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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2014 (2014), No. 07, 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}
\thanks{\copyright 2014 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2014/07\hfil Existence of solutions]
{Existence of solutions for iterative differential equations}

\author[P. Zhang, X. Gong \hfil EJDE-2014/07\hfilneg]
{Pingping Zhang, Xiaobing Gong}  

\address{Pingping Zhang \newline
Department of Mathematics and Information Science,
Binzhou University, Shandong 256603, China}
\email{zhangpingpingmath@163.com}

\address{Xiaobing Gong \newline
Department of Mathematics, Neijiang Normal University,
Sichuan 641100, China}
\email{xbgong@163.com}

\thanks{Submitted August 21, 2013. Published January 7, 2014.}
\thanks{Supported by grants J12L59, 12ZA086 and 2013Y04}
\subjclass[2000]{34A12, 39B12, 47H10}
\keywords{Existence; nonautonomous; iteration differential
equation; \hfill\break\indent Schauder's fixed point theorem}

\begin{abstract}
 The presence of self-mapping increases the difficulty
 in proving the existence of solutions for general
 iterative differential equation. In this article we
 provide conditions for the existence of solutions for the initial
 value problem, in which the conditions are natural
 and easily verifiable. We generalize the relevant results
 and point out the mistake in some references.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

   Differential equations with state-dependent delays
attract interests of specialists since they widely
arise from application models, such as
two-body problem of classical electrodynamics
\cite{Dr1,Dr2}, position control \cite{MB1,MB2},
mechanical models \cite{RAJ}, infection disease transmission
\cite{PW}, population models \cite{Bj91,RMN},  the dynamics of
economical systems \cite{Bj89}, etc.
As special type of state-dependent delay-differential equations,
iterative differential equations have distinctive characteristics and have
been investigated in  recent years, e.g.
smoothness \cite{ChengSiWang,SiWang},
equivariance \cite{YZ}, analyticity
\cite{SiZh2}, \cite{zhang-2012}-\cite{zhang-2009}, monotonicity
\cite{Eder,SS2}), convexity \cite{SiCheng}
as well as numerical solution \cite{NO}.
In the theory of differential equations, one of the fundamental and
important problems is the initial value problem, there are many existence
results \cite{AP,Berinde-2010}, \cite{Buica-1995},
 \cite{Eder}-\cite{Lauran-2011}, \cite{Wk}
on special iterative differential equations.
In 1984  Eder \cite{Eder}
proved the existence of the unique monotone solution for the $2$-th iterative
differential equation
\begin{equation}
x'(t)=x(x(t))\label{Eder}
\end{equation}
associated with $x(t_0)=t_0\ (t_0\in[-1,1])$ by
Contraction Principle. Later, M. Fe$\check{c}$kan (\cite{FM})
investigated the generally $2$-th iterative differential equation
\begin{equation}
x'(t)=f(x(x(t)))
\label{FM}
\end{equation}
with the initial value $x(0)=0$ and
obtained the local solution applying Contraction
Principle. By using Schauder's fixed point theorem,  Wang \cite{Wk}
obtained the strong solutions of equation \eqref{FM}
associated with $x(a)=a$, where $a$ is an
endpoint of the well-defined interval. Consequently,  Ge and  Mo
\cite{GM} provided the sufficient conditions for
the initial value problem of \eqref{FM} associated with
\begin{equation}
x(t_0)=x_0
\label{initial value}
\end{equation}
on a given compact interval, where the endpoints of
the interval are two adjacent null points of $f$. 
The $2$-th nonautonomous equation
\begin{equation}
x'(t)=f(t,x(t),x(x(t))), \label{AP}
\end{equation}
together with initial value
$$
x(0)=c\quad (c>0)
$$
was investigated by P. Andrzej (\cite{AP}) using Picard's successive
approximation, where $0$ is the left end point of the domain.

In 2010 Berinde \cite{Berinde-2010} applied
the nonexpansive operators to investigate \eqref{FM}
associated with \eqref{initial value}
and extended the existence results in \cite{Buica-1995}.
Subsequently,  Lauran
investigated the nonautonomous equation \eqref{AP} together with
\eqref{initial value} in \cite{Lauran-2011}.
We see that the existence of solutions for
the general iterative differential equation
\begin{equation}
x'(t)=f(t,x^{[1]} (t),x^{[2]} (t),\dots ,x^{[n]} (t))
\label{consider}
\end{equation}
associated with \eqref{initial value} is still open,
$x^{[i]} (t):=x(x^{[i-1]}(t))$ indicates
the $i$-th iterate of self-mapping $x$, where $i=1,2,\dots ,n$.
In this paper we provide two existence results for the initial
value problem, in which the conditions are natural and easily verifiable.
We generalize the relevant results and point out
the mistake in \cite{Berinde-2010} and \cite{Lauran-2011}.
As the application, we consider the smooth solutions of the equation 
discussed in \cite{SiWang} by Theorem \ref{thm1} and
give an example to verify Theorem \ref{thm2}.

\section{Main results}

For the continuous function $\varphi(x)$, we use the supremum norm
$$
\|\varphi\|_{P}=\sup_{x\in P\subset\mathbb{R}^n}\|\varphi(x)\|
$$
and need the following lemma (the statement is slightly different
from the original one presented in \cite{Zhang} but perfectly equivalent):

\begin{lemma}[\cite{Zhang}] \label{lem1}
 Let
$$
\Phi_M=\{x\in\,C^{0}([t_0-h,t_0+h]):
  |x(t)-x(s)|\le M|t-s|, \forall t, s \in [t_0-h,t_0+h]\},
$$
where $M<1$. If $f, g\in \Phi_M$, then
\begin{equation}
 \|f^{[j]}-g^{[j]}\|_{[t_0-h,t_0+h]}\leq \frac{1-M^j}{1-M}
\|f-g\|_{[t_0-h,t_0+h]},\quad j=1,2,\dots .
\label{2.8}
\end{equation}
\end{lemma}


\begin{theorem} \label{thm1}
Suppose that $f:{\mathbb{R}}^{n+1}\rightarrow{\mathbb{R}}$ is
continuous. If there exists a positive $r$ such that
\begin{equation}
(1-M_1)\ r\geq l_0,
\label{main condition}
\end{equation}
where $M_1=\|f\|_{\bar{B}(y_0,r)}\leq1$ and $l_0 =|x_0-t_0|$ and
$\bar{B}(y_0,r)$ denotes the closed ball centered at
$y_0=(t_0,x_0,\dots ,x_0)$ with radius $r$.
Then equation \eqref{consider} associated with \eqref{initial value}
 has a solution defined on $[t_0-l,t_0+l]$ for any $l\in[\ l_0 /(1-M_1),r]$.
\end{theorem}

\begin{proof}
 The existence of solutions of equation \eqref{consider}
associated with \eqref{initial value} is equivalent to find a
continuous solution of the integral equation
\begin{equation}
x(t)=x_0+\int_{t_0}^t f(s,x^{[1]}(s),x^{[2]}(s),\dots ,x^{[n]}(s))ds.
\label{2.3}
\end{equation}
Define
\begin{align*}
\Phi_{M_1} =
  \big\{&x\in C^{0}([t_0-l, t_0+l]): x(t_0)=x_0,  |x(t)-x(s)|\le M_1|t-s|,\\
  &\forall t, s \in [t_0-l, t_0+l]\big\}.
\end{align*}
for any $l\in[l_0 /(1-M_1),r]$.
Then for $x\in \Phi_{M_1}$,
we show that $x^{[i]} (t)$ $(i=2,3,\dots ,n)$ are well defined on 
$[t_0 -l,t_0 +l]$.
It is suffices to prove
\begin{equation}
|x^{[i]}(t)-t_0|\leq l \label{2.4}
\end{equation}
for $i\in\mathbb{N}$ by induction. In fact
\begin{align*}
|x(t)-t_0|&\leq  |x(t)-x(t_0)|+|x(t_0)-t_0| \\
&\leq  M_1 l+|x_0-t_0| 
\leq  l,
\end{align*}
we assume that
$|x^{[i]}(t)-t_0|\le l$ for positive integer $i\geq1$, then
\begin{align*}
|x^{[i+1]}(t)-t_0|&\leq  |x^{[i+1]}(t)-x(t_0)|+|x(t_0)-t_0|
\\
&\leq  M_1 |x^{[i]}(t)-t_0|+|x_0-t_0|
\\
&\leq  M_1 l+|x_0-t_0|
\leq  l.
\end{align*}
Hence it follows by induction that \eqref{2.4} holds and
$x^{[i]}([t_0-l,t_0+l])$ are well defined for any $x\in \Phi_{M_1}$.

In the sequel we apply the Schauder's fixed point theorem to
prove the existence of the continuous solution of \eqref{2.3}. To this end,
we define the integral operator
${\mathcal{G}}:\ \Phi_{M_1}\rightarrow C^{0}([t_0-l,t_0+l])$ by
\begin{equation}
{\mathcal{G}}x(t):=x_{0}+\int_{t_{0}}^t
f(s,x^{[1]}(s),x^{[2]}(s),\dots ,x^{[n]}(s))ds. \label{2.5}
\end{equation}
Clearly
\begin{equation}
{\mathcal{G}}x(t_{0}) = x_{0}+\int_{t_{0}}^{t_{0}}
f(s,x^{[1]}(s),x^{[2]}(s),\dots ,x^{[n]}(s))ds=x_{0}
\label{2.6}
\end{equation}
for any $x\in \Phi_{M_1}$. In view of
\begin{align*}
&\|(t,x^{[1]}(t),x^{[2]}(t),\dots ,x^{[n]}(t))-(t_0,x_0,x_0,\dots ,x_0)\|
\\
 &= 
\max\{|t-t_0|,|x^{[1]}(t)-x_0|,|x^{[2]}(t)-x_0|,\dots ,|x^{[n]}(t)-x_0|\}
\\
&\leq 
\max\{|t-t_0|,M_1 |t-t_0|,M_1 |x^{[1]}(t)-t_0|,\dots ,M_1 |x^{[n-1]}(t)-t_0|\}
\\
 &\leq
\max\{l,M_1 l,M_1 l,\dots ,M_1 l\}
\\
&\leq  l \leq  r,
\end{align*}
we get
\begin{equation}
\begin{aligned}
|{\mathcal{G}}x(t_1)-{\mathcal{G}}x(t_2)|
 &\leq
|\int_{t_2}^{t_1}|f(s,x^{[1]}(s),x^{[2]}(s),\dots ,x^{[n]}(s))|ds| \\
 &\leq
M_1 |t_1-t_2|
\end{aligned} \label{2.7}
\end{equation}
for any $t_1,t_2\in[t_0-l,t_0+l]$. Thus
\eqref{2.5}, \eqref{2.6} and \eqref{2.7} yield
${\mathcal{G}}x\in \Phi_{M_1}$; i.e., ${\mathcal{G}}$ is a self-mapping
operator.

It remains to show that ${\mathcal{G}}$ is continuous. 
For this purpose, take any
$x_1, x_2\in \Phi_{M_1}$, we have
\begin{align*}
&|{\mathcal{G}}x_1(t)-{\mathcal{G}}x_2(t)|\\
&\leq
|\int_{t_{0}}^t|f(s,x_{{1}}^{[1]}(s),x_1^{[2]}(s),\dots ,x_1^{[n]}(s))
-f(s,x_2^{[1]}(s),x_2^{[2]}(s),\dots ,x_2^{[n]}(s))|ds|.
\end{align*}
By Lemma \ref{lem1},
\begin{align*}
&\|(s,x_{{1}}^{[1]}(s),x_1^{[2]}(s),\dots ,x_1^{[n]}(s))
-(s,x_2^{[1]}(s),x_2^{[2]}(s),\dots ,x_2^{[n]}(s))\|\\
 &=
\max\{|x_{{1}}^{[1]}(s)-x_{{2}}^{[1]}(s)|,|x_{{1}}^{[2]}(s)-x_{{2}}^{[2]}(s)|,
\dots ,|x_{{1}}^{[n]}(s)-x_{{2}}^{[n]}(s)|\}\\
 &\leq
\max\{\|x_1-x_2\|_{[t_0-l,
t_0+l]},\frac{1-{M_1}^2}{1-M_1}\|x_1-x_2\|_{[t_0-l,
t_0+l]},\\
&\quad \dots ,\frac{1-{M_1}^n}{1-M_1}\|x_1-x_2\|_{[t_0-l,
t_0+l]}\}\\
 &=
\frac{1-{M_1}^n}{1-M_1}\|x_1-x_2\|_{[t_0-l,
t_0+l]}\\
&< 
\frac{1}{1-M_1}\|x_1-x_2\|_{[t_0-l, t_0+l]}.
\end{align*}
Because of the uniform continuity of $f$ on
$\bar{B}(y_0,r)$, for any $\varepsilon>0$ there exist
$\delta(\varepsilon)>0$, the inequality
$$\|{\mathcal{G}}x_1-{\mathcal{G}}x_2\| <\varepsilon l$$
holds for $\|x_1-x_2\|_{[t_0-l,t_0+l]}<\delta$, which implies
${\mathcal{G}}$ is continuous.

$\Phi_{M_1}$ is a convex, compact subset of Banach space $C^{0}([t_0-l,
t_0+l])$ and ${\mathcal{G}}$ is a continuous operator, which satisfy all
conditions of the Schauder's fixed point theorem, so
${\mathcal{G}}$ has a fixed point $g\in \Phi_{M_1}$ and $g$ is a solution
for equation \eqref{consider} associated with \eqref{initial value}
on the interval $[t_{0}-l,t_{0}+l]$. This completes the proof.
\end{proof}


\begin{theorem} \label{thm2}
Suppose that $f:\mathbb{R}^{n+1}\rightarrow\mathbb{R}$ is
continuous and any compact interval $[a,b]$ includes $t_0$ and $x_0$. If
\begin{equation}
M_2 A_{t_0} \leq B_{x_0},
\label{main condition 2}
\end{equation}
where $A_{t_0}=\max\{t_0 -a,b-t_0\},\
B_{x_0}=\min\{x_0 -a,b-x_0\},\ M_2 =\|f\|_{[a,b]^{n+1}}$
and $[a,b]^{n+1}$ denotes the product of $n+1$ intervals $[a,b]$.
Then equation \eqref{consider} associated with \eqref{initial value}
 has a solution defined on $[a,b]$.
\end{theorem}

\begin{proof} 
As in the proof of Theorem \ref{thm1}, we apply
the Schauder fixed point theorem to prove the result. Let
\begin{equation}
\begin{aligned}
\Phi_{M_2} =
  \big\{&x\in\,C^{0}([a,b],[a,b]): x(t_0)=x_0,\\
 & |x(t)-x(s)|\le M_2 |t-s|,\; \forall t, s \in [a,b]\big\},
\end{aligned} \label{condition 2}
\end{equation}
then $\Phi_{M_2}$ is a non-empty convex and compact subset of the Banach
space $C^{0}([a,b])$. We consider the mapping 
${\mathcal{T}}: \Phi_{M_2}\rightarrow C^{0}([a,b])$
defined by
\begin{equation}
{\mathcal{T}}x(t):=x_{0}+\int_{t_{0}}^t
f(s,x^{[1]}(s),x^{[2]}(s),\dots ,x^{[n]}(s))ds.
\label{2.11}
\end{equation}
To prove $\mathcal{T}$ is a self-mapping, we note that
\begin{equation}
\begin{aligned}
{\mathcal{T}}x(t) 
&\leq  x_0 +|\int_{t_0}^t f(s,x^{[1]}(s),x^{[2]}(s),\dots ,x^{[n]}(s))ds| 
\\
&\leq  x_0 +M_2 |t-t_0|
\\
&\leq  x_0 +M_2 A_{t_0}
\\
&\leq  x_0 +B_{t_0}
\leq  b,
\label{<}
\end{aligned}
\end{equation}

\begin{equation}
\begin{aligned}
{\mathcal{T}}x(t) 
&\geq  x_0 -|\int_{t_0}^t f(s,x^{[1]}(s),x^{[2]}(s),\dots ,x^{[n]}(s))ds|
\\
&\geq  x_0 -M_2 |t-t_0|
\\
&\geq  x_0 -M_2 A_{t_0}
\\
&\geq  x_0 -B_{x_0}
\geq  a.
\end{aligned} \label{>}
\end{equation}
Clearly,
\begin{equation}
{\mathcal{T}}x(t_0)=x_0. \label{x_0}
\end{equation}
Moreover, for any $t_1,t_2\in[a,b]$, we have
\begin{equation}
\begin{aligned}
|{\mathcal{T}}x(t_1)-{\mathcal{T}}x(t_2)| 
 &\leq
|\int_{t_2}^{t_1}|f(s,x_{{1}}^{[1]}(s),x_1^{[2]}(s),\dots ,x_1^{[n]}(s))|ds| \\
&\leq M_2 |t_1 -t_2|.
\end{aligned} \label{M_2}
\end{equation}
Thus \eqref{<}, \eqref{>}, \eqref{x_0} and \eqref{M_2}
imply that ${\mathcal{T}}$ maps $\Phi_{M_2}$ into itself.

The definitions of $A_{t_0}$ and
$B_{x_0}$ show that $M_2 \leq1$, then for any $x_1,x_2\in \Phi_{M_2}$,
according to Lemma \ref{lem1}, we have
\begin{align*}
&\|(s,x_{{1}}^{[1]}(s),x_1^{[2]}(s),\dots ,x_1^{[n]}(s))
-(s,x_2^{[1]}(s),x_2^{[2]}(s),\dots ,x_2^{[n]}(s))\|\\
&=
\max\{|x_{{1}}^{[1]}(s)-x_{{2}}^{[1]}(s)|,|x_{{1}}^{[2]}(s)-x_{{2}}^{[2]}(s)|,
\dots ,|x_{{1}}^{[n]}(s)-x_{{2}}^{[n]}(s)|\}\\
&\leq
 \max\{\|x_1-x_2\|_{[a,b]},\frac{1-{M_2}^2}{1-M_2}\|x_1-x_2\|_{[a,b]},
\dots ,\frac{1-{M_2}^n}{1-M_2}\|x_1-x_2\|_{[a,b]}\}\\
&= \frac{1-{M_2}^n}{1-M_2}\|x_1-x_2\|_{[a,b]}\\
&< \frac{1}{1-M_2}\|x_1-x_2\|_{[a,b]}.
\end{align*}
By the uniform continuity of $f$ on $[a,b]^{n+1}$,
for any $\varepsilon>0$ there exists
$\delta(\varepsilon)>0$, when $\|x_1-x_2\|_{[a,b]}<\delta$
we have
$$
|f(s,x_{{1}}^{[1]}(s),x_1^{[2]}(s),\dots ,x_1^{[n]}(s))
-f(s,x_2^{[1]}(s),x_2^{[2]}(s),\dots ,x_2^{[n]}(s))|<\varepsilon.
$$
Consequently,
\begin{align*}
&|{\mathcal{T}}x_1(t)-{\mathcal{T}}x_2(t)|\\
&\leq
|\int_{t_{0}}^t|f(s,x_{{1}}^{[1]}(s),x_1^{[2]}(s),\dots ,x_1^{[n]}(s))
-f(s,x_2^{[1]}(s),x_2^{[2]}(s),\dots ,x_2^{[n]}(s))|ds|\\
&< \varepsilon (b-a),
\end{align*}
which means that ${\mathcal{T}}$ is a continuous operator.

It follows that $\Phi_{M_2}$ is a convex, compact subset of Banach 
space $C^{0}([a,b])$
and ${\mathcal{T}}$ is a continuous operator.
By the Schauder's fixed point theorem,
${\mathcal{T}}$ has a fixed point $h\in \Phi_{M_2}$ and $h$ is a solution
of equation \eqref{consider} associated with \eqref{initial value}
on the interval $[a,b]$. This completes the proof.
\end{proof}

\section{Examples and remarks}


In this section our theorems are demonstrated by the following two examples.
Firstly, we prove the existence of smooth solutions
of the equation, discussed in \cite{SiWang},
together with the general initial value \eqref{initial value} by using 
Theorem \ref{thm1}. Here, smooth function $g\in C^n$ means the function $g$ has
a number of continuous derivatives and
its $n$-th continuous derivative also is Lipschtzian. We need the following
lemma introduced in \cite{SiWang}.


\begin{lemma}[\cite{SiWang}] \label{lem2}
Let
\begin{align*}
\Omega(N_1,\dots ,N_{n+1}; I)
=\big\{&g\in\,C^{n}(I,I): |g^{(i)}(t)|\leq\,N_{i},\, i=1,2,\dots ,n;\\
& |g^{(n)}(t)-g^{(n)}(s)|\leq N_{n+1}|t-s|,\;t,s\in I\big\}.
\end{align*}
For any
$x(t)\in \Omega(N_1,\dots ,N_{n+1}; I)$,
there is
$$
x_{*jk}(t)=P_{jk}(x_{10}(t),\dots ,x_{1,j-1}(t);\dots ;x_{k0}(t),\dots
 ,x_{k,j-1}(t))
$$
and exist positive constants $N_{uv}^{jk}$ such that
$$
|P_{jk}(\bar{\lambda}_{10},\dots ,\bar{\lambda}_{k,j-1})-P_{jk}
(\tilde{\lambda}_{10},\dots ,\tilde{\lambda}_{k,j-1})|
\leq\sum_{u=1}^{k}\sum_{v=0}^{j-1}N_{uv}^{jk}
|\bar{\lambda}_{uv}-\tilde{\lambda}_{uv}|
$$
for $(\bar{\lambda}_{10},\dots ,\bar{\lambda}_{k,j-1}),
(\tilde{\lambda}_{10},\dots ,\tilde{\lambda}_{k,j-1})$ belong to
compact set $[0,N_1]^{j}\times[0,N_2]^{j}\times\dots \times[0,N_{k}]^{j}$,
where $x_{ij}(t)=x^{(i)}(x^{[j]}(t))$,
$x_{*jk}(t)=(x^{[j]}(t))^{(k)}$ and $P_{jk}$ is a uniquely defined 
multivariate polynomial with
nonnegative coefficients and $1\leq u \leq k$, $0\leq v\leq j-1$.
\end{lemma}


\begin{example} \label{exam1} \rm
Consider the equation
\begin{equation}
x'(t)=\sum_{j=1}^{m}a_{j}(t)x^{[j]}(t)+F(t) \label{smooth}
\end{equation} 
associated with \eqref{initial value}, where $a_{j}(t), F(t)\in C^n $ 
are given smooth functions.
\end{example}


For $R>0$, by the smoothness of the given functions,
we have positive $M_{a_{j}}$ and $M_{F}$ such that
$$
|a_{j}(t)|\leq M_{a_{j}},\quad |F(t)|\leq M_{F},\quad t\in[t_{0}-R,
t_{0}+R],\; j=1,2,\dots ,m.
$$
Denote
$$
M_{a}=\max_{1\leq j\leq m}\{M_{a_{j}}\},\quad
N_1=mM_{a}(|t_{0}|+R)+M_{F}.
$$
If $(1-N_1)R\geq |x_0 -t_0|$, the equation \eqref{smooth} associated 
with \eqref{initial value} has a solution in the function set
\begin{align*}
\Phi_{N_1}=\big\{&x\in C^{0}([t_0-l_1, t_0+l_1]): x(t_0)=x_0,\\
 & |x(t)-x(s)|\le N_1|t-s|,\,  \forall t, s \in [t_0-l_1, t_0+l_1]\big\}
\end{align*}
by Theorem \ref{thm1}, where arbitrary $l_1 \in[|x_0 -t_0|/(1-N_1),R]$.
In fact, for any $x\in\Phi_{N_1}$, we see that
the function
$$
f(t,x^{[1]}(t),x^{[2]}(t),\dots ,x^{[m]}(t))=\sum_{j=1}^{m}a_{j}(t)x^{[j]}(t)+F(t)
$$
is continuous on $[t_{0}-l_1,t_{0}+l_1]$ and
\begin{align*}
|f(t,x^{[1]}(t),x^{[2]}(t),\dots ,x^{[m]}(t))| &=
|\sum_{j=1}^{m}a_{j}(t)x^{[j]}(t)+F(t)| \\
&\leq
\sum_{j=1}^{m}M_{a}(|t_{0}|+R)+M_{F} \\
&=
mM_{a}(|t_{0}|+R)+M_{F} 
= N_1.
\end{align*}
Since $(1-N_1)R\geq |x_0 -t_0|$, the condition of Theorem \ref{thm1}
is satisfied, there exists a solution
$x=\varphi(t)$ of equation \eqref{smooth}
together with \eqref{initial value} in the functional set $\Phi_{N_1}$.


The form of equation \eqref{smooth}
and $a_{j}(t), F(t)\in C^{n}([t_0-l_1,t_0+l_1])$
show that $\varphi(t)\in C^{(n+1)}([t_0-l_1,t_0+l_1])$.
In the sequel, we prove $\varphi^{(n+1)}(t)$ also is Lipschtzian
on the compact interval $[t_0-l_1,t_0+l_1]$.
From Lemma \ref{lem2}, we have
\begin{align*}
x_{*jk}({t})
&= P_{jk}(x_{10}({t}),\dots ,x_{1,j-1}({t});\dots ;x_{k0}({t}),\dots ,
 x_{k,j-1}({t}))\\
&= P_{jk}(x'({t}),x'(x_1),\dots ,x'(x_{j-1});\dots ;x^{(k)}({t}),
 x^{(k)}(x_1),\dots ,x^{(k)}(x_{j-1})),
\end{align*}
where $x_{m}=x^{[m]}({t}),\ m=1,2,\dots ,j-1$.
Denote
\begin{gather*}
H_{jk}=P_{jk}(\stackrel{j\ terms}{\overbrace{N_1,\dots ,N_1}};
\stackrel{j\ terms}{\overbrace{N_2,\dots ,N_2}};\dots ; \stackrel{j\
terms}{\overbrace{N_{k},\dots ,N_{k}}}),
\\
a_{j}(t)\in\Omega(L_{j1},\dots ,L_{j(n+1)}; [t_{0}-l_1, t_{0}+l_1]),
\\
F(t)\in\Omega(M_1,\dots ,M_{n+1}; [t_{0}-l_1, t_{0}+l_1]).
\end{gather*}
Then for any $t_1, t_2\in[t_{0}-l_1, t_{0}+l_1]$,
we get
\begin{align*}
& |\varphi^{(n+1)}(t_1)-\varphi^{(n+1)}(t_2)| \\
&\leq
\sum_{j=1}^{m}\sum_{s=0}^{n}C_{n}^{s}|a_{j}^{(n-s)}(t_1)
(\varphi^{[j]}(t_1))^{(s)}-a_{j}^{(n-s)}(t_2)(\varphi^{[j]}(t_2))^{(s)}|\\
&\quad+ |F^{(n)}(t_1)-F^{(n)}(t_2)|
\\
&\leq
\sum_{j=1}^{m}\{|a_{j}^{(n)}(t_1)-a_{j}^{(n)}(t_2)|\cdot
|\varphi^{[j]}(t_1)|+|a_{j}^{(n)}(t_2)|\cdot|\varphi^{[j]}(t_1)
-\varphi^{[j]}(t_2)|\}
\\
&\quad +\sum_{j=1}^{m}\sum_{s=1}^{n}C_{n}^{s}(|a_{j}^{(n-s)}
 (t_1)-a_{j}^{(n-s)}(t_2)|\cdot|(\varphi^{[j]}(t_1))^{(s)}|\\
&\quad +|a_{j}^{(n-s)}(t_2)|\cdot|p_{js}(\varphi_{10}(t_1),\dots ,
\varphi_{s,j-1}(t_1))-p_{js}(\varphi_{10}(t_2),\dots ,\varphi_{s,j-1}(t_2))|)\\
&\quad+M_{n+1}|t_1-t_2|
\\
&\le \sum_{j=1}^{m}(L_{j(n+1)}(|t_{0}|+l_1)+L_{jn}{N_1}^j)|t_1-t_2|\\
&\quad +\sum_{j=1}^{m}\sum_{s=1}^{n}C_{n}^{s}(L_{j(n+1-s)}H_{js}|t_1-t_2|+L_{j(n-s)}\sum_{u=1}^{s}\sum_{v=0}^{j-1}N_{uv}^{js}|\varphi_{uv}(t_1)-\varphi_{uv}(t_2)|)\\
&\quad +M_{n+1}|t_1-t_2|.
\end{align*}
Since
$$
|\varphi_{uv}(t_1)-\varphi_{uv}(t_2)|\leq N_{u+1}|\varphi^{[v]}(t_1)
-\varphi^{[v]}(t_2)|\leq N_{u+1}{N_1}^v|t_1-t_2|,
$$
we have
\begin{align*}
&|\varphi^{(n+1)}(t_1)-\varphi^{(n+1)}(t_2)| \\
&\le \sum_{j=1}^{m}(L_{j(n+1)}(|t_{0}|+l_1)+L_{jn}{N_1}^j)|t_1-t_2|\\
&\quad+
\sum_{j=1}^{m}\sum_{s=1}^{n}C_{n}^{s}(L_{j(n+1-s)}H_{js}|t_1-t_2|+L_{j(n-s)}\sum_{u=1}^{s}\sum_{v=0}^{j-1}N_{uv}^{js}|\varphi_{uv}(t_1)-\varphi_{uv}(t_2)|)\\
&\quad+M_{n+1}|t_1-t_2|\\
&= \{(\sum_{j=1}^{m}L_{j(n+1)}(|t_{0}|+l_1)+L_{jn}{N_1}^j)\\
&\quad+
(\sum_{j=1}^{m}\sum_{s=1}^{n}C_{n}^{s}(L_{j(n+1-s)}H_{js}+L_{j(n-s)}\sum_{u=1}^{s}\sum_{v=0}^{j-1}N_{uv}^{js}N_{u+1}{N_1}^v))+M_{n+1} \}|t_1-t_2|.
\end{align*}
That is, $\varphi^{(n+1)}(t)$ is Lipschtzian.

\begin{remark} \rm
The existence and uniqueness of smooth solutions through 
$(t_{0},t_{0})$, with $ |t_{0}|<1$, for \eqref{smooth} was 
studied in \cite{SiWang}. 
According to Theorem \ref{thm1}, we have the similar conclusion for \eqref{smooth}
through general point $(t_{0},x_{0})$ even for $|t_{0}|\geq1$ provided
$M_a$ and $M_F$ are small enough, which generalizes the results in \cite{SiWang}.
The similar discussion can be applied for the equation in
\cite{ChengSiWang}.
\end{remark}

\begin{example} \rm
Consider the equation
\begin{equation}
x'(t)=\frac{1}{5}x(x(t))-\frac{1}{4} \label{equation}
\end{equation}
associated with
\begin{equation}
x(-1)=-\frac{1}{2}. \label{initial}
\end{equation}

For the compact interval $[-1,0]$ including $t_0 =-1$ and $x_0 =- 1/2$,
it is clear that $M_2 =1/5 +1/4 =9/20,\ A_{t_0}=1,\ B_{x_0}=1/2$,
which satisfy the conditions of Theorem \ref{thm2}. Then the
equation \eqref{equation} associated with \eqref{initial} has a solution.
\label{exam2}
\end{example}

\begin{remark} \rm
In the proof of invariant set in \cite{Berinde-2010}
and \cite{Lauran-2011}, they require the inequalities
\begin{gather}
|(Fy)(t)|\leq |y_0 | + |\int_{x_0}^t f(s,y(s),y(y(s)))ds|\leq |y_0 | 
+ M\cdot |t - x_0|\leq b, \label{example21}
\\
|(Fy)(t)|\geq |y_0 | - |\int_{x_0}^t f(s,y(s),y(y(s)))ds|\geq y_0  
- C_{y_0}\geq a. \label{example22}
\end{gather}
The right-most inequality of \eqref{example22} contradicts the definition 
of $C_{y_0}$. We overcome this difficulty by defining $B_{x_0}$.
Furthermore, \eqref{example21} implies that $b$ is a nonnegative number,
which is given up in Theorem \ref{thm2}
such as Example \ref{exam2}.
\end{remark}


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\end{document}