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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 203, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/203\hfil Rapid convergence of approximate solutions]
{Rapid convergence of approximate solutions for singular differential systems}

\author[P. Wang, X. Liu \hfil EJDE-2015/203\hfilneg]
{Peiguang Wang, Xiang Liu}

\address{Peiguang Wang \newline
College of Electronic and Information Engineering,
Hebei University, Baoding 071002, China}
\email{pgwang@hbu.edu.cn}

\address{Xiang Liu \newline
College of Mathematics and Information Science,
 Hebei University, Baoding 071002, China}
\email{1065265114@qq.com}

\thanks{Submitted February 12, 2015. Published August 10, 2015.}
\subjclass[2010]{34A09, 34A45}
\keywords{Singular system; generalized quasilinearization; rapid convergence}

\begin{abstract}
 In this article we show the existence and approximation of solutions
 for a class of singular differential system with initial value condition.
 We present a generalized quasilinearization method for obtain monotone
 iterative sequences of approximate solutions converging uniformly
 to a solution at a rate higher than quadratic.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks


\section{Introduction}

In 1974, Rosenbrock \cite{r1} introduced the concept of singular systems,
which is more complicated than the ordinary ones, and
its qualitative analysis involve greater difficulty than those of the
ordinary systems. A systematic development of the basic theory of the linear
singular systems has been provided by Campbell \cite{c3,c4}.
The results of qualitative properties for nonlinear singular differential
systems can be found in \cite{k1,m2,u1,v1,w1,y1,z1}.
However, we noticed that most of the previous results focused on
stability problems. In fact, the convergence of the solution has an
important function in the development of qualitative theory, and the
rapid convergence of solutions is also very important in practical
applications.

Generalized quasilinearization is an efficient
method for constructing approximate solutions of nonlinear problems.
Bellman and Kalaba \cite{b1}, Lakshmikantham and Vatsala \cite{l1} gave a
systematic development of the method to ordinary differential
equations. Up till now, only some rapid convergence
results have been found on ordinary differential equations,
for initial value problems \cite{c1,m1,m3,
m4}, for boundary value problem 
\cite{c2,m5,w2}.
The applications of the method of quasilinearization in singular differential
systems are rare \cite{a1,e1,e2,o1}.
 For example, in \cite{a1}, the authors investigated the uniform and quadratic
convergence of singular differential systems. We do not find any results on the
rapid convergence of solutions for singular systems.

In this paper, we discuss the existence and uniqueness of solutions
and give rapid convergence result of solutions for singular
differential systems by using the method of generalized
quasilinearization and the order relation of lower and upper solutions.

\section{Preliminaries and Lemmas}

We consider the following initial value problem for singular differential system
\begin{equation} \label{e2.1}
\begin{gathered}
 Ax' =f(t,x),\quad t\in J,\\
 x(0)=x_0.
\end{gathered}
\end{equation}
where $A$ is a singular $n\times n$ matrix, $x\in R^n$,
$f\in C(J\times R^{n},R^{n})$, $J=[0,a]$, $a>0$ is a fixed constant.

\begin{definition} \label{def2.1} \rm
The function $\alpha_0\in C^{1}(J,R^{n})$ is called a lower
solution of \eqref{e2.1}, if the following inequalities are satisfied:
\begin{equation} \label{e2.2}
\begin{gathered}
 A\alpha'_0 \leq f(t,\alpha_0),\quad t\in J,\\
 \alpha_0(0)\leq x_0.
\end{gathered}
\end{equation}
\end{definition}

\begin{definition} \label{def2.2} \rm
 The function $\beta_0\in C^{1}(J,R^{n})$ is called an upper solution
of IVP \eqref{e2.1}, if the following inequalities are satisfied:
\begin{equation} \label{e2.3}
\begin{gathered}
 A\beta'_0\ge f(t,\beta_0),\quad t\in J,\\
 \beta_0(0)\ge x_0.
\end{gathered}
\end{equation}
\end{definition}

In our further investigations we will need some results on linear
singular differential inequalities and linear singular differential
systems.
 Consider the singular differential inequality
\begin{equation} \label{e2.4}
Ax'+M(t)x\leq0,\quad x(0)\leq0,\quad t\in J,
\end{equation}
where $A$, $M(t)$ are $n\times n$ matrices, $A$ is singular and
$M(t)$ is continuous for $t\in J$.

\begin{lemma}[\cite{a1}] \label{lem2.1} 
Assume that
\begin{itemize}
\item[(A1)] There exists a constant $\lambda$ such that,
$L(t)=[\lambda A+M(t)]^{-1}$ exists and $\hat{A}=AL(t)$ is a
constant matrix.

\item[(A2)] There exists a nonsingular matrix $T$ such that
$T^{-1}$, $(LT)^{-1}$ exist and $T^{-1}$, $(LT)$, $(LT)^{-1}\geq 0$,
satisfying
\[
{T^{ - 1}}\hat{A}T = \begin{pmatrix}
C & 0\\
0 & 0\end{pmatrix},\quad
 T^{- 1}[ I - \lambda\hat{A}]T = \begin{pmatrix}
I_1 - \lambda C & 0\\
0 & I_2
\end{pmatrix},
\]
where $C$ is a diagonal matrix with $C^{-1}\geq 0$.
\end{itemize}
Then $x(0)\leq 0$ implies $x(t)\leq 0$ for $t\in J$.
 \end{lemma}

For the singular linear initial value problem
\begin{equation} \label{e2.5}
Ax'+M(t)x=g(t),\quad x(0)=y_0,
\end{equation}
we have the following result.

\begin{lemma}[\cite{c4}] \label{lem2.2}
Assume that condition {\rm (A1)} holds, $\operatorname{index}(A)=1$ and
\begin{itemize}
\item[(A3)] $y_0$ satisfies
$(I-\hat{A}{\hat{A}}^{D})(y_0-w(0))=0$, where
$w(t)={\hat{M}}^{D}g(t)$,
$\hat{M}=M(t)L(t)$.
\end{itemize}
Then the unique solution $y(t)$ of
\begin{equation} \label{e2.6}
\hat{A}y'+\hat{M}y=g(t),\quad y(0)=y_0.
\end{equation}
is given by
\[
y(t)=e^{-{\hat{A}}^{D}\hat{M}t}\hat{A}{\hat{A}}^{D}y_0
+e^{-{\hat{A}}^{D}\hat{M}t}\int_0^{t}e^{{\hat{A}}^{D}\hat{M}s}{\hat{A}}^{D}g(s)ds
+(I-\hat{A}{\hat{A}}^{D}){\hat{M}}^{D}g(t),
\]
where the notation ${\hat{A}}^{D},{\hat{M}}^{D}$ mean the Drazin
inverse of the matrices $\hat{A},\hat{M}$ respectively. Furthermore,
we note that $x(t)=L(t)y(t)$, then we obtain the solution $x(t)$ of
\eqref{e2.5}.
\end{lemma}

To prove our main result, we need the following comparison result.

\begin{lemma} \label{lem2.3}
Assume that the conditions {\rm (A1), (A2)} hold, and
\begin{itemize}
\item[(A4)] The functions $\alpha_0$, $\beta_0\in C^1(J, R^{n})$
 are lower and upper solutions of \eqref{e2.1}, $f_{x}$ exists and are continuous.
\end{itemize}
Then $\alpha_0(0)\leq \beta_0(0)$ implies that
$\alpha_0(t)\leq \beta_0(t)$ on $J$.
\end{lemma}

 \begin{proof}
It can be noted from the condition (A4) that
\begin{align*}
A\alpha'_0-A\beta'_0
&\leq f(t,\alpha_0)-f(t,\beta_0)\\
&=\Big(\int_0^{1}f_{x}(t,\sigma\alpha_0+(1-\sigma)\beta_0)d\sigma\Big)
(\alpha_0-\beta_0).
\end{align*}
Taking
\[
M(t)=-\Big(\int_0^{1}f_{x}(t,\sigma\alpha_0+(1-\sigma)\beta_0)d\sigma\Big),
\]
we have
$$
A(\alpha_0-\beta_0)'+M(t)(\alpha_0-\beta_0)\leq0.
$$
Noting that $\alpha_0(0)\leq \beta_0(0)$, therefore, by Lemma \ref{lem2.1},
we have $\alpha_0(t)\leq \beta_0(t)$ on $J$.
\end{proof}

The following existence result is needed for
our main result. For convenience we define the set
$$
S(\alpha_0,\beta_0)=\{u\in C(J,R^{n}):\alpha_0(t)\leq u(t)\leq
\beta_0(t),\, t\in J\}.
$$

\begin{lemma} \label{lem2.4}
Assume that the conditions {\rm (A1)--(A3)} hold, and
\begin{itemize}
\item[(A5)] The functions $\alpha_0$,
$\beta_0\in C^{1}(J,R^{n})$ are lower and upper solutions of \eqref{e2.1} with
$\alpha_0\leq \beta_0$ on $J$.

\item[(A6)] The function $f\in C(S(\alpha_0,\beta_0),R^{n})$
satisfies the inequality
$$
f(t,x)-f(t,y)\geq -M_0(x-y)
$$
for $x\geq y$, $M(t_0)=M_0$, and $t_0\in J$.
\end{itemize}
Then \eqref{e2.1} has a solution $x(t)$ that satisfies
$\alpha_0(t)\leq x(t)\leq \beta_0(t)$ on $J$.
 \end{lemma}

\begin{proof}
Let $\alpha_{n+1}$ and $\beta_{n+1}$ be the solutions of the
singular linear systems
\begin{equation} \label{e2.7}
\begin{gathered}
A\alpha'_{n+1}=f(t,\alpha_{n})-M_0(\alpha_{n+1}-\alpha_{n}),\quad t\in J,\\
\alpha_{n+1}(0)=x_0,
\end{gathered}
\end{equation}
and
\begin{equation} \label{e2.8}
\begin{gathered}
A\beta'_{n+1}=f(t,\beta_{n})-M_0(\beta_{n+1}-\beta_{n}),\quad t\in J,\\
\beta_{n+1}(0)=x_0,
\end{gathered}
\end{equation}
where $\alpha_{n+1}$ and $\beta_{n+1}$ exist because of Lemma \ref{lem2.2}.
According to the iterative schemes \eqref{e2.7} and \eqref{e2.8}, we obtain the
sequences $\{\alpha_{n}(t)\}$ and $\{\beta_{n}(t)\}$ which were
generated by the initial conditions $\alpha_0(t)$ and
$\beta_0(t)$, respectively.

Let $n=0$, we first show that
$\alpha_0(t)\leq\alpha_1(t)\leq\beta_1(t)\leq\beta_0(t)$ on
$J$.

For this purpose, setting $p(t)=\alpha_0(t)-\alpha_1(t)$. Using
the condition (A5), we obtain
\begin{align*}
Ap'&=A(\alpha_0-\alpha_1)'\\
&\leq f(t,\alpha_0)-[f(t,\alpha_0)-M_0(\alpha_1-\alpha_0)]\\
&=-M_0p,\quad t\in J.
\end{align*}
 Noting that $p(0)\leq0$, by Lemma \ref{lem2.1}, we have
$p(t)\leq0$, that is, $\alpha_0(t)\leq \alpha_1(t)$ on $J$.
similarly, letting $p(t)=\beta_1(t)-\beta_0(t)$, we can show
that $\beta_1(t)\leq \beta_0(t)$ on $J$.

To prove $\alpha_1(t)\leq \beta_1(t)$. Setting
$p(t)=\alpha_1(t)-\beta_1(t)$ so that $p(0)=0$. From the
condition (A6), we have
\begin{align*}
Ap'&=A(\alpha_1-\beta_1)'\\
&=f(t,\alpha_0)-M_0(\alpha_1-\alpha_0)
-[f(t,\beta_0)-M_0(\beta_1-\beta_0)]\\
&\leq M_0(\beta_0-\alpha_0)-M_0(\alpha_1-\alpha_0)
+M_0(\beta_1-\beta_0)\\
&=-M_0p,\quad t\in J.
\end{align*}
 As before, this implies that
$\alpha_1(t)\leq \beta_1(t)$ on $J$. Thus, we conclude that
$$
\alpha_0(t)\leq\alpha_1(t)\leq \beta_1(t)\leq \beta_0(t),\ \ t\in J.$$

The process can be continued successively to obtain that
$$
\alpha_0(t)\leq \alpha_1(t)\leq\dots\alpha_{n}(t)\leq
\beta_{n}(t)\leq \dots\leq \beta_1(t)\leq \beta_0(t),\quad
 t\in J.
$$
It is easy to see that the sequence $\{\alpha_{n}(t)\}$ is uniformly
bounded and equicontinuous, employing the Ascoli-Arzela Theorem, the
nondecreasing sequence $\{\alpha_{n}(t)\}$ has a pointwise limit
$x(t)$ that satisfies $\alpha_0(t)\leq x(t)\leq \beta_0(t)$. A
passage to the limit based on the Dominated Convergence Theorem
shows that $x(t)$ is a solution of
\begin{gather*}
Ax'=f(t,x)-M_0(x-x),\quad t\in J,\\
x(0)=x_0;
\end{gather*}
that is, $x(t)$ is a solution of \eqref{e2.1}. Thus, we conclude that there
exists a solution $x(t)$ of IVP \eqref{e2.1} satisfies
$\alpha_0(t)\leq x(t)\leq \beta_0(t)$ on $J$. The proof is complete.
\end{proof}

\section{Main results}

 In this section we prove the convergence of the sequence
of successive approximations is of order $k\geq2$.

\begin{theorem} \label{thm3.1}
 Assume that the conditions {\rm (A1)--(A3), (A6)} hold, and
\begin{itemize}
\item[(A7)] The functions $\alpha_0,\ \beta_0\in C^{1}(J,R^{n})$
are lower and upper solutions of \eqref{e2.1} with $\alpha_0\leq
\beta_0$ on $J$.

\item[(A8)] The Frechet derivatives $\frac{\partial^{i}f(t,x)}{\partial x^{i}}$
$(i=0,1,2,\dots,k)$ exist and are continuous satisfying
$f(t,x)+Mx^{k}$ is $(k-1)$-hyperconvex and $f(t,x)-Nx^{k}$ is
$(k-1)$-hyperconcave in $x$; that is,
$\frac{\partial^{k}(f(t,x)+Mx^{k})}{\partial x^{k}}\geq0$ and
$\frac{\partial^{k}(f(t,x)-Nx^{k})}{\partial x^{k}}\leq0$, where the
$n\times n$ matrices $M$, $N>0$, $k\geq1$.
\end{itemize}
Then there exist monotone sequences $\{\alpha_{n}(t)\}$,
$\{\beta_{n}(t)\}$
 which convergence is of order $k$, that is, there exist constant matrices
$\lambda_1$ and $\mu_1>0$ such that for the solution $x(t)$ of
 \eqref{e2.1} in $S(\alpha_0,\beta_0)$, the inequalities
\begin{gather*}
\max_{t\in J}|x(t)-\alpha_{n+1}(t)|
 \leq\lambda_1\max_{t\in J}|x(t)-\alpha_{n}(t)|^{k},
 \\
\max_{t\in J}|\beta_{n+1}(t)-x(t)|
\leq\mu_1\max_{t\in J}|\beta_{n}(t)-x(t)|^{k}
\end{gather*}
hold, where 
\[
\max_{t\in J}|u(t)|=(\max_{t\in J}|u_1(t)|,
\dots,\max_{t\in J}|u_{n}(t)|)^T,
\]
$|u|^{k}=(|u_1|^{k},\dots, |u_{n}|^{k})^T$ for any function
$u\in C(J, R^{n})$.
 \end{theorem}

\begin{proof}
Firstly, for $t\in J$, applying Taylor mean value theorem yields
\begin{equation} \label{e3.1}
\begin{aligned}
 f(t,x)&=\sum_{i=0}^{k-1}\frac{\partial^if(t,y)}{\partial x^i}\frac{(x-y)^i}{i!}\\
 &\quad +\Big(\int_0^{1}(1-\sigma)^{k-1}\frac{\partial^k
 f(t,\sigma x+(1-\sigma)y)}{\partial x^k}d\sigma\Big)
 \frac{(x-y)^k}{(k-1)!},
\end{aligned}
\end{equation}
with $\alpha_0\leq x,y \leq\beta_0$,
$\frac{\partial^{0}f}{\partial x^{0}}=f$. Furthermore, for
$\alpha_0\leq y\leq x\leq\beta_0$, in view of the condition
(A8), we have
\begin{equation} \label{e3.2}
f(t,x) \geq\sum_{i=0}^{k-1}\frac{\partial^if(t,y)}{\partial x^i}\frac{(x-y)^i}{i!}
 -M(x-y)^{k} \equiv F(t,x,y),
\end{equation}
and $F(t,x,x)= f(t,x)$, where
$x^{i}=(x_1^{i},x_2^{i},\dots,x_{n}^{i})^{T}$, $x\in R^{n}$,
 $i=0,1,2,\dots,k$. Similarly, for a giving $t\in J$,
$\alpha_0\leq x\leq y\leq\beta_0$, we obtain
 \begin{equation} \label{e3.3}
 \begin{aligned}
 f(t,x)&\leq
\begin{cases}
 \sum_{i=0}^{k-1}\frac{\partial^if(t,y)}{\partial x^i}\frac{(x-y)^i}{i!}
 -M(x-y)^{k}, & k=2n+1.\\
\sum_{i=0}^{k-1}\frac{\partial^if(t,y)}{\partial x^i}\frac{(x-y)^i}{i!}
 +N(x-y)^{k}, & k=2n.
 \end{cases}\\
& \equiv G(t,x,y)
\end{aligned}
\end{equation}
and $G(t,x,x)= f(t,x)$.

Consider the singular differential system
\begin{equation} \label{e3.4}
\begin{gathered}
Ax' =\sum_{i=0}^{k-1}\frac{\partial^if(t,\alpha_0)}{\partial x^i}
\frac{(x-\alpha_0)^i}{i!}-M(x-\alpha_0)^{k}
\equiv F(t,x,\alpha_0),\quad t\in J,\\
x(0) =x_0.
\end{gathered}
\end{equation}
We will show that $\alpha_0$ and $\beta_0$ are lower and upper
solutions of \eqref{e3.4} respectively. The condition (A7) and the
inequality \eqref{e3.2} imply
\begin{gather*}
A\alpha'_0\leq f(t,\alpha_0)=F(t,\alpha_0,\alpha_0),\quad t\in J,\\
\alpha_0(0)\leq x_0,
\end{gather*}
and
\begin{gather*}
 A\beta'_0\geq f(t,\beta_0)\geq
F(t,\beta_0,\alpha_0),\quad t\in J,\\
 \beta_0(0)\geq x_0.
\end{gather*}
Hence by Lemma \ref{lem2.4}, there exists a solution $\alpha_1(t)$ of
\eqref{e3.4} such that $\alpha_0(t)\leq\alpha_1(t)\leq\beta_0(t)$ on
$J$. Furthermore, we can prove that $\alpha_1(t)$ is the unique
solution of \eqref{e3.4}. For this purpose, we assume $x_1(t)$ and
$x_2(t)$ are two solutions of \eqref{e3.4} and $\alpha_0\leq
x_2\leq x_1\leq \beta_0$ holds. Then, from  \eqref{e3.4},
and noting that $x_1(0)-x_2(0)=0$, and
$$
A^{i}-B^{i}=(A-B)\sum_{j=0}^{i-1}A^{i-1-j}B^{j},
$$
 we have
\begin{align*}
Ax'_1-Ax'_2
&= F(t,x_1,\alpha_0)-F(t,x_2,\alpha_0)\\
&=\sum_{i=0}^{k-1}\frac{\partial^if(t,\alpha_0)}{\partial
x^i}\frac{(x_1-\alpha_0)^i}{i!}
 -M(x_1-\alpha_0)^{k}\\
&\quad -\sum_{i=0}^{k-1}\frac{\partial^if(t,\alpha_0)}
{\partial x^i}\frac{(x_2-\alpha_0)^i}{i!}
 +M(x_2-\alpha_0)^{k}\\
&=\Big\{\sum_{i=1}^{k-1}\frac{\partial^if(t,\alpha_0)}{\partial
x^i}\frac{1}{i!}\sum_{j=0}^{i-1}(x_1-\alpha_0)^{i-1-j}(x_2-\alpha_0)^{j}\Big\}
 (x_1-x_2)\\
&\quad -M\sum_{j=0}^{k-1}(x_1-\alpha_0)^{k-1-j}(x_2-\alpha_0)^{j}(x_1-x_2)\\
&\leq
\Big\{\sum_{i=1}^{k-1}\frac{\partial^if(t,\alpha_0)}{\partial
x^i}\frac{1}{i!}\sum_{j=0}^{i-1}(x_1-\alpha_0)^{i-1-j}(x_2-\alpha_0)^{j}\big\}
 (x_1-x_2)\\
&\leq L_1(x_1-x_2),
\end{align*}
where
\[
\sum_{i=1}^{k-1}\frac{\partial^if(t,\alpha_0)}{\partial
x^i}\frac{1}{i!}\sum_{j=0}^{i-1}(x_1-\alpha_0)^{i-1-j}(x_2-\alpha_0)^{j}\leq
L_1,
\]
$M(\bar{t})=-L_1$, $\bar{t}\in J$. Using Lemma \ref{lem2.1}, we can
get $x_1(t)\leq x_2(t)$. Then we have $x_1(t)\equiv x_2(t)$,
that is, $\alpha_1(t)$ is the unique solution of \eqref{e3.4}.

Furthermore, we consider the singular differential system
\begin{equation} \label{e3.5}
\begin{gathered}
\begin{aligned}
Ay'&= \begin{cases}
 \sum_{i=0}^{k-1}\frac{\partial^if(t,\beta_0)}{\partial
x^i}\frac{(y-\beta_0)^i}{i!}
 -M(y-\beta_0)^{k},& k=2n+1,\\
\sum_{i=0}^{k-1}\frac{\partial^if(t,\beta_0)}{\partial
x^i}\frac{(y-\beta_0)^i}{i!}
 +N(y-\beta_0)^{k}, & k=2n,
\end{cases}
\\
&\equiv G(t,y,\beta_0),\quad t\in J,
\end{aligned}\\
y(0) =x_0.
\end{gathered}
\end{equation}

Now we show that $\alpha_0$ and $\beta_0$ are lower and
upper solutions of \eqref{e3.5} respectively. For this purpose, the
condition (A7) and the inequality \eqref{e3.3} yield
\begin{gather*}
 A\alpha'_0\leq f(t,\alpha_0) \leq G(t,\alpha_0,\beta_0),\quad t\in J,\\
\alpha_0(0)\leq x_0,
\end{gather*}
and
\begin{gather*}
A\beta'_0\geq f(t,\beta_0)=
G(t,\beta_0,\beta_0),\quad t\in J,\\
\beta_0(0)\geq x_0.
\end{gather*}

Thus from Lemma \ref{lem2.4}, we see that there exists a solution
$\beta_1(t)$ of \eqref{e3.5} such that
$\alpha_0(t)\leq \beta_1(t)\leq \beta_0(t)$ on $J$. Similarly, we can prove that
$\beta_1(t)$ is the unique solution of \eqref{e3.5}. To do this, we talk
about two cases. If $k=2n+1$, the uniqueness of the solution of
\eqref{e3.5} is the same as \eqref{e3.4}.
 If $k=2n$, taking two solutions $x_1$, $x_2$ of \eqref{e3.5} such that
$\alpha_0\leq x_2\leq x_1\leq \beta_0$.
 Then, we obtain
\begin{align*}
Ax'_1-Ax'_2
&=G(t,x_1,\beta_0)-G(t,x_2,\beta_0)\\
&=\Big\{\sum_{i=1}^{k-1}\frac{\partial^if(t,\beta_0)}{\partial
x^i}\frac{1}{i!}\sum_{j=0}^{i-1}(x_1-\beta_0)^{i-1-j}(x_2-\beta_0)^{j}\Big\}
(x_1-x_2) \\
&\quad +N\sum_{j=0}^{k-1}(x_1-\beta_0)^{k-1-j}(x_2-\beta_0)^{j}(x_1-x_2)\\
&\leq \Big\{\sum_{i=1}^{k-1}\frac{\partial^if(t,\beta_0)}{\partial
x^i}\frac{1}{i!}\sum_{j=0}^{i-1}(x_1-\beta_0)^{i-1-j}(x_2-\beta_0)^{j}\Big\}(x_1-x_2)\\
&\leq L_1(x_1-x_2).
\end{align*}
Noting that $x_1(0)-x_2(0)=0$, by Lemma \ref{lem2.1}, we obtain $x_1(t)\leq x_2(t)$. Thus, \eqref{e3.5} has a
unique solution.

Next, we prove that $\alpha_1(t)\leq\beta_1(t)$ on $J$. From
 \eqref{e3.2}, we obtain
\begin{gather*}
 A\alpha'_1=F(t,\alpha_1,\alpha_0)\leq f(t,\alpha_1),\quad t\in J,\\
 \alpha_1(0) = x_0.
\end{gather*}

Proceeding as before, one can obtain that $\beta_1(t)$
is an upper solution of \eqref{e2.1}. It then follows from Lemma \ref{lem2.3} that
$\alpha_1(t)\leq\beta_1(t)$ on $J$. Consequently,
$$
\alpha_0(t)\leq\alpha_1(t)\leq\beta_1(t)\leq\beta_0(t),\quad t\in J.
$$
Continuing this process by induction, we obtain two monotone sequences
$\{\alpha_{n}(t)\}$ and $\{\beta_{n}(t)\}$ satisfying
\[
 \alpha_0(t)\leq\alpha_1(t)\leq \dots \leq\alpha_{n}(t)
 \leq\beta_{n}(t)\leq\dots\leq\beta_1(t)\leq\beta_0(t),\quad t\in J.
\]
Let $\alpha_{n}$, $\beta_{n}$ be lower and upper solutions of
\eqref{e2.1} respectively with $\alpha_{n}\leq\beta_{n}$ on $J$. Then we
consider the singular differential system
\begin{equation} \label{e3.6}
\begin{gathered}
Ax'=\sum_{i=0}^{k-1}\frac{\partial^if(t,\alpha_{n})}
{\partial x^i}\frac{(x-\alpha_{n})^i}{i!}
 -M(x-\alpha_{n})^{k}
\equiv F(t,x,\alpha_{n}),\quad t\in J,\\
x(0) =x_0.
\end{gathered}
\end{equation}

In this case, we can show easily that $\alpha_{n}$ and $\beta_{n}$
are lower and upper solutions of \eqref{e3.6}.
Therefore, by Lemma \ref{lem2.4}, there exists a solution $\alpha_{n+1}(t)$
of \eqref{e3.6} such that
$\alpha_{n}(t)\leq\alpha_{n+1}(t)\leq\beta_{n}(t)$ on $J$. The
uniqueness of $\alpha_{n+1}(t)$ is analogous to $\alpha_1(t)$, we
omit the details.

Next, consider the singular differential system
\begin{equation} \label{e3.7}
\begin{gathered}
\begin{aligned}
Ay'&= \begin{cases}
\sum_{i=0}^{k-1}\frac{\partial^if(t,\beta_{n})}{\partial x^i}
 \frac{(y-\beta_{n})^i}{i!}
 -M(y-\beta_{n})^{k}, & k=2n+1,\\
\sum_{i=0}^{k-1}\frac{\partial^if(t,\beta_{n})}{\partial x^i}
\frac{(y-\beta_{n})^i}{i!}
 +N(y-\beta_{n})^{k}, & k=2n,
\end{cases} \\
&\equiv G(t,y,\beta_{n}),\quad t\in J,
\end{aligned}\\
y(0) =x_0.
\end{gathered}
\end{equation}

Similarly, we can show that $\alpha_{n}$ and $\beta_{n}$ are lower
and upper solutions of \eqref{e3.7}, respectively. Consequently,
by Lemma \ref{lem2.4}, we obtain a solution $\beta_{n+1}(t)$ of \eqref{e3.7} exists
such that $\alpha_{n}(t)\leq\beta_{n+1}(t)\leq \beta_{n}(t)$ on $J$.
Furthermore, we can show that $\alpha_{n+1}(t)\leq\beta_{n+1}(t)$ on
$J$. By induction, we have that for all $n$,
$$
\alpha_0(t)\leq\alpha_1(t)\leq\dots \leq\alpha_{n}(t)
\leq\beta_{n}(t)\leq\dots\leq\beta_1(t)\leq\beta_0(t),\ \ t\in J.
$$

Employing the Ascoli-Arzela Theorem, it can be shown that they have
pointwise limits $\rho(t)$ and $r(t)$. Taking the limit as
$n\to\infty$, we obtain
$$
\lim_{n\to\infty}\alpha_{n}(t)
=\rho(t)\leq r(t)=\lim_{n\to\infty}\beta_{n}(t).
$$
We can show easily that $\rho(t)$ and $r(t)$ are solutions of \eqref{e2.1}.

Finally, we show that the order of convergence is $k\geq 2$. For
that, let $x(t)$ be a solution of \eqref{e2.1} in
$S(\alpha_0,\beta_0)$, we define
$$
e_{n}(t)=x(t)-\alpha_{n}(t),\
a_{n}(t)=\alpha_{n+1}(t)-\alpha_{n}(t),\ \ t\in J,
$$
so that, $e_{n}\geq 0$, $a_{n}\geq 0$, $e_{n}(0)=0$, $a_{n}(0)=0$.
From the equality \eqref{e3.1}, we have
\begin{align*}
Ax'&=\sum_{i=0}^{k-1}\frac{\partial^if(t,\alpha_{n})}{\partial
x^i}\frac{(x-\alpha_{n})^i}{i!}\\
&\quad +\Big(\int_0^{1}(1-\sigma)^{k-1}\frac{\partial^kf(t,\sigma
x+(1-\sigma)\alpha_{n})}{\partial x^k}d\sigma \Big
)\frac{(x-\alpha_{n})^k}{(k-1)!}.
\end{align*}
 On the other hand, by \eqref{e3.6}, we have
$$
A\alpha'_{n+1}
=\sum_{i=0}^{k-1}\frac{\partial^if(t,\alpha_{n})}{\partial
x^i}\frac{(\alpha_{n+1}-\alpha_{n})^i}{i!}
-M(\alpha_{n+1}-\alpha_{n})^{k}.
$$
Therefore,
\begin{align*}
Ae'_{n+1}
&=\sum_{i=0}^{k-1}\frac{\partial^if(t,\alpha_{n})}{\partial
x^i}\frac{(x-\alpha_{n})^i}{i!}\\
&\quad +\Big(\int_0^{1}(1-\sigma)^{k-1}
 \frac{\partial^kf(t,\sigma
x+(1-\sigma)\alpha_{n})}{\partial
 x^k}d\sigma \Big )\frac{(x-\alpha_{n})^k}{(k-1)!}\\
&\quad -\sum_{i=0}^{k-1}\frac{\partial^if(t,\alpha_{n})}{\partial
x^i}\frac{(\alpha_{n+1}-\alpha_{n})^i}{i!}
 +M(\alpha_{n+1}-\alpha_{n})^{k}\\
&\leq \sum_{i=1}^{k-1}\frac{\partial^if(t,\alpha_{n})}{\partial
x^i}\frac{(e_{n}^{i}-a_{n}^{i})}{i!}+Ne_{n}^{k}+Ma_{n}^{k}\\
&\leq
\Big\{\sum_{i=1}^{k-1}\frac{\partial^if(t,\alpha_{n})}{\partial
x^i}\frac{1}{i!}\sum_{j=0}^{i-1}e_{n}^{i-1-j}a_{n}^{j}\Big\}
 e_{n+1}+ce_{n}^{k}\\
&\leq L_1e_{n+1}+ce_{n}^{k},
\end{align*}
where $c=N+M$, $a_{n}\leq e_{n}$. Furthermore, we have
$$
Ae'_{n+1}\leq -M(\bar{t})e_{n+1}+ce_{n}^{k}.
$$
Lemma \ref{lem2.1} implies $e_{n+1}(t)\leq x(t)$ on $J$, where $x(t)$ is
the solution of
$$
Ax'+M(\bar{t})x=ce_{n}^{k},\ \ x(0)=0.
$$
Thus, using the expression of $x(t)$ in Lemma \ref{lem2.2}, we obtain
$$
x(t)=[\lambda
A+M(\bar{t})]^{-1}\big[e^{-{\hat{A}}^{D}\hat{M}t}
\int_0^{t}e^{{\hat{A}}^{D}\hat{M}s}{\hat{A}}^{D}
ce_{n}^{k}(s)ds+(I-\hat{A}{\hat{A}}^{D}){\hat{M}}^{D}ce_{n}^{k}(t)\big],
$$
we arrive at after taking suitable estimates
$$
\max_{t\in J}|x(t)-\alpha_{n+1}(t)|\leq
\lambda_1\max_{t\in J}|x(t)-\alpha_{n}(t)|^{k},
$$
 where $\lambda_1$ is an appropriate positive matrix.

Similarly, we define
$$
g_{n}(t)=x(t)-\beta_{n}(t),\ b_{n}(t)=\beta_{n+1}(t)-\beta_{n}(t),\ \ t\in J,
$$
so that, $e_{n}(t)\leq0$, $b_{n}(t)\leq0$, $g_{n}(0)=0$, $b_{n}(0)=0$.
In view of \eqref{e3.1}, we have
\begin{align*}
Ax'
&=\sum_{i=0}^{k-1}\frac{\partial^if(t,\beta_{n})}{\partial x^i}
 \frac{(x-\beta_{n})^i}{i!}\\
&\quad +\Big(\int_0^{1}(1-\sigma)^{k-1}\frac{\partial^{k}f(t,\sigma
 x+(1-\sigma)\beta_{n})}{\partial
 x^k}d\sigma\Big)\frac{(x-\beta_{n})^k}{(k-1)!}.
\end{align*}
Furthermore, by \eqref{e3.7}, we obtain
\[
A\beta'_{n+1}
= \begin{cases}
\sum_{i=0}^{k-1}\frac{\partial^if(t,\beta_{n})}{\partial
x^i}\frac{(\beta_{n+1}-\beta_{n})^i}{i!}
 -M(\beta_{n+1}-\beta_{n})^{k}, & k=2n+1,\\
\sum_{i=0}^{k-1}\frac{\partial^if(t,\beta_{n})}{\partial
x^i}\frac{(\beta_{n+1}-\beta_{n})^i}{i!}
 +N(\beta_{n+1}-\beta_{n})^{k}, & k=2n.
\end{cases}
\]
Hence, if $k=2n+1$, we obtain
\begin{align*}
&-Ag'_{n+1}\\
&=A\beta'_{n+1}-Ax'\\
&=\sum_{i=0}^{k-1}\frac{\partial^if(t,\beta_{n})}{\partial
x^i}\frac{(\beta_{n+1}-\beta_{n})^i}{i!}-M(\beta_{n+1}-\beta_{n})^{k}
-\sum_{i=0}^{k-1}\frac{\partial^if(t,\beta_{n})}{\partial x^i}
 \frac{(x-\beta_{n})^i}{i!}\\
&\quad -\Big(\int_0^{1}(1-\sigma)^{k-1}\frac{\partial^{k}f(t,\sigma
x+(1-\sigma)\beta_{n})}{\partial x^k}d\sigma\Big)\frac{(x-\beta_{n})^k}{(k-1)!}\\
&\leq \sum_{i=1}^{k-1}\frac{\partial^if(t,\beta_{n})}{\partial
x^{i}}\frac{b_{n}^{i}-g_{n}^{i}}{i!}-Mb_{n}^{k}+N(-g_{n})^k\\
&\leq\Big\{\sum_{i=1}^{k-1}\frac{\partial^if(t,\beta_{n})}{\partial
x^i}\frac{1}{i!}\sum_{j=0}^{i-1}b_{n}^{i-1-j}
g_{n}^j\Big\}(-g_{n+1})+(M+N)(-g_{n})^k\\
&\leq L_1(-g_{n+1})+(M+N)(-g_{n})^k.
\end{align*}
Furthermore, we obtain
$$
A(-g_{n+1})'\leq -M(\bar{t})(-g_{n+1})+(M+N)(-g_{n})^k.
$$
According to Lemma \ref{lem2.1}, we have $-g_{n+1}(t)\leq x(t)$ on $J$, where
$x(t)$ is the solution of
$$
Ax'+M(\bar{t})x=(M+N)(-g_{n})^k,\ \ x(0)=0.
$$
Thus, using the expression of $x(t)$ in Lemma \ref{lem2.2} and taking
suitable estimates, we obtain
$$
\max_{t\in J}|\beta_{n+1}(t)-x(t)|\leq
\mu_1\max_{t\in J}|\beta_{n}(t)-x(t)|^{k},
$$
 where $\mu_1$ is a positive matrix.

 If $k=2n$, we obtain
\begin{align*}
&-Ag'_{n+1}\\
&=\sum_{i=0}^{k-1}\frac{\partial^{i}f(t,\beta_{n})}{\partial
x^{i}}\frac{(\beta_{n+1}-\beta_{n})^{i}}{i!}
+N(\beta_{n+1}-\beta_{n})^{k}
-\sum_{i=0}^{k-1}\frac{\partial^{i}f(t,\beta_{n})}{\partial
x^{i}}\frac{(x-\beta_{n})^i}{i!}\\
&\quad -\Big(\int_0^{1}(1-\sigma)^{k-1}\frac{\partial^{k}f(t,\sigma
x+(1-\sigma)\beta_{n})}{\partial
 x^k}d\sigma\Big)\frac{(x-\beta_{n})^k}{(k-1)!}\\
&\leq \sum_{i=1}^{k-1}\frac{\partial^{i}f(t,\beta_{n})}{\partial
x^{i}}\frac{b_{n}^{i}-g_{n}^{i}}{i!}+Nb_{n}^{k}+Mg_{n}^{k}\\
&\leq\Big\{\sum_{i=1}^{k-1}\frac{\partial^{i}f(t,\beta_{n})}{\partial
x^{i}}\frac{1}{i!}\sum_{j=0}^{i-1}b_{n}^{i-1-j}
g_{n}^{j}\Big\}(-g_{n+1})+(M+N)g_{n}^{k}\\
&\leq L_1(-g_{n+1})+(M+N)g_{n}^{k}.
\end{align*}
Then, we obtain
$$
A(-g_{n+1})'\leq -M(\bar{t})(-g_{n+1})+(M+N)g_{n}^{k}.
$$
Now applying Lemma \ref{lem2.1}, we have $-g_{n+1}(t)\leq x(t)$ on $J$, where
$x(t)$ is the solution of
$$
Ax'+M(\bar{t})x=(M+N)g_{n}^k,\quad x(0)=0.
$$

Analogous to the above discussion, after taking suitable
estimates, we obtain
$$
\max_{t\in J}|\beta_{n+1}(t)-x(t)|
\leq \mu_1\max_{t\in J}|\beta_{n}(t)-x(t)|^{k}.
$$
The proof is complete.
\end{proof}

The following corollaries are immediate results of Theorem \ref{thm3.1}.

\begin{corollary} \label{coro3.1}
Assume that conditions {\rm (A1)--(A3), (A6), (A7)} hold, and
\begin{itemize}
\item[(A9)] The Frechet derivatives $\frac{\partial^{i}f(t,x)}{\partial x^{i}}$
$(i=0,1,2,3)$ exist and are continuous, also $f(t,x)+Mx^{3}$ is
$(2)$-hyperconvex and $f(t,x)-Nx^{3}$ is $(2)$-hyperconcave in $x$;
that is, 
\[
\frac{\partial^{3}(f(t,x)+Mx^{3})}{\partial x^{3}}\geq0, \quad 
\frac{\partial^{3}(f(t,x)-Nx^{3})}{\partial x^{3}}\leq0,
\]
where the $n\times n$ matrices $M$, $N>0$.
\end{itemize}
Then there exist monotone sequences $\{\alpha_{n}(t)\}$,
$\{\beta_{n}(t)\}$ which converge uniformly to the solution of
\eqref{e2.1} and the convergence is cubic.
\end{corollary}

\begin{corollary} \label{coro3.2}
Assume that conditions {\rm (A1)--(A3), (A6), (A7)} hold, and
\begin{itemize}
\item[(A10)] The Frechet derivatives $\frac{\partial^{i}f(t,x)}{\partial x^{i}}$
$(i=0,1,2,3,4)$ exist and are continuous satisfying $f(t,x)+Mx^{4}$
is $(3)$-hyperconvex and $f(t,x)-Nx^{4}$ is $(3)$-hyperconcave in
$x$, that is, $\frac{\partial^{4}(f(t,x)+Mx^{4})}{\partial
x^{4}}\geq0$ and $\frac{\partial^{4}(f(t,x)-Nx^{4})}{\partial
x^{4}}\leq0$, where the $n\times n$ matrices $M$, $N>0$.
\end{itemize}
Then there exist monotone sequences $\{\alpha_{n}(t)\}$,
$\{\beta_{n}(t)\}$ which converge uniformly to the solution of
\eqref{e2.1} and the convergence is quartic.
\end{corollary}

\begin{remark} \label{rmk3.1} \rm
If  $f(t,x)$ is $(k-1)$-hyperconvex or $(k-1)$-hyperconcave in $x$,
 by the method of quasilinearization, we also can obtain the convergence of the
monotone sequences is of order $k$ ($k$ is odd or even).
\end{remark}


\subsection{Acknowledgments}
This research is supported by the National Natural Science Foundation of
China (11271106) and
the Natural Science Foundation of Hebei Province of China (A2013201232).
The authors would like to thank the reviewers for their valuable
suggestions and comments.


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\end{document}