\documentclass[reqno]{amsart}

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2005(2005), No. 03, 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  (login: ftp)}
\thanks{\copyright 2005 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}

\title[\hfilneg EJDE-2005/03\hfil Existence and approximation of solutions]
{Existence and approximation of solutions of second order
nonlinear Neumann problems}

\author[R. A. Khan\hfil EJDE-2005/03\hfilneg]
{Rahmat Ali Khan}

\address{Rahmat Ali Khan \hfill\break
Department of Mathematics, University of
Glasgow, Glasgow G12 8QW, UK}
\email{rak@maths.gla.ac.uk}

\date{}
\thanks{Submitted November 6, 2004. Published January 2, 2005.}
\thanks{Partially supported by MoST, Pakistan.}
\subjclass[2000]{34A45, 34B15}
\keywords{Neumann problems; quasilinearization;  quadratic convergence}

\begin{abstract}
 We study existence and approximation of solutions of some Neumann
 boundary-value problems in the presence of an upper solution
 $\beta$ and a lower solution $\alpha$ in the reversed order
 ($\alpha\geq \beta$). We use the method of quasilinearization for
 the existence and approximation of solutions.
 We also discuss quadratic convergence of the sequence of approximants.
\end{abstract}

\maketitle
\newtheorem{thm}{Theorem}[section]
\newtheorem{rem}[thm]{Remark}
\numberwithin{equation}{section}

\section{Introduction}

In this paper, we study existence and approximation of solutions of some second
order nonlinear Neumann problem of the form
\begin{gather*}
 -x''(t)=f(t,x(t)),\quad t\in[0,1],
\\x'(0)=A,\quad x'(1)=B,
\end{gather*}
in the presence of a lower solution $\alpha$ and an upper solution
$\beta$ with $\alpha\geq \beta$ on $[0,1]$. We use the
quasilinearization technique for the existence and approximation
of solutions. We show that under suitable conditions the sequence
of approximants obtained by the method of quasilinearization
converges quadratically to a solution of the original problem.

There is a  vast literature dealing with the solvability of
nonlinear boundary-value problems with the method of upper and
lower solution and the quasilinearization technique in the case
where the lower solution $\alpha$ and the upper solution $\beta$
are ordered by $\alpha\leq \beta$. Recently, the case where the
upper and lower solutions are in the reversed order has also
received some attention. Cabada, et al. \cite{c1, c2}, Cherpion,
et al. \cite{m} have studied existence results for Neumann
problems in the presence of lower and upper solutions in the
reversed order. In these papers, they developed the monotone
iterative technique for existence of a solution $x$ such that
$\alpha\geq x\geq \beta$.

The purpose of this paper is to develop the quasilinearization
technique for the solution of the original problem in the case
upper and lower solutions are in the reversed order. The main idea
of the method of quasilinearization as developed by Bellman and
Kalaba \cite{bk}, and generalized by Lakshmikantham \cite{v1, v2},
has recently been studied and extended extensively to a variety of
nonlinear problems \cite{br1, br2, pe2, v, n}. In all these quoted
papers, the key assumption is that the upper and lower solutions
are ordered with $\alpha\leq \beta$. When $\alpha$ and $\beta$ are
in the reverse order, the quasilinearization technique seems not
to have studied previously.

In section 2, we discuss some basic known existence results for a
solution of the BVP \eqref{1.1}. The key assumption is that the
function $f(t,x)-\lambda x$ is non-increasing in $x$ for some
$\lambda$. In section 3, we approximate our problem by a sequence
of linear problems by the method of quasilinearization and prove
that under some suitable conditions there exist monotone sequences
of solutions of linear problems converging to a solution of the
BVP \eqref{1.1}. Moreover, we prove that the convergence of the
sequence of approximants is quadratic. In section 4, we study the
generalized quasilinearization method by allowing weaker
hypotheses on $f$ and prove that the conclusion of section 3 is
still valid.

 \section{ Preliminaries}

We know that the linear Neumann boundary value problem
\begin{gather*}
-x''(t)+Mx(t)=0,\quad t\in[0,1]\\
x'(0)=0,\quad x'(1)=0,
\end{gather*}
 has only the trivial solution if $M\neq -n^{2}\pi^{2}, \,n\in \mathbb{Z}$.
 For $M\neq -n^{2}\pi^{2}$ and any $\sigma \in C[0,1]$,
 the unique solution of the linear problem
 \begin{equation}\label{A}
\begin{gathered} -x''(t)+Mx(t)=\sigma(t),\quad t\in[0,1]\\
x'(0)=A,\quad x'(1)=B
\end{gathered}
\end{equation}
is given by
$$
x(t)=P_{\lambda}(t)+\int^{1}_{0}G_{\lambda}(t,s)\sigma(s)ds,
$$
where
$$
P_{\lambda}(t)=\begin{cases}
\frac{1}{\sqrt\lambda\sin\sqrt\lambda}(A\cos\sqrt\lambda(1-t)-B\cos\sqrt\lambda t),
&\text{ if $ M =-\lambda,\, \lambda >0 $, }  \\
\frac{1}{\sqrt\lambda\sinh\sqrt\lambda}(B\cosh\sqrt\lambda t-A\cosh\sqrt\lambda(1-t))
&\text{ if $ M =\lambda,\, \lambda >0 $, }
\end{cases}
$$
and (for $M=-\lambda$),
$$
G_{\lambda}(t,s)=-\frac{1}{\sqrt\lambda\sin\sqrt\lambda}
\begin{cases}
\cos\sqrt\lambda(1-s)\cos\sqrt\lambda t,
&\text{ if $0\leq t \leq s \leq 1$, }  \\
\cos\sqrt\lambda(1-t)\cos\sqrt\lambda s,
&\text{ if $0\leq s \leq t \leq 1$, } \end{cases}$$ and (\text{for $M=\lambda$}),
$$G_{\lambda}(t,s)=\frac{1}{\sqrt\lambda\sinh\sqrt\lambda}
\begin{cases}
\cosh\sqrt\lambda(1-s)\cosh\sqrt\lambda t,
&\text{ if $0\leq t \leq s \leq 1$, }  \\
\cosh\sqrt\lambda(1-t)\cosh\sqrt\lambda s,
&\text{ if $0\leq s \leq t \leq 1$, }
\end{cases}
$$
is the Green's function of the problem. For $M=-\lambda$, we note that
 $G_{\lambda}(t,s)\leq 0$ if $0<\sqrt\lambda \leq \pi/2$.
Moreover, for such values of $M$ and $\lambda$, we have,
 $P_{\lambda}(t)\leq 0$ if $A\leq 0 \leq B$, and  $P_{\lambda}(t)\geq 0$
if  $A\geq 0 \geq B$.
  Thus we have the following anti-maximum principle

\noindent\textbf{Anti-maximum Principle.}
Let $-\pi^{2}/4\leq M<0$. If $A\leq 0 \leq B$ and $\sigma(t)\geq 0$,
then a solution $x(t)$ of \eqref{A} is such that $x(t) \leq 0$. If $A\geq 0 \geq B$
and $\sigma(t)\leq 0$, then $x(t)\geq 0$.
\smallskip

Consider the nonlinear Neumann problem
\begin{equation}\label{1.1}
\begin{gathered}
-x''(t)=f(t,x(t)),\quad t\in[0,1],\\
x'(0)=A,\quad x'(1)=B,
\end{gathered}
\end{equation}
where $f:[0,1]\times \mathbb{R}\to \mathbb{R}$ is continuous and $A,\, B\in \mathbb{R}$.
We recall the concept of lower and upper solutions.

\noindent\textbf{Definition.}
Let $ \alpha \in C^{2} [0,1]$. We say that $ \alpha$ is a lower
solution of \eqref{1.1}, if
\begin{gather*}
-\alpha''(t) \leq f(t,\alpha(t)),\quad t\in [0,1], \\
\alpha'(0)\geq A,\quad \alpha'(1)\leq B.
\end{gather*}
An upper solution $\beta \in C^{2} [0,1]$ of the BVP \eqref{1.1}
is defined similarly by reversing the inequalities.


\begin{thm}[Upper and Lower solutions method]\label{thm1}
Let $0<\lambda \leq \pi^{2}/4$.
 Assume that  $\alpha$ and $\beta$ are respectively lower and
upper solutions of \eqref{1.1} such that $\alpha(t)\geq\beta(t),
\, t\in [0,1]$. If $ f(t,x)-\lambda x$ is non-increasing in $x$,
then there exists a solution $x$ of the boundary value problem
\eqref{1.1} such that
$$
\alpha(t)\geq x(t)\geq \beta(t), \quad t\in [0,1].
$$
\end{thm}

\begin{proof}
This result is known \cite{c1} and we provide a proof for completeness.
Define $p(\alpha (t), x,\beta(t))=\min\big\{\alpha(t),
\max\{x,\beta(t)\}\big\}$, then $p(\alpha (t), x,\beta(t))$
satisfies $\beta(t)\leq p(\alpha (t), x,\beta(t))\leq
\alpha(t),\,x\in \mathbb{R},\,t\in [0,1]$. Consider the modified boundary
value problem
 \begin{equation}
\begin{gathered}\label{1.2}
-x''(t)-\lambda x(t)= F(t,x(t)), \quad t \in  [0,1],   \\
 x'(0)=A,\quad x'(1)=B,
\end{gathered}
\end{equation}
 where
$$
F(t,x) = f(t,p(\alpha(t),x,\beta(t)))-\lambda
p(\alpha(t),x,\beta(t)).
$$
This is equivalent to the integral equation
  \begin{equation}\label{1.3}
x(t)= P_{\lambda}(t)+\int^{1}_{0}G_{\lambda}(t,s)F(s,x(s))ds.
\end{equation}
Since $P_{\lambda}(t)$ and $F(t,x(t))$ are continuous and bounded,
this integral equation has a fixed point by the Schauder fixed point theorem.
Thus, problem (\ref{1.2}) has a solution. Moreover,
\begin{gather*}
F(t,\alpha(t))=f(t,\alpha(t))-\lambda \alpha (t)\geq -\alpha''(t)
-\lambda\alpha (t),\quad t\in [0,1],\\
F(t,\beta(t))=f(t,\beta(t))-\lambda \beta(t) \leq
-\beta''(t)-\lambda\beta(t),\quad t\in [0,1].
\end{gather*}
Thus, $\alpha,\,\beta$ are lower and upper solutions of \eqref{1.2}.
 Further, we note that any solution $x(t)$ of \eqref{1.2} with the property
  $\beta(t)\leq x(t)\leq \alpha(t), t\in [0,1]$, is also a solution of \eqref{1.1}.
   Now, we show that any solution $x$ of \eqref{1.2} does satisfy
  $\beta(t)\leq x(t)\leq \alpha(t),\, t\in [0,1]$. For this, set  $v(t)=\alpha(t)-x(t)$, then $v'(0)\geq 0,\, v'(1)\leq 0$.
In view of the non-increasing property of the function $
f(t,x)-\lambda x$ in $x$, the definition of lower solution and the
fact that $p(\alpha(t),x,\beta(t))\leq \alpha(t)$, we have
\begin{align*}
&-v''(t)- \lambda v(t)\\
&=(-\alpha''(t)- \lambda \alpha(t))-(-x''(t)-\lambda x(t))\\
& \leq (f(t,\alpha(t))-\lambda
\alpha(t))-(f(t,p(\alpha(t),x(t),\beta(t)))-\lambda
p(\alpha(t),x(t),\beta(t)) )\leq 0.
\end{align*}
By the anti-maximum principle, we obtain $v(t)\geq 0$, $t\in [0,1]$.
Similarly, $x(t)\geq \beta(t)$, $t\in [0,1]$.
\end{proof}

\begin{thm}\label{thm2}
Assume that $\alpha$ and $\beta$ are lower and upper solutions of
the boundary value problem \eqref{1.1} respectively. If
$f:[0,1]\times \mathbb{R}\to \mathbb{R} $ is continuous and
\begin{equation}\label{1.4}
f(t,\alpha(t))-\lambda \alpha(t) \leq
f(t,\beta(t))-\lambda \beta(t) \quad  \text{for some } 0<\lambda\leq
\pi^{2}/4,\; t\in [0,1],
\end{equation}
 then $ \alpha(t)\geq \beta(t)$, $t\in[0,1]$.
\end{thm}

\begin{proof}
Define  $ m(t)=\alpha(t)-\beta(t)$, $t\in [0,1]$, then $ m(t) \in
C^{2}[0,1]$ and $m'(0)\geq 0$, $m'(1)\leq 0$. In view of
\eqref{1.4} and the definition of upper and lower solution, we
have
\begin{align*}
-m''(t)- \lambda m(t)&=(-\alpha''(t)- \lambda
\alpha(t))-(-\beta''(t)- \lambda \beta(t))\\
& \leq (f(t,\alpha(t))-\lambda \alpha(t))-(f(t,\beta(t))-\lambda
\beta(t)) \leq 0.
\end{align*} Thus, by anti-maximum principle, $ m(t)\geq 0$,  $t \in [0,1]$.
\end{proof}

\section{Quasilinearization Technique}

We now approximate our problem by the method of quasilinearization.
Lets state the following assumption.
\begin{itemize}
\item[(A1)] $\alpha ,\,\beta \in C^{2}[0,1]$ are respectively
lower and upper solutions of \eqref{1.1} such that $\alpha(t) \geq
\beta(t)$, $t\in [0,1]=I$. \item[(A2)]  $f(t,x),\, f_{x}(t,x),\,
f_{xx}(t,x)$ are continuous on $I\times \mathbb{R}$ and are such
that $ 0<f_{x}(t,x)\leq \frac{\pi^{2}}{4}$ and $f_{xx}(t,x)\leq 0$
for $(t,x)\in I\times [\min\beta(t),\max\alpha(t)]$.
\end{itemize}

\begin{thm}\label{thm3b}
Under assumptions (A1)-(A2), there exists a monotone sequence $\{
w_{n}\}$ of solutions converging uniformly and quadratically to a
solution of the problem \eqref{1.1}.
\end{thm}

\begin{proof} Taylor's theorem and the condition $f_{xx}(t,x)\leq 0 $ imply that
\begin{equation}\label{3.1}
f(t,x)\leq f(t,y)+f_{x}(t,y)(x-y),\end{equation} for
$(t,x),\,(t,y)\in I\times [\min\beta(t),\max\alpha(t)]$. Define
\begin{equation}\label{3.2}
F(t,x,y)=f(t,y)+f_{x}(t,y)(x-y),
\end{equation}
$x,y\in \mathbb{R}$, $t\in I$. Then,
$F(t,x,y)$ is continuous and  satisfies the  relations
 \begin{equation}\label{3.3}
 \begin{gathered}
f(t,x)\leq F(t,x,y)\\
f(t,x)=F(t,x,x),
\end{gathered}
\end{equation}
for $(t,x),\,(t,y)\in I\times [\min\beta(t),\max\alpha(t)]$.
Let $\lambda=\max\{f_{x}(t,x):\, (t,x)\in I\times
[\min\beta(t),\max\alpha(t)]\}$, then $0<\lambda\leq
\frac{\pi^{2}}{4}$.
Now, set $w_{0}=\beta$ and consider the linear problem
\begin{equation}\label{3.4}
\begin{gathered}
-x''(t)-\lambda x(t)=F(t,p(\alpha(t),x(t),w_{0}(t)),w_{0}(t))
-\lambda p(\alpha(t),x(t),w_{0}(t)),\quad t \in I,   \\
 x'(0)=A,\quad x'(1)=B.
 \end{gathered}
 \end{equation}
This is equivalent to the integral equation
\begin{align*}
&x(t)\\
&= P_{\lambda}(t)+\int^{1}_{0}G_{\lambda}(t,s)
\big[F(s,p(\alpha(s),x(s),w_{0}(s)),w_{0}(s))-\lambda
p(\alpha(s),x(s),w_{0}(s))\big]ds.
\end{align*}
Since $P_{\lambda}(t)$ and $F(t,p(\alpha,x,w_{0}),w_{0})-\lambda
p(\alpha,x,w_{0})$ are continuous and bounded, this integral
equation has a fixed point $w_{1}$ (say) by the Schauder fixed
point theorem. Moreover,
 \begin{align*}
&F(t,p(\alpha(t),w_{0}(t),w_{0}(t)),w_{0}(t))-\lambda p(\alpha(t),w_{0}(t),w_{0}(t))\\
&=f(t,w_{0}(t))-\lambda w_{0}(t)\\
&\leq -w''_{0}(t)-\lambda w_{0}(t),\quad t\in I,
\end{align*}
and
\begin{align*}
&F(t,p(\alpha(t),\alpha(t),w_{0}(t)),w_{0}(t))
-\lambda p(\alpha(t),\alpha(t),w_{0}(t))\\
&\geq f(t,\alpha(t))-\lambda \alpha(t)\\
&\geq -\alpha''(t)-\lambda\alpha(t),\quad t\in I.
\end{align*}
This implies that $\alpha,\,w_{0}$ are lower and upper solutions
of \eqref{3.4}. Now, we show that
$$
w_{0}(t)\leq w_{1}(t)\leq \alpha(t)\text{ on }I.
$$
For this, set $v(t)=w_{1}(t)-w_{0}(t)$,
then the boundary conditions imply that
$v'(0)\geq 0,\, v'(1)\leq 0$. Further, in view of the condition
$f_{x}(t,x)\leq \lambda$  for $(t,x)\in I\times [\min\beta(t),\max\alpha(t)]$ and
\eqref{3.2}, we have
\begin{align*}
-v''(t)- \lambda v(t)
&=(-w_{1}''(t)- \lambda
w_{1}(t))-(-w_{0}''(t)- \lambda w_{0}(t))\\
& \leq (f_{x}(t,w_{0}(t))- \lambda
)(p(\alpha(t),w_{1}(t),w_{0}(t))-w_{0}(t)) \leq 0.
\end{align*}
Thus, by anti-maximum principle, we obtain $v(t)\geq 0, \, t\in I$.
 Similarly, $\alpha(t)\geq w_{1}(t)$.  Thus,
\begin{equation}\label{3.5}
w_{0}(t)\leq w_{1}(t)\leq \alpha(t), \quad  t\in I.
\end{equation}
In view of \eqref{3.3} and the fact that $w_{1}$ is a solution of
\eqref{3.4} with the property \eqref{3.5}, we have
\begin{equation}\label{3.6}
\begin{gathered}
-w_{1}''(t) =F(t,w_{1}(t),w_{0}(t))\geq f(t,w_{1}(t))\\
w'_{1}(0)=A,\,w'_{1}(1)=B,
\end{gathered}\end{equation}
 which implies that $w_{1}$ is an upper solution of \eqref{1.1}.

 Now, consider the problem
  \begin{equation}\label{3.7}
\begin{gathered}
 -x''(t)-\lambda x (t) =F(t,p(\alpha(t),x(t),w_{1}(t)),w_{1}(t))
 -\lambda p(\alpha(t),x(t),w_{1}(t)),\quad t \in I,   \\
 x'(0)=A,\quad x'(1)=B.
 \end{gathered}
 \end{equation}
Denote by $w_{2}$ a solution of \eqref{3.7}. In order to show that
\begin{equation}\label{3.8}
 w_{1}(t)\leq w_{2}(t)\leq \alpha(t), \quad  t\in I,
\end{equation}
set $v(t)=w_{2}(t)-w_{1}(t)$, then  $v'(0)= 0,\, v'(1)=0$.
Further, in view of \eqref{3.2} and the condition
$f_{x}(t,x)\leq \lambda$  for $(t,x)\in I\times [\min\beta(t),\max\alpha(t)]$,
we obtain
\begin{align*}
&-v''(t)- \lambda v(t)\\
&\leq F(t,p(\alpha(t),w_{2}(t),w_{1}(t)),w_{1}(t))-\lambda
p(\alpha(t),w_{2}(t),w_{1}(t))-(f(t,w_{1}(t))-\lambda
w_{1}(t))\\
& \leq (f_{x}(t,w_{1}(t))-\lambda)(p(\alpha(t),w_{2}(t),w_{1}(t))-w_{1}(t))
\leq 0,\quad t\in I.
\end{align*}
Hence $w_{2}(t)\geq w_{1}(t)$ follows from the anti-maximum
principle. Similarly, we can show that $w_{2}(t)\leq \alpha(t)$ on
$I$.

 Continuing this process, we obtain a monotone
 sequence $\{w_{n}\}$ of solutions satisfying
 \begin{equation}\label{3.7a}
 w_{0}(t)\leq w_{1}(t)\leq w_{2}(t)\leq \dots \leq w_{n}(t)\leq
 \alpha(t), \quad t\in I,
\end{equation}
 where, the element $w_{n}$ of the sequence
 $\{w_{n}\}$ that for $t \in I$, satisfies
 \begin{gather*}
 -x''(t)-\lambda x(t)  =F(t,p(\alpha(t),x(t),w_{n-1}(t)),w_{n-1}(t))
 -\lambda p(\alpha(t),x(t),w_{n-1}(t)),   \\
 x'(0)=A,\quad x'(1)=B.
 \end{gather*}
That is,
 \begin{gather*}
-w_{n}''(t) =F(t,w_{n}(t),w_{n-1}(t)),\quad t \in I,   \\
 w_{n}'(0)=A,\quad w_{n}'(1)=B.
 \end{gather*}
Employing the standard argument \cite{gl}, it follows that the convergence of
the sequence is uniform.
  If $x(t)$ is the limit point of the sequence, since $F$ is continuous,
  we have
$$
\lim_{n\to \infty}F(t,w_{n}(t),w_{n-1}(t))=F(t,x(t),x(t))=f(t,x(t))
$$
which implies that, $x$ is a solution the boundary value problem \eqref{1.1}.

Now, we show that the convergence of the sequence is quadratic.
For this, set $e_{n}(t)=x(t)-w_{n}(t)$, $t\in I,\,n\in \mathbb{N}$, where
$x$ is a solution of \eqref{1.1}. Note that,
$e_{n}(t)\geq 0$ on $I$ and $e'_{n}(0)=0$,  $e'_{n}(1)=0$. Let
$\rho=\min\big\{f_{x}(t,x): (t,x)\in I\times
[\min\beta(t),\max\alpha(t)]\big\}$, then
$0<\rho<\frac{\pi^{2}}{4}$. Using Taylor's theorem and
\eqref{3.2}, we obtain
\begin{equation}\label{3.9}
\begin{aligned}
&-e_{n}''(t) \\
& =- x''(t)+w_{n}''(t)= f(t,x(t))-F(t,w_{n}(t),w_{n-1}(t)) \\
&=f(t,w_{n-1}(t))+f_{x}(t,w_{n-1}(t))(x(t)-w_{n-1}(t))
  +\frac{f_{xx}(t,\xi(t))}{2!}(x(t)-w_{n-1}(t))^{2}\\
&\quad - [f(t,w_{n-1}(t))+f_{x}(t,w_{n-1}(t))(w_{n}(t)-w_{n-1}(t))]\\
&=f_{x}(t,w_{n-1}(t))e_{n}(t)+\frac{f_{xx}(t,\xi(t))}{2!}e^{2}_{n-1}(t)\\
&\geq \rho
e_{n}(t)+\frac{f_{xx}(t,\xi(t))}{2!}\|e_{n-1}\|^{2},\quad t\in I
\end{aligned}
\end{equation}
where, $w_{n-1}(t)<\xi(t)<x(t)$.
 Thus, by comparison results, the error function $e_{n}$ satisfies
$e_{n}(t)\leq r(t),\,t\in I$, where $r$ is the unique solution of the
boundary-value problem
\begin{equation}\label{d}
\begin{gathered}
-r''(t)-\rho r(t)=\frac{f_{xx}(t,\xi(t))}{2!}\|e_{n-1}\|^{2},\quad t\in I\\
r'(0)=0,\quad r'(1)=0,
\end{gathered}
\end{equation}
and
\[
r(t)=\int^{1}_{0}G_{\rho}(t,s)\frac{f_{xx}(t,\xi(s))}{2!}\|e_{n-1}\|^{2}ds
\leq \delta \| e_{n-1}\|^{2},
\]
where
 $\delta=\max \{\frac{1}{2}|G_{\rho}(t,s)f_{xx}(s,x)|: (t,x)\in I\times
 [\min\beta(t),\max\alpha(t)]\}$. Thus
 $\| e_{n}\|\leq \delta \| e_{n-1}\|^{2}$.
 \end{proof}

 \begin{rem} \rm
In (A2), if we replace the concavity
 assumption $f_{xx}(t,x)\leq 0$ on $I\times [\min\beta(t),\max\alpha(t)]$
by the convexity
 assumption $f_{xx}(t,x)\geq 0$ on $I\times [\min\beta(t),\max\alpha(t)]$.
Then we  have the relations
\begin{gather*}
f(t,x)\geq F(t,x,y)\\
f(t,x)=F(t,x,x),
\end{gather*}
for $x,y\in [\min\beta(t),\max\alpha(t)],\,t\in [0,1]$, instead of \eqref{3.3}
and we obtain a monotonically nonincreasing sequence
\[
\alpha(t)\geq w_{1}(t)\geq w_{2}(t)\geq \dots \geq w_{n}(t)\geq
 \beta(t), \quad t\in I,
\]
 of solutions of linear problems which converges uniformly and quadratically
to a solution of  \eqref{1.1}.
 \end{rem}

\section{Generalized quasilinearization technique}

Now we introduce an auxiliary function $\phi$ to relax the
concavity(convexity) conditions on the function $f$ and hence
prove results on the generalized quasilinearization. Let
\begin{itemize}
\item[(B1)] $\alpha, \, \beta \in C^{2}(I)$ are lower
and upper solutions of \eqref{1.1} respectively, such that
$\alpha(t)\geq \beta(t)$ on $I$.
\item[(B2)] $ f \in C^{2}(I\times \mathbb{R})$ and is such that
$0<f_{x}(t,x)\leq \frac{\pi^{2}}{4}$ for
$(t,x)\in I\times [\min\beta(t),\max\alpha(t)]$ and
$$
\frac{\partial^{2}}{\partial^{2}x}(f(t,x)+\phi(t,x))\leq 0
$$
on $I\times [\min\beta(t),\max\alpha(t)]$,
for some function $\phi\in C^{2}(I\times \mathbb{R})$ satisfies
$\phi_{xx}(t,x)\leq 0\text{ on } I\times [\min\beta(t),\max\alpha(t)]$.
\end{itemize}

\begin{thm} \label{thm4} Under assumptions (B1)-(B2),
 there exists a monotone sequence $\{ w_{n}\}$ of solutions
converging uniformly and quadratically to a solution of the
problem \eqref{1.1}.
\end{thm}
\begin{proof}
 Define $F:I\times \mathbb{R}\to \mathbb{R}$ by
\begin{equation}\label{4.1a}
F(t,x)=f(t,x)+\phi(t,x).
\end{equation}
Then, in view of (B2), we have  $F(t,x)\in C^{2}(I\times \mathbb{R})$
and
\begin{equation}\label{4.1}
F_{xx}(t,x)\leq 0\quad \text{on } I\times [\min\beta(t),\max\alpha(t)],
\end{equation}
which implies
\begin{equation}\label{4.2}
f(t,x)\leq F(t,y)+ F_{x}(t,y)(x-y)-\phi (t,x),
\end{equation}
for $(t,x),\,(t,y) \in I\times[\min\beta(t),\max\alpha(t)]$. Using
Taylor's theorem on $\phi$, we obtain
$$
\phi(t,x)=\phi(t,y) +
\phi_{x}(t,y)(x-y)+\frac{\phi_{xx}(t,\eta)}{2!}(x-y)^{2},$$
where
$x,\,y\in \mathbb{R},\,t\in I$ and $\eta$ lies between $x$ and $y$. In
view of (B2), we have
\begin{equation}\label{4.3}
\phi(t,x)\leq \phi(t,y) + \phi_{x}(t,y)(x-y),
\end{equation}
for $(t,x),\,(t,y) \in I\times [\min\beta(t),\max\alpha(t)]$ and
\begin{equation}\label{4.3t}
\phi(t,x)\geq \phi(t,y) +
\phi_{x}(t,y)(x-y)-\frac{M}{2}\|x-y\|^{2},
\end{equation}
for $(t,x),\,(t,y)\in I\times [\min\beta(t),\max\alpha(t)]$, where
$$
M=\max\{|\phi_{xx}(t,x)|:(t,x)\in I\times
[\min\beta(t),\max\alpha(t)]\}.
$$
Using \eqref{4.3t} in
\eqref{4.2}, we obtain
\begin{equation}\label{4.2t}
f(t,x)\leq f(t,y)+ f_{x}(t,y)(x-y)+\frac{M}{2}\|x-y\|^{2},
\end{equation}
for $(t,x),\,(t,y)\in I\times [\min\beta(t),\max\alpha(t)]$. Define
 \begin{equation}\label{4.4}
F^{*}(t,x,y)=f(t,y)+ f_{x}(t,y)(x-y)+\frac{M}{2}\|x-y\|^{2},
\end{equation}
for $t\in I,\,x,\,y \in \mathbb{R}$.
then, $F^{*}(t,x,y)$ is continuous and for
$(t,x),\,(t,y)\in I\times [\min\beta(t),\max\alpha(t)]$, satisfies the following
relations
\begin{equation}\label{4.5}
\begin{gathered}
 f(t,x) \leq F^{*}(t,x,y)\\
 f(t,x) = F^{*}(t,x,x).
\end{gathered}
\end{equation}
Now, we set $\beta =w_{0}$ and consider the Neumann problem
\begin{equation}\label{4.6}
\begin{gathered}
 -x''(t)-\lambda x(t)= F^{*}(t,p(\alpha(t),x(t),w_{0}(t)),w_{0}(t))
 -\lambda p(\alpha(t),x(t),w_{0}(t)),\quad t \in I,  \\
 x'(0)=A,\quad x'(1)=B,
\end{gathered}
\end{equation}
where $\lambda$  and $p$ are the same as
defined in Theorem \ref{thm3b}. Since
$F^{*}(t,p(\alpha,x,w_{0}),w_{0})-\lambda p(\alpha,x,w_{0})$ is
continuous and bounded, it follows that the problem \eqref{4.6}
has a solution. Also, we note that any solution $x$ of \eqref{4.6}
which satisfies
 \begin{equation}\label{B}
 w_{0}(t)\leq x(t) \leq \alpha(t), \quad t \in I,
\end{equation}
is a solution of
 \begin{gather*}
 -x''(t) = F^{*}(t,x(t),w_{0}(t)),\quad t \in I,\\
 x'(0)=A,\quad x'(1)=B,
\end{gather*}
and in view of \eqref{4.5},  $F^{*}(t,x(t),w_{0}(t))\geq f(t,x(t))$.
It follows that any solution $x$ of \eqref{4.6} with the property
\eqref{B} is an upper solution of \eqref{1.1}.
 Now, set $v(t)=\alpha(t)-x(t)$, where $x$ is a solution of \eqref{4.6},
then $v'(0)\geq 0$, $v'(1)\leq 0$.
 Moreover, using (B2) and  \eqref{4.5}, we obtain
  \begin{align*}
&-v''(t)-\lambda v(t)\\
&=(-\alpha''(t)-\lambda \alpha(t))-(-x''(t)-\lambda x(t))\\
&\leq (f(t,\alpha(t))-\lambda
\alpha(t))-[F^{*}(t,p(\alpha(t),x(t),w_{0}(t)),w_{0}(t))-\lambda
p(\alpha(t),x(t),w_{0}(t))]\\
& \leq (f(t,\alpha(t))-\lambda
\alpha(t))-[f(t,p(\alpha(t),x(t),w_{0}(t)))-\lambda
p(\alpha(t),x(t),w_{0}(t))]\leq 0.
\end{align*}
Hence, by anti-maximum principle, $\alpha(t)\geq x(t),\,t\in I$.
Similarly,  $w_{0}(t)\leq x(t)$, $t\in I$.
Continuing this process we obtain a monotone sequence $\{w_{n}\}$ of
solutions satisfying
$$
w_{0}(t)\le w_{1}(t)\le w_{2}(t)\le w_{3}(t)\le \dots\le w_{n-1}(t)
\le w_{n}(t)\le \alpha(t),  \quad t \in I .
$$
 The same arguments as in Theorem \ref{thm3b}, shows that the sequence
converges to a
solution $x$ of the boundary value problem \eqref{1.1}.

 Now we show that the convergence of the sequence of solutions is quadratic.
 For this,
we set $e_{n}(t)=x(t)-w_{n}(t)$, $t\in I$, where $x$ is a
solution of the boundary-value problem \eqref{1.1}. Note that,
$e_{n}(t)\geq 0$ on $I$ and, $e'_{n}(0)=0,\, e'_{n}(1)=0$.
Using Taylor's theorem, \eqref{4.3} and the fact that
$\|w_{n}-w_{n-1}\|\leq \|e_{n-1}\|$, we obtain
\begin{align*}
&-e_{n}''(t) \\
& =- x''(t)+w_{n}''(t)\\
&=(F(t,x(t))-\phi(t,x(t)))-F^{*}(t,w_{n}(t),w_{n-1}(t)) \\
&=F(t,w_{n-1}(t))+F_{x}(t,w_{n-1}(t))(x(t)-w_{n-1}(t))
+\frac{F_{xx}(t,\xi(t))}{2}(x(t)-w_{n-1}(t))^{2}\\
&\quad -[\phi(t,w_{n-1}(t))+\phi_{x}(t,w_{n-1}(t))(x(t)-w_{n-1}(t))]\\
&\quad -
[f(t,w_{n-1}(t))+f_{x}(t,w_{n-1}(t))(w_{n}(t)-w_{n-1}(t))+\frac{M}{2}\|w_{n}-w_{n-1}\|^{2}]\\&
=f_{x}(t,w_{n-1}(t))e_{n}(t)+\frac{F_{xx}(t,\xi(t))}{2}e^{2}_{n-1}(t)
-\frac{M}{2}\|w_{n}-w_{n-1}\|^{2}\\
&\geq f_{x}(t,w_{n-1}(t))e_{n}(t)-(\frac{|F_{xx}(t,\xi(t))|}{2}+\frac{M}{2})\|e_{n-1}
\|^{2}\\
&\geq \rho e_{n}(t)-Q\|e_{n-1}\|^{2},\quad t\in I,
\end{align*}
where, $w_{n-1}(t)\leq \xi(t) \leq x(t)$,
$$
Q=\max\{\frac{|F_{xx}(t,x)|}{2}+\frac{M}{2}:(t,x)\in I\times
[\min\beta(t),\max\alpha(t)]\}
$$
and $\rho$ is defined as in Theorem \ref{thm3b}. Thus, by comparison results
$e_{n}(t)\leq r(t)$,   $t\in I$, where $r$ is a unique solution of the linear
problem
\begin{gather*}
-r''(t)-\rho r(t)=-Q\|e_{n-1}\|^{2},\quad t\in I\\
 r'(0)=0,\quad r'(1)=0,
\end{gather*}
and
\[
r(t)=Q\int^{1}_{0}|G_{\rho}(t,s)|\|e_{n-1}\|^{2}ds
 \leq \sigma \| e_{n-1}\|^{2},
\]
where  $\sigma=Q\max \{|G_{\rho}(t,s)|: (t,s)\in I\times
 I\}$. Thus $\| e_{n}\|\leq \sigma \| e_{n-1}\|^{2}$.
\end{proof}

\subsection*{Acknowledgment}
The author thanks the referee for his valuable comments which lead
to improve the original manuscript.


\begin{thebibliography}{99}

\bibitem{br1} Bashir Ahmad, Rahmat Ali Khan, and S. Sivasundaram;
\emph{Generalized quasilinearization method for nonlinear
functional differential equations},  J. Appl. Math. Stochastic
Anal. 16(2003), no 1, 33-43.

\bibitem{br2} Bashir Ahmad, Rahmat Ali Khan, and P. W. Eloe;
\emph{Generalized quasilinearization method for a second order
three-point boundary-value problem with nonlinear boundary conditions},
 Electron. J. of Diff. Eqns. Vol 2000(2002), No 90, 1-12.

\bibitem{bk}  R. Bellman and R. Kalaba;
\emph{Quasilinearisation and Nonlinear Boundary Value Problems},
American Elsevier, New York, 1965.

\bibitem{m} M. Cherpion, C. De Coster, P. Habets;
\emph{A constructive monotone iterative method for second order BVP
 in the presence of lower and upper solutions},
Appl. Math. Comput. 123(2001), 75-91.

\bibitem{c2} A. Cabada, P. Habets, S. Lois;
\emph{Monotone method of the Neumann problem with lower and upper
solutions in the reverse order},
 Appl. Math. Comput. 117(2001), 1-14.

\bibitem{c1} A. Cabada, L. Sanchez;
\emph{A positive operator approach to the Neumann problem for second order
ordinary differential equation},
J. Math. Anal. Appl. 204(1996), 774-785.

\bibitem{pe2}  P. W. Eloe;
\emph{The quasilinearization method on an unbounded domain}, Proc.
Amer. Math. Soc., 131(2002)5 1481-1488.

\bibitem{gl} G. S. Ladde, V. Lakshmikantham, A.S. Vatsala;
\emph{Monotone iterative technique for nonlinear differential equations},
Pitman, Boston, 1985.

\bibitem{v1}  V. Lakshmikantham;  \emph{An extension of the method of
quasilinearization}, J. Optim. Theory Appl., 82(1994) 315-321.

\bibitem{v2}  V. Lakshmikantham;
\emph{Further improvement of generalized quasilinearization},
Nonlinear Analysis, 27(1996) 315-321.

\bibitem{v}  V. Lakshmikantham and A.S. Vatsala; \emph{Generalized
quasilinearization for nonlinear problems}, Kluwer Academic
Publishers, Boston (1998).

\bibitem{n}  J. J. Nieto;
\emph{Generalized quasilinearization method for a second order ordinary differential
 equation with Dirichlet boundary conditions},
Proc. Amer. Math. Soc., 125(1997) 2599-2604.

\end{thebibliography}
\end{document}