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\markboth{\hfil Generalized quasilinearization method \hfil EJDE--2002/90}
{EJDE--2002/90\hfil Bashir Ahmad, Rahmat Ali Khan, \& Paul W. Eloe \hfil}
\begin{document}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2002}(2002), No. 90, pp. 1--12. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)}
 \vspace{\bigskipamount} \\
 %
  Generalized quasilinearization method for a second order three point
  boundary-value \\ problem with nonlinear boundary conditions
 %
\thanks{ {\em Mathematics Subject Classifications:} 34B10, 34B15.
\hfil\break\indent
{\em Key words:} Generalized quasilinearization, boundary value problem,
\hfil\break\indent
          nonlinear boundary conditions, quadratic and rapid convergence.
\hfil\break\indent
\copyright 2002 Southwest Texas State University. \hfil\break\indent
Submitted March 14, 2002. Published October 22, 2002.} }
\date{}
%
\author{Bashir Ahmad, Rahmat Ali Khan, \& Paul W. Eloe}
\maketitle

\begin{abstract}
  The generalized quasilinearization technique is applied to
  obtain a monotone sequence of iterates converging uniformly and
  quadratically to a solution of three point boundary value problem
  for second order differential equations with nonlinear boundary
  conditions. Also, we improve the convergence of the sequence of
  iterates by establishing a convergence of order $k$.
\end{abstract}

\newtheorem{theorem}{Theorem}[section]
\numberwithin{equation}{section}

\section{Introduction}

The method of quasilinearization pioneered by Bellman and Kalaba \cite{b1}
and
generalized by Lakshmikantham \cite{l1,l2} has been applied to a variety of
problems \cite{c1, l3,l4,l5,l6,n1}.

Multipoint boundary value problems for second order differential equations
have also been receiving considerable attention recently. Kiguradze and
Lomtatidze \cite{k1} and Lomtatidze \cite{l7,l8} have studied closely
related
problems. Gupta et.al. \cite{c2,c3,c4} have studied problems related to
three
point boundary value problems. More recently, Paul Eloe and Yang Gao
\cite{e1}
discussed the method of quasilinearization for a three point boundary value
problem. In this paper, we develop the method of generalized
quasilinearization for a three point boundary value problem involving
nonlinear boundary conditions and obtain a monotone sequence of approximate
solutions converging uniformly and quadratically to a solution of the
problem. Also, we have discussed the convergence of order $k$.

\subsection*{Basic Results}

Consider the three point boundary value problem with nonlinear boundary
conditions
\begin{equation}\begin{gathered}
x'' =f(t,x(t)) ,\quad t\in [0,1]=J \\
x(0) =a,\quad x(1) =g(x(\frac{1}{2})),
\end{gathered}\label{a}
\end{equation}
where $f\in C[J\times \mathbb{R},\mathbb{R}]$ and $g:\mathbb{R}\to
\mathbb{R}$
is continuous. Let $G(t,s)$ denote the Green's function for the conjugate
or
Dirichlet boundary value problem and is given by
$$
G(t,s) =\begin{cases} t(s-1),& 0\leq t<s\leq 1\\
 s(t-1),& 0\leq s<t\leq 1.\end{cases}
$$
We note that $G(t,s) <0$ on $(0,1) \times (0,1) $. If $x(t) $ is the
solution of $(1.1) $
and $(1.2) $, then
\begin{equation}
x(t) =a(1-t) +g(x(\frac 12))
t+\int_0^1G(t,s) f(s,x(s)) ds.  \label{1.3}
\end{equation}
Let $\alpha ,\beta \in C^2[0,1]$. We say that $\alpha $ is a lower
solution of the BVP \eqref{a}, if
\begin{gather*}
\alpha '' \geq f(t,\alpha) ,\quad t\in [0,1]\\
\alpha (0) \leq a,\quad \alpha (1) \leq g(\alpha (\frac{1}{2})) ,
\end{gather*}
and $\beta $ be an upper solution of the BVP \eqref{a}, if
\begin{gather*}
\beta '' \leq f(t,\beta) ,\quad t\in [0,1] \\
\beta (0)  \geq a,\quad \beta (1) \geq g(\beta (\frac 12)) .
\end{gather*}
Now, we state the following theorems without proof \cite{e1}.

\begin{theorem} \label{thm1}
Assume that $f$ is continuous with $f_x>0$ on $[0,1] \times
\mathbb{R}$ and $g$ is continuous with $0\leq g'<1$ on $\mathbb{R}$.
Let $\beta $ and $\alpha $ be the upper and lower solutions of the BVP
\eqref{a} respectively. Then $\alpha (t) \leq \beta (t)$, $t\in [0,1]$.
\end{theorem}

\begin{theorem}[Method of upper and lower solutions] \label{thm2}
Assume that $f$ is continuous on $[0,1] \times \mathbb{R}$ and $g$ is
continuous on $\mathbb{R}$ satisfying $0\leq g'<1$. Further, we assume that
there exists an upper solution $\beta $ and a lower solution $\alpha $ of
the BVP \eqref{a} such that $\alpha (t) \leq \beta (t)$, $t\in [0,1]$.
Then there exists a solution $x$ of the BVP \eqref{a}
such that
\[
\alpha (t) \leq x\leq \beta (t) ,\quad t\in [0,1] .
\]
\end{theorem}

\section{Main Result}
\begin{theorem}[Generalized quasilinearization method] \label{thm3}\quad
\begin{enumerate}
\item[$(A_1)$]  $f,f_x$ are continuous on $[0,1]\times \mathbb{R}$ and
$f_{xx}$
exists on $[0,1]\times \mathbb{R}$. Further, $f_x>0$ and $f_{xx}+\phi
_{xx}\leq
0$, where $\phi ,\phi _x$ are continuous on $[0,1]\times \mathbb{R}$ and
$\phi
_{xx}\leq 0$.
\item[$(A_2)$]  $g$, $g'$ are continuous on $\mathbb{R}$ and $g''$ exists
and
$0\leq g'<1$, $g''(x) \geq 0$, $x\in \mathbb{R}$.
\item[$(A_3) $]  $\alpha $ and $\beta $ are lower and upper
solutions of the BVP \eqref{a} respectively.
\end{enumerate}
Then there exists a monotone sequence $\left\{ w_n\right\} $ of solutions
converging quadratically to the unique solution $x$ of the BVP \eqref{a}.
\end{theorem}

\paragraph{Proof.}
Define $F:[0,1] \times \mathbb{R}\to \mathbb{R}$ as
\[
F(t,x) =f(t,x) +\phi (t,x) .
\]
Then, in view of $(A_1) $, we note that $F$, $F_x$ are
continuous on $[0,1] \times \mathbb{R}$, and $F_{xx}$ exists such that
\begin{equation}
F_{xx}(t,x) \leq 0.  \label{2.1}
\end{equation}
Using the mean value theorem and the assumptions $(A_1) $ and
$(A_2) $, we obtain
\begin{gather}
f(t,x)  \leq F(t,y) +F_x(t,y) (x-y) -\phi (t,x) ,  \label{2.2} \\
g(x)  \geq g(y) +g'(y) (x-y) ,  \label{2.3}
\end{gather}
where $x,y\in \mathbb{R}$ such that $x\geq y$ and $t\in [0,1]$. Here, we
remark that
\eqref{2.2} and \eqref{2.3} are also valid independent of the requirement
$x\geq y$.
Define the functions $\stackrel *F (t,x,y) $ and $h(x,y) $ as
\begin{gather*}
\stackrel *F(t,x,y) =F(t,y) +F_x(t,y) (x-y) -\phi (t,x) , \\
h(x,y) =g(y) +g'(y) (x-y) .
\end{gather*}
We observe that
\begin{equation}
f(t,x) =\min_yF^* (t,x,y) . \label{2.4}
\end{equation}
Further
\begin{equation} \begin{aligned}
\stackrel *F_x(t,x,y) =& F_x(t,y) -\phi_x(t,x) \geq F_x(t,x) -\phi _x(t,x)
\\
=&f_x(t,x) >0, \end{aligned}\label{2.4a}
\end{equation}
implies that $\stackrel *F (t,x,y) $ is increasing in $x$ for
each fixed $(t,y) \in [0,1]\times \mathbb{R}$. Similarly
\begin{gather}
g(x) =\max_y h(x,y), \label{2.4b}  \\
0 \leq h'(x,y) <1. \label{2.4c}
\end{gather}
Now, set $\alpha =w_0$, and consider the three point BVP
\begin{equation} \begin{gathered}
x'' =\stackrel *F (t,x(t) ,w_0(t)) ,\quad t\in [0,1]=J  \\
x(0) =a,\quad x(1) =h(x(\frac12),w_0(\frac 12)) . \end{gathered}
\label{2.6}
\end{equation}
Using $(A_3) $ together with \eqref{2.4} and \eqref{2.4b}, we have
\begin{gather*}
w_0'' \geq f(t,w_0) =\stackrel *F (t,w_0,w_0) ,\quad t\in [0,1] \\
w_0(0) \leq a,\quad w_0(1) \leq g(w_0(\frac 12)) =h(w_0(\frac 12),
w_0(\frac 12)) ,
\end{gather*}
and
\begin{gather*}
\beta '' \leq f(t,\beta) \leq \stackrel *F (t,\beta ,w_0) ,\quad t\in
[0,1] \\
\beta (0) \geq a,\quad \beta (1) \geq g(\beta (\frac 12))
\geq h(\beta (\frac 12),w_0(\frac 12)) ,
\end{gather*}
which imply that $w_0$ and $\beta $ are lower and upper solutions of the
BVP
\eqref{2.6} respectively. In view of
\eqref{2.4a} \eqref{2.4c} and the fact that $w_0$ and
$\beta $ are lower and upper solutions of the BVP \eqref{2.6} respectively,
it follows by Theorems \ref{thm1} and \ref{thm2} that
there exists a unique solution $w_1$ of the BVP \eqref{2.6} such that
\[
w_0(t) \leq w_1(t) \leq \beta (t) ,\quad t\in [0,1].
\]
Now, consider the BVP
\begin{equation} \begin{gathered}
x'' =\stackrel *F (t,x(t),w_1(t)) ,\quad t\in [0,1]=J \\
x(0) =a,~~x(1) =h(x(\frac12),w_1(\frac 12)) .
\end{gathered} \label{2.8}
\end{equation}
Again, using $(A_3) $, \eqref{2.4} and \eqref{2.4b}, we find that $w_1$
and $\beta $ are lower and upper solutions
of \eqref{2.8} respectively, that is,
\begin{gather*}
w_1'' =\stackrel *F (t,w_1,w_0) \geq \stackrel *F (t,w_1,w_1) ,\quad t\in
[0,1] \\
w_1(0) =a,\quad
w_1(1) =h(w_1(\frac12),w_0(\frac 12)) \leq h(w_1(\frac 12),w_1(\frac 12)) ,
\end{gather*}
and
\begin{gather*}
\beta '' \leq f(t,\beta) \leq \stackrel *F (t,\beta ,w_1) ,\quad t\in
[0,1] \\
\beta (0) \geq a,\quad
\beta (1) \geq g(\beta (\frac 12)) \geq h(\beta (\frac 12),w_1(\frac 12)) .
\end{gather*}
Hence, by Theorems \ref{thm1} and \ref{thm2}, there exists a unique
solution
$w_2$ of \eqref{2.8} such that
\[
w_1(t) \leq w_2(t) \leq \beta (t) ,\quad t\in [0,1].
\]
Continuing this process successively, we obtain a monotone sequence
$\{w_n\} $ of solutions satisfying
\[
w_0(t) \leq w_1(t) \leq \dots \leq w_n(t) \leq \beta (t) ,\quad t\in [0,1],
\]
where each element $w_n$ of the sequence is a solution of the BVP
\begin{gather*}
x'' =\stackrel *F (t,x(t) ,w_{n-1}(t)) ,\quad t\in [0,1]=J \\
x(0) =a,\quad x(1) =h(x(\frac12),w_{n-1}(\frac 12)) ,
\end{gather*}
and
\begin{equation}
w_n(t) =a(1-t) +h(w_n(\frac 12),w_{n-1}(\frac
12)) t+\int_0^1G(t,s) \stackrel *F (s,w_n,w_{n-1}) ds.  \label{2.9}
\end{equation}
Employing the fact that $[0,1]$ is compact and the monotone convergence is
pointwise, it follows that the convergence of the sequence is uniform. If
$x(t) $ is the limit point of the sequence, then passing onto the
limit $n\to \infty $, \eqref{2.9} gives
\begin{eqnarray*}
x(t) &=&a(1-t) +h(x(\frac 12),x(\frac12)) t+\int_0^1G(t,s)
\stackrel *F (s,x(s) ,x(s)) ds \\
&=&a(1-t) +g(x(\frac 12)) t+\int_0^1G(t,s) f(s,x(s)) ds.
\end{eqnarray*}
Thus, $x(t) $ is the solution of the BVP \eqref{a}.
Now, we show that the convergence of the sequence is quadratic. For that,
set
\[
e_n(t) =x(t) -w_n(t) ,\quad t\in [0,1].
\]
Observe that
\begin{gather*}
e_n(t) \geq 0,\quad e_n(0) =0, \\
e_n(1) =g(x(\frac 12)) -h(w_n(\frac 12),w_{n-1}(\frac 12)) .
\end{gather*}
Using the mean value theorem repeatedly, $(A_1) $ and the
nonincreasing property of $F_x$, we have
\begin{eqnarray*}
e_{n+1}''(t)  &=&x''(t) -w_{n+1}''(t)  \\
&=&f(t,x) -[F(t,w_n) +F_x(t,w_n) (w_{n+1}-w_n) -\phi (t,w_{n+1}) ] \\
&=&F_x(t,c_1) (x-w_n) -F_x(t,w_n)
(x-w_n)+F_x(t,w_n) (x-w_{n+1}) \\
&&-\phi _x(t,c_2) (x-w_{n+1}) \\
&=&(F_{xx}(t,c_3) (c_1-w_n)(x-w_n) +(F_x(t,w_n) -\phi _x(t,c_2)
)(x-w_{n+1}) \\
&\geq &F_{xx}(t,c_3) (x-w_n) ^2+(F_x(t,c_2) -\phi _x(t,c_2) )(x-w_{n+1}) \\
&=&F_{xx}(t,c_3) (e_n) ^2+f_x(t,c_2)
e_{n+1} \\
&\geq &F_{xx}(t,c_2) (e_n) ^2\geq -M\parallel
e_n\parallel ^2,
\end{eqnarray*}
where $M$ is a bound on $F_{xx}(t,x) $ for $t\in [0,1]$,
$w_n<c_3<c_1<x(t) $, $w_{n+1}<c_2<x(t) $, and $||\cdot ||$
denotes the supremum norm on $C[0,1]$. Thus, we have
\begin{eqnarray*}
e_{n+1}(t)  &=&[g(x(\frac 12))-h(w_{n+1}(\frac 12),w_n(\frac
12))]t+\int_0^1G(t,s) e_{n+1}''(s) ds
\\
&\leq &[g(x(\frac 12))-g(w_n(\frac 12))-g'(w_n(\frac
12))(w_{n+1}(\frac 12)-w_n(\frac 12))]t \\
&&+\int_0^1G(t,s)M\|e_n\|^2ds \\
&\leq &[g'(c_o)(x(\frac 12)-w_n(\frac 12))-g'(w_n(\frac
12))(w_{n+1}(\frac 12)-w_n(\frac 12))]t \\
&&+M\|e_n\|^2\int_0^1\left| G(t,s)\right| ds \\
&=&[g''(c_1)(c_o-w_n(\frac 12))(x(\frac 12)-w_n(\frac
12))+g'(w_n(\frac 12))e_{n+1}]t \\
&&+M_1\|e_n\|^2 \\
&\leq &[g''(c_1)e_n^2(t)+g'(w_n(\frac
12))e_{n+1}]t+M_1\parallel e_n\parallel ^2,
\end{eqnarray*}
where $w_n(\frac 12)<c_1<c_o<x(\frac 12)$. Taking the maximum over the
interval $[0,1]$, we get
\[
\|e_{n+1}\|\leq M_2\|e_n\|^2+\lambda \|e_{n+1}\|+M_1\|e_n\|^2.
\]
Solving algebraically, we get
\[
\|e_{n+1}\|\leq \frac{M_3}{1-\lambda }\|e_n\|^2,
\]
where, $| g'| \leq \lambda <1$, $M_1$ provides a bound
on $M\int_0^1\mid G(t,s) \mid ds$, $M_2$ provides a bound for
$| g''| $ on $[w_n(\frac 12),x(\frac 12)]$ and
$M_3=M_1+M_2$. This establishes the quadratic convergence.

\section{Rapid Convergence}

\begin{theorem} \label{thm4} Assume that
\begin{enumerate}
\item[$(B_1)$]  $\frac{\partial ^i}{\partial x^i}f(t,x) $ $(i=0,1,2,\dots
k)$
are continuous on $[0,1]\times \mathbb{R}$ satisfying
\begin{gather*}
\frac{\partial ^i}{\partial x^i}f(t,x) \geq 0,\quad (i=0,1,2,\dots k-1) \\
\frac{\partial ^k}{\partial x^k}(f(t,x) +\phi (t,x)) \leq 0,
\end{gather*}
where $\frac{\partial ^i}{\partial x^i}\phi (t,x) $
$(i=0,1,2,\dots k)$ are continuous and $\frac{\partial ^k}{\partial
x^k}\phi
(t,x)<0$ for some function $\phi (t,x) $.

\item[$(B_{2}) $]  $\alpha ,\beta \in C^2[J,\mathbb{R}]$ are lower and
upper solutions of the BVP \eqref{a}.

\item[$(B_{3}) $]  $\frac{d^i}{dx^i}g(x) $
$(i=0,1,2,\dots k)$ are continuous on $\mathbb{R}$ satisfying
\[
0\leq \frac{d^i}{dx^i}g(x) <\frac M{(\beta -\alpha)
^{i-1}},
\]
with
$0<M<\frac 13$ and $\frac{d^k}{dx^k}g(x) \geq 0$.
\end{enumerate}
Then there exists a monotone sequence of solutions $\{w_n\}$
that converge to the unique solution, $x$, of the BVP \eqref{a} with
the order of convergence $k\geq 2$.
\end{theorem}

\paragraph{Proof.}
Define $F:[0,1] \times \mathbb{R}\to \mathbb{R}$ as
\[
F(t,x) =f(t,x) +\phi (t,x) .
\]
Using $(B_1) $, $(B_3) $ and the generalized mean value theorem,
we obtain
\begin{gather*}
f(t,x)  \leq \sum_{i=0}^{k-1}\frac{\partial ^i}{\partial x^i}
F(t,y) \frac{(x-y) ^i}{i!}-\phi (t,x) , \\
g(x)  \geq \sum_{i=0}^{k-1}\frac{d^i}{dx^i}g(y)\frac{(x-y) ^i}{i!}.
\end{gather*}
Define
\begin{equation}
\stackrel{**}{F}(t,x,y) =\sum_{i=0}^{k-1}\frac{\partial ^i}{
\partial x^i}F(t,y) \frac{(x-y) ^i}{i!}-\phi (t,x) ,  \label{3.1}
\end{equation}
and
\begin{equation}
\stackrel *h (x,y) =\sum_{i=0}^{k-1}\frac{d^i}{dx^i}g(y)
\frac{(x-y) ^i}{i!}.  \label{3.2}
\end{equation}
Observe that $\stackrel{**}{F}(t,x,y) $ and $\stackrel *h (x,y) $ are
continuous and further
\begin{gather}
f(t,x) = \min_y \stackrel{**}{F}(t,x,y) , \label{3.1a} \\
g(x)  = \max_y \stackrel *h (x,y) . \label{3.2a}
\end{gather}
Using generalized mean value theorem, \eqref{3.1} can be written as
\begin{equation}
\stackrel{**}{F}(t,x,y)
=\sum_{i=0}^{k-1}\frac{\partial ^i}{\partial x^i}f(t,y)
\frac{(x-y) ^i}{i!}-\frac{\partial ^k}{\partial x^k}\phi (t,\xi)
\frac{(x-y)^k}{k!}.  \label{3.3}
\end{equation}
Differentiating \eqref{3.3} and using $(B_1)$, we get
\begin{equation}
\stackrel{\ast *}{F}_x(t,x,y) >\sum_{i=0}^{k-1}\frac{\partial ^i%
}{\partial x^i}f(t,y) \frac{(x-y) ^{i-1}}{(i-1)!}%
\geq 0,  \label{3.4}
\end{equation}
which implies that $\stackrel{**}{F}(t,x,y) $ is increasing in
$x$ for each $(t,y) \in [0,1] \times \mathbb{R}$. Similarly,
differentiation of \eqref{3.2}, in view of $(B_3)$, yields
\[
\stackrel{*}{h}{}'(x,y) =\sum_{i=0}^{k-1}\frac{d^i}{%
dx^i}g(y)\frac{(x-y) ^{i-1}}{(i-1)!}.
\]
Clearly $\stackrel *h{} '(x,y) \geq 0$ and
\begin{eqnarray*}
\stackrel *h{}'(x,y)  &=&\sum_{i=0}^{k-1}\frac{d^i}{dx^i}g(y)
\frac{(x-y) ^{i-1}}{(i-1)!}\leq \sum_{i=0}^{k-1}
\frac{d^i}{dx^i}g(y)\frac{(\beta -\alpha) ^{i-1}}{(i-1)!} \\
&\leq &\sum_{i=0}^{k-1}\frac M{(i-1)!}<M(1+\sum_{i=0}^{k-2}\frac
1{2^{k-1}})=M(3-\frac 1{2^{k-3}}) \\
&<&3M<1.
\end{eqnarray*}
Now, set $\alpha =w_0$, and consider the linear BVP
\begin{equation} \begin{gathered}
x'' =\stackrel{**}{F}(t,x(t),w_0(t)) ,\quad t\in [0,1]=J  \\
x(0)  =a,\quad x(1) =\stackrel *h (x(\frac 12),w_0(\frac 12)) .
\end{gathered} \label{3.6}
\end{equation}
Using $(B_2) $, \eqref{3.1a} and \eqref{3.2a}, we find that
\begin{gather*}
w_0'' \geq f(t,w_0) =\stackrel{**}{F}(t,w_0,w_0) ,\quad t\in [0,1] \\
w_0(0) \leq a,\quad
w_0(1) \leq g(w_0(\frac 12)) =\stackrel *h (w_0(\frac 12),w_0(\frac 12)) ,
\end{gather*}
and
\begin{gather*}
\beta '' \leq f(t,\beta) \leq \stackrel{**}{F}(t,\beta ,w_0) ,\quad t\in
[0,1] \\
\beta (0) \geq a,\quad \beta (1) \geq g(\beta (\frac 12)) \geq
\stackrel *h (\beta (\frac 12),w_0(\frac 12)) ,
\end{gather*}
imply that $w_0$ and $\beta $ are lower and upper solutions of the BVP
\eqref{3.6} respectively. It follows by Theorems \ref{thm1} and
\ref{thm2} that there exists a unique solution $w_1$ of the BVP
\eqref{3.6} such that
\[
w_0(t) \leq w_1(t) \leq \beta (t) ,\quad t\in [0,1].
\]
Continuing this process successively, we obtain a monotone sequence
$\{w_n\} $ of solutions satisfying
\[
w_0(t) \leq w_1(t) \leq w_2(t) \leq
\dots \leq w_n(t) \leq \beta (t) ,\quad t\in [0,1],
\]
where each element $w_n$ of the sequence is a solution of the BVP
\begin{gather*}
x'' =\stackrel{**}{F}(t,x(t),w_{n-1}(t)) ,\quad t\in [0,1]=J \\
x(0)  =a,\quad x(1) =\stackrel *h (x(\frac 12),w_{n-1}(\frac 12)) ,
\end{gather*}
and is given by
\begin{equation}
w_n(t) =a(1-t) +\stackrel *h (w_n(\frac
12),w_{n-1}(\frac 12)) t+\int_0^1G(t,s) \stackrel{**}{F}%
(s,w_n,w_{n-1}) ds.  \label{3.7}
\end{equation}
Again, using the standard arguments employed in the last section, it
follows
that
\begin{eqnarray*}
x(t)  &=&a(1-t) +\stackrel *h (x(\frac
12),x(\frac 12)) t+\int_0^1G(t,s) \stackrel{**}{F}(s,x(s) ,x(s)) ds \\
&=&a(1-t) +g(x(\frac 12)) t+\int_0^1G(t,s) f(s,x(s)) ds.
\end{eqnarray*}
Hence $x(t) $ is the solution of the BVP \eqref{a}.
Now, we show that the convergence of the sequence of iterates is of order
$k\geq 2$. For that, we set
\[
e_n(t) =x(t) -w_n(t) ,\quad a_n(t) =w_{n+1}(t) -w_n(t) ,\quad t\in [0,1].
\]
Note that $e_n(t) \geq 0$, $a_n(t) \geq 0$,
$e_n(t) -a_n(t) =e_{n+1}(t) $, and
\[
e_n(0) =0,\quad e_n(1) =g(x(\frac 12))-h(w_n(\frac 12),w_{n-1}(\frac 12)) .
\]
Also, $e_n(t) \geq $ $a_n(t) $ and hence by
induction $e_n^k(t) \geq a_n^k(t)$.
Using the generalized mean value theorem, we have
\begin{eqnarray}
\lefteqn{e_{n+1}''(t)}\nonumber\\
  &=&x''-w_{n+1}''  \nonumber \\
&=&\Big[\sum_{i=0}^{k-1}\frac{\partial ^i}{\partial x^i}f(t,w_n)
\frac{(x-w_n) ^i}{i!}+\frac{\partial ^k}{\partial
x^k}f(t,\xi) \frac{(x-w_n) ^k}{k!}\Big]
\nonumber \\
&&-\Big[\sum_{i=0}^{k-1}\frac{\partial ^i}{\partial x^i}f(t,w_n)
\frac{(w_{n+1}-w_n) ^i}{i!}-\frac{\partial ^k}{%
\partial x^k}\phi (t,\xi) \frac{(w_{n+1}-w_n) ^k}{k!
}\Big]   \nonumber \\
&=&\sum_{i=0}^{k-1}\frac{\partial ^i}{\partial x^i}f(t,w_n)
\frac{(e_n^i-a_n^i) }{i!}+\frac{\partial ^k}{\partial x^k}
f(t,\xi) \frac{(e_n) ^k}{k!}+\frac{\partial ^k}{%
\partial x^k}\phi (t,\xi) \frac{(a_n) ^k}{k!}
\nonumber \\
&\geq &\big(\sum_{i=0}^{k-1}\frac{\partial ^i}{\partial x^i}f(t,w_n)
\frac 1{i!}\sum_{i=0}^{k-1}e_n^ja_n^{i-1-j}\big) e_{n+1}
+(\frac{\partial ^k}{\partial x^k}f(t,\xi) +\frac{
\partial ^k}{\partial x^k}\phi (t,\xi)) \frac{(e_n) ^k}{k!}  \nonumber \\
&\geq &\frac{\partial ^k}{\partial x^k}F(t,\xi) \frac{(e_n) ^k}{k!}
\geq -M\|e_n\|^k,  \label{3.8}
\end{eqnarray}
where $M$ is a bound on $\frac 1{k!}\frac{\partial ^k}{\partial
x^k}F(t,\xi) $
for $t\in [0,1] $. Thus, in view of \eqref{3.8}, we have
\begin{eqnarray}
\lefteqn{e_{n+1}(t)}\nonumber \\
&=&\big(g(x(\frac 12))-h(w_{n+1}(\frac
12),w_n(\frac 12))\big) t+\int_0^1G(t,s) e_{n+1}''(t) ds  \nonumber \\
&\leq &\big(g(x(\frac 12))-h(w_{n+1}(\frac 12),w_n(\frac 12))\big)
t+M\|e_n\|^k\int_0^1\left| G(t,s) \right| ds
\nonumber \\
&=&\Big[\sum_{i=0}^{k-1}\frac{d^i}{dx^i}g(w_n(\frac 12))\frac{(x(\frac
12)-w_n(\frac 12)) ^i}{i!}+\frac{d^k}{dx^k}g(\xi (\frac 12))\frac{%
(x(\frac 12)-w_n(\frac 12)) ^k}{k!}  \nonumber \\
&&-\sum_{i=0}^{k-1}\frac{d^i}{dx^i}g(w_n(\frac 12))
\frac{(w_{n+1}(\frac 12)-w_n(\frac 12)) ^i}{i!}\Big]t+M_1\|e_n\|^k
\nonumber \\
&=&\Big[\sum_{i=0}^{k-1}\frac{d^i}{dx^i}g(w_n(\frac 12))\frac{(e_n^i(\frac
12)-a_n(\frac 12)) ^i}{i!}+\frac{d^k}{dx^k}g(\xi (\frac 12))\frac{%
(e_n(\frac 12)) ^k}{k!}\Big]t   +M_1\|e_n\|^k  \nonumber \\
&=&\Big[\sum_{i=0}^{k-1}\frac{d^i}{dx^i}g(w_n(\frac 12))\frac
1{i!}\sum_{i=0}^{k-1}e_n^j(\frac 12)a_n^{i-1-j}(\frac 12))
e_{n+1}(\frac 12)  \nonumber \\
&&+\frac{d^k}{dx^k}g(\xi (\frac 12))\frac{(e_n(\frac 12)) ^k}{k!}%
\Big]t+M_1\|e_n\|^k   \label{3.9} \\
&\leq &\Big[\sum_{i=1}^{k-1}\frac M{(\beta -\alpha )^{i-1}}\frac
1{i!}\sum_{j=0}^{i-1}e_n^{i-1-j}(\frac 12)a_n^j(\frac 12)\Big]
e_{n+1}(\frac 12) +M_2\|e_n\|^k+M_1\|e_n\|^k. \nonumber
\end{eqnarray}
Letting
\[
P_n(t) =\sum_{i=1}^{k-1}\frac M{(\beta -\alpha )^{i-1}}\frac
1{i!}\sum_{j=0}^{i-1}e_n^{i-1-j}(\frac 12)a_n^j(\frac 12),
\]
we observe that
\[
\lim_{n\to \infty }P_n(t) =\lim_{n\to \infty }
\sum_{i=1}^{k-1}\frac M{(\beta -\alpha
)^{i-1}}\frac 1{i!}\sum_{j=0}^{i-1}e_n^{i-1-j}(\frac 12)a_n^j(\frac 12)
=M<\frac 13.
\]
Therefore, we can choose $\lambda <1/3$ and $n_0\in N$ such that for
$n\geq n_0$, we have $P_n(t) <\lambda $ and consequently \eqref{3.9}
becomes
\[
\|e_{n+1}\|<\lambda \|e_{n+1}\|+M_3\|e_n\|^k.
\]
Solving algebraically, we obtain
\[
\|e_{n+1}\|\le \frac{M_3}{1-\lambda }\|e_n\|^k,
\]
where $M_3=M_1+M_2,\;M_1$ provides bound for $M\int_0^1| G(t,s)|ds$, and
$M_2$ provides bound for
$$\frac{d^k}{dx^k}g(\xi(\frac 12))\frac 1{k!}.
$$

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\noindent\textsc{Bashir Ahmad}\\
Department of Mathematics, King Abdulaziz University,\\
Jeddah-21589, Saudi Arabia\\
e-mail:bashir\_qau@yahoo.com
\smallskip

\noindent\textsc{Rahmat Ali Khan}\\
Department of Mathematics, Quaid-i-Azam University, \\
Islamabad, Pakistan\\
e-mail:rahmat\_alipk@yahoo.com
\smallskip

\noindent\textsc{Paul W. Eloe}\\
Department of Mathematics, University of Dayton,\\
Dayton, OH 45469-2316, USA \\
e-mail: Paul.Eloe@notes.udayton.edu

\end{document}