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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 245, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/245\hfil Existence of solutions]
{Existence of solutions for two-point boundary-value problems
with singular differential equations of variable order}

\author[S. Zhang \hfil EJDE-2013/245\hfilneg]
{Shuqin Zhang}  % in alphabetical order

\address{Shuqin Zhang \newline
Department of Mathematics,
China University of Mining and Technology, \newline
 Beijing 100083, China}
\email{zsqjk@163.com, Tel +86 10 62331118, Fax +86 10 62331465}

\thanks{Submitted May 22, 2013. Published November 12, 2013.}
\subjclass[2000]{26A33, 34B15}
\keywords{Derivatives and integrals of variable order; singular;
\hfill\break\indent differential equations
of variable order;  Arzela-Ascoli theorem}

\begin{abstract}
 In this work, we show the existence of a solution for
 a two-point boundary-value problem having a singular differential
 equation of variable order. We use some analysis techniques and
 the Arzela-Ascoli theorem, and then illustrate our results with
 examples.
\end{abstract}

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

\section{Introduction}

Fractional calculus (fractional derivatives and integrals) refer
to the differential and integral operators of arbitrary order, and
fractional differential equations refer to those containing
fractional derivatives. The former are the generalization of
integer-order differential and integral operators and the latter,
the generalization of differential equations of integer order. The
derivatives and integrals of variable-order, which fall into a
more complex category, are those whose orders are the functions of
certain variables. Recently, derivatives and integrals and
differential equations of variable-order have been considered, see
the references in this article.
In these works, authors consider the applications of
variable-order derivatives in various topics, such as anomalous
diffusion modeling, mechanical applications, multifractional
Gaussian noises. Moreover, a physical experimental study of
calculus of variable-order has been considered in \cite{r1},
a comparative study of constant-order and variable-order
models has been considered in \cite{s5}.

The nonlinear functional analysis methods (such as some fixed point
theorems) have played a very important role in considering
existence of solutions to differential equations of integer order
and fractional order (constant order, such as 1/3). For such
applications, because differential equations can be transformed
into integral equations, by means of some fundamental properties
of differential and integral calculus of integer order and
fractional calculus (constant order). But, in general, we find that
calculus of variable-order lacks these fundamental properties,
thereby making it difficult to apply nonlinear functional analysis
methods to consider existence of solution to problems for
differential equations of variable-order. The following are
several definitions of derivatives and integrals of variable-order
for a function $f$, which can be founded in for example in \cite{r1,v1},
\begin{equation}
I_{a+}^{p(t)}f(t)=\int_{a}^{t}\frac{(t-s)^{p(t)-1}}{\Gamma(p(t))}f(s)ds,\quad
 p(t)>0,\; t>a,\label{e1.1}
\end{equation}
where $\Gamma(\cdot)$ denotes the Gamma function, $-\infty<a<+\infty$,
provided that the right-hand side is pointwise defined.
\begin{equation}
I_{a+}^{p(t)}f(t)=\int_{a}^{t}\frac{(t-s)^{p(s)-1}}{\Gamma(p(s))}f(s)ds, \quad
 p(t)>0,\; t>a,\label{e1.2}
\end{equation}
provided that the right-hand side is pointwise defined.
\begin{equation}
I_{a^{+}}^{p(t)}f(t)=\int_{a}^{t}\frac{(t-s)^{p(t-s)-1}}{\Gamma(p(t-s))}f(s)ds,
\quad p(t)>0,\; t>a,\label{e1.3}
\end{equation}
provided that the right-hand side is pointwise defined.
\begin{equation}
D_{a+}^{p(t)}f(t)=\frac{ d^n}{dt^n}I_{a+}^{n-p(t)}f(t)
=\frac{d^n}{dt^n}\int_{a}^{t}\frac{(t-s)^{n-1-p(t)}}{\Gamma(n-p(t))}f(s)ds,
\quad t>a,\label{e1.4}
\end{equation}
where $n-1<p(t)<n, t>a, n\in \mathbb{N}$, provided that the right-hand side is pointwise
defined.
\begin{equation}
D_{a+}^{p(t)}f(t)=\frac {d^n}{dt^n}I_{a+}^{n-p(t)}f(t)
=\frac{ d^n}{dt^n}\int_{a}^{t}\frac{(t-s)^{n-1-p(s)}}{\Gamma(n-p(s))}f(s)ds, \quad
 t>a,\label{e1.5}
\end{equation}
where $n-1<p(t)<n, t>a, n\in \mathbb{N}$, provided that the right-hand side is pointwise
defined.
\begin{equation}
D_{a^{+}}^{p(t)}f(t)=\frac{ d^n}{dt^n}I_{a+}^{n-p(t)}f(t)
=\frac {d^n}{dt^n}\int_{a}^{t}\frac{(t-s)^{n-1-p(t-s)}}{\Gamma(n-p(t-s))}f(s)ds,
\quad t>a,\label{e1.6}
\end{equation}
where $n-1<p(t)<n$, $t>a$, $n\in \mathbb{N}$, provided that the right-hand side is pointwise
defined.

In particular, when $p(t)$ is a constant function, $p(t)\equiv q$,
where $q$ is a finite positive constant, then $I_{a+}^{p(t)}, D_{a+}^{p(t)}$
are usual Riemann-Liouville fractional integral $I_{a+}^q$ and derivative
$D_{a+}^q$, see \cite{k1}. It is well known that fractional calculus
$I_{a+}^q, D_{a+}^q$ have the following very important properties,
which play a very important role in considering existence of solutions
of fractional differential equation denoted by $D_{a+}^q$, by means
of some fixed point theorems.

 \begin{proposition}[\cite{k1}] \label{prop1.1}
The equality $I_{a+}^\gamma I_{a+}^\delta
f(t)=I_{a+}^{\gamma+\delta}f(t)$, $\gamma>0$, $\delta>0$ holds for $f\in L(a,b)$.
\end{proposition}

\begin{proposition}[\cite{k1}] \label{prop1.2}
 The equality $D_{a+}^\gamma I_{a+}^\gamma f(t)=f(t)$, $\gamma>0$
 holds for $f\in L(a,b)$.
\end{proposition}

\begin{proposition}[\cite{k1}] \label{prop1.3}
Let $\alpha>0$. Then the differential equation
$D_{a+}^\alpha u=0$
has unique solution
$$
u(t)=c_1(t-a)^{\alpha-1}+c_2(t-a)^{\alpha-2}+\dots+c_n(t-a)^{\alpha-n},
$$
where $c_i\in\mathbb{R}$, $i=1,2,\dots,n$, and $n-1<\alpha\leq n$.
\end{proposition}

\begin{proposition}[\cite{k1}] \label{prop1.4}
Let $\alpha>0$, $u\in L(a,b)$,
$D_{a+}^\alpha u\in L(a,b)$. Then the following equality holds
$$
I_{a+}^\alpha D_{a+}^\alpha
u(t)=u(t)+c_1(t-a)^{\alpha-1}+c_2(t-a)^{\alpha-2}+\dots+c_n(t-a)^{\alpha-n},
$$
where $c_i\in\mathbb{R}$, $i=1,2,\dots,n$, and $n-1<\alpha\leq n$.
\end{proposition}

In general, these properties do not hold for derivatives and integrals
of variable-order $D_{a+}^{p(t)}, I_{a+}^{p(t)}$ defined by
\eqref{e1.1}--\eqref{e1.6}. For example, when $p(t),q(t)$ are not constant
functions, we have that
\begin{equation}
I_{a+}^{p(t)}I_{a+}^{q(t)}f(t)\neq I_{a+}^{p(t)+q(t)}f(t), p(t)>0, q(t)>0,\quad
 f\in L(a,b).\label{e1.7}
\end{equation}

\begin{example} \label{examp1.1}\rm
 Let $p(t)=t$, $0\leq t\leq 6$,
\[
q(t)=\begin{cases}
 2,& 0\leq t\leq 2\\
1, & 2<t\leq 3,\\
t, & 3<t\leq 6,\end{cases}
\]
$f(t)=1$, $0\leq t\leq 6$. We calculate $I_{0+}^{p(t)}f(t)$ and
$I_{0+}^{p(t)+q(t)}$ defined by \eqref{e1.3}.
 \begin{align*}
&I_{0+}^{p(t)}I_{0+}^{q(t)}f(t)\\
&= \int_0^t\frac{(t-s)^{p(t)-1}}{\Gamma(p(t))}
\int_0^s\frac{(s-\tau)^{q(s)-1}}{\Gamma(q(s))}f(\tau)d\tau ds
\\
&=\int_0^2\frac{(t-s)^{p(t)-1}}{\Gamma(p(t))}
\int_0^s\frac{(s-\tau)^{q(s)-1}}{\Gamma(q(s))}d\tau ds
 +\int_2^t\frac{(t-s)^{p(t)-1}}{\Gamma(p(t))}
\int_0^s\frac{(s-\tau)^{q(s)-1}}{\Gamma(q(s))}d\tau ds
\\
&=\int_0^2\frac{(t-s)^{p(t)-1}}{\Gamma(p(t))}
\int_0^s\frac{(s-\tau)^{2-1}}{\Gamma(2)}d\tau ds
+\int_2^t\frac{(t-s)^{p(t)-1}}{\Gamma(p(t))}
\int_0^s\frac{(s-\tau)^{q(s)-1}}{\Gamma(q(s))}d\tau ds
\\
&=\int_0^2\frac{(t-s)^{p(t)-1}s^2}{2\Gamma(p(t))}ds
+\int_2^t\frac{(t-s)^{p(t)-1}}{\Gamma(p(t))}
\int_0^s\frac{(s-\tau)^{q(s)-1}}{\Gamma(q(s))}d\tau ds,
\end{align*}
\[
I_{0+}^{p(t)+q(t)}f(t)
=\int_0^t\frac{(t-s)^{p(t)+q(t)-1}}{\Gamma(p(t)+q(t))}f(s)ds,
\]
we see that
\begin{align*}
I_{0+}^{p(t)}I_{0+}^{q(t)}f(t)|_{t=3}
&= \int_0^2\frac{(3-s)^{3-1}s^2}{2\Gamma(3)}ds+\int_2^3\frac{(3-s)^{3-1}}{\Gamma(3)}
\int_0^s\frac{(s-\tau)^{1-1}}{\Gamma(1)}d\tau ds
\\
&= \frac 85+\int_2^3\frac{(3-s)^{3-1}s}{\Gamma(3)}ds
=\frac 85+\frac 9{24}=\frac{79}{40},
\end{align*}
\[
 I_{0+}^{p(t)+q(t)}f(t)|_{t=3}
 =\int_0^3\frac{(3-s)^{p(3)+q(3)-1}}{\Gamma(p(3)+q(3))}f(s)ds
 =\int_0^3\frac{(3-s)^{3+1-1}}{\Gamma(3+1)}ds=\frac {27}8
\]
we see easily that
$$
I_{0+}^{p(t)}I_{0+}^{q(t)}f(t)|_{t=3}\neq I_{0+}^{p(t)+q(t)}f(t)|_{t=3}.
$$
According to \eqref{e1.7}, we can see that Propositions
\ref{prop1.2}--\ref{prop1.4}
 do not hold for $D_{a+}^{p(t)}$ and $I_{a+}^{p(t)}$
defined by \eqref{e1.1}-\eqref{e1.6}.
\end{example}

\begin{remark} \label{rmk1.1} \rm
For integral of variable-order defined by \eqref{e1.5}-\eqref{e1.6}, we
can not easily calculate out fractional integral $I_{a+}^{p(t)}$
of some functions $f(t)$, for example, we do not know that
 what $I_{a+}^{p(t)}1=\int_a^t\frac{(t-s)^{p(s)-1}}{\Gamma(p(s))}ds$
and $I_{a+}^{p(t)}1=\int_a^t\frac{(t-s)^{p(t-s)-1}}{\Gamma(p(t-s))}ds$ equal.
\end{remark}

There also has more complex integrals and derivatives of variable-order,
whose order function $p(t)$ of \eqref{e1.1}--\eqref{e1.6}
is replaced by $p(t,f(t))$; see \cite{k1,o1,r1}.
 For example, for given  a function $f$, its integral and
derivative of variable order
$p(t,f(t))$ ($1<p(t,f(t))<2$) can be defined as follows:
\begin{gather}
I_{a+}^{p(t,f(t))}f(t)=\int_a^t\frac{(t-s)^{p(s,f(s))-1}}{\Gamma(p(s,f(s)))}
 f(s)ds, \quad t>a, \label{e1.8}
\\
D_{a+}^{p(t,f(t))}f(t)=\frac{d^2}{dt^2}I_{a+}^{2-p(t,f(t))}f(t)
=\frac{d^2}{dt^2}\int_a^t\frac{(t-s)^{1-p(s,f(s))}}{\Gamma(2-p(s,f(s)))}f(s)ds,
\quad t>a, \label{e1.9}
\end{gather}
provided that the right-hand side is pointwise defined.

Of course, Propositions \ref{prop1.1}--\ref{prop1.4}
 do not usually hold for integral and
derivative of variable-order defined by \eqref{e1.8}, \eqref{e1.9}.
Therefore, without those properties,
a variable-order differential equation cannot be transformed into an
equivalent integral equation, so that one can consider existence
of solutions of a differential equation of variable-order,
by means of some fixed point theorems.

In this paper, we will consider the existence of solutions to the following
singular two-point boundary-value problem for differential equation
of variable order
\begin{gather}
D_{0+}^{q(t,x(t))}x(t)=f(t,x),\quad  0<t<T,\quad  0<T<+\infty, \label{e1.10}\\
x(0)=0, \quad x(T)=0,\label{e1.11}
\end{gather}
where $D_{0+}^{q(t,x(t))}$ denotes derivative of variable-order defined
by \eqref{e1.9}, $1<q(t,x(t))\leq q^{*}<2$, $0\leq t\leq T$, $x\in\mathbb{R}$,
and $t^rf:[0,T]\times R\to R$ is a continuous function, here $0\leq r<1$.

Due to the properties of variable-order calculus, we do not transform
 problem $\eqref{e1.10}-\eqref{e1.11}$ to an integral equation,
 but, through the use of analysis techniques and the Arzela-Ascoli
theorem to consider existence of solution to \eqref{e1.10}--\eqref{e1.11}.

\section{Preliminaries}

Through this paper, we assume that:
\begin{itemize}
\item[(H1)] $q: [0,T]\times R\to (1,q^{*}]$ is a continuous function,
here $1<q^{*}<2$;

\item[(H2)] $t^rf:[0,T]\times R\to R$ is a continuous function, $0\leq r<1$.
\end{itemize}
It follows from the continuity of compose functions that $\Gamma(q(t,x(t)))$
is continuous on $[0,T]\times R$, when $q$ satisfies assumption condition (H1).

We assume $\delta>0$ to be an arbitrary small number, which is  important
for the next step in the analysis.

  \begin{lemma} \label{lem2.1}
Let {\rm (H1)} hold. And let $x_n, x\in C[0,T]$, assume that
$x_n(t)\to x(t), t\in [0,T]$ as $n\to \infty$, then
\begin{equation}
\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x_n(s))}}{\Gamma(2-q(s,x_n(s)))}x_n(s)ds
\to \int_0^{t-\delta}\frac{(t-s)^{1-q(s,x(s))}}{\Gamma(2-q(s,x(s)))}x(s)ds, 
\label{e2.1}
\end{equation}
for $t\in [\delta,T]$, as $n\to \infty$.
\end{lemma}

\begin{proof} For $x_n, x\in C[0,T]$, we see that
\begin{gather}
\text{if $0<T\leq 1$, then $T^{1-q(s,x_n(s))}\leq T^{1-q^{*}}$,
$T^{1-q(s,x(s))}\leq T^{1-q^{*}}$}, \label{e2.2}
\\
\text{if $1<T<+\infty$, then $T^{1-q(s,x_n(s))}<1$,
$T^{1-q(s,x(s))}<1$.} \label{e2.3}
\end{gather}
Thus, for $0<T<+\infty$, we let
\begin{equation}
T^{*}=\max\{T^{1-q^{*}}, 1\}. \label{e2.4}
\end{equation}
Let
\begin{gather*}
M=\max_{0\leq t\leq T}|x(t)|+1, \quad
M_1=\max_{0\leq t\leq T}|x_n(t)|+1, \\
L=\max_{0\leq t\leq T, \|x_n\|\leq M_1}|\frac 1{\Gamma(2-q(t,x_n(t)))}|+1.
\end{gather*}
By the convergence of $x_n$, for $\frac{(2-q^{*})\varepsilon}{3LT^{*}T}$
($ \varepsilon$ is arbitrary small positive number),
there exists $N_0\in \mathbb{N}$ such that
  $$
|x_n(t)-x(t)|<\frac{(2-q^{*})\varepsilon}{3LT^{*}T}, \quad t\in [0,T],\;
 n\geq N_0.
$$

 Since $(t-s)^{1-q(s,x(s))}, \delta\leq t-s\leq T$, is continuous with
respect to its exponent $1-q(s,x(s))$, for $\frac\varepsilon{3MLT}$,
when $n\geq N_0$, it holds
\begin{equation}
|(t-s)^{1-q(s,x_n(s))}-(t-s)^{1-q(s,x(s))}|<\frac\varepsilon{3MLT},
\delta\leq t-s\leq T, \label{e2.5}
\end{equation}
also, by continuity of $\frac 1{\Gamma(2-q(s,x(s)))}$, for
$\frac{(2-q^{*})\varepsilon}{3MT^{2-q^{*}}}$, when $n\geq N_0$, it holds
\begin{equation}
|\frac 1{\Gamma(2-q(s,x_n(s)))}-\frac 1{\Gamma(2-q(s,x(s)))}|
<\frac{(2-q^{*})\varepsilon}{3MT^{*}T}, 0\leq s\leq T. \label{e2.6}
\end{equation}
Hence, from \eqref{e2.2}, \eqref{e2.3}, \eqref{e2.4}, \eqref{e2.5}, \eqref{e2.6},
 for $\forall \varepsilon>0$, when $n>N_0$, we have that
\begin{align*}
&\big|\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x_n(s))}}{\Gamma(2-q(s,x_n(s)))}x_n(s)ds
-\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x(s))}}{\Gamma(2-q(s,x(s)))}x(s)ds\big|
  \\
&\leq \int_0^{t-\delta}|\frac{(t-s)^{1-q(s,x_n(s))}}{\Gamma(2-q(s,x_n(s)))}|
 |x_n(s)-x(s)|ds\\
&\quad +\int_0^{t-\delta}|\frac{(t-s)^{1-q(s,x_n(s))}-
  (t-s)^{1-q(s,x(s))}}{\Gamma(2-q(s,x_n(s)))}||x(s)|ds
  \\
&\quad +\int_0^{t-\delta}|(t-s)^{1-q(s,x(s))}||\frac 1{\Gamma(2-q(s,x_n(s)))}
-\frac 1{\Gamma(2-q(s,x(s)))}||x(s)|ds
  \\
&\leq \frac{L(2-q^{*})\varepsilon}{3LT^{*}T}
 \int_0^{t-\delta}(t-s)^{1-q(s,x_n(s))}ds
 +  \frac{ML\varepsilon}{3MLT}\int_0^{t-\delta}ds
\\
&\quad  +\frac{M(2-q^{*})\varepsilon}{3MT^{*}T}\int_0^{t-\delta}(t-s)^{1-q(s,x(s))}ds
  \\
&= \frac{(2-q^{*})\varepsilon}{3T^{*}T}\int_0^{t-\delta}T^{1-q(s,x_n(s))}
 (\frac{t-s}T)^{1-q(s,x_n(s))}ds+\frac{\varepsilon}{3T}\int_0^{t-\delta}ds
  \\
&\quad +\frac{(2-q^{*})\varepsilon}{3T^{*}T}\int_0^{t-\delta}T^{1-q(s,x(s))}
 (\frac{t-s}T)^{1-q(s,x(s))}ds
\\
&\leq \frac{(2-q^{*})\varepsilon}{3T^{*}T}
 \int_0^{t-\delta}T^{*}(\frac{t-s}T)^{1-q^{*}}ds
 +\frac{\varepsilon}{3T}\int_0^{t-\delta}ds
\\
&\quad + \frac{(2-q^{*})\varepsilon}{3T^{*}T}
 \int_0^{t-\delta}T^{*}(\frac{t-s}T)^{1-q^{*}}ds
  \\
&= \frac{(2-q^{*})\varepsilon}{3T^{2-q^{*}}}
 \int_0^{t-\delta}(t-s)^{1-q^{*}}ds+\frac{\varepsilon}{3T}\int_0^{t-\delta}ds+
  \frac{(2-q^{*})\varepsilon}{3T^{2-q^{*}}}
  \int_0^{t-\delta}(t-s)^{1-q^{*}}ds
  \\
&= \frac{\varepsilon}{3T^{2-q^{*}}}(t^{2-q^{*}}-\delta^{2-q^{*}})
 +\frac{\varepsilon}{3T}(t-\delta)+
  \frac{\varepsilon }{3T^{2-q^{*}}}(t^{2-q^{*}}-\delta^{2-q^{*}})
  \\
&< \frac{\varepsilon T^{2-q^{*}}}{3T^{2-q^{*}}}+\frac{T\varepsilon}{3T}
 +\frac{\varepsilon T^{2-q^{*}}}{3T^{2-q^{*}}}
  \\
&= \frac\varepsilon 3+\frac\varepsilon 3+\frac\varepsilon 3=\varepsilon,
\end{align*}
which implies that \eqref{e2.1} holds.
\end{proof}

By a similar argument, we can show the following result.

\begin{lemma} \label{lem2.2}
Let {\rm (H1), (H2)} hold. And let $x_n, x\in C[0,T]$, assume that
$x_n(t)\to x(t), t\in [0,T]$ as $n\to \infty$, then
\begin{equation}
\int_0^{t-\delta}(t-s)f(s,x_n(s))ds\to \int_0^{t-\delta}(t-s)f(s,x(s))ds,
\quad t\in [\delta,T], \label{e2.7}
\end{equation}
as $n\to \infty$.
\end{lemma}

\begin{proof}
By the convergence of $x_n$, for $\zeta>0$, there exists $N_0\in \mathbb{N}$ such that
$$
|x_n(t)-x(t)|<\zeta, \quad t\in [0,T],\; n\geq N_0,
$$
  by the continuity of $tf$, for $\frac {2\varepsilon}{T^2}$
(where $\varepsilon$ is arbitrary small number), when $n\geq N_0$, it holds
  $$
s^r|f(s,x_n(s))-f(s,x(s))|<\frac {\Gamma(3-r)\varepsilon}{T^{2-r}\Gamma(1-r)},
\quad s\in [0,T].
$$
  Thus, we have
\begin{align*}
&|\int_0^{t-\delta}(t-s)(f(s,x_n(s))-f(s,x(s)))ds|
  \\
&\leq \int_0^{t-\delta}(t-s)s^{-r}s^r|f(s,x_n(s))-f(s,x(s))|ds
  \\
&<\frac{\Gamma(3-r)\varepsilon}{T^{2-r}\Gamma(1-r)}\int_0^{t-\delta}(t-s)s^{-r}ds
  \\
&\leq \frac{\Gamma(3-r)\varepsilon}{T^{2-r}\Gamma(1-r)}\int_0^t(t-s)s^{-r}ds
  \\
&= \frac{\Gamma(3-r)\Gamma(1-r)\varepsilon}{T^{2-r}\Gamma(1-r)\Gamma(3-r)}t^{2-r}
 \leq\varepsilon,
\end{align*}
  which implies that \eqref{e2.7} holds.
\end{proof}

\begin{lemma}[\cite{k1}] \label{lem2.3}
Let $[a, b]$ be a finite interval and let $AC[a, b]$ be the
space of functions which are absolutely continuous on $[a, b]$. It
is known that $AC[a,b]$ coincides with the space of primitives of
Lebesgue summable functions:
$$
f(t)\in AC[a,b]\Leftrightarrow f(t)
=c+ \int_0^t\varphi(s)ds, \quad\varphi\in L(a,b),\;c\in\mathbb{R},
$$
and therefore the absolutely continuous function $f(t)$ has a summable
 derivative $f'(t) = \varphi(t)$ almost everywhere on $[a,b]$.
\end{lemma}

\section{Existence result}

By the definition of derivative of variable order, defined by \eqref{e1.9},
we see that problem \eqref{e1.10}-\eqref{e1.11} is equivalent to the
equation
\begin{equation} \label{e3.1}
\int_0^t\frac{(t-s)^{1-q(s,x(s))}}{\Gamma(2-q(s,x(s)))}x(s)ds
=c_1+c_2t+\int_0^t(t-s)f(s,x(s))ds,
\end{equation}
for $t\in [0,T]$,
  where $c_1,c_2\in\mathbb{R}$ such that $x(0)=x(T)=0$ holds.

\begin{theorem} \label{thm3.1}
 Assume that {\rm (H1), (H2)} hold. Then problem \eqref{e1.10}-\eqref{e1.11}
 exists one solution $x^{*}\in C[0,T]$.
\end{theorem}

\begin{proof}
To obtain the existence result for \eqref{e1.10}-\eqref{e1.11},
we firstly verify the following sequence has convergent subsequence,
\begin{equation}
x_k(t)=   \begin{cases}
  0, & 0\leq t\leq \delta,
  \\
  x_{k-1}(t)+\int_0^{t-\delta}
\frac{(t-s)^{1-q(s,x_{k-1}(s))}}{\Gamma(2-q(s,x_{k-1}(s)))}x_{k-1}(s)ds
\\
-c_{2,k-1}(t-\delta) -\int_0^{t-\delta}(t-s)f(s,x_{k-1}(s))ds, & \delta<t\leq T,
   \end{cases}
  \label{e3.2}
\end{equation}
for  $k=1,2,\dots$, where $x_0(t)=0$, $t\in [\delta,T]$, $\delta$ is an arbitrary
small number, and
 \begin{equation}
c_{2,k-1}=\frac{\int_0^{T-\delta}
\frac{(T-s)^{1-q(s,x_{k-1}(s))}}{\Gamma(2-q(s,x_{k-1}(s)))}x_{k-1}(s)ds
-  \int_0^{T-\delta}(T-s)f(s,x_{k-1}(s))ds}{T-\delta},\label{e3.3}
\end{equation}
such that
 \begin{equation}
x_k(\delta)=x_k(T)=0,\quad  k=1,2,\dots.\label{e3.4}
\end{equation}

To apply the Arzela-Ascoli theorem to consider the existence of
convergent subsequence of sequence $x_k$ defined by \eqref{e3.2}, firstly,
we prove the uniformly bounded of sequence $x_k$ on $[0,T]$.

We find that $x_k$ is uniformly bounded on $[0,\delta]$. Now, we will
verify sequence $x_k$ is uniformly bounded on $[\delta,T]$.
Since $x_0=0$ is uniformly bounded on $[0,T]$, we have that
  \begin{align*}
&\big|\int_0^{T-\delta}\frac{(T-s)^{1-q(s,x_0(s))}}
{\Gamma(2-q(s,x_0(s)))}x_0(s)ds-
  \int_0^{T-\delta}(T-s)f(s,x_0(s))ds\big|
  \\
&= |\int_0^{T-\delta}(T-s)f(s,0)ds|
  \\
&= |\int_0^{T-\delta}(T-s)s^{-r}s^rf(s,0)ds|
  \\
&\leq M\int_0^{T-\delta}(T-s)s^{-r}ds
  \\
&\leq M\int_0^T(T-s)s^{-r}ds
  \\
&= \frac{M\Gamma(1-r)}{\Gamma(3-r)}T^{2-r},
  \end{align*}
where $M=\max_{0\leq t\leq T}t^r|f(t,0)|+1$, which implies that
$|c_{2,0}|\leq\frac{M\Gamma(1-r)}{(T-\delta)\Gamma(3-r)}T^{2-r}$.
Then, for $t\in [\delta,T]$, we have
\begin{align*}
  |x_1(t)|
&= |x_0(t)+\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x_0(s))}}{\Gamma(2-q(s,x_0(s)))}
x_0(s)ds-   c_{2,0}(t-\delta)
\\
&\quad -\int_0^{t-\delta}(t-s)f(s,0)ds|
  \\
&= |c_{2,0}(t-\delta)-\int_0^{t-\delta}(t-s)f(s,0)ds|
  \\
&\leq  |c_{2,0}|(T-\delta)+M\int_0^{t-\delta}(t-s)s^{-r}ds
  \\
&\leq |c_{2,0}|(T-\delta)+M\int_0^t(t-s)s^{-r}ds
  \\
&\leq \frac{M\Gamma(1-r)}{\Gamma(3-r)}T^{2-r}
 +\frac{M\Gamma(1-r)}{\Gamma(3-r)}T^{2-r}\doteq M_1,
\end{align*}
which implies that $x_1$ is uniformly bounded on $[\delta,T]$,
together with $x_1(t)=0$ for $t\in [0,\delta]$, we obtain that $x_1$
is uniformly bounded on $[0,T]$.

 From \eqref{e2.2}, \eqref{e2.3}, \eqref{e2.4}, it holds that
\begin{align*}
&| \int_0^{T-\delta}\frac{(T-s)^{1-q(s,x_1(s))}}{\Gamma(2-q(s,x_1(s)))}x_1(s)ds-
  \int_0^{T-\delta}(T-s)f(s,x_1(s))ds|
  \\
&\leq M_1\int_0^{T-\delta}|\frac{T^{1-q(s,x_1(s))}}{\Gamma(2-q(s,x_1(s)))}|
|(\frac{T-s}T)^{1-q(s,x_1(s))}|ds+M_f\int_0^{T-\delta}(T-s)s^{-r}ds
  \\
&\leq M_1L\int_0^{T-\delta}T^{*}(\frac{T-s}T)^{1-q^{*}}ds
 +M_f\int_0^T(T-s)s^{-r}ds
  \\
&= \frac{M_1LT^{*}T^{q^{*}-1}}{2-q^{*}}(T^{2-q^{*}}-\delta^{2-q^{*}})
 +\frac{M_f\Gamma(1-r)}{\Gamma(3-r)}T^{2-r}
  \\
&\leq \frac{M_1LT^{*}T}{2-q^{*}}+\frac{M_f\Gamma(1-r)}{\Gamma(3-r)}T^{2-r}
:=\widetilde{M},
\end{align*}
where
\[
L=\max_{0\leq t\leq T, \|x_1\|\leq M_1}|\frac 1{\Gamma(2-q(t,x_1(t)))}|+1, \quad
M_f=\max_{0\leq t\leq T, \|x_1\|\leq M_1}t^r|f(t,x_1(t))|+1,
\]
which implies that $|c_{2,1}|\leq \frac{\widetilde{M}}{T-\delta}$. Also,
for $t\in [\delta,T]$, by \eqref{e2.2}, \eqref{e2.3}, \eqref{e2.4},
we have that
\begin{align*}
  |x_2(t)|&\leq |x_1(t)|+|c_{2,1}|(T-\delta)+\int_0^{t-\delta}|
 \frac{(t-s)^{1-q(s,x_1(s))}}{\Gamma(2-q(s,x_1(s)))}||x_1(s)|ds
  \\
&\quad +\int_0^{t-\delta}(t-s)|f(s,x_1(s))|ds
  \\
&\leq M_1+|c_{2,1}|(T-\delta)+M_1L\int_0^{t-\delta}
 T^{1-q(s,x_1(s))}(\frac{t-s}T)^{1-q(s,x_1(s))}ds
\\
&\quad +\frac{M_f\Gamma(1-r)}{\Gamma(3-r)}T^{2-r}
  \\
&\leq M_1+|c_{2,1}|(T-\delta)+M_1L\int_0^{t-\delta}T^{*}(\frac{t-s}T)^{1-q^{*}}ds
+\frac{M_f\Gamma(1-r)}{\Gamma(3-r)}T^{2-r}
  \\
&= M_1+|c_{2,1}|(T-\delta)+\frac{M_1LT^{*}T^{q^{*}-1}}{2-q^{*}}(t^{2-q^{*}}
 -\delta^{2-q^{*}})+\frac{M_f\Gamma(1-r)}{\Gamma(3-r)}T^{2-r}
  \\
&\leq M_1+\widetilde{M}+\frac{M_1LT^{*}T}{2-q^{*}}
 +\frac{M_f\Gamma(1-r)}{\Gamma(3-r)}T^{2-r}=:M_2,
\end{align*}
which implies that $x_2$ is uniformly bounded on $[\delta,T]$,
together with $x_2(t)=0$ for $t\in [0,\delta]$, we obtain that $x_2$
is uniformly bounded on $[0,T]$. Continuous this process,
we can obtain that sequence $x_k$ is uniformly bounded on $[0,T]$.

Now, we consider the equicontinuous of sequence $x_k$ on $[0,T]$.
Firstly, we can know that
\begin{equation}
\text{the function $k(t)=a^t-b^t$ is decreasing for
$t\in (-1,0)$ and $0<a<b<1$}.\label{e3.5}
\end{equation}
Indeed, since $\ln a<\ln b<0$, $a^t>b^t>0$, we have that
   $$
k'(t)=a^t\ln a-b^t\ln b<b^t\ln a-b^t\ln b=b^t(\ln a-\ln b)<0,
$$
which implies that $k(t)$ is decreasing function.
Thus, for
\[
l(s)=(\frac{t_1-s}T)^{1-q(s,x(s))}-(\frac{t_2-s}T)^{1-q(s,x(s))}
\]
where $0<\frac{t_1-s}T<\frac{t_2-s}T<1$, we may look $l(s)$ as the same
type as $k(s)$, then $l(s)$ is decreasing with respect to its exponent
$1-q(s,x(s))$.

In the next analysis, we will use the Minkowsk's inequality:
for $a,b$ non-negative, and any $R\geq 0$, it holds
$$
(a+b)^R\leq c_R(a^R+b^R),\quad\text{where } c_R=\max\{1,2^{R-1}\}.
$$
  As a result, for $a,b$ non negative, and any $0<\mu<1$, it holds
\begin{equation}
(a+b)^\mu\leq c_\mu(a^\mu+b^\mu)=\max\{1,2^{\mu-1}\}(a^\mu+b^\mu)
=a^\mu+b^\mu.\label{e3.6}
\end{equation}
  Obviously, $x_0$ is equicontinuous on $[0,T]$. We let
$M=\max_{0\leq t\leq T}s^r|f(s,0)|+1$. For all $\varepsilon>0$,  and all
$t_1, t_2\in [0,T]$, $t_1<t_2$. we consider result in two cases.

\noindent\textbf{Case I:} $0\leq t_1\leq \delta<t_2\leq T$.
We take $\eta_{1,I}=\min\{\frac\varepsilon{2(|c_{2,0}|+1)},
(\frac{\varepsilon(1-r)}{2MT})^{\frac 1{1-r}}\}$, when
$t_2-t_1<\eta_{1,I}$, we have
\begin{align*}
|x_1(t_2)-x_1(t_1)|
&= |c_{2,0}(t_2-\delta)+\int_0^{t_2-\delta}(t_2-s)f(s,0)ds|
\\
&\leq |c_{2,0}|(t_2-\delta)+M\int_0^{t_2-\delta}(t_2-s)s^{-r}ds
\\
&\leq |c_{2,0}|(t_2-\delta)+MT\int_0^{t_2-\delta}s^{-r}ds
\\
&= |c_{2,0}|(t_2-\delta)+\frac{MT}{1-r}(t_2-\delta)^{1-r}
\\
&\leq (|c_{2,0}|+1)|(t_2-t_1)+\frac{MT}{1-r}(t_2-t_1)^{1-r}
\\
&< (|c_{2,0}|+1)|\eta_{1,I}+\frac{MT}{1-r}\eta_{1,I}^{1-r}
\\
&\leq \frac\varepsilon 2+\frac\varepsilon 2=\varepsilon.
\end{align*}

\noindent\textbf{Case II:} $\delta\leq t_1<t_2\leq T$.
 We take
\[
\eta_{1,II}=\min\big\{\frac{\varepsilon(1-r)}{2((|c_{2,0}|+1)(1-r)+MT^{1-r}},
(\frac{\varepsilon(1-r)}{2MT})^{\frac 1{1-r}}\big\},
\]
 when $t_2-t_1<\eta_{1,II}$, by \eqref{e3.6}, we have
 \begin{align*}
&|x_1(t_2)-x_1(t_1)|\\
&= |c_{2,0}(t_1-t_2)+\int_0^{t_1-\delta}(t_1-s)f(s,0)ds
 -\int_0^{t_2-\delta}(t_2-s)f(s,0)ds|
\\
&\leq |c_{2,0}|(t_2-t_1)+\int_0^{t_1-\delta}|t_1-t_2||f(s,0)|ds
 +\int_{t_1-\delta}^{t_2-\delta}(t_2-s)|f(s,0)|ds
\\
&\leq |c_{2,0}|(t_2-t_1)+M\int_0^{t_1-\delta}(t_2-t_1)s^{-r}ds
 +M\int_{t_1-\delta}^{t_2-\delta}(t_2-s)s^{-r}ds
\\
&\leq \frac{(|c_{2,0}|+1)(1-r)+MT^{1-r}}{1-r}(t_2-t_1)
 +\frac{MT}{1-r}((t_2-\delta)^{1-r}-(t_1-\delta)^{1-r})
\\
&= \frac{(|c_{2,0}|+1)(1-r)+MT^{1-r}}{1-r}(t_2-t_1)
 +\frac{MT}{1-r}((t_2-t_1+t_1-\delta)^{1-r}
\\
&\quad -(t_1-\delta)^{1-r})
\\
&\leq \frac{(|c_{2,0}|+1)(1-r)+MT^{1-r}}{1-r}(t_2-t_1)
 +\frac{MT}{1-r}((t_2-t_1)^{1-r}+(t_1-\delta)^{1-r}
\\
&\quad -(t_1-\delta)^{1-r})
\\
&= \frac{(|c_{2,0}|+1)(1-r)+MT^{1-r}}{1-r}(t_2-t_1)
 +\frac{MT}{1-r}(t_2-t_1)^{1-r}
\\
&< \frac{(|c_{2,0}|+1)(1-r)+MT^{1-r}}{1-r}\eta_{1,II}
 +\frac{MT}{1-r}\eta_{1,II}^{1-r}
\\
&\leq \frac\varepsilon 2+\frac\varepsilon 2=\varepsilon.
\end{align*}
These imply that $x_1(t)$ is equicontinuous on $[0,T]$, the same result
can be obtained when $t_2<t_1$.

We let
\[
M_f=\max_{0\leq s\leq T, \|x_1\|\leq M_1}s^r|f(s,x_1)|+1, \quad
L=\max_{0\leq s\leq T, \|x_1\|\leq M_1}|\frac 1{\Gamma(2-q(s,x_1(s)))}|+1.
\]
For all $\varepsilon>0$, and all $t_1, t_2\in [0,T]$, $t_1<t_2$.
 We consider result in two cases.

\noindent\textbf{Case I:}  $0\leq t_1\leq \delta<t_2\leq T$.
We take
$$
\eta_{2,I}=\min\big\{\eta_{1,I},\frac\varepsilon{4(|c_{2,0}|+1)},
\big(\frac{2-q^{*}}{4M_1LT^{*}T^{q^{*}-1}}\big)^{\frac 1{2-q^{*}}},
\big(\frac{\varepsilon(1-r)}{4M_fT}\big)^{\frac 1{1-r}}\big\},
$$
when $t_2-t_1<\eta_{2,I}$, by \eqref{e2.2}, \eqref{e2.3}, \eqref{e3.6}
 and the previous arguments, we have
  \begin{align*}
&|x_2(t_2)-x_2(t_1)|
\\
&= |x_1(t_2)-c_{2,0}(t_2-\delta)
+\int_0^{t_2-\delta}\frac{(t_2-s)^{1-q(s,x_1(s))}}{\Gamma(2-q(s,x_1(s)))}x_1(s)ds
\\
&\quad -\int_0^{t_2-\delta}(t_2-s)f(s,x_1)ds|
\\
&\leq |x_1(t_2)|+|c_{2,0}|(t_2-\delta)+M_1L\int_0^{t_2-\delta}
 (t_2-s)^{1-q(s,x_1(s))}ds
\\
&\quad +M_f\int_0^{t_2-\delta}(t_2-s)s^{-r}ds
\\
&\leq |x_1(t_2)|+|c_{2,0}|(t_2-\delta)
+M_1L\int_0^{t_2-\delta}T^{1-q(s,x_1(s))}(\frac{t_2-s}T)^{1-q(s,x_1(s))}ds
\\
&\quad +M_fT\int_0^{t_2-\delta}s^{-r}ds
\\
&\leq |x_1(t_2)|+|c_{2,0}|(t_2-\delta)
+M_1L\int_0^{t_2-\delta}T^{*}(\frac{t_2-s}T)^{1-q^{*}}ds
\\
&\quad +\frac{M_fT}{1-r}(t_2-\delta)^{1-r}
\\
&= |x_1(t_2)|+|c_{2,0}|(t_2-\delta)
+\frac{M_1LT^{*}T^{q^{*}-1}}{2-q^{*}}(t_2^{2-q^{*}}-\delta^{2-q^{*}})
+\frac{M_fT}{1-r}(t_2-\delta)^{1-r}
\\
&= |x_1(t_2)-x_1(t_1)|+|c_{2,0}|(t_2-\delta)
 +\frac{M_1LT^{*}T^{q^{*}-1}}{2-q^{*}}((t_2-\delta+\delta)^{2-q^{*}}
 -\delta^{2-q^{*}})
\\
&\quad +\frac{M_fT}{1-r}(t_2-\delta)^{1-r}
\\
&\leq |x_1(t_2)-x_1(t_1)|+|c_{2,0}|(t_2-\delta)
 +\frac{M_1LT^{*}T^{q^{*}-1}}{2-q^{*}}((t_2-\delta)^{2-q^{*}}
 +\delta^{2-q^{*}}-\delta^{2-q^{*}})
\\
&\quad +\frac{M_fT}{1-r}(t_2-\delta)^{1-r}
\\
&\leq |x_1(t_2)-x_1(t_1)|+(|c_{2,0}|+1)(t_2-t_1)
 +\frac{M_1LT^{*}T^{q^{*}-1}}{2-q^{*}}(t_2-t_1)^{2-q^{*}}
\\
&\quad +\frac{M_fT}{1-r}(t_2-t_1)^{1-r}
\\
&< |x_1(t_2)-x_1(t_1)|+(|c_{2,0}|+1)\eta_{2,I}
 +\frac{M_1LT^{*}T^{q^{*}-1}}{2-q^{*}}\eta_{2,I}^{2-q^{*}}
 +\frac{M_fT}{1-r}\eta_{2,I}^{1-r}
\\
&< \varepsilon+\frac\varepsilon 4+\frac\varepsilon 4+\frac\varepsilon 4
= \frac{7\varepsilon}4.
\end{align*}

\noindent\textbf{Case II:} $\delta\leq t_1<t_2\leq T$. We take
$$
\eta_{2,II}=\min\big\{\eta_{1,II},\frac\varepsilon{4(|c_{2,0}|+1)},
\big(\frac{(2-q^{*})\varepsilon}{8M_1LT^{*}T^{q^{*}-1}}\big)^{\frac 1{2-q^{*}}},
\frac{\varepsilon(1-r)}{4M_fT^{1-r}},
\big(\frac{\varepsilon(1-r)}{4M_fT}\big)^{\frac 1{1-r}}\},
$$
when $t_2-t_1<\eta_{2,I}$, by \eqref{e2.2}, \eqref{e2.3}, \eqref{e2.4},
\eqref{e3.5}, \eqref{e3.6} and the previous arguments, we have
  \begin{align*}
&|x_2(t_2)-x_2(t_1)|
\\
&= |x_1(t_2)-x_1(t_1)-c_{2,1}(t_2-t_1)
+\int_0^{t_2-\delta}\frac{(t_2-s)^{1-q(s,x_1(s))}}{\Gamma(2-q(s,x_1(s)))}x_1(s)ds
\\
&\quad -\int_0^{t_1-\delta}\frac{(t_1-s)^{1-q(s,x_1(s))}}
 {\Gamma(2-q(s,x_1(s)))}x_1(s)ds-\int_0^{t_2-\delta}(t_2-s)f(s,x_1)ds
\\
&\quad +\int_0^{t_1-\delta}(t_1-s)f(s,x_1)ds|
\\
&\leq |x_1(t_2)-x_1(t_1)|+|c_{2,1}|(t_2-t_1)
 +\int_{t_1-\delta}^{t_2-\delta}|\frac{(t_2-s)^{1-q(s,x_1(s))}}
 {\Gamma(2-q(s,x_1(s)))}||x_1(s)|ds
\\
&\quad +\int_0^{t_1-\delta}|\frac 1{\Gamma(2-q(s,x_1(s)))}|
 |(t_2-s)^{1-q(s,x_1(s))}-(t_1-s)^{1-q(s,x_1(s))}||x_1(s)|ds
\\
&\quad +\int_0^{t_1-\delta}|t_2-t_1||f(s,x_1(s))|ds
 +\int_{t_1-\delta}^{t_2-\delta}(t_2-s)|f(s,x_1(s))|ds
\\
&\leq |x_1(t_2)-x_1(t_1)|+|c_{2,1}|(t_2-t_1)
\\
&\quad  +M_1L\int_0^{t_1-\delta}((t_1-s)^{1-q(s,x_1(s))}-(t_2-s)^{1-q(s,x_1(s))})ds
\\
&\quad + M_1L\int_{t_1-\delta}^{t_2-\delta}(t_2-s)^{1-q(s,x_1(s))}ds
+M_f\int_0^{t_1-\delta}(t_2-t_1)s^{-r}ds
\\
&\quad +M_fT\int_{t_1-\delta}^{t_2-\delta}s^{-r}ds
\\
&= |x_1(t_2)-x_1(t_1)|+|c_{2,1}|(t_2-t_1)
\\
&\quad +M_1L\int_{t_1-\delta}^{t_2-\delta}T^{1-q(s,x_1(s))}
(\frac{t_2-s}T)^{1-q(s,x_1(s))}ds
\\
&\quad +M_1L\int_0^{t_1-\delta}T^{1-q(s,x_1(s))}((\frac{t_1-s}T)^{1-q(s,x_1(s))}
-(\frac{t_2-s}T)^{1-q(s,x_1(s))})ds
\\
&\quad +\frac{M_f(t_1-\delta)^{1-r}}{1-r}(t_2-t_1)
 +\frac{M_fT}{1-r}((t_2-\delta)^{1-r}-(t_1-\delta)^{1-r})
\\
&\leq |x_1(t_2)-x_1(t_1)|+|c_{2,1}|(t_2-t_1)
\\
&\quad +M_1L\int_0^{t_1-\delta}T^{*}((\frac{t_1-s}T)^{1-q^{*}}
 -(\frac{t_2-s}T)^{1-q^{*}})ds
\\
&\quad +M_1L\int_{t_1-\delta}^{t_2-\delta}T^{*}(\frac{t_2-s}T)^{1-q^{*}}ds
+\frac{M_fT^{1-r}}{1-r}(t_2-t_1)
\\
&\quad +\frac{M_fT}{1-r}((t_2-\delta)^{1-r} -(t_1-\delta)^{1-r})
\\
&= |x_1(t_2)-x_1(t_1)|+|c_{2,1}|(t_2-t_1)
 +\frac{M_1LT^{*}T^{q^{*}-1}}{2-q^{*}}(t_1^{2-q^{*}}
 -\delta^{2-q^{*}}
\\
&\quad +2(t_2-t_1+\delta)^{2-q^{*}}
 -t_2^{2-q^{*}}-\delta^{2-q^{*}})+\frac{M_fT^{1-r}}{1-r}(t_2-t_1)
\\
&\quad  +\frac{M_fT}{1-r}((t_2-\delta)^{1-r}-(t_1-\delta)^{1-r})
\\
&\leq |x_1(t_2)-x_1(t_1)|+|c_{2,1}|(t_2-t_1)+\frac{M_1LT^{*}T^{q^{*}-1}}{2-q^{*}}
\Big(t_2^{2-q^{*}}-2\delta^{2-q^{*}}
\\
&\quad +2(t_2-t_1)^{2-q^{*}}  +2\delta^{2-q^{*}}-t_2^{2-q^{*}}\Big)
 +\frac{M_fT^{1-r}}{1-r}(t_2-t_1)
 +\frac{M_fT}{1-r}((t_2-t_1)^{1-r}\\
&\quad +(t_1-\delta)^{1-r}-(t_1-\delta)^{1-r})
\\
&= |x_1(t_2)-x_1(t_1)|+|c_{2,1}|(t_2-t_1)+\frac{2M_1LT^{*}T^{q^{*}-1}}{2-q^{*}}
 (t_2-t_1)^{2-q^{*}}
\\
&\quad +\frac{M_fT^{1-r}}{1-r}(t_2-t_1) +\frac{M_fT}{1-r}(t_2-t_1)^{1-r}
\\
&< |x_1(t_2)-x_1(t_1)|+(|c_{2,1}|+1)\eta_{2,II}
 +\frac{2M_1LT^{*}T^{q^{*}-1}}{2-q^{*}}\eta_{2,II}^{2-q^{*}}+
\frac{M_fT^{1-r}}{1-r}\eta_{2,II}
\\
&\quad +\frac{M_fT}{1-r}\eta_{2,II}^{1-r}
\\
&< \varepsilon+\frac\varepsilon 4+\frac\varepsilon 4
+\frac\varepsilon 4+\frac\varepsilon 4 = 2\varepsilon.
\end{align*}
These imply that $x_2(t)$ is equicontinuous on $[0,T]$, the same result
can be obtained when $t_2<t_1$. Continue these process, we can obtain
that $x_k$, $k=1,2 \dots$, is equaicontinuous on $[0,T]$.

By the arguments of equicontinuity of $x_k$, we can show that $x_k\in C[0,T]$,
for $k=1,2,\dots$. Then, from the Arzela-Ascoli theorem, sequence $x_k$ exists
 a convergent subsequence $x_{m_k}$. From \eqref{e3.2}, $x_{m_k}$
should satisfy
\begin{equation}
x_{m_k}(t)= \begin{cases}
  0, &0\leq t\leq \delta,
\\
  x_{m_{k-1}}(t)+\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x_{m_{k-1}}(s))}}
 {\Gamma(2-q(s,x_{m_{k-1}}(s)))}x_{m_{k-1}}(s)ds
\\
-c_{2,m_{k-1}}(t-\delta) -\int_0^{t-\delta}(t-s)f(s,x_{m_{k-1}}(s))ds, 
& \delta<t\leq T,
   \end{cases}
\label{e3.7}
\end{equation}
 where
\begin{equation}
c_{2,m_{k-1}}=\frac{\int_0^{T-\delta}
\frac{(T-s)^{1-q(s,x_{m_{k-1}}(s))}}{\Gamma(2-q(s,x_{m_{k-1}}(s)))}x_{m_{k-1}}
 (s)ds-  \int_0^{T-\delta}(T-s)f(s,x_{m_{k-1}}(s))ds}{T-\delta},
\label{e3.8}
\end{equation}
such that
\begin{equation}
x_{m_k}(\delta)=x_{m_k}(T)=0,\quad  k=1,2,\dots.\label{e3.9}
\end{equation}
 Now, we prove that the continuous limit of $x_{m_k}$, denoted
by $x^{*}$ is one solution of problem \eqref{e1.10}-\eqref{e1.11}.

Let $k\to +\infty$ in \eqref{e3.7}, \eqref{e3.8}, \eqref{e3.9},
by Lemmas \ref{lem2.1}, \ref{lem2.2}, we have
\begin{gather}
x^{*}(t)=   \begin{cases}
  0,&0\leq t\leq \delta,
  \\
  x^{*}(t)+\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x^{*}(s))}}
 {\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds-c_2(t-\delta)
  \\
-\int_0^{t-\delta}(t-s)f(s,x^{*}(s))ds, & \delta<t\leq T,
   \end{cases}
 \label{e3.10}
\\
x^{*}(\delta)=x^{*}(T)=0.\label{e3.11}
\end{gather}
where
\begin{equation}
c_2=\frac{\int_0^{T-\delta}\frac{(T-s)^{1-q(s,x^{*}(s))}}
{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds-
  \int_0^{T-\delta}(T-s)f(s,x^{*}(s))ds}{T-\delta},\label{e3.12}
\end{equation}
Thus, we find that, for $t\in [0,\delta]$, $x^{*}=0$; for
$t\in [\delta,T]$, $x^{*}$ satisfies relation
\begin{equation}
\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x^{*}(s))}}{\Gamma(2-q(s,x^{*}(s)))}
x^{*}(s)ds-c_2(t-\delta)-\int_0^{t-\delta}(t-s)f(s,x^{*}(s))ds=0,
\label{e3.13}
\end{equation}
for $\delta\leq t\leq T$.

To verify $x^{*}$ is one solution of problem \eqref{e1.10}-\eqref{e1.11},
we let $\delta\to 0$ in \eqref{e3.11}, \eqref{e3.12}, \eqref{e3.13}.
 Now, for all $\varepsilon>0$, take
\[
\delta_0=\min\{(\frac{\varepsilon(2-q^{*})}
{MLT^{*}T^{q^{*}-1}})^{\frac 1{2-q^{*}}},
(\frac{\varepsilon(1-r)}{M_fT})^{\frac 1{1-r}}\}
\]
where
\[
M=\max_{0\leq t\leq T}|x^{*}(t)|+1, \quad
L=\max_{0\leq t\leq T, \|x^{*}\|\leq M}
 |\frac 1{\Gamma(2-q(t,x^{*}(t)))}|+1,
\]
when $\delta<\delta_0$, by \eqref{e2.2}, \eqref{e2.3}, \eqref{e2.4},
\eqref{e3.6}, we have
\begin{align*}
&\big|\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x^{*}(s))}}
 {\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds
 -\int_0^t\frac{(t-s)^{1-q(s,x^{*}(s))}}{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds\big|
\\
&= \big|\int_{t-\delta}^t\frac{(t-s)^{1-q(s,x^{*}(s))}}
 {\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds\big|
\\
&= |\int_{t-\delta}^t\frac{T^{1-q(s,x^{*}(s))}}
 {\Gamma(2-q(s,x^{*}(s)))}(\frac{t-s}T)^{1-q(s,x^{*}(s))}x^{*}(s)ds|
\\
&\leq ML\int_{t-\delta}^tT^{*}(\frac{t-s}T)^{1-q^{*}}ds
\\
&= \frac{MLT^{*}T^{q^{*}-1}}{2-q^{*}}\delta^{2-q^{*}}
\\
&< \frac{MLT^{*}T^{q^{*}-1}}{2-q^{*}}\delta_0^{2-q^{*}}
= \varepsilon,
\end{align*}
which implies that
\begin{equation}
\lim_{\delta\to 0}\int_0^{t-\delta}\frac{(t-s)^{1-q(s,x^{*}(s))}}
{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds
=\int_0^t\frac{(t-s)^{1-q(s,x^{*}(s))}}{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds.
\label{e3.14}
\end{equation}
By the same arguments, we have that
\begin{equation}
\lim_{\delta\to 0}\int_0^{T-\delta}
\frac{(T-s)^{1-q(s,x^{*}(s))}}{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds
=\int_0^T\frac{(T-s)^{1-q(s,x^{*}(s))}}{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds.
\label{e3.15}
\end{equation}
Similarly, we have
\begin{align*}
&\big|\int_0^{t-\delta}(t-s)f(s,x^{*}(s))-\int_0^t(t-s)f(s,x^{*}(s))ds\big|
\\
&= |\int_{t-\delta}^t(t-s)f(s,x^{*}(s))ds|
\\
&\leq M_f\int_{t-\delta}^t(t-s)s^{-r}ds
\\
&\leq M_fT\int_{t-\delta}^ts^{-r}ds
\\
&= \frac{M_fT}{1-r}(t^{1-r}-(t-\delta)^{1-r})
\\
&= \frac{M_fT}{1-r}((t-\delta+\delta)^{1-r}-(t-\delta)^{1-r})
\\
&\leq \frac{M_fT}{1-r}((t-\delta)^{1-r}+\delta^{1-r}-(t-\delta)^{1-r})
\\
&= \frac{M_fT}{1-r}\delta^{1-r}
\\
&< \frac{M_fT}{1-r}\delta_0^{1-r}<\varepsilon,
\end{align*}
which implies
\begin{equation}
\lim_{\delta\to 0}\int_0^{t-\delta}(t-s)f(s,x^{*}(s))ds
=\int_0^t(t-s)f(s,x^{*}(s))ds.\label{e3.16}
\end{equation}
By the same arguments, we also have
\begin{equation}
\lim_{\delta\to 0}\int_0^{T-\delta}(T-s)f(s,x^{*}(s))ds
=\int_0^T(T-s)f(s,x^{*}(s))ds.\label{e3.17}
\end{equation}

Now, we let $\delta\to 0$ in \eqref{e3.11}, \eqref{e3.12}, \eqref{e3.13},
by \eqref{e3.14}, \eqref{e3.15}, \eqref{e3.16} and \eqref{e3.17},
we obtain
\begin{gather}
x^{*}(0)=x^{*}(T)=0,\label{e3.18}\\
\int_0^t\frac{(t-s)^{1-q(s,x^{*}(s))}}{\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds
=\widetilde{c}t+\int_0^t(t-s)f(s,x^{*}(s))ds, \quad 0\leq t\leq T.\label{e3.19}
\end{gather}
where
\[
\widetilde{c}=\frac{\int_0^T\frac{(T-s)^{1-q(s,x^{*}(s))}}
 {\Gamma(2-q(s,x^{*}(s)))}x^{*}(s)ds-\int_0^T(T-s)f(s,x^{*}(s))ds}T.
\]
Differentiating on both sides of \eqref{e3.19}, we obtain
\begin{equation}
\frac d{dt}I_{0+}^{2-q(t,x^{*}(t))}x^{*}(t)=\widetilde{c}+\int_0^tf(t,x^{*}),
\quad 0<t<T,\label{e3.20}
\end{equation}
From the continuity of $t^rf$ and Lemma \ref{lem2.3} it follows
that $\int_0^tf(s,x^{*}(s))ds$ is in $AC[0,T]$; consequently, from
\eqref{e3.20}, we obtain
\begin{equation}
\int_0^tf(s,x^{*}(s))ds=\frac d{dt}I_{0+}^{2-q(t,x^{*}(t))}x^{*}(t)
-\widetilde{c}\in AC[0,T].\label{e3.21}
\end{equation}
As a result, differentiating on both sides of \eqref{e3.21}, by definition
of derivative of variable-order \eqref{e1.9}, we obtain
\begin{equation}
D_{0+}^{q(t,x^{*}(t))}x^{*}(t)=f(t,x^{*}), 0<t\leq T,\label{e3.22}
\end{equation}
which together with \eqref{e3.18} yields that $x^{*}$ is a solution
of  \eqref{e1.10}-\eqref{e1.11}. Thus the proof is complete.
\end{proof}


\begin{example} \label{examp2} \rm
Consider the  problem
\begin{equation}
  \begin{gathered}
  D_{0+}^{q(t,x(t))}x(t)=f(t,x),\quad  0<t<1,
  \\
  x(0)=x(1)=0,
\end{gathered} \label{e3.23}
\end{equation}
where $q(t,x)=1+\frac{t^3}3+\frac 1{3(1+x^2)}$ is a continuous function
on $[0,1]\times R$, $f(t,x)=t^{-\frac 12}+x^3$ is a continuous function
on $(0,1]\times R$.
Clearly, for $(t,x)\in [0,1]\times R$, we have
$1<q(t,x)<1+\frac 13+\frac 13=\frac 53$.
Therefore Theorem \ref{thm3.1} implies that \eqref{e3.22} has one solution
 $x^{*}\in C[0,1]$.
\end{example}

\subsection*{Acknowledgments}
The author was supported by the NNSF of China (11371364), and by
the Fundamental Research Funds for the Central Universities
(2009QS06).


 \begin{thebibliography}{99}

\bibitem{c1} C. H. Chan, J. J. Shyu, R. H. H. Yang;
\emph{A new structure for the design of wideband variable fractional-order
 FIR differentiator}, Signal Processing, 90, 2594-2604, 2010.

\bibitem{c2} C. H. Chan, J. J. Shyu, R. H. H. Yang;
\emph{An iterative method for the design of variable fractional-order
FIR differintegrators}, Signal Processing, 89, 320-327, 2009.

\bibitem{c3} C.H. Chan, J. J. Shyu, R. H. H. Yang;
\emph{Iterative design of variable fractional-order IIR differintegrators},
Signal Processing, 90, 670-678, 2010.

\bibitem{c4} C. F. M. Coimbra;
\emph{Mechanics with variable-order differential operators},
Annalen der Physik, 12, 692-703, 2003.

\bibitem{h1} T. T. Hartley, C. F. Lorenzo;
\emph{Fractional system identilcation: An approach using continuous
order distributions}, NASA/TM, 40, 1999-2096, 1999.

\bibitem{k1} A. A. Kilbas, H. M. Srivastava, J. J. Trujillo;
\emph{Theory and Applications of Fractional
Differential Equations}, Elsevier B. V., Amsterdam, 2006.

\bibitem{l1}  R. Lin, F. Liu, V. Anh, I. Turner;
\emph{Stability and convergence of a new explicit finite difference
approximation for the variable-order nonlinear fractional diffusion equation},
Applied Mathematics and Computation, 2, 435-445, 2009.

\bibitem{l2}  C. F. Lorenzo, T. T. Hartley;
\emph{Variabe order and distributed order fractional operators},
 Nonlinear Dynamics, 29, 57-98, 2002.

\bibitem{o1} T. Odzijewicz, A. B. Malinowska, D. F. M. Torres;
\emph{Fractional varibleavle calculus of variable order,
 Operator Theory: Advances and Applications}, Birkh \"{a}user Verlag,
http: //www.springer.com/series/4850.

\bibitem{r1} L. E. S. Ramirez, C. F. M. Coimbra;
\emph{On the Selection and Meaning of Variable Order Operators
for Dynamic Modeling},
Hindawi Publishing Corporation International Journal of
Differential Equations Volume 2010, Article ID 846107, 16 pages,
doi:10.1155/2010/846107.

\bibitem{r2} A. Razminia, A. F. Dizaji, V. J. Majd;
\emph{Solution existence for non-autonomous variable-order fractional
differential equations}, Mathematical and Computer Modelling, 55, 1106-1117,
2012.

 \bibitem{r3} B. Ross;
\emph{Fractional integration operator of variable order in the
H\"{o}lder spaces $H^\mu(x)$}, Internat. J. Math. and Math. Sci.
Vol. 18, 4, 777-788, 1995.

\bibitem{s1} H. Sheng, H. G. Sun, Y. Q. Chen, T. S. Qiu;
\emph{Synthesis of multifractional Gaussian noises based on variable-order
fractional operators}, Signal Processing, 91, 1645-1650, 2011.

\bibitem{s2} H. Sheng, H. Sun, C. Coopmans, Y. Q. Chen, G. W. Bohannan;
\emph{Physical experimental study of variable-order fractional
integrator and differentiator}, in: Proceedings of FDA'10. The 4th
IFAC Workshop Fractional Differentiation and its Applications,
2010.

\bibitem{s3} C. M. Soon, C. F. M. Coimbra, M. H. Kobayashi;
\emph{The variable Viscoelasticity Oscillator}, Annalen der Physik, 14, 6,
378-389, 2005.

\bibitem{s4} H. G. Sun, W. Chen, Y. Q. Chen;
\emph{Variable-order fractional differential operators in anomalous
diffusion modeling}, Physica A, 388, 4586-4592, 2009.

\bibitem{s5}14] H. G. Sun, W. Chen, H. Wei, Y. Q. Chen;
\emph{A comparative study of constant-order and variable-order fractional
models in characterizing memory property of systems}, Eur. Phys. J. Special
Topics Perspectives on Fractional Dynamics and Control, 193,
185-192, 2011.

\bibitem{t1} C. C. Tseng;
\emph{Design of variable and adaptive fractional order FIR differentiators},
Signal Processing, 86, 2554-2566, 2006.

\bibitem{t2} C. C.Tseng;
\emph{Series expansion design for variable
fractional order integrator and Differentiator using logarithm},
Signal Processing, 88, 278-2292, 2008.

\bibitem{v1}  D. Val\'{e}rio, G. Vinagre, J. Domingues, J. S. Costa;
\emph{Variable fractional derivatives and their numberical
approximation I-real orders}, Symposium on Fractional Signals and
Systems Lisbon09, Lisbon, Portugal, November 4-6, 2009.

\end{thebibliography}
\end{document}