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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 293, pp. 1--24.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/293\hfil Generalized uniformly continuous solution operators]
{Generalized uniformly continuous solution operators and inhomogeneous
fractional evolution equations with variable coefficients}

\author[M. Japund\v{z}i\'c, D. Rajter-\'Ciri\'c \hfil EJDE-2017/293\hfilneg]
{Milo\v{s} Japund\v{z}i\'c, Danijela Rajter-\'Ciri\'c}

\address{Milo\v{s} Japund\v{z}i\'c \newline
Novi Sad Business School
- Higher Education Institution for Applied Studies,
Vladimira Peri\'ca-Valtera 4, 21000 Novi Sad, Serbia}
\email{milos.japundzic@gmail.com}

\address{Danijela Rajter-\'Ciri\'c \newline
Department of Mathematics and Informatics,
Faculty of Science, University of Novi Sad,
Trg Dositeja Obradovi\'ca 4, 21000 Novi Sad, Serbia}
\email{rajter@dmi.uns.ac.rs}

\dedicatory{Communicated by Mokhtar Kirane}

\thanks{Submitted October 21, 2016. Published November 27, 2017.}
\subjclass[2010]{35R11, 46F30, 26A33}
\keywords{Fractional evolution equation; fractional Duhamel principle;
\hfill\break\indent generalized Colombeau solution operator;
  fractional derivative;  Mittag-Leffler type function}

\begin{abstract}
 We consider Cauchy problem for inhomogeneous fractional evolution
 equations with Caputo fractional derivatives of order $0<\alpha<1$
 and variable coefficients depending on $x$. In order to solve this
 problem we introduce generalized uniformly continuous solution operators
 and use them to obtain the unique solution on a certain Colombeau space.
 In our solving procedure, instead of the original problem we solve a
 certain approximate problem, but therefore we also prove that the
 solutions of these two problems are associated. At the end, we illustrate
 the applications of the developed theory by giving some appropriate examples.
\end{abstract}

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

\section{Introduction}

Fractional evolution equations have been studied very often in the previous
decades because of their numerous applications. Many well known problems are,
in fact special cases of fractional evolution equations. For instance,
 both time fractional diffusion problem and time fractional
reaction-advection-diffusion problem are of that type. In literature,
authors have mainly considered several cases: homogeneous case, case when $f$
is linear or case with constant coefficients. In this paper, we want
to study semilinear problem which also includes space variable coefficients,
i.e. the equation of the type
\begin{equation}\label{uvodna-1a}
 ^C\mathcal{D}_t^{\alpha}u(t)=Au(t)+f(\cdot,t,u),
 \hspace{5mm} u(0)=u_{0},
\end{equation}
where $^C\mathcal{D}_t^{\alpha}$ is the Caputo's fractional derivative of
 order $0<\alpha<1$ and $A$ is a linear, closed operator densely defined on
some Banach space.

Semilinear fractional Cauchy problem with variable coefficients in general
case has been solved approximately, usually applying different numerical methods.
 One of the reasons why we have considered fractional equations in the
framework of the Colombeau theory of generalized functions is the intention
that these equations be treated using operator's approach, that is, applying
the solution operators as generalization of semigroup of operators.

In our solving procedure, instead of the original problem \eqref{uvodna-1a}
we consider the approximate problem
\begin{equation}\label{uvodna-1}
 ^C\mathcal{D}_t^{\alpha}u(t)=\widetilde{A}u(t)+f(\cdot,t,u),
 \hspace{5mm} u(0)=u_{0},
\end{equation}
where $\widetilde{A}$ is a generalized linear bounded operator associated
(in certain sense) to the original operator $A$. Therefore we will pay
special attention to comparison analysis of these two problems.
To solve the approximate problem, we introduce a notion of generalized uniformly
continuous solution operator generated by $\widetilde A$.
This generalized solutions operator is, in fact, a generalization of
generalized uniformly continuous semigroup of operators. (For $\alpha =1$
a generalized solution operator is defined as a generalized semigroup
of operators). Generalized uniformly continuous semigroups were introduced
in \cite{Rajter} and the theory has been developed
later in \cite{Nedeljkov} in order to use the theory of semigroups in
solving some partial differential equations with singularities in some
generalized function spaces.

Solution operators as a generalization of $C_{0}$ semigroups and cosine
families of operators are introduced by Bazhlekova in \cite{Bazhlekova}.
 In \cite{Bazhlekova1} and \cite{Bazhlekova} the corresponding solution
operator theory was developed for solving some homogeneous fractional
evolution problems. We remark that in some literature the solution
operator is also called fractional resolvent family or fractional
resolvent operator function (see e.g. \cite{Chen,Miao}).

In this article, we solve problem \eqref{uvodna-1} in the framework of
the Colombeau theory. The theory of Colombeau generalized functions is
developed in order to make possible studying some nonlinear
differential equations that can not be treated
neither classically (there is no classical solution) nor in distributional
sense (nonlinear problems include the
multiplication and the multiplication of
distribution is not well defined). For the Colombeau
theory in general we refer, for example, to
\cite{Biagoni,Colombeau,Nedeljkovb,Oberguggenberger}.

In \cite{Japundzic} we considered a special case of \eqref{uvodna-1} for
$\alpha =1$ and with Colombeau generalized operator $\widetilde A$ defined
by space fractional derivatives. In this paper we make a step further
 by considering the problem with fractional time derivative of order
$0<\alpha<1$. We obtain the unique solution to the problem \eqref{uvodna-1}
in a certain Colombeau space. In case when $A$ is a differential operator
(integer of fractional order) the regularization is necessary in order
to obtain bounded operators. Our method admits variable coefficients
in both $\tilde A$ and $A$.

This article is organized as follows. Fractional derivatives and some useful
estimates involving them are investigated in Section 2.
A part of this section is devoted to the Mittag-Leffler function since
it has an important role in defining the solution operator. Colombeau spaces
that we use later in the paper are defined in Section 3.
In Section 4 we define uniformly continuous solution operators and prove
some basic properties. In Section 5 we introduce the Colombeau uniformly
continuous solution operators and develop the corresponding theory.
After setting the framework theory, in Section 6 we investigate the
inhomogeneous problem \eqref{uvodna-1}. We prove that the problem has
a unique solution in a certain Colombeau space. Since in the whole paper,
 instead of the original problem \eqref{uvodna-1a} we study the
corresponding approximate problem \eqref{uvodna-1}, Section 7 is devoted to
a comparison analysis of these two problems. Finally, in the last section
we illustrate how one can use our theory in solving some fractional evolution
problems appearing in applications, such as time and time-space fractional
diffusion equation and also time-space fractional reaction-advection-diffusion
equation. In these problems the corresponding differential operators will be
in the form of regularized operators, in order to transform unbounded
differential operators into (integral) bounded operators.

\section{Time fractional derivatives and some useful estimates}

In this section we recall definitions of fractional derivatives with respect
to time variable and give some useful estimates for fractional derivatives
 and Mittag-Leffler function that we will use later.

\subsection{Fractional derivatives with respect to time variable}
The Caputo fractional derivative of order $\alpha$, $m-1<\alpha\leq m$,
$m\in\mathbb{N}$, has the form (see, for example,
\cite{Atanackovic,Atanackovic1,Kilbas,Podlubny,Samko})
\begin{equation}\label{24}
 ^C\mathcal{D}_t^{\alpha}f(t)=J_t^{m-\alpha}f^{(m)}(t),
\end{equation}
where $J_t^{\alpha}$, $\alpha\geq0$, is a fractional integral
for function $f(t)$ given by
\begin{equation*}
 J_t^{\alpha}f(t)=\frac{1}{\Gamma(\alpha)}\int_{0}^{t}(t-\tau)^{\alpha-1}
f(\tau)d\tau,
\end{equation*}
with $J^{0}=I$, $I$ is identity operator.

The Riemann-Liouville fractional derivative of order $\alpha$,
 $m-1<\alpha\leq m$, $m\in\mathbb{N}$, is given by
\begin{equation}\label{17}
 ^{RL}\mathcal{D}_t^{\alpha}f(t)=\frac{d^{m}}{dt^{m}}J_t^{m-\alpha}f(t).
\end{equation}

Recall that $^C\mathcal{D}_t^{\alpha}$ is a left inverse of
$J_t^{\alpha}$, i.e. $^C\mathcal{D}_t^{\alpha}J_t^{\alpha}f=f$
for a continuous function $f$, but in general it is not a right
inverse \cite[Theorems 3.7, 3.8]{Diethelm}. In general, for an absolutely
continuous function $f$ and $0<\alpha<1$, the following holds
$J_t^{\alpha C}\mathcal{D}_t^{\alpha}f(t)=f(t)-f(0)$.

The following holds for a Riemann-Liouville fractional derivative:

\begin{proposition}[{\cite[Lemma 2.1]{Umarov}}] \label{28}  
For all $\alpha\in (m-1,m]$ and $\beta\geq0$, it holds
\begin{equation}\label{25}
 J_t^{\beta+\alpha}f(t)=J_t^{\beta+m \;RL}\mathcal{D}_t^{m-\alpha}f(t).
\end{equation}
\end{proposition}




\begin{remark} \label{rmk2.1} \rm
Taking into account the properties of Mittag-Leffler function and its integer
 order derivatives (especially at zero), the Colombeau space from which we
choose generalized solution operators will be defined using the space
$C^{m-1}([0,\infty):\mathcal{L}(E))\cap C^{m}((0,\infty):\mathcal{L}(E))$
and supposing some additional properties (see Definition \ref{Colombeau}
and Definition \ref{Colombeau1}). It is the space of continuously
differentiable functions with respect to $t$ and with values in space
$\mathcal{L}(E)$, where $(E, \|\cdot\|)$ is a Banach space and
$\mathcal{L}(E)$ is the space of all linear continuous mappings from $E$
into $E$ with the norm
$$
\|A\|_{\mathcal{L}(E)}=\sup_{x\in E, \;x\neq 0}\frac{\|Ax\|_{E}}{\|x\|_{E}}.
$$
\end{remark}

The following lemma will play an important role in later proofs.

\begin{lemma} \label{lem2.1}
Let $(E, \|\cdot\|)$ be a Banach space and $\mathcal{L}(E)$ the space of all linear
continuous mappings from $E$ into $E$. Let $m-1<\alpha<m$, $m\in \mathbb{N}$.
Suppose that $(\cdot,t)\to f(\cdot,t)\in C^{m-1}([0,\infty):\mathcal{L}(E))
\cap C^{m}((0,\infty):\mathcal{L}(E))$ is such that
$\lim_{t\to0^{+}} 
\big\|\frac{f^{(m)}(\cdot,t)} {t^{\alpha-m}} \big\|_{\mathcal{L}(E)}
=C<+\infty$.
 Then
\begin{equation}\label{eta}
 ^C\mathcal{D}_t^{\alpha}f(\cdot,t)
=\lim_{\eta\to0^{+}}{^C_{\eta}\mathcal{D}}_t^{\alpha}f(\cdot,t) \quad
\text{in } \mathcal{L}(E),
\end{equation}
where
\begin{equation}
\label{etafrac}
{^C_{\eta}\mathcal{D}}_t^{\alpha}f(\cdot,t)
=\frac{1}{\Gamma(m-\alpha)}\int_{\eta}^{t}\frac{f^{(m)}(\cdot,\tau)}
{(t-\tau)^{\alpha-m+1}}d\tau.
\end{equation}
\end{lemma}

\begin{proof}
Fix $m\in \mathbb{N}$ and $\alpha$ such that $m-1<\alpha<m$. Then
\begin{align*}
&\|^C\mathcal{D}_t^{\alpha}f(\cdot,t)
-{^C_{\eta}\mathcal{D}}_t^{\alpha}f(\cdot,t)\|_{\mathcal{L}(E)} \\
&\leq \frac{1}{\Gamma(m-\alpha)}\int_{0}^{\eta}
 \frac{\|f^{(m)}(\cdot,\tau)\|_{\mathcal{L}(E)}}{(t-\tau)^{\alpha-m+1}}d\tau \\
&= \frac{1}{\Gamma(m-\alpha)}\int_{0}^{\eta}
 \frac{\|f^{(m)}(\cdot,\tau)\|_{\mathcal{L}(E)}}{\tau^{\alpha-m}}
 \frac{\tau^{\alpha-m}}{(t-\tau)^{\alpha-m+1}}d\tau \\
&\leq \frac{1}{\Gamma(m-\alpha)} \sup_{\tau\in [0,\eta]}
 \|\frac{f^{(m)}(\cdot,\tau)}{\tau^{\alpha-m}}\|_{\mathcal{L}(E)}\int_{0}^{\eta}
 \frac{\tau^{\alpha-m}}{(t-\tau)^{\alpha-m+1}}d\tau \\
 &\leq \frac{1}{\Gamma(m-\alpha)} \sup_{\tau\in [0,\eta]}
\|\frac{f^{(m)}(\cdot,\tau)}{\tau^{\alpha-m}}
\|_{\mathcal{L}(E)}\frac{\eta^{\alpha-m+1}}{(\alpha-m+1)(t-\eta)^{\alpha-m+1}}.
\end{align*}
Letting $\eta\to0^{+}$ one easily gets \eqref{eta}.
\end{proof}


The following assertion is so-called fractional mean value theorem.

\begin{theorem}[\cite{Odi}] \label{meanvalue}
Let $0< \alpha <1$. For $t\to f(t)\in C[a,b]$ and 
${^C_{a}\mathcal{D}}_t^{\alpha}f\in C(a,b]$, the following holds
$$
f(t)=f(a)+\frac{1}{\Gamma(1+\alpha)}({^C_{a}\mathcal{D}}_t^{\alpha}f)
(\xi)(t-a)^{\alpha}, \quad a \leq\xi\leq t, \; t\in (a,b],
$$
where ${^C_{a}\mathcal{D}}_t^{\alpha}f$ is defined as in \eqref{etafrac}.
\end{theorem}

\subsection{Mittag-Leffler function}
The two-parameter Mittag-Leffler function $E_{\alpha,\beta}$ is given by
 $$
E_{\alpha,\beta}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{\Gamma(\beta+n\alpha)},\quad
 z\in \mathbb{C}, \; \alpha>0,\; \beta\in \mathbb{C}.
$$ 
When $\beta=1$ we shortly write $E_{\alpha,1}(z)\equiv E_{\alpha}(z)$. 

If $0<\alpha<2$ and $\beta>0$ then, for $|z|\to\infty$,
\begin{equation}\label{27}
 E_{\alpha,\beta}(z)=\frac{1}{\alpha}z^{(1-\beta)/\alpha}
\exp(z^{1/\alpha})+\varepsilon_{\alpha,\beta}(z),\quad
 |argz|\leq\frac{\alpha\pi}{2},
\end{equation}
where 
$$
\varepsilon_{\alpha,\beta}(z)=-\sum_{n=1}^{N-1}\frac{z^{-n}}{\Gamma(\beta-\alpha n)}
+\mathcal{O}(|z|^{-N}),
$$
for some $N\in\mathbb{N}$, $N\neq1$ (see \cite{Erdelyi}).

Using the previous asymptotic expansion when $|z|\to\infty$,
 one can get a very useful estimation for the two-parameter Mittag-Leffler 
function.

\begin{proposition}
Let $0<\alpha<2$ and $\beta>0$. Then
\begin{equation}\label{19}
 E_{\alpha,\beta}(\omega t^{\alpha})
\leq C_{\alpha,\beta}(1+\omega^{(1-\beta)/\alpha})
(1+t^{1-\beta})\exp(\omega^{1/\alpha}t), \quad \omega\geq0, \; t\geq0.
\end{equation}
\end{proposition}

\begin{proof}
For $\omega=0$ and all $t\geq0$, the inequality is trivially satisfied. 
Fix $0<\alpha<2$, $\beta>0$ and $\omega>0$. Choose an arbitrarily large $T>0$. 
Then from \eqref{27}, for all $t>(\frac{T}{\omega})^{\frac{1}{\alpha}}$, 
follows that there exists a constant $C_1>0$ such that
\begin{align*}
 E_{\alpha,\beta}(\omega t^{\alpha}) 
&\leq C_1(\omega t^{\alpha})^{(1-\beta)/\alpha}
\exp\big((\omega t^{\alpha})^{1/\alpha}\big) \\
 &= C_1\omega^{(1-\beta)/\alpha}t^{1-\beta}\exp(\omega^{1/\alpha}t) \\
 &\leq C_1(1+\omega^{(1-\beta)/\alpha})(1+t^{1-\beta})\exp(\omega^{1/\alpha}t).
\end{align*}
Since $E_{\alpha,\beta}$ is a continuous function, for all $t\in[0,\infty)$, 
we have that, for $t\in[0,(\frac{T}{\omega})^{\frac{1}{\alpha}}]$, 
there exists a constant $C_{2}$ such that 
$$
E_{\alpha,\beta}(\omega t^{\alpha})\leq C_{2}
\leq C_{2}(1+\omega^{(1-\beta)/\alpha})(1+t^{1-\beta})\exp(\omega^{1/\alpha}t).
$$
Taking $C_{\alpha,\beta}=\max\{C_1,C_{2}\}$ we obtain the inequality \eqref{19}.
\end{proof}

 The linear Cauchy problem \eqref{uvodna-1a} with Caputo fractional derivatives 
has been considered in \cite{Umarov}, but in some special spaces of $L^{p}$ 
functions whose Fourier transforms are compactly supported in a some domain $G$, 
and the following result was obtained.

\begin{proposition}[Fractional Duhamel principle \cite{Umarov}]\label{29-U}
The solution of the Cauchy problem \eqref{uvodna-1a} is given by
\begin{equation}\label{30-U}
 u(t)=E_{\alpha}(t^{\alpha}A)u_{0} 
+ \int_{0}^{t}E_{\alpha}((t-\tau )^{\alpha}A)^{RL}\mathcal{D}_{\tau}^{1-\alpha}
f(\tau)d\tau.
\end{equation}
\end{proposition}

\section{Colombeau spaces}

 Let $(E, \|\cdot\|)$ be a Banach space and $\mathcal{L}(E)$ the space of all linear
continuous mappings from $E$ into $E$.

\begin{definition} \label{Colombeau} \rm
Let $m-1<\alpha<m$, $m\in \mathbb{N}$. 
$\mathcal{S}E_{M}^{\alpha,m}([0,\infty):\mathcal{L}(E))$ 
is the space of nets 
$$
(S_{\alpha})_{\varepsilon}:[0,\infty)\to
\mathcal{L}(E), \quad \varepsilon\in (0,1),
$$
 with the following properties:
 \begin{itemize}
 \item[(i)] $t\to (S_{\alpha})_{\varepsilon}(t)\in C^{m-1}([0,\infty):
\mathcal{L}(E))\cap C^{m}((0,\infty):\mathcal{L}(E))$.

 \item[(ii)] $\lim_{t\to0^{+}}\|\frac{\frac{d^{m}}{dt^{m}}
(S_{\alpha})_{\varepsilon}(t)}{t^{\alpha-m}}\|_{\mathcal{L}(E)}=C<+\infty$.

 \item[(iii)] For every $T>0$ there exist $N\in \mathbb{N}$, $M>0$ and 
$\varepsilon_{0}\in (0,1)$ such that
\begin{gather*}
 \sup_{t\in [0,T)} \|^C\mathcal{D}_t^{\gamma}(S_{\alpha})_{\varepsilon}(t)
\|_{\mathcal{L}(E)}\leq  M\varepsilon^{-N},
 \quad \varepsilon<\varepsilon_{0},\; \gamma\in \{0,\ldots,m-1, \alpha\}, \\
 \sup_{t\in (0,T)} \|\frac{d^{m}}{dt^{m}}(S_{\alpha})_{\varepsilon}(t)
\|_{\mathcal{L}(E)}\leq  M\varepsilon^{-N}, \quad 
 \varepsilon<\varepsilon_{0}.
\end{gather*}
 \end{itemize}
 \end{definition}

Similarly we define the following space.

\begin{definition} \label{Colombeau1} \rm
Let $m-1<\alpha<m$, $m\in \mathbb{N}$. 
$\mathcal{S}N_{\alpha,m}([0,\infty):\mathcal{L}(E))$ is the space of nets
$$
(N_{\alpha})_{\varepsilon}:[0,\infty)\to \mathcal{L}(E), \quad 
 \varepsilon\in (0,1)
$$ 
with the following properties:
 \begin{itemize}
 \item[(i)] $t\to(N_{\alpha})_{\varepsilon}(t)\in C^{m-1}([0,\infty)
:\mathcal{L}(E))\cap C^{m}((0,\infty):\mathcal{L}(E))$.

 \item[(ii)] $\lim_{t\to0^{+}}
\|\frac{\frac{d^{m}}{dt^{m}}(N_{\alpha})_{\varepsilon}(t)}{t^{\alpha-m}}
\|_{\mathcal{L}(E)}=C<+\infty$.

 \item[(iii)] For every $T>0$ and $a\in\mathbb{R}$ there exist $M>0$ and 
$\varepsilon_{0}\in (0,1)$ such that
\begin{equation} \label{e3.2}
\begin{gathered}
 \sup_{t\in [0,T)} \|^C\mathcal{D}_t^{\gamma}(N_{\alpha})_{\varepsilon}(t)
\|_{\mathcal{L}(E)}\leq  M\varepsilon^{a},\quad 
 \varepsilon<\varepsilon_{0},\; \gamma\in \{0,\ldots,m-1, \alpha\},\\
 \sup_{t\in (0,T)} \|\frac{d^{m}}{dt^{m}}(N_{\alpha})_{\varepsilon}(t)
\|_{\mathcal{L}(E)}
\leq  M\varepsilon^{a}, \quad \varepsilon<\varepsilon_{0}.
\end{gathered}
\end{equation}
 \end{itemize}
 \end{definition}

When $m=1$ we  denote
\begin{gather*}
\mathcal{S}E_{M}^{\alpha, 1}([0,\infty):\mathcal{L}(E))
=\mathcal{S}E_{M}^{\alpha}([0,\infty):\mathcal{L}(E)) ,\\
\mathcal{S}N_{\alpha,1}([0,\infty):\mathcal{L}(E))
=\mathcal{S}N_{\alpha}([0,\infty):\mathcal{L}(E)).
\end{gather*}
In this article, Caputo's fractional derivative in the problem of 
consideration is of order $0<\alpha <1$ and therefore, from now on, 
we will consider only that case. Hence, further we investigate spaces
 $\mathcal{S}E_{M}^{\alpha}([0,\infty):\mathcal{L}(E))$ and 
$\mathcal{S}N_{\alpha}([0,\infty):\mathcal{L}(E))$, although all the 
assertions we give can be extended for all $m\in {\mathbb N}$.

\begin{proposition} \label{algebra}
The space $\mathcal{S}E_{M}^{\alpha}([0,\infty):\mathcal{L}(E))$ 
is an algebra with respect to composition of operators, and 
$\mathcal{S}N_{\alpha}([0,\infty):\mathcal{L}(E))$ is an ideal of 
$\mathcal{S}E_{M}^{\alpha}([0,\infty):\mathcal{L}(E))$.
\end{proposition}

\begin{proof}
Fix $0<\alpha<1$ and let $S_{\alpha}$ and $T_{\alpha}$ are from the 
space $\mathcal{S}E_{M}^{\alpha}([0,\infty):\mathcal{L}(E))$. 
Then, it easily follows that $S_{\alpha}(t)T_{\alpha}(t)$ satisfies 
the properties (i) and (ii) from Definition \eqref{Colombeau}. 
The fact that $S_{\alpha}(t)T_{\alpha}(t)$ satisfies the property (iii) 
for $\gamma\in \{0,1\}$, can be proved in the usual way as in the case 
of Colombeau spaces with integer order derivatives.

Let us prove that (iii) is satisfied for $\gamma=\alpha$, too. 
Indeed, for $t\in (\eta,T)$, where $\eta >0$ is arbitrarily small and 
$T>0$, using the property \eqref{eta} we have
\begin{align*}
 & \|^C\mathcal{D}_t^{\alpha}((S_{\alpha})_{\varepsilon}(t)
(T_{\alpha})_{\varepsilon}(t))\|_{\mathcal{L}(E)}\\ 
&\leq  \frac{1}{\Gamma(1-\alpha)}\lim_{\eta\to0^{+}}\int_{\eta}^{t}
 \frac{\|((S_{\alpha})_{\varepsilon}(\tau)(T_{\alpha})_{\varepsilon}
(\tau))'\|_{\mathcal{L}(E)}}{(t-\tau)^{\alpha}}d\tau\\
 &\leq  \frac{1}{\Gamma(1-\alpha)}\lim_{\eta\to0^{+}}\int_{\eta}^{t}
 \frac{\|((S_{\alpha})_{\varepsilon}(\tau))'\|_{\mathcal{L}(E)}
 \|(T_{\alpha})_{\varepsilon}(\tau)\|_{\mathcal{L}(E)}}{(t-\tau)^{\alpha}}d\tau \\
 &\quad +\frac{1}{\Gamma(1-\alpha)}\lim_{\eta\to0^{+}}\int_{\eta}^{t}
 \frac{\|(S_{\alpha})_{\varepsilon}(\tau)\|_{\mathcal{L}(E)}
 \|((T_{\alpha})_{\varepsilon}(\tau))'\|_{\mathcal{L}(E)}}{(t-\tau)^{\alpha}}d\tau\\
 &\leq  \lim_{\eta\to0^{+}}\frac{(t-\eta)^{1-\alpha}}{\Gamma(2-\alpha)}
M_1\varepsilon^{-N}\\
 &\leq  \frac{T^{1-\alpha}}{\Gamma(2-\alpha)}M_1\varepsilon^{-N}.
\end{align*}
Thus, we obtain the moderate bound for $t\in (0,T)$, i.e.
$$
\sup_{t\in (0,T)} \|^C\mathcal{D}_t^{\alpha}((S_{\alpha})_{\varepsilon}
(t)(T_{\alpha})_{\varepsilon}(t))\|_{\mathcal{L}(E)}
\leq  M\varepsilon^{-N}.
$$
 It remains to prove the moderate bound for 
$\| ^C\mathcal{D}_t^{\alpha}((S_{\alpha})_{\varepsilon}(t)(T_{\alpha})_{\varepsilon}
(t))\big|_{t=0}\|$. From Theorem \ref{meanvalue} we obtain
 \begin{align*}
 &{}^C\mathcal{D}_t^{\alpha}((S_{\alpha})_{\varepsilon}(t)
 (T_{\alpha})_{\varepsilon}(t))\big|_{t=0}\\
 & =\frac{1}{\Gamma(1+\alpha)}\lim_{t\to0^{+}}
 \frac{(S_{\alpha})_{\varepsilon}(t)(T_{\alpha})_{\varepsilon}(t)-(S_{\alpha})_{\varepsilon}(0)(T_{\alpha})_{\varepsilon}(0)}{t^{\alpha}} \\
 &=\frac{1}{\Gamma(1+\alpha)}\lim_{t\to0^{+}}
 \frac{((S_{\alpha})_{\varepsilon}(t)-(S_{\alpha})_{\varepsilon}(0))(T_{\alpha})_{\varepsilon}(t)}{t^{\alpha}}\\
 &\quad+\frac{1}{\Gamma(1+\alpha)}\lim_{t\to0^{+}}
 \frac{(S_{\alpha})_{\varepsilon}(0))((T_{\alpha})_{\varepsilon}(t)-(T_{\alpha})_{\varepsilon}(0))}{t^{\alpha}}\\
 &={}^C\mathcal{D}_t^{\alpha}((S_{\alpha})_{\varepsilon}(t))\big|_{t=0}
(T_{\alpha})_{\varepsilon}(0)+
 (S_{\alpha})_{\varepsilon}(0)^C\mathcal{D}_t^{\alpha}
((T_{\alpha})_{\varepsilon}(t))\big|_{t=0}.
 \end{align*}
Estimating in norm, we obtain a moderate bound for 
$\| ^C\mathcal{D}_t^{\alpha}((S_{\alpha})_{\varepsilon}(t)
(T_{\alpha})_{\varepsilon}(t))\big|_{t=0}\|$. 
Thus, (iii) is satisfied.

 Similarly, one can prove that 
$(T_{\alpha})_{\varepsilon}(t)(S_{\alpha})_{\varepsilon}(t)$, 
also satisfies all properties from Definition \eqref{Colombeau}. 
Thus, the space $\mathcal{S}E_{M}^{\alpha}([0,\infty):\mathcal{L}(E))$ 
is an algebra.
One can similarly prove that the space 
$\mathcal{S}N_{\alpha}([0,\infty):\mathcal{L}(E))$ is an ideal of 
$\mathcal{S}E_{M}^{\alpha}([0,\infty):\mathcal{L}(E))$.
 \end{proof}

Now we can define a Colombeau-type space as a factor algebra by 
\begin{equation}
 \mathcal{S}G_{\alpha}([0,\infty):\mathcal{L}(E))
=\frac{\mathcal{S}E_{M}^{\alpha}([0,\infty):
\mathcal{L}(E))}{\mathcal{S}N_{\alpha}([0,\infty):\mathcal{L}(E))}.
\end{equation}
For every $0<\alpha<1$ elements of 
$\mathcal{S}G_{\alpha}([0,\infty):\mathcal{L}(E))$ will be denoted by 
$S=[(S_{\alpha})_{\varepsilon}]$,
where $(S_{\alpha})_{\varepsilon}$ is a representative of the
class.

Similarly, one can define the following spaces:
 $\mathcal{S}E_{M}(E)$ is the space of nets of linear
continuous mappings 
$$
A_{\varepsilon}: E\to E,\quad \varepsilon\in (0,1),
$$ 
with the property that there exists constants $N\in \mathbb{N}$, $M>0$ and
$\varepsilon_{0}\in (0,1)$ such that
$$
\|A_{\varepsilon}\|_{\mathcal{L}(E)}\leq M\varepsilon^{-N}, \quad
\varepsilon<\varepsilon_{0}.
$$
$\mathcal{S}N(E)$ is the space of nets of linear
continuous mappings $A_{\varepsilon}: E\to E$, $\varepsilon\in (0,1)$, 
with the property that
for every $a\in \mathbb{R}$, there exist $M>0$
and $\varepsilon_{0}\in (0,1)$ such that 
$$
\|A_{\varepsilon}\|_{\mathcal{L}(E)}\leq M\varepsilon^{a}, \quad
 \varepsilon<\varepsilon_{0}.
$$ The Colombeau
space of generalized linear operators on $E$ is defined by
$$
\mathcal{S}G(E)=\frac{\mathcal{S}E_{M}(E)}{\mathcal{S}N(E)}.
$$ 
Elements of $\mathcal{S}G(E)$ will be
denoted by $A=[A_{\varepsilon}]$, where $A_{\varepsilon}$ is a 
representative of the class.

Finally, we introduce the Colombeau space within which we will 
solve \eqref{uvodna-1}. We give the definitions for arbitrary $m\in {\mathbb N}$.
Let $m-1<\alpha<m$, $m\in \mathbb{N}$. 
$\mathcal{E}_{M}^{\alpha}([0,\infty):H^{m}(\mathbb{R}))$ is the
space of nets 
$$
G_{\varepsilon}:[0,\infty)\times\mathbb{R}\to \mathbb{C}, \quad
 \varepsilon \in (0,1),
$$
 with the following properties:
 \begin{itemize}
 \item[(i)] $G_{\varepsilon}(\cdot,\cdot)\in C^{m-1}([0,\infty):H^{m}(\mathbb{R}))
\cap C^{m}((0,\infty):H^{m}(\mathbb{R}))$.

 \item[(ii)] $\lim_{t\to0^{+}}\|\frac{\frac{d^{m}}{dt^{m}}
G_{\varepsilon}(t,\cdot)}{t^{\alpha-m}}\|_{H^{m}}=C<+\infty$.

 \item[(iii)] For every $T>0$ there exist $M>0, N\in \mathbb{N}$ and
$\varepsilon_{0}>0$ such that
\begin{equation} \label{e3.4}
\begin{gathered}
 \sup_{t\in [0,T)} \|^C\mathcal{D}_t^{\gamma}G_{\varepsilon}(t,\cdot)\|_{H^{m}}
\leq  M\varepsilon^{-N}, \quad
 \varepsilon<\varepsilon_{0},\; \gamma\in \{0,\ldots,m-1, \alpha\}, \\
 \sup_{t\in (0,T)} \|\frac{d^{m}}{dt^{m}}G_{\varepsilon}(t,\cdot)\|_{H^{m}}\leq
 M\varepsilon^{-N}, \quad \varepsilon<\varepsilon_{0}.
\end{gathered}
\end{equation}
 \end{itemize}
It is an algebra with respect to multiplication.

Similarly, for $m-1<\alpha<m$, $m\in \mathbb{N}$,
 $\mathcal{N}_{\alpha}([0,\infty):H^{m}(\mathbb{R}))$ is the
space of nets $G_{\varepsilon}\in
\mathcal{E}_{M}^{\alpha}([0,\infty):H^{m}(\mathbb{R}))$ 
with the following properties:
 \begin{itemize}
 \item[(i)] $G_{\varepsilon}(\cdot,\cdot)\in C^{m-1}([0,\infty)
:H^{m}(\mathbb{R}))\cap C^{m}((0,\infty):H^{m}(\mathbb{R}))$.
 \item[(ii)] $\lim_{t\to0^{+}}\|\frac{\frac{d^{m}}{dt^{m}}G_{\varepsilon}
(t,\cdot)}{t^{\alpha-m}}\|_{H^{m}}=C<+\infty$.
 \item[(iii)] For every $T>0$ and $a\in\mathbb{R}$ there exist $M>0$ and 
$\varepsilon_{0}>0$ such that
\begin{equation} \label{e3.5}
\begin{gathered}
 \sup_{t\in [0,T)} \|^C\mathcal{D}_t^{\gamma}G_{\varepsilon}(t,\cdot)
\|_{H^{m}}
\leq  M\varepsilon^{a}, \quad \varepsilon<\varepsilon_{0},\;
 \gamma\in \{0,\ldots,m-1, \alpha\}, \\
 \sup_{t\in (0,T)} \|\frac{d^{m}}{dt^{m}}G_{\varepsilon}(t,\cdot)\|_{H^{m}}\leq
 M\varepsilon^{a}, \quad \varepsilon<\varepsilon_{0}.
\end{gathered}
\end{equation}
 \end{itemize}
The space $\mathcal{N}_{\alpha}([0,\infty):H^{m}(\mathbb{R}))$ is an ideal 
of $\mathcal{E}_{M}^{\alpha}([0,\infty):H^{m}(\mathbb{R}))$.

The quotient space
$$
\mathcal{G}_{\alpha}([0,\infty):H^{m}(\mathbb{R}))
=\frac{\mathcal{E}_{M}^{\alpha}([0,\infty):H^{m}(\mathbb{R}))}
{\mathcal{N}_{\alpha}([0,\infty):H^{m}(\mathbb{R}))}
$$ 
is the corresponding Colombeau generalized function space
related to the Sobolev space $H^{m}$. Again, in this paper we will 
consider only the case $m=1$ and $m=2$, i.e. the solution of our fractional
 evolution problem will be an element of
$\mathcal{G}_{\alpha}([0,\infty):H^{1}(\mathbb{R}))$ or 
$\mathcal{G}_{\alpha}([0,\infty):H^2(\mathbb{R}))$.

In a similar way, by omitting variable $t$, one can define spaces 
$\mathcal{E}_{M}^{\alpha}(H^{m}(\mathbb{R}))$,
$\mathcal{N}_{\alpha}(H^{m}(\mathbb{R}))$, and
$\mathcal{G}_{\alpha}(H^{m}(\mathbb{R}))$.

\section{Uniformly continuous solution operators}

Consider the Cauchy problem for the fractional evolution equation of order 
$\alpha$ with $0<\alpha<1$,
\begin{equation}\label{1}
 ^C{\mathcal{D}}_t^{\alpha}u(t)=Au(t), \quad t>0; \quad u(0)=x,
\end{equation}
where $^C{\mathcal{D}}_t^{\alpha}$ is the Caputo fractional derivative
of order $\alpha$, and $A$ is a linear and bounded operator defined on a 
Banach space $E$. The more general case when $A$ is a closed linear 
operator densely defined in a Banach space $E$ was considered in \cite{Bazhlekova}. 
As it is pointed out in \cite{Bazhlekova}, the problem \eqref{1} is well-posed 
if and only if the  Volterra integral equation
\begin{equation}\label{2}
 u(t)=x+\int_{0}^{t}g_{\alpha}(t-\tau)Au(\tau)d\tau
\end{equation}
is well-possed, where $g_{\alpha}(t)$ is defined for $\alpha>0$, by
\begin{equation*}
g_{\alpha}(t)= \begin{cases}
 t^{\alpha-1}/\Gamma(\alpha), & t>0, \\
 0, & t\leq 0.
 \end{cases}
\end{equation*}

In the general case when $A$ is a closed linear operator densely defined 
in a Banach space $E$, strongly continuous solution operator for \eqref{1} 
is introduced in \cite{Bazhlekova}. Similarly, when $A$ is linear and bounded,
 we introduce uniformly continuous solution operator.

\begin{definition} \label{def4.1} \rm
A family $S_{\alpha}(t), \; t\geq0$, of linear and bounded operators on 
Banach space $E$ is called a uniformly continuous solution operator 
for \eqref{1} if the following conditions are satisfied:
\begin{itemize}
 \item[(i)] $S_{\alpha}(t)$ is a uniformly continuous function for 
$t\geq0$ and $S_{\alpha}(0)=I$, where $I$ is identity operator on $E$.
 \item[(ii)] $AS_{\alpha}(t)x=S_{\alpha}(t)Ax$, for all $x\in E$, $t\geq0$.
 \item[(iii)] $S_{\alpha}(t)x$ is a solution of \eqref{2} for all 
$x\in E$, $t\geq0$.
\end{itemize}
\end{definition}

\begin{definition}\label{4} \rm
The infinitesimal generator $A$ of a uniformly continuous solution operator 
$S_{\alpha}(t)$, $\alpha>0$, $t\geq0$, for \eqref{1} is defined by
\begin{equation}\label{3}
 Ax=\Gamma(1+\alpha)\lim_{t\downarrow 0}\frac{S_{\alpha}(t)x-x}{t^{\alpha}},
\end{equation}
for all $x\in E$.
\end{definition}

The generator $A$ could also be defined as 
$$
Ax=(^C{\mathcal{D}}_t^{\alpha}S_{\alpha})(t)x\big|_{t=0},
$$ 
since $J_t^{\alpha}{^C\mathcal{D}}_t^{\alpha}S_{\alpha}(t)x=S_{\alpha}(t)x-x$
and for all functions $v\in C(\mathbb{R}_{+};E)$ holds 
$$
\lim_{t\downarrow 0}\frac{J_t^{\alpha}v(t)}{g_{\alpha+1}(t)}=v(0)
$$ 
(see \cite{Bazhlekova}).

\begin{remark} \label{rmk4.1} \rm
In the case $0<\alpha\leq1$, the definition given by \eqref{3} also follows 
from Theorem \ref{meanvalue}.
\end{remark}

\begin{definition}[\cite{Bazhlekova}] \rm
The solution operator $S_{\alpha}(t)$ is called exponentially bounded 
if there exist constants $M\geq1$ and $\omega\geq0$ such that 
$$
\|S_{\alpha}(t)\|\leq Me^{\omega t}, \; t\geq0.
$$
\end{definition}

\begin{theorem}[{\cite[Theorem 2.5]{Bazhlekova}}]\label{8}
Let $\alpha>0$. Then exponentially bounded uniformly continuous solution 
operator $S_{\alpha}(t)$ is the solution operator for the Cauchy
 problem \eqref{1} if and only if $A\in \mathcal{L}(E)$.
\end{theorem}

From Definition \ref{4} it follows that every solution operator has a unique 
infinitesimal generator. If $S_{\alpha}(t)$ is a uniformly continuous solution 
operator satisfying $\|S_{\alpha}(t)\|\leq Me^{\omega t}$, for some 
$M\geq1$ and $\omega\geq0$, its infinitesimal generator is a bounded 
linear operator.

On the other hand, every bounded linear operator $A$ is the infinitesimal 
generator of a uniformly continuous solution operator given by 
$$
S_{\alpha}(t)=E_{\alpha}(t^{\alpha}A)
=\sum_{n=0}^{\infty}\frac{t^{n\alpha}A^{n}}{\Gamma(1+n\alpha)}, \quad 
\alpha>0,\;t\geq 0. 
$$
For every $0<\alpha\leq1$ this solution operator is unique as asserted in 
the following theorem.

\begin{theorem}\label{16}
Let $0<\alpha\leq1$ and let $S_{\alpha}(t)$ and $T_{\alpha}(t)$ be 
exponential bounded uniformly continuous solution operators with infinitesimal 
generators $A$ and $B$, respectively. If $A=B$ then $S_{\alpha}(t)=T_{\alpha}(t)$, 
for every $t\geq0$.
\end{theorem}

\begin{proof}
Since $S_{\alpha}(t)$ is exponential bounded there exist constants $M\geq1$ 
and $\omega_1\geq0$ such that
$$
\|S_{\alpha}(t)\|\leq Me^{\omega_1 t}, \; t\geq0.
$$ 
Then for $Re\lambda>\omega_1$ and $x\in E$ we have
$$
\lambda^{\alpha-1}R(\lambda^{\alpha},A)x
=\int_{0}^{\infty}e^{-\lambda t}S_{\alpha}(t)xdt,
$$
where $R(\lambda,A)=(\lambda I-A)^{-1}$ stands for the resolvent operator of $A$.
Similarly, for $T_{\alpha}(t)$ there exists $\omega_{2}\geq0$ such that 
for $Re\lambda>\omega_{2}$ and $x\in E$ we have 
$$
\lambda^{\alpha-1}R(\lambda^{\alpha},A)x=\int_{0}^{\infty}
e^{-\lambda t}T_{\alpha}(t)xdt,
$$ 
and $S_{\alpha}(t)=T_{\alpha}(t)$ follows from the uniqueness of the 
Laplace transform.
\end{proof}


\begin{proposition}\label{5}
Let $S_{\alpha}(t)$, $0<\alpha\leq1$, $t\geq0$, be a uniformly continuous 
solution operator satisfying $\|S_{\alpha}(t)\|\leq Me^{\omega t}$, 
for some $M\geq1$ and $\omega\geq0$. Then
\begin{itemize}
 \item[(i)] There exists a unique bounded linear operator $A$ such that 
$$
S_{\alpha}(t)=E_{\alpha}(t^{\alpha}A), \quad t\geq0.
$$
 \item[(ii)] The operator $A$ in (i) is the infinitesimal generator of 
solution operator $S_{\alpha}(t)$.
 \item[(iii)] For every $t\geq0$,
$$
{}^C{\mathcal{D}}_t^{\alpha}S_{\alpha}(t)=AS_{\alpha}(t)=S_{\alpha}(t)A.
$$
\end{itemize}
\end{proposition}

\begin{proof}
Fix $0<\alpha\leq1$. From Theorem \ref{8} we know that the infinitesimal 
generator of $S_{\alpha}(t)$ is a bounded linear operator $A$. Also, $A$ is 
the infinitesimal generator of $E_{\alpha}(t^{\alpha}A)$ and therefore 
by Theorem \ref{16}, $S_{\alpha}(t)=E_{\alpha}(t^{\alpha}A)$. 
All others assertions of the proposition follow from (i).
\end{proof}


Integral representation stated in the next proposition will often be used 
in proving some auxiliary results as well as in proving our main result.

\begin{proposition}\label{15}
Let $0<\alpha<1$ and let $S_{\alpha}(t)$ be a solution operator generated by $A$. 
Then
\begin{equation}\label{22}
 \int_{0}^{t}S_{\alpha}(t-\tau)^{RL}\mathcal{D}_{\tau}^{1-\alpha}f(\tau)d\tau
=\int_{0}^{t}(t-\tau)^{\alpha-1}
 E_{\alpha,\alpha}((t-\tau)^{\alpha}A)f(\tau)d\tau.
\end{equation}
\end{proposition}

\begin{proof}
Fix $0<\alpha<1$. Taking into account the relation \ref{25} in 
Proposition \ref{28} one gets
\begin{align*}
\int_{0}^{t}S_{\alpha}(t-\tau)^{RL}\mathcal{D}_{\tau}^{1-\alpha}f(\tau)d\tau 
&= \int_{0}^{t}\sum_{n=0}^{\infty} \frac{1}{\Gamma(1+n\alpha)}
 (t-\tau)^{n\alpha}A^{n\; RL}\mathcal{D}_{\tau}^{1-\alpha}f(\tau)d\tau \\
 &= \sum_{n=0}^{\infty}\int_{0}^{t}\frac{1}{\Gamma(1+n\alpha)}
(t-\tau)^{n\alpha}A^{n\; RL}\mathcal{D}_{\tau}^{1-\alpha}f(\tau)d\tau \\
 &= \sum_{n=0}^{\infty}J_t^{n\alpha+1}A^{n\; RL}\mathcal{D}_t^{1-\alpha}f(t)
= \sum_{n=0}^{\infty}J_t^{n\alpha+\alpha}A^{n}f(t)\\
 &= \sum_{n=0}^{\infty}\frac{1}{\Gamma(n\alpha+\alpha)}\int_{0}^{t}
(t-\tau)^{n\alpha+\alpha-1}A^{n}f(\tau)d\tau \\
 &= \int_{0}^{t}(t-\tau)^{\alpha-1}\sum_{n=0}^{\infty}
\frac{1}{\Gamma(n\alpha+\alpha)}(t-\tau)^{n\alpha}A^{n}f(\tau)d\tau \\
 &= \int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}
((t-\tau)^{\alpha}A)f(\tau)d\tau.
\end{align*}
\end{proof}

Similarly, the first order derivative of the previously integral representation 
has the following form.

\begin{proposition}\label{18}
Let $0<\alpha<1$ and let $S_{\alpha}(t)$ be a solution operator generated by $A$. 
Then
\begin{equation}\label{21}
\begin{aligned}
\frac{d}{dt}\int_{0}^{t}S_{\alpha}(t-\tau)^{RL}
\mathcal{D}_{\tau}^{1-\alpha}f(\tau)d\tau 
&= \int_{0}^{t}(t-\tau)^{\alpha-1}
 E_{\alpha,\alpha}((t-\tau)^{\alpha}A)\partial_{\tau}fd\tau \\
 &\quad +t^{\alpha-1}E_{\alpha,\alpha}(t^{\alpha}A)f(0).
\end{aligned}
\end{equation}
\end{proposition}

\begin{proof}
Fix $0<\alpha<1$. From the proof of Proposition \ref{15} it follows that
\begin{equation*}
\frac{d}{dt}\int_{0}^{t}S_{\alpha}(t-\tau)^{RL}
\mathcal{D}_{\tau}^{1-\alpha}f(\tau)d\tau 
= \sum_{n=0}^{\infty}\frac{d}{dt}J_t^{n\alpha+\alpha}A^{n}f(t).
\end{equation*}
Further, since
\begin{equation*}
 \frac{d}{dt}J_t^{\alpha}f(t)={}^{RL}\mathcal{D}_t^{1-\alpha}f(t)
={} ^C\mathcal{D}_t^{1-\alpha}f(t)+\frac{f(0)t^{\alpha-1}}{\Gamma(\alpha)}
= J_t^{\alpha}\frac{d}{dt}f(t)+\frac{f(0)t^{\alpha-1}}{\Gamma(\alpha)},
\end{equation*}
we have
\begin{align*}
 \sum_{n=0}^{\infty}\frac{d}{dt}J_t^{n\alpha+\alpha}A^{n}f(t) 
&= \sum_{n=0}^{\infty}J_t^{n\alpha+\alpha}A^{n}\frac{d}{dt}f(t)+
 \sum_{n=0}^{\infty}\frac{J_t^{n\alpha}t^{\alpha-1}A^{n}f(0)}{\Gamma(\alpha)} \\
 &= \sum_{n=0}^{\infty}J_t^{n\alpha+\alpha}A^{n}\frac{d}{dt}f(t)+
 \sum_{n=0}^{\infty}\frac{t^{n\alpha+\alpha-1}}{\Gamma(\alpha+n\alpha)}A^{n}f(0),
\end{align*}
and similarly to the proof of Proposition \ref{15} one finally gets the 
relation \eqref{21}.
\end{proof}

Motivated by Proposition \ref{29-U} we give the fractional Duhamel principle 
in the case of solution operator.

\begin{proposition}\label{29}
The solution of the Cauchy problem \eqref{uvodna-1a} with Caputo fractional
 derivative is given by
\begin{equation}\label{30}
 u(t)=S_{\alpha}(t)u_{0} 
+ \int_{0}^{t}S_{\alpha}(t-\tau)^{RL}{\mathcal{D}}_{\tau}^{1-\alpha}
f(\cdot,\tau,u)d\tau,
\end{equation}
where $S_{\alpha}(t)$ is a solution operator generated by $A$. 
The solution above is called mild solution to the problem \eqref{uvodna-1a}.
\end{proposition}

\begin{proof}
Since $^C{\mathcal{D}}_t^{\alpha}S_{\alpha}(t)=AS_{\alpha}(t)$,
 for a continuous function its fractional integral $J_t^{\alpha}$ 
is a continuous function too and $^C{\mathcal{D}}_t^{\alpha}$
is a left inverse of fractional integral $J_t^{\alpha}$ for all 
$\alpha\geq0$ and all continuous functions, it can be easily shown that $u(t)$ 
given by \eqref{30} satisfies the Cauchy problem \eqref{uvodna-1a}.
\end{proof}

\begin{remark} \label{rmk4.2} \rm
The solution of the Cauchy problem \eqref{uvodna-1a} can also be represented
 by Caputo fractional derivative, but in that case one must additionally 
suppose that $f(\cdot,0,u_{0})=0$.
\end{remark}

\section{Generalized uniformly continuous solution operators}

First, recall that every linear and bounded operator on Banach space $E$ 
is a closed and densely defined operator in $E$. Therefore, all results 
in the previous section continue to be valid in the case of linear 
and bounded operators on Banach space.

Instead of the Cauchy problem \eqref{1} with closed and densely defined operator $A$,
 let us now consider fractional Cauchy problem given by
\begin{equation}\label{1a}
 {}^C{\mathcal{D}}_t^{\alpha}u(t)=\widetilde{A}u(t), \quad t>0; \,u(0)=x,
\end{equation}
where $\widetilde{A}$ is a generalized linear bounded operator.

\begin{definition} \label{def5.1} \rm
Let $0<\alpha<1$. $S_{\alpha}\in \mathcal{SG}_{\alpha}([0,\infty):\mathcal{L}(E))$ 
is called a Colombeau uniformly continuous solution operator for \eqref{1a} 
if it has a representative $(S_{\alpha})_{\varepsilon}$ which is a uniformly 
continuous solution operator for \eqref{1a} and for every $\varepsilon$ small enough.
\end{definition}

\begin{proposition} \label{prop5.1}
Let $0<\alpha<1$ and let $(S_{\alpha})_{1\varepsilon}$ and 
$(S_{\alpha})_{2\varepsilon}$ be representatives of a generalized uniformly 
continuous solution operator $S_{\alpha}$, with infinitesimal generators 
$\widetilde{A}_{1\varepsilon}$ and $\widetilde{A}_{2\varepsilon}$, 
respectively, for $\varepsilon$ small enough. Then 
$$
\widetilde{A}_{1\varepsilon}-\widetilde{A}_{2\varepsilon}\in \mathcal{SN}(E).
$$
\end{proposition}

\begin{proof}
Fix $0<\alpha<1$. Then we have 
\begin{align*}
\widetilde{A}_{1\varepsilon}-\widetilde{A}_{2\varepsilon} 
&= (^C{\mathcal{D}}_t^{\alpha}(S_{\alpha})_{1\varepsilon})(t)
\big|_{t=0}-(^C{\mathcal{D}}_t^{\alpha}(S_{\alpha})_{2\varepsilon})(t)\big|_{t=0}\\
&={}^C{\mathcal{D}}_t^{\alpha}((S_{\alpha})_{1\varepsilon}
-(S_{\alpha})_{2\varepsilon})(t)\big|_{t=0}. 
\end{align*}
Since 
$$
(S_{\alpha})_{1\varepsilon}-(S_{\alpha})_{2\varepsilon} 
\in \mathcal{SN}_{\alpha}([0,\infty):\mathcal{L}(E)),
$$
we have that, for every $a\in\mathbb{R}$, there exists $M>0$ such that 
$$
\|^C{\mathcal{D}}_t^{\alpha}((S_{\alpha})_{1\varepsilon}
-(S_{\alpha})_{2\varepsilon})(t)\big|_{t=0}\|_{\mathcal{L}(E)}\leq M\varepsilon^{a}.
$$
It implies that for every $a\in\mathbb{R}$ there exists $M>0$ such that 
$\|\widetilde{A}_{1\varepsilon}-\widetilde{A}_{2\varepsilon}\|
\leq M\varepsilon^{a}$. Thus, 
$\widetilde{A}_{1\varepsilon}-\widetilde{A}_{2\varepsilon}\in \mathcal{SN}(E)$.
\end{proof}

\begin{definition} \label{def5.2} \rm
$\widetilde A\in \mathcal{SG}(E)$ is called the infinitesimal generator of 
a Colombeau uniformly continuous solution operator 
$S_{\alpha}\in \mathcal{SG}_{\alpha}([0,\infty):\mathcal{L}(E))$, 
$0<\alpha<1$, if $\widetilde A_{\varepsilon}$ is the infinitesimal 
generator of the representative $(S_{\alpha})_{\varepsilon}$, 
for every $\varepsilon$ small enough.
\end{definition}

\begin{proposition}\label{20}
Let $0<\alpha<1$. Let $\widetilde A$ be the infinitesimal generator of a
 Colombeau uniformly continuous solution operator $S_{\alpha}$, and $\widetilde B$ the infinitesimal generator of a Colombeau uniformly continuous solution operator $T_{\alpha}$. If $\widetilde A=\widetilde B$, then $S_{\alpha}=T_{\alpha}$.
\end{proposition}

\begin{proof}
Fix $0<\alpha<1$ and let 
$\widetilde N_{\varepsilon}=\widetilde A_{\varepsilon}
-\widetilde B_{\varepsilon}\in \mathcal{SN}(E)$. 
Then from the property (iii) in Proposition \ref{5} we obtain 
$$
{}^C{\mathcal{D}}_t^{\alpha}((S_{\alpha})_{\varepsilon}
-(T_{\alpha})_{\varepsilon})(t)x
=\widetilde A_{\varepsilon}((S_{\alpha})_{\varepsilon}
-(T_{\alpha})_{\varepsilon})(t)x 
+ \widetilde N_{\varepsilon}(T_{\alpha})_{\varepsilon}(t)x.
$$
By using fractional Duhamel principle \eqref{30} and since
 $(S_{\alpha})_{\varepsilon}(0)=(T_{\alpha})_{\varepsilon}(0)=I$, one gets
\begin{equation}\label{23}
 ((S_{\alpha})_{\varepsilon}-(T_{\alpha})_{\varepsilon})(t)x
 = \int_{0}^{t}(S_{\alpha})_{\varepsilon}(t-\tau)^{RL}
{\mathcal{D}}_{\tau}^{1-\alpha}\widetilde N_{\varepsilon}(T_{\alpha})_{\varepsilon}
(\tau)xd\tau.
\end{equation}
Then, from the integral representation given in Proposition \ref{15} we have
\begin{align*}
 ((S_{\alpha})_{\varepsilon}-(T_{\alpha})_{\varepsilon})(t)x 
&= \int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}((t-\tau)^{\alpha}
 \widetilde A_{\varepsilon})\widetilde N_{\varepsilon}(T_{\alpha})_{\varepsilon}
 (\tau)xd\tau \\
 &= \int_{0}^{t}(t-\tau)^{\alpha-1}\sum_{n=0}^{\infty}
\frac{(t-\tau)^{n\alpha}\widetilde A_{\varepsilon}^{n}}{\Gamma(\alpha+n\alpha)}
 \widetilde N_{\varepsilon}(T_{\alpha})_{\varepsilon}(\tau)xd\tau \\
 &= \int_{0}^{t}\sum_{n=0}^{\infty}
 \frac{(t-\tau)^{(n+1)\alpha-1}\widetilde A_{\varepsilon}^{n}}
 {\Gamma((n+1)\alpha)}\widetilde N_{\varepsilon}(T_{\alpha})_{\varepsilon}
 (\tau)xd\tau \\
 &= \sum_{n=1}^{\infty}\int_{0}^{t}\frac{(t-\tau)^{n\alpha-1}
 \widetilde A_{\varepsilon}^{n-1}}{\Gamma(n\alpha)}
 \widetilde N_{\varepsilon}(T_{\alpha})_{\varepsilon}(\tau)xd\tau.
 \end{align*}
For $t\in[0,T)$, $T>0$, we obtain estimate
\begin{align*}
 \|((S_{\alpha})_{\varepsilon}-(T_{\alpha})_{\varepsilon})(t)\|
 &\leq \sum_{n=1}^{\infty}\frac{1}{\Gamma(n\alpha)}
 \int_{0}^{t}(t-\tau)^{n\alpha-1}\|\widetilde A_{\varepsilon}^{n-1}
 \widetilde N_{\varepsilon}(T_{\alpha})_{\varepsilon}(\tau)\|d\tau \\
 &\leq \|\widetilde N_{\varepsilon}\|\sup_{t\in[0,T)}
 \|(T_{\alpha})_{\varepsilon}(t)\|
 \sum_{n=1}^{\infty}\|\widetilde A_{\varepsilon}\|^{n-1}
\frac{1}{\Gamma(n\alpha)}\frac{T^{n\alpha}}{n\alpha} \\
 &\leq \|\widetilde N_{\varepsilon}\|\sup_{t\in[0,T)}
 \|(T_{\alpha})_{\varepsilon}(t)\|\frac{T^{\alpha}}{\alpha}
 \sum_{n=0}^{\infty}\frac{T^{n\alpha}\|\widetilde A_{\varepsilon}\|^{n}}
{\Gamma(\alpha+n\alpha)} \\
 &= \|\widetilde N_{\varepsilon}\|\sup_{t\in[0,T)}
 \|(T_{\alpha})_{\varepsilon}(t)\|\frac{T^{\alpha}}{\alpha}
 E_{\alpha,\alpha}(T^{\alpha}\|\widetilde A_{\varepsilon}\|),
\end{align*}
and using the estimate \eqref{19} for $E_{\alpha,\alpha}$ we have
\begin{align*}
 &\|((S_{\alpha})_{\varepsilon}-(T_{\alpha})_{\varepsilon})(t)\| \\
&\leq  \|\widetilde N_{\varepsilon}\|\sup_{t\in[0,T)}
 \|(T_{\alpha})_{\varepsilon}(t)\|\frac{T^{\alpha}}{\alpha}C_{\alpha}
 (1+\|\widetilde A_{\varepsilon}\|^{(1-\alpha)/\alpha})
 (1+T^{1-\alpha})\exp({T\|\widetilde A_{\varepsilon}\|^{1/\alpha}}) \\
 &= \frac{C_{\alpha}}{\alpha}\|\widetilde N_{\varepsilon}
 \|\sup_{t\in[0,T)}\|(T_{\alpha})_{\varepsilon}(t)\|
 (1+\|\widetilde A_{\varepsilon}\|^{(1-\alpha)/\alpha})
 (T+T^{\alpha})\exp({T\|\widetilde A_{\varepsilon}\|^{1/\alpha}}).
\end{align*}

Now, we consider the case $\gamma=\alpha$. For $t\in[0,T)$, $T>0$, one 
similarly gets
\begin{align*}
 \|^C{\mathcal{D}}_t^{\alpha}((S_{\alpha})_{\varepsilon}
-(T_{\alpha})_{\varepsilon})(t)\| 
&\leq \|\widetilde N_{\varepsilon}\|\sup_{t\in[0,T)}\|(T_{\alpha})_{\varepsilon}(t)\|
 \sum_{n=0}^{\infty}\frac{\|\widetilde A_{\varepsilon}\|^{n}}
 {\Gamma(n\alpha)}\cdot\frac{T^{n\alpha}}{n\alpha} \\
&= \|\widetilde N_{\varepsilon}\|\sup_{t\in[0,T)}
 \|(T_{\alpha})_{\varepsilon}(t)\|\cdot E_{\alpha}(T^{\alpha}
 \|\widetilde A_{\varepsilon}\|).
\end{align*}

Differentiation of integral representation \eqref{23} with respect to $t$, 
using integral representation \eqref{21} one gets
\begin{align*}
&\frac{d}{dt}((S_{\alpha})_{\varepsilon}-(T_{\alpha})_{\varepsilon})(t)x \\
&= \int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}
((t-\tau)^{\alpha}\widetilde A_{\varepsilon})
 \widetilde N_{\varepsilon}\frac{d}{d\tau}
(T_{\alpha})_{\varepsilon}(\tau)xd\tau + t^{\alpha-1}
E_{\alpha,\alpha}(t^{\alpha}\widetilde A_{\varepsilon})\widetilde N_{\varepsilon}x.
\end{align*}
Then, for every $T_1>0$ and $t\in [T_1,T)$, the estimate in norm is
\begin{align*}
&\|\frac{d}{dt}((S_{\alpha})_{\varepsilon}-(T_{\alpha})_{\varepsilon})(t)\| \\
&\leq  \lim_{\eta\to 0^{+}}\int_{\eta}^{t}(t-\tau)^{\alpha-1}
 \|E_{\alpha,\alpha}((t-\tau)^{\alpha}\widetilde A_{\varepsilon}) 
 \widetilde N_{\varepsilon}\frac{d}{d\tau}(T_{\alpha})_{\varepsilon}(\tau)\|d\tau \\
&\quad + t^{\alpha-1}E_{\alpha,\alpha}(t^{\alpha}\|\widetilde A_{\varepsilon}\|)
 \|\widetilde N_{\varepsilon}\|\\
&\leq \lim_{\eta\to 0^{+}}\sup_{\tau\in[\eta,T)}E_{\alpha,\alpha}
 ((T-\tau)^{\alpha}\|\widetilde A_{\varepsilon}\|)
 \|\widetilde N_{\varepsilon}\|\sup_{\tau\in[\eta,T)}
 \|\frac{d}{d\tau}(T_{\alpha})_{\varepsilon}(\tau)
\|\frac{(T-\eta)^{\alpha}}{\alpha} \\
&\quad + T_1^{\alpha-1}E_{\alpha,\alpha}(T^{\alpha}\|\widetilde A_{\varepsilon}\|)
\|\widetilde N_{\varepsilon}\|.
\end{align*}

Finally, since $\widetilde N_{\varepsilon}\in\mathcal{SN}(E)$ it follows that 
for every $a\in\mathbb{R}$ there exists $M>0$ such that 
\begin{gather*}
\sup_{t\in[0,T)}\|^C{\mathcal{D}}_t^{\gamma}((S_{\alpha})_{\varepsilon}
-(T_{\alpha})_{\varepsilon})(t)\|_{\mathcal{L}(E)}
\leq M\varepsilon^{a},\; \gamma \in \{0, \alpha\}, \\
 \sup_{t\in (0,T)} \|\frac{d}{dt}((S_{\alpha})_{\varepsilon}
-(T_{\alpha})_{\varepsilon})(t)\|_{\mathcal{L}(E)}\leq
 M\varepsilon^{a},
\end{gather*}
i.e. $ (S_{\alpha})_{\varepsilon}-(T_{\alpha})_{\varepsilon}
\in \mathcal{SN}_{\alpha}([0,\infty):\mathcal{L}(E))$.
\end{proof}

\begin{definition} \label{def5.3} \rm
Let $h_{\varepsilon}$ be a positive net satisfying 
$h_{\varepsilon}\leq \varepsilon^{-1}$. It is said that 
$\widetilde A\in \mathcal{SG}(E)$ is of $h_{\varepsilon}$-type if it has 
a representative $\widetilde A_{\varepsilon}$ such that 
$$
\|\widetilde A_{\varepsilon}\|_{\mathcal{L}(E)}=\mathcal{O}(h_{\varepsilon}), \quad
 \varepsilon\to0.
$$
An element  $G\in \mathcal{G}_{\alpha}([0,\infty):H^{1}(\mathbb{R}))$ is said to 
be of $h_{\varepsilon}$-type if it has a representative $G_{\varepsilon}$ 
such that 
$$
\|G_{\varepsilon}\|_{H^{1}}=\mathcal{O}(h_{\varepsilon}), \quad \varepsilon\to0.
$$
\end{definition}

The following proposition holds for generalized operators.

\begin{proposition}\label{9}
Let $0<\alpha<1$. Every $\widetilde A\in \mathcal{SG}(E)$ of $h_{\varepsilon}$-type,
 where $h_{\varepsilon}\leq C(\log1/\varepsilon)^{\alpha}$, is the infinitesimal 
generator of some generalized uniformly continuous solution operator 
$S_{\alpha} \in \mathcal{SG}_{\alpha}([0,\infty):{\mathcal{L}(E)})$.
\end{proposition}

\begin{proof}
Fix $0<\alpha<1$. From Theorem \ref{8} one knows that every linear and bounded 
operator $\widetilde A_{\varepsilon}$ is the infinitesimal generator of some 
uniformly continuous solution operator $(S_{\alpha})_{\varepsilon}(t)$ defined by 
$$
(S_{\alpha})_{\varepsilon}(t)=E_{\alpha}(t^{\alpha}\widetilde A_{\varepsilon})
=\sum_{n=0}^{\infty}\frac{t^{n\alpha}\widetilde A_{\varepsilon}^{n}}
{\Gamma(1+n\alpha)}.
$$
Let us show that 
$(S_{\alpha})_{\varepsilon}\in \mathcal{S}E_{M}^{\alpha}([0,\infty):{\mathcal{L}(E)})$. From the inequality for Mittag-Leffler function it follows that there exists constant $M>0$ such that $$\|(S_{\alpha})_{\varepsilon}(t)\|\leq M\exp({t\|\widetilde A_{\varepsilon}\|^{1/\alpha}}).$$
Since $h_{\varepsilon}\leq C(\log1/\varepsilon)^{\alpha}$, we have
$$
\sup_{t\in[0,T)}\|(S_{\alpha})_{\varepsilon}(t)\|
\leq M\varepsilon^{-TC^{1/{\alpha}}},
$$
for $\varepsilon$ small enough. Also, since
 $^C\mathcal{D}_t^{\alpha}(S_{\alpha})_{\varepsilon}(t)
=\widetilde A_{\varepsilon}(S_{\alpha})_{\varepsilon}(t)$, 
for every $t\geq0$, we have for every $\varepsilon$ small enough
\[
\|^C\mathcal{D}_t^{\alpha}(S_{\alpha})_{\varepsilon}(t)\|
\leq \|\widetilde A_{\varepsilon}\|\|(S_{\alpha})_{\varepsilon}(t)\| 
\leq C(\log\frac{1}{\varepsilon})^{\alpha}M\varepsilon^{-TC^{1/{\alpha}}} 
\leq CM\varepsilon^{-\alpha-TC^{1/{\alpha}}}.
\]

It remains to prove the moderate bound for
 $\|\frac{d}{dt}(S_{\alpha})_{\varepsilon}(t)\|$. First, we have
\begin{align*}
 \frac{d}{dt}(S_{\alpha})_{\varepsilon}(t)
&= \sum_{n=0}^{\infty}\frac{t^{(n+1)\alpha-1}}{\Gamma(\alpha+n\alpha)}
 \widetilde A_{\varepsilon}^{n+1}\\
&=t^{\alpha-1}\widetilde A_{\varepsilon}
\sum_{n=0}^{\infty}\frac{t^{n\alpha}}{\Gamma(\alpha+n\alpha)} 
\widetilde A_{\varepsilon}^{n} \\
&=t^{\alpha-1}
\widetilde A_{\varepsilon}E_{\alpha,\alpha}(t^{\alpha}\widetilde A_{\varepsilon}).
\end{align*}
Then, for every $T_1>0$ and $t\in [T_1,T)$, the estimate in norm is
\begin{align*}
 \|\frac{d}{dt}(S_{\alpha})_{\varepsilon}(t)\| 
&\leq T_1^{\alpha-1}\|\widetilde A_{\varepsilon}\|E_{\alpha,\alpha}
 (t^{\alpha}\|\widetilde A_{\varepsilon}\|) \\
 &\leq T_1^{\alpha-1}\|\widetilde A_{\varepsilon}
\|C_{\alpha}(1+\|\widetilde A_{\varepsilon}\|^{(1-\alpha)/{\alpha}})\cdot
 \exp({\|\widetilde A_{\varepsilon}\|^{1/\alpha}T})(1+T^{1-\alpha}) \\
 &\leq T_1^{\alpha-1}C_{\alpha}(\|\widetilde A_{\varepsilon}\|
+\|\widetilde A_{\varepsilon}\|^{1/\alpha})\cdot
 \exp({\|\widetilde A_{\varepsilon}\|^{1/\alpha}T})(1+T^{1-\alpha}) \\
 &\leq T_1^{\alpha-1}C_{\alpha}((\log\frac{1}{\varepsilon})^{\alpha}
+\log\frac{1}{\varepsilon})
 \cdot\exp({C^{1/{\alpha}}T\log\frac{1}{\varepsilon}})(1+T^{1-\alpha}) \\
 &\leq 2T_1^{\alpha-1}C_{\alpha}(1+T^{1-\alpha})\varepsilon^{-1-C^{1/\alpha}T}.
\end{align*}

Thus finally we have 
$(S_{\alpha})_{\varepsilon}\in \mathcal{S}E_{M}^{\alpha}([0,\infty):
{\mathcal{L}(E)})$.
\end{proof}

Note that a Colombeau uniformly continuous solution operator always possess
 an infinitesimal generator and it is unique. 
That follows from the fact that its representative is a classical uniformly 
continuous solution operator for which there exists a unique infinitesimal generator.

\section{Existence and uniqueness result}

In this section we specify the Banach space, i.e. we take 
$E=L^2(\mathbb{R})$. Instead of the Cauchy problem \eqref{1} with closed
 and densely defined operator $A$ on $L^2(\mathbb{R})$ with domain
$D(A)=H^{1}(\mathbb{R})$, we will consider fractional Cauchy problem given by
$$
 ^C{\mathcal{D}}_t^{\alpha}u(t)=\widetilde{A}u(t), \, t>0; \,u(0)=x,
$$
where $\widetilde{A}$ is a generalized linear bounded operator $L^2$-associated
with $A$, i.e., for every $u\in H^{1}(\mathbb{R})$, the following holds 
$$
\|(A-\widetilde{A}_{\varepsilon})u\|_{L^2}\to 0,\quad \varepsilon\to 0.
$$

\begin{theorem}\label{14}
Let $0<\alpha<1$. Suppose that $u_{0}\in \mathcal{G}_{\alpha}(H^{1}(\mathbb{R}))$ 
and let the function $f(x,t,u)$ be continuously differentiable with respect to $t$, 
globally Lipschitz with respect to $x$ and $u$ with bounded second order 
derivative with respect to $u$ and $f(x,t,0)=0$. Also, suppose that 
$\partial_{x}f(x,t,u)$ and $\partial_tf(x,t,u)$ are globally Lipschitz
 function with respect to $u$. Let $g_1(x,t,u):=\partial_{u}f(x,t,u)$ and 
$g_2(x,t,u):=\partial_tf(x,t,u)$ satisfy the same conditions as $f(x,t,u)$.

Let the operator $\widetilde{A}\in \mathcal{SG}(H^{1}(\mathbb{R}))$ be of 
$h_{\varepsilon}$-type, with 
$h_{\varepsilon}= o\big((\log(\log1/\varepsilon)^{\alpha})^{\alpha}\big)$, 
such that $\|\widetilde{A}_{\varepsilon}u_{\varepsilon}\|_{L^2}
\leq h_{\varepsilon}\|u_{\varepsilon}\|_{L^2}$, for
$u_{\varepsilon}\in H^{1}(\mathbb{R})$.

Then for every $0<\alpha<1$ there exists a unique generalized solution 
$u\in \mathcal{G}_{\alpha}([0,\infty):H^{1}(\mathbb{R}))$ to the Cauchy problem
\begin{equation}\label{6}
{}^C\mathcal{D}_t^{\alpha}u(t)=\widetilde{A}u(t)+f(\cdot,t,u), \quad
 u(0)=u_{0}.
\end{equation}

An equivalent integral equation for the solution (i.e. mild solution) is given by
 \begin{equation}\label{7}
 u_{\varepsilon}(t)=(S_{\alpha})_{\varepsilon}(t)u_{0\varepsilon}
+\int_{0}^{t}(S_{\alpha})_{\varepsilon}(t-\tau)^{RL}
\mathcal{D}_{\tau}^{1-\alpha}f(\cdot,\tau,u_{\varepsilon})d\tau,
\end{equation}
where $S_{\alpha}\in \mathcal{SG}_{\alpha}([0,\infty):
\mathcal{L}(H^{1}(\mathbb{R})))$ is a Colombeau uniformly continuous
 solution operator generated by $\widetilde{A}$.
\end{theorem}

\begin{remark} \label{rmk6.1} \rm
The existence of a solution for integral equation \eqref{7} can be proved
 using a Banach principle of a fixed point.
\end{remark}

\begin{proof}[Proof of Theorem \ref{14}]
Fix $0<\alpha<1$. Since the operator $\widetilde{A}$ is of $h_{\varepsilon}$-type,
 with $h_{\varepsilon}= o((\log\log1/\varepsilon)^{\alpha})$, 
it is obvious that the operator $\widetilde{A}$ is the infinitesimal generator 
of a Colombeau solution operator $S_{\alpha}\in
\mathcal{SG}_{\alpha}([0,\infty):\mathcal{L}(H^{1}(\mathbb{R})))$ given by
 $S_{\alpha}(t)=E_{\alpha}(t^{\alpha}\widetilde{A})$ 
(see Proposition \ref{9}). Also, from \eqref{30} we know that \eqref{7} represents 
a solution to \eqref{6}.

Let us show that this solution is an element of 
$\mathcal{G}_{\alpha}([0,\infty):H^{1}(\mathbb{R}))$. 
First, we show that the solution satisfies
\begin{equation}\label{26}
 \lim_{t\to0^{+}}\|\frac{\frac{d}{dt}u_{\varepsilon}(t,\cdot)}{t^{\alpha-1}}
\|_{H^{1}}=C<+\infty.
\end{equation}
Indeed, after differentiation of \eqref{7} with respect to $t$, using the first 
order derivative of integral representation \eqref{21} one gets
 \begin{equation}\label{33}
\begin{aligned}
&\frac{d}{dt}u_{\varepsilon}(t,\cdot) \\
&= \frac{d}{dt}(S_{\alpha}^{})_{\varepsilon}(t)u_{0\varepsilon}+\int_{0}^{t}
 (t-\tau)^{\alpha-1}
 E_{\alpha,\alpha}((t-\tau)^{\alpha}\widetilde A_{\varepsilon})
\partial_{\tau} f({\cdot,\tau,u_{\varepsilon}(\tau)})d\tau \\
 &\quad +t^{\alpha-1}E_{\alpha,\alpha}(t^{\alpha}\widetilde A_{\varepsilon})
f(\cdot,0,u_{0\varepsilon}),
\end{aligned}
 \end{equation}
and by to the notation $g_1(x,t,u)=\partial_{u}f(x,t,u)$ and 
$g_{2}(x,t,u)=\partial_tf(x,t,u)$, we have
 \begin{equation}\label{34}
\begin{aligned}
\|\frac{d}{dt}u_{\varepsilon}(t,\cdot)\|_{L^2} 
&\leq \|\frac{d}{dt}(S_{\alpha}^{})_{\varepsilon}(t)u_{0\varepsilon}\|_{L^2}\\
&\quad +\int_{0}^{t}(t-\tau)^{\alpha-1}
 E_{\alpha,\alpha}((t-\tau)^{\alpha}\|\widetilde A_{\varepsilon}\|)
 \|\partial_{\tau} f({\cdot,\tau,u_{\varepsilon}(\tau)})\|_{L^2}d\tau \\
&\quad +t^{\alpha-1}E_{\alpha,\alpha}(t^{\alpha}\|\widetilde A_{\varepsilon}\|)
 \|f(\cdot,0,u_{0\varepsilon})\|_{L^2}\\
 &\leq \|\frac{d}{dt}(S_{\alpha})_{\varepsilon}(t)u_{0\varepsilon}\|_{L^2}\\
&\quad +\int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}((t-\tau)^{\alpha}
 \|\widetilde A_{\varepsilon}\|)
 \|g_1\|_{L^{\infty}}\|\partial_{\tau}u_{\varepsilon}(\tau)\|_{L^2}d\tau\\
 &\quad + \int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}
 ((t-\tau)^{\alpha}\|\widetilde A_{\varepsilon}\|)
 \|g_{2}\|_{L^{\infty}}\|u_{\varepsilon}(\tau)\|_{L^2}d\tau\\
 &\quad + t^{\alpha-1}E_{\alpha,\alpha}(t^{\alpha}
 \|\widetilde A_{\varepsilon}\|)\|f(\cdot,0,u_{0\varepsilon})\|_{L^2}.
\end{aligned}
 \end{equation}
 After applying the Gronwall's inequality to \eqref{34} one gets
 \begin{equation*}
 \lim_{t\to0^{+}}\|\frac{\frac{d}{dt}u_{\varepsilon}(t,\cdot)}{t^{\alpha-1}}
\|_{L^2}=C<+\infty.
\end{equation*}

 Further,  differentiation of \eqref{33} with respect to $x$, we have
 \begin{equation}\label{35}
\begin{aligned}
&\|\partial_{x}\frac{d}{dt}u_{\varepsilon}(t,\cdot)\|_{L^2} \\
&\leq \|\frac{d}{dt}(S_{\alpha}^{})_{\varepsilon}(t)\partial_{x}u_{0\varepsilon}
\|_{L^2}\\
&\quad +\int_{0}^{t}(t-\tau)^{\alpha-1}
 E_{\alpha,\alpha}((t-\tau)^{\alpha}\|\widetilde A_{\varepsilon}\|)
\|\partial_{x}\partial_{\tau} f({\cdot,\tau,u_{\varepsilon}(\tau)})\|_{L^2}d\tau \\
&\quad +t^{\alpha-1}E_{\alpha,\alpha}(t^{\alpha}\|\widetilde A_{\varepsilon}\|)
 \|\partial_{x}f(\cdot,0,u_{0\varepsilon})\|_{L^2}\\
&\leq \|\frac{d}{dt}(S_{\alpha}^{})_{\varepsilon}(t)\partial_{x}u_{0\varepsilon}
 \|_{L^2} \\
&\quad +\int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}((t-\tau)^{\alpha}
 \|\widetilde A_{\varepsilon}\|)
 \|\partial_{x}g_1\|_{L^{\infty}}\|\partial_{\tau}u_{\varepsilon}(\tau)
 \|_{L^2}d\tau\\
&\quad + \int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}
 ((t-\tau)^{\alpha}\|\widetilde A_{\varepsilon}\|)
 \|g_1\|_{L^{\infty}}\|\partial_{x}\partial_{\tau}u_{\varepsilon}(\tau)
 \|_{L^2}d\tau\\
&\quad +\int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}
 ((t-\tau)^{\alpha}\|\widetilde A_{\varepsilon}\|)
 \|\partial_{x}g_{2}\|_{L^{\infty}}\|u_{\varepsilon}(\tau)\|_{L^2}d\tau\\
&\quad +\int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}((t-\tau)^{\alpha}
 \|\widetilde A_{\varepsilon}\|)
 \|g_{2}\|_{L^{\infty}}\|\partial_{x}u_{\varepsilon}(\tau)\|_{L^2}d\tau\\
&\quad  + t^{\alpha-1}E_{\alpha,\alpha}(t^{\alpha}\|\widetilde A_{\varepsilon}\|)
 [\|\partial_{u}f(\cdot,0,u_{0\varepsilon})\|_{L^{\infty}}\|\partial_{x}
 u_{0\varepsilon}\|_{L^2}\\
&\quad + \|\partial_{x}f(\cdot,0,u_{0\varepsilon})\|_{L^{\infty}}
\|u_{0\varepsilon}\|_{L^2}].
\end{aligned}
 \end{equation}
Again, after applying the Gronwall's inequality one gets
 \begin{equation*}
 \lim_{t\to0^{+}}\|\frac{\partial_{x}\frac{d}{dt}u_{\varepsilon}(t,\cdot)}
{t^{\alpha-1}}\|_{L^2}=C<+\infty,
\end{equation*}
 and finally we have that property \eqref{26} is satisfied.

Further, we prove that one has the moderate bound for 
$\|^C\mathcal{D}_t^{\gamma}u_{\varepsilon}(t,\cdot)\|_{H^{1}}$,
$\gamma \in \{0,\alpha\}$, and $\|\frac{d}{dt}u_{\varepsilon}(t,\cdot)\|_{H^{1}}$. 
First, we prove the moderate bound for $\|^C\mathcal{D}_t^{\gamma}u_{\varepsilon}(t,\cdot)\|_{H^{1}}$,
 and consider the cases:
\smallskip

\noindent\textbf{Case 1:  $\gamma=0$.}
 From the representation \eqref{7} and Proposition \ref{15} we obtain
$$
\|u_{\varepsilon}(t)\|_{L^2}\leq \|(S_{\alpha})_{\varepsilon}(t)u_{0\varepsilon}
\|_{L^2}
+\int_{0}^{t}(t-\tau)^{\alpha-1}\|E_{\alpha,\alpha}
((t-\tau)^{\alpha}\widetilde{A}_{\varepsilon})\|\,
\|f(\cdot,\tau,u_{\varepsilon})\|_{L^2}d\tau.
$$
Next, using the estimate for $E_{\alpha,\alpha}$ one gets
\begin{align*}
 \|E_{\alpha,\alpha}(t^{\alpha}\widetilde{A}_{\varepsilon})\| 
&\leq \sum_{n=0}^{\infty}\frac{t^{n\alpha}
 \|\widetilde{A}_{\varepsilon}\|^{n}}{\Gamma(\alpha+n\alpha)}
 =E_{\alpha,\alpha}(t^{\alpha}\|\widetilde{A}_{\varepsilon}\|) \\
&\leq C_{\alpha}(1+\|\widetilde{A}_{\varepsilon}\|^{(1-\alpha)/\alpha})
 (1+t^{1-\alpha})\exp(t\|\widetilde{A}_{\varepsilon}\|^{1/\alpha}).
\end{align*}
Denote
\begin{equation}\label{31}
\widetilde{M}_{T}:=\sup_{t\in[0,T)}\|E_{\alpha,\alpha}(t^{\alpha}
\widetilde{A}_{\varepsilon})\|.
\end{equation}
 Note that for $\alpha=1$ it follows $\widetilde{M}_{T}:=\sup_{t\in[0,T)}\|S(t)\|$,
 where $S(t)$ is a generalized uniformly continuous semigroup of operators 
generated by the operator $\widetilde{A}$ (see \cite{Japundzic}).
Next,
\begin{equation}\label{32}
\begin{aligned}
 \widetilde{M}_{T}
& \leq C_{\alpha}(1+o((\log\log1/\varepsilon)^{1-\alpha}))
(1+T^{1-\alpha})\exp(T\cdot o(\log\log1/\varepsilon)) \\
&= \mathcal{O}(\log1/\varepsilon),
\end{aligned}
\end{equation}
by the well known properties of Landau's symbol $o$.
From
\begin{align*}
 \|u_{\varepsilon}(t)\|_{L^2}
&\leq \|(S_{\alpha})_{\varepsilon}(t)u_{0\varepsilon}\|_{L^2}+\widetilde{M}_{T}
\int_{0}^{t}(t-\tau)^{\alpha-1}\|f(\cdot,\tau,u_{\varepsilon})\|_{L^2}d\tau \\
 &\leq \|(S_{\alpha})_{\varepsilon}(t)u_{0\varepsilon}\|_{L^2}
+C\widetilde{M}_{T}\int_{0}^{t}(t-\tau)^{\alpha-1}\|u_{\varepsilon}
(\tau)\|_{L^2}d\tau,
 \end{align*}
 using  Gronwall's inequality we obtain the moderate bound for 
$\|u_{\varepsilon}(t)\|_{L^2}$.

 After differentiation of \eqref{7} with respect to $x$, using similar integral 
representation as the one in Proposition \ref{15} we have 
$$
\partial_{x}u_{\varepsilon}(t)=(S_{\alpha})_{\varepsilon}(t)\partial_{x}
u_{0\varepsilon}+\int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}
((t-\tau)^{\alpha}\widetilde{A}_{\varepsilon})\partial_{x}
f(\cdot,\tau,u_{\varepsilon})d\tau
$$
 and
 \begin{align*}
&\|\partial_{x}u_{\varepsilon}(t)\|_{L^2}\\
&\leq \|(S_{\alpha})_{\varepsilon}(t)\partial_{x}u_{0\varepsilon}\|_{L^2}
+ \int_{0}^{t}(t-\tau)^{\alpha-1}\|E_{\alpha,\alpha}((t-\tau)^{\alpha}
\widetilde{A}_{\varepsilon})\|\|\partial_{x}(f(\cdot,\tau,u_{\varepsilon}))
\|_{L^2}d\tau \\
&\leq \|(S_{\alpha})_{\varepsilon}(t)\partial_{x}u_{0\varepsilon}\|_{L^2}
  +\widetilde{M}_{T}\int_{0}^{t}(t-\tau)^{\alpha-1}
 \|\partial_{u}f\|_{L^{\infty}}\|\partial_{x}u_{\varepsilon}(\tau)\|_{L^2}d\tau\\
&\quad +\widetilde{M}_{T}\int_{0}^{t}(t-\tau)^{\alpha-1}
\|\partial_{x}f\|_{L^{\infty}}\|u_{\varepsilon}(\tau)\|_{L^2}d\tau.
 \end{align*}
Since $f$ is Lipschitz with respect to $u$ and $x$ the moderate bound for
 $\|\partial_{x}u_{\varepsilon}(t)\|_{L^2}$ again follows from the Gronwall's
 inequality.
\smallskip

\noindent\textbf{Case 2: $\gamma=\alpha$.}
 From \eqref{6} we have 
$$
\|^C\mathcal{D}_t^{\alpha}u_{\varepsilon}(t)\|_{L^2}
\leq\|\widetilde{A}_{\varepsilon}u_{\varepsilon}(t)\|_{L^2}
+\|f(\cdot,t,u_{\varepsilon})\|_{L^2}.
$$
 Since $f$ is globally Lipschitz with respect to $u$ and $f(x,t,0)=0$,
 it follows the moderate bound for 
$\|^C\mathcal{D}_t^{\alpha}u_{\varepsilon}(t)\|_{L^2}$.

Differentiation of \eqref{6} with respect to $x$ we have
 \begin{align*}
 \|\partial_{x}^C\mathcal{D}_t^{\alpha}u_{\varepsilon}(t)\|_{L^2}
 &\leq \|\partial_{x}(\widetilde{A}_{\varepsilon}u_{\varepsilon}(t))\|_{L^2}
 +\|\partial_{x}(f(\cdot,t,u_{\varepsilon}))\|_{L^2} \\
 &\leq C(\log1/\varepsilon)^{\alpha} \|u_{\varepsilon}(t)\|_{H^{1}}
 +\|\partial_{u}f\|_{L^{\infty}}\|\partial_{x}u_{\varepsilon}(t)\|_{L^2} \\
 &\quad + \|\partial_{x}f\|_{L^{\infty}}\|u_{\varepsilon}(t)\|_{L^2},
 \end{align*}
 and the moderate bound for 
$\|\partial_{x}^C\mathcal{D}_t^{\alpha}u_{\varepsilon}(t)\|_{L^2}$
immediately follows.


The moderate bound for $\|\frac{d}{dt}u_{\varepsilon}(t,\cdot)\|_{H^{1}}$ 
follows after applying the Gronwall's inequality to inequalities \eqref{34} 
and \eqref{35}.

To prove that this solution is unique in Colombeau space 
$\mathcal{G}_{\alpha}([0,\infty):H^{1}(\mathbb{R}))$, suppose that there 
exist two solutions $u$ and $v$ to \eqref{6} and set
 $\omega_{\varepsilon}=u_{\varepsilon}-v_{\varepsilon}$. This difference satisfies
\begin{equation}\label{11}
 ^C\mathcal{D}_t^{\alpha}\omega_{\varepsilon}(t)
=\widetilde{A}_{\varepsilon}\omega_{\varepsilon}(t)+f(\cdot,t,u_{\varepsilon})
-f(\cdot,t,v_{\varepsilon}) + \widetilde N_{\varepsilon}(t), \quad
 \omega_{\varepsilon}(0)=\omega_{0\varepsilon},
\end{equation}
where $\widetilde N_{\varepsilon}(t)\in \mathcal{N}_{\alpha}([0,\infty):H^{1}
(\mathbb{R}))$ and 
$\omega_{0\varepsilon}\in \mathcal{N}_{\alpha}(H^{1}(\mathbb{R}))$. Then
\begin{equation}\label{12}
\begin{aligned}
 \omega_{\varepsilon}(t) 
&= (S_{\alpha})_{\varepsilon}(t)\omega_{0\varepsilon}
+\int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}((t-\tau)^{\alpha}
 \widetilde{A}_{\varepsilon})(f(\cdot,\tau,u_{\varepsilon})\\
&\quad -f(\cdot,\tau,v_{\varepsilon}))d\tau 
 +\int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}((t-\tau)^{\alpha}
\widetilde{A}_{\varepsilon})\widetilde N_{\varepsilon}(\tau)d\tau,
\end{aligned}
 \end{equation}
and
\begin{align*}
\|\omega_{\varepsilon}(t)\|_{L^2} 
&\leq \|(S_{\alpha})_{\varepsilon}(t)\omega_{0\varepsilon}\|_{L^2}
 +\int_{0}^{t}(t-\tau)^{\alpha-1}\|E_{\alpha,\alpha}((t-\tau)^{\alpha}
 \widetilde{A}_{\varepsilon})\|\cdot\|f(\cdot,\tau,u_{\varepsilon})\\
&\quad -f(\cdot,\tau,v_{\varepsilon})\|_{L^2}d\tau
 +\int_{0}^{t}(t-\tau)^{\alpha-1}\|E_{\alpha,\alpha}((t-\tau)^{\alpha}
 \widetilde{A}_{\varepsilon})\|\cdot\|\widetilde N_{\varepsilon}
(\tau)\|_{L^2}d\tau.
\end{align*}
 Since $\|E_{\alpha,\alpha}((t-\tau)^{\alpha}\widetilde{A}_{\varepsilon})\|
\leq \widetilde{M}_{T}$, $0\leq t\leq T$, $0\leq\tau\leq t$, where 
$\widetilde{M}_{T}$ is estimated by \eqref{32} and since $f$ is a Lipschitz 
function with respect to $u$, we obtain the $\mathcal{N}$-bound for 
$\|\omega_{\varepsilon}(t)\|_{L^2}$.

Equation \eqref{11} implies 
$$
\|^C\mathcal{D}_t^{\alpha}\omega_{\varepsilon}(t)\|_{L^2}
\leq\|\widetilde{A}_{\varepsilon}\omega_{\varepsilon}(t)\|_{L^2}
+\|f(\cdot,t,u_{\varepsilon})-f(\cdot,t,v_{\varepsilon})
 \|_{L^2}+\|\widetilde N_{\varepsilon}(t)\|_{L^2},
$$
and the $\mathcal{N}$-bound for 
$\|^C\mathcal{D}_t^{\alpha}\omega_{\varepsilon}(t)\|_{L^2}$
immediately follows.

Differentiation of \eqref{12} with respect to $x$ we have
 \begin{align*}
&\|\partial_{x}\omega_{\varepsilon}(t)\|_{L^2} \\
&\leq\|(S_{\alpha})_{\varepsilon}(t)\partial_{x}\omega_{0\varepsilon}\|_{L^2} \\
 &\quad +\int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}((t-\tau)^{\alpha}
 \|\widetilde{A}_{\varepsilon}\|)\|\partial_{u}f(\cdot,\tau,u_{\varepsilon})
 \partial_{x}u_{\varepsilon}-\partial_{u}f(\cdot,\tau,v_{\varepsilon})
 \partial_{x}v_{\varepsilon}\|_{L^2}d\tau \\
&\quad +\int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}((t-\tau)^{\alpha}
 \|\widetilde{A}_{\varepsilon}\|)\|\partial_{x}f(\cdot,\tau,u_{\varepsilon})
 u_{\varepsilon}-\partial_{x}f(\cdot,\tau,v_{\varepsilon})v_{\varepsilon}
 \|_{L^2}d\tau \\
&\quad +\int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}((t-\tau)^{\alpha}
 \|\widetilde{A}_{\varepsilon}\|)\|\partial_{x}\widetilde N_{\varepsilon}(\tau)
 \|_{L^2}d\tau.
\end{align*}
However, $f$ has bounded second order derivative with respect to $u$, 
and similarly to \cite{Japundzic} we have
 \begin{align*}
&\|\partial_{u}f(\cdot,\tau,u_{\varepsilon})\partial_{x}u_{\varepsilon}
-\partial_{u}f(\cdot,\tau,v_{\varepsilon})
\partial_{x}v_{\varepsilon}\|_{L^2}  \\
&\leq C_1\|\partial_{x}u_{\varepsilon}(\tau)\|_{L^2}
 \|\omega_{\varepsilon}(\tau)\|_{H^{1}}
 +C_{2}\|\partial_{x}\omega_{\varepsilon}(\tau)\|_{L^2}.
 \end{align*}
Also, since $\partial_{x}f$ is Lipschitz with respect to $u$ we have
\begin{align*}
&\|\partial_{x}f(\cdot,\tau,u_{\varepsilon})u_{\varepsilon}
-\partial_{x}f(\cdot,\tau,v_{\varepsilon})v_{\varepsilon}\|_{L^2} \\
&\leq \|\partial_{x}f(\cdot,\tau,u_{\varepsilon})\|_{L^{\infty}} 
 \|\omega_{\varepsilon}(\tau)\|_{L^2}
 +\|v_{\varepsilon}(\tau)\|_{H^{1}}
 \|\partial^2_{u}f(\cdot,\tau,\widetilde{y})\|_{L^{\infty}}
 \|\omega_{\varepsilon}(\tau)\|_{L^2},
 \end{align*}
for some function $\widetilde{y}\in H^{1}(\mathbb{R})$, and the 
$\mathcal{N}$-bound for $\|\partial_{x}\omega_{\varepsilon}(t)\|_{L^2}$
follows from the Gronwall's inequality.

 Differentiation of \eqref{11} with respect to $x$ yields
\[
 \|\partial_{x}^C\mathcal{D}_t^{\alpha}\omega_{\varepsilon}(t)\|_{L^2}
\leq \|\partial_{x}(\widetilde{A}_{\varepsilon}\omega_{\varepsilon}(t))\|_{L^2}
+\|\partial_{x}(f(\cdot,t,u_{\varepsilon})-f(\cdot,t,v_{\varepsilon})) \|_{L^2}
 +\|\partial_{x}\widetilde N_{\varepsilon}(t)\|_{L^2},
\]
 and the $\mathcal{N}$-bound for 
$\|\partial_{x} ^C\mathcal{D}_t^{\alpha}\omega_{\varepsilon}(t)\|_{L^2}$
immediately follows.

The $\mathcal{N}$-bound for $\|\frac{d}{dt}\omega_{\varepsilon}(t)\|_{H^{1}}$ 
can be obtained in a similar manner: first by differentiating equation 
\eqref{12} with respect to $t$, then differentiating this new equation with 
respect to $x$, and, at the end, by applying the Gronwall's inequality.

Finally, it follows that 
$\omega_{\varepsilon}:=u_{\varepsilon}-v_{\varepsilon}\in 
\mathcal{N}_{\alpha}([0,\infty):H^{1}(\mathbb{R}))$, i.e. the solution is unique.
 \end{proof}

\begin{remark} \label{rmk6.2} \rm
If $\widetilde{A}\in \mathcal{SG}(H^2(\mathbb{R}))$ is an operator of
 $h_{\varepsilon}$-type with 
\[
h_{\varepsilon}= o\Big((\log(\log1/\varepsilon)^{\alpha})^{\alpha}\Big), 
\]
similarly one can prove that solution to \eqref{6} is also represented 
by \eqref{7} and this unique solution belongs to 
$\mathcal{G}_{\alpha}([0,\infty):H^2(\mathbb{R}))$.
\end{remark}

\begin{definition} \rm
The solution $u$ of  problem \eqref{6} introduced in Theorem \ref{14} 
is called generalized solution of the equation
\begin{equation*}
 ^C\mathcal{D}_t^{\alpha}u(t)=\widetilde{A}u(t)+f(\cdot,t,u)
\end{equation*}
with generalized operators.
\end{definition}

\section{Comparison of solutions to the original and approximate problems}

In this section we prove that, under certain additional conditions, the 
solutions of problem \eqref{1} and corresponding approximate problem \eqref{1a} 
are $L^2$-associated.

\begin{theorem} \label{comparison}
Let $0<\alpha<1$. Assume that there exists the solution, 
$u_{\varepsilon}\in H^2(\mathbb{R})$, of the equation
\begin{equation}\label{res1}
 ^C\mathcal{D}_t^{\alpha}u_{\varepsilon}(x,t)
=Au_{\varepsilon}(x,t)+f(x,t,u_{\varepsilon}(x,t)), \quad t>0, \;
 x\in \mathbb{R}, \quad u_{\varepsilon}(0)=u_{0 \varepsilon},
\end{equation}
where $A$ is a closed linear operator densely defined in the Banach space 
$L^2(\mathbb{R})$ with domain $D(A)=H^{1}(\mathbb{R})$ and property
$A: H^2(\mathbb{R})\to H^{1}(\mathbb{R})$. Let $v_{\varepsilon}$ be a
solution of the corresponding approximate equation with the same initial data:
\begin{equation}\label{res2}
 {}^C\mathcal{D}_t^{\alpha}v_{\varepsilon}(x,t)
=\widetilde{A}_{\varepsilon} v_{\varepsilon}(x,t)
+f(x,t,v_{\varepsilon}(x,t)), \quad t>0, \; x\in \mathbb{R}, \;
 v_{\varepsilon}(0)=u_{0 \varepsilon},
\end{equation}
where $f$ and $\widetilde{A}\in \mathcal{S}G(H^{1}(\mathbb{R}))$ are given as 
in Theorem \ref{14}. Additionally, let the generalized operator 
$\widetilde{A}$ satisfies:
\begin{itemize}
 \item[(i)] $\|\widetilde{A}_{\varepsilon}u_{\varepsilon}\|_{L^2}
\leq C\|u_{\varepsilon}\|_{H^{1}}$, for $u_{\varepsilon}\in H^2(\mathbb{R})$,
 where C does not depend on $\varepsilon$.
 \item[(ii)] $\|(A-\widetilde{A}_{\varepsilon})u_{\varepsilon}\|_{H^{1}}\to 0$, 
for $u_{\varepsilon}\in H^2(\mathbb{R})$, when $\varepsilon\to 0$.
\end{itemize}
Then the solutions $u_{\varepsilon}$ and $v_{\varepsilon}$ are $L^2-$associated,
i.e., for every $T>0$,
$$
\sup_{t\in [0,T)}\|u_{\varepsilon}(t)-v_{\varepsilon}(t)\|_{L^2}\to 0, \quad
 \text{as }  \varepsilon \to 0.
$$
\end{theorem}

\begin{remark} \label{rmk7.1} \rm
 The generalized operator $\widetilde{A}$ satisfying properties (i) and (ii) 
can be obtained, for instance, by regularization of space fractional or integer 
order derivatives appearing in the operator $A$.
 For details we refer to \cite{Japundzic}.
\end{remark}

\begin{proof}[Proof of Theorem \ref{comparison}]
Fix $0<\alpha<1$. Since $u_{\varepsilon}$ and $v_{\varepsilon}$ satisfy 
the equations \eqref{res1} and \eqref{res2}, respectively, one gets
\begin{equation}
\begin{aligned}
{}^C\mathcal{D}_t^{\alpha}(u_{\varepsilon}(x,t)-v_{\varepsilon}(x,t))
&= \widetilde{A}_{\varepsilon}(u_{\varepsilon}(x,t)-v_{\varepsilon}(x,t)) 
 + (A-\widetilde{A}_{\varepsilon})u_{\varepsilon}(x,t)  \\
&\quad +  f(x,t,u_{\varepsilon})-f(x,t,v_{\varepsilon}).
\end{aligned} \label{res}
\end{equation}
Put $\omega_{\varepsilon}=u_{\varepsilon}-v_{\varepsilon}$.
 Then \eqref{res} becomes
\begin{equation*}
 ^C\mathcal{D}_t^{\alpha}\omega_{\varepsilon}(t)
=\widetilde{A}_{\varepsilon}\omega_{\varepsilon}(t)+f(\cdot,t,u_{\varepsilon})
-f(\cdot,t,v_{\varepsilon})+N_{\varepsilon}(t),\quad
\omega_{\varepsilon}(0)=0, 
\end{equation*}
where $N_{\varepsilon}(t)=(A-\widetilde{A}_{\varepsilon})u_{\varepsilon}(\cdot,t)$.
Then $\omega_{\varepsilon}$ satisfies
\begin{align*}
 \omega_{\varepsilon}(t)
&= \int_{0}^{t}(S_{\alpha})_{\varepsilon}(t-\tau)^{RL}\mathcal{D}_{\tau}^{1-\alpha}
 (f(\cdot,\tau,u_{\varepsilon})
-f(\cdot,\tau,v_{\varepsilon}))d\tau  \\
&\quad + \int_{0}^{t}(S_{\alpha})_{\varepsilon}
(t-\tau)^{RL}\mathcal{D}_{\tau}^{1-\alpha}
N_{\varepsilon}(\tau)d\tau,
\end{align*}
and from integral representation \eqref{22} it follows that
\begin{align*}
 \omega_{\varepsilon}(t)
&= \int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}((t-\tau)^{\alpha}
\widetilde{A}_{\varepsilon})
 (f(\cdot,\tau,u_{\varepsilon})-f(\cdot,\tau,v_{\varepsilon}))d\tau \\
&\quad + \int_{0}^{t}(t-\tau)^{\alpha-1}E_{\alpha,\alpha}
((t-\tau)^{\alpha}\widetilde{A}_{\varepsilon})N_{\varepsilon}(\tau)d\tau.
\end{align*}
The estimation in the norm gives
\begin{align*}
 \|\omega_{\varepsilon}(t)\|_{L^2}
&\leq  \int_{0}^{t}(t-\tau)^{\alpha-1}\|E_{\alpha,\alpha}
 ((t-\tau)^{\alpha}\widetilde{A}_{\varepsilon})
 (f(\cdot,\tau,u_{\varepsilon})-f(\cdot,\tau,v_{\varepsilon}))\|_{L^2}d\tau \\
&\quad + \int_{0}^{t}(t-\tau)^{\alpha-1}\|E_{\alpha,\alpha}
((t-\tau)^{\alpha}\widetilde{A}_{\varepsilon})
 N_{\varepsilon}(\tau)\|_{L^2}d\tau.
\end{align*}

By  assumption (i) we have 
$\|\widetilde{A}_{\varepsilon}u_{\varepsilon}\|_{L^2}
\leq C\|u_{\varepsilon}\|_{H^{1}}$, where $C$ does not depend on $\varepsilon$. 
Therefore 
$$
\|E_{\alpha,\alpha}(t^{\alpha}\widetilde{A}_{\varepsilon})u_{\varepsilon}\|_{L^2}
\leq E_{\alpha,\alpha}(t^{\alpha}C)\|u_{\varepsilon}\|_{H^{1}},
$$
for $u_{\varepsilon}\in H^2(\mathbb{R})$. Further, from the assumption (ii)
it follows 
$$
\|E_{\alpha,\alpha}((t-\tau)^{\alpha}\widetilde{A}_{\varepsilon})
 N_{\varepsilon}(\tau)\|_{L^2}\to 0,
$$ 
as $\varepsilon \to 0$.
Using that $\|\partial_{u}f\|_{L^{\infty}}\leq C_1<\infty$ and the estimate
$$
\|f(\cdot,s,u_{\varepsilon})-f(\cdot,s,v_{\varepsilon})\|_{L^2}
\leq\|\partial_{u}f\|_{L^{\infty}}\cdot\|u_{\varepsilon}(s)
-v_{\varepsilon}(s)\|_{L^2}\leq C_1\|\omega_{\varepsilon}(s)\|_{L^2},
$$
Gronwall's inequality gives 
$\sup_{t\in [0,T)}\|\omega_{\varepsilon}(t)\|_{L^2}\to 0$, as $\varepsilon \to 0$.
\end{proof}

\begin{remark}\label{second} \rm
The similar result can be obtained in the case when $A$ is a closed linear 
operator densely defined in the Banach space $L^2(\mathbb{R})$ with domain
 $D(A)=H^2(\mathbb{R})$ and property $A: H^{4}(\mathbb{R})\to H^2(\mathbb{R})$,
assuming that there exists the solution $u_{\varepsilon}\in H^{4}(\mathbb{R})$.
\end{remark}

\section{Applications to fractional differential equations with space 
variable coefficients}

In this section we give the explicit examples and illustrate how one can obtain 
the approximate operator $\tilde A$ for a given (integer or fractional) 
differential operator $A$. In these examples, the corresponding generalized 
operators will be in the form of regularized operators. 
The regularization is necessary in order to transform unbounded differential 
operators into bounded operators. In all examples that we list below, 
one can prove that the operators $A$ and $\widetilde A$ satisfy similar properties 
(i) and (ii) from Theorem \ref{comparison} (for details we refer \cite{Japundzic}).

\subsection{Time fractional reaction-diffusion equation}

Let $0<\alpha<1$ and let $f(x,t,u(x,t))$ describes the outer force in the Cauchy 
problem for equation with space variable coefficients, i.e.
\begin{equation*}
 ^C\mathcal{D}_t^{\alpha}u(x,t)=\lambda(x)
\partial_{x}^2u(x,t)+f(x,t,u(x,t), \quad  t>0, \; x\in \mathbb{R},
\end{equation*}
where the function $f$ satisfy conditions from the Theorem \ref{14} and 
$\lambda(x)$ is a such that the operator 
$A=\lambda(x) \partial_{x}^2$ satisfies the conditions from
Remark \ref{second} (for example, one can choose 
$\lambda \in L^{\infty}(\mathbb{R})$). 
The equation of this type is very important in the theory of fractional
 Brownian motion and anomalous transport of premises \cite{Saydamatov}. 
Also, this equation is used in population biology to model the spread of 
invasive species. In that case, $u(x, t)$ is the population density
at location $x\in\mathbb{R}$ and time $t > 0$. The first term on the 
right-hand side is the diffusion term ($\lambda(x)$ is a diffusion 
coefficient) and it models migration, while the second term $f(x,t,u(x,t))$ 
is the reaction term that models population growth.

Instead of the previous problem let us consider the corresponding approximate
 problem
\begin{equation*}
 {}^C\mathcal{D}_t^{\alpha}u(x,t)=\widetilde{A} u(x,t)+f(x,t,u(x,t)), \quad t>0, 
\;  x\in \mathbb{R},
\end{equation*}
where the operator $\widetilde{A}\in \mathcal{SG}(H^2(\mathbb{R}))$
is represented by the nets of operators
\begin{gather*}
\widetilde{A}_{\varepsilon}:H^2(\mathbb{R})\to H^2(\mathbb{R}), \\
\widetilde{A}_{\varepsilon}u_{\varepsilon}=\lambda_{\varepsilon}(x)(
\partial_{x}^2u_{\varepsilon}*\phi_{h_{\varepsilon}}),
\end{gather*}
such that
$\lambda_{\varepsilon}\in H^2(\mathbb{R})$,
$\|\lambda_{\varepsilon}\|_{H^2(\mathbb{R})}
=\mathcal{O}\Big((\log(\log1/\varepsilon)^{\alpha})^{\alpha/2}\Big)$,
 $\phi_{h_{\varepsilon}}(x)=h_{\varepsilon}\phi(xh_{\varepsilon})$, where 
$h_{\varepsilon}= o\Big((\log(\log1/\varepsilon)^{\alpha})^{\alpha/5}\Big)$, 
$\phi\in C_{0}^{\infty}(\mathbb{R})$,  $\phi(x)\geq 0 $ and $\int\phi(x)dx=1$.

Then, the mild solution is given by \eqref{7} and the solution belongs 
to Colombeau space $\mathcal{G}_{\alpha}([0,\infty):H^2(\mathbb{R}))$.


\subsection{Time-space fractional reaction-diffusion equation}

Instead of the Cauchy problem for the time-space fractional equation with 
variable coefficients and with $f$ satisfying conditions from the 
Theorem \ref{14}, let us consider the corresponding approximate problem, i.e.
\begin{equation*}
 {}^C\mathcal{D}_t^{\alpha}u(t)=\widetilde{A}_{\beta}u(t)+f(\cdot,t,u),
\end{equation*}
where $0<\alpha<1$, $1<\beta<2$, the operator 
$\widetilde{A}_{\beta}\in \mathcal{SG}(H^2(\mathbb{R}))$ is represented
by the nets of operators
\begin{gather*}
(\widetilde{A}_{\beta})_{\varepsilon}:H^2(\mathbb{R})\to H^2(\mathbb{R}), \\
(\widetilde{A}_{\beta})_{\varepsilon}u_{\varepsilon}=\lambda_{\varepsilon}(x)(
\mathcal{D}_{+}^{\beta}u_{\varepsilon}*\phi_{h_{\varepsilon}}),
\end{gather*}
where $\mathcal{D}_{+}^{\beta}$ is the left Liouville fractional derivative of 
order $\beta$ on the whole axis $\mathbb{R}$ given by
$$
(\mathcal{D}_{+}^{\beta}u)(x)=\frac{1}{\Gamma(2-\beta)}
\big(\frac{d}{dx}\big)^2 \int_{-\infty}^{x}\frac{u(\xi)}{(x-\xi)^{\beta-1}}d\xi,
$$
$\lambda_{\varepsilon}\in
H^2(\mathbb{R})$ and $\phi_{h_{\varepsilon}}(x)$ satisfies the same properties 
as in the case of time fractional diffusion equation.

Then, the mild solution is given by \eqref{7} and the solution belongs 
to Colombeau space $\mathcal{G}_{\alpha}([0,\infty):H^2(\mathbb{R}))$. 
The same result holds if instead of left $\beta$th Liouville fractional 
derivative in the fractional operator $\widetilde{A}_{\beta}$, $1<\beta<2$, 
one uses right $\beta$th Liouville fractional derivative or Riesz 
$\beta$th fractional derivative.

\subsection{Time-space fractional reaction-advection-diffusion equation}

Let $0<\alpha<1$, $0<\beta\leq1$, $1<\gamma\leq2$ and consider fractional equation
\begin{align*}
 & &
 ^C\mathcal{D}_t^{\alpha}u(t)=-a(x)
\mathcal{D}_{+}^{\beta}u(t)+b(x)
\mathcal{D}_{+}^{\gamma}u(t)+f(\cdot,t,u),
\end{align*}
where $a(x)$ and $b(x)$ are such that corresponding differential operator 
$A$ again satisfies the conditions from Remark \ref{second}.

Such equation has a physical meaning, since it is an appropriate model for
 many interesting phenomena. For example, it models the transport of a chemical
 or biological tracer carried by water through a medium that is uniform, porous and
saturated. In that case, $u$ is a solute concentration,
$a(x)$ and $b(x)$ represent fluid velocity and the dispersion,
respectively, while $f$ is a given contaminant source which is common
in hydrogeological phenomena.

Again we consider the corresponding approximate Cauchy problem for the time-space 
fractional reaction-advection-diffusion equation with variable coefficients 
and with $f$ satisfying conditions from the Theorem \ref{14}, i.e.
\begin{equation*}
 ^C\mathcal{D}_t^{\alpha}u(t)=\widetilde{A}_{\beta,\gamma}u(t)+f(\cdot,t,u),
\end{equation*}
where $0<\alpha<1$, $0<\beta\leq1$, $1<\gamma\leq2$ and the operator 
$\widetilde{A}_{\beta,\gamma}\in \mathcal{SG}(H^2(\mathbb{R}))$ 
is represented by the nets of operators
\begin{gather*}
(\widetilde{A}_{\beta,\gamma})_{\varepsilon}:H^2(\mathbb{R})\to H^2(\mathbb{R}), \\
(\widetilde{A}_{\beta,\gamma})_{\varepsilon}u_{\varepsilon}=-a_{\varepsilon}(x)(
\mathcal{D}_{+}^{\beta}u_{\varepsilon}*\phi_{h_{\varepsilon}})+b_{\varepsilon}(x)(
\mathcal{D}_{+}^{\gamma}u_{\varepsilon}*\phi_{h_{\varepsilon}}),
\end{gather*}
assuming that functions $a_{\varepsilon}$ and $b_{\varepsilon}$ satisfy
similar conditions as $\lambda_{\varepsilon}$ in time fractional diffusion equation.

The mild solution is given by \eqref{7}, the solution belongs to Colombeau 
space $\mathcal{G}_{\alpha}([0,\infty):H^2(\mathbb{R}))$ and one can 
uses right Liouville fractional derivative or Riesz fractional derivative 
instead of the left Liouville fractional derivative.


\subsection*{Acknowledgements}

The authors would like to thank to the anonymous referees for their valuable
suggestions that improved the paper.
This research was supported by the projects OI174024, III44006 financed by
the Ministry of Science, Republic of Serbia. 
This research was also partially supported by the Project 142-451-2489 of 
the Provincial Secretariat for Higher Education and Scientific Research.

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\end{document}