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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2012 (2012), No. 04, pp. 1--10.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2012 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2012/04\hfil Monotone iterative method]
{Monotone iterative method and regular singular
nonlinear BVP in the presence of reverse ordered upper
and lower solutions}

\author[A. K. Verma\hfil EJDE-2012/04\hfilneg]
{Amit K. Verma}

\address{Amit K. Verma \newline
Department of Mathematics, BITS Pilani
Pilani - 333031, Rajasthan, India\newline
Phone +919413789285; fax: +911596244183}
\email{amitkverma02@yahoo.co.in,  akverma@bits-pilani.ac.in}

\thanks{Submitted October 19, 2011. Published January 9, 2012.}
\subjclass[2000]{34B16}
\keywords{Monotone iterative technique;
 lower and upper solutions; \hfill\break\indent
Neumann boundary conditions}

\begin{abstract}
 Monotone iterative technique is employed for studying the existence
 of  solutions to the second-order nonlinear singular boundary
 value problem
 $$
 -\big(p(x)y'(x)\big)'+p(x)f\big(x,y(x),p(x)y'(x)\big)=0
 $$
 for $0<x<1$ and $y'(0)=y'(1)=0$. Here $p(0)=0$ and
 $x p'(x)/p(x)$ is analytic at $x=0$. The source function
 $f(x,y,py')$ is Lipschitz in $py'$ and one sided Lipschitz in $y$.
 The initial approximations are upper solution $u_0(x)$ and lower
 solution $v_0(x)$ which can be ordered in one way
 $v_0(x)\leq u_0(x)$ or the other $u_0(x)\leq v_0(x)$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}\label{intro}

Recently, there have been a lot of activity as far as upper and lower
solutions technique is considered (see
\cite{DOR-MAEG-IMAJAM-2008,MC-CDC-PH-AMC-2001,AKV-NATMA-2011} and
the references therein). In most of the results upper and
solutions are well ordered; i.e., $u_0(x)\geq v_0(x)$. But
literature is not rich for the case of reverse ordered upper and
lower solutions; i.e., $u_0(x)\leq v_0(x)$. Though results are
available for nonsingular  boundary value problems
$(p(x)$=constant$)$ but singular boundary value problems require
attention. The details of the work done for the nonsingular
problem when upper and lower solutions are in reverse order can be
seen in \cite{MC-CDC-PH-AMC-2001} and the references therein. To
fill this gap in our recent result (\cite{AKV-NATMA-2011}) we
consider the singular boundary value problem of the form
$-\big(x^{\alpha}y'(x)\big)'+x^{\alpha}
f\big(x,y(x),x^{\alpha}y'(x)\big)=0$, $\alpha\geq1$
for $0<x<1$ and $y'(0)=y'(1)=0$ with upper and lower solution in
one order ($u_0\geq v_0$) or the other ($u_0\leq v_0$). We prove
the existence of the solutions under quite general conditions.
This problem is simple and its advantage is that for the
corresponding linear problem we obtain the solutions in terms of
Bessel functions. Bessel functions have some in built simplicity
which helps us in proving the results very easily. In the present
paper we consider a generalized problem
\begin{equation}\label{1.3}
-\big(p(x)y'(x)\big)'+p(x)f\big(x,y(x),p(x)y'(x)\big)=0, \quad 0<x<1.
\end{equation}
We replace the term $x^\alpha$ in the boundary value problem considered
in \cite{AKV-NATMA-2011} with a general function $p(x)$.
So in built simplicity due to Bessel functions is not there
with us in this paper. Here the source function $f(x,y,py')$
is derivative dependent and boundary conditions are
again of Neumann type and is written as
\begin{equation} \label{1.4}
y'(0)=0,\quad y'(1)=0.
\end{equation}
Let $p(x)$ satisfy the following conditions:
\begin{itemize}
\item[(A1)] \begin{itemize}
\item[(i)] $p(0)=0$ and $p>0$ in $(0,1)$.
\item[(ii)] $p\in C[0,1]\cap C^1(0,1)$ and
\item[(iii)] for some $r>1$, $x\frac{p'(x)}{p(x)}$ is analytic in $\{z:|z|<r\}$.
\item[(iv)] $\int_0^1\frac{dt}{p(t)}=\infty$.
\end{itemize}
\end{itemize}
In this work we consider a computationally simple iterative
scheme defined by
\begin{gather}\label{1.5}
-\big(p(x)y_{n}'(x)\big)'+\lambda p(x)y_{n}(x)
=-p(x)f\big(x,y_{n-1},p y_{n-1}'\big)+\lambda p(x)y_{n-1}(x)\\
\label{1.6} y_n'(0)=0,\quad y_n'(1)=0.
\end{gather}
The main aim of this work is to extend our earlier work
in \cite{AKV-NATMA-2011}.
Now we do not know the solutions explicitly in terms of
Bessel functions. Instead we have singular linear boundary-value
problems which we have to analyze and properties of the
solutions is to be extracted. We have arranged the paper in 4 sections.
In Section \ref{PR} we discuss some elementary results, e.g.,
maximum principles and existence of two differential inequalities.
Then using these elementary results we establish existence results
for well ordered upper and lower solutions in Section \ref{wouls}
and for reverse ordered upper and lower solutions in
Section \ref{rouls}. We conclude this article
 with some remarks.

\section{Preliminaries}\label{PR}

Let $h(x)\in C[0,1]$ and $\lambda\in \mathbb{R}_0$,
$\mathbb{R}_0=\mathbb{R}\setminus \{0\}$,
$A\in \mathbb{R}$ and $B\in \mathbb{R}$. Now, consider
the  class of linear singular problems
\begin{gather}\label{2.1}
-\left(p(x)y'(x)\right)'+\lambda p(x)y(x)= p(x)h(x),\quad 0<x<1,\\
\label{2.2} y'(0)=A,\quad y'(1)=B.
\end{gather}
The corresponding homogeneous system (eigenvalue problem) is given by
\begin{gather}\label{2.3}
-\big(p(x)y'(x)\big)'+\lambda p(x)y(x)=0,\quad 0<x<1,\\
\label{2.4} y'(0)=0,\quad y'(1)=0.
\end{gather}

\begin{remark}\label{RM-1} \rm
Since $xp'/p$ is analytic at $x=0$, the point $x=0$ is a regular
singular point of \eqref{2.3}. Thus using Frobenius series method
two linearly independent solutions can be computed (see
\cite[Lemma 2]{RKP-JDE-1996,RKP-JMAA-1997}). If $p=x^\alpha$ these
Frobenius series solutions can be written in terms of Bessel
functions.

It is easy to verify that all the eigenvalues of the
 Sturm-Liouville problem \eqref{2.3}--\eqref{2.4} are real,
negative and simple.
\end{remark}

The solution of the nonhomogeneous problem \eqref{2.1}-\eqref{2.2}
can be written as
\begin{equation} \label{2.11}
\begin{split}
w(x)
&=z_1(x)\Big[\int_0^x\frac{p(t)h(t)z_0(t)}{W_{p}(z_1,z_0)}dt
+\frac{A}{z_1'(0)}\Big]\\
&\quad +z_0(x)\Big[\int_x^1\frac{p(t)h(t)z_1(t)}{W_{p}(z_1,z_0)}dt
+\frac{B}{z_0'(1)}\Big]
\end{split}
\end{equation}
where $z_0(x,\lambda)$ is the solution of
\begin{equation} \label{2.12}
\begin{gathered}
-\big(p(x)z_0'(x)\big)'+\lambda p(x)z_0(x)=0,\quad 0<x<1,\\
 z_0(0)=1,\quad z_0'(0)=0,
\end{gathered}
\end{equation}
and $z_1(x,\lambda)$ is
the solution of
\begin{equation} \label{2.13}
\begin{gathered}
-\big(p(x)z_1'(x)\big)'+\lambda p(x)z_1(x)=0,\quad 0<x<1,\\
 z_1(1)=1,\quad z_1'(1)=0
\end{gathered}
 \end{equation}
and $W_{p}(z_1,z_0)=p(t)\left(z_1z_0'-z_1'z_0\right)$. By replacing
$x$ with $1-x$ in \eqref{2.12} it is easy to verify that
$$z_1(x)=z_0(1-x)$$ for both positive and negative values of
$\lambda$.

\begin{remark}\label{RM0} \rm
We have $z_0$ and $z_1$ as two linearly independent solutions
(Frobenius Series Solution) of \eqref{2.3}. Eigenvalues of the
eigenvalue problem \eqref{2.3}--\eqref{2.4} are the zeros of
$z'_0(1,\lambda)$. $z_0'(1,\lambda)$ is an analytic function of
$\lambda$ so its zeros are isolated and they all will be negative.
Let them be $-\lambda_0,-\lambda_1,-\lambda_2,\dots$ such that
$\lambda_i>0$ for $i=0,1,2,\dots$. Where $-\lambda_0$ is the
first negative zero of $z_0'(1,\lambda)$ or in other words first
negative eigenvalue of \eqref{2.3}--\eqref{2.4}.

Since $z_0(x,\lambda)$ does not change its sign for
$-\lambda_0<\lambda<0$ and $z_0(0,\lambda)=1$ therefore
$z_0(x,\lambda)>0$ for all $x\in[0,1]$ and for all
$-\lambda_0<\lambda<0$.
\end{remark}

\begin{remark}\label{RM1}\rm
Using \eqref{2.12} and the fact that $z_1(x)=z_0(1-x)$ it is easy
to prove that if $\lambda>0$ then for all $x\in(0,1]$, $z_0(x)>1$
and $z'_0(x)>0$ and for all $x\in[0,1)$ we have $z_1(x)>1$ and
$z_1'(x)<0$.
\end{remark}

\begin{remark}\label{RM2}\rm
Using Remark \ref{RM0}, $z_1(x)=z_0(1-x)$ and the differential
equation \eqref{2.13} it is easy to prove that if
$-\lambda_0<\lambda<0$ for all $x\in[0,1)$, $z_0(x)>0$ and
$z'_1(x)>0$ and for all $x\in(0,1]$ we have $z_0'(x)<0$ and
$z_1(x)>0$.
\end{remark}

\begin{remark}\label{R1} \rm
Let $\lambda>0$ and $h\in C[0,1]$. If $h\geq0$ $(or~h\leq0)$ then
\[
\int_{0}^{x}\frac{p(t)h(t)z_0(t)}{W_{p}(z_1,z_0)}dt
\quad\text{and}\quad
\int_x^1 \frac{p(t)h(t)z_1(t)}{W_{p}(z_1,z_0)}dt
\]
are non-negative (or non-positive).
\end{remark}

\begin{remark}\label{R2} \rm
Let $-\lambda_0<\lambda<0$ and $h\in C[0,1]$. If $h\geq0$
(or $h\leq0$) then
\[
\int_{0}^{x}\frac{p(t)h(t)z_0(t)}{W_{p}(z_1,z_0)}dt\quad\text{and}\quad
\int_x^1\frac{p(t)h(t)z_1(t)}{W_{p}(z_1,z_0)}dt
\]
are non-positive (or non-negative).
\end{remark}

\begin{proposition}[Maximum~Principle]\label{MP}
Let $\lambda>0$. If $A\leq 0$, $B\geq0$ (or $A\geq 0$, $B\leq0$)
and $h\in C[0,1]$ such that $h\geq0$ (or $h\leq0$),
then $w(x)\geq0$ (or $w(x)\leq0$), where $w(x)$ is the solution
of \eqref{2.1}-\eqref{2.2}.
\end{proposition}

\begin{proposition}[Anti-maximum~Principle]\label{AMP}
 Let $-\lambda_0<\lambda<0$. If $A\leq 0$, $B\geq0$
(or $A\geq 0$, $B\leq0$) and $h\in C[0,1]$ such that $h\geq0$
(or $h\leq0$), then $w(x)\leq0$ (or $w(x)\geq0$),
 where $w(x)$ is the solution of \eqref{2.1}-\eqref{2.2}.
\end{proposition}

Now we derive conditions on $\lambda$ which will help us to prove
the monotonicity of the solutions generated by iterative scheme
\eqref{1.5}-\eqref{1.6}.

\begin{lemma}\label{DI1}
Let $M$ and $N$ $\in\mathbb{R}^{+}$. If $\lambda>0$ such that
\[
\lambda\geq M \Big(1-N\int_0^1p(t)dt\Big)^{-1},
\]
then for all $x\in [0,1]$,
\begin{equation} \label{2.14}
(M-\lambda)z_0(x)+Np(x)z'_0(x)\leq0.
\end{equation}
\end{lemma}

\begin{proof}
 Integrating \eqref{2.12} from $0$ to $x$ and using
 that $z_0'(x)>0$ in $(0,1]$ we obtain
 \[
p(x)z_0'(x)\leq \lambda z_0(x)\int_0^1p(t)dt.
\]
Therefore we obtain
$(M-\lambda)z_0(x)+Np(x)z'_0(x)\leq (M-\lambda)z_0+N\lambda
z_0(x)\int_0^1p(t)dt$. Hence \eqref{2.14} will hold if
$(M-\lambda)+N\lambda\int_0^1p(t)dt\leq 0$. Hence the result.
\end{proof}

\begin{lemma}\label{DI2}
Let $M$ and $N$ $\in\mathbb{R}^{+}$. If $-\lambda_0<\lambda<0$ is
such that
$$
-\Big(\int_0^1\frac{1}{p(x)}\int_0^xp(t)\,dt\,dx\Big)^{-1}<\lambda<-M
$$
and
\[
(M+\lambda)\Big(1+\lambda\int_0^1\frac{1}{p(x)}\int_0^xp(t)\,dt\,dx\big)
-N\lambda \int_0^1p(x)dx\leq0
\]
then for all $x\in [0,1]$,
\begin{equation} \label{2.15}
(M+\lambda)z_0(x)-Np(x)z'_0(x)\leq0.
\end{equation}
\end{lemma}

\begin{proof}
Using \eqref{2.12} and Remark \ref{RM2} it can be
deduced that $z_0(x)$ and $p(x)z_0'(x)$ are decreasing functions
of $x$ for $-\lambda_0<\lambda<0$, thus
\[
(M+\lambda)z_0(x)-Np(x)z'_0(x)\leq (M+\lambda)z_0(1)-Np(1)z'_0(1).
\]
Now using \eqref{2.12} we obtain
\[
-p(1)z_0'(1)\leq
(-\lambda)\int_0^1p(x)dx\quad\text{and}\quad
z_0(1)>1+\lambda \int_0^1\frac{1}{p(x)}\int_0^xp(t)\,dt\,dx.
\]
 Which completes the proof.
\end{proof}

\section{Well-ordered upper and lower solutions}\label{wouls}

Let us define upper and lower solutions:

A function $u_0\in C^2[0,1]$ is an upper solution of
\eqref{1.3}-\eqref{1.4} if
\begin{equation} \label{3.1}
\begin{gathered}
-(pu_0')'+p(x)f(x,u_0,pu_0')\geq0,\quad 0<x<1;\\
u_0'(0)\leq 0\leq u_0'(1).
\end{gathered}
\end{equation}
A function $v_0\in C^2[0,1]$ is a lower solution of
\eqref{1.3}-\eqref{1.4} if
\begin{equation} \label{3.2}
\begin{gathered}
-(pv_0')'+p(x)f(x,v_0,pv_0')\leq0,\quad 0<x<1;\\
v_0'(0)\geq0\geq v_0'(1).
\end{gathered}
\end{equation}

Now, for every $n$, problem \eqref{1.5}-\eqref{1.6} has a
unique solution $y_{n+1}$ given by \eqref{2.11} with $h(x)=
-f(x,y'_n,py'_n)+\lambda y_n$, $A=0$ and $B=0$.

In this section we show that for the proposed scheme
\eqref{1.5}-\eqref{1.6} a good choice of $\lambda$ is possible so
that the solutions generated by the approximation scheme converge
monotonically to solutions of \eqref{1.3}-\eqref{1.4}. We require
a number of results.

\begin{lemma}\label{U-L-1}
Let $\lambda>0$. If $u_n$ is an upper solution of
\eqref{1.3}-\eqref{1.4} and $u_{n+1}$ is defined by
\eqref{1.5}-\eqref{1.6} then $u_{n+1}\leq u_{n}$.
\end{lemma}

\begin{proof}
Let $w_{n}=u_{n+1}-u_{n}$, then
\begin{gather*}
-(pw_{n}')'+\lambda pw_{n} =(pu'_n)'-pf(x,u_n,pu'_n)\leq0,\\
w_{n}'(0)\geq0,\quad w_{n}'(1)\leq0,
\end{gather*}
and using Proposition \ref{MP} we have $u_{n+1}\leq u_n$.
\end{proof}

For the next proposition we use the following assumptions:
\begin{itemize}
\item[(H1)] there exists upper solution $(u_0)$ and lower
solution $(v_0)$ in $C^2[0,1]$ such that $v_0\leq u_0$
for all $x\in[0,1]$;

\item[(H2)] the function $f:D\to \mathbb{R}$ is continuous
on
\[
D:=\{(x,y,py')\in[0,1]\times R\times R: v_0\leq y\leq u_0\};
\]

\item[(H3)] there exists $M\geq0$ such that for all
$(x,\tau,pv'), (x,\sigma,pv')\in D$,
\[
f(x,\tau,pv')-f(x,\sigma,pv')\geq M(\tau-\sigma),\quad
(\tau\leq \sigma);
\]

\item[(H4)] there exist $N\geq0$ such that for all
$(x,u_,pv'_1),(x,u,pv'_2)\in D$,
\[
|f(x,u,pv'_1)-f(x,u,pv'_2)|\leq N|pv'_2-pv'_1|.
\]
\end{itemize}

\begin{proposition}\label{U-L-2}
Assume {\em (H1)--(H4)}, and let $\lambda>0$ such that
$$
\lambda\geq M \big(1-N\int_0^1p(t)dt\big)^{-1}.
$$
Then the functions $u_{n+1}$
defined recursively by \eqref{1.5}-\eqref{1.6} are such that for
all $n\in \mathbb{N}$,
\begin{itemize}
\item[(i)] $u_n$ is an upper solution of \eqref{1.3}-\eqref{1.4}.
\item[(ii)] $u_{n+1}\leq u_{n}$.
\end{itemize}
\end{proposition}

\begin{proof}
We prove the claims by the principle of mathematical induction.
Since $u_0$ is an upper solution and by Lemma \ref{U-L-1}
$u_0\geq u_1$, therefore both the claims are true for $n=0$.

Further, let the claims be true for $n-1$; i.e., $u_{n-1}$ is
an upper solution and $u_{n-1}\geq u_{n}$. Now we are required
to prove that $u_{n}$ is an upper solution and $u_{n+1}\leq u_{n}$.
To prove this let $w=u_n-u_{n-1}$, then we have
\[
-(pu_n')'+pf(x,u_n,pu_n')\geq  p[(M-\lambda)w-N
(\operatorname{sign}w') pw'].
\]
Thus to prove that $u_n$ is an upper solution we require to prove that
\begin{equation} \label{ineq}
(M-\lambda)w-N (\operatorname{sign}w') pw'\geq 0.
\end{equation}
Now, since $w$ satisfies
\[
-(pw')'+\lambda pw=(pu'_{n-1})'-pf(x,u_{n-1},pu_{n-1}')\leq 0,
\quad w'(0)\geq0,\; w'(1)\leq0,
\]
from Proposition \ref{MP} we have $w\leq0$ for $\lambda>0$. Now,
putting the value of $w$ from \eqref{2.11} in \eqref{ineq} and in
view of $h=(pu'_{n-1})'-pf(x,u_{n-1},pu_{n-1}')\leq 0$ we deduce
that to prove \eqref{ineq} it is sufficient to prove that
\[
(M-\lambda)z_0-N(sign~w')pz'_0\leq0
\]
and
\[
(M-\lambda)z_1-N(sign~w')pz'_1\leq0
\]
for all $x\in[0,1]$. Since $z_1=z_0(1-x)$, using Remark \ref{R1},
above inequalities will be true if for all $x\in[0,1]$ we have
\[
(M-\lambda)z_0(x)+N p(x)z_0'(x)\leq0,
\]
and which is true (by Lemma \ref{DI1}). Therefore \eqref{ineq}
holds and hence $u_n$ is an upper solution.

Now applying Lemma \ref{U-L-1} we deduce that $u_{n+1}\leq u_n$.
This completes the proof.
\end{proof}

Similarly we can prove the following two results (Lemma \ref{L-L-1},
Proposition \ref{L-L-2}) for lower solutions.

\begin{lemma}\label{L-L-1}
Let $\lambda>0$. If $v_n$ is a lower solution of
\eqref{1.3}-\eqref{1.4} and $v_{n+1}$ is defined by
\eqref{1.5}-\eqref{1.6} then $v_{n}\leq v_{n+1}$.
\end{lemma}

\begin{proposition}\label{L-L-2}
Assume that {\rm (H1)--(H4)} hold and let
$\lambda>0$ be such that
$\lambda\geq M \big(1-N\int_0^1p(t)dt\big)^{-1}$.
Then the functions $v_{n+1}$ defined recursively by
\eqref{1.5}-\eqref{1.6} are such that for
all $n\in \mathbb{N}$,
\begin{itemize}
\item[(i)] $v_n$ is a lower solution of \eqref{1.3}-\eqref{1.4}.
\item[(ii)] $v_{n}\leq v_{n+1}$.
\end{itemize}
\end{proposition}

In the next result we prove that upper solution $u_n$ is larger
than lower solution $v_n$ for all $n$.

\begin{proposition}\label{U-L-L-WO}
Assume that {\rm (H1)--(H4)} hold and let $\lambda>0$ such that
$\lambda\geq M \big(1-N\int_0^1p(t)dt\big)^{-1}$ and for all
$x\in [0,1]$
\[
f(x,v_0,pv_0')-f(x,u_0,pu_0')+\lambda(u_0-v_0)\geq0.
\]
Then for all $n\in\mathbb{N}$, the functions $u_n$ and $v_n$
defined recursively by \eqref{1.5}-\eqref{1.6} satisfy
$v_n\leq u_n$.
\end{proposition}

\begin{proof}
We define a function
\[
h_i(x)=f(x,v_ipv'_i)-f(x,u_i,pu'_i)+\lambda (u_i-v_i),\quad
i\in\mathbb{N}.
\]
It is easy to see that for all $i\in\mathbb{N}$, $w_i=u_i-v_i$
satisfies the  differential equation
\begin{align*}
-(pw_i')'+\lambda p w_i
&=p\{f(x,v_{i-1},pv'_{i-1})-f(x,u_{i-1},pu'_{i-1})
 +\lambda (u_{i-1}-v_{i-1})\}\\
&= p h_{i-1}\,.
\end{align*}
To prove this proposition  we use again the principle of
mathematical induction. For $i=1$ we have $h_0\geq0$ and $w_1$ is
the solution of \eqref{2.1}-\eqref{2.2} with $A=0$ and $B=0$.
Using Proposition \ref{MP} we deduce that $w_1\geq0$; i.e.,
$u_1\geq v_1$.

Now, let $n\geq2$, $h_{n-2}\geq0$ and $u_{n-1}\geq v_{n-1}$,
then we are required to prove that $h_{n-1}\geq0$ and
$u_{n}\geq v_{n}$. First we show that for all $x\in [0,1]$
the function $h_{n-1}$ is non-negative. Indeed we have
\begin{align*}
h_{n-1}&= f(x,v_{n-1},pv'_{n-1})-f(x,u_{n-1},pu'_{n-1})
 +\lambda (u_{n-1}-v_{n-1})\\
&\geq -[(M-\lambda)w_{n-1}+N(sign~w'_{n-1})pw'_{n-1}].
\end{align*}
Here $w_{n-1}$ is a solution of \eqref{2.1} with
$h(x)=h_{n-2}\geq0$, $A=0$ and $B=0$. Arguments similar to the
Proposition \ref{U-L-2} can be used to prove that $h_{n-1}\geq0$.
Now, we have $h_{n-1}\geq0$, $w'_n(0)=0$ and $w'_{n}(1)=0$ thus
from Proposition \ref{MP} we deduce that $w_n\geq0$, i.e.,
$u_n\geq v_n$.

\end{proof}
For the next lemma we use the assumption
\begin{itemize}
\item[(H5)] For all $(x,u,pu')\in D$, $|f(x,u,pu')|\leq \varphi(|pu'|)$
where  $\varphi:[0,\infty)\to(0,\infty)$ is continuous and
satisfies
\[
\int_{0}^{\infty}\frac{ds}{\varphi(s)}>\int_0^1p(x)dx.
\]
\end{itemize}

\begin{lemma}\label{U-L-3}
If $f(x,u,pu')$ satisfies {\rm(H1), (H2), (H5)},
then there exists $R_0>0$ such that any solution of
\[
-(pu')'+pf(x,u,pu')\geq0,\quad 0<x<1,\quad u'(0)=0=u'(1)
\]
with $u\in[v_0,u_0]$ for all $x\in[0,1]$, satisfies
$\| pu'\|_{\infty}< R_0$.
\end{lemma}

\begin{proof}
Consider an interval $[x,x_0]\subset[0,1]$ such that
for all $s\in [x,x_0)$,
\[
 u'(s)<0 \quad \text{and}\quad u'(x_0)=0.
\]
Now using (H5) we have
\[
(pu')'\leq p\varphi(|pu'|)
\]
and after integrating it from $x$ to $x_0$ and using $(H5)$ we
have
$-pu'\leq R_0$.
Similarly for the interval $[x_0,x]$ we have
$pu'\leq R_0 $.
Thus $\|pu'\|_{\infty}\leq R_0$.
\end{proof}

In the same way we can prove the following result for lower solutions.

\begin{lemma}\label{L-L-3}
If $f(x,v,pv')$ satisfies {\rm (H1), (H2), (H5)},
 then there exists $R_0>0$ such that any solution of
\[
-(pv')'+pf(x,v,pv')\leq0,\quad 0<x<1,\quad v'(0)=0=v'(1)
\]
with $v\in[v_0,u_0]$ for all $x\in[0,1]$, satisfies
$\| pv'\|_{\infty}< R_0$.
\end{lemma}

Now we are in a situation to prove the our final result for
the case when upper and lower solutions are well ordered.

\begin{theorem}\label{WOT}
Assume {\rm (H1)--(H5)} hold. Let $\lambda>0$ be such that
\[
\lambda\geq M \Big(1-N\int_0^1p(t)dt\Big)^{-1}
\]
 and for all the $x\in [0,1]$,
\[
f(x,v_0,pv_0')-f(x,u_0,pu_0')+\lambda(u_0-v_0)\geq0.
\]
Then the sequences $u_n$ and $v_n$ defined by
\eqref{1.5}--\eqref{1.6} converge monotonically to solutions
$\widetilde{u}(x)$ and $\widetilde{v}(x)$ of
\eqref{1.3}-\eqref{1.4}. Any solution $z(x)$ of
\eqref{1.3}-\eqref{1.4} in $D$ satisfies
\[
\widetilde{v}(x)\leq z(x)\leq \widetilde{u}(x).
\]
\end{theorem}

\begin{proof}
Using Lemma \ref{U-L-1} to Lemma \ref{L-L-3} and
Proposition \ref{U-L-2} to Proposition \ref{U-L-L-WO}
we deduce that sequences $\{u_n\}$ and $\{v_n\}$ are monotonic
$(u_0\geq u_1\geq u_2\dots \geq u_n\geq v_n\dots
 \geq v_2\geq v_1\geq v_0)$ and are bounded by $v_0$ and
$u_0$ in $C[0,1]$ and by Dini's theorem they converge
uniformly to $\widetilde{u}$ and $\widetilde{v}$ (say).
We can also deduce that the sequences $\{pu'_n\}$ and
$\{pv'_n\}$ are uniformly bounded and equi-continuous in
$C[0,1]$ and by Arzela-Ascoli theorem there exists uniformly
convergent subsequences $\{pu'_{n_{k}}\}$ and
$\{pv'_{n_{k}}\}$ in $C[0,1]$. It is easy to observe
that $u_n\to \widetilde{u}$ and $v_n \to \widetilde{v}$
implies $pu'_n\to p\widetilde{u}'$ and
$p\widetilde{v}'_n\to p\widetilde{v}'$.

Solution of \eqref{1.5}-\eqref{1.6} is given by \eqref{2.11} where
$h(x)=-pf(x,y_{n-1},py_{n-1}')+\lambda py_{n-1}$. Since
the sequences are uniformly convergent taking limit as
$n\to \infty$ we obtain $\widetilde{u}$ and $\widetilde{v}$
as the solutions of the nonlinear boundary value problem
\eqref{1.3}-\eqref{1.4}. Any solution $z(x)$ in $D$ plays the role
of $u_0$. Hence $z(x)\geq \widetilde{v}(x)$. Similarly one
concludes that $z(x)\leq \widetilde{u}(x)$.
\end{proof}

\begin{remark} \rm
When the source function is derivative independent; i.e., $N=0$.
In this case we can choose $\lambda=M$.
\end{remark}

\section{Upper and lower solutions in reverse order}\label{rouls}

In this section we consider the case when the upper and lower
solutions are in reverse order; i.e., $u_0(x)\leq v_0(x)$.
For this we require opposite one-sided Lipschitz condition
and we assume that
\begin{itemize}
\item[(F1)] there exists upper solution ($u_0$) and lower solution
($v_0$) in $C^2[0,1]$ such that $u_0\leq v_0$ for all $x\in[0,1]$;

\item[(F2)] the function $f:D_0\to \mathbb{R}$ is continuous on
\[
D_0:=\{(x,y,py')\in[0,1]\times R\times R: u_0\leq y\leq v_0\};
\]

\item[(F3)] there exists $M\geq0$ such that for all
$(x,\widetilde{\tau},pv'),~(x,\widetilde{\sigma},pv')\in D_0$,
\[
f(x,\widetilde{\sigma},pv')-f(x,\widetilde{\tau},pv')
\geq -M(\widetilde{\sigma}-\widetilde{\tau}),
\quad (\widetilde{\tau}\leq \widetilde{\sigma});
\]

\item[(F4)] there exist $N\geq0$ such that for all
$(x,u_,pv'_1), (x,u,pv'_2)\in D_0$,
\[
|f(x,u,pv'_1)-f(x,u,pv'_2)|\leq N|pv'_2-pv'_1|.
\]
\end{itemize}
Here we define the approximation scheme by
\eqref{1.5}-\eqref{1.6} and use Anti-maximum principle.
We make a good choice of $\lambda$ so that the sequences thus
generated converge to the solution of the nonlinear problem.
Similar to the Section \ref{wouls} we require the following
Lemmas and Propositions.

\begin{lemma}\label{R-U-L-1}
Let $-\lambda_0<\lambda<0$. If $u_n$ is an upper solution of
\eqref{1.3}-\eqref{1.4} and $u_{n+1}$ is defined by
\eqref{1.5}-\eqref{1.6} then $u_{n+1}\geq u_{n}$.
\end{lemma}

\begin{proof}
Let $w_{n}=u_{n+1}-u_{n}$, then
\begin{gather*}
-(pw_{n}')'+\lambda pw_{n} =(pu'_n)'-pf(x,u_n,pu'_n)\leq0,\\
w_{n}'(0)\geq0,\quad w_{n}'(1)\leq0,
\end{gather*}
and using Proposition \ref{MP} we have $u_{n+1}\geq u_n$.
\end{proof}

\begin{proposition}\label{R-U-L-2}
Assume that {\rm (F1)--(F4)} hold. Let
$-\lambda_0<\lambda<0$ be such that $M+\lambda\leq0$ and
$(M+\lambda)\big(1+\lambda\int_0^1\frac{1}{p(x)}\int_0^xp(t)
\,dt\,dx\big)-N\lambda \int_0^1p(x)dx\leq0$.
Then the functions $u_{n+1}$ defined
recursively by \eqref{1.5}-\eqref{1.6} are such that for all
$n\in \mathbb{N}$,
\begin{itemize}
\item[(i)] $u_n$ is an upper solution of \eqref{1.3}-\eqref{1.4}.
\item[(ii)] $u_{n+1}\geq u_{n}$.
\end{itemize}
\end{proposition}

\begin{proof}
Using Remark \ref{RM2}, Remark \ref{R2}, Lemma \ref{DI2},
 Lemma \ref{R-U-L-1} and on the lines of the proof of
Proposition \ref{U-L-2} this proposition can be deduced easily.
\end{proof}

In the same way we can prove the following results for the
lower solutions.

\begin{lemma}\label{R-L-L-1}
Let $-\lambda_0<\lambda<0$. If $v_n$ is a lower solution of
\eqref{1.3}-\eqref{1.4} and $v_{n+1}$ is defined by
\eqref{1.5}-\eqref{1.6} then $v_{n}\geq v_{n+1}$.
\end{lemma}

\begin{proposition}\label{R-L-L-2}
Assume that {\rm (F1)--(F4)} hold. Let
$-\lambda_0<\lambda<0$ be such that $M+\lambda\leq0$ and
$(M+\lambda)\big(1+\lambda\int_0^1\frac{1}{p(x)}
\int_0^xp(t)\,dt\,dx\big)-N\lambda \int_0^1p(x)dx\leq0$.
Then the functions $v_{n+1}$ defined
recursively by \eqref{1.5}-\eqref{1.6} are such that for all
$n\in \mathbb{N}$,
\begin{itemize}
\item[(i)] $v_n$ is a lower solution of \eqref{1.3}-\eqref{1.4}.
\item[(ii)] $v_{n}\geq v_{n+1}$.
\end{itemize}
\end{proposition}

In the next result we prove that lower solution $v_n$ is larger
than upper solution $u_n$ for all $n$.

\begin{proposition}\label{R-U-L-L-WO}
Assume that {\rm (F1)--(F4)} hold.
Let $-\lambda_0<\lambda<0$ be such that $M+\lambda\leq0$ and
\[
(M+\lambda)\Big(1+\lambda\int_0^1\frac{1}{p(x)}\int_0^xp(t)
\,dt\,dx\Big)-N\lambda \int_0^1p(x)dx\leq0
\]
and for all $x\in [0,1]$
\[
f(x,v_0,pv_0')-f(x,u_0,pu_0')+\lambda(u_0-v_0)\geq0.
\]
Then for all $n\in\mathbb{N}$, the functions $u_n$ and $v_n$
defined recursively by \eqref{1.5}-\eqref{1.6} satisfy
$v_n\geq u_n$.
\end{proposition}

Now similar to the Lemma \ref{U-L-3} and Lemma \ref{L-L-3}
we state the following two results. These results establish
a bound on $p(x) u'(x)$ and $p(x) v'(x)$. We use the assumption
\begin{itemize}
\item[(F5)] For all $(x,u,pu')\in D_0$,
$|f(x,u,pu')|\leq \varphi(|pu'|)$
where  $\varphi:[0,\infty)\to(0,\infty)$ is continuous and
satisfies
\[
\int_{0}^{\infty}\frac{ds}{\varphi(s)}>\int_0^1p(x)dx.
\]
\end{itemize}

\begin{lemma}\label{R-U-L-3}
 If $f(x,u,pu')$ satisfies {\rm (F1), (F2), (F5)},
then there exists $R_0>0$ such that any solution of
\[
-(pu')'+pf(x,u,pu')\geq0,\quad 0<x<1,\quad u'(0)=0=u'(1)
\]
with $u\in[u_0,v_0]$ for all $x\in[0,1]$, satisfies
$\| pu'\|_{\infty}< R_0$.
\end{lemma}

\begin{lemma}\label{R-L-L-3}
If $f(x,v,pv')$ satisfies {\rm (F1), (F2),(F5)},
 then there exists $R_0>0$ such that any solution of
\[
-(pv')'+pf(x,v,pv')\leq0,\quad 0<x<1,\quad v'(0)=0=v'(1)
\]
with $v\in[u_0,v_0]$ for all $x\in[0,1]$, satisfies
$\| pv'\|_{\infty}< R_0$.
\end{lemma}

Finally we arrive at the theorem similar to the Theorem \ref{WOT}.

\begin{theorem}\label{ROT}
Assume {\rm (F1)--(F5)} hold.
Let $-\lambda_0<\lambda<0$ be such that $M+\lambda\leq0$ and
$(M+\lambda)\big(1+\lambda\int_0^1\frac{1}{p(x)}\int_0^xp(t)
\,dt\,dx\big)-N\lambda \int_0^1p(x)dx\leq0$ and for all
$x\in [0,1]$,
\[
f(x,v_0,pv_0')-f(x,u_0,pu_0')+\lambda(u_0-v_0)\geq0.
\]
Then the sequences $u_n$ and $v_n$ defined by
\eqref{1.5}--\eqref{1.6} converge monotonically to solutions
$\widetilde{u}(x)$ and $\widetilde{v}(x)$ of
\eqref{1.3}-\eqref{1.4}. Any solution $z(x)$ of
\eqref{1.3}-\eqref{1.4} in $D_0$ satisfies
\[\widetilde{u}(x)\leq z(x)\leq \widetilde{v}(x).\]
\end{theorem}

\begin{proof}
Using Lemma \ref{R-U-L-1} to Lemma \ref{R-L-L-3} and
Proposition \ref{R-U-L-2} to Proposition \ref{R-U-L-L-WO}
we deduce that
\[
u_0\leq u_1\leq u_2\leq\dots
\leq u_n\leq v_n \dots \leq v_1\leq v_0.
\]
Now similar to the proof of Theorem \ref{WOT} the result of
this theorem can be deduced.
\end{proof}

\begin{remark} \rm
When the source function is derivative independent; i.e., $N=0$.
In this case we can choose $\lambda=-M$.
\end{remark}

\subsection*{Conclusion}%\label{CONCLUSION}
This work fills the gap existing in the literature for reverse
ordered upper and lower solutions. Some new existence results
have been established. This work also generalize our earlier
work \cite{AKV-NATMA-2011}. We establish existence results
under quite general conditions on $p(x)$ and $f(x,y,py')$.
In this work we do not have Bessel functions and therefore we
have to analyze the differential equation with general
function $p(x)$. We prove some differential inequalities which
enables us to prove the monotonicity of the sequences
$\{u_n\}$ and $\{v_n\}$. As future scope of the present work
we can further consider the following differential equation and
generalize the present work even further.
$$
-\big(p(x)y'(x)\big)'+q(x)f\big(x,y(x),p(x)y'(x)\big)=0,\quad 0<x<1.
$$
Here $q(x)$ is an integrable function on $[0,1]$ such that
 $q(x)>0$ in $(0,1)$.

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\end{document}