\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