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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 240, pp. 1--14.\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-2013/240\hfil Solutions to nonlocal DEs]
{Solutions to nonlocal fractional differential equations using
a noncompact semigroup}

\author[S. Ji, G. Li\hfil EJDE-2013/240\hfilneg]
{Shaochun Ji, Gang Li}  % in alphabetical order

\address{Shaochun Ji (Corresponding author) \newline
Faculty of Mathematics and Physics, Huaiyin Institute
of Technology, Huaian 223003, China}
\email{jiscmath@gmail.com}

\address{Gang Li \newline
School of Mathematical Science, Yangzhou University,
Yangzhou 225002, China}
\email{gli@yzu.edu.cn}

\thanks{Submitted August 22, 2013. Published October 29, 2013.}
\subjclass[2000]{26A33, 34G20}
\keywords{Fractional differential equations;
 nonlocal conditions; \hfill\break\indent measure of noncompactness}

\begin{abstract}
 This article concerns the existence of solutions to nonlocal fractional
 differential equations in Banach spaces. By using a type of
 newly-defined measure of noncompactness, we discuss this problem
 in general Banach spaces without any compactness assumptions to
 the operator semigroup. Some existence results are obtained when
 the nonlocal term is compact and when is Lipschitz continuous.
\end{abstract}

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

\section{Introduction}

In this article, we study the following fractional differential
equations with nonlocal conditions
\begin{equation} \label{e1.1}
  \begin{gathered}
    {}^CD^\alpha u(t) =Au(t)+ f(t,u(t)), \quad t\in J=[0,b], \\
   u(0)=g(u),
  \end{gathered}
\end{equation}
where $A:D(A)\subseteq X\to X $ is the infinitesimal
generator of a strongly continuous semigroup of bounded linear
operators $\{T(t)\}_{t\geq 0}$ in a Banach space $X$; ${}^CD^\alpha$
is the Caputo fractional derivative operator of order $\alpha$ with
$0<\alpha\leq 1$; $f$ and $g$ are appropriate continuous functions
to be specified later.

Fractional differential equations arise in many engineering and
scientific problems, such as diffusion process, control theory,
signal and image processing. Compared with the classical
integer-order models, the fractional-order models are more realistic
and practical to describe many phenomena in nature (see \cite{bon}).
For some recent development on this topic, we refer to the
monographs of Kilbas et al. \cite{kil}, Podlubny \cite{pod},
Lakshmikantham et al. \cite{lak1}, and
\cite{aga,chang,kum,lak,peng1,yang}. By using some probability
density functions, El-Borai \cite{Bor} introduced fundamental
solutions of fractional evolution equations in a Banach space. Wang
et al. \cite{wang} obtained the existence and uniqueness of $\alpha$-mild
solutions by means of fractional calculus and Leray-Schauder fixed
point theorem with a compact analytic semigroup. Ren et al.
\cite{ren1,ren2} established the existence of mild solutions for a
class of semilinear integro-differential equations of fractional
order with delays.

The study of nonlocal semilinear differential equation in Banach spaces was
initiated by Byszewski \cite{bys} and the importance of the problem
consists in the fact that it is more general and has better effect
than the classical initial conditions $u(0)=u_0$ alone. For example,
Deng \cite{den} defined the function $g$ by 
$$
g(u)=\sum_{j=1}^q c_ju(s_j),
$$ 
where $c_j$ are given constants and $0<s_1<s_2<\dots<s_q\leq b$. 
This allows measurements to be made at $t=0, s_1,\dots,s_q$,
rather than just at $t=0$ and more information can be obtained.
Subsequently several authors have investigated some different types
of differential equations and integrodifferential equations in
Banach spaces\cite{nto,bal1,zhu}. Most of the previous results are
obtained with the assumption that semigroup $T(t)$ is compact. Then
one of the difficulties on the nonlocal problems is how to deal with
the compactness on the solution operator. By using the measure of
noncompactness, Xue\cite{xue} discussed integral solutions of the
nonlinear nonlocal initial value problem and Fan et al.\cite{fan},
Ji et al.\cite{ji} discussed nonlocal impulsive differential
equations when the semigroup $T(t)$ is equicontinuous.

From the viewpoint of theory and practice, it is natural for
mathematics to combine fractional differential equations and
nonlocal conditions. Mophou et al.\cite{mop} and Balachandran et
al.\cite{bal2} investigated the existence of solutions of fractional
abstract differential equations with nonlocal conditions. Zhou and
Jiao\cite{zho} discussed the problem based on Krasnoselskii's fixed
point theorem with the assumption that semigroup $T(t)$ is compact
and nonlocal item $g$ is Lipschitz continuous. Very recently, Li,
Peng and Gao\cite{peng2} study the existence of mild solutions to
the problem \eqref{e1.1} by using the Hausdorff measure of noncompactness
when the semigroup $T(t)$ is equicontinuous and $g$ is compact.  In
this paper, we study \eqref{e1.1} without assuming $T(t)$ is compact or
equicontinuous and do not require additional conditions compared 
with those in\cite{peng2}. Therefore, the existence
 theorems of mild solutions given here are quite general, even in the case
 of integer-order differential equations. The work is
based on a type of newly-defined measure of noncompactness 
(see Lemma \ref{lem3.1}), which can be seen as a generalization of classical
Hausdorff measure of noncompactness. Moreover, we do not need the
assumption of separability on the Banach space $X$.

The article is organized as follows. In Section 2 we recall some
preliminary facts that we need in the sequel. In Section 3 we prove
our results when nonlocal item $g$ is compact. In Section 4 we get
our results when nonlocal item $g$ is Lipschitz continuous. The
conclusions and applications of the paper are given in Section 5.

\section{Preliminaries}

 Let $(X,\|\cdot\|)$ be a real
Banach space and $\mathbb{N}$ be the set of positive integers.  We
denote by $C([0,b];X)$ the space of $X$-valued continuous functions
on $[0,b]$ with the norm $\|x\|=\sup\{\|x(t)\|, t\in[0,b]\}$ and by
$L^1([0,b];X)$ the space of $X$-valued Bochner integrable functions
on $[0,b]$ with the norm
$\|f\|_{L^1}=\int_0^b\|f(t)\|\,\mathrm{d}t$.

Let us recall the following definitions. For basic facts about
fractional derivatives and fractional calculus, one can refer to the
books \cite{kil,pod}.

\begin{definition} \label{def2.1} \rm
The Riemann-Liouville fractional integral of order $\alpha>0$ with
the lower limit zero for a function $f$ can be defined as
$$ 
I^\alpha f(t)=\frac{1}{\Gamma(\alpha)}\int_0^t(t-s)^{\alpha-1}f(s)\,\mathrm{d}s,
\quad t>0, 
$$
provided the right-hand side is pointwise defined on $[0,\infty)$,
where $\Gamma(\cdot)$ is the gamma function.
\end{definition}

\begin{definition} \label{def2.2} \rm
The Riemann-Liouville derivative of order $\alpha$ with the lower
limit zero for a function $f$ can be written as
$$ 
{}^{R-L}\!D^\alpha f(t)=
 \frac{1}{\Gamma(n-\alpha)}\frac{\mathrm{d}^n}{\mathrm{d}t^n}
\int_0^t(t-s)^{n-\alpha-1}f(s)\,\mathrm{d}s,\quad t>0,
$$
where $\alpha\in (n-1,n)$, $n\in \mathbb{N}$.
\end{definition}

\begin{definition} \label{def2.3} \rm
The Caputo derivative of order $\alpha$ with the lower limit zero
for a function $f$ can be written as
$$ 
{}^C D^\alpha f(t)=
 \frac{1}{\Gamma(n-\alpha)}\int_0^t(t-s)^{n-\alpha-1}f^{(n)}(s)\,\mathrm{d}s ,
\quad t>0, 
$$
where $\alpha\in (n-1,n)$, $n\in \mathbb{N}$.
\end{definition}

If $f$ takes values in a Banach space $X$, the integrals which
appear in the above three definitions are taken in Bochner's sense.
Especially, when $0<\alpha< 1$, we have
$$ 
{}^C D^\alpha f(t)=\frac{1}{\Gamma(1-\alpha)}\int_0^t 
\frac{f'(s)}{(t-s)^\alpha}\,\mathrm{d}s.
 $$

We firstly recall the concept of mild solutions to equation \eqref{e1.1}
developed in \cite{Bor,zho}.

\begin{definition}\label{def2.4} \rm
A function $u\in C(J;X)$ is said to be a mild solution of \eqref{e1.1} if
$u$ satisfies
$$
u(t)=S_\alpha(t)g(u)+\int_0^t(t-s)^{\alpha-1}T_\alpha(t-s)f(s,u(s))\,\mathrm{d}s,
$$
for $t\in J$, where
\begin{gather*}
S_\alpha(t) = \int_0^\infty
\xi_\alpha(\theta)T(t^\alpha\theta)\,\mathrm{d}\theta,\quad
T_\alpha(t)=\alpha\int_0^\infty
\theta\xi_\alpha(\theta)T(t^\alpha\theta)\,\mathrm{d}\theta, 
\\
 \xi_\alpha(\theta)= \frac{1}{\alpha}\theta^{-1-\frac{1}{\alpha}}
\varpi_\alpha(\theta^{-\frac{1}{\alpha}})\geq 0, 
 \\
  \varpi_\alpha(\theta)= \frac{1}{\pi}\sum_{n=1}^\infty (-1)^{n-1}
 \theta^{-\alpha n-1}\frac{\Gamma(n\alpha+1)}{n!}\sin(n\pi \alpha),\quad 
 \theta\in   (0,\infty).
\end{gather*}
Here $\xi_\alpha(\theta)$ is a probability density function defined on
$(0,\infty)$ satisfying
$$
\int_0^\infty \xi_\alpha(\theta)\,\mathrm{d}\theta=1,\quad 
\int_0^\infty \theta^v \xi_\alpha(\theta)\,\mathrm{d}\theta
=\frac{\Gamma(1+v)}{\Gamma(1+\alpha v)},\quad v\in [0,1]. 
$$
\end{definition}

\begin{lemma}[\cite{wang}] \label{lem2.5}
For any fixed $t\geq 0$, the operators $S_\alpha(t)$ and
$T_\alpha(t)$ are linear and bounded operators; i.e., for any $x\in
X$, $\|S_\alpha(t)x\|\leq M\|x\|$ and $\|T_\alpha(t)x\|\leq
\frac{M\alpha}{\Gamma(1+\alpha)  }\|x\|$, 
where $M$ is the constant such that $\|T(t)\|\leq M$ for
 all $t\in [0,b]$.
\end{lemma}

Now we give some facts on measure of noncompactness, see Banas and
Goebel\cite{ban}.

\begin{definition} \label{def2.6} \rm
Let $E^+$ be the positive cone of an ordered Banach space
$(E,\leq)$. A function $\Phi$ defined on the set of all bounded
subsets of the Banach space $X$ with values in $E^+$ is called a
measure of noncompactness (in short MNC) on $X$ if
$\Phi(\overline{{\rm co}}\Omega)=\Phi(\Omega)$ for all bounded subsets
$\Omega \subset X$, where $\overline{{\rm co}}\Omega$ stands for the
closed convex hull of $\Omega$.
A measure of noncompactness $\Phi$ is said to be:
\begin{itemize}
\item[(1)] \emph{monotone} if for all bounded subsets $\Omega_1,\Omega_2$
of $X$ we have: $(\Omega_1 \subseteq \Omega_2) \Rightarrow
(\Phi(\Omega_1)\leq \Phi(\Omega_2))$;

\item[(2)] \emph{nonsingular} if $\Phi(\{a\}\cup \Omega)=\Phi(\Omega)$
for every $a\in X$, $\Omega \subset X$;

\item[(3)] \emph{regular} if $\Phi(\Omega)=0$ if and only if $\Omega $ is
relatively compact in $X$.
\end{itemize}
\end{definition}

One of the most important examples of MNC is the Hausdorff measure
of noncompactness  $\beta(\cdot)$ defined by
$$
\beta(B)=\inf\{\varepsilon>0: B \text{has a finite $\varepsilon$-net in }  X\},
$$
for each bounded subset $B$ in a Banach space $X$.

It is well known that the Hausdorff measure of noncompactness
$\beta$ enjoys the above properties.

\begin{lemma}[\cite{ban}] \label{lem2.7}
Let $X$ be a real Banach space and $B,C\subseteq X$ be bounded.
Then the following properties are satisfied:
\begin{enumerate}
  \item $B $ is relatively compact if and only if $\beta(B)=0$;
  \item $\beta(B)=\beta(\overline{B})=\beta(\operatorname{conv} B)$, where
$\overline{B}$ and $\operatorname{conv}B$ mean the closure and convex
hull of $B$, respectively;
  \item $\beta(B)\leq \beta(C)$ when $B\subseteq C$;
  \item $\beta(B+C)\leq \beta(B) + \beta(C)$, where $B + C=\{x+y: x\in
B, y\in C\}$;
  \item $\beta(B\cup C)\leq \max\{\beta(B), \beta(C)\}$;
  \item $\beta(\lambda B )\leq |\lambda|\beta(B)$ for any $\lambda\in
R$;
  \item If the map $Q:D(Q)\subseteq X \to Z$ is Lipschitz
continuous with constant $k$, then $\beta_{Z}(QB)\leq k\beta(B)$ for
any bounded subset $B \subseteq D(Q)$, where $Z$ is a Banach space.
  \item If $\{ W_n\}_{n=1}^\infty$ is a decreasing sequence of bounded closed
  nonempty subsets of $X$ and $\lim_{n\to
  \infty}\beta(W_n)=0$, then $\cap_{n=1}^\infty{W_n}$ is nonempty
  and compact in $X$.
\end{enumerate}
\end{lemma}

We will also use the sequential MNC $\beta_0$ generated by $\beta$,
that is, for any bounded subset $B\subset X$, we define
$$
\beta_0(B)=\sup\big\{\beta(\{x_n :n\geq 1\}):\{x_n \}_{n=1}^\infty 
\text{ is a sequence in }B \big\}.  
$$
It follows that
\begin{equation}\label{e2.1}
\beta_0(B)\leq \beta(B)\leq 2\beta_0(B).
\end{equation}
If $X$ is a separable space, we have $\beta_0(B)=\beta(B)$.

\begin{lemma}[\cite{kam}]\label{lem2.8}
If $\{u_n\}_{n=1}^\infty\subset L^1(J;X)$ satisfies
 $\|u_n(t)\|\leq \varphi (t)$ a.e. on $[0,b]$ for all 
$n\geq 1$ with some $\varphi\in L^1(J;\mathbb{R_+})$, then for $t\in [0,b]$, 
we have
$$
\beta\Big(\big\{\int_0^t u_n(s)\,\mathrm{d}s\big\}_{n=1}^\infty \Big)
\leq  2\int_0^t \beta(\{u_n(s)\}_{n=1}^\infty ) \,\mathrm{d}s. 
$$
\end{lemma}

\begin{lemma}[\cite{hen}]\label{lem2.9}
Suppose $b\geq 0$, $\sigma>0$ and $a(t)$ is nonnegative function
locally integrable on $0\leq t<b$ ($b\leq +\infty$), and suppose
$c(t)$ is nonnegative and locally integrable on $0\leq t<b$ with
$$
c(t)\leq a(t)+b\int_0^t (t-s)^{\sigma-1}c(s)\, \mathrm{d}s  
$$
on this interval. Then
$$
c(t)\leq a(t)+\mu \int_0^t E'_\sigma(\mu(t-s))a(s)\, \mathrm{d}s,
\quad 0\leq t\leq b,   
$$
where $\mu=(b\Gamma(\sigma))^{1/\sigma}$,
$E_\sigma(z)=\sum_{n=0}^\infty z^{n\sigma}/\Gamma(n\sigma+1)$,
$E'_\sigma(z)=\frac{\mathrm{d}}{\mathrm{d}z}E_\sigma(z) $.
\end{lemma}

\section{$T(t)$ strongly continuous and $g$ compact}

 Let $r$ be a positive constant and $B_r=\{x\in
X:\|x\|\leq r \} $, $W_r=\{x\in C(J;X):x(t)\in B_r, t\in J\}$. We
define the solution operator $G:C(J;X)\to C(J;X)$ by
$$
Gu(t)=S_\alpha(t)g(u)+\int_0^t(t-s)^{\alpha-1}T_\alpha(t-s)f(s,u(s))\,\mathrm{d}s 
$$
with 
\begin{gather*}
G_1u(t)=S_\alpha(t)g(u), \\
G_2u(t)=\int_0^t(t-s)^{\alpha-1}T_\alpha(t-s)f(s,u(s))\,\mathrm{d}s,
\end{gather*}
for all $t\in J$. It is easy to see that $u$ is the mild solution of
the problem \eqref{e1.1} if and only if $u$ is a fixed point of the map
$G$.

Now we introduce some noncompact measures. For any bounded set
$B\subset C(J;X)$, we define 
$$
\chi_1(B)=\sup_{t\in J}\beta(B(t)),
$$ 
where $\beta$ is the Hausdorff MNC on $X$ and it is easy to see 
that $\chi_1$ coincides with the the
Hausdorff MNC $\beta_C$ on equicontinuous sets. We also define
$$
\chi_2(B)=\sup_{t\in J}\operatorname{mod}_C(B(t)), 
$$
where $\operatorname{mod}_C(B(t))$ is the modulus of equicontinuity of the set $B$
at $t\in J$, given by the formula
$$
\operatorname{mod}_C(B(t))=\lim_{\delta\to 0}\sup\{\|x(t_1)-x(t_2)\|:
t_1,t_2\in (t-\delta,t+\delta), x\in B\}.
$$
Then the MNC $\chi_1,\chi_2$ are well-defined and are both monotone
and nonsingular, but in general they are not necessarily regular.
Similar definitions with $\chi_1,\chi_2$ can be found in Kamenskii
\cite{kam} or Fan\cite{fan2}. Now we define
$$
\chi(B)=\chi_1(B)+\chi_2(B).
$$ 
Then we have the following result.

\begin{lemma}\label{lem3.1}
$\chi$ is a monotone, nonsingular and regular measure of
noncompactness defined on bounded subsets of $C(J;X)$.
\end{lemma}

\begin{proof}
 It is easy to check that $\chi$ is
well-defined, monotone and nonsingular. Now we shall show that
$\chi$ is regular. If $B$ is relatively compact in $C(J;X)$, then by
the abstract version of the Ascoli-Arzela theorem, we have
$\chi(B)=0$.

On the other hand, if $\chi(B)=0$, by the definition of $\chi$, we
have
$$ 
\chi_1(B)=0,~\chi_2(B)=0.  
$$
That is, for every $t\in J$, $B(t)$ is precompact in $X$. Then it
remains to prove that $B$ is equicontinuous on $J$. Suppose not,
then there exist $\varepsilon_0>0$ and sequences
 $\{u_n\}\subseteq B$, $\{t_n\},~\{\overline{t}_n\}\subseteq [0,b]$, such that
$t_n\to t_0$, $\overline{t}_n\to t_0$ as
$n\to \infty$ and
$$
\|u_n(t_n)-u_n(\overline{t}_n) \|\geq \varepsilon_0, 
$$
for all $n\geq 1$. Note that
$$ 
\|u_n(t_n)-u_n(\overline{t}_n) \|
\leq \sup\{ \|u(t_n)-u(\overline{t}_n)\|:u\in B\}. 
$$
We take the upper limit for $n$ and get that
$$
\overline{\lim_n} \|u_n(t_n)-u_n(\overline{t}_n) \|
\leq \operatorname{mod}_C(B(t_0))\leq \chi_2(B)=0, 
$$
which gives the contradiction $0<\varepsilon_0\leq 0$. Thus
$B\subseteq C(J;X)$ is equicontinuous on $J$. This completes the
proof. 
\end{proof}

We will use the following hypotheses:
\begin{itemize}
\item[(HA)] The operator $A$ generates a strongly continuous semigroup
$\{T(t)\}_{t\geq 0}$ in $X$. Moreover, there exists a positive
constant $M > 0$ such that $M=\sup_{0\leq t\leq b}\|T(t)\|$ (see
Pazy\cite{paz}).

\item[(HG1)] $g:C(J;X)\to X$ is continuous and compact. There
exists a positive constant $N$ such that $\|g(u)\|\leq N$ for all
$u\in C(J;X)$.

\item[(HF1)] $f:[0,b]\times X\to X$ is continuous.

\item[(HF2)] there exists a constant $L>0$, such that for any bounded
set $D\subset X$, 
$\beta(f(t,D))\leq L\beta(D)$, for a.e. $t\in J$.
\end{itemize}

The following lemma is useful for our proofs.

\begin{lemma}\label{lem3.2}
Suppose that the semigroup $\{T(t)\}_{t\geq 0}$ is strongly
continuous and hypotheses {\rm (HF1), (HF2)} are satisfied. Then for
any bounded set $B\subset C(J;X)$, we have
$$
\beta(G_2B(t))\leq \frac{4\alpha ML}{\Gamma(1+\alpha)}\int_0^t
   (t-s)^{\alpha-1}\beta(B(s))\,\mathrm{d}s  ,
$$
for $t\in [0,b]$.
\end{lemma}

\begin{proof} 
 For $t\in [0,b]$, due to the inequality \eqref{e2.1},
we obtain that for arbitrary $\varepsilon >0$, there exists a
sequence $\{v_k\}_{k=1}^\infty \subset B$ such that
\begin{equation}\label{e3.1}
   \beta(G_2B(t))\leq 2\beta(\{G_2v_k(t)\}_{k=1}^\infty)+\varepsilon.
\end{equation}
It follows from Lemma \ref{lem2.8} and hypotheses $(Hf_1),~(Hf_2)$ that
\begin{align*}
%  to remove numbering (before each equation)
   \beta(\{G_2v_k(t)\}_{k=1}^\infty )&\leq  2\int_0^t (t-s)^{\alpha-1}\beta(\{T_\alpha (t-s)f(s,v_k(s))\}_{k=1}^\infty)\,\mathrm{d}s  \\
   &\leq  2\int_0^t (t-s)^{\alpha-1}\frac{\alpha M}{\Gamma(1+\alpha)}L
   \beta(\{v_k(s)\}_{k=1}^\infty)\,\mathrm{d}s\\
   &\leq  \frac{2\alpha ML}{\Gamma(1+\alpha)}\int_0^t
   (t-s)^{\alpha-1}\beta(B(s))\,\mathrm{d}s.
\end{align*}
According to {e3.1}, we can derive that
$$
\beta(G_2B(t))\leq \frac{4\alpha ML}{\Gamma(1+\alpha)}\int_0^t
   (t-s)^{\alpha-1}\beta(B(s))\,\mathrm{d}s+\varepsilon. 
$$
Since the above inequality holds for arbitrary $\varepsilon >0$, it
follows that
$$
\beta(G_2B(t))\leq \frac{4\alpha ML}{\Gamma(1+\alpha)}\int_0^t
   (t-s)^{\alpha-1}\beta(B(s))\,\mathrm{d}s. 
$$
This completes the proof. 
\end{proof}

Now, we give the main existence result of this section.

\begin{theorem} \label{thm3.3}
Assume that the hypotheses {\rm (HA), (HG1), (HF1), (HF2)} are
satisfied. Then the nonlocal fractional differential  system \eqref{e1.1}
has at least one mild solution on $[0,b]$, provided that there
exists a constant $r>0$ such that
\begin{equation}\label{e3.2}
    MN+\frac{M b^\alpha}{\Gamma(1+\alpha)} \sup_{s\in [0,b],u\in
  W_r}\|f(s,u(s))\| \leq r.
\end{equation}
\end{theorem}

\begin{proof}
 We shall prove this result by using the
Schauder's fixed point theorem.

Step 1. We shall prove that $G$ is continuous on $C(J;X)$. Let
$\{u_m\}_{m=1}^\infty$ be a sequence in $C(J;X)$ with
$\lim_{m\to \infty}u_m=u $ in $C(J;X)$. By the continuity of
$f$, we deduce that for each $s\in [0,b]$, $f(s,u_m(s))$ converges
to $f(s,u(s))$ in $X$ uniformly for $s\in [0,b]$. And we have
\begin{align*}
  \|Gu_m-Gu\| &\leq \|S_\alpha(t)g(u_m)-g(u)\|\\
  &\quad +\int_0^t(t-s)^{\alpha-1}\|T_\alpha(t-s)[f(s,u_m(s))-f(s,u(s))]
  \|\,\ \mathrm{d}s\\
  &\leq M\|g(u_m)-g(u)\|+\frac{M b^\alpha}{\Gamma(1+\alpha)} 
\sup_{s\in [0,b]}\|f(s,u_m(s))-f(s,u(s))\|.
\end{align*}
Then by the continuity of $g$, we get 
$\lim_{m\to \infty}G u_m=Gu$ in $C(J;X)$, which implies that $G$ 
is continuous on $C(J;X)$.

Step 2. We construct a bounded convex and closed set 
$W\subset C(J;X)$ such that $G$ maps $W$ into itself. 
Let $W_0=\{u\in C(J;X):\|u(t)\|\leq r, t\in J \}$, where 
$r$ satisfies the condition
\eqref{e3.2}. For any $u\in W_0$, by hypotheses (HG1), (HF1) and
\eqref{e3.2}, we have
\begin{align*}
  \|Gu(t)\|
&\leq   \|S_\alpha(t)g(u)\|+\| \int_0^t(t-s)^{\alpha-1}T_\alpha(t-s)f(s,u(s))\,
 \mathrm{d}s\\
&\leq  MN+\frac{M b^\alpha}{\Gamma(1+\alpha)} \sup_{s\in [0,b],u\in
  W_0}\|f(s,u(s))\| 
 \leq  r,
\end{align*}
for $t\in [0,b]$, which implies that $GW_0\subseteq W_0$.

Define $W_1=\overline{\mathrm{conv}}\{G(W_0),u_0\}$, where
$\overline{\mathrm{conv}}$ means the closure of convex hull, $u_0\in
W_0 $. Then $W_1\subset W_0$ is nonempty bounded closed  and convex.
We define $W_n=\overline{\mathrm{conv}}\{G(W_{n-1}),u_0\}$ for
$n\geq 1$. It is easy to know that $\{W_n\}_{n=0}^\infty$ is a
decreasing sequence of $C(J;X)$. Moreover, set
$$
W=\cap_{n=0}^\infty W_n, 
$$
then $W$ is a nonempty, convex,
closed and bounded subset of $C(J;X)$ and $GW\subseteq W$.

Step 3. We claim that $W$ is compact in $C(J;X)$ by using the
newly-defined MNC $\chi$. As $\{W_n\}$ is a decreasing sequence of
$C(J;X)$, then $\{\beta(W_n(t))\}_{n=0}^\infty$ is nonnegative
decreasing sequence for any $t\in [0,b]$. From the compactness of
$g$ and Lemma \ref{lem3.2}, we get
\begin{equation}\label{e3.3}
\begin{aligned}
  \beta(W_{n+1}(t)) 
&\leq  \beta(\{S_\alpha(t)g(u):u\in W_n \}) \\
&\quad  +\beta(\{\int_0^t (t-s)^{\alpha-1}T_\alpha(t-s)f(s,u(s))\,\mathrm{d}s: 
 u\in W_n \} ) \\
&\leq  \frac{4\alpha ML}{\Gamma(1+\alpha)}\int_0^t
  (t-s)^{\alpha-1}\beta(W_n(s)) \mathrm{d}s.
\end{aligned}
\end{equation}
Taking $n\to \infty$ to both sides of \eqref{e3.3}, we have
\[
    \beta(W(t))\leq \frac{4\alpha ML}{\Gamma(1+\alpha)}\int_0^t
  (t-s)^{\alpha-1}\beta(W(s)) \mathrm{d}s.
\]
By Lemma \ref{lem2.9} we obtain that $\beta (W(t))=0$ for any $t\in [0,b]$.
Then according to the definition of $\chi_1$, we have
\begin{equation}\label{e3.4}
 \chi_1(W_n)\to 0,\quad  \text{as }n\to \infty.
\end{equation}

Next, we will estimate $\chi_2(W_n)$. Fix $t_0\in (0,b)$,
$0<p<\alpha$. Then for given $n\in \mathbb{N}$, according to the
continuity of $f$, we take $\delta_0$,
$0<\delta_0<\min\{t_0,b-t_0\}$ such that
\begin{equation}\label{e3.5}
   \big(\int_{t_0-\delta_0}^{t_0+\delta_0}\|f(s,u(s))
   \|^{1/p}\, \mathrm{d}s\big)^p \leq \frac{1}{n},\quad  \text{for any }u\in W_r.
\end{equation}
From the definition of Hausdorff MNC $\beta$, there exist finite
points $x_1,x_2,\dots,x_k\in X$ such that
$$
W_n(t_0-\delta_0)\subset \cup_{i=1}^k B(x_i, 2\beta(W_n(t_0-\delta_0))). 
$$
Moreover, for each $u\in GW_{n-1}$, there exists $v\in W_{n-1}$ such
that
\begin{gather}\label{e3.6}
\begin{aligned}
u(t_1)&=S_\alpha(t_1-(t_0-\delta_0))u(t_0-\delta_0)\\
&\quad +\int_{t_0-\delta_0}^{t_1}
(t_1-s)^{\alpha-1}T_\alpha(t_1-s)f(s,v(s))\,\mathrm{d}s,
\end{aligned}\\
\label{e3.7}
\begin{aligned}
u(t_2)&=S_\alpha(t_2-(t_0-\delta_0))u(t_0-\delta_0)\\
&\quad +\int_{t_0-\delta_0}^{t_2}
(t_2-s)^{\alpha-1}T_\alpha(t_2-s)f(s,v(s))\,\mathrm{d}s,
\end{aligned}
\end{gather}
for $t_1,t_2\in (t_0-\delta,t_0+\delta)$. For the above given $u$,
there exists $x_j$, $1\leq j\leq k$, such that
\begin{equation}\label{e3.8}
\|u(t_0-\delta_0)-x_j\|\leq 2\beta(W_n(t_0-\delta_0)).
\end{equation}
By the strong continuity of $S_\alpha(t)$, there exists $\delta$,
$0<\delta<\delta_0$, such that
\begin{equation}\label{e3.9}
\|S_\alpha(t_1-(t_0-\delta_0))x_i-S_\alpha(t_2-(t_0-\delta_0))x_i
\|\leq M\beta(W_n(t_0-\delta_0)).
\end{equation}
where $i=1,\dots,k$ and $t_1,t_2\in (t_0-\delta,t_0+\delta)$.

On the other hand, for $0<p<\alpha<1$, by the H\"{o}lder's
inequality and \eqref{e3.5}, we have
\begin{equation}\label{e3.10}
\begin{aligned}
&\|\int_{t_0-\delta_0}^{t_1}
(t_1-s)^{\alpha-1}T_\alpha(t_1-s)f(s,v(s))\,\mathrm{d}s
\\
&- \int_{t_0-\delta_0}^{t_2}
(t_2-s)^{\alpha-1}T_\alpha(t_2-s)f(s,v(s))\,\mathrm{d}s \|  
\\
&\leq  \frac{\alpha M}{\Gamma(1+\alpha)} \int_{t_0-\delta_0}^{t_1}
(t_1-s)^{\alpha-1}\|f(s,v(s))\|\,\mathrm{d}s \\
&\quad +\frac{\alpha M}{\Gamma(1+\alpha)} \int_{t_0-\delta_0}^{t_2}
(t_2-s)^{\alpha-1}\|f(s,v(s))\|\,\mathrm{d}s \\
&\leq  \frac{\alpha M}{\Gamma(1+\alpha)}
\Big(\int_{t_0-\delta_0}^{t_1}
(t_1-s)^{\frac{\alpha-1}{1-p}}\,\mathrm{d}s\Big)^{1-p}
\Big(\int_{t_0-\delta_0}^{t_1}\|f(s,v(s))\|^{1/p} \,\mathrm{d}s
\Big)^p   \\
&\quad +\frac{\alpha M}{\Gamma(1+\alpha)} \big(\int_{t_0-\delta_0}^{t_2}
(t_2-s)^{\frac{\alpha-1}{1-p}}\,\mathrm{d}s\big)^{1-p}
\Big(\int_{t_0-\delta_0}^{t_2}\|f(s,v(s))\|^{1/p} \,\mathrm{d}s
\Big)^p  \\
&\leq  \frac{2\alpha M}{\Gamma(1+\alpha)}
\big(\frac{1-p}{\alpha-p}\big)^{1-p}b^{\alpha-p}
\Big(\int_{t_0-\delta_0}^{t_0+\delta_0}\|f(s,v(s))\|^{1/p}
\,\mathrm{d}s \Big)^p \\
&\leq  \frac{2\alpha
M}{\Gamma(1+\alpha)}\big(\frac{1-p}{\alpha-p}\big)^{1-p}b^{\alpha-p}\frac{1}{n}.
\end{aligned}
\end{equation}
Therefore, for any given $u\in GW_{n-1}$, it follows from
\eqref{e3.6}--\eqref{e3.10} that
\begin{align*}
&\|u(t_1)-u(t_2)\| \\
&\leq  \|S_\alpha(t_1-(t_0-\delta_0))u(t_0-\delta_0)
 -S_\alpha(t_1-(t_0-\delta_0))x_j \|  \\
&\quad +\|S_\alpha(t_1-(t_0-\delta_0))x_j 
 -S_\alpha(t_2-(t_0-\delta_0))x_j \|  \\
&\quad +\|S_\alpha(t_2-(t_0-\delta_0))x_j
  -S_\alpha(t_2-(t_0-\delta_0))u(t_0-\delta_0)\|\\
&\quad +\|\int_{t_0-\delta_0}^{t_1}
(t_1-s)^{\alpha-1}T_\alpha(t_1-s)f(s,v(s))\,\mathrm{d}s\\
&\quad -\int_{t_0-\delta_0}^{t_2}
(t_2-s)^{\alpha-1}T_\alpha(t_2-s)f(s,v(s))\,\mathrm{d}s \|\\ 
&\leq 5M\beta(W_n(t_0-\delta_0))+\frac{2\alpha M}{\Gamma(1+\alpha)}
 \big(\frac{1-p}{\alpha-p}\big)^{1-p}b^{\alpha-p}\frac{1}{n},
\end{align*}
for $t_1,t_2\in (t_0-\delta_0,t_0+\delta_0)$. Then we have
\begin{align*}
\operatorname{mod}_C(GW_{n-1}(t_0))&\leq  5M\beta(W_n(t_0-\delta_0))
 +\frac{2\alpha M}{\Gamma(1+\alpha)}
 \big(\frac{1-p}{\alpha-p}\big)^{1-p}b^{\alpha-p}\frac{1}{n} \\
&\leq  5M\chi_1(W_n)+\frac{2\alpha
M}{\Gamma(1+\alpha)}\big(\frac{1-p}{\alpha-p}\big)^{1-p}
 b^{\alpha-p}\frac{1}{n}.
\end{align*}

For $t_0=0$ or $b$, we can also verify the above inequality. By the
definition of MNC $\chi_2$, we have
\begin{equation}\label{e3.11}
\chi_2(GW_{n-1})\leq 5M\chi_1(W_n) +\frac{2\alpha
M}{\Gamma(1+\alpha)}\big(\frac{1-p}{\alpha-p}\big)^{1-p}b^{\alpha-p}\frac{1}{n}.
\end{equation}
According to the property of MNC, we obtain
\begin{equation}\label{e3.12}
   \chi_2(W_n)=\chi_2(\overline{\mathrm{conv}}\{G(W_{n-1}),u_0\})
=\chi_2(GW_{n-1}).
\end{equation}
Thus from the definition of $\chi$ and \eqref{e3.4}, \eqref{e3.11}, 
\eqref{e3.12}, it follows that
\begin{equation*}
    \chi(W_n)= \chi_1(W_n)+\chi_2(W_n)
\leq (5M+1)\chi_1(W_n)+\frac{2\alpha M}{\Gamma(1+\alpha)}
\big(\frac{1-p}{\alpha-p}\big)^{1-p}b^{\alpha-p}\frac{1}{n}\to 0,
\end{equation*}
as $n\to \infty$. Therefore, the set $W=\cap_{n=1}^\infty W_n$ is a nonempty, 
convex, compact subset of $C(J,X)$, since $\chi$
is a monotone, nonsingular and regular MNC (see Lemma \ref{lem3.1}).
Moreover, $G$ maps $W$ into $W$.

Therefore, due to the Schauder's fixed point theorem, $G$ has at lest
one fixed point $u\in C(J;X)$, which is just the mild solution to
the problem \eqref{e1.1}. The proof of Theorem \ref{thm3.3} is complete.
\end{proof}

\begin{remark} \label{remar3.4} \rm
Li et al. \cite{peng2} discussed \eqref{e1.1} when $T(t)$ is
equicontinuous, $g$ is compact and the Banach space $X$ is
separable. Our existence results are more general than many previous
results in this field, where the compactness of $T(t)$ and $f$ are
needed. If $f$ is compact or Lipschitz continuous, then hypothesis
(HF2) is obviously satisfied. The regular MNC $\chi$ defined by
us plays a key role in the proof.
\end{remark}


\section{$T(t)$strongly continuous and $g$ Lipschitz continuous}

In this section, two existence results are given when $g$ is
Lipschitz continuous. The map $Q:D\subseteq X\to X$ is said
to be $\beta$-condensing if $Q$ is continuous, bounded and
for any nonprecompact bounded subset $B\subset D$, we have
$\beta(QB)< \beta(B)$, where $X$ is a Banach space.

\begin{lemma}[See Darbo-Sadovskii \cite{ban}] \label{lem4.1}
If $D\subset X $ is bounded closed and convex, the continuous map
$Q:D\to D $ is  $\beta$-condensing, then $Q$ has at least
one fixed point in $D$.
\end{lemma}

We will use the following hypotheses:
\begin{itemize}
\item[(HG2)] $g:C([0,b];X)\to X$ and there exists a constant
$l_g>0$ such that $\|g(x)-g(y)\|\leq l_g\|x-y\| $, $x,y\in
C([0,b];X)$.

\item[(HF3)] $f:[0,b]\times X\to X$ is continuous and compact.
\end{itemize}

\begin{theorem} \label{thm4.2}
Assume that  {\rm (HA), (HG2), (HF3)} are satisfied.
Then the nonlocal fractional differential  system \eqref{e1.1}  has at
least one mild solution on $[0,b]$, provided that \eqref{e3.2} and
\begin{equation}\label{e4.1}
Ml_g<1.
\end{equation}
\end{theorem}

\begin{proof} Define the solution operator $G:C(J;X)\to C(J;X)$ by
$$
Gu(t)=S_\alpha(t)g(u)+\int_0^t(t-s)^{\alpha-1}T_\alpha(t-s)f(s,u(s))\,\mathrm{d}s 
$$
with 
\begin{gather*}
G_1u(t)=S_\alpha(t)g(u), \\
G_2u(t)=\int_0^t(t-s)^{\alpha-1}T_\alpha(t-s)f(s,u(s))\,\mathrm{d}s,
\end{gather*}
for all $t\in J$. From the proof of Theorem \ref{thm3.3}, we have got that
the solution operator $G$ is continuous and maps $W_r$ into itself.
It remains to show that $G$ is $\beta$-condensing.

For $u,v\in W_r$, we have
\[
 \|G_1u-G_1v\|=\|S_\alpha(t)g(u)-S_\alpha(t)g(v) \|\leq  Ml_g\|u-v\|,
\]
which implies that $G_1$ is Lipschitz continuous with the constant
$Ml_g$. From Lemma \ref{lem2.7} (7), we have
\begin{equation}\label{e4.2}
    \beta(G_1W_r)\leq Ml_g\beta(W_r).
\end{equation}

Next, we shall show that $G_2$ is a compact operator. From the
Ascoli-Arzela theorem, we need prove that $G_2W_r$ is equicontinuous
and $G_2W_r(t)$ is precompact in $X$ for $t\in [0,b]$. For $u\in
W_r$ and $0\leq t_1<t_2\leq b$, we have
\begin{equation}\label{e4.3}
\begin{aligned}
&\|G_2u(t_2)-G_2u(t_1)\| \\ 
&\leq  \big\|\int_{t_1}^{t_2}(t_2-s)^{\alpha-1}T_\alpha(t_2-s)f(s,u(s))
 \,\mathrm{d}s \big\| \\
&\quad +\big\|\int_0^{t_1}[(t_2-s)^{\alpha-1}-(t_1-s)^{\alpha-1}  ]
 T_\alpha(t_2-s)f(s,u(s))\,\mathrm{d}s \big\|  \\
&\quad  +\big\|\int_0^{t_1}(t_1-s)^{\alpha-1}[T_\alpha(t_2-s)
 -T_\alpha(t_1-s)]f(s,u(s))\,\mathrm{d}s\big\|\\
&:= I_1+I_2+I_3,
\end{aligned}
\end{equation}
where
\begin{gather*}
I_1=\big\|\int_{t_1}^{t_2}(t_2-s)^{\alpha-1}T_\alpha(t_2-s)f(s,u(s))
 \,\mathrm{d}s \big\| ,  \\
I_2=\big\|\int_0^{t_1}[(t_2-s)^{\alpha-1}-(t_1-s)^{\alpha-1}  ]
 T_\alpha(t_2-s)f(s,u(s))\,\mathrm{d}s \big\| ,  \\
I_3=\big\|\int_0^{t_1}(t_1-s)^{\alpha-1}[T_\alpha(t_2-s)-T_\alpha(t_1-s)]
 f(s,u(s))\,\mathrm{d}s\big\|.
\end{gather*}
By direct calculation we obtain 
\begin{equation}\label{e4.4}
\begin{aligned}
I_1&\leq \frac{\alpha
        M}{\Gamma(1+\alpha)}\frac{1}{\alpha}(t_2-t_1)^\alpha\sup_{t\in J,u\in
        W_r}f(s,u(s))\\
&\leq  \frac{M}{\Gamma(1+\alpha)}\sup_{t\in J,u\in
        W_r}f(s,u(s))\cdot(t_2-t_1)^\alpha.
\end{aligned}
\end{equation}
For $t_1=0$, $0<t_2\leq b$, it is easy to see that $I_2=I_3=0$. For
$t_1>0$, we have
\begin{equation}\label{e4.5}
\begin{aligned}
I_2&\leq \frac{\alpha M}{\Gamma(1+\alpha)}\sup_{t\in J,u\in
    W_r}f(s,u(s))\cdot \int_0^{t_1} (t_2-s)^{\alpha-1}-(t_1-s)^{\alpha-1}\,\mathrm{d}s \\
 &\leq  \frac{M}{\Gamma(1+\alpha)}\sup_{t\in J,u\in
        W_r}f(s,u(s))\cdot(t_2^\alpha-t_1^\alpha).
\end{aligned}
\end{equation}
Since $f$ is compact, then
$\|[T_\alpha(t_2-s)-T_\alpha(t_1-s)]f(s,u(s))\|\to 0 $, as
$t_1\to t_2$, uniformly for $s\in J$ and $u\in W_r$. This
implies that, for any $\varepsilon>0$, there exists $\delta>0$, such
that 
$$
\|[T_\alpha(t_2-s)-T_\alpha(t_1-s)]f(s,u(s)) \|<\varepsilon,
$$ 
for $0<t_2-t_1<\delta$ and $u\in W_r$. Then we have
\begin{equation}\label{e4.6}
I_3\leq \varepsilon
    \int_0^{t_1}(t_1-s)^{\alpha-1}\,\mathrm{d}s\leq
    \frac{b^\alpha}{\alpha}\varepsilon.
\end{equation}
Thus, combining the above inequalities \eqref{e4.3}--\eqref{e4.6}, 
we obtain the equicontinuity of $GW_r$ on $[0,b]$.

The set $\{T_\alpha(t-s)f(s,u(s)):t,s\in J, u\in W_r\}\subset X$ is
precompact in $X$ as $f$ is compact and $T(t)$ is continuous.
Similarly with the proof of Lemma \ref{lem3.2}, for arbitrary 
$\varepsilon >0$, there exists a sequence $\{v_k\}_{k=1}^\infty\subset W_r$,
 such that
\begin{align*}
  \beta(G_2W_r(t)) 
&\leq  2\beta(\{ G_2v_k(t):k\geq 1\})+\varepsilon \\
&\leq  4\int_0^t (t-s)^{\alpha-1}\beta( \{T_\alpha(t-s)f(s,v_k(s)):k\geq
  1\})\,\mathrm{d}s+\varepsilon.
\end{align*}
Noticing that 
$$
\beta( \{T_\alpha(t-s)f(s,v_k(s)):k\geq
  1\})\leq  \beta( \{T_\alpha(t-s)f(s,u(s)):t,s\in J, u\in W_r\})=0, 
$$
we have $\beta(G_2W_r(t))=0 $ for $t\in J$. By the Ascoli-Arzela
theorem, we have that $G_2$ is a compact operator, which implies
\begin{equation}\label{e4.7}
 \beta(G_2W_r)=0.
\end{equation}
So, according to \eqref{e4.2} and \eqref{e4.7}, we can conclude that
\begin{equation*}
    \beta(GW_r)\leq \beta(G_1W_r)+\beta(G_2W_r)\leq Ml_g\beta(W_r).
\end{equation*}
From the condition $Ml_g<1$, $G$ is $\beta-$condensing in $W_r$. By
the Darbo-Sadovskii's fixed point theorem, $G$ has at least a fixed
point $u$ in $W_r$, which is just a mild solution of the problem
\eqref{e1.1}. The proof is complete.
 \end{proof}

By using Banach contraction principle, we also give the existence
theorem when $f$, $g$ are uniformly Lipschitz continuous. We give
the following hypothesis on  $f$.
\begin{itemize}
\item[(HF4)] $f:C(J;X)\to X$ is continuous and there exists a
constant $l_f>0$, such that
$$
\|f(t,x_1)-f(t,x_2)\|\leq l_f\|x_1-x_2\|,\quad x_1,x_2\in X. 
$$
\end{itemize}

\begin{theorem} \label{thm4.3}
Assume that the hypotheses {\rm (HA), (HG2), (HF4)} are satisfied.
Then the nonlocal fractional  system \eqref{e1.1} has a unique mild
solution on $[0,b]$, provided that
\begin{equation}\label{e4.8}
Ml_g+\frac{Mb^\alpha}{\Gamma(1+\alpha)}l_f<1.
\end{equation}
\end{theorem}

\begin{proof} 
For $u,v\in C(J;x)$, $t\in J$, we have that
\begin{align*}
\|Gu(t)-Gv(t)\|
&\leq  \|S_\alpha(t)[g(u)-g(v)]\| \\
&\quad +\int_0^{t}(t-s)^{\alpha-1}\|T_\alpha(t-s)[f(s,u(s))-f(s,v(s))] \|
   \,\mathrm{d}s\\
&\leq  Ml_g\|u-v\|_C +\frac{\alpha    M}{\Gamma(1+\alpha)}
 \int_0^{t}(t-s)^{\alpha-1}\,\mathrm{d}s\cdot    l_f\|u-v\|_C\\
&\leq  \big(Ml_g+\frac{Mb^\alpha}{\Gamma(1+\alpha)}l_f\big)\|u-v\|_C.
\end{align*}
Then 
$$
\|Gu-Gv\|_C \leq
\big(Ml_g+\frac{Mb^\alpha}{\Gamma(1+\alpha)}l_f\big) \|u-v\|_C.
$$
According to \eqref{e4.8}, we find that $G$ is a contraction operator in
$C(J;X)$. Thus $G$ has a unique fixed point $u$, which is the unique
mild solution to the problem \eqref{e1.1}. The proof is complete.
\end{proof}

\subsection*{Conclusions}
This article is motivated by some recent
papers \cite{mop,zho,peng2}, where some fractional nonlocal
differential equations are discussed when $T(t)$ is compact or
equicontinuous. Since it is difficult to determine whether an
operator semigroup is compact (see Pazy\cite{paz}), we do not assume
that  $A$ generates a compact semigroup. It allow us to discuss some
differential equations which contain a linear operator that
generates a noncompact semigroup. We give a simple example. Let
$X=L^2(-\infty,+\infty)$. The ordinary differential operator
$A=\mathrm{d}/\mathrm{d}x$ with $D(A)=H^1(-\infty,+\infty)$,
generates a semigroup $T(t)$ defined by
$T(t)u(s)=u(t+s)$, for every $u\in X$. The 
$C_0$-semigroup $T(t)$ is not compact on $X$.

Another motivation of this paper is the control problem of
fractional differential system. Exact controllability for fractional
order systems have been discussed by many authors. Some
controllability results are obtained with the assumptions that the
associated semigroup $T(t)$ is compact and the inverse of control
operator is bounded. However, Hern\'{a}ndez and O'Regan \cite{her}
have pointed out that in this case the application of
controllability result is restricted to the finite dimensional
space. Here we can also deal with the fractional control problem in
the similar way and give a way to remove the compactness assumptions
for the nonlocal control problems.


\subsection*{Acknowledgments}
Research is partially supported by the National Natural Science
Foundation of China (11271316, 11101353), the Foundation of Huaiyin
Institute of Technology (HGC1229).

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