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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 168, 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 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/168\hfil Fractional semilinear differential equations]
{Existence and uniqueness of mild solutions for fractional semilinear
 differential equations}

\author[B. H. Guswanto, T. Suzuki \hfil EJDE-2015/168\hfilneg]
{Bambang Hendriya Guswanto, Takashi Suzuki}

\address{Bambang Hendriya Guswanto \newline
Department of Mathematics, Faculty of Mathematics and Natural Sciences,
Jenderal Soedirman University (UNSOED), Purwokerto, Indonesia}
\email{bambanghg\_unsoed@yahoo.com; bambang.guswanto@unsoed.ac.id}

\address{Takashi Suzuki \newline
Division of Mathematical Science, Department of Systems Innovation,
Graduate School of Engineering Science, Osaka University, Osaka, Japan}
\email{suzuki@sigmath.es.osaka-u.ac.jp}

\thanks{Submitted April 2, 2015. Published June 18, 2015.}
\subjclass[2010]{34A08, 34A12}
\keywords{Fractional semilinear differential equation; sectorial operator;
\hfill\break\indent Caputo fractional derivative; fractional power; 
 mild solution}

\begin{abstract}
 In this article, we study the existence and uniqueness of a local mild solution
 for a class of semilinear differential equations involving the Caputo
 fractional time derivative of order $\alpha$ $(0<\alpha<1)$ and, in the
 linear part, a sectorial linear operator $A$. We put some conditions on a
 nonlinear term $f$ and an initial data $u_0$ in terms of the fractional
 power of $A$. By applying Banach's Fixed Point Theorem, we obtain a unique
 local mild solution with smoothing effects, estimates, and a behavior at $t$
 close to 0. An example as an application of our results is also given.
\end{abstract}

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

\section{Introduction}

Some existing researches showed that, in diffusion process, there are particle's
movements that can be no longer modelled by the (normal) diffusion equation.
To see these phenomenons, one can refer to
\cite{Adams92,Berkowitz06,Hatano98,Laffaldano05} observing the dispersion in a
heterogeneous aquifer, the transport of contaminants in geological formations,
the dispersive transport of ions in column experiments, and the diffusion of
water in sand, respectively. All of these processes follow the pattern
 \begin{equation}\label{subdiff}
 \langle x^{2}(t)\rangle\sim t^{\alpha},\quad 0<\alpha<1,
 \end{equation}
where $\langle x^{2}(t)\rangle$ is the mean square displacement at time $t$. 
These processes are called subdiffusion and can be modelled by the equation
 \begin{equation}\label{subdiffeq}
 D_{t}^{\alpha}u(x,t)=D_{\alpha}\Delta u(x,t),\quad x\in\mathbb{R}^{n},\;t>0,
 \end{equation}
where $0<\alpha<1$, $D_{\alpha}$ is a subdiffusion coeficient, and $D_{t}^{\alpha}$ 
is the Caputo fractional derivative of order $\alpha$. Reaction subdiffusion 
equation was also derived 
(see \cite{Barkai00,Henry10,Henry06,Henry00,Langlands10,Langlands08,Metzler99,
Seki03,Sung02}). Subdifusion model can also be a formula for memory phenomenon 
(see \cite{Hilfer00,Mainardi10}). In \cite{Du13}, Du et al. also found that the 
order of fractional derivative is an index of memory. Thus a study to investigate 
a solution to this model is very useful. Recently, there are some researches 
studying a solution to fractional evolution equations, for instance, 
see \cite{Chen13,Fan12,Li09,Li13,Peng12,Raheem13,Wang12,Zhang12,Zhang14,Zhou13}.

In this article, we show the existence and uniqueness of a local mild solution 
to the fractional abstract Cauchy problem
 \begin{equation}\label{fracnonlinpr}
 \begin{gathered}
 D_{t}^{\alpha}u=Au+f(u),\;t>0,\;0<\alpha<1,\\
 u(0)=u_{0},
 \end{gathered}
 \end{equation}
where $H$ is a Banach space, $D_{t}^{\alpha}$ is the Caputo fractional 
derivative of order $\alpha$, $A:D(A)\to  H$ is a sectorial linear
operator, $u_{0}\in H$, and $f:H\to  H$. We use some conditions on
$f$ and $u_{0}$ in terms of the fractional power of $A$. The conditions are
 \begin{itemize}
 \item[(i)] $f(0)=0$,
 \item[(ii)] there exist $C_{0}>0$, $\vartheta>1$, and $0<\beta<1$ such that
 $$
 \|f(u)-f(v)\|\leq C_{0}(\|A^{\beta}u\|+\|A^{\beta}v\|)^{\vartheta-1}
 \|A^{\beta}u-A^{\beta}v\|
 $$
 for all $u,v\in D(A^{\beta})$,
 \item[(iii)] $u_{0}\in D(A^{\nu})$ for some $0<\nu<1$.
 \end{itemize}
These conditions are used to study the solvability and smoothing effect 
for some class of semilinear parabolic equations (see \cite{Hoshino91}). 
As in \cite{Hoshino91}, we apply Banach's Fixed Point Theorem to construct 
a local mild solution to the problem \eqref{fracnonlinpr} by employing the 
properties of solution operators generated by $A$ and the fractional power of $A$. 
In this paper, we obtain the existence and uniqueness of the local mild solution 
with smoothing effects, estimates, and a behaviour at $t$ close to 0 as the 
advantages of our results compared with the preceding related results.

This article is composed of four sections. 
In section 2, we introduce briefly the fractional integration and differentiation 
of Riemann-Liouville and Caputo operators. In this section, we also provides 
some properties of analytic solution operators for fractional evolution 
equations including some estimates involving the fractional power of sectorial 
operators. In next section, our main results are showed. Finally, in the 
last section, an application of our main results is given.

\section{Preliminaries}

\subsection{Fractional time derivative}

Let $0<\alpha<1$, $a\geq0$ and $I=(a,T)$ for some $T>0$. 
The \textit{Riemann-Liouville fractional integral} of order $\alpha$ is defined by
 \begin{equation}\label{fracint}
 J_{a,t}^{\alpha}f(t)=\int_{a}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}f(s)ds,
 \quad f\in L^{1}(I),\;t>a.
 \end{equation}
We set $J_{a,t}^{0}f(t)=f(t)$. The fractional integral operator \eqref{fracint}
 obeys the semigroup property
 \begin{equation}\label{fracintsemprop}
 J_{a,t}^{\alpha}J_{a,t}^{\beta}=J_{a,t}^{\alpha+\beta},
 \quad 0\leq\alpha,\;\beta<1.
 \end{equation}
The \textit{Riemann-Liouville fractional derivative} of order $\alpha$ is defined by
 \begin{equation}\label{RLfracder}
 \mathcal{D}_{a,t}^{\alpha}f(t)=D_{t}\int_{a}^{t}\frac{(t-s)^{-\alpha}}
 {\Gamma(1-\alpha)}f(s)ds,\quad f\in L^{1}(I),\;t^{-\alpha}\ast f\in W^{1,1}(I),\;t>a,
 \end{equation}
where $\ast$ denotes the convolution of functions
 $$
 (f\ast g)(t)=\int_{a}^{t}f(t-\tau)g(\tau)d\tau,\quad t>a,
 $$
and $W^{1,1}(I)$ is the set of all functions $u\in L^{1}(I)$ such that the 
distributional derivative of $u$ exists and belongs to $L^{1}(I)$. 
The operator $\mathcal{D}_{a,t}^{\alpha}$ is a left inverse of $J_{a,t}^{\alpha}$;
that is,
 $$
 \mathcal{D}_{a,t}^{\alpha}J_{a,t}^{\alpha}f(t)=f(t),\quad t>a,
 $$
but it is not a right inverse, that is
 $$
 J_{a,t}^{\alpha}\mathcal{D}_{a,t}^{\alpha}f(t)=
 f(t)-\frac{(t-a)^{\alpha-1}}{\Gamma(\alpha)}J_{a,t}^{1-\alpha}f(a),\quad t>a.
 $$
The \textit{Caputo fractional derivative} of order $\alpha$ is defined by
 \begin{equation}\label{Capfracder1}
 D_{a,t}^{\alpha}f(t)=D_{t}\int_{a}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)}
 (f(s)-f(0))ds,\;t>a,
 \end{equation}
if $f\in L^{1}(I),\;t^{-\alpha}\ast f\in W^{1,1}(I)$, or
 \begin{equation}\label{Capfracder2}
 D_{a,t}^{\alpha}f(t)=\int_{a}^{t}\frac{(t-s)^{-\alpha}}{\Gamma(1-\alpha)}
 D_{s}f(s)ds,\;t>a,
 \end{equation}
if $f\in W^{1,1}(I)$. The operator $D_{a,t}^{\alpha}$ is also a left inverse of $J_{a,t}^{\alpha}$, that is
 \begin{equation}\label{Capleftinv}
 D_{a,t}^{\alpha}J_{a,t}^{\alpha}f(t)=f(t),\quad t>a,
 \end{equation}
but it is not also a right inverse, that is
 \begin{equation}\label{Caprightinv}
 J_{a,t}^{\alpha}D_{a,t}^{\alpha}f(t)=f(t)-f(a),\quad t>a.
 \end{equation}
The relation between the Riemann-Liouville and Caputo fractional derivative is
 \begin{equation}\label{RelRL-C}
 D_{a,t}^{\alpha}f(t)=\mathcal{D}_{a,t}^{\alpha}f(t)-\frac{(t-a)^{-\alpha}}
 {\Gamma(1-\alpha)}f(a),\quad t>a.
 \end{equation}
For $a=0$, we set $J_{a,t}^{\alpha}=J_{t}^{\alpha}$, 
$\mathcal{D}_{a,t}^{\alpha}=\mathcal{D}_{t}^{\alpha}$, and 
$D_{a,t}^{\alpha}=D_{t}^{\alpha}$. We refer to Kilbas et al \cite{Kilbas06} 
or Podlubny \cite{Podlubny99} for more details concerning the fractional 
integrals and derivatives.

\subsection{Analytic solution operators}

In this section, we provide briefly some results concerning solution operators 
for the fractional Cauchy problem
 \begin{equation}\label{fracpr}
 \begin{gathered}
 D_{t}^{\alpha}u(t)=Au(t)+f(t),\;t>0,\\
 u(0)=u_{0}.
 \end{gathered}
 \end{equation}
For more details, we refer to Guswanto \cite{Guswanto}.

Henceforth, we assume that the linear operator $A:D(A)\subset H\to  H$
satisfies the properties that there is a constant $\theta\in(\pi/2,\pi)$ such that
 \begin{gather}\label{assforA1}
 \rho(A)\supset S_{\theta}=\{\lambda\in \mathbb{C}:
\lambda\neq0,|\arg (\lambda)|<\theta\}, \\
\label{assforA2}
 \|R(\lambda;A)\|\leq\frac{M}{|\lambda|},\quad \lambda\in S_{\theta},
 \end{gather}
where $R(\lambda;A)=(\lambda-A)^{-1}$ and $\rho(A)$ are the resolvent 
operator and resolvent set of $A$, respectively. We call $A$ as a sectorial operator. 
Every operator satisfying this property is closed since its resolvent set 
is not empty. The linear operator $A$ generates solution operators for the 
problem \eqref{fracpr}, those are
 \begin{gather}\label{homsolopt}
 S_{\alpha}(t)=\frac{1}{2\pi i}\int_{\Gamma_{r,\omega}}
e^{\lambda t}\lambda^{\alpha-1}R(\lambda^{\alpha};A)d\lambda,\quad t>0, \\
\label{inhomsolopt}
 P_{\alpha}(t)=\frac{1}{2\pi i}\int_{\Gamma_{r,\omega}}e^{\lambda t}
R(\lambda^{\alpha};A)d\lambda,\quad t>0,
 \end{gather}
where $r>0$, $\pi/2<\omega<\theta$, and
 $$
 \Gamma_{r,\omega}=\{\lambda\in\mathbb{C}:|\arg (\lambda)|=\omega,
 |\lambda|\geq r\}\cup\{\lambda\in\mathbb{C}:|\arg (\lambda)|\leq\omega,
 |\lambda|=r\}
 $$
is oriented counterclockwise. By the Cauchy's theorem, the integral form 
\eqref{homsolopt} and \eqref{inhomsolopt} are independent of $r>0$ and 
$\omega\in (\pi/2,\theta)$.

Let $B(H)$ be the set of all bounded linear operators on $H$. The properties 
of the families $\{S_{\alpha}(t)\}_{t>0}$ and $\{P_{\alpha}(t)\}_{t>0}$ are given 
in the following theorems.

 \begin{theorem}\label{fracres}
 Let $A$ be a sectorial operator and $S_{\alpha}(t)$ be an operator defined 
by \eqref{homsolopt}. Then the following statements hold.
 \begin{itemize}
 \item[(i)] $S_{\alpha}(t)\in B(H)$ and there exists a constant 
$C_{1}=C_{1}(\alpha)>0$ such that
 $$
 \|S_{\alpha}(t)\|\leq C_{1},\quad t>0,
 $$

 \item[(ii)] $S_{\alpha}(t)\in B(H;D(A))$ for $t>0$, and if $x\in D(A)$ then 
$AS_{\alpha}(t)x=S_{\alpha}(t)Ax$. Moreover, there exists a constant 
$C_{2}=C_{2}(\alpha)>0$ such that
 $$
 \|AS_{\alpha}(t)\|\leq C_{2}t^{-\alpha},\quad t>0,
 $$

 \item[(iii)] The function $t\mapsto S_{\alpha}(t)$ belongs to 
$C^{\infty}((0,\infty);B(H))$ and it holds that
 $$
 S_{\alpha}^{(n)}(t)=\frac{1}{2\pi i}\int_{\Gamma_{r,\omega}}e^{t\lambda}
 \lambda^{\alpha+n-1}R(\lambda^{\alpha};A)d\lambda,\;n=1,2,\ldots
 $$
 and there exist constants $M_{n}=M_{n}(\alpha)>0, n=1,2,\ldots$ such that
 $$
 \|S_{\alpha}^{(n)}(t)\|\leq M_{n}t^{-n},\quad t>0,
 $$
 Moreover, it has an analytic continuation $S_{\alpha}(z)$ to the sector 
$S_{\theta-\pi/2}$ and, for $z\in S_{\theta-\pi/2}$, $\eta\in(\pi/2,\theta)$, 
it holds that
 $$
 S_{\alpha}(z)=\frac{1}{2\pi i}\int_{\Gamma_{r,\eta}}e^{\lambda z}\lambda^{\alpha-1}R(\lambda^{\alpha};A)d\lambda.
 $$
 \end{itemize}
 \end{theorem}

 \begin{theorem}\label{fracrep}
 Let $A$ be a sectorial operator and $P_{\alpha}(t)$ be an operator defined by 
\eqref{inhomsolopt}. Then the following statements hold.
 \begin{itemize}
 \item[(i)] $P_{\alpha}(t)\in B(H)$ and there exists a constant 
$L_{1}=L_{1}(\alpha)>0$ such that
 $$
 \|P_{\alpha}(t)\|\leq L_{1}t^{\alpha-1},\quad t>0,
 $$

 \item[(ii)] $P_{\alpha}(t)\in B(H;D(A))$ for all $t>0$, and if 
$x\in D(A)$ then $AP_{\alpha}(t)x=P_{\alpha}(t)Ax$. Moreover, there exists 
a constant $L_{2}=L_{2}(\alpha)>0$ such that
 $$
 \|AP_{\alpha}(t)\|\leq L_{2}t^{-1},\quad t>0,
 $$

 \item[(iii)] The function $t\mapsto P_{\alpha}(t)$ belongs to 
$C^{\infty}((0,\infty); B(H))$ and it holds that
 $$
 P_{\alpha}^{(n)}(t)=\frac{1}{2\pi i}\int_{\Gamma_{r,\omega}}e^{t\lambda}
 \lambda^{n}R(\lambda^{\alpha};A)d\lambda,\;n=1,2,\ldots
 $$
 and there exist constants $K_{n}=K_{n}(\alpha)>0, n=1,2,\ldots$ such that
 $$
 \|P_{\alpha}^{(n)}(t)\|\leq K_{n}t^{\alpha-n-1},\quad t>0,
 $$
 Moreover, it has an analytic continuation $P_{\alpha}(z)$ to the sector 
$S_{\theta-\pi/2}$ and, for $z\in S_{\theta-\pi/2}$, $\eta\in(\pi/2,\theta)$, 
it holds that
 $$
 P_{\alpha}(z)=\frac{1}{2\pi i}\int_{\Gamma_{r,\eta}}e^{\lambda z}
R(\lambda^{\alpha};A)d\lambda.
 $$
 \end{itemize}
 \end{theorem}

The following theorem states some identities concerning the operators 
$S_{\alpha}(t)$ and $P_{\alpha}(t)$ including the semigroup-like property.

 \begin{theorem} \label{thm2.3}
 Let $A$ be a sectorial operator, $S_{\alpha}(t)$ and $P_{\alpha}(t)$
 be operators defined by \eqref{homsolopt} and \eqref{inhomsolopt}, 
respectively. Then the following statements hold.
 \begin{itemize}
 \item[(i)] For $x\in H$ and $t>0$,
 \begin{equation*}
 S_{\alpha}(t)x=J_{t}^{1-\alpha}P_{\alpha}(t)x,\quad
 D_{t}S_{\alpha}(t)x=AP_{\alpha}(t)x,
 \end{equation*}

 \item[(ii)] For $x\in D(A)$ and $s,t>0$,
 \begin{gather*}
 D_{t}^{\alpha}S_{\alpha}(t)x=AS_{\alpha}(t)x,\\
 S_{\alpha}(t+s)x=S_{\alpha}(t)S_{\alpha}(s)x-A\int_{0}^{t}
 \int_{0}^{s}\frac{(t+s-\tau-r)^{-\alpha}}
 {\Gamma(1-\alpha)}P_{\alpha}(\tau)P_{\alpha}(r)x\,dr\,d\tau.
 \end{gather*}
 \end{itemize}
 \end{theorem}

The next theorem shows us the behavior of the operator $S_{\alpha}(t)$ 
at $t$ close to $0^{+}$.

 \begin{theorem}\label{bhvcls0}
 Let $A$ be a sectorial operator and $S_{\alpha}(t)$ be an operator defined 
by \eqref{homsolopt}. Then the following statements hold.
 \begin{itemize}
\item[(i)] If $x\in\overline{D(A)}$ then $\lim_{t\to 0^{+}}S_{\alpha}(t)x=x$.

\item[(ii)] For every $x\in D(A)$ and $t>0$,
 \begin{gather*}
 \int_{0}^{t}\frac{(t-\tau)^{\alpha-1}}
 {\Gamma(\alpha)}S_{\alpha}(\tau)
 xd\tau\in D(A),\\
 \int_{0}^{t}\frac{(t-\tau)^{\alpha-1}}
 {\Gamma(\alpha)}AS_{\alpha}(\tau)
 xd\tau=S_{\alpha}(t)x-x,
 \end{gather*}

\item[(iii)]If $x\in D(A)$ and $Ax\in\overline{D(A)}$ then
 $$
 \lim_{t\mapsto0^{+}}
 \frac{S_{\alpha}(t)x-x}{t^{\alpha}}
 =\frac{1}{\Gamma(\alpha+1)}Ax.
 $$
 \end{itemize}
 \end{theorem}

The representation of the solution to \eqref{fracpr} in term of 
$S_{\alpha}(t)$ and $P_{\alpha}(t)$ is given in the following theorem.

\begin{theorem} \label{thm2.5}
 Let $u\in C^{1}((0,\infty);H)\cap L^{1}((0,\infty);H)$, $u(t)\in D(A)$ for
 $t\in[0,\infty)$, $Au\in L^{1}((0,\infty);H)$, $f\in L^{1}((0,\infty);D(A))$, 
and $Af\in L^{1}((0,\infty);H)$. If $u$ is a solution to the problem \eqref{fracpr} 
then
 \begin{equation}\label{solinhopr}
 u(t)=S_{\alpha}(t)u_{0}+\int_{0}^{t}P_{\alpha}(t-s)f(s)ds,\quad t>0.
 \end{equation}
 \end{theorem}

Now, we consider the fractional power of operator $A$
 $$
 A^{-\beta}x=\frac{1}{2\pi i}\int_{\Gamma_{r,\omega}}
\lambda^{-\beta}R(\lambda;A)xd\lambda,\quad x\in H,\;\beta>0,
 $$
and
 $$
 A^{\beta}x=A(A^{\beta-1}x)=\frac{1}{2\pi i}
\int_{\Gamma_{r,\omega}}\lambda^{\beta-1}R(\lambda;A)Axd\lambda,\quad
x\in D(A),\;0<\beta<1.
 $$
Some estimates involving $A^{\beta}$ and the operators families 
$\{S_{\alpha}(t)\}_{t>0}$, $\{P_{\alpha}(t)\}_{t>0}$ generated by the sectorial 
operator $A$ are provided by the following theorem. These estimates are analogous 
to those as stated in \cite[Theorem 6.13]{Pazy83} for analytic semigroups.

 \begin{theorem}
 For each $0<\beta<1$, there exist positive constants $C_{1}'=C_{1}'(\alpha,\beta)$, 
$C_{2}'=C_{2}'(\alpha,\beta)$, and $C_{3}'=C_{3}'(\alpha,\beta)$ such that for all
 $x\in H$,
 \begin{gather}\label{AfracS}
 \|A^{\beta}S_{\alpha}(t)x\|\leq C_{1}'t^{-\alpha}(t^{-\alpha(\beta-1)}+1)\|x\|,
\quad t>0, \\
\label{AfracP}
 \|A^{\beta}P_{\alpha}(t)x\|\leq C_{2}'t^{-\alpha(\beta-1)-1}\|x\|,\quad t>0.
 \end{gather}
Moreover, for all $x\in D(A^{\beta})$,
 \begin{equation}\label{est-Sx-x}
 \|S_{\alpha}(t)x-x\|\leq C_{3}'t^{\alpha\beta}\|A^{\beta}x\|,\quad t>0.
 \end{equation}
 \end{theorem}

Now, let $\xi_{\zeta}=\alpha(\zeta-1)+1$, for $0<\zeta<1$, and $x^{+}=\max\{0,x\}$, 
for $x\in \mathbb{R}$. Thus we have the following result.

 \begin{corollary} \label{coro2.1}
 For each $\beta>(2-1/\alpha)^{+}$ or $\beta=2-1/\alpha>0$ and $x\in H$,
 \begin{gather}\label{est-smallt-AfracS}
 t^{\xi_{\beta}}\|A^{\beta}S_{\alpha}(t)x\|\leq 2C_{1}'\|x\|,\quad0<t\leq1, \\
\label{est-bigt-AfracS}
 t^{\xi_{\beta}}\|A^{\beta}S_{\alpha}(t)x\|\leq2C_{1}'t^{1-\alpha}\|x\|,\quad t>1, 
\\
 t^{\xi_{\beta}}\|A^{\beta}P_{\alpha}(t)x\|\leq C_{2}'\|x\|,\quad t>0,\\
\label{lim-t-AfracS}
 t^{\xi_{\beta}}\|A^{\beta}S_{\alpha}(t)x\|\to 0,\quad\text{as } t\to 0^{+}.
 \end{gather}
 \end{corollary}

Furthermore, we have the same result as Theorem \ref{thm2.3} (ii) with weaker condition.

\begin{theorem}
 Let $0<\beta<1$. Then, for $x\in D(A^{\beta})$ and $s,t>0$,
 \begin{gather}\label{solopt}
 D_{t}^{\alpha}S_{\alpha}(t)x=AS_{\alpha}(t)x, \\
\label{semprop}
 S_{\alpha}(t+s)x=S_{\alpha}(t)S_{\alpha}(s)x-A\int_{0}^{t}
 \int_{0}^{s}\frac{(t+s-\tau-r)^{-\alpha}}
 {\Gamma(1-\alpha)}P_{\alpha}(\tau)P_{\alpha}(r)x\,dr\,d\tau.
 \end{gather}
 \end{theorem}

\section{Main results}

In this section, we show the existence and uniqueness of a mild solution for 
the problem \eqref{fracnonlinpr} under certain conditions by applying Banach's 
Fixed Point Theorem. Based on Theorem \ref{thm2.5}, we define a mild solution to the 
problem \eqref{fracnonlinpr} as follows.

 \begin{definition} \label{def3.1} \rm
 A continuous function $u:(0,T]\to  H$ is a mild solution to the problem 
\eqref{fracnonlinpr} if it satisfies
 $$
 u(t)=S_{\alpha}(t)u_{0}+\int_{0}^{t}P_{\alpha}(t-s)f(u(s))ds,\quad0<t
 \leq T.
 $$
 \end{definition}

The conditions on $f$ are:
 \begin{itemize}
 \item[(i)] $f(0)=0$,
 \item[(ii)] there exist $C_{0}>0$, $\vartheta>1$, and $0<\beta<1$ such that
 $$
 \|f(u)-f(v)\|\leq C_{0}(\|A^{\beta}u\|+\|A^{\beta}v\|)^{\vartheta-1}
 \|A^{\beta}u-A^{\beta}v\|,
 $$
 for all $u,v\in D(A^{\beta})$.
 \end{itemize}
Let $BC((0,T];D(A^{\beta}))$ be the set of all bounded and continuous 
functions $w:(0,T]\to  D(A^{\beta})$. Under the conditions on $f$ above, 
we obtain the following results.

\begin{theorem} \label{thm3.1}
 Let $u_{0}\in D(A^{\nu})$ with
 \begin{equation}\label{par-ass}
 \beta-\nu>(2-1/\alpha)^{+},\quad1-\alpha\nu-\vartheta\xi_{\beta-\nu}
 \geq0,\quad0<\vartheta\xi_{\beta-\nu}<1,
 \end{equation}
 where
 $$
 \xi_{\zeta}=\alpha(\zeta-1)+1,\quad0<\zeta<1;\quad x^{+}=\max\{0,x\},\quad 
x\in\mathbb{R}.
 $$
 Then there exits $T>0$ sufficiently small such that the problem 
\eqref{fracnonlinpr} has a unique mild solution $u$ satisfying
\begin{gather*}
 t^{\xi_{\eta-\nu}}u\in BC((0,T];D(A^{\eta})),\quad
\lim_{t\to  0^{+}}t^{\xi_{\eta-\nu}}A^{\eta}u(t)=0, \\
 \|A^{\eta}u(t)\|\leq Ct^{-\xi_{\eta-\nu}}\|A^{\nu}u_{0}\|,\quad t\in(0,T],
\end{gather*}
 for every $\eta\in(\nu+(2-1/\alpha)^{+},\beta]$.
\end{theorem}

\begin{theorem} \label{thm3.2}
 Let $u$ be the mild solution to the problem \eqref{fracnonlinpr} 
in Theorem \ref{thm3.1}. If $f(u(t))\in D(A)$, for $t\in(0,\infty)$, then
 $$
 t^{\xi_{1-\nu}}u\in BC((0,T];D(A))
 $$
 with
 $$
 \|Au(t)\|\leq Ct^{-\xi_{1-\nu}}\|A^{\nu}u_{0}\|,\quad t\in(0,T].
 $$
\end{theorem}

\subsection{Proof of Theorem \ref{thm3.1}}
We define first the Banach space
 $$
 E_{\beta,T}=\{u:[0,T]\to  H:t^{\xi_{\beta-\nu}}u\in BC((0,T];D(A^{\beta}))\}
 $$
equipped with the norm
 \begin{equation}\label{norm-E}
 \||u|\|_{\beta,T}=\sup_{0<t\leq T}t^{\xi_{\beta-\nu}}\|A^{\beta}u(t)\|,
 \end{equation}
and define a closed ball $B_{\beta,T}$ in $E_{\beta,T}$ by
 $$
 B_{\beta,T}=\{u\in E_{\beta,T}:\||u|\|_{\beta,T}\leq K\},
 $$
where $T$ and $K$ are some constants which will be specified later.

Next, we define a mapping $F$ on $B_{\beta,T}$ by
 $$
 Fu(t)=S_{\alpha}(t)u_{0}+\int_{0}^{t}P_{\alpha}(t-s)f(u(s))ds.
 $$
First, we prove the continuity of $A^{\beta}Fu(t)$ with respect to
 $t$ in $(0,T]$. Since $A^{\beta}$ is a bounded operator on $D(A)$ and, 
for each $x\in H$, $S_{\alpha}(t)x$ is continuous with respect to $t$ 
in $(0,\infty)$, then, for each $x\in H$, $A^{\beta}S_{\alpha}(t)x$ 
is continuous with respect to $t$ in $(0,\infty)$. Thus it remains to show 
the continuity of
 $$
 A^{\beta}\int_{0}^{t}P_{\alpha}(t-s)f(u(s))ds,\quad0<t\leq T.
 $$
Note that
\begin{align*}
& A^{\beta}\int_{0}^{t+h}P_{\alpha}(t+h-s)f(u(s))ds-A^{\beta}\int_{0}^{t}
 P_{\alpha}(t-s)f(u(s))ds\\
 &=A^{\beta}\int_{-h}^{t}P_{\alpha}(t-s)f(u(s+h))ds-A^{\beta}\int_{0}^{t}
 P_{\alpha}(t-s)f(u(s))ds\\
 &=A^{\beta}\int_{0}^{t}P_{\alpha}(t-s)(f(u(s+h))-f(u(s)))ds\\
 &\quad+A^{\beta}\int_{0}^{h}P_{\alpha}(t+h-s)f(u(s))ds.
\end{align*}
Observe that, for $u\in E_{\beta,T}$,
 \begin{equation}
 \|f(u(t+h))-f(u(t))\|\leq C_{0}2^{\vartheta-1}K^{\vartheta-1}
 t^{-(\vartheta-1)\xi_{\beta-\nu}}\|A^{\beta}u(t+h)-A^{\beta}u(t)\|
 \end{equation}
and
 \begin{equation}\label{est-f}
 \|f(u(t))\|\leq C_{0}\|A^{\beta}u(t)\|^{\vartheta}\leq C_{0}
 t^{-\vartheta\xi_{\beta-\nu}}\||u|\|_{\beta,T}^{\vartheta}
 \leq C_{0}K^{\vartheta}
 t^{-\vartheta\xi_{\beta-\nu}},
 \end{equation}
for $0<t\leq T$. Next, we have 
 \begin{align*}
& \int_{0}^{t}\|A^{\beta}P_{\alpha}(t-s)(f(u(s+h))-f(u(s)))\|ds\\
&\leq2^{\vartheta-1}C_{0}C_{2}K^{\vartheta-1}\int_{0}^{t}(t-s)^{-\xi_{\beta}}
 s^{-(\vartheta-1)\xi_{\beta-\nu}}\|A^{\beta}u(s+h)-A^{\beta}u(s)\|ds.
\end{align*}
Now, consider that, for $0<s<t\leq T$,
 \begin{gather*}
 (t-s)^{-\xi_{\beta}}s^{-(\vartheta-1)\xi_{\beta-\nu}}
 \|A^{\beta}u(s+h)-A^{\beta}u(s)\|\leq2K(t-s)^{-\xi_{\beta}}
 s^{-\vartheta\xi_{\beta-\nu}},\\
 s\mapsto2K(t-s)^{-\xi_{\beta}}s^{-\vartheta\xi_{\beta-\nu}}\in L^{1}((0,t);H),
\quad0<t\leq T, \\
 \|A^{\beta}u(s+h)-A^{\beta}u(s)\|\to 0,\,\text{as}\,h\to 0.
\end{gather*}
Hence, by the Dominated Convergence theorem, 
 $$
 \int_{0}^{t}(t-s)^{-\xi_{\beta}}s^{-(\vartheta-1)\xi_{\beta-\nu}}
 \|A^{\beta}u(s+h)-A^{\beta}u(s)\|ds\to 0,\quad \text{as }h\to 0.
 $$
This implies
 $$
 \int_{0}^{t}\|A^{\beta}P_{\alpha}(t-s)(f(u(s+h))-f(u(s)))\|ds
 \to 0,\quad \text{as }h\to 0.
 $$
Next, observe that
 \begin{align*}
&\int_{0}^{h}\|A^{\beta}P_{\alpha}(t+h-s)\|\|f(u(s)\|)ds\\
 &\leq C_{0}C_{2}'(\alpha,\beta)K^{\vartheta}
 \int_{0}^{h}(t+h-s)^{-\xi_{\beta}}s^{-\vartheta\xi_{\beta-\nu}}ds\\
&=C_{0}C_{2}'(\alpha,\beta)K^{\vartheta}(t+h)^{1-\xi_{\beta}
 -\vartheta\xi_{\beta-\nu}}
 \int_{0}^{\frac{h}{t+h}}(1-r)^{-\xi_{\beta}}r^{-\vartheta\xi_{\beta-\nu}}dr\\
&=C_{0}C_{2}'(\alpha,\beta)K^{\vartheta}(t+h)^{1-\xi_{\beta}
 -\vartheta\xi_{\beta-\nu}} \frac{1}{1-\vartheta\xi_{\beta-\nu}}\\
 &\quad\times \Big(\frac{h}{t+h}\Big)^{1-\vartheta\xi_{\beta-\nu}}
 H\Big(1-\vartheta\xi_{\beta-\nu},\xi_{\beta};2-\vartheta\xi_{\beta-\nu};
 \frac{h}{t+h}\Big)\\
 &=\frac{C_{0}C_{2}'(\alpha,\beta)K^{\vartheta}}{1-\vartheta\xi_{\beta-\nu}}
 h^{1-\vartheta\xi_{\beta-\nu}}(t+h)^{-\xi_{\beta}}
 H\Big(1-\vartheta\xi_{\beta-\nu},\xi_{\beta};2-\vartheta\xi_{\beta-\nu};
 \frac{h}{t+h}\Big),
\end{align*}
where
 $$
 H(a,b;c;x)=\frac{\Gamma(c)}{\Gamma(b)\Gamma(c-b)}
 \int_{0}^{1}\frac{t^{b-1}(1-t)^{c-b-1}}{(1-xt)^{a}},\quad c-b-a>0,\;|x|\leq1
 $$
is hypergeometric function (see \cite{Kilbas06}). Thus
 $$
 \int_{0}^{h}\|A^{\beta}P_{\alpha}(t+h-s)\|\|f(u(s)\|ds
 \to 0,\quad \text{as }h\to 0.
 $$
Therefore the continuity of $A^{\beta}Fu(t)$ with respect to $t$ in $(0,T]$
 is obtained.

Next, we prove that the mapping $F$ is well-defined and maps $B_{\beta,T}$ 
into itself. Consider 
 \begin{align*}
&\int_{0}^{t}\|A^{\beta}P_{\alpha}(t-s)\|\|f(u(s))\|ds\\
&\leq C_{0}C_{2}'(\alpha,\beta)K^{\vartheta-1}\||u|\|_{\beta,T}
 \int_{0}^{t}(t-s)^{-\xi_{\beta}}s^{-\vartheta\xi_{\beta-\nu}}ds\\
&\leq C_{0}C_{2}'(\alpha,\beta)K^{\vartheta-1}
 B(1-\vartheta\xi_{\beta-\nu},1-\xi_{\beta})
 \||u|\|_{\beta,T}t^{1-\xi_{\beta}-\vartheta\xi_{\beta-\nu}},
\end{align*}
where
 $$
 B(a,b)=\int_{0}^{1}r^{a-1}(1-r)^{b-1}dr,\quad a,b>0,
 $$
is Beta function. Therefore
 \begin{equation}
 t^{\xi_{\beta-\nu}}\|A^{\beta}Fu(t)\|
\leq t^{\xi_{\beta-\nu}}\|A^{\beta}S_{\alpha}(t)u_{0}\|
 +C_{4}K^{\vartheta-1}\||u|\|_{\beta,T}
 t^{1-\xi_{\beta}-\vartheta\xi_{\beta-\nu}+\xi_{\beta-\nu}},
 \end{equation}
where $C_{4}=C_{0}C_{2}'(\alpha,\beta)B(1-\xi_{\beta},1-\vartheta\xi_{\beta-\nu})$, 
implying
 \begin{equation}\label{est-F0}
\||Fu|\|_{\beta,T}\leq\sup_{0<t\leq T}t^{\xi_{\beta-\nu}}
 \|A^{\beta}S_{\alpha}(t)u_{0}\|+C_{4}
 K^{\vartheta-1}T^{1-\xi_{\beta}-\vartheta\xi_{\beta-\nu}+\xi_{\beta-\nu}}
 \||u|\|_{\beta,T}.
\end{equation}
Note that $1-\xi_{\beta}-\vartheta\xi_{\beta-\nu}+\xi_{\beta-\nu}=1-\alpha\nu-
\vartheta\xi_{\beta-\nu}\geq0$ by \eqref{par-ass}. By \eqref{est-smallt-AfracS},
 we can find $0<T\leq1$ such that
 $$
 t^{\xi_{\beta-\nu}}\|A^{\beta-\nu}S_{\alpha}(t)A^{\nu}u_{0}\|
\leq 2C_{1}'(\alpha,\beta-\nu)\|A^{\nu}u_{0}\|,\quad 0<t\leq T.
 $$
Then,  for $u\in B_{\beta,T}$, we have
 \begin{equation}\label{est-F}
 \begin{split}
 \||Fu|\|_{\beta,T}
&\leq\sup_{0<t\leq T}t^{\xi_{\beta-\nu}}\|A^{\beta-\nu}
 S_{\alpha}(t)A^{\nu}u_{0}\|+C_{4}
 K^{\vartheta}T^{1-\alpha\nu-\vartheta\xi_{\beta-\nu}}\\
&\leq2C_{1}'(\alpha,\beta-\nu)\|A^{\nu}u_{0}\|+C_{4}
 K^{\vartheta}T^{1-\alpha\nu-\vartheta\xi_{\beta-\nu}}.
 \end{split}
 \end{equation}
Next, we choose $K>0$ such that
 \begin{equation}\label{est-F1}
 2C_{1}'(\alpha,\beta-\nu)\|A^{\nu}u_{0}\|+C_{4}
 K^{\vartheta}T^{1-\alpha\nu-\vartheta\xi_{\beta-\nu}}\leq K.
 \end{equation}
For the case $1-\alpha\nu-\vartheta\xi_{\beta-\nu}>0$, we can get such 
a $K$ by taking $T$ sufficiently small. 
For the case $1-\alpha\nu-\vartheta\xi_{\beta-\nu}=0$, we choose $K>0$ 
sufficiently small such that
 $$
 C_{4}K^{\vartheta}<K,
 $$
and then take $T$ such that
 \begin{equation}\label{est-F2}
 \sup_{0<t\leq T}t^{\xi_{\beta-\nu}}\|A^{\beta-\nu}
 S_{\alpha}(t)A^{\nu}u_{0}\|\leq K-C_{4}K^{\vartheta}.
 \end{equation}
Note that in both cases, we can find $C=C(\alpha,\beta)>0$ such that
 \begin{equation}\label{est-K}
 K\leq C\|A^{\nu}u_{0}\|.
 \end{equation}
Hence $\||Fu|\|_{\beta,T}\leq K$. Thus the mapping $F$ is well-defined 
and maps $B_{\beta,T}$ into itself.

Next, we show that the mapping $F:B_{\beta,T}\to  B_{\beta,T}$ is a strict 
contraction. Note that, if $u,v\in B_{\beta,T}$, we have
 \begin{align*}
& \|A^{\beta}Fu(t)-A^{\beta}Fv(t)\| \\
&\leq\int_{0}^{t}\|A^{\beta}
 P_{\alpha}(t-s)\|\|f(u(s))-f(v(s))\|ds\\
&\leq C_{0}C_{2}'(\alpha,\beta)\int_{0}^{t}(t-s)^{-\xi_{\beta}}
 \big(\||u|\|_{\beta,T}+\||v|\|_{\beta,T}\big)^{\vartheta-1}\\*
&\quad\times  s^{-(\vartheta-1)\xi_{\beta-\nu}}\||u-v|\|_{\beta,T}s^{-\xi_{\beta-\nu}}ds\\
&\leq C_{0}C_{2}'(\alpha,\beta)2^{\vartheta-1}K^{\vartheta-1}\int_{0}^{t}
 (t-s)^{-\xi_{\beta}}s^{-\vartheta\xi_{\beta-\nu}}ds\||u-v|\|_{\beta,T}\\
&\leq C_{4}2^{\vartheta-1}K^{\vartheta-1}\||u-v|\|_{\beta,T}
 t^{1-\xi_{\beta}-\vartheta\xi_{\beta-\nu}}.
 \end{align*}
Then
\begin{align*}
t^{\xi_{\beta-\nu}}\|A^{\beta}Fu(t)-A^{\beta}Fv(t)\|
&\leq C_{4}2^{\vartheta-1}K^{\vartheta-1}\||u-v|\|_{\beta,T}
 t^{1-\alpha\nu-\vartheta\xi_{\beta-\nu}}\\
&\leq C_{4}2^{\vartheta-1}K^{\vartheta-1}\||u-v|\|_{\beta,T}
 T^{1-\alpha\nu-\vartheta\xi_{\beta-\nu}}.
\end{align*}
Note that we can select $K>0$ and $T>0$ sufficiently small such that
 \begin{equation}\label{est-C5}
 C_{5}=C_{4}2^{\vartheta-1}K^{\vartheta-1}T^{1-\alpha\nu-\vartheta\xi_{\beta-\nu}}<1.
 \end{equation}
Consequently,
 $$
 \||Fu-Fv|\|\leq C_{5}\||u-v|\|_{\beta,T}.
 $$
It means the mapping $F:B_{\beta,T}\to  B_{\beta,T}$ is a strict contraction. 
Thus, by Banach's Fixed Point Theorem, we can get a unique $u\in B_{\beta,T}$ 
which is a mild solution to the problem \eqref{fracnonlinpr}. 
Furthermore, by \eqref{est-F}, \eqref{est-F1}, \eqref{est-F2}, and \eqref{est-K}, 
for this $u$, we have
 $$
 \||u|\|_{\beta,T}\leq\sup_{0<t\leq T}t^{\xi_{\beta-\nu}}
 \|A^{\beta-\nu}S_{\alpha}(t)A^{\nu}u_{0}\|+C_{4}
 K^{\vartheta}T^{1-\xi_{\beta}-\vartheta\xi_{\beta-\nu}+\xi_{\beta-\nu}}
 \leq C\|A^{\nu}u_{0}\|.
 $$
Then, by \eqref{norm-E},
 $$
 \|A^{\beta}u(t)\|\leq Ct^{-\xi_{\beta-\nu}}\|A^{\nu}u_{0}\|,\quad0<t\leq T.
 $$
\\
Now, we check the continuity of $u$ at $t=0$. Note that
\begin{equation}\label{est-t-Afrac-u}
 \begin{split}
 t^{\xi_{\beta-\nu}}\|A^{\beta}u(t)\|
&\leq t^{\xi_{\beta-\nu}}\|A^{\beta}
 S_{\alpha}(t)u_{0}\|+t^{\xi_{\beta-\nu}}\int_{0}^{t}\|A^{\beta}P_{\alpha}
 (t-s)\|\|f(u(s))\|\\
&\leq t^{\xi_{\beta-\nu}}\|A^{\beta-\nu}
 S_{\alpha}(t)A^{\nu}u_{0}\|+C_{4}K^{\vartheta}t^{1-\alpha\nu-\vartheta
 \xi_{\beta-\nu}}.
 \end{split}
\end{equation}
Thus, if $1-\alpha\nu-\vartheta\xi_{\beta-\nu}>0$, letting $t\to  0^{+}$ 
on both sides of \eqref{est-t-Afrac-u}, we obtain
 $$
 \lim_{t\to  0^{+}}t^{\xi_{\beta-\nu}}A^{\beta}u(t)=0.
 $$
For the case $1-\alpha\nu-\vartheta\xi_{\beta-\nu}=0$, consider first that,
 from \eqref{est-F0}, we have
 $$
 \||u|\|_{\beta,T'}\leq\sup_{0<t\leq T'}
 t^{\xi_{\beta-\nu}}\|A^{\beta-\nu}S_{\alpha}(t)A^{\nu}u_{0}\|+C_{4}
 K^{\vartheta-1}\||u|\|_{\beta,T'},
 $$
for any $0<T'\leq T$. Since $C_{5}<1$, then $C_{4}K^{\vartheta-1}<1$. Hence there exists $C_{6}>0$ such that
 $$
 \||u|\|_{\beta,T'}\leq C_{6}\sup_{0<t\leq T'}
 t^{\xi_{\beta-\nu}}\|A^{\beta-\nu}S_{\alpha}(t)A^{\nu}u_{0}\|.
 $$
By taking $T'\to 0$, thus we also have
 $$
 \lim_{t\to  0^{+}}t^{\xi_{\beta-\nu}}A^{\beta}u(t)=0,
 $$
for the case $1-\alpha\nu-\vartheta\xi_{\beta-\nu}=0$. 
We can also conclude that the results above also hold for every 
$\eta\in(\nu+(2-1/\alpha)^{+},\beta)$ since such a $\eta$ satisfies the 
condition \eqref{par-ass}.

\begin{remark}\label{rmk3.1} \rm
 From \eqref{est-bigt-AfracS}, for $T>1$, we have
 $$
 t^{\xi_{\beta-\nu}}\|A^{\beta}S_{\alpha}(t)u_{0}\|\leq2C_{1}'
 (\alpha,\beta-\nu)t^{1-\alpha}\|A^{\nu}u_{0}\|,\quad t\in(0,T].
 $$
Then, it follows that \eqref{est-F1} becomes
\begin{equation}\label{est-F3}
 2C_{1}'(\alpha,\beta-\nu)\|A^{\nu}u_{0}\|T^{1-\alpha}+C_{4}
 K^{\vartheta}T^{1-\alpha\nu-\vartheta\xi_{\beta-\nu}}\leq K.
\end{equation}
 Observe that we can not get $K>0$ satisfying \eqref{est-C5} and \eqref{est-F3} 
for $T$ sufficiently large although $K$ is taken to be sufficiently small. 
Thus the problem \eqref{fracnonlinpr} has no a global mild solution $u$ on 
$(0,\infty)$.
\end{remark}

\begin{remark} \label{rmk3.2} \rm
 If we assume that $f$ is a nonlinear operator in $H$ satisfying
 \begin{itemize}
 \item[(i)] $f(0)=0$,
 \item[(ii)] there exist $C_{0}>0$, $\vartheta>1$, and $0<\beta<1$ such that
 $$
 \|f(u)-f(v)\|\leq C_{0}(1+(\|A^{\beta}u\|+\|A^{\beta}v\|)^{\vartheta-1})
 \|A^{\beta}u-A^{\beta}v\|,
 $$
 for all $u,v\in D(A^{\beta})$,
 \end{itemize}
 then Theorem \ref{thm3.1} remains valid.
\end{remark}

\subsection{Proof of Theorem \ref{thm3.2}}

We verify first the following lemma.

\begin{lemma} \label{lem3.1}
 Let $u\in B_{\beta,T}$ be a mild solution to \eqref{fracnonlinpr}. 
Then, by the condition \eqref{par-ass}, $A^{\beta}u(t)$ is H\"older continuous 
in $[\varepsilon,T]$ for each $\varepsilon>0$.
\end{lemma}

\begin{proof}
First, consider that, by \eqref{semprop},
 \begin{align*}
& A^{\beta}S_{\alpha}(t+h)u_{0}-A^{\beta}S_{\alpha}(t)u_{0}\\
&=A^{\beta}(S_{\alpha}(h)-I)S_{\alpha}(t)u_{0}
-A^{\beta}\int_{0}^{t}
 \int_{0}^{h}\frac{(t+h-\tau-r)^{-\alpha}}{\Gamma(1-\alpha)}
 AP_{\alpha}(\tau)P_{\alpha}(r)u_{0}\,dr\,d\tau
\end{align*}
and
 \begin{align*}
& A^{\beta}\int_{0}^{t+h}P_{\alpha}(t+h-s)f(u(s))ds-A^{\beta}\int_{0}^{t}
 P_{\alpha}(t-s)f(u(s))ds\\
&=A^{\beta}\int_{-h}^{t}P_{\alpha}(t-s)f(u(s+h))ds-A^{\beta}\int_{0}^{t}
 P_{\alpha}(t-s)f(u(s))ds\\
&=A^{\beta}\int_{0}^{t}P_{\alpha}(t-s)(f(u(s+h))-f(u(s)))ds\\
&\quad+A^{\beta}\int_{0}^{h}P_{\alpha}(t+h-s)f(u(s))ds.
\end{align*}
Now, let $\varepsilon\leq t<t+h\leq T$ with $\varepsilon>0$. Observe that
\begin{align*}
&\int_{0}^{h}\frac{(t+h-\tau-r)^{-\alpha}}
 {\Gamma(1-\alpha)}\tau^{-\xi_{1-\delta}}d\tau\\
&=\frac{h^{1-\xi_{1-\delta}
 }(t+h-r)^{-\alpha}}{\Gamma(1-\alpha)(1-\xi_{1-\delta})}
 H\Big(1-\delta,\alpha;2-\delta;\frac{h}{t+h-r}\Big)\\
&\leq\frac{\Gamma(2-\xi_{1-\delta})B(\alpha,1-\alpha)}{
 (1-\xi_{1-\delta})\Gamma(1-\alpha)\Gamma(\alpha)
 \Gamma(2-\xi_{1-\delta}-\alpha)}h^{1-\xi_{1-\delta}
 }(t+h-r)^{-\alpha}
\end{align*}
implying
 \begin{align*}
&\int_{0}^{t}\int_{0}^{h}\frac{(t+h-\tau-r)^{-\alpha}}
 {\Gamma(1-\alpha)}\tau^{-\xi_{1-\delta}}r^{-\xi_{\beta+\delta-\nu}}\,dr\,d\tau\\
&\leq\frac{\Gamma(2-\xi_{1-\delta})B(\alpha,1-\alpha)h^{1-\xi_{1-\delta}}}
 {(1-\xi_{1-\delta})\Gamma(1-\alpha)\Gamma(\alpha)
 \Gamma(2-\xi_{1-\delta}-\alpha)}\int_{0}^{t}(t+h-r)^{-\alpha}
 r^{-\xi_{\beta+\delta-\nu}}dr\\
&=\frac{\Gamma(2-\xi_{1-\delta})B(\alpha,1-\alpha)h^{1-\xi_{1-\delta}}
 (t+h)^{1-\alpha-\xi_{\beta+\delta-\nu}}}{
 (1-\xi_{1-\delta})\Gamma(1-\alpha)
 \Gamma(\alpha)\Gamma(2-\xi_{1-\delta}-\alpha)}\int_{0}^{\frac{t}{t+h}}
 (1-s)^{-\alpha}s^{-\xi_{\beta+\delta-\nu}}ds\\
&\leq C_{7}h^{1-\xi_{1-\delta}}(t+h)^{1-\alpha-\xi_{\beta+\delta-\nu}}
\end{align*}
where
$$
 C_{7}=\frac{\Gamma(2-\xi_{1-\delta})B(\alpha,1-\alpha)
 B(1-\xi_{\beta+\delta-\nu},1-\alpha)}{(1-\xi_{1-\delta})\Gamma(1-\alpha)
 \Gamma(\alpha)\Gamma(2-\xi_{1-\delta}-\alpha)}.
$$
Then, for every $0<\delta<1-\beta$,
 \begin{align*}
&\|A^{\beta}S_{\alpha}(t+h)u_{0}-A^{\beta}S_{\alpha}(t)u_{0}\|\\
&\leq  \|(S_{\alpha}(h)-I)A^{\beta}S_{\alpha}(t)u_{0}\|\\
&\quad+\int_{0}^{t}\int_{0}^{h}\frac{(t+h-\tau-r)^{-\alpha}}{\Gamma(1-\alpha)}
 \|A^{1-\delta}P_{\alpha}(\tau)A^{\beta+\delta-\nu}P_{\alpha}(r)A^{\nu}u_{0}\|
 \,dr\,d\tau\\
 &\leq C_{3}'(\alpha,\delta)h^{\alpha\delta}
 \|A^{\beta+\delta-\nu}S_{\alpha}(t)A^{\nu}u_{0}\|+C_{2}'(\alpha,1-\delta)
 C_{2}'(\alpha,\beta+\delta-\nu)\\
 &\quad\times \int_{0}^{t}\int_{0}^{h}\frac{(t+h-\tau-r)^{-\alpha}}
 {\Gamma(1-\alpha)}\tau^{-\xi_{1-\delta}}r^{-\xi_{\beta+\delta-\nu}}\,dr\,d\tau\|
 A^{\nu}u_{0}\|\\
 &\leq C_{1}'(\alpha,\beta+\delta-\nu)
 C_{3}'(\alpha,\delta)h^{1-\xi_{1-\delta}}
 t^{-\alpha}(t^{1-\xi_{\beta+\delta-\nu}}+1)\|A^{\nu}u_{0}\|\\
 &\quad +C_{2}'(\alpha,1-\delta)
 C_{2}'(\alpha,\beta+\delta-\nu)C_{7}h^{1-\xi_{1-\delta}}t^{1-\alpha
 -\xi_{\beta+\delta-\nu}}\|A^{\nu}u_{0}\|\\
 &\leq C_{1}'(\alpha,\beta+\delta-\nu)
 C_{3}'(\alpha,\delta)h^{1-\xi_{1-\delta}}
 t^{-\alpha}(t^{1-\xi_{\beta+\delta-\nu}}+1)\|A^{\nu}u_{0}\|\\
 &\quad+C_{8}h^{1-\xi_{1-\delta}}
 t^{-\alpha}(t^{1-\xi_{\beta+\delta-\nu}}+1)\|A^{\nu}u_{0}\|\\
 &\leq C_{9}h^{1-\xi_{1-\delta}}
 t^{-\alpha}(t^{1-\xi_{\beta+\delta-\nu}}+1)\|A^{\nu}u_{0}\|,
\end{align*}
for some constants $C_{8},C_{9}>0$. Next, note that
 \begin{align*}
&\int_{0}^{t}\|A^{\beta}P_{\alpha}(t-s)(f(u(s+h))-f(u(s)))\|ds\\
&\leq2^{\vartheta-1}C_{0}C_{2}'(\alpha,\beta)
 K^{\vartheta-1}\int_{0}^{t}(t-s)^{-\xi_{\beta}}
 s^{-(\vartheta-1)\xi_{\beta-\nu}}\|A^{\beta}u(s+h)-A^{\beta}u(s)\|ds
\end{align*}
and
\begin{align*}
&\int_{0}^{h}\|A^{\beta}P_{\alpha}(t+h-s)f(u(s))\|ds\\
&\leq C_{0}C_{2}'(\alpha,\beta)K^{\vartheta}\int_{0}^{h}(t+h-s)^{-\xi_{\beta}}
 s^{-\vartheta\xi_{\beta-\nu}}ds\\
&\leq C_{10}(t+h)^{1-\xi_{\beta}-\vartheta\xi_{\beta-\nu}}
 \int_{0}^{\frac{h}{t+h}}(1-r)^{-\xi_{\beta}}r^{-\vartheta\xi_{\beta-\nu}}dr\\
&\leq \frac{C_{10}}{1-\vartheta\xi_{\beta-\nu}}(t+h)^{1-\xi_{\beta}
 -\vartheta\xi_{\beta-\nu}}
 \Big(\frac{h}{t+h}\Big)^{1-\vartheta\xi_{\beta-\nu}}
 H\Big(1-\vartheta\xi_{\beta-\nu},\xi_{\beta};
 2-\vartheta\xi_{\beta-\nu};\frac{h}{t+h}\Big)\\
&\leq C_{11}h^{1-\vartheta\xi_{\beta-\nu}}
 t^{-\xi_{\beta}},
 \end{align*}
for some constants $C_{10}$, $C_{11}>0$. Thus we obtain
 \begin{align*}
&\|A^{\beta}u(t+h)-A^{\beta}u(t)\|\\
&\leq C_{9}h^{1-\xi_{1-\delta}}
 t^{-\alpha}(t^{1-\xi_{\beta+\delta-\nu}}+1)\|A^{\nu}u_{0}\|
 +C_{11}h^{1-\vartheta\xi_{\beta-\nu}} t^{-\xi_{\beta}}\\
&\quad+2C_{0}C_{2}
 K^{\vartheta-1}\int_{0}^{t}(t-s)^{-\xi_{\beta}}
 s^{-(\vartheta-1)\xi_{\beta-\nu}}\|A^{\beta}u(s+h)-A^{\beta}u(s)\|ds
\end{align*}
By the Gronwall's inequality, it implies that $A^{\beta}u(t)$ 
is H\"older continuous on $[\varepsilon,T]$ for any $\varepsilon>0$.
\end{proof}

Next, by the Lemma \ref{lem3.1}, $f(u(t))$ is also H\"older continuous on 
$[\varepsilon,T]$ for any $\varepsilon>0$; that is,
 \begin{align*}
\|f(u(t+h))-f(u(t))\|
&\leq C_{12}\big\{h^{1-\xi_{1-\delta}}t^{-\alpha-(\vartheta-1)\xi_{\beta-\nu}}(t^{1-
 \xi_{\beta+\delta-\nu}}+1)\|A^{\nu}u_{0}\|\\
&\quad+h^{1-\vartheta\xi_{\beta-\nu}}t^{-\xi_{\beta}
 -(\vartheta-1)\xi_{\beta-\nu}}\big\},
\end{align*}
for some constant $C_{12}>0$. Note that the assumption \eqref{par-ass} 
assures that $0<1-\vartheta\xi_{\beta-\nu}$ and, for each $0<\delta<1-\beta$, 
it holds that $0<1-\xi_{1-\delta}$. Furthermore, consider that, for $t\in(0,T]$, 
we have
 $$
 t^{\xi_{1-\nu}}\|AS_{\alpha}(t)u_{0}\|\leq C_{1}'(\alpha,1-\nu)t^{\xi_{1-\nu}
-\alpha}(t^{1-\xi_{1-\nu}}+1)\|A^{\nu}u_{0}\|
 $$
with
 $$
 \xi_{1-\nu}-\alpha>0,\quad \xi_{1-\nu}-\alpha+1-\xi_{1-\nu}=1-\alpha>0.
 $$
It follows, for $T$ sufficiently small, that
 $$
 t^{\xi_{1-\nu}}\|AS_{\alpha}(t)u_{0}\|\leq 2C_{1}'(\alpha,1-\nu)\|A^{\nu}u_{0}\|.
 $$
Now, observe that
 \begin{align*}
& A\int_{0}^{t}P_{\alpha}(t-s)f(u(s))ds\\
&=\int_{0}^{t/2}AP_{\alpha}(t-s)f(u(s))ds\\
&\quad +\int_{t/2}^{t}AP_{\alpha}(t-s)(f(u(s))-f(u(t)))ds
 +(S_{\alpha}(t/2)-I)f(u(t))\\
&=I_{1}+I_{2}+I_{3}.
\end{align*}
Next, we note that
\[
 t^{\xi_{1-\nu}}\|(S_{\alpha}(t/2)-I)f(u(t))\|
 \leq C_{13}t^{\xi_{1-\nu}}\|f(u(t))\|\\
 \leq C_{0}C_{13}K^{\vartheta}t^{\xi_{1-\nu}
 -\vartheta\xi_{\beta-\nu}},
\]
for some constant $C_{13}>0$, and
 $$
 \xi_{1-\nu}-\vartheta\xi_{\beta-\nu}=1-\alpha\nu
 -\vartheta\xi_{\beta-\nu}.
 $$
Therefore, for $t\in(0,T]$ with $T>0$ sufficiently small, we have
 $$
 t^{\xi_{1-\nu}}\|(S_{\alpha}(t/2)-I)f(u(t))\|
\leq 2C_{0}C_{14}K^{\vartheta}t^{1-\alpha\nu-\vartheta\xi_{\beta-\nu}},
 $$
for some constant $C_{14}>0$. Hence, by \eqref{est-K}, we obtain
 $$
 t^{\xi_{1-\nu}}\|I_{3}\|\leq C_{15}\|A^{\nu}u_{0}\|,
 $$
for some constant $C_{15}>0$. Furthermore,
 \begin{align*}
t^{\xi_{1-\nu}}\|I_{1}\|
&\leq L_{2}(\alpha)C_{0}K^{\vartheta} t^{\xi_{1-\nu}}\int_{0}^{t/2}(t-s)^{-1}
 s^{-\vartheta\xi_{\beta-\nu}}ds\\
&\leq C_{16}t^{\xi_{1-\nu}-\vartheta\xi_{\beta-\nu}}
\leq C_{16}t^{1-\alpha\nu-\vartheta\xi_{\beta-\nu}},
\end{align*}
for some constant $C_{16}>0$. Thus, for $T$ sufficiently small, we find that
 $$
 t^{\xi_{1-\nu}}\|I_{1}\|\leq C_{17}\|A^{\nu}u_{0}\|,
 $$
for some constant $C_{17}>0$. Now, consider 
 \begin{align*}
&\|AP_{\alpha}(t-s)(f(u(s))-f(u(t)))\|\\
&\leq L_{2}(\alpha)C_{12}\Big\{ (t-s)^{-\xi_{1-\delta}}s^{-\alpha-(\vartheta-1)
 \xi_{\beta-\nu}}(s^{1-\xi_{\beta+\delta-\nu}}+1)\|A^{\nu}u_{0}\| \\*
&\quad+(t-s)^{-\vartheta\xi_{\beta-\nu}}
 s^{-\xi_{\beta}-(\vartheta-1)\xi_{\beta-\nu}}\Big\}.
\end{align*}
Therefore,
\begin{align*}
&\int_{t/2}^{t}\|AP_{\alpha}(t-s)(f(u(s))-f(u(t)))\|ds\\
 &\leq C_{18}\big\{ t^{1-\xi_{1-\delta}-\alpha-(\vartheta-1)
 \xi_{\beta-\nu}}(t^{1-\xi_{\beta+\delta-\nu}}+1)\|A^{\nu}u_{0}\|
 +t^{1-\vartheta\xi_{\beta-\nu}
 -\xi_{\beta}-(\vartheta-1)\xi_{\beta-\nu}}\big\},
\end{align*}
for some constant $C_{18}>0$. Furthermore, by using the assumption 
\eqref{par-ass},
 \begin{gather*}
 \xi_{1-\nu}+1-\xi_{1-\delta}-\alpha-(\vartheta-1)
 \xi_{\beta-\nu}>1-\alpha\nu-\vartheta\xi_{\beta-\nu}\geq0, 
\\
 1-\vartheta\xi_{\beta-\nu}-\xi_{\beta}-(\vartheta-1)\xi_{\beta-\nu}
 =2(1-\alpha\nu-\vartheta\xi_{\beta-\nu})\geq0.
\end{gather*}
Note also that $1-\xi_{\beta+\delta-\nu}>0$. Then, for $T$ sufficiently small,
 $$
 t^{\xi_{1-\nu}}\|I_{2}\|\leq C_{19}\|A^{\nu}u_{0}\|,
 $$
for some constant $C_{19}>0$. Thus we conclude that
 $$
 \|Au(t)\|\leq C_{20}t^{-\xi_{1-\nu}}\|A^{\nu}u_{0}\|,\quad t\in(0,T],
 $$
for some constant $C_{20}>0$.

\section{Applications}

We consider the parabolic initial-value problem
 \begin{equation}\label{fractparivp}
\begin{gathered}
 D_{t}^{\alpha}u=\Delta u+|u|^{p-1}u,\quad\text{in}\;\Omega\times(0,T)\\
 u|_{\partial\Omega}=0,\\
 u(0)=u_{0},\quad\text{in}\;\Omega
\end{gathered}
 \end{equation}
where $\Omega\in\mathbb{R}^{N}$ with $C^{2}$ boundary and $p>1$. 
The abstract formulation of the problem \eqref{fractparivp} is
 \begin{equation}
 \begin{gathered}
 D_{t}^{\alpha}u=Au+f(u),\quad\text{in}\;\Omega\times(0,T)\\
 u(0)=u_{0},\quad\text{in}\;\Omega,
 \end{gathered}
 \end{equation}
where
 $$
 A=\Delta,\quad f(u)=|u|^{p-1}u.
 $$
Here, we set $H=L^{2}(\Omega)$ and 
$D(A)=H^{2}_{D}=\{u\in H^{2}(\Omega):u=0\;\text{on}\;\partial\Omega\}$. 
Note that $A$ is sectorial in $H$.

Next, for $\beta\geq N(1-1/p)/4$ and $p>1$, we have
 $$
 \|u\|_{2p}\leq C\|A^{\beta}u\|_{2},\quad u\in D(A^{\beta})
 $$
(see \cite{Henry81} for more details). By the mean value theorem 
and the H\"older inequality, for $u,v\in D(A^{\beta})$, one can obtain that
 $$
 \|f(u)-f(v)\|_{2}^{2}\leq p^{2}(\|u\|_{(p-1)q}
+\|v\|_{(p-1)q})^{2(p-1)}\|u-v\|_{r}^{2}
 $$
where $2/p+2/r=1$. It implies
 $$
 \|f(u)-f(v)\|_{2}\leq p(\|u\|_{2p}+\|v\|_{2p})^{p-1}\|u-v\|_{2p}
 $$
by taking $r=2p$ such that $(p-1)q=2p$. Thus we get
 $$
 \|f(u)-f(v)\|_{2}\leq p(\|A^{\beta}u\|_{2}
+\|A^{\beta}v\|_{2})^{p-1}\|A^{\beta}u-A^{\beta}v\|_{2}
 $$
for $u,v\in D(A^{\beta})$. We find that
 $$
 D(A^{\beta})=H^{2\beta}_{D},\quad H^{2\beta}_{D}=\{u\in H^{2\beta}(\Omega):u|_{\partial\Omega}=0\},\quad1/4<\beta<1
 $$
(see \cite{Yagi10} for more details). Thus, for
 \begin{gather*}
 \frac{1}{4}<\beta<1,
 \quad\text{if }N\big(1-\frac{1}{p}\big)\leq1,\\
 \frac{N}{4}\big(1-\frac{1}{p}\big)\leq\beta<1,
 \quad\text{if } 1<N\big(1-\frac{1}{p}\big)<4,
 \end{gather*}
and $u_{0}\in D(A^{\nu})$ with
 \begin{gather*}
 \frac{p\xi_{\beta}-1}{\alpha(p-1)}\leq\nu<\beta-(2-\frac{1}{\alpha})^{+},
 \quad\text{if } p\xi_{\beta}>1,\\
 0<\nu<\beta-(2-\frac{1}{\alpha})^{+},
 \quad\text{if }p\xi_{\beta}\leq1,
 \end{gather*}
by Theorem \ref{thm3.1},  problem \eqref{fractparivp} has a unique mild solution $u$ 
satisfying
\begin{gather*}
 t^{\xi_{\eta-\nu}}u\in BC((0,T];D(A^{\eta})),\quad
\lim_{t\to  0^{+}}t^{\xi_{\eta-\nu}}A^{\eta}u(t)=0, \\
 \|A^{\eta}u(t)\|_{H}\leq Ct^{-\xi_{\eta-\nu}}\|A^{\nu}u_{0}\|_{H},\quad t\in(0,T]
\end{gather*}
for every $\eta\in(\nu+(2-1/\alpha)^{+},\beta]$ with $T$ sufficiently small.

\subsection*{Acknowledgments}
The first author would like to thank the Directorate General of Higher Education, 
the Ministry of Research, Technology, and Higher Education of the Republic of 
Indonesia for their support.

\begin{thebibliography}{99}

\bibitem{Adams92} E. E. Adams, L. W. Gelhar;
 \emph{Field Study of Dispersion in a Heterogeneous Aquifer 2. Spatial 
Moments Analysis}, Water Resources Research, Vol. 28 No. 12 (1992), 3293-3307.

\bibitem{Barkai00} E. Barkai, R. Metzler, J. Klafter;
 \emph{From Continuous Random Walks to the Fractional Fokker-Planck Equation}, 
Physical Review E, Vol. 61(1) (2000).

\bibitem{Berkowitz06} B. Berkowitz, A. Cortis, M. Dentz, H. Scher;
\emph{Modeling Non-Fickian Transport In Geological Formations as a Continuous
 Time Random Walk}, Reviews of Geophysics, 44 (2006), RG 2003, 1-49.

\bibitem{Chen13} P. Chen, Y. Li;
\emph{Existence of Mild Solutions for Fractional Evolution Equations with Mixed 
Monotone Nonlocal Conditions}, Z. Angew. Math. Phys., Springer Basel, 2013.

\bibitem{Du13} M. Du, Z. Wang,  H. Hu;
\emph{Measuring Memory with The Order of Fractional Derivative},
Scientific Reports 3 (2013), Article Number: 3431.

\bibitem{Fan12} H. Fan, J. Mu;
\emph{Initial Value Problem for Fractional Evolutions Equations}, 
Advance in Difference Equations, 49 (2012).

\bibitem{Guswanto} B. H. Guswanto;
\emph{On the Properties of Solution Operators of Fractional Evolution Equations}, 
Journal of Fractional Calculus and Applications, Vol. 6(1) (2015), 131-159.

\bibitem{Hatano98} Y. Hatano, N. Hatano;
\emph{Dispersive transport of ions in column experiments: 
An explanation of long-tailed profiles}, Water Resources Research,
 Vol. 34 and No. 5 (1998), 1027-1033.

\bibitem{Henry10} B. I. Henry, T. A. M. Langlands, P. Straska;
 \emph{Fractional Fokker-Planck Equations for Subdiffusion with Space-and 
Time-Dependent Forces}, Physical Review Letters 105 (2010), 170602.

\bibitem{Henry06} B. I. Henry, T. A. M. Langlands, S. L. Wearne;
\emph{Anomalous Diffusion with Linear Reaction Dynamics: 
From Continuous Time Random Walks to Fractional Reaction-Diffusion Equations},
 Physical Review E 74 (2006), 031116.

\bibitem{Henry00} B. I. Henry, S. L. Wearne;
\emph{Fractional Reaction Diffusion}, Physical Review A 276 (2000), 448-455.

\bibitem{Henry81} D. Henry; 
\emph{Geometric Theory of Semilinear Parabolic Equations}, 
Lecture Notes in Math., Vol. 840, Springer Verlag, Berlin, 1981.

\bibitem{Hilfer00} R. Hilfer;
\emph{Applications of Fractional Calculus in Physics}, World Scientific Publishing 
Co. Pte. Ltd., Singapore, 2000.

\bibitem{Hoshino91} H. Hoshino, Y. Yamada;
\emph{Solvability and Smoothing Effect for Semilinear Parabolic Equations}, 
Funkcialaj Exvacioj, 34 (1991), 475-494.

\bibitem{Kilbas06} A. A. Kilbas, H. M. Srivastava, J. J. Trujilo;
\emph{Theory and Application of Fractional Differential Equations}, 
North Holland Mathematics Studies, Elsevier, 2006.

\bibitem{Laffaldano05} G. Laffaldano, M. Caputo,  S. Martino;
\emph{Experimental and Theoretical Memory Diffusion of Water in Sand},
 Hydrol. Earth Sys. Sci. Discuss., 2 (2005), 1329-1357.

\bibitem{Langlands10} T. A. M. Langlands, B. I. Henry,  S. L. Wearne;
 \emph{Anomalous Diffusion with Multispecies Linear Reaction Dynamics}, 
Physical Review E 77 (2008), 021111.

\bibitem{Langlands08} T. A. M. Langlands, B. I. Henry;
\emph{Fractional Chemotaxis Diffusion Equations}, Physical Review E 81 (2010), 
051102.

\bibitem{Li09} F.-B. Li, M. Li,  Q. Zheng;
\emph{Fractional Evolution Equations Governed by Coercive Differential Operators},
 Abstract and Applied Analysis, Vol. 2009, Hindawi Publishing Corporation, 2009.

\bibitem{Li13} K. Li, J. Peng,  J. Gao;
 \emph{Nonlocal Fractional Semilinear Differential Equations in Separable 
Banach Space}, Electronic Journal of Differential Equations, 07 (2013), 1-7.

\bibitem{Mainardi10} F. Mainardi;
\emph{Fractional Calculus and Waves in Linear Viscoelasticity}, 
Imperial College Press, London, 2010.

\bibitem{Metzler99} R. Metzler, E. Barkai, J. Klafter;
\emph{Deriving Fractional Fokker-Planck Equations from a Generalised Master Equation},
 Europhys. Lett., 46 (4) (1999), pp. 431-436.

\bibitem{Pazy83} A. Pazy;
\emph{Semigroup of Linear Operators and Applications to Partial Differential 
Equations}, Springer Verlag, New York, 1983.

\bibitem{Peng12} J. Peng, K. Li;
\emph{A Novel Characteristic of Solution Operator for the Fractional Abstract 
Cauchy Problem}, J. Math. Anal. Appl. 385 (2012), 786-796.

\bibitem{Podlubny99} I. Podlubny;
 \emph{Fractional Differential Equations}, Academic Press 198, 1999.

\bibitem{Raheem13} A. Raheem, D. Bahuguna;
 \emph{Existence and Uniqueness of a Solution for a Fractional Differential 
Equations by Rothe's Methode}, Journal of Nonlinear Evolution Equations 
and Applications, 4 (2013), 43-54.

\bibitem{Seki03} K. Seki, M.Wojcik, M. Tachiya;
\emph{Fractional reaction-diffusion equation}, J. Chem. Phys. 119 (2003), 2165.

\bibitem{Sung02} J. Sung, E. Barkai, R. J. Silbey, S. Lee;
 \emph{Fractional dynamics approach to diffusion-assisted reactions in disordered 
 media}, J. Chem. Phys. 116 (2002), 2338.

\bibitem{Wang12} R. N. Wang, D. H. Chen, T. J. Xiao;
 \emph{Abstract Fractional Cauchy Problems with Almost Sectorial Operators}, 
 J. Differential Equations 252 (2012), 202-235.

\bibitem{Yagi10} A. Yagi;
 \emph{Abstract Parabolic Evolution Equations and their Applications}, 
 Springer Verlag, Berlin, 2010.

\bibitem{Zhang12} Z. Zhang, B. Liu;
 \emph{Existence of Mild Solutions for Fractional Evolution Equations}, 
  Journal Of Fractional Calculus and Applications, Vol. 2 No. 10 (2012), 1-10.

\bibitem{Zhang14} Z. Zhang, Q. Ning, H. Wang;
  \emph{Mild Solution of Fractional Evolution Equations on an Unbounded Interval}, 
 Advance in Difference Equations, 27 (2014).

\bibitem{Zhou13} Y. Zhou, X. H. Shen,  L. Zhang;
  \emph{Cauchy Problem for Fractional Evolution Equations with Caputo Derivative}, 
Eur. Phys. J. Special Topics 222 (2013), 1749-1765.

\end{thebibliography}

\end{document}