# Perturbation theory for fractional evolution equations in a Banach space

Arzu Ahmadova<EMAIL_ADDRESS>Ismail T. Huseynov
<EMAIL_ADDRESS>Nazim I. Mahmudov<EMAIL_ADDRESS>Department of Mathematics, Eastern Mediterranean University, Mersin 10, 99628,
T.R. North Cyprus, Turkey

###### Abstract

A strong inspiration for studying perturbation theory for fractional evolution
equations comes from the fact that they have proven to be useful tools in
modeling many physical processes. In this paper, we study fractional evolution
equations of order $\alpha\in(1,2]$ associated with the infinitesimal
generator of an operator fractional cosine function generated by bounded time-
dependent perturbations in a Banach space. We show that the abstract
fractional Cauchy problem associated with the infinitesimal generator $A$ of a
strongly continuous fractional cosine function remains uniformly well-posed
under bounded time-dependent perturbation of $A$. We also provide some
necessary special cases.

###### keywords:

Perturbation theory, fractional evolution equation, strongly continuous
fractional cosine and sine families, well-posedness

##  1 Introduction

Some partial differential equations arising in the transverse motion of an
extensible beam [1], the vibration of hinged bars [2], damped McKean-Vlasov
equations [3] and many other physical phenomena can be formulated as the
second-order abstract differential equations with cosine families in infinite-
dimensional spaces. The most fundamental and extensive work on cosine families
of linear operators is that of Fattorini in [4, 5].

Moreover, Travis and Webb [6] have investigated the following semilinear
second-order Cauchy problem in a Banach space $\mathbb{X}$ via the theory of
strongly continuous cosine families of linear bounded operators:

$\begin{cases}u^{\prime\prime}(t)=Au(t)+f(t,u(t),u^{\prime}(t)),\quad
t\in\mathbb{R},\\\ u(t_{0})=x\in\mathbb{X},\quad
u^{\prime}(t_{0})=y\in\mathbb{X}.\end{cases}$ (1.1)

A systematic and general treatment of the abstract semilinear second-order
Cauchy problem from the stand point of existence, uniqueness, continuous
dependence and smoothness of solutions have been provided by Travis and Webb
in [6, 7]. Also, Bochenek in [8] has investigated the existence of a solution
of the initial value problem for the second-order abstract evolution equation
under more general hypotheses than in [6]. The fractional analogue of the same
problem (1.1) with Caputo time-derivative is considered by Kexue [9] and
general results are attained by using the theory of fractional cosine
families.

The existence and uniqueness of the following inhomogeneous $\alpha$-order
Cauchy problem is studied by Kexue et al. [10] in a Banach space $\mathbb{X}$:

$\begin{cases*}\left(\prescript{}{}{\mathcal{D}^{\alpha}_{t}}u\right)(t)=Au(t)+f(t),\quad
t>0,\\\
\left(\mathcal{I}^{2-\alpha}u\right)(0)=0,\quad\left(\mathcal{I}^{1-\alpha}u\right)(0)=x\in\mathbb{X},\end{cases*}$

where $1<\alpha<2$, $A:\mathscr{D}(A)\subseteq\mathbb{X}\to\mathbb{X}$ is a
closed, densely-defined linear operator, $\mathscr{D}(A)$ is the domain of $A$
endowed with the graph norm $\left\lVert
x\right\rVert_{\mathscr{D}(A)}=\left\lVert x\right\rVert+\left\lVert
Ax\right\rVert$, and $\prescript{}{}{\mathcal{D}^{\alpha}_{t}}$ is the
$\alpha$-order Riemann-Liouville fractional derivative operator. Furthermore,
in [11] Chen and Li have established the properties of three kinds of
resolvent families defined by purely algebraic equations which extend the
classical cosine functional equation. In [12], Henríquez et al. have studied
the differentiability of mild solutions for a class of fractional abstract
Cauchy problems of order $\alpha\in(1,2)$.

The perturbation theory has long been a very useful tool in the hands of both
the analyst and physicist. A considerable amount of research has been done on
the perturbation theory of linear operators in a Banach space, principally by
Phillips [13], Lutz [14], Travis and Webb [15].

In [13], Philips has obtained the closed-form of the classical solution to the
following perturbed Cauchy problem in a Banach space $\mathbb{X}$:

$\begin{cases*}u^{\prime}(t)=(A+B(t))u(t)+f(t),\quad t>0,\\\
u(0)=x\in\mathscr{D}(A),\end{cases*}$

where $B:\mathbb{R_{+}}\to\mathcal{L}(\mathbb{X})$,
$f:\mathbb{R_{+}}\to\mathbb{X}$ be strongly continuously differentiable
functions on $\mathbb{R_{+}}$.

In [14], Lutz has investigated for the first time the effect on the
homogeneous Cauchy problem associated with the infinitesimal generator of an
operator cosine function produced by bounded time-dependent perturbations:

$\begin{cases}u^{\prime\prime}(t)=(A+B(t))u(t),\quad t\in\mathbb{R},\\\
u(0)=x\in\mathscr{D}(A),\quad u^{\prime}(0)=y\in\mathscr{D}(A).\end{cases}$
(1.2)

Lin [16] has studied the time-dependent perturbation theory for abstract
second-order evolution equations and their applications to partial
differential equation in the form of the following problem:

$\begin{cases}u^{\prime\prime}(t)=\left(A+B(t)\right)u(t)+f(t),\quad
t\in(0,\tau],\quad\tau>0,\\\ u(0)=u_{0},\quad u^{\prime}(0)=v_{0}.\end{cases}$
(1.3)

On the other hand, Serizawa and Watanabe [17] have investigated the same
problem stated in (1.3) with Neumann boundary condition.

The fractional analogue of the abstract problem (1.2) has been established by
Bazhlekova in [18, 19]. Bazhlekova [19] has proposed uniquely determined a
classical solution of the following homogeneous Cauchy problem:

$\begin{cases*}\left(\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}u\right)(t)=(A+B(t))u(t),\quad
t>0,\\\ u(0)=x\in\mathscr{D}(A),\quad u^{\prime}(0)=0,\end{cases*}$

where $B:\mathbb{R_{+}}\to\mathcal{L}(\mathbb{X})$ is a continuous function on
$\mathbb{R_{+}}$.

Motivated by above articles, we consider an abstract Cauchy problem of
fractional order $\alpha\in(1,2]$ in a Banach space $\mathbb{X}$:

$\left(\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}u\right)(t)=\left(A+B(t)\right)u(t)+f(t),\quad
t>0,$ (1.4)

with initial conditions

$\displaystyle u(0)=x\in\mathscr{D}(A),\qquad
u^{\prime}(0)=y\in\mathscr{D}(A),$ (1.5)

which is uniformly well-posed. Here
$\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}$ is a fractional-order Caputo
differentiation operator and
$A:\mathscr{D}(A)\subseteq\mathbb{X}\to\mathbb{X}$ be a densely-defined,
closed linear operator in a Banach space $\mathbb{X}$. Let
$\mathscr{D}(A)\subseteq\mathbb{X}$ denote the domain of $A$ which will be
specified later, $B:\mathbb{R}_{+}\to\mathcal{L}(\mathbb{X})$ and
$f:\mathbb{R}_{+}\to\mathbb{X}$ be strongly continuously differentiable
functions on $\mathbb{R_{+}}$.

Therefore, the aim of this paper is to develop the perturbation theory to
study fractional-order abstract Cauchy problems of order $\alpha\in(1,2]$
which generalizes classical case (1.2). The pioneering work on (1.4)-(1.5) in
classical sense was done by Phillips [13], Lutz [14] and our development
follows these approaches. To the best of our knowledge, the presented results
extend those of [14, 19] in several aspects. First, we allow for more general
case, in that we assume $u^{\prime}(0)=y\in\mathscr{D}(A)$, i.e., the second
initial value is non-zero and as a result of this choice we derive the
corresponding perturbed fractional sine family $S_{\alpha}(t;A+B)$, however
this condition is zero in [19]. Second, we consider both homogeneous and
inhomogeneous abstract Cauchy problems for fractional evolution equations,
however Bazhlekova [19] and Lutz [14] have studied only homogeneous cases in
fractional and classical senses, respectively. Furthermore, we have attained
new closed-form of solutions whenever $A,B\in\mathcal{L}(\mathbb{X})$ are non-
permutable linear bounded operators. Thus, in this work it is shown that the
inhomogeneous Cauchy problem with the infinitesimal generator $A$ of a
strongly continuous fractional cosine families remains uniformly well-posed
under bounded time-dependent perturbations.

In this paper, we study perturbation properties of problem (1.4) with initial
conditions (1.5), generalizing some facts concerning fractional cosine
operator functions and also distinguishing some new features. Therefore, the
structure of our paper is organized as follows. In Section 2, we provide main
definitions and relations from fractional calculus, fractional evolution
equations and operator theory. Section 3 is devoted to studying homogeneous
and inhomogeneous abstract fractional-order perturbed Cauchy problem in a
Banach space $\mathbb{X}$. In Section 4, we investigate some necessary
particular cases of the main problem (1.4)-(1.5) and we give some comparisons
between existing and our results.

##  2 Preliminaries

We embark on this section by briefly introducing the essential structure of
fractional calculus, fractional evolution equations and operator theory for
linear operators. For the more salient details on these matters, see the
textbooks [20, 21, 22, 23, 24, 25, 26].

Let $\mathbb{R_{+}}=[0,\infty)$ and $\mathbb{N}$ denote the set of natural
numbers with $\mathbb{N}_{0}=\mathbb{N}\cup\left\\{0\right\\}$. Let
$\mathbb{\mathbb{X}}$ be Banach space equipped with the norm
$\left\lVert\cdot\right\rVert$. We donote by $\mathcal{L}(\mathbb{X})$ the
Banach algebra of all bounded linear operators on $\mathbb{X}$ and becomes a
Banach space with regard to the norm $\left\lVert
T\right\rVert=\sup\left\\{\left\lVert Tx\right\rVert:\left\lVert
x\right\rVert\leq 1\right\\}$, for any $T\in\mathcal{L}(\mathbb{X})$. The
identity and zero operators on $\mathbb{X}$ are denoted by
$I\in\mathcal{L}(\mathbb{X})$ and $0\in\mathcal{L}(\mathbb{X})$, respectively.

We will use the following functional spaces [27] through the paper:

  * 1.

$\mathbb{C}\left(\mathbb{R_{+}},\mathbb{X}\right)$ denotes the Banach space of
continuous $\mathbb{X}$-valued functions $g:\mathbb{R_{+}}\to\mathbb{X}$
equipped with an infinity norm
$\|g\|_{\infty}=\sup\limits_{t\in\mathbb{R_{+}}}\left\lVert g(t)\right\rVert$;

  * 2.

$\mathbb{L}^{1}(\mathbb{R_{+}},\mathbb{X})$ denotes a Banach space of
equivalence classes of all measurable $\mathbb{X}$-valued functions
$g:\mathbb{R_{+}}\to\mathbb{X}$ which are Bochner integrable and normed by
$\left\lVert g\right\rVert_{\mathbb{L}_{1}}=\int\limits_{0}^{T}\left\lVert
g(s)\right\rVert\mathrm{d}s<\infty$;

  * 3.

$\mathbb{C}^{n}(\mathbb{R_{+}},\mathbb{X})$, $n\in\mathbb{N}$ denotes the
Banach space of $n$-times continuously differentiable $\mathbb{X}$-valued
functions defined by

$\mathbb{C}^{n}(\mathbb{R_{+}},\mathbb{X})=\left\\{g\in\mathbb{C}^{n}(\mathbb{R_{+}},\mathbb{X}):g^{(n)}\in\mathbb{C}(\mathbb{R_{+}},\mathbb{X}),n\in\mathbb{N}\right\\}$

and equipped with an infinity norm
$\|g\|_{\infty}=\sum\limits_{k=0}^{n}\sup\limits_{t\in\mathbb{R_{+}}}\left\lVert
g^{(k)}(t)\right\rVert$. In addition,
$\mathbb{C}^{n}(\mathbb{R_{+}},\mathbb{X})\subset\mathbb{C}(\mathbb{R_{+}},\mathbb{X})$,
$n\in\mathbb{N}$.

Let $f:\mathbb{R_{+}}\to\mathbb{R}$ be an integrable scalar-valued function
and let $g:\mathbb{R_{+}}\to\mathbb{X}$ be a continuous $\mathbb{X}$-valued
function. We denote by

$(f\ast g)(t)=\int\limits_{0}^{t}f(t-s)g(s)\mathrm{d}s,\quad
t\in\mathbb{R_{+}},$

the convolution operator of $f$ and $g$. Furthermore, if
$T:\mathbb{R_{+}}\to\mathscr{L}(\mathbb{X})$ is a strongly continuous
operator-valued map, we define

$(T\ast
g)(t)=\int\limits_{0}^{t}T(t-s)g(s)\mathrm{d}s=\int\limits_{0}^{t}T(s)g(t-s)\mathrm{d}s,\quad
t\in\mathbb{R_{+}}.$

######  Lemma 2.1.

[22] Let $\mathbb{J}$ be some real interval and
$P,Q:\mathbb{J}\to\mathcal{L}(\mathbb{X})$ be two strongly continuous
operator-valued functions defined on $\mathbb{J}$. Furthermore, suppose that
$P(\cdot)x:\mathbb{J}\to\mathbb{X}$ and $Q(\cdot)x:\mathbb{J}\to\mathbb{X}$
are differentiable for all $x\in\mathscr{D}$ for some subspace $\mathscr{D}$
of $\mathbb{X}$, which is invariant under $Q$. Then
$(PQ)(\cdot)x:\mathbb{J}\to\mathbb{X}$, defined by $(PQ)(t)x\coloneqq
P(t)Q(t)x$, is differentiable for every $x\in\mathscr{D}$ and

$\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}P(\cdot)Q(\cdot)x\Big{)}(t_{0})=\frac{\mathrm{d}}{\mathrm{d}t}\Big{(}P(\cdot)Q(t_{0})x\Big{)}(t_{0})+P(t_{0})\Big{(}\frac{\mathrm{d}}{\mathrm{d}t}Q(\cdot)x\Big{)}(t_{0}).$
(2.1)

######  Definition 2.1.

[20, 21] The Riemann-Liouville fractional integral of order $\alpha>0$ for a
function
$f\in\mathbb{L}^{1}\left(\mathbb{R_{+}},\mathbb{X}\right)$ is defined by

$\prescript{}{}{(}\mathcal{I}^{\alpha}_{t}f)(t)=\left(g_{\alpha}\ast
f\right)(t)=\frac{1}{\Gamma(\alpha)}\int\limits_{0}^{t}(t-s)^{\alpha-1}f(s)\,\mathrm{d}s\\\
,\quad t>0.$ (2.2)

By setting $\left(\mathcal{I}^{0}_{t}f\right)(t)=f(t)$. For the sake of
brevity, we use the following notation for $\alpha\geq 0$:

$\displaystyle
g_{\alpha}(t)=\begin{cases*}\frac{t^{\alpha-1}}{\Gamma(\alpha)},\quad t>0,\\\
0,\qquad t\leq 0,\end{cases*}$

where $\Gamma:\mathbb{R_{+}}\to\mathbb{R}$ is the well-known Euler’s gamma
function. Note that $g_{0}(t)=0$, since $\frac{1}{\Gamma(0)}=0$,
$t\in\mathbb{R}$. These functions satisfy the semigroup property:

$(g_{\alpha}\ast g_{\beta})(t)=g_{\alpha+\beta}(t),\quad t\in\mathbb{R}.$
(2.3)

Moreover, the Riemann-Liouville fractional integral operators
$\left\\{\mathcal{I}^{\alpha}_{t}\right\\}_{\alpha\geq 0}$ satisfy the
semigroup property

$\mathcal{I}^{\alpha}_{t}\mathcal{I}^{\beta}_{t}=\mathcal{I}^{\alpha+\beta}_{t},\quad\alpha,\beta\geq
0,\quad t>0.$ (2.4)

######  Definition 2.2.

[20, 21] The Riemann-Liouville fractional derivative of order $n-1<\alpha\leq
n$ for a function
$\ f\in\mathbb{C}^{n}\left(\mathbb{R_{+}},\mathbb{X}\right)$ is defined by

$\left(\prescript{}{}{\mathcal{D}}^{\alpha}_{t}f\right)(t)=\prescript{}{}{\mathcal{D}^{n}_{t}}\left(g_{n-\alpha}\ast
f\right)(t)=\prescript{}{}{\mathcal{D}^{n}_{t}}\left(\mathcal{I}^{n-\alpha}_{t}f\right)(t),\quad
t>0,$ (2.5)

where
$\prescript{}{}{\mathcal{D}^{n}_{t}}=\frac{\mathrm{d}^{n}}{\mathrm{d}t^{n}}$
for $n\in\mathbb{N}$.

We sometimes adopt the convention

$\left(\prescript{}{}{\mathcal{D}}^{-\alpha}_{t}f\right)(t)=\left(\mathcal{I}^{\alpha}_{t}f\right)(t),\quad\alpha>0,\quad
t>0,$

under which the Riemann-Liouville fractional derivative is the analytic
continuation of the Riemann-Liouville fractional integral for positive
$\alpha$.

######  Definition 2.3.

[20, 21] The Caputo fractional derivative of a function
$f\in\mathbb{C}^{n}\left(\mathbb{R_{+}},\mathbb{X}\right)$ with fractional
order
$n-1<\alpha\leq n$, $n\in\mathbb{N}$ is defined by

$\left(\prescript{C}{}{\mathcal{D}}^{\alpha}_{t}f\right)(t)=\Big{(}g_{n-\alpha}\ast\mathcal{D}^{n}_{t}f\Big{)}(t)=\mathcal{I}^{n-\alpha}_{t}\prescript{}{}{\mathcal{D}^{n}_{t}}f(t),\quad
t>0.$

In particular, for $\alpha\in(1,2]$, the definition for
$f\in\mathbb{C}^{2}\left(\mathbb{R_{+}},\mathbb{X}\right)$ is given by:

$\left(\prescript{C}{}{\mathcal{D}}^{\alpha}_{t}f\right)(t)=\left(g_{2-\alpha}\ast\mathcal{D}^{2}_{t}f\right)(t)=\frac{1}{\Gamma(2-\alpha)}\int\limits_{0}^{t}(t-s)^{1-\alpha}f^{\prime\prime}(s)\,\mathrm{d}s,\quad
t>0.$

######  Lemma 2.2.

[20, 21] The relationship between Riemann-Liouville and Caputo fractional
differential operators of order $n-1<\alpha\leq n$, $n\in\mathbb{N}$ for
$f\in\mathbb{C}^{n}\left(\mathbb{R_{+}},\mathbb{X}\right)$ is given by:

$\left(\prescript{C}{}{\mathcal{D}}^{\alpha}_{t}f\right)(t)=\prescript{}{}{\mathcal{D}}^{\alpha}_{t}\left(f(t)-\sum_{i=0}^{n-1}\frac{f^{(i)}(0)t^{i}}{\Gamma(i+1)}\right).$
(2.6)

By making use of the above property, we attain the relation of order
$1<\alpha\leq 2$ for
$f\in\mathbb{C}^{2}\left(\mathbb{R_{+}},\mathbb{X}\right)$:

$\prescript{}{}{\mathcal{I}}^{\alpha}_{t}\left(\prescript{C}{}{\mathcal{D}}^{\alpha}_{t}f\right)(t)=\left(g_{\alpha}\ast
g_{2-\alpha}\ast\mathcal{D}^{2}_{t}f\right)(t)=f(t)-f(0)-tf^{\prime}(0).$
(2.7)

######  Remark 2.1.

Since $f$ is an abstract function with values in $\mathbb{X}$, the integrals
which appear in Definition 2.1, 2.2, 2.3 and Lemma 2.2 are taken in Bochner’s
sense.

######  Definition 2.4.

[22] If $f:\mathbb{R_{+}}\to\mathbb{X}$ is measurable and exponentially
bounded function of exponent $\omega\in\mathbb{R}$, i.e., $\|f(t)\|\leq
Me^{\omega t}$ for all $t\geq 0$ and some constant $M>0$. Then the Laplace
transform
$\mathscr{L}f:\left\\{\lambda\in\mathscr{C}:Re(\lambda)>\omega\right\\}\to\mathbb{X}$
is defined by

$\left(\mathscr{L}f\right)(\lambda)=\hat{f}(\lambda)=\int\limits_{0}^{\infty}e^{-\lambda
t}f(t)\mathrm{d}t,$

where $Re(\lambda)$ represents the real part of the complex number $\lambda$.
The Laplace transform of the Caputo’s fractional differentiation operator [20,
21] is defined by

$\left(\mathscr{L}(\prescript{C}{}{\mathcal{D}}^{\alpha}_{t}f)\right)(\lambda)=\lambda^{\alpha}\hat{f}(\lambda)-\sum_{k=1}^{n}\lambda^{\alpha-k}f^{(k-1)}(0),\quad
n-1<\alpha\leq n,\quad n\in\mathbb{N}.$ (2.8)

In the particular case, the Laplace integral transform of the Caputo
fractional derivative for $\alpha\in(1,2]$ is:

$\left(\mathscr{L}(\prescript{C}{}{\mathcal{D}}^{\alpha}_{t}f)\right)(\lambda)=\lambda^{\alpha}\hat{f}(\lambda)-\lambda^{\alpha-1}f(0)-\lambda^{\alpha-2}f^{\prime}(0).$

The Mittag-Leffler function is a natural generalization of the exponential
function, first proposed as a single and two parameter functions of one
variable by using an infinite series.

######  Definition 2.5.

[28] The classical Mittag-Leffler function is defined by

$E_{\alpha}(t)=\sum_{k=0}^{\infty}\frac{t^{k}}{\Gamma(k\alpha+1)},\quad\alpha>0,\quad
t\in\mathbb{R}.$ (2.9)

The two-parameter Mittag-Leffler function is given by

$E_{\alpha,\beta}(t)=\sum_{k=0}^{\infty}\frac{t^{k}}{\Gamma(k\alpha+\beta)},\quad\alpha>0,\quad\beta\in\mathbb{R},\quad
t\in\mathbb{R}.$ (2.10)

It is important to note that

$E_{\alpha,1}(t)=E_{\alpha}(t),\quad E_{1}(t)=\exp(t),\quad t\in\mathbb{R}.$

$\star$ Strongly continuous cosine and sine families of linear operators

######  Definition 2.6.

[6] 1) A one parameter family
$\left\\{C(t;A):t\in\mathbb{R}\right\\}\in\mathcal{L}(\mathbb{X})$ into itself
is called a strongly continuous cosine family in the Banach space $\mathbb{X}$
if and only if

  * 1.

$C(0;A)=I$;

  * 2.

$C(s+t;A)+C(s-t;A)=2C(s;A)C(t;A)$ for all $s,t\in\mathbb{R}$;

  * 3.

$C(t;A)x$ is continuous in $t$ on $\mathbb{R}$ for each fixed
$x\in\mathbb{X}$.

2) The corresponding strongly continuous sine family
$\left\\{S(t;A):t\in\mathbb{R}\right\\}\in\mathcal{L}(\mathbb{X})$ is defined
by

$S(t;A)x=\int\limits_{0}^{t}C(s;A)x\mathrm{d}s,\quad x\in\mathbb{X},\quad
t\in\mathbb{R}.$ (2.11)

3) The linear operator $A:\mathscr{D}(A)\subseteq\mathbb{X}\to\mathbb{X}$
defined by

$Ax=2\lim\limits_{t\to 0_{+}}\frac{C(t;A)x-x}{t^{2}},\quad
x\in\mathscr{D}(A)=\left\\{x\in\mathbb{X}:C(\cdot;A)x\in\mathbb{C}^{2}(\mathbb{R},\mathbb{X})\right\\},$
(2.12)

is called the infinitesimal generator of a strongly continuous cosine family
$\left\\{C(t;A):t\in\mathbb{R}\right\\}$ and $\mathscr{D}(A)$ is a domain of
$A$.

It is known that the infinitesimal generator $A$ is closed, densely-defined
operator on $\mathbb{X}$. We will also make use of the set

$\mathcal{E}=\left\\{x\in\mathbb{X}:C(\cdot;A)x\in\mathbb{C}^{1}\left(\mathbb{R},\mathbb{X}\right)\right\\}.$

The families $\left\\{C(t;A):t\in\mathbb{R}\right\\}$ and
$\left\\{S(t;A):t\in\mathbb{R}\right\\}$ have the following properties:

######  Theorem 2.1.

[6] Let $\\{C(t;A):t\in\mathbb{R}\\}$ be a strongly continuous cosine family
in $\mathbb{X}$ with infinitesimal generator $A$. The following assertions
hold true:

  * 1.

There exists constants $M\geq 1$ and $\omega\geq 0$ such that $\left\lVert
C(t;A)\right\rVert\leq Me^{\omega|t|}$ for all $t\in\mathbb{R}$;

  * 2.

If $x\in\mathbb{X}$, then $S(t;A)x\in\mathcal{E}$;

  * 3.

If $x\in\mathcal{E}$, then $S(t;A)x\in\mathscr{D}(A)$ and
$\mathcal{D}^{1}_{t}C(t;A)x=AS(t;A)x$;

  * 4.

If $x\in\mathscr{D}(A)$, then $C(t;A)x\in\mathscr{D}(A)$ and
$\mathcal{D}^{2}_{t}C(t;A)x=AC(t;A)x=C(t;A)Ax$;

  * 5.

If $x\in\mathcal{E}$, then $\lim\limits_{t\to 0}AS(t;A)x=0$;

  * 6.

If $x\in\mathcal{E}$, then $S(t;A)x\in\mathscr{D}(A)$ and
$\mathcal{D}^{2}_{t}S(t;A)x=AS(t;A)x$;

  * 7.

If $x\in\mathscr{D}(A)$, then $S(t;A)x\in\mathscr{D}(A)$ and
$AS(t;A)x=S(t;A)Ax$.

######  Theorem 2.2.

[6] Let $\\{C(t;A):t\in\mathbb{R}\\}$ be a strongly continuous cosine family
in $\mathbb{X}$ satisfying $\left\lVert C(t;A)\right\rVert\leq Me^{\omega|t|}$
for all $t\in\mathbb{R}$ and let $A$ be the infinitesimal generator of
$\\{C(t;A):t\in\mathbb{R}\\}$. Then for $Re(\lambda)>\omega$, $\lambda^{2}$ is
in the resolvent set of $A$ and

$\displaystyle\lambda\mathcal{R}(\lambda^{2};A)x=\int\limits_{0}^{\infty}e^{-\lambda
t}C(t;A)x\mathrm{d}t,\quad x\in\mathbb{X},$ (2.13)
$\displaystyle\mathcal{R}(\lambda^{2};A)x=\int\limits_{0}^{\infty}e^{-\lambda
t}S(t;A)x\mathrm{d}t,\quad x\in\mathbb{X}.$ (2.14)

######  Theorem 2.3.

[6] Let $\\{C(t;A):t\in\mathbb{R}\\}$ be a strongly continuous cosine family
in $\mathbb{X}$ with infinitesimal generator $A$. If
$f:\mathbb{R}\to\mathbb{X}$ is continuously differentiable,
$x_{0}\in\mathscr{D}(A)$, $x^{\prime}_{0}\in\mathcal{E}$, and

$x(t)\coloneqq
C(t;A)x_{0}+S(t;A)x^{\prime}_{0}+\int\limits_{0}^{t}S(t-s;A)f(s)\mathrm{d}s,\quad
t\in\mathbb{R},$ (2.15)

then $x(t)\in\mathscr{D}(A)$ for $t\in\mathbb{R}$, $x$ is twice continuously
differentiable, and $x$ satisfies

$\begin{cases}\mathcal{D}^{2}_{t}x(t)=Ax(t)+f(t),\quad t\in\mathbb{R},\\\
x(0)=x_{0},\quad x^{\prime}(0)=x^{\prime}_{0}.\end{cases}$ (2.16)

$\star$ Strongly continuous fractional cosine, sine and Riemann-Liouville
families of linear operators

######  Definition 2.7.

[9] Let $1<\alpha<2$. A family
$C_{\alpha}(\cdot;A):\mathbb{R_{+}}\to\mathcal{L}(\mathbb{X})$ of all bounded
linear operators on $\mathbb{X}$ is called a fractional cosine family if it
satisfies the following assumptions:

  * 1.

$C_{\alpha}(t;A)$ is strongly continuous for all $t\in\mathbb{R_{+}}$ and
$C_{\alpha}(0;A)=I$;

  * 2.

$C_{\alpha}(s;A)C_{\alpha}(t;A)=C_{\alpha}(t;A)C_{\alpha}(s;A)$ for all
$s,t\in\mathbb{R_{+}}$;

  * 3.

The functional equation

$C_{\alpha}(s;A)\mathcal{I}_{t}^{\alpha}C_{\alpha}(t;A)-\mathcal{I}_{s}^{\alpha}C_{\alpha}(s;A)C_{\alpha}(t;A)=\mathcal{I}_{t}^{\alpha}C_{\alpha}(t;A)-\mathcal{I}_{s}^{\alpha}C_{\alpha}(s;A)\quad\text{holds
for all}\quad s,t\in\mathbb{R_{+}}.$ (2.17)

The linear operator $A$ is defined by

$\displaystyle Ax\coloneqq\Gamma(\alpha+1)\lim_{t\to
0_{+}}\frac{C_{\alpha}(t;A)x-x}{t^{\alpha}},\quad
x\in\mathscr{D}(A)\coloneqq\left\\{x\in\mathbb{X}:C_{\alpha}(\cdot;A)\in\mathbb{C}^{2}\left(\mathbb{R_{+}},\mathbb{X}\right)\right\\},$
(2.18)

where $A$ is the generator of the strongly continuous fractional cosine family
$\\{C_{\alpha}(t;A):t\in\mathbb{R_{+}}\\}$.

######  Definition 2.8.

[9] A strongly continuous fractional cosine family
$\\{C_{\alpha}(t;A):t\in\mathbb{R_{+}}\\}$ is said to be exponentially bounded
if there exists constants $M\geq 1$, $\omega\geq 0$ such that

$\left\lVert C_{\alpha}(t;A)\right\rVert\leq Me^{\omega t},\quad
t\in\mathbb{R_{+}}.$ (2.19)

######  Theorem 2.4.

[11] Let $\\{C_{\alpha}(t;A):t\in\mathbb{R_{+}}\\}$ be a strongly continuous
fractional cosine family generated by the operator $A$. The following
assertions hold true:

  * 1.

$C_{\alpha}(t;A)$ commutes with $A$ which means that
$C_{\alpha}(t;A)\mathscr{D}(A)\subset\mathscr{D}(A)$ and
$AC_{\alpha}(t;A)x=C_{\alpha}(t;A)Ax$ for all $x\in\mathscr{D}(A)$ and
$t\in\mathbb{R_{+}}$;

  * 2.

For all $x\in\mathbb{X}$,
$\mathcal{I}_{t}^{\alpha}C_{\alpha}(t;A)x\in\mathscr{D}(A)$ and

$C_{\alpha}(t;A)x=x+A\mathcal{I}_{t}^{\alpha}C_{\alpha}(t;A)x,\quad
t\in\mathbb{R_{+}}.$ (2.20)

  * 3.

For all $x\in\mathscr{D}(A)$,

$C_{\alpha}(t;A)x=x+\mathcal{I}_{t}^{\alpha}C_{\alpha}(t;A)Ax,\quad
t\in\mathbb{R_{+}}.$ (2.21)

  * 4.

$A$ is closed, densely-defined operator on $\mathbb{X}$;

######  Definition 2.9.

[9] The strongly continuous fractional sine family
$S_{\alpha}(\cdot;A):\mathbb{R_{+}}\to\mathcal{L}(\mathbb{X})$ associated with
$C_{\alpha}$ is defined by

$S_{\alpha}(t;A)x=\int\limits_{0}^{t}C_{\alpha}(s;A)x\mathrm{d}s,\quad
x\in\mathbb{X},\quad t\in\mathbb{R_{+}}.$ (2.22)

This implies that

$\mathcal{D}^{1}_{t}S_{\alpha}(t;A)x=C_{\alpha}(t;A)x,\quad
x\in\mathbb{X},\quad t\in\mathbb{R_{+}}.$ (2.23)

######  Definition 2.10.

[9] The strongly continuous fractional Riemann-Liouville family
$T_{\alpha}(\cdot;A):\mathbb{R_{+}}\to\mathcal{L}(\mathbb{X})$ associated with
$C_{\alpha}$ is defined by

$T_{\alpha}(t;A)x=\mathcal{I}_{t}^{\alpha-1}C_{\alpha}(t;A)x,\quad
x\in\mathbb{X},\quad t\in\mathbb{R_{+}}.$ (2.24)

For a strongly continuous fractional cosine family
$\left\\{C_{\alpha}(t;A):t\in\mathbb{R_{+}}\right\\}$, we define

$\mathcal{E}=\left\\{x\in\mathbb{X}:C_{\alpha}(\cdot;A)x\in\mathbb{C}^{1}\left(\mathbb{R_{+}},\mathbb{X}\right)\right\\}.$

It is known that $T_{\alpha}(t;A)\mathcal{E}\subset\mathscr{D}(A)$ for all
$t\in\mathbb{R_{+}}$ such that

$\mathcal{D}^{1}_{t}C_{\alpha}(t;A)x=AT_{\alpha}(t;A)x,\quad
x\in\mathcal{E},\quad t\in\mathbb{R_{+}}.$ (2.25)

From (2.25) (or (2.20) and (2.24)), for $x\in\mathbb{X}$,
$\mathcal{I}_{t}^{1}T_{\alpha}(t;A)x\in\mathscr{D}(A)$ such that

$A\mathcal{I}_{t}^{1}T_{\alpha}(t;A)x=C_{\alpha}(t;A)x-x.$ (2.26)

######  Theorem 2.5.

[9] Let $\\{C_{\alpha}(t;A):t\in\mathbb{R_{+}}\\}$ be a strongly continuous
fractional cosine family in $\mathbb{X}$ satisfying $\left\lVert
C_{\alpha}(t;A)\right\rVert\leq Me^{\omega t}$ for all $t\in\mathbb{R_{+}}$
and let $A$ be the infinitesimal generator of
$\\{C_{\alpha}(t;A):t\in\mathbb{R_{+}}\\}$. Then for $Re(\lambda)>\omega$,
$\lambda^{\alpha}$ for $\alpha\in(1,2]$ is in the resolvent set of $A$ and

$\displaystyle\lambda^{\alpha-1}\mathcal{R}(\lambda^{\alpha};A)x=\int\limits_{0}^{\infty}e^{-\lambda
t}C_{\alpha}(t;A)x\mathrm{d}t,\quad x\in\mathbb{X},$ (2.27)
$\displaystyle\lambda^{\alpha-2}\mathcal{R}(\lambda^{\alpha};A)x=\int\limits_{0}^{\infty}e^{-\lambda
t}S_{\alpha}(t;A)x\mathrm{d}t,\quad x\in\mathbb{X},$ (2.28)
$\displaystyle\mathcal{R}(\lambda^{\alpha};A)x=\int\limits_{0}^{\infty}e^{-\lambda
t}T_{\alpha}(t;A)x\mathrm{d}t,\quad x\in\mathbb{X}.$ (2.29)

######  Theorem 2.6.

[9] Let $\\{C_{\alpha}(t;A):t\in\mathbb{R_{+}}\\}$ be a strongly continuous
fractional cosine family in $\mathbb{X}$ with the infinitesimal generator $A$.
If $f:\mathbb{R_{+}}\to\mathbb{X}$ is continuously differentiable,
$x_{0}\in\mathscr{D}(A)$, $x^{\prime}_{0}\in\mathcal{E}$, and

$x(t)\coloneqq
C_{\alpha}(t;A)x_{0}+S_{\alpha}(t;A)x^{\prime}_{0}+\int\limits_{0}^{t}T_{\alpha}(t-s;A)f(s)\mathrm{d}s,\quad
t\in\mathbb{R_{+}},$ (2.30)

then $x(t)\in\mathscr{D}(A)$ for $t\in\mathbb{R}_{+}$, $x$ is twice
continuously differentiable, and $x$ satisfies

$\begin{cases}\prescript{C}{}{\mathcal{D}}^{\alpha}_{t}x(t)=Ax(t)+f(t),\quad
t\in\mathbb{R_{+}},\\\ x(0)=x_{0},\quad
x^{\prime}(0)=x^{\prime}_{0}.\end{cases}$ (2.31)

##  3 Perturbation theory for fractional evolution equations

We first consider the following Cauchy problem for homogeneous fractional
evolution equation in a Banach space $\mathbb{X}$:

$\begin{cases}\left(\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}u\right)(t)=\left(A+B(t)\right)u(t),\quad
t>0,\\\ u(0)=x,\qquad u^{\prime}(0)=y.\end{cases}$ (3.1)

######  Theorem 3.1.

Let $A$ be the infinitesimal generator of the fractional cosine family
$\left\\{C_{\alpha}(t;A):t\in\mathbb{R_{+}}\right\\}$ with
$\|C_{\alpha}(t;A)\|\leq Me^{\omega t}$ for all $t\in\mathbb{R_{+}}$, and let
$\left\\{S_{\alpha}(t;A):t\in\mathbb{R_{+}}\right\\}$ denote the fractional
sine family associated with $C_{\alpha}$. Furthermore, suppose that
$B:\mathbb{R_{+}}\to\mathcal{L}(\mathbb{X})$ is a strongly continuously
differentiable function on $\mathbb{R_{+}}$. Then

  * 1.

The Cauchy problem for the $\mathbb{X}$-valued differential equation (3.1) is
uniformly well-posed; more precisely, for each pair $x,y\in\mathscr{D}(A)$
there is a uniquely determined solution
$u\in\mathbb{C}^{2}\left(\mathbb{R_{+}};\mathscr{D}(A)\right)$ of (3.1)
fulfilling initial conditions and $u\in\mathscr{D}(A)$ depends continuously
(with respect to the topology of uniform convergence) on compact subsets of
$\mathbb{R}_{+}$ upon $(x,y)$.

  * 2.

$u\in\mathscr{D}(A)$ is given by

$u(t)=C_{\alpha}(t;A+B)x+S_{\alpha}(t;A+B)y,\quad t\in\mathbb{R_{+}},$ (3.2)

where for $x\in\mathscr{D}(A)$, $t\mapsto C_{\alpha}(t;A+B)x$ and $t\mapsto
S_{\alpha}(t;A+B)x$ are $2$-times strongly continuously differentiable
solutions of (3.1) fulfilling

$\displaystyle C_{\alpha}(0;A+B)x=x,\quad C^{\prime}_{\alpha}(0;A+B)x=0,$
(3.3) $\displaystyle S_{\alpha}(0;A+B)x=0,\quad
S^{\prime}_{\alpha}(0;A+B)x=x.$ (3.4)

  * 3.

$C_{\alpha}(\cdot;A+B),S_{\alpha}(\cdot;A+B)\in\mathcal{L}(\mathbb{X})$ are
defined by

$\displaystyle
C_{\alpha}(t;A+B)x\coloneqq\sum_{n=0}^{\infty}C_{\alpha,n}(t,A)x,\quad
x\in\mathbb{X},$ (3.5) $\displaystyle
S_{\alpha}(t;A+B)x\coloneqq\sum_{n=0}^{\infty}S_{\alpha,n}(t;A)x,\quad
x\in\mathbb{X},$ (3.6)

where

$\displaystyle C_{\alpha,0}(t;A)x\coloneqq C_{\alpha}(t;A)x,\quad
S_{\alpha,0}(t;A)x\coloneqq S_{\alpha}(t;A)x,$ $\displaystyle
C_{\alpha,n}(t;A)x\coloneqq\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)C_{\alpha,n-1}(s;A)x\mathrm{d}s,$
(3.7) $\displaystyle
S_{\alpha,n}(t;A)x\coloneqq\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)S_{\alpha,n-1}(s;A)x\mathrm{d}s,$
(3.8) $\displaystyle
T_{\alpha}(t;A)x=\int\limits_{0}^{t}g_{\alpha-1}(t-s)C_{\alpha}(s;A)x\mathrm{d}s,\quad
n\in\mathbb{N},\quad x\in\mathbb{X}.$ (3.9)

  * 4.

With $K_{t}\coloneqq\sup\limits_{0\leq s\leq t}\Biggl{\\{}\left\lVert
B(s)\right\rVert,\left\lVert B^{\prime}(s)\right\rVert\Biggr{\\}}$, we have
for all $s\in[0,t]$, the bounds

$\displaystyle\left\lVert C_{\alpha}(s;A+B)\right\rVert\leq Me^{\omega
s}E_{\alpha}\left(MK_{t}s^{\alpha}\right),$ $\displaystyle\left\lVert
S_{\alpha}(s;A+B)\right\rVert\leq Me^{\omega
s}sE_{\alpha,2}\left(MK_{t}s^{\alpha}\right),$ $\displaystyle\left\lVert
C_{\alpha}(s;A+B)-C_{\alpha}(s;A)\right\rVert\leq Me^{\omega
s}\Big{[}E_{\alpha}\left(MK_{t}s^{\alpha}\right)-1\Big{]},$
$\displaystyle\left\lVert S_{\alpha}(s;A+B)-S_{\alpha}(s;A)\right\rVert\leq
Me^{\omega s}s\Big{[}E_{\alpha,2}\left(MK_{t}s^{\alpha}\right)-1\Big{]}.$

###### Proof.

For simplicity, we split our proof into several steps.

  1. 1.

From $\left\lVert C_{\alpha}(t;A)\right\rVert\leq Me^{\omega t}$, it follows
that $\left\lVert S_{\alpha}(t;A)\right\rVert\leq Mte^{\omega t}$. By using
the formula (3.9), we have

$\displaystyle\left\lVert T_{\alpha}(t;A)\right\rVert$
$\displaystyle\leq\int\limits_{0}^{t}g_{\alpha-1}(t-s)\left\lVert
C_{\alpha}(s;A)\right\rVert\mathrm{d}s\leq
M\int\limits_{0}^{t}\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}e^{\omega
s}\mathrm{d}s$ $\displaystyle\leq Me^{\omega
t}\int\limits_{0}^{t}\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}\mathrm{d}s=Me^{\omega
t}g_{\alpha}(t).$ (3.10)

Put $K_{t}\coloneqq\sup\limits_{0\leq s\leq t}\Biggl{\\{}\left\lVert
B(s)\right\rVert,\left\lVert B^{\prime}(s)\right\rVert\Biggr{\\}}$. Then, it
is true that for $n\in N_{0}$

$\displaystyle\left\lVert C_{\alpha,n}(t;A)\right\rVert\leq
M^{n+1}K_{t}^{n}e^{\omega t}g_{n\alpha+1}(t),$ (3.11)
$\displaystyle\left\lVert S_{\alpha,n}(t;A)\right\rVert\leq
M^{n+1}K_{t}^{n}e^{\omega t}g_{n\alpha+2}(t).$ (3.12)

In the case of (3.11), this is true by our remark above for $n=0$. By
mathematical induction principle, we verify

$\displaystyle\left\lVert
C_{\alpha,n+1}(t;A)x\right\rVert\leq\int\limits_{0}^{t}\left\lVert
T_{\alpha}(t-s;A)\right\rVert\left\lVert B(s)\right\rVert\left\lVert
C_{\alpha,n}(s;A)x\right\rVert\mathrm{d}s$ $\displaystyle\leq
M^{n+2}K_{t}^{n+1}\left\lVert
x\right\rVert\int\limits_{0}^{t}g_{\alpha}(t-s)e^{w(t-s)}e^{\omega
s}g_{n\alpha+1}(s)\mathrm{d}s$ $\displaystyle\leq M^{n+2}K_{t}^{n+1}e^{\omega
t}\left\lVert
x\right\rVert\int\limits_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\frac{s^{n\alpha}}{\Gamma(n\alpha+1)}\mathrm{d}s$
$\displaystyle=M^{n+2}K_{t}^{n+1}e^{\omega t}\left\lVert x\right\rVert
g_{(n+1)\alpha+1}(t),\quad x\in\mathbb{X}.$ (3.13)

The analogues procedure gives the bound for $S_{\alpha,n}(t;A)$,
$t\in\mathbb{R_{+}}$, as follows:

$\displaystyle\left\lVert
S_{\alpha,n+1}(t;A)x\right\rVert\leq\int\limits_{0}^{t}\left\lVert
T_{\alpha}(t-s;A)\right\rVert\left\lVert B(s)\right\rVert\left\lVert
S_{\alpha,n}(s;A)x\right\rVert\mathrm{d}s$ $\displaystyle\leq
M^{n+2}K_{t}^{n+1}\left\lVert
x\right\rVert\int\limits_{0}^{t}g_{\alpha}(t-s)e^{w(t-s)}e^{\omega
s}g_{n\alpha+2}(s)\mathrm{d}s$ $\displaystyle\leq M^{n+2}K_{t}^{n+1}e^{\omega
t}\left\lVert
x\right\rVert\int\limits_{0}^{t}\frac{(t-s)^{\alpha-1}}{\Gamma(\alpha)}\frac{s^{n\alpha+1}}{\Gamma(n\alpha+2)}\mathrm{d}s$
$\displaystyle=M^{n+2}K_{t}^{n+1}e^{\omega t}\left\lVert x\right\rVert
g_{(n+1)\alpha+2}(t),\quad x\in\mathbb{X}.$ (3.14)

From these bounds it follows that the series representing $C_{\alpha}(t;A+B)$
and $S_{\alpha}(t;A+B)$ in (3.5) and (3.6), respectively are uniformly
convergent on compact subsets of $\mathbb{R}_{+}$ with respect to the operator
norm topology. Hence, $C_{\alpha}(t;A+B)$ and $S_{\alpha}(t;A+B)$ are strongly
continuous families on $\mathbb{R}_{+}$ with values in
$\mathcal{L}(\mathbb{X})$.

Furthermore, the bounds for $C_{\alpha}(s;A+B)$, $S_{\alpha}(s;A+B)$,
$C_{\alpha}(s;A+B)-C_{\alpha}(s;A)$, and $S_{\alpha}(s;A+B)-S_{\alpha}(s;A)$,
$s\in[0,t]$ derive as the following direct consequences:

$\displaystyle\left\lVert
C_{\alpha}(s;A+B)\right\rVert\leq\sum_{n=0}^{\infty}\left\lVert
C_{\alpha,n}(s;A)\right\rVert\leq Me^{\omega
s}\sum_{n=0}^{\infty}M^{n}K_{t}^{n}\frac{s^{n\alpha}}{\Gamma(n\alpha+1)}=Me^{\omega
s}E_{\alpha}\left(MK_{t}s^{\alpha}\right),$

$\displaystyle\left\lVert
S_{\alpha}(s;A+B)\right\rVert\leq\sum_{n=0}^{\infty}\left\lVert
S_{\alpha,n}(s;A)\right\rVert\leq Mse^{\omega
s}\sum_{n=0}^{\infty}M^{n}K_{t}^{n}\frac{s^{n\alpha}}{\Gamma(n\alpha+2)}=Me^{\omega
s}sE_{\alpha,2}\left(MK_{t}s^{\alpha}\right),$ $\displaystyle\left\lVert
C_{\alpha}(s;A+B)-C_{\alpha}(s;A)\right\rVert\leq\sum_{n=1}^{\infty}\left\lVert
C_{\alpha,n}(s;A)\right\rVert\leq Me^{\omega
s}\sum_{n=1}^{\infty}M^{n}K_{t}^{n}\frac{s^{n\alpha}}{\Gamma(n\alpha+1)}=Me^{\omega
s}\Big{[}E_{\alpha}\left(MK_{t}s^{\alpha}\right)-1\Big{]},$
$\displaystyle\left\lVert
S_{\alpha}(s;A+B)-S_{\alpha}(s;A)\right\rVert\leq\sum_{n=1}^{\infty}\left\lVert
S_{\alpha,n}(s;A)\right\rVert\leq Mse^{\omega
s}\sum_{n=1}^{\infty}M^{n}K_{t}^{n}\frac{s^{n\alpha}}{\Gamma(n\alpha+2)}=Me^{\omega
s}s\Big{[}E_{\alpha,2}\left(MK_{t}s^{\alpha}\right)-1\Big{]}.$

  2. 2.

We show that, for $x\in\mathscr{D}(A)$, the function $t\mapsto
C_{\alpha}(t;A+B)x$ is twice continuously differentiable. Assuming that this
is true for $t\mapsto C_{\alpha,n}(t;A)x$, then as in Phillips [13], it is
easily shown to be true for $t\mapsto B(t)C_{\alpha,n}(t;A)x$ and thus, for
$t\mapsto C_{\alpha,n+1}(t;A)x$. Furthermore, for $x\in\mathscr{D}(A)$, we
have the following formulas:

$\displaystyle
C^{\prime}_{\alpha}(t;A)x=\prescript{}{}{\mathcal{I}^{1}_{t}}\prescript{}{}{\mathcal{D}^{2}_{t}}C_{\alpha}(t;A)x=\prescript{}{}{\mathcal{I}^{\alpha-1}_{t}}\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}\prescript{}{}{\mathcal{D}^{2}_{t}}C_{\alpha}(t;A)x$
$\displaystyle\hskip
42.67912pt=\prescript{}{}{\mathcal{I}^{\alpha-1}_{t}}\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}C_{\alpha}(t;A)x=\prescript{}{}{\mathcal{I}^{\alpha-1}_{t}}AC_{\alpha}(t;A)x=\prescript{}{}{\mathcal{I}^{\alpha-1}_{t}}C_{\alpha}(t;A)Ax=T_{\alpha}(t;A)Ax=AT_{\alpha}(t;A)x,$
$\displaystyle
C^{\prime}_{\alpha,n+1}(t;A)x=\int\limits_{0}^{t}T^{\prime}_{\alpha}(t-s;A)B(s)C_{\alpha,n}(s;A)x\mathrm{d}s,$
$\displaystyle
C^{\prime\prime}_{\alpha}(t;A)x=T^{\prime}_{\alpha}(t;A)Ax=AT^{\prime}_{\alpha}(t;A)x,$
$\displaystyle
C^{\prime\prime}_{\alpha,n+1}(t;A)x=\int\limits_{0}^{t}T^{\prime}_{\alpha}(t-s;A)B^{\prime}(s)C_{\alpha,n}(s;A)x\mathrm{d}s+\int\limits_{0}^{t}T^{\prime}_{\alpha}(t-s;A)B(s)C^{\prime}_{\alpha,n}(s;A)x\mathrm{d}s,$
$\displaystyle T^{\prime}_{\alpha}(t;A)x=g_{\alpha-1}(t)\ast
T_{\alpha}(t;A)Ax+g_{\alpha-1}(t)x=g_{\alpha-1}(t)\ast
AT_{\alpha}(t;A)x+g_{\alpha-1}(t)x.$ (3.15)

Furthermore, from $\left\lVert T_{\alpha}(t;A)\right\rVert\leq Me^{\omega
t}g_{\alpha}(t)$, it follows that

$\displaystyle\left\lVert T^{\prime}_{\alpha}(t;A)x\right\rVert$
$\displaystyle\leq\int\limits_{0}^{t}g_{\alpha-1}(t-s)\left\lVert
T_{\alpha}(s;A)\right\rVert\mathrm{d}s\left\lVert
Ax\right\rVert+g_{\alpha-1}(t)\|x\|$ $\displaystyle\leq
M\int\limits_{0}^{t}g_{\alpha-1}(t-s)g_{\alpha}(s)e^{\omega
s}\mathrm{d}s\left\lVert Ax\right\rVert+g_{\alpha-1}(t)\|x\|$
$\displaystyle\leq Me^{\omega t}g_{2\alpha-1}(t)\left\lVert
Ax\right\rVert+g_{\alpha-1}(t)\left\lVert x\right\rVert,\quad
x\in\mathscr{D}(A).$ (3.16)

Using (3.11) and (2), we obtain by induction from these equations (3.15) the
following bounds for $x\in\mathscr{D}(A)$:

$\displaystyle\|C^{\prime}_{\alpha}(t;A)x\|\leq Me^{\omega
t}g_{\alpha}(t)\|Ax\|,$ $\displaystyle\|C^{\prime}_{\alpha,1}(t;A)x\|\leq
M^{2}K_{t}e^{\omega t}g_{2\alpha}(t)\|Ax\|+MK_{t}e^{\omega
t}g_{\alpha}(t)\left\lVert x\right\rVert,$
$\displaystyle|C^{\prime}_{\alpha,n}(t;A)x\|\leq M^{n+1}K^{n}_{t}e^{\omega
t}g_{n\alpha+\alpha}(t)\|Ax\|+M^{n}K^{n}_{t}e^{\omega
t}g_{n\alpha}(t)\|x\|,\quad n\in\mathbb{N},$
$\displaystyle\|C^{\prime\prime}_{\alpha}(t;A)x\|\leq Me^{\omega
t}g_{2\alpha-1}(t)\|A^{2}x\|+g_{\alpha-1}(t)\|Ax\|,$
$\displaystyle\|C^{\prime\prime}_{\alpha,1}(t;A)x\|\leq MK_{t}e^{\omega
t}g_{\alpha}(t)\|x\|+MK_{t}e^{\omega
t}\Big{(}Mg_{2\alpha}(t)+g_{2\alpha-1}(t)\Big{)}\|Ax\|+M^{2}K_{t}e^{\omega
t}g_{3\alpha-1}(t)\|A^{2}x\|,$
$\displaystyle|C^{\prime\prime}_{\alpha,n}(t;A)x\|\leq M^{n}K^{n}_{t}e^{\omega
t}\Big{(}Mg_{n\alpha+\alpha}(t)+ng_{n\alpha+\alpha-1}(t)\Big{)}\|Ax\|+M^{n+1}K^{n}_{t}e^{\omega
t}g_{n\alpha+2\alpha-1}(t)\|A^{2}x\|$ $\displaystyle\hskip
56.9055pt+M^{n-1}K^{n}_{t}e^{\omega
t}\Big{(}Mg_{n\alpha}(t)+g_{n\alpha-1}(t)\Big{)}\|x\|,\quad n\geq 2.$

In a completely analogues manner $t\mapsto S_{\alpha}(t;A+B)$ is seen to be
twice continuously differentiable on $\mathbb{R_{+}}$ for every
$x\in\mathscr{D}(A)$, as follows:

$\displaystyle S^{\prime}_{\alpha}(t;A)x=C_{\alpha}(t;A)x,$ $\displaystyle
S^{\prime}_{\alpha,n+1}(t;A)x=\int\limits_{0}^{t}T^{\prime}_{\alpha}(t-s;A)B(s)S_{\alpha,n}(s;A)x\mathrm{d}s,$
$\displaystyle
S^{\prime\prime}_{\alpha}(t;A)x=T_{\alpha}(t;A)Ax=AT_{\alpha}(t;A)x,$
$\displaystyle
S^{\prime\prime}_{\alpha,n+1}(t;A)x=\int\limits_{0}^{t}T^{\prime}_{\alpha}(t-s;A)B^{\prime}(s)S_{\alpha,n}(s;A)x\mathrm{d}s$
$\displaystyle\hskip
56.9055pt+\int\limits_{0}^{t}T^{\prime}_{\alpha}(t-s;A)B(s)S^{\prime}_{\alpha,n}(s;A)x\mathrm{d}s.$
(3.17)

Using (3.12) and (2), we obtain by induction from these equations (3.17) the
following bounds:

$\displaystyle\|S^{\prime}_{\alpha}(t;A)x\|\leq Me^{\omega t}\|x\|,$
$\displaystyle\|S^{\prime}_{\alpha,1}(t;A)x\|\leq M^{2}K_{t}e^{\omega
t}g_{2\alpha+1}(t)\|Ax\|+MK_{t}e^{\omega t}g_{\alpha+1}(t)\|x\|,$
$\displaystyle|S^{\prime}_{\alpha,n}(t;A)x\|\leq M^{n+1}K^{n}_{t}e^{\omega
t}g_{n\alpha+\alpha+1}(t)\|Ax\|+M^{n}K^{n}_{t}e^{\omega
t}g_{n\alpha+1}(t)\|x\|,\quad n\in\mathbb{N},$
$\displaystyle\|S^{\prime\prime}_{\alpha}(t;A)x\|\leq Me^{\omega
t}g_{\alpha}(t)\|Ax\|,$
$\displaystyle\|S^{\prime\prime}_{\alpha,1}(t;A)x\|\leq M^{2}K_{t}e^{\omega
t}\Big{(}g_{2\alpha}(t)+g_{2\alpha+1}(t)\Big{)}\|Ax\|+MK_{t}e^{\omega
t}\Big{(}g_{2\alpha}(t)+g_{\alpha+1}(t)\Big{)}\|x\|,$
$\displaystyle\|S^{\prime\prime}_{\alpha,n}(t;A)x\|\leq
M^{n}K^{n}_{t}e^{\omega
t}\Big{(}Mg_{n\alpha+\alpha+1}(t)+ng_{n\alpha+\alpha}(t)\Big{)}\|Ax\|+M^{n+1}K^{n}_{t}e^{\omega
t}g_{n\alpha+2\alpha}(t)\|A^{2}x\|$ $\displaystyle\hskip
56.9055pt+M^{n-1}K^{n}_{t}e^{\omega
t}\Big{(}g_{n\alpha}(t)+Mg_{n\alpha+1}(t)\Big{)}\|x\|,\quad n\geq 2.$

Therefore, for $x\in\mathscr{D}(A)$, the series
$\sum\limits_{n=0}^{\infty}C^{\prime}_{\alpha,n}(t;A)x$,
$\sum\limits_{n=0}^{\infty}C^{\prime\prime}_{\alpha,n}(t;A)x$ and
$\sum\limits_{n=0}^{\infty}S^{\prime}_{\alpha,n}(t;A)x$,
$\sum\limits_{n=0}^{\infty}S^{\prime\prime}_{\alpha,n}(t;A)x$ converge
uniformly in every compact interval of $\mathbb{R_{+}}$ to continuous
functions which are $C^{\prime}_{\alpha}(t;A)x$,
$C^{\prime\prime}_{\alpha}(t;A)x$ and $S^{\prime}_{\alpha}(t;A)x$,
$S^{\prime\prime}_{\alpha}(t;A)x$, respectively.

  3. 3.

Next we prove that $u\in\mathscr{D}(A)$ with initial conditions $u(0)=x$,
$u^{\prime}(0)=y$, where $x,y\in\mathscr{D}(A)$ satisfies the abstract Cauchy
problem (3.1). Since

$\displaystyle C_{\alpha}(0;A)=I,\quad C_{\alpha,n}(0;A)=0,\quad
C^{\prime}_{\alpha}(0;A)=0,\quad C^{\prime}_{\alpha,n}(0;A)=0,\quad
n\in\mathbb{N},$ $\displaystyle S_{\alpha}(0;A)=0,\quad
S_{\alpha,n}(0;A)=0,\quad S^{\prime}_{\alpha}(0;A)=I,\quad
S^{\prime}_{\alpha,n}(0;A)=0,\quad n\in\mathbb{N},$

we have (3.3) and (3.4), i.e. initial conditions (1.5) are satisfied. Applying
(3.2), (3.5)-(3.8), it follows that

$\displaystyle u(t)$
$\displaystyle=C_{\alpha}(t;A)x+S_{\alpha}(t;A)y+\sum_{n=1}^{\infty}\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)\Big{(}C_{\alpha,n-1}(s;A)x+S_{\alpha,n-1}(s;A)y\Big{)}\mathrm{d}s$
$\displaystyle=C_{\alpha}(t;A)x+S_{\alpha}(t;A)y+\sum_{n=0}^{\infty}\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)\Big{(}C_{\alpha,n}(s;A)x+S_{\alpha,n}(s;A)y\Big{)}\mathrm{d}s$
$\displaystyle=C_{\alpha}(t;A)x+S_{\alpha}(t;A)y+\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)u(s)\mathrm{d}s,$
(3.18)

where the interchanging of the summation and integration is justified by the
uniform convergence of the series. Then integro-differentiating (3) term-by-
term and by using the formulas (2.23)-(2.26), we obtain for
$x,y\in\mathscr{D}(A)$:

$\displaystyle\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}C_{\alpha}(t;A)x$
$\displaystyle=\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}{\mathcal{D}}_{t}^{2}C_{\alpha}(t;A)x=A\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}{\mathcal{D}}_{t}^{1}T_{\alpha}(t;A)x$
$\displaystyle=A\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}{\mathcal{D}}_{t}^{1}\left({\mathcal{I}}_{t}^{\alpha-1}C_{\alpha}(t;A)x\right)=A\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}\Big{(}{\mathcal{I}}_{t}^{\alpha-1}AT_{\alpha}(t;A)x+g_{\alpha-1}(t)x\Big{)}$
$\displaystyle=A\Big{(}\prescript{}{}{\mathcal{I}^{1}_{t}}AT_{\alpha}(t;A)x+x\Big{)}=A\Big{(}A\prescript{}{}{\mathcal{I}^{1}_{t}}T_{\alpha}(t;A)x+x\Big{)}=AC_{\alpha}(t;A)x,$
(3.19)

$\displaystyle\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}S_{\alpha}(t;A)y$
$\displaystyle=\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}{\mathcal{D}}_{t}^{2}S_{\alpha}(t;A)y=\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}{\mathcal{D}}_{t}^{1}C_{\alpha}(t;A)y$
$\displaystyle=A\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}T_{\alpha}(t;A)x=A\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}{\mathcal{I}}_{t}^{\alpha-1}C_{\alpha}(t;A)y$
$\displaystyle=A\prescript{}{}{\mathcal{I}^{1}_{t}}C_{\alpha}(t;A)x=AS_{\alpha}(t;A)y.$
(3.20)

Then, it follows that

$\displaystyle\left(\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}u\right)(t)=AC_{\alpha}(t;A)x+AS_{\alpha}(t;A)y+\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}\left(\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)u(s)\mathrm{d}s\right).$
(3.21)

Making use of (3.21) and (2.6),
$\left(T_{\alpha}\ast(Bu)\right)(0)=\left(T_{\alpha}\ast(Bu)\right)^{\prime}(0)=0$,
the property of $\mathcal{I}^{\alpha}_{t}\left(f\ast
g\right)=\left(\mathcal{I}^{\alpha}_{t}f\right)\ast g$ and the semigroup
property for operators of Riemann-Liouville fractional integration, we have

$\displaystyle\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}\Big{(}T_{\alpha}(t;A)\ast\Big{(}B(t)u(t)\Big{)}\Big{)}$
$\displaystyle=\prescript{}{}{\mathcal{D}^{\alpha}_{t}}\Big{(}T_{\alpha}(t;A)\ast\Big{(}B(t)u(t)\Big{)}\Big{)}=\prescript{}{}{\mathcal{D}^{2}_{t}}\mathcal{I}^{2-\alpha}_{t}\Big{(}\mathcal{I}^{\alpha-1}_{t}C_{\alpha}(t;A)\ast\Big{(}B(t)u(t)\Big{)}\Big{)}$
$\displaystyle=\prescript{}{}{\mathcal{D}^{2}_{t}}\Big{(}\mathcal{I}^{2-\alpha}_{t}\mathcal{I}^{\alpha-1}_{t}C_{\alpha}(t;A)\Big{)}\ast\Big{(}B(t)u(t)\Big{)}=\prescript{}{}{\mathcal{D}^{1}_{t}}\Big{(}C_{\alpha}(t;A)\ast\Big{(}B(t)u(t)\Big{)}\Big{)}$
$\displaystyle=\Big{(}\prescript{}{}{\mathcal{D}^{1}_{t}}C_{\alpha}(t;A)\Big{)}\ast\Big{(}B(t)u(t)\Big{)}+B(t)u(t),$
(3.22)

where

$\displaystyle\prescript{}{}{\mathcal{D}^{1}_{t}}C_{\alpha}(t;A)=T_{\alpha}(t;A)A=AT_{\alpha}(t;A),$
(3.23)

which together with (3.21), (3), and using the closedness of $A$ implies that
$u(t)$ satisfies (3.1) for any $t\in\mathbb{R_{+}}$.

Uniqueness: To prove the uniqueness, we let
$v:\mathbb{R}_{+}\to\mathscr{D}(A)$ be a solution of (3.1) with zero initial
conditions $v(0)=v^{\prime}(0)=0$. Then, by using the well-known property
(2.7), we have
$v(t)=\prescript{}{}{\mathcal{I}^{\alpha}_{t}}Av(t)+\prescript{}{}{\mathcal{I}^{\alpha}_{t}}\left(B(t)v(t)\right)$
and applying the variation of parameters formula, $v(t)$ satisfies the
Volterra integral equation of second-kind:

$v(t)=\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)v(s)\mathrm{d}s.$

Setting $m_{t}=\sup\limits_{0\leq s\leq t}\|v(s)\|$, we see that for $m_{t}>0$

$m_{t}\leq\frac{MK_{t}m_{t}}{\Gamma(\alpha)}\int\limits_{0}^{t}(t-s)^{\alpha-1}e^{w(t-s)}\mathrm{d}s\leq
MK_{t}m_{t}g_{\alpha+1}(t)e^{\omega t}<m_{t},$

if $t>0$ is chosen sufficiently small. Thus, $v(t)=0$ on $[0,t_{0}]$ with
$t_{0}>0$. Iteration of this argument leads to $v(t)\equiv 0$ for any
$t\in\mathbb{R}_{+}$. The proof is complete. ∎

The following lemma plays a crucial role in the proof of Theorem 3.2.

######  Lemma 3.1.

Let $A$ be infinitesimal generator of the strongly continuous fractional
cosine family $\left\\{C_{\alpha}(t;A),t\in\mathbb{R_{+}}\right\\}$ and
$T_{\alpha}(\cdot;A):\mathbb{R_{+}}\to\mathcal{L}(\mathbb{X})$ be the
corresponding fractional Riemann-Liouville family on $\mathbb{X}$. For $f(t)$
strongly continuous on $\mathbb{R_{+}}$ to $\mathbb{X}$,
$g(t)=\int\limits_{0}^{t}T_{\alpha}(t-s;A)f(s)\mathrm{d}s=\int\limits_{0}^{t}T_{\alpha}(s;A)f(t-s)\mathrm{d}s$
exists and is itself strongly continuous on $\mathbb{R_{+}}$ to $\mathbb{X}$.
If $f(t)$ is strongly continuously differentiable on $\mathbb{R_{+}}$ to
$\mathbb{X}$, then $g(t)$ is strongly continuously differentiable for any
$t\in\mathbb{R_{+}}$ and

$\displaystyle\mathcal{D}_{t}^{1}g(t)$
$\displaystyle=T_{\alpha}(t;A)f(0)+\int\limits_{0}^{t}T_{\alpha}(t-s;A)f^{\prime}(s)\mathrm{d}s$
$\displaystyle=\int\limits_{0}^{t}T^{\prime}_{\alpha}(t-s;A)f(s)\mathrm{d}s,\quad
t\in\mathbb{R_{+}},$ (3.24)

where $T^{\prime}_{\alpha}(t;A)$ is defined as in (3.15).

###### Proof.

Since $\left\lVert T_{\alpha}(t;A)\right\rVert\leq Me^{\omega t}g_{\alpha}(t)$
for any $t\in\mathbb{R_{+}}$, it is obvious that $T_{\alpha}(t-s;A)f(s)$ is
strongly continuous in $s\in[0,t]$ whenever the same is true of $f(s)$. In
this case, $g(t)=\int\limits_{0}^{t}T_{\alpha}(t-s;A)f(s)\mathrm{d}s$ will
exist in the strong topology and be equal to
$\int\limits_{0}^{t}T_{\alpha}(s;A)f(t-s)\mathrm{d}s$. Moreover, since
$f:\mathbb{R_{+}}\to\mathcal{L}(\mathbb{X})$ is strongly continuously
differentiable for any $t\in\mathbb{R_{+}}$, by using the well-known
integration by parts formula for the operator-valued functions, we obtain the
desired result:

$\displaystyle\int\limits_{0}^{t}T_{\alpha}(t-s;A)f^{\prime}(s)\mathrm{d}s=-T_{\alpha}(t;A)f(0)+\int\limits_{0}^{t}T^{\prime}_{\alpha}(t-s;A)f(s)\mathrm{d}s,\quad
t\in\mathbb{R_{+}}.$

∎

######  Theorem 3.2.

Let $\left\\{C_{\alpha}(t;A):t\in\mathbb{R_{+}}\right\\}$ be a strongly
continuous fractional cosine family with infinitesimal generator $A$ and
$\left\\{T_{\alpha}(t;A):t\in\mathbb{R_{+}}\right\\}$ be fractional Riemann-
Liouville family corresponding to the $C_{\alpha}$. Let $B(t)$ and $f(t)$ be
strongly continuously differentiable functions on $\mathbb{R_{+}}$ to
$\mathcal{L}(\mathbb{X})$ and $\mathbb{X}$, respectively. Then there exists a
unique $2$-times continuously differentiable function
$w:\mathbb{R_{+}}\to\mathscr{D}(A)$ which is a particular solution of equation
(1.4) with zero initial conditions $w(0)=w^{\prime}(0)=0$. This solution has
the closed-form

$w(t)=\sum_{n=0}^{\infty}w_{n}(t),$ (3.25)

where

$\displaystyle w_{0}(t)=\int\limits_{0}^{t}T_{\alpha}(t-s;A)f(s)\mathrm{d}s,$
$\displaystyle
w_{n}(t)=\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)w_{n-1}(s)\mathrm{d}s,\quad
t>0,\quad n\in\mathbb{N}.$

###### Proof.

It follows since $f\in\mathbb{C}^{1}(\mathbb{R_{+}},\mathbb{X})$, $w_{0}(t)$
is $2$-times strongly continuously differentiable and hence by induction that
$w_{n}(t)$ is likewise for any $n\in\mathbb{N}$. In fact, by Lemma 2.1, we
have

$\displaystyle
w^{\prime}_{0}(t)=\int\limits_{0}^{t}T_{\alpha}^{\prime}(t-s;A)f(s)\mathrm{d}s,$
$\displaystyle
w^{\prime\prime}_{0}(t)=T^{\prime}_{\alpha}(t;A)f(0)+\int\limits_{0}^{t}T_{\alpha}^{\prime}(t-s;A)f^{\prime}(s)\mathrm{d}s,$
$\displaystyle
w^{\prime}_{n}(t)=\int\limits_{0}^{t}T_{\alpha}^{\prime}(t-s;A)B(s)w_{n-1}(s)\mathrm{d}s,$
$\displaystyle
w^{\prime\prime}_{n}(t)=\int\limits_{0}^{t}T_{\alpha}^{\prime}(t-s;A)B^{\prime}(s)w_{n-1}(s)\mathrm{d}s+\int\limits_{0}^{t}T_{\alpha}^{\prime}(t-s;A)B(s)w^{\prime}_{n-1}(s)\mathrm{d}s,\quad
n\in\mathbb{N}.$

By using (1) and (2) it is easy to acquire the estimations for
$x\in\mathscr{D}(A)$:

$\displaystyle\|w_{n}(t)x\|\leq M^{n+1}K_{t}^{n}N_{t}e^{\omega
t}g_{n\alpha+\alpha+1}(t)\left\lVert x\right\rVert,\quad n\in\mathbb{N}_{0},$
$\displaystyle\|w_{0}^{\prime}(t)x\|\leq MN_{t}e^{\omega
t}g_{2\alpha}(t)\left\lVert Ax\right\rVert+N_{t}g_{\alpha}(t)\left\lVert
x\right\rVert,$ $\displaystyle\|w_{1}^{\prime}(t)x\|\leq
M^{2}K_{t}N_{t}e^{\omega t}g_{3\alpha}(t)\left\lVert
Ax\right\rVert+MK_{t}N_{t}e^{\omega t}g_{2\alpha}(t)\left\lVert
x\right\rVert,$ $\displaystyle\|w_{n}^{\prime}(t)x\|\leq
M^{n+1}K_{t}^{n}N_{t}e^{\omega t}g_{n\alpha+2\alpha}(t)\left\lVert
Ax\right\rVert+M^{n}K_{t}^{n}N_{t}e^{\omega t}g_{n\alpha+\alpha}(t)\left\lVert
x\right\rVert,\quad n\in\mathbb{N},$
$\displaystyle\|w_{0}^{\prime\prime}(t)x\|\leq MN_{t}e^{\omega
t}\Big{[}g_{2\alpha-1}(t)+g_{2\alpha}(t)\Big{]}\left\lVert
Ax\right\rVert+N_{t}\Big{[}g_{\alpha-1}(t)+g_{\alpha}(t)\Big{]}\left\lVert
x\right\rVert,$ $\displaystyle\|w_{1}^{\prime\prime}(t)x\|\leq
M^{2}K_{t}N_{t}e^{\omega t}g_{4\alpha-1}(t)\left\lVert
A^{2}x\right\rVert+MK_{t}N_{t}e^{\omega
t}\Big{[}2g_{3\alpha-1}(t)+Mg_{3\alpha}(t)\Big{]}\left\lVert Ax\right\rVert$
$\displaystyle\hskip 41.25641pt+K_{t}N_{t}\Big{[}Me^{\omega
t}g_{2\alpha}(t)+g_{2\alpha-1}(t)\Big{]}\left\lVert x\right\rVert,$
$\displaystyle\|w_{n}^{\prime\prime}(t)\|\leq M^{n+1}K_{t}^{n}N_{t}e^{\omega
t}g_{n\alpha+3\alpha-1}(t)\left\lVert
A^{2}x\right\rVert+M^{n}K_{t}^{n}N_{t}e^{\omega
t}\Big{[}(n+1)g_{n\alpha+2\alpha-1}(t)+Mg_{n\alpha+2\alpha}(t)\Big{]}\left\lVert
Ax\right\rVert$ $\displaystyle\hskip
39.83368pt+K_{t}^{n}N_{t}\Big{[}M^{n}e^{\omega
t}g_{n\alpha+\alpha}(t)+g_{n\alpha+\alpha-1}(t)\Big{]}\left\lVert
x\right\rVert,\quad n\in\mathbb{N},$

where $N_{t}\coloneqq\sup\limits_{0\leq s\leq t}\Biggl{\\{}\left\lVert
f(s)\right\rVert,\left\lVert f^{\prime}(s)\right\rVert\Biggr{\\}}$.

If we set $w(t)=\sum\limits_{n=0}^{\infty}w_{n}(t)$, then as in Theorem 3.1,
$w(t)$ is 2-times strongly continuously differentiable on $\mathbb{R_{+}}$ to
$\mathscr{D}(A)$ and
$w^{\prime}(t)=\sum\limits_{n=0}^{\infty}w^{\prime}_{n}(t)$,
$w^{\prime\prime}(t)=\sum\limits_{n=0}^{\infty}w^{\prime\prime}_{n}(t)$.
Furthermore, $w(0)=w^{\prime}(0)=0$. From the definition of $w_{n}(t)$,
$n\in\mathbb{N}_{0}$ and uniform convergence of the series
$\sum\limits_{n=0}^{\infty}w_{n}(t)$ in every finite interval of
$\mathbb{R_{+}}$, it follows that

$\displaystyle
w(t)=\sum_{n=0}^{\infty}w_{n}(t)=\sum_{n=1}^{\infty}w_{n}(t)+w_{0}(t)$
$\displaystyle=w_{0}(t)+\sum_{n=1}^{\infty}\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)w_{n-1}(s)\mathrm{d}s$
$\displaystyle=w_{0}(t)+\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)\sum_{n=1}^{\infty}w_{n-1}(s)\mathrm{d}s$
$\displaystyle=w_{0}(t)+\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)\sum_{n=0}^{\infty}w_{n}(s)\mathrm{d}s$
$\displaystyle=w_{0}(t)+\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)w(s)\mathrm{d}s,$
(3.26)

where the interchanging summation and integration is justified by the uniform
convergence of the series. Since $w(t)$ is $2$-times strongly continuously
differentiable, we may differentiate (3.26) term-wise in Caputo’s sense by
considering $C_{\alpha}(0;A)=I$, using (2.6) and (3) as follows:

$\displaystyle\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}w(t)$
$\displaystyle=\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}\Big{(}w_{0}(t)+\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)w(s)\mathrm{d}s\Big{)}$
$\displaystyle=\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}\Big{(}T_{\alpha}(t;A)\ast
f(t)\Big{)}+\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}\Big{(}T_{\alpha}(t;A)\ast\Big{(}B(t)w(t)\Big{)}\Big{)}$
$\displaystyle=\prescript{}{}{\mathcal{D}^{\alpha}_{t}}\Big{(}T_{\alpha}(t;A)\ast
f(t)\Big{)}+\prescript{}{}{\mathcal{D}^{\alpha}_{t}}\Big{(}T_{\alpha}(t;A)\ast\Big{(}B(t)w(t)\Big{)}\Big{)}$
$\displaystyle=AT_{\alpha}(t;A)\ast
f(t)+f(t)+AT_{\alpha}(t;A)\ast\Big{(}B(t)w(t)\Big{)}+B(t)w(t)$
$\displaystyle=Aw_{0}(t)+f(t)+A\int\limits_{0}^{t}T_{\alpha}(t-s;A)B(s)w(s)\mathrm{d}s+B(t)w(t)$
$\displaystyle=\Big{(}A+B(t)\Big{)}w(t)+f(t).$

Note that since initial values are zero we have used Riemann-Liouville
fractional derivative instead of Caputo one. This shows immediately that
$w(t)$ is a particular solution of (1.4). The uniqueness of solution follows
precisely as in the uniqueness proof of Theorem 3.1. The proof is complete. ∎

######  Remark 3.1.

Considering the homogeneous equation with the second initial condition equal
to zero, we derive the special case published in [19] by Bazhlekova. The
reason is that in some models (e.g. for waves in viscoelastic media) there are
some physical arguments to choose the second initial condition equal to zero
in her thesis [18]. Therefore, our theory generalizes those results.

  * 1.

In this special case, if $B(t)\equiv B\in\mathcal{L}(\mathbb{X})$, then for
any $x,y\in\mathscr{D}(A)$, the solution to the following abstract Cauchy
problem for fractional evolution equation in a Banach space $\mathbb{X}$:

$\displaystyle\begin{cases}\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}u(t)=(A+B)u(t)+f(t),\quad
t>0,\\\ u(0)=x,\quad u^{\prime}(0)=y,\end{cases}$ (3.27)

can be written as follows:

$\displaystyle u(t)=C_{\alpha}(t;A+B)x+S_{\alpha}(t;A+B)y+w(t),\quad t>0,$

where

$\displaystyle C_{\alpha}(t;A+B)x=\sum_{n=0}^{\infty}C_{\alpha,n}(t,A)x,$
$\displaystyle S_{\alpha}(t;A+B)x=\sum_{n=0}^{\infty}S_{\alpha,n}(t;A)x,\quad
x\in\mathbb{X},$ $\displaystyle w(t)=\sum_{n=0}^{\infty}w_{n}(t),$

with

$\displaystyle C_{\alpha,0}(t;A)x=C_{\alpha}(t;A)x,\quad
S_{\alpha,0}(t;A)x=S_{\alpha}(t;A)x,\quad
w_{0}(t)=\int\limits_{0}^{t}T_{\alpha}(t-s)f(s)\mathrm{d}s,$ $\displaystyle
C_{\alpha,n}(t;A)x=\int\limits_{0}^{t}T_{\alpha}(t-s;A)BC_{\alpha,n-1}(s;A)x\mathrm{d}s,$
$\displaystyle
S_{\alpha,n}(t;A)x=\int\limits_{0}^{t}T_{\alpha}(t-s;A)BS_{\alpha,n-1}(s;A)x\mathrm{d}s,$
$\displaystyle
w_{n}(t)=\int\limits_{0}^{t}T_{\alpha}(t-s)Bw_{n-1}(s)\mathrm{d}s,\quad
n\in\mathbb{N},\quad x\in\mathbb{X}.$

In this case, the particular solution $w(t)$ of (3.27) can also be put in a
more suggestive form by stating and proving the following theorem.

######  Theorem 3.3.

Let $A$ be infinitesimal generator of $2$-times strongly continuous fractional
family of linear operators $C_{\alpha}(t;A)$ on $\mathbb{R_{+}}$. Let
$B\in\mathcal{L}(\mathbb{X})$ and $f(t)$ be strongly continuously
differentiable function on $\mathbb{R_{+}}$ to $\mathbb{X}$. Then there exists
a unique $2$-times continuously differentiable function
$w:\mathbb{R_{+}}\to\mathscr{D}(A)$ which is a particular solution of equation
(3.27) with zero initial conditions $w(0)=w^{\prime}(0)=0$. This solution has
a variation of constants formula

$w(t)=\int\limits_{0}^{t}T_{\alpha}(t-s;A+B)f(s)\mathrm{d}s,\quad t>0,$ (3.28)

where

$T_{\alpha}(t;A+B)=\int\limits_{0}^{t}g_{\alpha-1}(t-s)C_{\alpha}(s;A+B)\mathrm{d}s.$
(3.29)

###### Proof.

From $\left\lVert C_{\alpha}(t;A+B)\right\rVert\leq Me^{\omega
t}E_{\alpha}\left(M\left\lVert B\right\rVert t^{\alpha}\right)$ and (3.29), we
have

$\displaystyle\left\lVert T_{\alpha}(t;A+B)\right\rVert$
$\displaystyle\leq\int\limits_{0}^{t}g_{\alpha-1}(t-s)\left\lVert
C_{\alpha}(s;A+B)\right\rVert\mathrm{d}s$ $\displaystyle\leq
M\int\limits_{0}^{t}\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}e^{\omega
s}E_{\alpha}\left(M\left\lVert B\right\rVert s^{\alpha}\right)\mathrm{d}s$
$\displaystyle\leq e^{\omega t}\sum_{n=0}^{\infty}M^{n+1}\left\lVert
B\right\rVert^{n}\int\limits_{0}^{t}\frac{(t-s)^{\alpha-2}}{\Gamma(\alpha-1)}\frac{s^{n\alpha}}{\Gamma(n\alpha+1)}\mathrm{d}s$
$\displaystyle=e^{\omega t}\sum_{n=0}^{\infty}M^{n+1}\left\lVert
B\right\rVert^{n}\frac{t^{n\alpha+\alpha-1}}{\Gamma(n\alpha+\alpha)}$
$\displaystyle=Me^{\omega t}t^{\alpha-1}E_{\alpha,\alpha}\left(M\left\lVert
B\right\rVert t^{\alpha}\right).$ (3.30)

In fact, by Lemma 3.1 and formula (3.15), we have

$\displaystyle
w^{\prime}(t)=\int\limits_{0}^{t}T_{\alpha}(t-s;A+B)f^{\prime}(s)\mathrm{d}s+T_{\alpha}(t;A+B)f(0),$
$\displaystyle
w^{\prime\prime}(t)=T^{\prime}_{\alpha}(t;A+B)f(0)+\int\limits_{0}^{t}T^{\prime}_{\alpha}(t-s;A+B)f^{\prime}(s)\mathrm{d}s,$
$\displaystyle T^{\prime}_{\alpha}(t;A+B)=g_{\alpha-1}(t)\ast
C^{\prime}_{\alpha}(t;A+B)+g_{\alpha-1}(t),$ $\displaystyle
C^{\prime}_{\alpha}(t;A+B)=AT_{\alpha}(t;A)+\int\limits_{0}^{t}T^{\prime}_{\alpha}(t-s;A)BC_{\alpha}(s;A+B)\mathrm{d}s.$

It is now easy to acquire the estimations for $n\in\mathbb{N}_{0}$:

$\displaystyle\|w(t)x\|\leq MN_{t}e^{\omega
t}t^{\alpha}E_{\alpha,\alpha+1}(M\left\lVert B\right\rVert
t^{\alpha})\left\lVert x\right\rVert,$ $\displaystyle\|w^{\prime}(t)x\|\leq
MN_{t}e^{\omega t}\Big{[}t^{\alpha-1}E_{\alpha,\alpha}(M\left\lVert
B\right\rVert t^{\alpha})+t^{\alpha}E_{\alpha,\alpha+1}(M\left\lVert
B\right\rVert t^{\alpha})\Big{]}\left\lVert x\right\rVert,$
$\displaystyle\|w^{\prime\prime}(t)x\|\leq MN_{t}e^{\omega
t}\Big{[}g_{2\alpha-1}(t)+g_{2\alpha}(t)\Big{]}\left\lVert
Ax\right\rVert+N_{t}\Big{[}g_{\alpha-1}(t)+g_{\alpha}(t)\Big{]}\left\lVert
x\right\rVert$ $\displaystyle+M^{2}\left\lVert B\right\rVert N_{t}e^{\omega
t}\Big{[}t^{3\alpha-1}E_{\alpha,3\alpha}(M\left\lVert B\right\rVert
t^{\alpha})+t^{3\alpha-2}E_{\alpha,3\alpha-1}(M\left\lVert B\right\rVert
t^{\alpha})\Big{]}\left\lVert Ax\right\rVert$ $\displaystyle+M\left\lVert
B\right\rVert N_{t}e^{\omega
t}\Big{[}t^{2\alpha-1}E_{\alpha,2\alpha}(M\left\lVert B\right\rVert
t^{\alpha})+t^{2\alpha-2}E_{\alpha,2\alpha-1}(M\left\lVert B\right\rVert
t^{\alpha})\Big{]}\left\lVert x\right\rVert,\quad x\in\mathscr{D}(A),$

where we have used the following estimations:

$\displaystyle\|C_{\alpha}^{\prime}(t;A+B)x\|$ $\displaystyle\leq
M^{2}\left\lVert B\right\rVert e^{\omega
t}t^{2\alpha-1}E_{\alpha,2\alpha}(M\left\lVert B\right\rVert
t^{\alpha})\left\lVert Ax\right\rVert$ $\displaystyle+M\left\lVert
B\right\rVert e^{\omega t}t^{\alpha-1}E_{\alpha,\alpha}(M\left\lVert
B\right\rVert t^{\alpha})\left\lVert x\right\rVert+Me^{\omega
t}g_{\alpha}(t)\left\lVert Ax\right\rVert,$
$\displaystyle\|T_{\alpha}^{\prime}(t;A+B)x\|\leq M^{2}\left\lVert
B\right\rVert e^{\omega t}t^{3\alpha-2}E_{\alpha,3\alpha-1}(M\left\lVert
B\right\rVert t^{\alpha})\left\lVert Ax\right\rVert$
$\displaystyle+M\left\lVert B\right\rVert e^{\omega
t}t^{2\alpha-2}E_{\alpha,2\alpha-1}(M\left\lVert B\right\rVert
t^{\alpha})\left\lVert x\right\rVert+Me^{\omega t}g_{2\alpha-1}(t)\left\lVert
Ax\right\rVert+g_{\alpha-1}(t)\left\lVert x\right\rVert,\quad
x\in\mathscr{D}(A),$

with $N_{t}\coloneqq\sup\limits_{0\leq s\leq t}\Biggl{\\{}\left\lVert
f(s)\right\rVert,\left\lVert f^{\prime}(s)\right\rVert\Biggr{\\}}$.

Therefore, $w(t)$ is 2-times strongly continuously differentiable function on
$\mathbb{R_{+}}$. Next, we prove that $w(t)$ satisfies (3.27) with
$w(0)=w^{\prime}(0)=0$. By making use of the formulas (2.6),(3.28) and (3.29),
we derive that

$\displaystyle\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}w(t)$
$\displaystyle=\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}\Big{(}\int\limits_{0}^{t}T_{\alpha}(t-s;A+B)f(s)\mathrm{d}s\Big{)}=\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}\Big{(}T_{\alpha}(t;A+B)\ast
f(t)\Big{)}=\prescript{}{}{\mathcal{D}^{\alpha}_{t}}\Big{(}T_{\alpha}(t;A+B)\ast
f(t)\Big{)}$
$\displaystyle=\prescript{}{}{\mathcal{D}^{2}_{t}}\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}\Big{(}\prescript{}{}{\mathcal{I}^{\alpha-1}_{t}}C_{\alpha}(t;A+B)\Big{)}\ast
f(t)=\prescript{}{}{\mathcal{D}^{2}_{t}}\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}\prescript{}{}{\mathcal{I}^{\alpha-1}_{t}}\Big{(}C_{\alpha}(t;A+B)\ast
f(t)\Big{)}$
$\displaystyle=\prescript{}{}{\mathcal{D}^{2}_{t}}\prescript{}{}{\mathcal{I}^{1}_{t}}\Big{(}C_{\alpha}(t;A+B)\ast
f(t)\Big{)}\ =\prescript{}{}{\mathcal{D}^{1}_{t}}\Big{(}C_{\alpha}(t;A+B)\ast
f(t)\Big{)}$
$\displaystyle=\prescript{}{}{\mathcal{D}^{1}_{t}}C_{\alpha}(t;A+B)\ast
f(t)+C_{\alpha}(0;A+B)f(t).$

In a similar to (3), we have

$\displaystyle\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}C_{\alpha}(t;A+B)=(A+B)C_{\alpha}(t;A+B)=C_{\alpha}(t;A+B)(A+B).$
(3.31)

Since $C_{\alpha}(0;A+B)=I$ and using (3.31), we attain

$\displaystyle\prescript{}{}{\mathcal{D}^{1}_{t}}C_{\alpha}(t;A+B)$
$\displaystyle=\prescript{}{}{\mathcal{I}^{1}_{t}}\prescript{}{}{\mathcal{D}^{2}_{t}}C_{\alpha}(t;A+B)=\prescript{}{}{\mathcal{I}^{\alpha-1}_{t}}\prescript{}{}{\mathcal{I}^{2-\alpha}_{t}}\prescript{}{}{\mathcal{D}^{2}_{t}}C_{\alpha}(t;A+B)=\prescript{}{}{\mathcal{I}^{\alpha-1}_{t}}\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}C_{\alpha}(t;A+B)$
$\displaystyle=\prescript{}{}{\mathcal{I}^{\alpha-1}_{t}}\left(A+B\right)C_{\alpha}(t;A+B)=\prescript{}{}{\mathcal{I}^{\alpha-1}_{t}}C_{\alpha}(t;A+B)\left(A+B\right)$
$\displaystyle=(A+B)T_{\alpha}(t;A+B)=T_{\alpha}(t;A+B)(A+B).$

So, we derive a desired result:

$\displaystyle\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}w(t)$
$\displaystyle=\Big{(}(A+B)T_{\alpha}(t;A+B)\Big{)}\ast
f(t)+f(t)=\left(A+B\right)\int\limits_{0}^{t}T_{\alpha}(t-s;A+B)f(s)\mathrm{d}s+f(t)$
$\displaystyle=\Big{(}A+B\Big{)}w(t)+f(t).$

The proof is complete. ∎

  * 2.

In the particular case, if $B=0$, then the solution of the following abstract
initial value problem for fractional evolution equation in a Banach space
$\mathbb{X}$ which is fulfilling $u(0)=x$, $u^{\prime}(0)=y$ for
$x,y\in\mathscr{D}(A)$:

$\left(\prescript{C}{}{\mathcal{D}^{\alpha}_{t}}u\right)(t)=Au(t),\quad t>0,$
(3.32)

where a linear operator $A:\mathscr{D}(A)\subseteq\mathbb{X}\to\mathbb{X}$ is
the infinitesimal generator of an operator fractional cosine function
$C_{\alpha}$,

$u(t)=C_{\alpha}(t;A)x+S_{\alpha}(t;A)y,\quad t\in\mathbb{R}_{+},$

where $S_{\alpha}:\mathbb{R}_{+}\to\mathcal{L}(\mathbb{X})$ denotes the
fractional sine function associated with $C_{\alpha}$, defined by

$\displaystyle
S_{\alpha}(t;A)x\coloneqq\int\limits_{0}^{t}C_{\alpha}(s;A)x\mathrm{d}s.$

It is important to note that this case have considered by Li in [9].
Furthermore, the same particular case has studied by Li et al. in [10] with
the first initial condition is zero, i.e., $u(0)=0$ and by Chen and Li in [11]
with the second initial condition is zero, i.e., $u^{\prime}(0)=0$.

##  4 Special cases

In this section, we consider several special cases and remarks.

Case 1. Classical case: $\alpha=2$. We consider the following abstract Cauchy
problem for a second-order linear inhomogeneous evolution equation in a Banach
space $\mathbb{X}$:

$\begin{cases}u^{\prime\prime}(t)=\left(A+B(t)\right)u(t)+f(t),\quad
t\in\mathbb{R},\\\ u(0)=x,\quad u^{\prime}(0)=y.\end{cases}$ (4.1)

It should be note that in the case of $\alpha=2$, $T_{\alpha}(t;A)$ and
$S_{\alpha}(t;A)$ coincide with the strongly continuous sine function
$S(t;A)$. Furthermore, for $\alpha=2$, one and two parameter Mittag-Leffler
type functions are converting to the hyperbolic cosine and sine functions,
respectively:

$\displaystyle
E_{2}(MK_{t}t^{2})=\sum_{k=0}^{\infty}\frac{M^{k}K^{k}_{t}t^{2k}}{(2k)!}=\cosh\Big{(}\sqrt{MK_{t}}t\Big{)},\quad
t\in\mathbb{R},$ (4.2) $\displaystyle
tE_{2,2}(MK_{t}t^{2})=\sum_{k=0}^{\infty}\frac{M^{k}K^{k}_{t}t^{2k+1}}{(2k+1)!}=\frac{1}{\sqrt{MK_{t}}}\sinh\Big{(}\sqrt{MK_{t}}t\Big{)},\quad
t\in\mathbb{R}.$ (4.3)

######  Theorem 4.1.

Let $A$ be the infinitesimal generator of the an operator cosine function
$\left\\{C(t;A):t\in\mathbb{R}\right\\}$ with
$\|C(t;A)\|\leq Me^{w|t|}$ for all $t\in\mathbb{R}$, and let
$\left\\{S(t;A):t\in\mathbb{R}\right\\}$ denote the sine function associated
with $C(t;A)$. Furthermore, suppose that
$B:\mathbb{R}\to\mathcal{L}(\mathbb{X})$ and $f:\mathbb{R}\to\mathbb{X}$ are
strongly continuously differentiable. Then

  * 1.

The Cauchy problem for the $\mathbb{X}$-valued differential equation (4.1) is
uniformly well-posed; more precisely, for each pair $x,y\in\mathscr{D}(A)$
there is a uniquely determined solution
$u\in\mathbb{C}^{2}\left(\mathbb{R};\mathscr{D}(A)\right)$ of (4.1) fulfilling
initial conditions and $u\in\mathscr{D}(A)$ depends continuously (with respect
to the topology of uniform convergence) on compact subsets of $\mathbb{R}$
upon $(x,y)$.

  * 2.

$u\in\mathscr{D}(A)$ is given by

$u(t)=C(t;A+B)x+S(t;A+B)y+w(t),\quad t\in\mathbb{R},$ (4.4)

where for $x\in\mathscr{D}(A)$, $t\mapsto C(t;A+B)x$ and $t\mapsto S(t;A+B)x$
are $2$-times strongly continuously differentiable solutions of (4.1)
fulfilling

$\displaystyle C(0;A+B)x=x,\quad C^{\prime}(0;A+B)x=0,$ (4.5) $\displaystyle
S(0;A+B)x=0,\quad S^{\prime}(0;A+B)x=x.$ (4.6)

  * 3.

$C(\cdot;A+B),S(\cdot;A+B)\in\mathcal{L}(\mathbb{X})$ are defined by

$\displaystyle C(t;A+B)x\coloneqq\sum_{n=0}^{\infty}C_{n}(t,A)x,\quad
x\in\mathbb{X},$ (4.7) $\displaystyle
S(t;A+B)x\coloneqq\sum_{n=0}^{\infty}S_{n}(t;A)x,\quad x\in\mathbb{X},$ (4.8)

where

$\displaystyle C_{0}(t;A)\coloneqq C(t;A),\quad S_{0}(t;A)\coloneqq S(t;A),$
$\displaystyle
C_{n}(t;A)x\coloneqq\int\limits_{0}^{t}S(t-s;A)B(s)C_{n-1}(s;A)x\mathrm{d}s,$
(4.9) $\displaystyle
S_{n}(t;A)x\coloneqq\int\limits_{0}^{t}S(t-s;A)B(s)S_{n-1}(s;A)x\mathrm{d}s,\quad
n\in\mathbb{N},\quad x\in\mathbb{X}.$ (4.10)

  * 4.

A particular solution $w\in\mathscr{D}(A)$ of (4.1) is defined by

$w(t)=\sum_{n=0}^{\infty}w_{n}(t),$ (4.11)

where

$\displaystyle w_{0}(t)=\int\limits_{0}^{t}S(t-s;A)f(s)\mathrm{d}s,$
$\displaystyle
w_{n}(t)=\int\limits_{0}^{t}S(t-s;A)B(s)w_{n-1}(s)\mathrm{d}s,\quad
n\in\mathbb{N}.$

  * 5.

With $K_{t}\coloneqq\sup\limits_{0\leq s\leq t}\Biggl{\\{}\left\lVert
B(s)\right\rVert,\left\lVert B^{\prime}(s)\right\rVert\Biggr{\\}}$, we have
for all $s\in[0,t]$, the bounds

$\displaystyle\left\lVert C(s;A+B)\right\rVert\leq
Me^{w|s|}\cosh\Big{(}\sqrt{MK_{t}}s\Big{)},$ $\displaystyle\left\lVert
S(s;A+B)\right\rVert\leq
Me^{w|s|}\frac{1}{\sqrt{MK_{t}}}\sinh\Big{(}\sqrt{MK_{t}}|s|\Big{)},$
$\displaystyle\left\lVert C(s;A+B)-C(s;A)\right\rVert\leq
Me^{w|s|}\Big{[}\cosh\Big{(}\sqrt{MK_{t}}s\Big{)}-1\Big{]},$
$\displaystyle\left\lVert S(s;A+B)-S(s;A)\right\rVert\leq
Me^{w|s|}\Big{[}\frac{1}{\sqrt{MK_{t}}}\sinh\Big{(}\sqrt{MK_{t}}|s|\Big{)}-|s|\Big{]}.$

In the special case, if $B(t)\coloneqq B\in\mathcal{L}(\mathbb{X})$ in
$\mathbb{R}$, a particular solution of (4.1) can be represented by as below:

$\displaystyle w(t)=\int\limits_{0}^{t}S(t-s;A+B)f(s)\mathrm{d}s.$ (4.12)

Note that the homogeneous case of (4.1) have been considered by Lutz in [14]
and we have added a particular solution to (4.1) as a consequence of the above
results in fractional-order sense.

Case 2. If we consider $A$ and $B$ are bounded linear operators, then two
subcases arise in this direction. In this case, for a bounded system operator
$\mathscr{D}(A)$ coincide with the state space $\mathbb{X}$.

Case 2a: Non-permutable case $\left(AB\neq BA\right)$. It is known that
fractional cosine, sine and Riemann-Liouville families can also be
characterized via the Laplace transform of their generators.

It should be note that the following lemma will be helpful to derive a
solution to (3.27).

######  Lemma 4.1.

For $A,B\in\mathcal{L}(\mathbb{X})$ satisfying $AB\neq BA$, we have:

$\displaystyle\mathscr{L}^{-1}\Biggl{\\{}\lambda^{\gamma}\left[(\lambda^{\alpha}I-A)^{-1}B\right]^{k}(\lambda^{\alpha}I-A)^{-1}\Biggr{\\}}(t)=\sum_{m=0}^{\infty}\frac{\mathcal{Q}^{A,B}_{k,m}}{\Gamma(k\alpha+m\alpha+\alpha-\gamma)}t^{k\alpha+m\alpha+\alpha-\gamma-1},\quad
k\in\mathbb{N}_{0},$

where $\mathcal{Q}^{A,B}_{k,m}\in\mathcal{L}(\mathbb{X})$ is defined by

$\displaystyle\mathcal{Q}^{A,B}_{k,m}=\sum_{l=0}^{k}A^{k-l}B\mathcal{Q}^{A,B}_{l,m-1},k,m\in\mathbb{N},\quad\mathcal{Q}^{A,B}_{k,0}=A^{k},k\in\mathbb{N}_{0},\quad\mathcal{Q}^{A,B}_{0,m}=B^{m},m\in\mathbb{N}_{0}.$
(4.13)

###### Proof.

This lemma is a special case of Lemma 3.2 in [29, 30] with $\beta=0$ which can
be proved via mathematical induction principle. So, the proof of this lemma is
omitted here. ∎

Then, we apply Laplace transform method and solve the equation (3.27) with
bounded linear operators $A,B$. After solving the equation with respect to
$U(\lambda)$, we obtain

$\displaystyle
U(\lambda)=\lambda^{\alpha-1}(\lambda^{\alpha}I-A-B)^{-1}x+\lambda^{\alpha-2}(\lambda^{\alpha}I-A-B)^{-1}y+(\lambda^{\alpha}I-A-B)^{-1}F(\lambda),$
(4.14)

where $U(\lambda)=\left(\mathscr{L}u\right)(\lambda)$ and
$F(\lambda)=\left(\mathscr{L}f\right)(\lambda)$.

Then for non-permutable linear operators $A,B\in\mathcal{L}(\mathbb{X})$ and
sufficiently large $\lambda$, we have

$\displaystyle(\lambda^{\alpha}I-A-B)^{-1}$
$\displaystyle=\Big{[}(\lambda^{\alpha}I-A)(I-(\lambda^{\alpha}I-A)^{-1}B)\Big{]}^{-1}$
$\displaystyle=(I-(\lambda^{\alpha}I-A)^{-1}B)^{-1}(\lambda^{\alpha}I-A)^{-1}$
$\displaystyle=\sum_{k=0}^{\infty}\left[(\lambda^{\alpha}I-A)^{-1}B\right]^{k}(\lambda^{\alpha}I-A)^{-1},$

where $\|(\lambda^{\alpha}I-A)^{-1}B\|<I$.

Taking inverse Laplace transform of (4.14), we have

$\displaystyle u(t)$
$\displaystyle=\mathscr{L}^{-1}\Biggl{\\{}\sum_{k=0}^{\infty}\lambda^{\alpha-1}\left[(\lambda^{\alpha}I-A)^{-1}B\right]^{k}(\lambda^{\alpha}I-A)^{-1}\Biggr{\\}}(t)x$
$\displaystyle+\mathscr{L}^{-1}\Biggl{\\{}\sum_{k=0}^{\infty}\lambda^{\alpha-2}\left[(\lambda^{\alpha}I-A)^{-1}B\right]^{k}(\lambda^{\alpha}I-A)^{-1}\Biggr{\\}}(t)y$
$\displaystyle+\mathscr{L}^{-1}\Biggl{\\{}\sum_{k=0}^{\infty}\left[(\lambda^{\alpha}I-A)^{-1}B\right]^{k}(\lambda^{\alpha}I-A)^{-1}F(\lambda)\Biggr{\\}}(t)$

According to the Lemma 4.1, we derive the explicit analytical representation
of the solution $u(t)$ to (3.27) with linear bounded operators $A$ and $B$ as
follows:

$\displaystyle u(t)$
$\displaystyle=\sum_{n=0}^{\infty}\sum_{\begin{subarray}{c}k+m=n\\\ k,m\geq
0\end{subarray}}\mathcal{Q}^{A,B}_{k,m}\frac{t^{n\alpha}}{\Gamma(n\alpha+1)}x$
$\displaystyle+\sum_{n=0}^{\infty}\sum_{\begin{subarray}{c}k+m=n\\\ k,m\geq
0\end{subarray}}\mathcal{Q}^{A,B}_{k,m}\frac{t^{n\alpha+1}}{\Gamma(n\alpha+2)}y$
(4.15)
$\displaystyle+\int\limits_{0}^{t}\sum_{n=0}^{\infty}\sum_{\begin{subarray}{c}k+m=n\\\
k,m\geq
0\end{subarray}}\mathcal{Q}^{A,B}_{k,m}\frac{(t-s)^{n\alpha+\alpha-1}}{\Gamma(n\alpha+\alpha)}f(s)\mathrm{d}s.$

Thus, we proved the following theorem:

######  Theorem 4.2.

Let $A,B\in\mathcal{L}(\mathbb{X})$ with non-zero commutator, i.e., $[A,B]=AB-
BA\neq 0$. Then the closed-form of a solution $u\in\mathbb{X}$ of the abstract
Cauchy problem (3.27) can be expressed as

$\displaystyle u(t)=\sum_{n=0}^{\infty}\sum_{\begin{subarray}{c}k+m=n\\\
k,m\geq
0\end{subarray}}\mathcal{Q}^{A,B}_{k,m}\frac{t^{n\alpha}}{\Gamma(n\alpha+1)}x$
$\displaystyle+\sum_{n=0}^{\infty}\sum_{\begin{subarray}{c}k+m=n\\\ k,m\geq
0\end{subarray}}\mathcal{Q}^{A,B}_{k,m}\frac{t^{n\alpha+1}}{\Gamma(n\alpha+2)}y$
$\displaystyle+\int\limits_{0}^{t}\sum_{n=0}^{\infty}\sum_{\begin{subarray}{c}k+m=n\\\
k,m\geq
0\end{subarray}}\mathcal{Q}^{A,B}_{k,m}\frac{(t-s)^{n\alpha+\alpha-1}}{\Gamma(n\alpha+\alpha)}f(s)\mathrm{d}s.$

Case 2b: Permutable case $\left(AB=BA\right)$. This case is obtained directly
from (4) using the following identity for $\mathcal{Q}^{A,B}_{k,m}$ [29, 30]:

$\mathcal{Q}^{A,B}_{k,m}=\binom{k+m}{m}A^{k}B^{m},$

and the eminent Cauchy product formula, we acquire

$\displaystyle u(t)$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}\binom{k+m}{m}A^{k}B^{m}\frac{t^{(k+m)\alpha}}{\Gamma((k+m)\alpha+1)}x$
$\displaystyle+\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}\binom{k+m}{m}A^{k}B^{m}\frac{t^{(k+m)\alpha+1}}{\Gamma((k+m)\alpha+2)}y$
$\displaystyle+\int\limits_{0}^{t}\sum_{k=0}^{\infty}\sum_{m=0}^{\infty}\binom{k+m}{m}A^{k}B^{m}\frac{(t-s)^{(k+m)\alpha+\alpha-1}}{\Gamma((k+m)\alpha+\alpha)}f(s)\mathrm{d}s$
$\displaystyle=\sum_{k=0}^{\infty}\sum_{m=0}^{k}\binom{k}{m}A^{k-m}B^{m}\frac{t^{k\alpha}}{\Gamma(k\alpha+1)}x$
$\displaystyle+\sum_{k=0}^{\infty}\sum_{m=0}^{k}\binom{k}{m}A^{k-m}B^{m}\frac{t^{k\alpha+1}}{\Gamma(k\alpha+2)}y$
$\displaystyle+\int\limits_{0}^{t}\sum_{k=0}^{\infty}\sum_{m=0}^{k}\binom{k}{m}A^{k-m}B^{m}\frac{(t-s)^{k\alpha+\alpha-1}}{\Gamma(k\alpha+\alpha)}f(s)\mathrm{d}s.$

Now, we apply the well-known binomial theorem for permutable linear bounded
operators $A,B\in\mathcal{L}(\mathbb{X})$:

$\sum_{m=0}^{k}\binom{k}{m}A^{k-m}B^{m}=(A+B)^{k}.$

Then we acquire an explicit analytical representation of the solution to
(3.27) with the help of Mittag-Leffler type functions generated by linear
bounded operators:

$\displaystyle
u(t)=E_{\alpha,1}((A+B)t^{\alpha})x+tE_{\alpha,2}((A+B)t^{\alpha})y+\int\limits_{0}^{t}(t-s)^{\alpha-1}E_{\alpha,\alpha}((A+B)(t-s)^{\alpha})f(s)\mathrm{d}s.$

Therefore, we proved the following theorem.

######  Theorem 4.3.

Let $A,B\in\mathcal{L}(\mathbb{X})$ with zero commutator, i.e., $[A,B]=AB-
BA=0$. A classical solution $u\in\mathbb{X}$ of the abstract initial value
problem (3.27) can be expressed as

$\displaystyle
u(t)=E_{\alpha,1}((A+B)t^{\alpha})x+tE_{\alpha,2}((A+B)t^{\alpha})y+t^{\alpha-1}E_{\alpha,\alpha}((A+B)t^{\alpha})\ast
f(t).$

######  Remark 4.1.

A classical case with non-permutable and permutable linear bounded operators
can be derived as a result of above Case 2a and 2b. Therefore, we consider the
following abstract Cauchy problem:

$\begin{cases}u^{\prime\prime}(t)=\left(A+B\right)u(t)+f(t),\quad
t\in\mathbb{R},\\\ u(0)=x,\quad u^{\prime}(0)=y.\end{cases}$ (4.16)

######  Theorem 4.4.

Let $A,B\in\mathcal{L}(\mathbb{X})$ with non-zero commutator, i.e., $[A,B]=AB-
BA\neq 0$. Then the closed-form of a solution $u\in\mathbb{X}$ of the abstract
Cauchy problem (4.16) can be expressed as

$\displaystyle u(t)=\sum_{n=0}^{\infty}\sum_{\begin{subarray}{c}k+m=n\\\
k,m\geq 0\end{subarray}}\mathcal{Q}^{A,B}_{k,m}\frac{t^{2n}}{(2n)!}x$
$\displaystyle+\sum_{n=0}^{\infty}\sum_{\begin{subarray}{c}k+m=n\\\ k,m\geq
0\end{subarray}}\mathcal{Q}^{A,B}_{k,m}\frac{t^{2n+1}}{(2n+1)!}y$
$\displaystyle+\int\limits_{0}^{t}\sum_{n=0}^{\infty}\sum_{\begin{subarray}{c}k+m=n\\\
k,m\geq
0\end{subarray}}\mathcal{Q}^{A,B}_{k,m}\frac{(t-s)^{2n+1}}{(2n+1)!}f(s)\mathrm{d}s.$

######  Theorem 4.5.

Let $A,B\in\mathcal{L}(\mathbb{X})$ with zero commutator, i.e., $[A,B]=AB-
BA=0$. A classical solution $u\in\mathbb{X}$ of the abstract initial value
problem (4.16) can be expressed as

$\displaystyle
u(t)=\cos(\sqrt{A+B}t)x+\frac{1}{\sqrt{A+B}}\sin(\sqrt{A+B}t)y+\frac{1}{\sqrt{A+B}}\sin(\sqrt{A+B}t)\ast
f(t).$

## References

  * [1] W.E. Fitzgibbon, Global existence and boundedness of solutions to the extensible beam equation, SIAM J. Math. Anal., 13 (1982) 739-745.
  * [2] S. Woinowsky-Krieger, The effect of an axial force on the vibration of hinged bars, ASME J. Appl. Mech., 17 (1950) 35-36.
  * [3] N.I. Mahmudov, M.A. McKibben, Abstract second-order damped McKean-Vlasov stochastic evolution equations, Stoc. Anal. Appl., 242 (2006) 303-328.
  * [4] H.O. Fattorini, Ordinary differential equations in linear topological spaces, I, J. Differential Equations, 5 (1968) 72-105.
  * [5] H.O. Fattorini, Ordinary differential equations in linear topological spaces, II, J. Differential Equations, 6 (1969) 50-70.
  * [6] C.C Travis, G.F. Webb, Cosine families and abstract nonlinear second order differential equations, Acta Math. Acad. Scient. Hung., 32 (3-4) (1978) 75-96.
  * [7] C.C. Travis, G.F. Webb, Compactness, regularity and uniform continuity properties of strongly continuous cosine families, Houston J. Math., 3 (1977) 555-567.
  * [8] J. Bochenek, An abstract nonlinear second order differential equation, Annales Polonici Mathematici, 54 (1991) 155-166.
  * [9] K. Li, Fractional order semilinear Volterra integrodifferential equations in Banach spaces, Topol. Methods Nonlinear Anal., 47 (2) (2016) 439-455.
  * [10] K. Li, J. Peng, J. Jia, Cauchy problems for fractional differential equations with Riemann-Liouville derivatives, J. Funct. Anal., 263 (2012) 476-510.
  * [11] C. Chen, M. Li, On fractional resolvent operator functions, Semigroup forum, 80 (2010) 121-142.
  * [12] H.R. Henríquez, J.G. Mesquita, J.C. Poza, Existence of solutions of the abstract Cauchy problem of fractional order, J. Funct. Anal., 281 (2021) 109028.
  * [13] R.S. Phillips, Perturbation theory for semi-groups of linear operators. Trans. Am. Math. Soc., 74 (1954) 199-221.
  * [14] D. Lutz, On bounded time-dependent perturbations of operator cosine functions, Aequationes Mathematicae, 23 (1981) 197-203.
  * [15] C.C. Travis, G.F. Webb, Perturbation of strongly continuous cosine family generators, Colloquium Mathematicae, 45 (2) (1981) 277-285.
  * [16] Y. Lin, Time-dependent perturbation theory for abstract evolution equations of second order, Studio Mathematica, 130 (1998), 263-274.
  * [17] H. Serizawa, M. Watanabe, Time-dependent perturbation for cosine families in Banach spaces, Houstan Journal of Mathematics, 12 (4) (1986) 579-586.
  * [18] E. Bazhlekova, Fractional evolution equations in Banach spaces, Ph.D. Thesis, Eindhoven University of Technology, 2001.
  * [19] E. Bazhlekova, Perturbation properties for abstract evolution equations of fractional order, Fract. Cal. Appl. Anal., 2 (4) (1999) 359-366.
  * [20] A.A. Kilbas, H.M. Srivastava, J.J. Trujillo, Theory and applications of fractional differential equations, Elsevier Sceince B.V., 2006.
  * [21] I. Podlubny, Fractional differential equations, Academic Press, New York, 1999.
  * [22] K.-J. Engel, R. Nagel, One-Parameter semigroups for linear evolution equations, vol. 194, Graduate Texts in Mathematics, Springer-Verlag, New York, 2000.
  * [23] T. Kato, Perturbation theory for linear operators, Berlin, Heidelberg, New York, Springer, 1966.
  * [24] M. Watanabe, A Perturbation theory for abstract evolution equations of second order, vol. 58, Proc. Japan Acad., 1982.
  * [25] J. Voigt, On the perturbation theory for strongly continuous semigroups, Math. Ann., (229) (1977) 163-171.
  * [26] M. Shimizu, I. Miyadera, Perturbation Theory for Cosine Families on Banach Spaces, Tokyo J. Math. 1 (2) (1978) 333-343.
  * [27] E. Hille, R.S. Philips, Functional analysis and semigroups, vol. 31, Revised Ed. Providence: Am. Math. Soc. Colloq. Publ., 1957.
  * [28] R. Gorenflo, A.A. Kilbas, F. Mainardi, S.V. Rogosin, Mittag-Leffler functions, related topics and applications, Springer-Verlag, Berlin, 2014.
  * [29] N.I. Mahmudov, A. Ahmadova, I.T Huseynov, A new technique for solving Sobolev type fractional multi-order evolution equations, arXiv preprint arXiv:2102.10318.
  * [30] I.T. Huseynov, A. Ahmadova, N.I. Mahmudov, On a study of Sobolev type fractional functional evolution equations, Authorea, M2021, DOI: 10.22541/au.161562420.01059626/v1.