results are available 
(see \cite{pandey1}-\cite{pandey2} and the references there in).
 When the upper and lower solutions are in reverse order, that is 
$\beta_0\leq\alpha_0$ some results are available
 (see \cite{decoster,li,verma1,verma2}
and the references there in).

As far as three-point BVPs are concerned, different techniques are used to 
prove existence results. The case when source function is independent from 
first derivative results are available. But there are very few results when 
the first order derivative is involved explicitly in the nonlinear term $f$.

Guo et al \cite{guo} consider the  three point BVPs
\begin{gather}
\label{1D} y''(t)+f(t,y,y')=0,\quad 0<t<1,\\
\label{2D} y(0)=0,\quad y(1)=\delta y(\eta)
\end{gather}
where $f$ is a nonnegative continuous function, $\delta>0$, $\eta\in(0,1)$ 
and $\alpha \eta <1$. They used a new fixed point theorem in a cone.
 Bao et al \cite{bao} proved some existence results
for three-point BVPs \eqref{1D}-\eqref{2D}. They used fixed point index method 
under a non-well-ordered upper and lower solution condition.

Recently Singh et al \cite{singh1,singh2} used monotone
iterative method with upper and lower solutions when they are well ordered 
as well as reverse ordered, and proved some existence results for a class of 
three point non-linear BVPs.

The aim of this article is to explore the monotone iterative method with upper 
and lower solutions (for both well order as well as reverse order case) 
for the Neumann type nonlinear three point BVPs given by
\begin{gather}
\label{1.1} y''(t)+f(t,y,y')=0,\quad 0<t<1,\\
\label{1.2} y'(0)=0,\quad y'(1)=\delta y'(\eta),
\end{gather}
where $f\in C(I\times R,R)$, $I=[0,1]$, $0<\eta<1$, $0<\delta<1$.

This article is organized in six sections. In Section 2 we discuss 
the corresponding linear case of nonlinear three point BVPs and construct 
Green's function. In Sections 3 and 4  we discuss some other important 
Lemmas and maximum and anti-maximum principles. 
In Section 5 we derive sufficient conditions which guarantee the existence 
of solutions of nonsingular nonlinear three point BVPs for both case;
 i.e., when upper and lower solutions are  well ordered and also when reverse 
ordered. Finally in Section 6 two examples are constructed to validate our results.

\section{Linear Case and Green's Function}

This section deals with linear three-point BVPs with Neumann type conditions.
 We construct Green's function for these problems to be investigated in 
later sections.

\subsection{Construction of the Green's function}
Consider the corresponding linear three-point BVPs as given by
\begin{gather}
\label{2.1} Ly\equiv-y''(t)-\lambda y(t)=h(t),\quad 0<t<1,\\
\label{2.2} y'(0)=0,\quad y'(1)=\delta y'(\eta)+ b,
\end{gather}
where $h\in C(I)$ and $b$ is any constant. Based on the sign of $\lambda$ 
we can divide the construction of Green's function into two cases. 
In one case $\lambda>0$ we get Green's function in terms of $\cos$ and $\sin$.
 In the case when $\lambda<0$ we get Green's function in terms of $\cosh$ 
and $\sinh$.

\begin{lemma}\label{Lemma2.1}
Assume
\begin{itemize}
\item[(H0)] $\lambda\in(0,\pi^2/4)$,
$ \sin (\sqrt{\lambda })-\delta  \sin (\eta  \sqrt{\lambda })>0$ and 
 $ \cos (\sqrt{\lambda })-\delta  \cos (\eta  \sqrt{\lambda })\geq0$.
\end{itemize}
When $\lambda>0$, Green's function for the three-point BVPs 
\begin{gather}
\label{2.3} y''(t)+\lambda y(t)=0,\quad 0<t<1,\\
\label{2.4} y'(0)=0,\quad y'(1)=\delta y'(\eta),
\end{gather}
is
\begin{equation*}
G(t,s)=  \begin{cases}
\frac{\cos  (\sqrt{\lambda } t) (\cos  (\sqrt{\lambda } (s-1))
 -\delta  \cos  (\sqrt{\lambda } (s-\eta )))}{\sqrt{\lambda } 
 (\sin (\sqrt{\lambda })-\delta  \sin (\eta  \sqrt{\lambda }))}, 
& 0 \leq t\leq s\leq \eta, \\[4pt]
\frac{\cos  (\sqrt{\lambda } s) (\cos  (\sqrt{\lambda } (t-1))
-\delta  \cos  (\sqrt{\lambda } (t-\eta )))}{\sqrt{\lambda } 
 (\sin (\sqrt{\lambda })-\delta  \sin (\eta  \sqrt{\lambda }))}, 
& s \leq t, \; s \leq \eta,\\[4pt]
\frac{\cos  (\sqrt{\lambda } (s-1)) \cos  (\sqrt{\lambda } t)}{\sqrt{\lambda }
(\sin (\sqrt{\lambda })-\delta  \sin (\eta  \sqrt{\lambda }))}, 
& t\leq s,\; \eta \leq s, \\[4pt]
\frac{\delta  \sin (\eta  \sqrt{\lambda }) \sin (\sqrt{\lambda } (s-t))
+\cos (\sqrt{\lambda } s) \cos (\sqrt{\lambda } (1-t))}{\sqrt{\lambda }
 (\sin (\sqrt{\lambda })-\delta  \sin (\eta  \sqrt{\lambda }))}, 
& \eta \leq s\leq t\leq 1.
  \end{cases}
\end{equation*}
It can be checked that (H0) can be satisfied for some sub interval of 
$\lambda\in(0,\pi^2/4)$.
\end{lemma}

For a proof of the above lemma, see \cite[Lemma 2.1]{singh1}.

\begin{lemma}\label{Lemma2.2}
Let $\lambda>0$. If $y\in C^2(I)$ is the solution of the three-point BVPs \eqref{2.1} 
and \eqref{2.2}, then it can be expressed as 
\begin{equation}
\label{2.5} y(t)= \frac{b\cos\sqrt{\lambda}t}{\sqrt{\lambda}
(\delta \sin\sqrt{\lambda}\eta-\sin\sqrt{\lambda})}- \int^1_0{G(t,s)h(s)ds}.
\end{equation}
\end{lemma}

For a proof of the above lemma, see \cite[Lemma 2.2]{singh1}.

\begin{lemma}\label{Lemma2.3}
Assume 
\begin{itemize}
\item[(H0$'$)] $\lambda<0$, 
$\sinh\sqrt{|\lambda|}-\delta\sinh\sqrt{|\lambda|}\eta\ > 0$ and
$\delta \cosh\sqrt{|\lambda|}\eta-\cosh\sqrt{|\lambda|}\leq 0 $.
\end{itemize}
Then for $\lambda <0$, the Green's function of the three-point BVPs
\begin{gather*}
y''(t)+\lambda y(t)=0,\quad 0<t<1,\\
y'(0)=0,\quad y'(1)=\delta y'(\eta)
\end{gather*}
 is
\begin{equation*}
G(t,s)=  \begin{cases}
\frac{\cosh (\sqrt{|\lambda|} t) (\delta  \cosh (\sqrt{|\lambda|} (s-\eta ))
-\cosh (\sqrt{|\lambda|} (s-1)))}{\sqrt{|\lambda|} (\sinh (\sqrt{|\lambda|})
-\delta  \sinh (\eta  \sqrt{|\lambda|}))}, 
& 0 \leq t\leq s\leq \eta, \\[4pt]
\frac{\cosh (\sqrt{|\lambda|} s) (\delta  \cosh (\sqrt{|\lambda|} 
(t-\eta ))-\cosh (\sqrt{|\lambda|} (t-1)))}{\sqrt{|\lambda|} 
(\sinh (\sqrt{|\lambda|})-\delta  \sinh (\eta  \sqrt{|\lambda|}))}, 
& s \leq t,\; s \leq \eta,\\[4pt]
-\frac{\cosh (\sqrt{|\lambda|} (s-1)) 
\cosh (\sqrt{|\lambda|} t)}{\sqrt{|\lambda|} (\sinh (\sqrt{|\lambda|})
-\delta  \sinh (\eta  \sqrt{|\lambda|}))}, 
& t\leq s,\; \eta \leq s, \\[4pt]
\frac{\delta  \sinh (\eta  \sqrt{|\lambda|}) \sinh (\sqrt{|\lambda|} (s-t))
-\cosh (\sqrt{|\lambda|} s) \cosh (\sqrt{|\lambda|} (1-t))}{\sqrt{|\lambda|} 
(\sinh (\sqrt{|\lambda|})-\delta  \sinh (\eta  \sqrt{|\lambda|}))},
 & \eta \leq s\leq t\leq 1.
  \end{cases}
\end{equation*}
\end{lemma}

It can be easily checked that (H0$'$) can be satisfied for some values of 
$\lambda\in(-\infty,0)$.
For a proof of the above lemma, see \cite[Lemma 2.3]{singh1}.

\begin{lemma}\label{Lemma2.4}
Let $\lambda<0$. If $y\in C^2(I)$ is a solution of the three-point BVPs \eqref{2.1}
 and \eqref{2.2}, then it is given by
\begin{equation}
\label{2.6}y(t)=\frac{b\cosh\sqrt{|\lambda|}t}{\sqrt{|\lambda|} 
(\sinh (\sqrt{|\lambda|})-\delta  \sinh (\eta  \sqrt{|\lambda|}))}
- \int^1_0{G(t,s)h(s)ds}.
\end{equation}
\end{lemma}

For a proof of the above lemma, see \cite[Lemma 2.4]{singh1}.

\section{Some inequalities}

\begin{lemma}\label{Lemma2.5}
Let $\lambda\in(0,\pi^2/4)$ and $\lambda-M\geq 0$. Further if
\begin{eqnarray*}
&&(\lambda-M)\cos\sqrt{\lambda}-N\sqrt{\lambda}\sin\sqrt{\lambda}\geq 0,
\end{eqnarray*}
then for all $t\in [0,1]$
\begin{eqnarray*}
&&(\lambda-M)\cos\sqrt{\lambda}t-N\sqrt{\lambda}\sin\sqrt{\lambda}t\geq 0,
\end{eqnarray*}
where $M,N\in~\mathbb{R}^+$.
\end{lemma}

\begin{proof}
Using monotonicity of $\sin$ and $\cos$, we derive that for all $t\in[0,1]$,
\[
(\lambda-M)\cos\sqrt{\lambda}t-N\sqrt{\lambda}\sin\sqrt{\lambda}t 
\geq (\lambda-M)\cos\sqrt{\lambda}-N\sqrt{\lambda}\sin\sqrt{\lambda}\geq 0.
\]
Which completes the proof.
\end{proof}

\begin{lemma}\label{Lemma2.6}
If $\lambda<0$ is such that $M+\lambda\leq 0$, and
\[
\lambda \leq-M-\frac{N^2}{2}-\frac{N}{2}\sqrt{N^2+4M},
\]
then for all $t\in[0,1]$,
\[
(M+\lambda)\cosh\sqrt{|\lambda|} t + N\sqrt{|\lambda|}\sinh\sqrt{|\lambda|} t\leq 0,
\]
where $ M,N \in \mathbb{R}^+$.
\end{lemma}

\begin{proof} As
\[
(M+\lambda)\cosh\sqrt{|\lambda|}t+N\sqrt{|\lambda|}\sinh\sqrt{|\lambda|}t
\leq [(M+\lambda)+N\sqrt{|\lambda|}]\cosh\sqrt{|\lambda|}t.
\]
We will have $[(M+\lambda)+N\sqrt{|\lambda|}]\cosh\sqrt{|\lambda|}t\leq 0$ 
for all $t\in[0,1]$ if
\[
[(M+\lambda)+N\sqrt{|\lambda|}]\leq0.
\]
The above inequality will be satisfied if
\[
{\lambda}\leq-M-\frac{N^2}{2}-\frac{N}{2}\sqrt{N^2+4M}.
\]
This completes the proof.
\end{proof}

\begin{lemma}\label{Lemma2.7}
Let {\rm (H0)} be satisfied. Then
\begin{itemize}
\item[(i)] $G(t,s)\geq0$,
\item[(ii)] $\frac{\partial G(t,s)}{\partial t}\leq0$ and
\item[(iii)] $(\lambda-M)G(t,s)+N(\operatorname{sign} y')
\frac{\partial G(t,s)}{\partial t}\geq0$
\end{itemize}
for any $t,s\in[0,1]$ and $M,N\in \mathbb{R}^{+}$.
\end{lemma}

\begin{proof} 
The conditions assumed in (H0) ensure that $G(t,s)\geq0$.
 Since $G(t,s)$ is the solution of \eqref{2.3}--\eqref{2.4}, we deduce that
 $\frac{\partial G(t,s)}{\partial t}\leq0$. 
For part (iii) we prove that for all $t,s\in[0,1]$,
\[
(\lambda-M)G+ N \frac{\partial G(t,s)}{\partial t}\geq0.
\]
Which can easily be deduced by using Lemmas \ref{Lemma2.1}, 
 \ref{Lemma2.2} and  \ref{Lemma2.5}.
\end{proof}

\begin{lemma}\label{Lemma2.8}
Assume {\rm (H0$'$)}. Then for any $t,s \in[0,1]$, we have
\begin{itemize}
\item[(i)] $G(t,s)\leq0$,
\item[(ii)] $\frac{\partial G(t,s)}{\partial t}\leq0$ and
\item[(iii)] $(M+\lambda)G(t,s)+N(\operatorname{sign}y')
\frac{\partial G(t,s)}{\partial t}\geq0$ whenever we have
 $M+\lambda-N\lambda\leq0$
\end{itemize}
where $M,N\in \mathbb{R}^{+}$.
\end{lemma}

\begin{proof}
 Parts (i) and (ii) follow the analysis of Lemma \ref{Lemma2.7}. 
For part (iii), it will be sufficient to prove that for all $t,s\in[0,1]$,
\[
(M+\lambda)G(t,s)+N \frac{\partial G(t,s)}{\partial t}\geq0.
\]
Since $G(t,s)$ is the the Green function for \eqref{2.1}--\eqref{2.2},
 we have 
$$
\frac{\partial G(t,s)}{\partial t}\geq -\lambda G(t,s).
$$ 
The above inequality along with condition $M+\lambda-N\lambda \leq0$ gives
\[
(M+\lambda)G(t,s)+N\frac{\partial G(t,s)}{\partial t}
\geq (M+\lambda-N\lambda)G(t,s)\geq0.
\]
\end{proof}

\section{Anti-maximum and maximum principle}

\begin{proposition}\label{Proposition2.1} 
Let $b\geq 0$, $h(t)\in C[0,1] $ be such that $h(t)\geq 0$, and {\rm (H0)} hold. 
Then the solution of $Ly=h$ and \eqref{2.2} is non-positive.
\end{proposition}

\begin{proposition}\label{Proposition2.2}
Let $ b\geq 0$, $h(t)\in C[0,1] $ be such that $h(t)\geq 0$ and {\rm(H0$'$)} hold. 
Then the solution of $Ly=h$ and \eqref{2.2} is non-negative.
\end{proposition}

\section{Existence results for nonlinear three-point BVP}

In this section, we prove two existence results for the nonlinear three-point
 Neumann type BVPs. On the basis of the order of upper and lower solutions, 
we divide this section into the following subsections.

\subsection{Reverse ordered case}
\begin{definition} \rm
The functions $\alpha,\beta\in C^2[0,1]$ are called lower and upper solutions 
for the class of three-point BVPs \eqref{1.1}-\eqref{1.2} if they satisfy 
the following inequalities:
\begin{gather*}
-\alpha''(t)\leq f(t,\alpha,\alpha'),\quad 0<t<1,\\
\alpha'(0)=0, \quad \alpha'(1)\leq\delta \alpha'(\eta).\\
-\beta''(t)\geq f(t,\beta,\beta'),\quad 0<t<1,\\
\beta'(0)=0,\quad \beta'(1)\geq \delta \beta'(\eta).
\end{gather*}
\end{definition}

The sequences $(\alpha_n)_n$ and $(\beta_n)_n$  are defined by the following 
iterative schemes
\begin{gather}
\label{3.1}\alpha_0=\alpha,\quad 
-\alpha_{n+1}''(t)-\lambda \alpha_{n+1}(t)=f(t,\alpha_{n},\alpha_{n}')
-\lambda \alpha_{n},\\
\label{3.2} \alpha_{n+1}'(0)=0,\quad 
\alpha_{n+1}'(1)=\delta \alpha_{n+1}'(\eta),\\
\label{3.3} \beta_0=\beta,\quad
-\beta_{n+1}''(t)-\lambda \beta_{n+1}(t) 
= f(t,\beta_{n},\beta_{n}')-\lambda \beta_{n},\\
\label{3.4} \beta_{n+1}'(0)=0,\quad 
\beta_{n+1}'(1)=\delta \beta_{n+1}'(\eta),
\end{gather}
where $\lambda\neq0$.

\begin{theorem}\label{Theorem3.1}
Assume that {\rm (H0)}and following hypothesis hold: 
\begin{itemize}
\item[(H1)] there exist $\alpha$ and $ \beta\in C^2[0,1]$ lower and upper 
solutions of \eqref{1.1} and \eqref{1.2} such that for all 
$t\in[0,1]$, $\alpha \geq \beta$;

\item[(H2)] the function $f: D \to R$ is continuous on 
$D:=\{(t,u,v) \in [0,1] \times \mathbb{R}^2 :\beta(t)\leq u \leq \alpha(t)\}$;

\item[(H3)] there exist $M\geq 0$ such that for all $(t,u_1,v),(t,u_2,v)\in D$
$$
u_1\leq u_2 \to  f(t,u_2,v)- f(t,u_1,v)\leq M (u_2 - u_1);
$$

\item[(H4)] there exist $N\geq 0$ such that for all $(t,u,v_1),(t,u,v_2)\in D$
\[
|f(t,u,v_2)- f(t,u,v_1)|\leq N|v_2 - v_1|;
\]

\item[(H5)] for all $(t,u,v)\in D,~|f(t,u,v)|\leq \varphi(|v|)$,
such that $\varphi:\mathbb{R}^+ \to \mathbb{R}^+$ is continuous and satisfies
\[
\max_{t\in[0,1]}\alpha-\min_{t\in[0,1]}\beta \leq  \int_{l_{0}}^\infty 
\frac{s\ ds}{\varphi(s)}
\]
where $l_0=\sup_{[0,1]}[2|\alpha(t)|]$.
\end{itemize}
Further if $(\lambda-M)\cos\sqrt{\lambda}-N\sqrt{\lambda}\sin\sqrt{\lambda}\geq0$ 
and for all $t\in[0,1]$
\[
f(t,\beta(t),\beta'(t))- f(t,\alpha(t),\alpha'(t))-\lambda(\beta-\alpha)\geq 0
\]
then the sequences $(\alpha_n)$ and $(\beta_n)$ defined by \eqref{3.1}, \eqref{3.2} 
and \eqref{3.3}, \eqref{3.4} converge monotonically in $C^1([0,1])$ 
to the solutions $v$ and $u$ of the nonlinear boundary-value problem 
\eqref{1.1} and \eqref{1.2}, such that for all $t\in[0,1]$
\[
\beta \leq u \leq v \leq \alpha.
\]
\end{theorem}

The proof of the above theorem can be divided into several small results 
stated as follows.

\begin{lemma}\label{Lemma3.1}
If $\alpha_n$ is a lower solution of \eqref{1.1} and \eqref{1.2}, $\alpha_{n+1}$ 
is defined by \eqref{3.1} and \eqref{3.2} where $\lambda\in(0,\pi^2/4)$, 
then $\alpha_{n+1}\leq \alpha_n$.
\end{lemma}

\begin{proof}
Since $y(t)=\alpha_{n+1}-\alpha_{n}$ satisfies $Ly\geq0$, \eqref{2.2} with 
$b\geq0$, the result can be concluded by Proposition \ref{Proposition2.1}.
\end{proof}

\begin{proposition}\label{Proposition3.1}
Let {\rm (H0)--(H4)} hold and 
$(\lambda-M)\cos\sqrt{\lambda}-N\sqrt{\lambda}\sin\sqrt{\lambda}\geq0$ 
then the function $\alpha_{n}$ defined by \eqref{3.1} and \eqref{3.2} 
are such that for all $n\in N$,
\begin{itemize}
\item[(i)] $\alpha_n$ is a lower solution of \eqref{1.1}-\eqref{1.2}; and
\item[(ii)] $\alpha_{n+1}\leq \alpha_n$.
\end{itemize}
\end{proposition}

\begin{proof} We prove it by recurrence.
By Lemma \ref{Lemma3.1} the claim holds for $n=0$.

Let $\alpha_{n-1}$ is a lower solution of \eqref{1.1} and \eqref{1.2} 
and $\alpha_{n}\leq \alpha_{n-1}$. 
Let $y=\alpha_n-\alpha_{n-1}$. Then we have
\[
-\alpha_{n}''-f(t,\alpha_{n},\alpha_{n}')
\leq (\lambda-M)y+N(\operatorname{sign}y')y'.
\]
Let $(\lambda-M)y+N(\operatorname{sign}y')y'=g$. Now to show $\alpha_n$ 
is a lower solution we have to show that $g\leq0$. 
Since $y$ is given by Lemma \ref{Lemma2.2} with 
$h(t)= \alpha_{n-1}''+ f(t,\alpha_{n-1},\alpha_{n-1}')\geq 0$. 
Thus to show $g\leq0$, it is enough to prove that
\begin{gather*}
(\lambda-M)\cos\sqrt{\lambda}t-N\sqrt{\lambda}
\sin\sqrt{\lambda}t\geq 0, \\
(\lambda-M)G(t,s)+N\frac{\partial G(t,s)}{\partial t}\geq 0
\end{gather*}
for all $t\in[0,1]$. Lemma \ref{Lemma2.5} and Lemma \ref{Lemma2.7} verify 
the existence of above two inequalities. Thus $\alpha_{n+1}\geq \alpha_n$.
\end{proof}

Similarly we can prove the following result.

\begin{proposition}\label{Proposition3.2}
Let {\rm (H0)--(H4)} be true and 
$(\lambda-M)\cos\sqrt{\lambda}-N\sqrt{\lambda}\sin\sqrt{\lambda}\geq0$. 
Then the function $\beta_{n}$ defined by \eqref{3.3}-\eqref{3.4} 
are such that for all $n\in N$
\begin{itemize}
\item[(i)] $\beta_n$ is an upper solution of \eqref{1.1}-\eqref{1.2};
\item[(ii)] $\beta_{n+1}\geq \beta_n $.
\end{itemize}
\end{proposition}

\begin{proposition}\label{Proposition3.3}
Let {\rm (H0)--(H4)} hold, 
$(\lambda-M)\cos\sqrt{\lambda}-N\sqrt{\lambda}\sin\sqrt{\lambda}\geq0$ 
and for all $t\in[0,1]$,
\[
f(t,\beta(t),\beta'(t))- f(t,\alpha(t),\alpha'(t))-\lambda(\beta - \alpha)\geq 0\,.
\]
Then for all $n\in N$, the functions $\alpha_n$ and $\beta_n$ defined by
 \eqref{3.1}-\eqref{3.2} and \eqref{3.3}-\eqref{3.4} satisfy $\alpha_n\geq \beta_n$.
\end{proposition}

\begin{proof} We define
\[
h_i(t)=f(t,\beta_i,\beta_i')-f(t,\alpha_i,\alpha_i')-\lambda(\beta_i-\alpha_i),
\quad \text{for all } i\in N.
\]
Now, for all $i\in N$, $y_i := \beta_i-\alpha_i$ satisfies
\[
-y_i''-\lambda y_i =f(t,\beta_{i-1},\beta_{i-1}')
-f(t,\alpha_{i-1},\alpha_{i-1}')-\lambda(\beta_{i-1}-\alpha_{i-1})=h_{i-1}.
\]

\noindent Claim 1. $\alpha_1\geq\beta_1$.
The function $y_1=\beta_1-\alpha_1$ is a solution of \eqref{2.1}-\eqref{2.2} 
with $h(t)=h_0(t)\geq0$ and $b=0$, by Proposition \ref{Proposition2.1}, 
$y_1(t)\leq 0$; i.e., $\alpha_1\geq\beta_1$.
\smallskip

\noindent Claim 2. Let $n\geq2$. If $h_{n-2}\geq 0$ and 
$\alpha_{n-1}\geq\beta_{n-1}$, then $h_{n-1}\geq 0$ and $\alpha_n\geq\beta_n$.
First we will prove that, for all $t\in[0,1]$, the function $h_{n-1}$ 
is non-negative, as we have
\[
h_{n-1} \geq -[(\lambda-M)y_{n-1}+N(\operatorname{sign} y_{n-1}')y_{n-1}'].
\]
Since $y_{n-1}$ is a solution of \eqref{2.1}-\eqref{2.2} with 
$h(t)=h_{n-2}(t)\geq 0 $, $b=0$. Hence we can proceed similar to the proof of 
Proposition \ref{Proposition3.1} to show that $h_{n-1}\geq0$.
 Now $y_n'(0)=0$ and $y_n'(1)=\delta y'(\eta)$; i.e., $b=0$, we deduce 
from Proposition \ref{Proposition2.1} that $y_n\leq 0$;
 i.e., $\alpha_n\geq\beta_n$.
\end{proof}

\begin{lemma}\label{Lemma3.2}
If $f(t,y,y')$ satisfies {\rm (H5)}, then there exists $R>0$ such that any 
solution of the differential inequality
\begin{gather}
\label{3.5} -y''(t)\geq f(t,y,y'),\quad 0<t<1,\\
\label{3.6} y'(0)=0,\quad y'(1)\geq\delta y'(\eta)
\end{gather}
with $y\in[\beta(t),\alpha(t)]$ for all $t\in[0,1]$ satisfies 
$\|y'\|_\infty\leq R$.
\end{lemma}

\begin{lemma}\label{Lemma3.3}
If $f(t,y,y')$ satisfies {\rm (H5)}, then there exists $R>0$ such that any 
solution of the differential inequality
\begin{gather}
\label{3.7} -y''(t)\leq f(t,y,y'),\quad 0<t<1,\\
\label{3.8}  y'(0)=0,\quad y'(1)\leq\delta y'(\eta)
\end{gather}
with $y\in[\beta(t),\alpha(t)]$ for all $t\in[0,1]$ satisfies $\|y'\|_\infty\leq R$.
\end{lemma}

The proof of above two Lemmas are similar to the proof of 
\cite[Lemma 3.2]{singh1} (Priory bound).
Now we can complete the proof of Theorem \ref{Theorem3.1}.

\begin{proof}[Proof of Theorem \ref{Theorem3.1}] 
Consider $(\alpha_n)_n$ and $(\beta_n)_n$ defined, respectively by 
\eqref{3.1}-\eqref{3.2} and \eqref{3.3}-\eqref{3.4}. By using the 
Propositions \ref{Proposition3.1}, \ref{Proposition3.2} and \ref{Proposition3.3} 
we deduce the following inequality
\begin{equation}
\label{3.10}
\alpha=\alpha_0\geq\alpha_1\geq \dots \geq\alpha_n\geq \dots \geq\beta_n\geq 
\dots \geq\beta_1\geq\beta_0=\beta.
\end{equation}
Since $(\alpha_n)_n$ and $(\beta_n)_n$ are monotonic in nature and bounded, 
they converge pointwise to functions
\[
v(t)= \lim_{n \to \infty}\alpha_n(t)\quad \mathrm{and}\quad 
u(t)= \lim_{n \to \infty} \beta_n(t)
\]
which are such that for all $n$, $\alpha_n\geq u \geq v\geq\beta_n$. 
Using \eqref{3.1}, \eqref{3.2} and \eqref{3.3}, \eqref{3.4} along with 
\eqref{3.10} and Lemma \ref{Lemma3.3} we verify that $(\alpha_n)_n$ is
 equibounded and equicontinuous in $C^1([0,1])$. Hence any subsequence 
$(\alpha_{n_k})_k$ of $(\alpha_n)_n$ is also equibounded and equicontinuous 
in $C^1([0,1])$ and from Arzela-Ascoli theorem we deduce that there exists 
a subsequence $(\alpha_{{n_k}_j})_j$ of $(\alpha_{n_k})_k$ which converge 
in $C^1([0,1])$. By monotonicity and uniqueness of the limit of the sequence 
$(\alpha_n)_n$, we have $\alpha_n \to v$ in $C^1([0,1])$. 
As any subsequence of $(\alpha_n)_n$ contains a subsequence $(\alpha_{{n_k}_j})_j$ 
which converge in $C^1([0,1])$ to $v$ it follows that $\alpha_n \to v$ in 
$C^1([0,1])$. In a similar way, using Proposition \ref{Proposition3.2} and 
Lemma \ref{Lemma3.2} it can be proved that $(\beta_n)_n$ converge to $u$ 
in $C^1([0,1])$.

The solution of the iterative schemes \eqref{3.1}-\eqref{3.2} and 
\eqref{3.3}-\eqref{3.4} can be written by equations given in 
Lemmas \ref{Lemma2.2} and  \ref{Lemma2.4} for corresponding $h(t)$ given 
in terms of nonlinear $f$. Now by using the uniform convergence of $\alpha_n$ 
and $\beta_n$ and taking Limit both sides we get solution of the nonlinear 
three point BVPs \eqref{1.1}-\eqref{1.2}. This proves existence of solution 
of the nonlinear three point BVPs \eqref{1.1}-\eqref{1.2}.
\end{proof}

\subsection{Well ordered case}

We state our main result as Theorem \ref{Theorem3.2}. 
Proof here again can be written as we did earlier for non well ordered case. 
We skip the proof for brevity.

\begin{theorem}\label{Theorem3.2}
Assume {\rm (H0$'$)} and the following hypothesis hold:
\begin{itemize}
\item[(H1$'$)] there exist $\alpha,\beta\in C^2[0,1]$ lower and upper solutions 
 of \eqref{1.1}-\eqref{1.2} such that $\alpha \leq \beta$;

\item[(H2$'$)] the function $f: \widetilde{D} \to R$ is continuous on 
$\widetilde{D} :=\{(t,u,v) \in [0,1] \times \mathbb{R}^2 :
 \alpha(t)\leq u \leq \beta(t)\}$;

\item[(H3$'$)] there exist $M\geq 0$ such that for all 
 $(t,u_1,v),(t,u_2,v)\in \widetilde{ D}$,
$$
u_1\leq u_2 \to  f(t,u_2,v)- f(t,u_1,v)\geq -M (u_2 - u_1);
$$

\item[(H4$'$)] there exist  $N\geq 0$ such that for all 
$(t,u,v_1),(t,u,v_2)\in \widetilde{D}$,
\[
|f(t,u,v_2)- f(t,u,v_1)|\leq N|v_2 - v_1|;
\]

\item[(H5$'$)] for all $(t,u,v)\in \widetilde{D},|f(t,u,v)|\leq \varphi(|v|)$;
 where $\varphi:\mathbb{R}^+ \to \mathbb{R}^+$ is continuous and satisfies
\[
\max_{t\in[0,1]}\beta-\min_{t\in[0,1]}\alpha 
\leq  \int_{l_{0}}^\infty \frac{s\ ds}{\varphi(s)}
\]
where $l_0=\sup_{[0,1]}[2|\beta(t)|]$.
\end{itemize}
Let $\lambda<0$ be such that 
$\lambda \leq \min\{-M,-\frac{M}{1-N},-M-\frac{N^2}{2}-\frac{N}{2}\sqrt{N^2+4M}\}$,
and for all $t\in[0,1]$,
\[
f(t,\beta(t),\beta'(t))- f(t,\alpha(t),\alpha'(t))-\lambda(\beta-\alpha)\geq 0
\]
then the sequences $(\alpha_n)$ and $(\beta_n)$ defined by \eqref{3.1}-\eqref{3.2} 
and \eqref{3.3}-\eqref{3.4} converges monotonically in $C^1([0,1])$ 
to solution $v$ and $u$ of \eqref{1.1}-\eqref{1.2}, such that for all 
$t\in[0,1]$, $\alpha \leq v \leq u \leq \beta$.
\end{theorem}

\section{Numerical illustrations}

In this section we consider two examples and verify that conditions derived 
in this paper can actually be verified and existence of solutions can be proved.

\begin{example} \rm
Consider the nonlinear three-point BVPs given by
\begin{gather}
\label{Ex1.1} -y''(t)=\frac{10y^3-9e^{y'}}{90},\quad 0<t<1,\\
\label{Ex1.2}  y'(0)=0,\quad y'(1)=0.6 y'(0.9).
\end{gather}
where $f(t,y,y')=\frac{10y^3-9e^{y'}}{90}$, $\delta=0.6$, $\eta =0.9$. 
Here $\alpha= 1$ and $\beta= -1$ are lower and upper solutions. 
It is a non well ordered case.
\end{example}

The priory bound can be computed as follows. 
$\varphi=(10+9e^{|v|})/90$. $|y'|\leq \sqrt{\frac{1}{5}}$;
 i.e., $R=\sqrt{1/5}$.
The Lipschitz constants are computed as $M=1/3$ and 
$N=e^R/10=0.156395$.

The inequality $f(t,\beta(t),\beta'(t))- f(t,\alpha(t),\alpha'(t))
-\lambda(\beta - \alpha)\geq 0$ is satisfied when $\lambda\geq 0.11111$.
Now we can find out a subinterval $(0.44,1.8)\subset (0.11111, 2.4674)$
 (approx) such that 
$(\lambda-M)\cos\sqrt{\lambda}-N\sqrt{\lambda}\sin\sqrt{\lambda}\geq 0$ 
and inequalities in (H0) are satisfied (cf. Figure \ref{Fig-1}).

Thus it is guaranteed that $\exists$ at least one $\lambda\in(0.44,1.8)$ 
such that sequences generated by iterative scheme converge uniformly to a
solution of the nonlinear three point boundary value problem \eqref{Ex1.1} 
and \eqref{Ex1.2}.

\begin{figure}[htb]
\begin{center}
  \includegraphics[width=8cm]{fig1}  % Fig-Example-1}
\end{center}
\caption{Plots of inequalities (H0) and $(\lambda-M)\cos\sqrt{\lambda}
-N\sin\sqrt{\lambda }\geq0$ for $\lambda\in(0,\pi^2/4)$}
\label{Fig-1} 
\end{figure}

\begin{example} \rm
Consider the nonlinear three-point BVPs
\begin{gather}
\label{Ex2.1} -y''(t)=\frac{[y'(t)]^2}{60}-5y(t)-\frac{e^2}{18},\\
\label{Ex2.2}  y'(0)=0,\quad y'(1)=0.7 y'(0.5).
\end{gather}
where $f(t,y,y')=\frac{[y'(t)]^2}{60}-5y(t)-\frac{e^2}{18}$,
$\delta=7/10$, $\eta =\frac{1}{2}$.
\end{example}

This example has $\alpha= -(t^2+\frac{1}{2})$ and $\beta= (t^2+\frac{1}{2})$ 
as lower and upper solutions; i.e., we are in well ordered case.
The priory bound can be computed as follows. 
$\varphi=  \frac{15}{2}+\frac{e^2}{18}+\frac{|v|^2}{60}$,
$|y'|\leq 2e^{\frac{3}{32}}$; i.e., $R=3e^{\frac{1}{20}}$.
The Lipschitz constants are give by $M=5$ and $N=R/30=0.105127$.

Now we can find out at least one $\lambda<0$ such that when 
$$
\lambda \leq \min\big\{-M,-\frac{M}{1-N},-M-\frac{N^2}{2}
-\frac{N}{2}\sqrt{N^2+4M}\big\},
$$
Assumption  (H0$'$) is satisfied (cf. Figure \ref{Fig-2}),
 we get two monotonic sequences.

Thus for any $\lambda<-5.58739$ the solutions of the iterative scheme
 converge uniformly to the solution of the nonlinear three point boundar- value 
problem \eqref{Ex2.1} and \eqref{Ex2.2}.

\begin{figure}[htbp]
\begin{center}
   \includegraphics[width=8cm]{fig2} % Fig-Example-2}\\
\end{center}
  \caption{Plots of inequalities (H0$'$) for $\lambda<0$}
\label{Fig-2}
\end{figure}

\subsection{Acknowledgements}
This work was partially supported by grant provided by UGC, New Delhi,
India File No. F.4-1/2006 (BSR)/7-203/2009(BSR).

\begin{thebibliography}{99}

\bibitem{aftabizadeh} A. R. Aftabizadeh, C. P. Gupta, Jian-Ming Xu;
\emph{Existence and uniqueness theorems for three-point boundary value problem},
SIAM J. Math. Anal., 20 (1989) 716--726.

\bibitem{bao} G. Bao, X. Xu, Y. Song;
\emph{Positive solutions for three-point boundary value problems with a
non-well-ordered upper and lower solution condition},
Appl. Math. Lett., 25 (2012) 767--770.

\bibitem{decoster} C. De Coster, P. Habets;
\emph{Two-point boundary value problems: lower and upper solutions},
Elsevier Science \& Technology, 2006.

\bibitem{gregus} M. Gregus, F. Neumann, F. M. Arscott;
\emph{Three-point boundary value problems for differential equations},
 J. London Math. Soc., 3 (1971) 429--436.

\bibitem{guo} Y. Guo, W. Ge;
\emph{Positive solutions for three-point boundary value problems with
dependence on the first order derivative}, J. Math. Anal. Appl., 290 (2004) 291--301.

\bibitem{henderson} J. Henderson, B. Karna, C. C. Tisdell;
\emph{Existence of solutions for three-point boundary value problems
for second order equations}, Proc. Amer. Math. Soc., 133 (2005) 1365--1369.

\bibitem{lazer} A. C. Lazer, P. J. McKenna;
\emph{Large-Amplitude Periodic Oscillations in Suspension Bridges:
Some New Connections with Nonlinear Analysis},
SIAM Review, Vol. 32, No. 4. (1990) 537--578.

\bibitem{li} F. Li, M. Jia, X. Liu, C. Li, G. Li;
\emph{Existence and uniqueness of solutions of second-order three-point
boundary value problems with upper and lower solutions in the reversed order},
Nonlinear Anal., 68 (2008) 2381--2388.

\bibitem{ma} R. Ma;
\emph{Existence of solutions of nonlinear m-point boundary-value problems},
J. Math. Anal. Appl., 256 (2001) 556--567.

\bibitem{pandey1} R. K. Pandey,  A. K. Verma;
\emph{Existence-uniqueness results for a class of singular boundary value problems-II},
 J. Math. Anal. Appl., 338 (2008) 1387--1396.

\bibitem{pandey2} R. K. Pandey,  A. K. Verma;
\emph{Existence-uniqueness results for a class of singular boundary value
problems arising in physiology}, Nonlinear Analysis: Real World Applications, 9 (2008) 40--52.

\bibitem{singh1} M. Singh, A. K. Verma;
\emph{On a monotone iterative method for a class of three point nonlinear
nonsingular BVPs with upper and lower solutions in reverse order},
J Appl. Math. Comput., 43 (2013) 99--114.

\bibitem{singh2} M. Singh, A. K. Verma;
\emph{Picard Type Iterative Scheme with Initial Iterates in Reverse Order for
a Class of Nonlinear Three Point BVPs},
International Journal of Differential Equations, Volume 2013,
Article ID 728149, 6 pages.

\bibitem{verma1} A. K. Verma, R. P. Agarwal;
\emph{Upper and lower solutions method for regular singular differential 
equations with quasi-derivative boundary conditions}, 
Commun. Nonlin. Sci. Numer. Simulat., 17 (2012) 4551--4558.

\bibitem{verma2} A. K. Verma, R. P. Agarwal, Lajja Verma;
\emph{Bessel functions and singular BVPs arising in physiology in the 
presence of upper and lower solutions in reverse order}, 
J. Appl. Math. Comput., 39 (2012) 445--457.

\bibitem{webb} J. R. L. Webb;
\emph{Existence of positive solutions for a thermostat model},
Nonlinear Analysis: Real World Applications, 13 (2012) 923--938.

\bibitem{wei} Z. Wei, C. Pang;
\emph{The method of lower and upper solutions for fourth order singular
m-point boundary value problems}, J. Math. Anal. Appl., 322 (2006) 675--692.

\end{thebibliography}

\end{document}