\documentclass[twoside]{article}
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\markboth{\hfil Quadratic Convergence of Approximate Solutions  \hfil}%
{\hfil V. Doddaballapur, P. W. Eloe \& Y. Zhang \hfil}
\begin{document}
\setcounter{page}{81}
\title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent
{\sc  Differential Equations and Computational Simulations III}\newline
J. Graef, R. Shivaji, B. Soni J. \& Zhu (Editors)\newline
Electronic Journal of Differential Equations, Conference~01, 1997, pp. 81--95. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  147.26.103.110 or 129.120.3.113 (login: ftp)}
 \vspace{\bigskipamount} \\
  Quadratic Convergence of Approximate Solutions of Two-Point  
Boundary Value Problems with Impulse 
\thanks{ {\em 1991 Mathematics Subject Classifications:} 34A37, 34B15.
\hfil\break\indent
{\em Key words and phrases:} Quasilinearization, boundary value problem with 
impulse, \hfil\break\indent 
quadratic convergence, Nagumo conditions.
\hfil\break\indent
\copyright 1998 Southwest Texas State University  and University of
North Texas. \hfil\break\indent
Published November 12, 1998.} }

\date{}
\author{Vidya Doddaballapur, Paul W. Eloe, \& Yongzhi Zhang}
\maketitle

\begin{abstract}
The method of quasilinearization, coupled with the method of
upper and lower solutions, is applied to a boundary value problem for an 
ordinary differential equation with impulse that has a unique solution.
The method generates sequences of approximate solutions which converge
monotonically and quadratically to the unique solution.  In this work, we 
allow nonlinear terms with respect to velocity; in particular, Nagumo
conditions are employed.
\end{abstract}


\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{proposition}[theorem]{Proposition}


\section{Introduction}

Let $0=t_{0}<t_{1}<\dots <t_{m}<t_{m+1}=1$ be given.  In this paper, we shall 
apply the method of quasilinearization to the two-point
conjugate boundary value problem (BVP) with impulse,
\begin{eqnarray} 
&x''(t)=f(t,x(t), x'(t)),\quad t_{k}< t< t_{k+1},\quad k=0,\dots ,m,&
\label{e1} \\
&x(0)=a,\quad x(1)=b, &\label{e2}
\end{eqnarray}
and for $k=1,\dots ,m$,
\begin{eqnarray}  
&\Delta x(t_{k})=u_{k}&  \label{e3} \\ 
&\Delta x'(t_{k})=v_{k}(x(t_{k}),x'(t_{k}))\,,& \nonumber
\end{eqnarray}
where $f:[0,1]\times {\mathbb R}^2 \rightarrow {\mathbb R}$ is continuous, 
$u_{k}\in {\mathbb R}$, $v_{k}: {\mathbb R}^2 \rightarrow {\mathbb R}$ is continuous,   
$k=1,\dots ,m$.  Define the impulse, $\Delta x(t_{k})=x(t_{k}^{+})-  
x(t_{k}^{-})$, and by convention, let $x(t_{k})=x(t_{k}^{-})$, 
$k=1,\dots ,m$.
We shall employ the method of 
upper and lower solutions and the method of quasilinearization to obtain a
bilateral iteration scheme in which the iterates converge quadratically to
the unique solution of the BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}).

The method of quasilinearization is described by Bellman \cite{l4,l5},
and has recently been generalized by Lakshmikantham, Leela  and various co-authors
to apply to a wide variety of problems.  See, for example, \cite{l14,l15},
and references therein.  
The method generates sequences of approximate solutions which converge
monotonically and quadratically to the problem of interest.  Recently, 
Vatsala, et. al., \cite {l16}, \cite{l17,l18}, have applied
the method of quasilinearization to families of two-point BVPs related to
(\ref{e1}), in the case that $f$ is independent of $x'$, and the boundary 
conditions are more general than (\ref{e2}).
 
More recently, Eloe and Zhang \cite{l9} extended the work of Vatsala, et. al.
\cite{l16,l17,l18} to the BVP, (\ref{e1}), (\ref{e2}), in the case 
where $f$ depends on $x'$. As pointed out in \cite{l9}, Knobloch \cite{l13}
and Jackson and Schrader \cite{l12} have obtained conditions
such that there exists a sequence
of solutions of (\ref{e1}) converging monotonically and in $C^{1}[0,1]$ to a 
solution of the BVP, (\ref{e1}), (\ref{e2}).  Neither Knoblach \cite{l13}
nor Jackson and Schrader \cite{l12} considered the rate of convergence.

Also, recently, Devi, Chandrakala and Vatsala \cite{l6} applied the method
of quasilinearization to initial value problems for scalar ordinary
differential equations with impulse.
Doddaballapur and Eloe \cite{l7} have extended the work of 
Vatsala, et. al.
\cite{l16, l17}, \cite{l18} to the BVP with impulse, (\ref{e1}), (\ref{e2}), 
(\ref{e3}), in the case that $f$ and each $v_{k}$ are independent of $x'$.
Thus, the primary contribution of this paper then is that we extend the work 
in \cite{l16}, \cite{l17}, \cite{l18}, \cite{l9} and \cite{l7} to the BVP with impulse, 
(\ref{e1}), (\ref{e2}), (\ref{e3}), when $f$ and each $v_{k}$ depend on $x'$.  

The paper is organized in the following manner.  We shall obtain
a preliminary result in Theorem 1 concerning the properties of upper and lower solutions
of the BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}).  In Theorem 2, we 
shall obtain a fundamental existence of solutions result for the BVP with
impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}).  The proof of this result employs
the Schauder fixed point theorem.  Due to the dependence on $x'$, technical 
difficulties arise which require the assumption of Nagumo type conditions and
extensions of the Kamke convergence theorem \cite{l10, l11}.  In Theorem 3,
we shall state a uniqueness of solutions result for the BVP with impulse,
(\ref{e1}), (\ref{e2}), (\ref{e3}).  We shall state and prove 
the main result of this paper in Theorem 4.  The proof of Theorem 4 
employs a clever manipulation of Theorems 1 and 2.  The iterative details 
in the proof of Theorem 4 are completely analogous to those found in 
\cite{l7, l9, l16, l17, l9} once Theorems 1 and 2 are obtained.  
Hence, we consider 
these details to be standard and only highlight those details in the proof 
of Theorem 4 that are particular to the BVP with impulse, (\ref{e1}),
(\ref{e2}), (\ref{e3}).

\section{Results}

We begin with the definition of an appropriate Banach space, $B$. 
Let $PC[0,1]$ denote the piecewise continuous functions on $[0,1]$ and let 
$PC^{1}[0,1]$ denote the functions, $x$, such that $x\in PC[0,1]$ and  
$x'\in PC[0,1]$.
Define
$$
B=\{ x\in PC^{1}[0,1]: x^{(i)}|_{[t_{k},\ t_{k+1}]}\in C^{i}[t_{k},t_{k+1}],\ 
k=0,\dots ,m,i=0,1\} ,
$$
with $\|x\|_{B}=\max_{k=0,\dots ,m}\|x\|_{k}$ and 
$\|x\|_{k}=\max_{i=0,1}\sup_{t_{k}\le
t\le t_{k+1}}|x^{(i)}(t)|$.
We shall say that $\alpha \in B$ is a lower solution of the BVP with
impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}), if 
\begin{eqnarray*}
&\alpha ''(t)\ge f(t,\alpha (t),\alpha '(t)),\quad t_{k}<t<t_{k+1},\quad
 k=0,\dots ,m,& \\
&\alpha (0)\le a,\quad \alpha (1)\le b,&
\end{eqnarray*}
and for $k=1,\dots ,m$,
\begin{eqnarray*}
&\Delta \alpha (t_{k})=u_{k}&\\ 
&\Delta \alpha '(t_{k})\ge v_{k}(\alpha (t_{k}), \alpha '(t_{k}))\,.&
\end{eqnarray*}
We shall say that $\beta \in B$ is an upper solution of the BVP with 
impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}), if 
\begin{eqnarray*}
&\beta ''(t)\le f(t,\beta (t), \beta '(t) ),\quad t_{k}<t<t_{k+1},\quad
 k=0,\dots ,m,& \\
&\beta (0)\ge a,\quad \beta (1)\ge b\,,
\end{eqnarray*}
and for $k=1,\dots ,m$, 
\begin{eqnarray*}
&\Delta \beta(t_{k})=u_{k}& \\ 
&\Delta \beta '(t_{k})\le v_{k}(\beta (t_{k}),\beta '(t_{k}) )\,.&
\end{eqnarray*}

For the remainder of this paper, we shall assume that
\begin{eqnarray}
& f\in C([0,1]\times {\mathbb R}^2),\quad
(\partial f/\partial x)=f_{x}\in C([0,1]\times {\mathbb R}^2),& \label{e4}\\
&f_{x}(t,x,y)>0,\ (t,x,y)\in [0,1]\times {\mathbb R}^2,& \label{e5}\\
&v_{k}\in C^{1}({\mathbb R}^2),&\label{e6}
\end{eqnarray}
and for $k=1,\dots ,m$,
\begin{equation} v_{kx}(x,y)>0,\ (x,y)\in {\mathbb R}^2,\quad
v_{ky}(x,y)>0,\ (x,y)\in {\mathbb R}^2. \label{e7}
\end{equation}

In order to obtain Theorem 2, we shall define an appropriate fixed point 
operator, $T$.  For $x\in B$, define an operator $T$ on $x$ by
\begin{equation} \label{e8}
Tx(t)=p(t)+I(t,x)+\int_{0}^{1}G(t,s)f(s,x(s),x'(s))\,ds,
\end{equation}
where $p(t)=a(1-t)+bt$, $I(t,x)=\sum_{k=1}^{m}I_{k}(t,x)$.
For  $k=1,\dots ,m$, let
$$
I_{k}(t,x)=\left\{ \begin{array}{ll}
t(-u_{k}-(1-t_{k})v_{k}(x(t_{k}),x'(t_{k})))&,0\le t\le t_{k},\\ 
(1-t)(u_{k}-t_{k}v_{k}(x(t_{k}),x'(t_{k})))&,t_{k}\le t\le 1\,.
\end{array}
\right. 
$$
Let
$$
G(t,s)=\left\{ \begin{array}{ll}
t(s-1)&,0\le t<s\le 1,\\
s(t-1)&,0\le s<t\le 1,
\end{array}
\right .
$$
denote the Green's function for the BVP, $x''(t)=0$, $0\le t\le 1$, 
$x(0)=0$, $x(1)=0$.
Eloe and Henderson \cite{l8} have argued that $x$ is a solution of the BVP
with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}), if, and only if,
$x\in B$ and $Tx=x$.  Finally, we shall define a partial order on $B$ as
follows:  for $\alpha$, $\beta$ $\in B$, we say that $\alpha\le \beta$ if,
and only if,
$$
\alpha |_{[t_{k},t_{k+1}]}(t)\le \beta |_{[t_{k},t_{k+1}]}(t)
,t_{k}\le t\le t_{k+1},k=0,\dots ,m.
$$

\begin{theorem} \label{t1}
Assume (\ref{e4}), (\ref{e5}), (\ref{e6}), and (\ref{e7}) hold.  
Let $\alpha, \beta$ be lower and upper solutions of the BVP with impulse, 
(\ref{e1}), (\ref{e2}), (\ref{e3}), respectively.  
Then $\alpha \le \beta $.
\end{theorem}

\paragraph{Proof.}  
Set $w(t)=\alpha (t)-\beta (t)$ and note that $w$ is continuous
on $[0,1]$ by (\ref{e3}).  Assume, for the sake of contradiction, that $w$ is 
positive on $[0,1]$.  Since $w(0)\le 0$, $w(1)\le 0$,  
$w$ has a positive maximum at some $\tau \in (0,1)$.  
Assume $\tau \in \cup_{k=0}^{m}(t_{k},t_{k+1})$.  Then
$w''(\tau )\le 0$ and 
$\alpha '(\tau )=\beta '(\tau)$.  However, employing that 
$\alpha$ and $\beta$ are lower and upper solutions of the BVP with impulse, 
(\ref{e1}), (\ref{e2}), (\ref{e3}), respectively, and employing (\ref{e5}), 
it follows that
$$
w''(\tau )=\alpha ''(\tau )-\beta ''(\tau )\ge 
f(\tau ,\alpha (\tau ),\alpha '(\tau ))-f(\tau ,\beta (\tau ), \beta '(\tau )) >0.
$$
This provides a contradiction and so, 
$\tau \notin \cup_{k=0}^{m}(t_{k},t_{k+1})$.
Now, assume that $\tau =t_{k}$ for some $k\in \{ 1,\dots ,m\}$.  By Taylor's
theorem, $w'(t_{k}^{-})\ge 0$ and $w'(t_{k}^{+})\le 0$, or
$\Delta w'(t_{k})\le 0$ and 
$$
\alpha '(t_{k}^{-})=\alpha '(t_{k})\ge \beta '(t_{k})=\beta '(t_{k}^{-}).  
$$
But 
$$
\Delta w'(t_{k})=\Delta \alpha '(t_{k})-\Delta \beta '(t_{k}) 
\ge v_{k}(\alpha (t_{k}),\alpha '(t_{k}))-v_{k}(\beta (t_{k}), \beta '(t_{k}))>0
$$
by (\ref{e7}).  Thus, $\tau \notin \{ t_{1},\dots t_{m}\}$, and
$w(t)\le 0$, $0\le t\le 1$.

\begin{theorem} \label{t2}
Assume $g\in C([0,1]\times {\mathbb R}^2)$, $z_{k}\in C({\mathbb R}^2)$, 
$k=1,\dots ,m$, and assume that each $z_{k}(x,y)$ is monotone increasing
in $y$ for fixed $x$.  Assume that each solution of 
$x''(t)=g(t,x(t),x'(t))$ extends to $[0,1]$, or becomes unbounded on its 
maximal interval of convergence. Let
$\alpha ,\beta $ be lower and upper solutions of the BVP,
\begin{eqnarray} 
&x''(t)=g(t,x(t),x'(t)),\quad t_{k}< t< t_{k+1},& \label{e9}\\
&\Delta x(t_{k})=u_{k} &  \nonumber \\ 
&\Delta x'(t_{k})=z_{k}(x(t_{k}),x'(t_{k}))\,,& \label{e10} 
\end{eqnarray}
with $k=1,\dots ,m$ and  boundary conditions given by (\ref{e2}),
respectively, such that
$$
\alpha \le \beta.
$$
Then, there exists a solution, $x$, of the
BVP with impulse, (\ref{e9}), (\ref{e2}), (\ref{e10}), satisfying
$$
\alpha \le x\le \beta .
$$
\end{theorem}

\paragraph{Proof.} Define
$$
\hat f (t,x,y)=\left\{  \begin{array}{ll}
g(t,\beta (t),y)+(x-\beta (t))/[1+(x-\beta (t))],&x>\beta (t),\\
g(t,x,y),&\alpha (t)\le x\le \beta (t),\\
g(t,\alpha (t),y)+(x-\alpha (t))/[1+|x-\alpha (t)|],&x<\alpha (t),
\end{array}\right .
$$
and for $k=1,\dots ,m$, define 
$$ \hat v_{k} (x,y)=
\left\{ \begin{array}{ll}
z_{k}(\beta (t_{k}),y)+(x-\beta (t_{k}))/[1+(x-\beta (t_{k}))],&x>\beta (t_{k}),\\
z_{k}(x,y),&\alpha (t_{k})\le x\le \beta (t_{k}),\\
z_{k}(\alpha (t_{k}),y)+(x-\alpha (t_{k}))/[1+|x-\alpha (t_{k})|],&x<\alpha
(t_{k}).
\end{array}\right .
$$

Let $N>0$ be such that $|\alpha '(t)|\le N$, $|\beta '(t)|\le N$,
$t\in [t_{k},t_{k+1}]$, $k=0,\dots ,m$.
For each positive integer, $l$, define 
$$
f_{l}(t,x,y) =\left\{ \begin{array}{ll}
\hat f(t,x,N+l),&y>N+l,\\
\hat f(t,x,y),&|y|\le N+l,\\
\hat f(t,x, -(N+l)),&y<-(N+l),
\end{array}\right .
$$
and
$$
v_{kl}(t,x,y) =\left\{ \begin{array}{ll}
\hat v_{k}(x,N+l),&y>N+l,\\
\hat v_{k}(x,y),&|y|\le N+l,\\
\hat v_{k}(x, -(N+l)),&y<-(N+l).
\end{array}\right .
$$
Notice that $f_{l}$ and each $v_{kl}$ are bounded and continuous.
With a standard application
of the Schauder fixed point theorem to the operator $T$, defined by (\ref{e8}),
one obtains a solution, $x_{l}\in B$, to the BVP with impulse, (\ref{e1}), 
(\ref{e2}), (\ref{e3}), with $f=f_{l}$ and each $v_{k}=v_{kl}$ 
bounded and continuous.

We now argue that each solution, $x_{l}$, satisfies $\alpha\le x_{l}\le \beta$. 
We shall show that $x_{l}\le \beta$.  As in the proof of Theorem 1,
assume for the sake of contradiction that $x_{l}-\beta$ has a positive maximum
at $\tau$.  As in the proof of Theorem 1, $\tau\in (0,1)$.  If 
$\tau \in \cup_{k=0}^{m}(t_{k},t_{k+1})$, then
$x_{l}''(\tau )\le \beta ''(\tau )$, and $|x_{l}'(\tau )|=|\beta '(\tau )|
\le N <N+l$.  Thus,
$$
(x_{l}-\beta )''(\tau )\ge (x_{l}-\beta )(\tau )/[1+(x-\beta )(\tau )]>0,
$$
which is a contradiction. 
If $\tau =t_{k}$, for some $k\in\{ 1,\dots ,m\}$, then $x_{l}'(t_{k})
\ge \beta '(t_{k})$.  Since each $z_{k}(x,y)$ is monotone increasing in $y$ 
for fixed $x$, it follows that each $v_{kl}(x,y)$ is monotone increasing 
in $y$ for fixed $x$.  Moreover, note that
$v_{kl}(\beta (t_{k}),\beta '(t_{k}))=$$z_{k}(\beta (t_{k}),\beta '(t_{k}))$.
Thus,
\begin{eqnarray*}
\Delta (x_{l}-\beta )'(t_{k})&\geq& 
v_{kl}(\beta (t_{k}),x_{l}'(t_{k}))
-v_{kl}(\beta (t_{k}),\beta '(t_{k})) \\
&&+ (x_{l}-\beta )(t_{k})/[1+(x_{l}-\beta (t_{k}))]\\
&\ge&(x_{l}-\beta )(t_{k})/[1+(x_{l}-\beta (t_{k}))]>0\,
\end{eqnarray*}
which is also a contradiction.  Therefore, $x_{l}\le \beta$.
To show that $\alpha\le x_{l}$ we follow a similar procedure.

For each $l$ there exists $t_{l}\in [0,t_{1}]$ such that
$$
t_{1}|x_{kl}^{'}(t_{l})|=|x_{kl}(t_{1})-a|
\le \max \{ |\beta (0)-\alpha (t_{1})|, |\beta (t_{1})-\alpha (0)| \} .
$$
Thus, each of the sequences $\{ x_{kl}(t_{l})\}$ and $\{ x_{kl}^{'}(t_{l})\}$
are bounded.  One can now apply the Kamke convergence theorem (see \cite{l11})
for solutions of initial value problems and obtain a subsequence of
$\{ x_{kl}\}$ which converges to a solution of
$x''(t)=\hat f(t,x(t),x'(t))$ on a maximal subinterval of $[0,t_{1}]$.
Clearly, $\alpha (t)\le x(t)\le \beta (t)$ and solutions of
$x''(t)=g(t,x(t),x'(t))$ extend to all of $[0,1]$ or become unbounded;
thus,
$x''(t)=\hat f(t,x(t),x'(t))$ on $[0,t_{1}]$.

Now, apply the impulse defined by (\ref{e10}) at $t_{1}$.  Apply the Kamke
theorem to the subsequence that was extracted in the preceding paragraph.
Because of (\ref{e10}) one can employ $t_{1}=t_{l}$ for each $l$.  Thus,
one obtains a further subsequence which converges to a solution, $x$, of
$x''(t)=\hat f(t,x(t),x'(t))$ on $(0,t_{1})\cup (t_{1},t_{2})$
such that $x$ satisfies (\ref{e10}) at $t_{1}$.

Continue inductively, first applying (\ref{e10}) at each $t_{j}$ and then applying the
Kamke convergence theorem on that subinterval $(t_{j},t_{j+1})$.  Finally, since
$\alpha \le x\le \beta$, $\hat f(t,x(t),x'(t))= f(t,x(t),x'(t))$ and
the proof of Theorem 2 is complete.

\paragraph{Remark.}  For simplicity, we can assume that $g$ satisfies a Nagumo condition
in $x'$ (\cite{l10}, \cite{l11}).  That is, assume that for each $M>0$ there exists a positive
continuous function, $h_{M}(s)$, defined on $[0,\infty )$ such that 
$$
|g(t,x,x')|\le h_{M}(|x'|)
$$
for all $(t,x,x')\in [0,1]\times [-M,M]\times {\mathbb R}$ and such that
$$
\int_{0}^{\infty}(s/h_{M}(s))ds=+\infty .
$$
The assumption that $g$ satisfies a Nagumo condition implies that each solution 
of the differential equation, 
$x''(t)=g(t,x(t),x'(t))$,
either extends to $[0,1]$
or becomes unbounded on its maximal interval of existence 
(\cite{l10}, \cite{l11}).  In our main result, Theorem 4, $g$ will
represent a modification of $f$.  Thus, we shall assume in Theorem 4 that
$f$ satisfies a Nagumo condition in $x'$.

\begin{theorem} \label{t3}
Assume that (\ref{e4}), (\ref{e5}), (\ref{e6}), and (\ref{e7}) hold. 
Then, solutions of the BVP with impulse,
(\ref{e1}), (\ref{e2}), (\ref{e3}), are unique.
\end{theorem}

\paragraph{Proof.}  The uniqueness of solutions result follows immediately from 
Theorem 1 and the observation that solutions are respectively upper and
lower solutions.

\begin{theorem} \label{t4}
Assume that (\ref{e4}), (\ref{e5}), (\ref{e6}), and (\ref{e7}) hold, 
and assume that
$$
(\partial ^2/\partial x^2)f\in C([0,1]\times {\mathbb R}^2),v_{k}''
\in C({\mathbb R}^2), k=1,\dots ,m\,.
$$
Assume that $f$ satisfies a Nagumo condition in $x'$.
Assume that $\alpha _{0}$ and $\beta_{0}$ are lower and upper solutions of the
BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}), respectively.  
Then there exist monotone sequences,
$\{ \alpha_{n}(t)\}$ and $\{ \beta_{n}(t)\}$, which converge in $B$
to the unique solution, $x(t)$, of the BVP with impulse, (\ref{e1}), 
(\ref{e2}), (\ref{e3}), and the convergence is quadratic.
\end{theorem}

\paragraph{Proof.}  
Let $F(t,x):  [0,1]\times {\mathbb R}\rightarrow {\mathbb R}$ be such that
$F, F_{x}, F_{xx}$ are continuous on $[0,1]\times {\mathbb R}$ and 
\begin{equation} \label{e11}
F_{xx}(t,x)\ge 0, (t,x)\in [0,1]\times {\mathbb R} \,.
\end{equation}
Set $\phi_{1} (t,x_{1},x_{2}) =F(t,x_{1})-f(t,x_{1},x_{2})$ on $[0,1]\times 
{\mathbb R}^2$.  From (\ref{e11}) it follows that, if $x_{1},y_{1}\in {\mathbb R}$,
then $F(t,x_{1})\ge F(t,y_{1})+F_{x}(t,y_{1})(x_{1}-y_{1})$.  In particular,
for $x_{1},y_{1},x_{2},y_{2}\in {\mathbb R}$, 
\begin{equation} \label{e12}
f(t,x_{1},x_{2})\ge f(t,y_{1},y_{2})+ 
F_{x}(t,y_{1})(x_{1}-y_{1})
-\phi_{1} (t,x_{1},x_{2})+\phi_{1} (t,y_{1},y_{2}).
\end{equation}

For each $k=1,\dots ,m$, let 
$V_{k}(x):  {\mathbb R}\rightarrow {\mathbb R}$ be such that
$V_{k}, V_{k}', V_{k}''$ are continuous on ${\mathbb R}$ and 
\begin{equation} \label{e13}
V_{k}''(x)\ge 0, \quad x\in {\mathbb R} \,.
\end{equation}
Set $\phi_{2k} (x_{1},x_{2}) =V_{k}(x_{1})-v_{k}(x_{1},x_{2})$ on 
${\mathbb R}^2$.  From (\ref{e13}) it follows that, if $x_{1},y_{1}\in {\mathbb R}$,
then $V_{k}(x_{1})\ge V_{k}(y_{1})+V_{k}'(y_{1})(x_{1}-y_{1})$.  
In particular, for $x_{1},y_{1},x_{2},y_{2}\in {\mathbb R}$, 
\begin{equation} \label{e14}
v_{k}(x_{1},x_{2})\geq v_{k}(y_{1},y_{2})+V_{k}'(y_{1})(x_{1}-y_{1})
-(\phi_{2k} (x_{1},x_{2})-\phi_{2k} (y_{1},y_{2})).
\end{equation}
Define
\begin{eqnarray*}
g(t,x_{1},x_{2};\alpha_{0},\beta_{0},\alpha '_{0})&=&
f(t,\alpha_{0}(t),\alpha '_{0}(t))+
F_{x}(t,\beta_{0}(t))(x_{1}-\alpha_{0}(t))\\
&&-\phi_{1}(t,x_{1},x_{2})+\phi_{1}(t, \alpha_{0}(t),\alpha '_{0}(t))\,,
\\
G(t,x_{1},x_{2};\beta_{0},\beta '_{0})&=&
f(t,\beta_{0}(t),\beta '_{0}(t))+
F_{x}(t,\beta_{0}(t))(x_{1}-\beta_{0}(t)) \\
&&-\phi_{1}(t,x_{1},x_{2})+\phi_{1}(t, \beta_{0}(t),\beta '_{0}(t))\,,
\\
h_{k}(x_{1},x_{2};\alpha_{0},\beta_{0},\alpha '_{0})&=&
v_{k}(\alpha_{0}(t_{k}),\alpha '_{0}(t_{k}))+
V'_{k}(\beta_{0}(t_{k}))(x_{1}-\alpha_{0}(t_{k}))\\
&&-(\phi_{2k}(x_{1},x_{2})-\phi_{2k}(\alpha_{0}(t_{k}),\alpha '_{0}(t_{k})))
\,, \\
H_{k}(x_{1},x_{2};\beta_{0},\beta '_{0})&=&
v_{k}(\beta_{0}(t_{k}),\beta '_{0}(t_{k}))+
V'_{k}(\beta_{0}(t_{k}))(x_{1}-\beta_{0}(t_{k}))\\
&&-(\phi_{2k}(x_{1},x_{2})-\phi_{2k}(\beta_{0}(t_{k}),\beta '_{0}(t_{k})))\,.
\end{eqnarray*}
First consider the BVP with impulse,
\begin{equation} \label{e15}
x''(t)=g(t,x(t),x'(t);\alpha_{0},\beta_{0},\alpha_{0}'),\ t_{k}<t<t_{k+1},
\ k=0,\dots ,m,
\end{equation}
and for $k=1,\dots ,m$, 
\begin{eqnarray} 
&\Delta x(t_{k})=u_{k} & \label{e16} \\ 
& \Delta x'(t_{k})=h_{k}(x(t_{k}),x'(t_{k});\alpha_{0},\beta_{0},\alpha_{0}')
\,,& \nonumber
\end{eqnarray}
with boundary conditions given by (\ref{e2}).
Each $h_{k}$ readily satisfies the hypotheses of Theorem 2.  A limit
comparison implies that $g$ satisfies a Nagumo condition in $x'$.

We now show that $\alpha_{0}$ and $\beta_{0}$ are lower and upper solutions, 
respectively, of the BVP with impulse, (\ref{e15}), (\ref{e2}), (\ref{e16}); 
thus, by Theorem 2, there exists a solution $\alpha_{1}(t)$ of the BVP with 
impulse, (\ref{e15}), (\ref{e2}), (\ref{e16}), satisfying
$$
\alpha_{0}\le \alpha_{1}\le \beta_{0}.
$$
To this end, note that for $t_{k}<t<t_{k+1}$, $k=0,\dots ,m$,
$$ 
\alpha_{0}''(t)\ge f(t,\alpha_{0}(t),\alpha '_{0}(t))
=g(t,\alpha_{0}(t),\alpha '_{0}(t);\alpha_{0},\beta_{0},\alpha '_{0} ), 
$$
and, for $k=1,\dots ,m$,
$$
\Delta \alpha_{0}'(t_{k})\ge v_{k}(\alpha_{0}(t_{k}),\alpha_{0}'(t_{k}))
=h_{k}(\alpha_{0}(t_{k}),\alpha_{0}'(t_{k}) ;\alpha_{0},\beta_{0},\alpha_{0}').
$$
Moreover, from (\ref{e12}) and (\ref{e14}), it follows that for
$t_{k}<t<t_{k+1}$,  $k=0,\dots ,m$,
\begin{eqnarray*}
\beta_{0}''(t)\le f(t,\beta_{0}(t),\beta '_{0}(t))
&\le& f(t,\alpha_{0}(t),\alpha '_{0}(t))- 
F_{x}(t,\beta_{0}(t))(\alpha_{0}(t)-\beta_{0}(t))\\
&&+\phi_{1}(t,\alpha_{0}(t),\alpha '_{0}(t))
-\phi_{1}(t,\beta_{0}(t),\beta '_{0}(t))\\
&=&g(t,\beta_{0}(t),\beta '_{0}(t);\alpha_{0},\beta_{0},\alpha '_{0} ), 
\end{eqnarray*}
and for $k=1,\dots ,m$, 
\begin{eqnarray*}
\Delta \beta_{0}'(t_{k})&\le& v_{k}(\beta_{0}(t_{k}),\beta_{0}'(t_{k}) )\\
&\le& v_{k}(\alpha_{0}(t),\alpha_{0}'(t) )-
V'_{k}(\beta_{0}(t_{k}))(\alpha_{0}(t_{k})-\beta_{0}(t_{k}))\\
&&+(\phi_{2k}(\alpha_{0}(t_{k}),\alpha_{0}'(t) )
-\phi_{2k}(\beta_{0}(t_{k}),\beta_{0}'(t_{k}) )\\
&=&h_{k}(\beta_{0}(t_{k}),\beta_{0}'(t_{k});\alpha_{0},\beta_{0},\beta_{0}').
\end{eqnarray*}
Since $\alpha_{0}$ and $\beta_{0} $ satisfy (\ref{e2}),
$\alpha_{0}$ and $\beta_{0}$ are lower and upper solutions, respectively, of 
the BVP with impulse, (\ref{e15}), (\ref{e2}), (\ref{e16}), and thus, 
by Theorem 2, there exists a solution $\alpha_{1}(t)$ of the BVP with impulse, 
(\ref{e15}), (\ref{e2}), (\ref{e16}),  such that
$$
\alpha_{0}\le \alpha_{1}\le \beta_{0}\,.
$$
Now, consider the BVP with impulse, 
\begin{eqnarray} 
&x''(t)=G(t,x(t),x'(t);\beta_{0},\beta '_{0} ),\quad t_{k}<t<t_{k+1},\quad
k=0,\dots ,m,&\nonumber \\
&x(0)=a,\quad x(1)=b,& \label{e17}
\end{eqnarray}
and for $k=1,\dots ,m$, 
\begin{eqnarray} 
&\Delta x(t_{k})=u_{k} & \label{e18}   \\ 
&\Delta x'(t_{k})=H_{k}(x(t_{k}),x'(t_{k});\alpha_{0},\beta_{0},\beta_{0}')
\,.& \nonumber
\end{eqnarray}
Again, $G$ and each $H_{k}$ satisfy the hypotheses of Theorem 2, and, again,
(\ref{e12}) and (\ref{e14}) are employed to show that 
$\alpha_{0}$ and $\beta_{0}$ are lower and upper solutions, respectively, of the 
BVP with impulse, (\ref{e17}), (\ref{e2}), (\ref{e18}); thus, there exists a 
solution $\beta_{1}(t)$ of the BVP with impulse, (\ref{e17}), (\ref{e2}), 
(\ref{e18}), such that
$$
\alpha_{0}\le \beta_{1}\le \beta_{0}\,.
$$

We now show that 
$\alpha_{1}$ and $\beta_{1}$ are lower and upper solutions, respectively, of the 
BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}).
Thus, it will follow by Theorem 1 that
$$
\alpha_{0}\le \alpha_{1}\le \beta_{1}\le \beta_{0}\,.
$$

Employ (\ref{e12}) and (\ref{e11}) to see that for
$t \in \cup_{k=0}^{m}(t_{k},t_{k+1})$,
\begin{eqnarray*}
\alpha_{1}''(t)&=& g(t,\alpha_{1}(t),\alpha_{1}'(t);
\alpha_{0},\beta_{0},\alpha '_{0} )\\
&=&f(t,\alpha_{0}(t),\alpha '_{0}(t))+
F_{x}(t,\beta_{0}(t))(\alpha_{1}(t)-\alpha_{0}(t)) \\
&&-(\phi (t,\alpha_{1}(t),\alpha_{1}'(t))-
\phi (t, \alpha_{0}(t),\alpha '_{0}(t))) \\
&\ge& f(t,\alpha_{1}(t),\alpha '_{1}(t))+F_{x}(t,\alpha_{1}(t))(\alpha_{0}(t)
-\alpha_{1}(t)) 
+\phi (t,\alpha_{1}(t),\alpha_{1}'(t))\\
&&-\phi (t, \alpha_{0}(t),\alpha '_{0}(t))
+F_{x}(t,\beta_{0}(t))(\alpha_{1}(t)-\alpha_{0}(t))\\
&&-(\phi (t,\alpha_{1}(t),\alpha_{1}'(t))
-\phi (t, \alpha_{0}(t),\alpha '_{0}(t)))\\
&=& f(t,\alpha_{1}(t),\alpha '_{1}(t))+
(F_{x}(t,\beta_{0}(t))-F_{x}(t,\alpha_{1}(t)))(\alpha_{1}(t)-\alpha_{0}(t))\\
&\ge& f(t,\alpha_{1}(t),\alpha_{1}'(t)).
\end{eqnarray*}
Similarly, for $k=1,\dots ,m$, employ (\ref{e14}) and (\ref{e13})
to see that 
\begin{eqnarray*}
\Delta \alpha '_{1}(t_{k})&=&
h_{k}(\alpha_{1}(t_{k}),\alpha '_{1}(t_{k}) );
\alpha_{0},\beta_{0},\alpha '_{0})\\
&=& v_{k}(\alpha_{0}(t_{k}),\alpha '_{0}(t_{k}) )+
V'_{k}(\beta_{0}(t_{k}))(\alpha_{1}(t_{k})-\alpha_{0}(t_{k}))\\
&&-(\phi_{2k}(\alpha_{1}(t_{k}),\alpha '_{1}(t_{k}) )
-\phi_{2k}(\alpha_{0}(t_{k}),\alpha '_{0}(t_{k})))\\
&\ge& v_{k}(\alpha_{1}(t_{k}),\alpha '_{1}(t_{k}))+ 
 V'_{k}(\alpha_{1}(t_{k}))(\alpha_{0}(t_{k})-\alpha_{1}(t_{k})) \\
&&+\phi_{2k}(\alpha_{1}(t_{k}),\alpha '_{1}(t_{k}))
 -\phi_{2k}(\alpha_{0}(t_{k}),\alpha '_{0}(t_{k}))\\
&&+V'_{k}(\beta_{0}(t_{k}))(\alpha_{1}(t_{k})-\alpha_{0}(t_{k}))
 -(\phi_{2k}(\alpha_{1}(t_{k}),\alpha '_{1}(t_{k})) \\
&& -\phi_{2k}(\alpha_{0}(t_{k}),\alpha '_{0}(t_{k})))\\
&=&v_{k}(\alpha_{1}(t_{k}))+ (V'_{k}(\beta_{0}(t_{k}))
-V'_{k}(\alpha_{1}(t_{k})))(\alpha_{1}(t_{k})-\alpha_{0}(t_{k}))\\
&\ge& v_{k}(\alpha_{1}(t_{k})). 
\end{eqnarray*}
Similarly, it follows by (\ref{e11})-(\ref{e14}) that 
for $t \in \cup_{k=0}^{m}(t_{k},t_{k+1})$,
$$
\beta_{1}''(t)\le f(t,\beta_{1}(t),\beta_{1}'(t)), 
$$
and for $k\in\{ 1,\dots ,m\}$,
$$
\Delta \beta '_{1}(t_{k})\le v_{k}(\beta_{1}(t_{k}),\beta_{1}'(t_{k})).
$$
In particular, 
$\alpha_{1}$ and $\beta_{1}$ are lower and upper solutions, respectively, of the 
BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}), and by Theorem 1,
$$
\alpha_{0}\le \alpha_{1}\le \beta_{1}\le \beta_{0}.
$$

Inductively, define sequences of functions $\{ g_{l}\}$, $\{ G_{l}\}$, 
$\{ h_{kl}\}$, and $\{ H_{kl}\}$ by
\begin{eqnarray*}
g_{l}(t,x_{1},x_{2})&=&g(t,x_{1},x_{2};\alpha_{l},\beta_{l},\alpha '_{l})\\
&=& f(t,\alpha_{l}(t),\alpha '_{l}(t))+F_{x}(t,\beta_{l}(t))(x_{1}-\alpha_{l}(t))\\
&& -\phi_{1} (t,x_{1},x_{2})+\phi_{1} (t, \alpha_{l}(t),\alpha '_{l}(t))\,\\
G_{l}(t,x_{1},x_{2})&=&G(t,x_{1},x_{2};\beta_{l},\beta '_{l})\\  
&=& f(t,\beta_{l}(t),\beta '_{l}(t))+F_{x}(t,\beta_{l}(t))(x_{1}-\beta_{l}(t))\\
&&-\phi_{1} (t,x_{1},x_{2})+\phi_{1} (t, \beta_{l}(t),\beta '_{l}(t))\,,\\
h_{kl}&=&h_{k}(x_{1},x_{2};\alpha_{l},\beta_{l},\alpha_{l}')\\
&=&v_{k}(\alpha_{l}(t_{k}),\alpha_{l}'(t_{k}))+
V'_{k}(\beta_{l}(t_{k}))(x_{1}-\alpha_{l}(t_{k}))\\
&&-(\phi_{2k}(x_{1},x_{2})-\phi_{2k}(\alpha_{l}(t_{k})))\,,\\
H_{kl}&=&H_{k}(x_{1},x_{2};\beta_{l},\beta_{l}')\\
&=&v_{k}(\beta_{l}(t_{k}),\beta_{l}'(t_{k}))+V'_{k}(\beta_{l}(t_{k}))
(x_{1}-\beta_{l}(t_{k}))\\
&&-(\phi_{2k}(x_{1},x_{2})-
\phi_{2k}(\beta_{l}(t_{k}),\beta_{l}'(t_{k})))\,.
\end{eqnarray*}
Inductively, Theorem 2 implies there exists a solution $\alpha_{l+1}(t)$
of the BVP with impulse, (\ref{e15}), (\ref{e2}), (\ref{e16}), with $g=g_{l}$
and each $h_{k}=h_{kl}$ satisfying
$$
\alpha_{0}\le \dots \le \alpha_{l}\le \alpha_{l+1}
\le \beta_{l}\le \dots \le \beta_{0}\,.
$$
Similarly, there exists a solution $\beta_{l+1}(t)$
of the BVP with impulse, (\ref{e17}), (\ref{e2}), (\ref{e18}), with $G=G_{l}$
and each $H_{k}=H_{kl}$ satisfying
$$
\alpha_{0}\le \dots \le \alpha_{l}\le \beta_{l+1}
\le \beta_{l}\le \dots \le \beta_{0}.
$$
Finally, inductively, $\alpha_{l+1}$ and $\beta_{l+1}$ are lower and
upper solutions, respectively, of the BVP with impulse, 
(\ref{e1}), (\ref{e2}), (\ref{e3}), and by Theorem 1,
$$
\alpha_{0}\le \dots \le \alpha_{l}\le \alpha_{l+1}
\le \beta_{l+1} \le \beta_{l}\le \dots \le \beta_{0}.
$$

We now show that each sequence $\{ \alpha_{l}\}$ and
$\{ \beta_{l}\}$ converge in $B$ to $x$, the unique solution of the
BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}).  
Recall
$$
B=\{ x\in PC^{1}[0,1]: x^{(i)}|_{[t_{k},t_{k+1}]}\in C^{i}[t_{k},t_{k+1}],
k=0,\dots ,m,i=0,1\} ,
$$
with $\|x\|_{B}=\max_{k=0,\dots ,m}\|x\|_{k}$ and 
$\|x\|_{k}=\max_{i=0,1}\sup_{t_{k}\le
t\le t_{k+1}}|x^{(i)}(t)|$.
The Kamke convergence theorem does not apply directly to either sequence,
$\{ \alpha_{l}\}$ or $\{ \beta_{l}\}$ since neither $g_{l}$ nor $G_{l}$ 
converge uniformly on compact sets to $f$.  To see this, note that 
$$
g_{l}(t,x_{1},x_{2})=f(t,x_{1},x_{2})+F_{x}(t,\beta_{l}(t))(x_{1}-\alpha_{l}(t))+
F(t,\alpha_{l}(t))-F(t,x_{1})
$$
and
$$
G_{l}(t,x_{1},x_{2})=f(t,x_{1},x_{2})+F_{x}(t,\beta_{l}(t))(x_{1}-\beta_{l}(t))+
F(t,\beta_{l}(t))-F(t,x_{1}).
$$
Define 
$$\hat g_{l}(t,x_{1},x_{2})=
f(t,x_{1},x_{2})+F_{x}(t,\beta_{l}(t))(\alpha_{l+1}-\alpha_{l})(t)+
F(t,\alpha_{l}(t))-F(t,\alpha_{l+1}(t))
$$
and 
$$
\hat G_{l}(t,x_{1},x_{2})=
f(t,x_{1},x_{2})+F_{x}(t,\beta_{l}(t))(\beta_{l+1}-\beta_{l})(t)+
F(t,\beta_{l}(t))-F(t,\beta_{l+1}(t)).
$$
Theorem 3 applies to the BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}),
with $f=\hat g_{l}$ and each $v_{k}=h_{kl}$ and note that 
$\alpha_{k+1}$ is the unique solution.  The Kamke convergence 
theorem now does apply and, with omitted details that are similar to those 
given in the proof of Theorem 2, $\{ \alpha_{l}\}$ converges in $B$ to $x$, 
the unique solution of the BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}).
Similarly, $\{ \beta_{l}\}$ converges in $B$ to $x$, the unique solution of 
the BVP with impulse, (\ref{e1}), (\ref{e2}), (\ref{e3}).

We now argue that the convergence is quadratic.  Let 
$q_{n}(t)=\beta _{n}(t)-x(t)$ and $p_{n}(t)=x(t)-\alpha _{n}(t)$,
where $x(t)$ denotes the unique solution of the BVP with impulse,
(\ref{e1}), (\ref{e2}), (\ref{e3}).
Set 
$$
e_{n}=\max \{ \|q_{n}\|_{B},\|p_{n}\|_{B}\}.
$$
First, consider $q_{n+1}(t)$ and note that $q_{n+1}\ge 0$.  For
$t \in \cup_{k=0}^{m}(t_{k},t_{k+1})$, 
\begin{eqnarray*}
q''_{n+1}(t)&=&
F(t,\beta_{n}(t))+F_{x}(t,\beta_{n}(t))(\beta_{n+1}-\beta_{n})(t)\\
&&-\phi_{1}(t, \beta_{n+1}(t),\beta_{n+1}'(t))
-F(t,x(t))+\phi_{1}(t,x(t),x'(t))\\
&=& F_{x}(t,c_{1}(t))q_{n}(t)-F_{x}(t,\beta_{n}(t))q_{n}(t)
+F_{x}(t,\beta_{n}(t))q_{n+1}(t)\\
&&-\phi_{1x}(t,c_{2}(t),c_{3}(t))q_{n+1}(t)
-\phi_{1x'}(t,c_{2}(t),c_{3}(t))q'_{n+1}(t)\,,
\end{eqnarray*}
where $x(t)\le c_{1}(t)\le \beta_{n}(t)$, 
$x(t)\le c_{2}(t)\le \beta_{n+1}(t)$,
and $c_{3}(t)$ is between $x'(t)$ and $\beta '_{n+1}(t)$.  Thus, there exists
$c_{1}(t)\le c_{4}(t)\le \beta_{n}(t)$ such that 
\begin{eqnarray*}
\lefteqn{q''_{n+1}(t)}\\
&=& F_{xx}(t,c_{4}(t))q_{n}(t)(c_{1}(t)-\beta_{n}(t))\\
&&+(F_{x}(t,\beta_{n}(t))-\phi_{1x}(t,c_{2}(t),c_{3}(t)))q_{n+1}(t)
-\phi_{1x'}(t,c_{2}(t),c_{3}(t))q'_{n+1}(t)\\
&\ge& -F_{xx}(t,c_{4}(t))q_{n}^2(t)+f_{x'}(t,c_{2}(t),c_{3}(t))
q_{n+1}'(t)\,.
\end{eqnarray*}
Note that to obtain this inequality, we have employed the monotonicity  
of $F_{x}$ in the second component.  In particular, there exists $M>0$,
such that
\begin{equation} \label{e19}
q''_{n+1}(t)-f_{x'}(t,c_{2}(t),c_{3}(t))q'_{n+1}(t)\ge -Me_{n}^2,
\end{equation}
where $M>\max_{i}\max_{(t,x)\in D_{i}}F_{xx}(t,x)$, and for $i=0,\dots m$,
$$
D_{i}=\{ (t,x):  t_{i}\le t\le t_{i+1},
\alpha_{0}(t)\le x\le \beta_{0}(t)\}\,.
$$ 
Similarly, there exist appropriate $c_{4}$ and $c_{5}$ such that for
$k=1,\dots ,m$,
\begin{equation} \label{e20}
\Delta q_{n+1}'(t_{k})-v_{ky}(c_{4},c_{5})q'_{n+1}(t_{k})\ge -Me_{n}^2\,.
\end{equation}
Let $m(t)=\exp\bigg(-\int_{0}^{t}f_{x'}(s,c_{2}(s),c_{3}(s))ds\bigg)$
denote the integrating factor associated with (\ref{e19}).  Then
\begin{equation} \label{e21}
(q'_{n+1}(t)m(t))'\ge -Mm(t)e_{n}^2\,.
\end{equation}
Thus, for $t_{m}\le t\le 1$,
$$
q'_{n+1}(1)m(1)-q'_{n+1}(t)m(t)\ge -Me_{n}^2\int_{t}^{1}m(s)ds\,.
$$
Since, $q'_{n+1}(1)\le 0$, it follows that 
$$
q'_{n+1}(t)\le Me_{n}^2\int_{t}^{1}m(s)ds/m(t)\,.
$$
Since $q_{n+1}$ converges to $0$ in $B$, eventually $(s,c_{2}(s),c_{3}(s))$ belongs
to
$$
\hat D=\{ (s,x_{1},x_{2}):t_{m}\le s\le 1, \alpha_{0}(s)\le x_{1}\le \beta_{0}(s),
x'(s)-1\le x_{2}\le x'(s)+1\}.
$$
Thus, we can bound $m(t)$ away from both $0$ and $\infty$ for $n$ 
sufficiently large; in particular,
there exists $N_{1}>0$ such that for $t_{m}\le t\le 1$ and $n$ sufficiently large,
\begin{equation} \label{e22}
q'_{n+1}(t)\le N_{1}e_{n}^2\,.
\end{equation}

Apply (\ref{e20}) at $t_{m}$.  Then
$$
q'_{n+1}(t_{m}^{+})-q'_{n+1}(t_{m})
-v_{my}(c_{4},c_{5})q'_{n+1}(t_{m})\ge -Me_{n}^2\,. 
$$
Employ (\ref{e7}) and also bound $v_{my}$ away from both $0$ and $\infty$ to
obtain some $\hat M >0$ such that
\begin{equation} \label{e23}
q'_{n+1}(t_{m}^{-})\ge -\hat M e_{n}^2\,. 
\end{equation}
Now, employ (\ref{e21}) and (\ref{e23}) to obtain (\ref{e22}) for $t_{m-1}
\le t\le t_{m}$ for some $N_{2}>0$.  Again, apply (\ref{e20}) to obtain
a suitable (\ref{e23}) at $t_{m-1}$.  Proceed inductively and obtain that
there exists $N>0$ such that for $t \in \cup_{k=0}^{m}[t_{k},t_{k+1}]$ and
$n$ sufficiently large,
\begin{equation} \label{e24}
q'_{n+1}(t)\le Ne_{n}^2.
\end{equation}


Recall that $q_{n+1}(t)\ge 0$, and that $q_{n+1}\in C[0,1]$.  Integrate 
(\ref{e24}) from $0$ to $t$; then for $n$ sufficiently large,
\begin{equation} \label{e25}
0\le q_{n+1}\le Ne_{n}^2\,.
\end{equation}
  
Beginning again at (\ref{e21}), integrate from $0$ to $t\le t_{1}$ to obtain
$$
q'_{n+1}(t)m(t)-q'_{n+1}(0)\ge -Me_{n}^2\int_{0}^{t}m(s)\,ds\,.
$$
Since, $q'_{n+1}(0)\ge 0$, it follows that for $0\le t\le t_{1}$,
there exists $N_{1}>0$, such that
$$
q'_{n+1}(t)\ge -Me_{n}^2\int_{0}^{t}m(s)\,ds/m(t)\ge -N_{1}e_{n}^2\,,
$$
for $n$ sufficiently large.
This is analogous to (\ref{e22}).  Proceed analogously, then, and choose
$N$ large enough such that for $t \in \cup_{k=0}^{m}[t_{k},t_{k+1}]$
for $n$ sufficiently large,
\begin{equation} \label{e26}
q'_{n+1}(t)\ge -Ne_{n}^2\,.
\end{equation}
It now follows from (\ref{e24}), (\ref{e25}), and (\ref{e26}) that 
$\beta_{n}$ converges to $x$ quadratically in $B$.

The argument that 
$\{ \alpha _{n}\}$ converges quadratically to $x$ in $B$
is similar and we provide some details.
\begin{eqnarray*}
\lefteqn{ p''_{n+1}(t)} \\ &=&
F(t,x(t))-\phi_{1}(t,x(t),x'(t))\\
&&-(F(t,\alpha_{n}(t))+F_{x}(t,\beta_{n}(t))(\alpha_{n+1}-\alpha_{n})(t)
-\phi_{1}(t, \alpha_{n+1}(t),\alpha '_{n+1}(t))) \\
&=& F_{x}(t,c_{1}(t))p_{n}(t)-F_{x}(t,\beta_{n}(t))p_{n}(t)
+F_{x}(t,\beta_{n}(t))p_{n+1}(t) \\
&&-\phi_{1x}(t,c_{2}(t),c_{3}(t))p_{n+1}(t)
-\phi_{1x'}(t,c_{2}(t),c_{3}(t))p'_{n+1}(t)\\
&=& F_{xx}(t,c_{4}(t))p_{n}(t)(c_{1}(t)-\beta_{n}(t))\\
&&+(F_{x}(t,\beta_{n}(t))-\phi_{1x}(t,c_{2}(t),c_{3}(t)))p_{n+1}(t)
-\phi_{1x'}(t,c_{2}(t),c_{3}(t))p'_{n+1}(t) \\
&\ge& -F_{xx}(t,c_{4}(t))p_{n}(t)(p_{n}(t)+q_{n}(t))+
f_{x'}(t,c_{2}(t),c_{3}(t))p_{n+1}'(t)\,.
\end{eqnarray*}
In particular, 
$$
p''_{n+1}(t)-f_{x'}(t,c_{2}(t),c_{3}(t))p '_{n+1}(t)\ge -2Me_{n}^2
$$
on an appropriate set and for sufficiently large $n$.  A similar inequality
is obtained with respect to the impulse and the details for
quadratic convergence follow as above.

\begin{corollary} \label{t5} 
The sequence $\{ \beta ''_{n}(t)-f(t,\beta_{n}(t), \beta '_{n}(t))\}$ 
converges quadratically to $0$ in $B$.
\end{corollary}

\paragraph{Proof:}  There exist $\beta_{n}\ge c_{2}\ge c_{1}\ge \beta_{n+1}$
such that
\begin{eqnarray*}
f(t,\beta_{n+1}(t),\beta_{n+1}'(t))&\ge& \beta_{n+1}''(t)\\
&=&f(t,\beta_{n}(t),\beta_{n}'(t))+F_{x}(t,\beta_{n}(t))(\beta_{n+1}(t)
-\beta_{n}(t))\\
&&-(\phi_{1}(t,\beta_{n+1}(t),\beta_{n+1}'(t))-\phi_{1}(t,\beta_{n}(t),\beta_{n}'(t)))
\\
&=&f(t,\beta_{n+1}(t),\beta_{n+1}'(t))\\
&&+F_{xx}(t,c_{2}(t))(\beta_{n+1}(t)
-\beta_{n}(t))(\beta_{n}(t)-c_{1}(t))\,.
\end{eqnarray*}
Thus, 
\begin{eqnarray*}
0&\le& f(t,\beta_{n+1}(t),\beta_{n+1}'(t))-\beta_{n+1}''(t)\\
&\le& F_{xx}(t,c_{2}(t))(\beta_{n+1}(t)-\beta_{n}(t))^2\\
&\le & F_{xx}(t,c_{2}(t))e_{n}^2\,.
\end{eqnarray*}
Similar inequalities are obtained for the impulse.
Quadratic convergence can also be obtained for the sequence
$$
\{ f(t,\alpha_{n}(t),\alpha '_{n}(t))-\alpha ''_{n}(t)\}.
$$

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\end{thebibliography}
\bigskip

\noindent{\sc Vidya Doddaballapur, Paul W. Eloe, \& Yongzhi Zhang}\\
Department of Mathematics\\ University of Dayton\\
Dayton, Ohio 45469-2316, USA\\
Email address: eloe@@saber.udayton.edu

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