Hille–Yosida-Type Theorem for Fractional Differential Equations with Dzhrbashyan–Nersesyan Derivative

Abstract

It is a well-known fact that the Dzhrbashyan–Nersesyan fractional derivative includes as particular cases the fractional derivatives of Riemann–Liouville, Gerasimov–Caputo, and Hilfer. The notion of resolving a family of operators for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative is introduced here. Hille–Yosida-type theorem on necessary and sufficient conditions of the existence of a strongly continuous resolving family of operators is proved using Phillips-type approximations. The conditions concern the location of the resolvent set and estimates for the resolvent of a linear closed operator A at the unknown function in the equation. The existence of a resolving family means the existence of a solution for the equation under consideration. For such equation with an operator A satisfying Hille–Yosida-type conditions the uniqueness of a solution is shown also. The obtained results are illustrated by an example for an equation of the considered form in a Banach space of sequences. It is shown that such a problem in a space of sequences is equivalent to some initial boundary value problems for partial differential equations. Thus, this paper obtains key results that make it possible to determine the properties of the initial value problem involving the Dzhrbashyan–Nersesyan derivative by examining the properties of the operator in the equation; the results prove the existence and uniqueness of the solution and the correctness of the problem.

1. Introduction

The concept of a resolving family of operators has an important role in the theory of differential equations in Banach spaces. Every operator of this family, corresponding to the fixed value of the parameter $t>0$, is a mapping of the initial data in the youngest initial condition of the problem to the state of the system at the moment t. In addition to studying the traditional issues of uniform well-posedness of an initial value problem for a differential equation, resolving families allow us to obtain representations of the solutions for linear equations, including inhomogeneous ones. These representations, in turn, are used in the study of the issues of the unique solvability of initial value problems for quasi-linear equations with the respective linear part by the methods of compressive maps.

The resolving family of a first-order linear equation has a semigroup property. The semigroups of the theory of operators are studied in many classical monographs; see, e.g., monographs of E. Hille and R Phillips [1] and K. Yosida [2], where necessary and sufficient conditions (Hille–Yosida conditions) for a generator of a $C_0$-semigroup of operators were obtained. K. Kato [3] studied the perturbations of operators’ semigroup generators. S. G. Krein [4] established the accordance between the resolving semigroup existence and the Cauchy problem of well-posedness for a first-order equation in a Banach space. Second-order equations have resolving families in the form of so-called cosine operator functions (see works of M. Sova [5], H.O. Fattorini [6], J. Goldstein [7]). Many authors consider resolving families of operators for integro-differential equations [8] for evolution integral equations [9]. A theorem on necessary and sufficient conditions of the existence of strongly continuous resolving families of operators (Hille–Yosida-type theorem) for equations with a Gerasimov–Caputo derivative was proven by E.G. Bazhlekova [10] (see also [11]). Analogous results were obtained for equations with a Riemann–Liouville derivative by A.V. Glushak [12] in the case of the equation order $\alpha \in (0,1)$ and in the work [13] for order $\alpha \in (0,2)$. For equations with distributed-order Gerasimov–Caputo derivatives, Hille–Yosida-type conditions were found in [14,15], and for equations with the Hilfer fractional derivative, the Hille–Yosida-type theorem was proven in [16].

In the present paper, we find the necessary and sufficient conditions of the existence of a strongly continuous resolving family for a linear equation with the Dzhrbashyan–Nersesyan fractional derivative [17] in a Banach space $\mathcal{Z}$

$$D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}z\left(t\right)=Az\left(t\right),t\in {\mathbb{R}}_{+}.$$

(1)

Here, A is a linear closed operator in a Banach space $\mathcal{Z}$; $0<{\alpha }_k\le 1,$$k=0,1,\dots ,n$; and the Dzhrbashyan–Nersesyan derivative has the form

$$D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}z\left(t\right)=D^{{\alpha }_n-1}{D}^{{\alpha }_{n-1}}{D}^{{\alpha }_{n-2}}\dots D^{{\alpha }_0}z\left(t\right)$$

(for more details, see below), where $D^{\beta }$ is the Riemann–Liouville derivative of the order $\beta >0$ or the Riemenn–Liouville fractional integral of the order $-\beta$, if $\beta \le 0$. It is proved that the resolving family of operators for Equation (1) exists, if and only if the proposed Hille–Yosida-type conditions on the resolvent set and the resolvent of the operator A hold. In such statements, it is particularly difficult to prove that the existence of a resolving family follows from the fulfillment of Hille–Yosida-type conditions. This is performed here using Phillips-type approximations.

Note that the works [18,19] consider some applied problems with Dzhrbashyan–Nersesyan fractional derivatives. A theorem on the unique solvability of the initial value problem for Equation (1) with $\mathcal{Z}={\mathbb{R}}^n$ and a matrix A was proved in [20]. Various partial differential equations with Dzhrbashyan–Nersesyan derivatives were studied in refs. [21,22,23,24,25].

Let us give a brief description of the content of this work. In the second section, we give a statement of the Dzhrbashyan–Nersesyan initial value problem [17], introduce the definition of a resolving family of operators for Equation (1), define a class of operators ${\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$, and prove some properties of such families and operators. It is shown that if a resolving family for Equation (1) exists, then $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$. In the third section, we prove that the existence of a resolving family of Equation (1) with ${\alpha }_0+{\alpha }_1+\dots +{\alpha }_n>3$ implies the boundedness of the operator A. Further, we prove that Phillips-type approximations in the case of $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$ converge to the resolving family of operators. Thus, we prove the Hille–Yosida-type theorem on the equivalence of the inclusion $A\in {\mathcal{C}}_{\alpha ,R}\left(\omega \right)$ and the existence of a resolving family of operators for Equation (1). In the corollary of this theorem, we also prove the uniqueness of a solution for the Dzhrbashyan–Nersesyan initial value problem to Equation (1). The last section contains an example of a set of operators $A_{\beta }$ in the Banach space $l_2$, which, for some of the values of parameter $\beta$, belong to the class ${\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$ and do not belong for other values of $\beta$. It is shown that the considered illustrative problem is equivalent to the initial boundary value problem for an equation with the Dzhrbashyan–Nersesyan time derivative and with a Laplace operator with respect to the spatial variables.

2. Definition and Properties of Resolving Families of Operators

Let $\mathcal{Z}$ be a Banach space. For $h:{\mathbb{R}}_{+}\to \mathcal{Z}$, the Riemann–Liouville fractional integral of an order $\beta >0$ has the form

$$J^{\beta }h\left(t\right):={\displaystyle \frac{1}{\mathrm{\Gamma }\left(\beta \right)}}\underset{0}{\overset{t}{\int }}{(t-s)}^{\beta -1}h\left(s\right)ds,t\in {\mathbb{R}}_{+},$$

$J^{0}h\left(t\right):=h\left(t\right)$. Take $m-1<\alpha \le m\in \mathbb{N}$, $D^m$ as a usual derivative of the m-th order; the Riemann–Liouville fractional derivative has the form $D^{\alpha }h\left(t\right):=D^m{J}^{m-\alpha }h\left(t\right)$. For $\alpha \le 0$, we will use the denotation $D^{\alpha }h\left(t\right):=J^{-\alpha }h\left(t\right)$.

Take a set ${\left\{{\alpha }_k\right\}}_0^n=\left\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\right\}$ of real numbers, such that $0<{\alpha }_k\le 1$, $k=0,1,\dots ,n,$$n\in \mathbb{N}\cup \left\{0\right\}.$ Denote Dzhrbashyan–Nersesyan fractional derivatives as follows:

$$D^{\left\{{\alpha }_0\right\}}z\left(t\right):=D^{{\alpha }_0-1}z\left(t\right),$$

(2)

$$D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}z\left(t\right):=D^{{\alpha }_k-1}{D}^{{\alpha }_{k-1}}{D}^{{\alpha }_{k-2}}\dots D^{{\alpha }_0}z\left(t\right),k=1,2,\dots ,n,$$

(3)

where ${\sigma }_n:={\alpha }_0+{\alpha }_1+\dots +{\alpha }_n-1>0$. The Dzhrbashyan–Nersesyan fractional derivative $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}z$ associated with the sequence ${\left\{{\alpha }_k\right\}}_{k=0}^n$ includes the Riemann–Liouville (${\alpha }_0=\alpha -n+1\in (0,1)$, ${\alpha }_k=1$, $k=1,2,\dots ,n$), the Gerasimov–Caputo [26,27,28] (${\alpha }_k=1$, $k=0,1,\dots ,n-1$, ${\alpha }_n=\alpha -n+1\in (0,1)$), and the Hilfer (${\alpha }_0=(1-\beta )(\alpha -n)+1\in (0,1)$, ${\alpha }_k=1$, $k=1,2,\dots ,n-1$, ${\alpha }_n=\beta (\alpha -n)+1$) fractional derivatives of the order ${\sigma }_n=\alpha \in (n-1,n]$ as special cases.

Denote by $\widehat{h}$ or $\mathfrak{L}\left[h\right]$ the Laplace transform of a function $h:{\mathbb{R}}_{+}\to \mathcal{Z}$. The inverse Laplace transform of $H:\{\mathrm{Re}\lambda >\omega \}\to \mathcal{Z}$ for some $\omega \in \mathbb{R}$ will be denoted by ${\mathfrak{L}}^{-1}\left[H\right]$. The Laplace transform of the Riemann–Liouville integral of an order $\beta >0$, the Riemann–Liouville derivative of an order $\alpha \in (m-1,m]$, $m\in \mathbb{N}$, and the Dzhrbashyan–Nersesyan derivative satisfies the equalities (see, e.g., [29,30,31])

$$\widehat{J^{\beta }h}\left(\lambda \right)={\lambda }^{-\beta }\widehat{h}\left(\lambda \right),\widehat{D^{\alpha }h}\left(\lambda \right)={\lambda }^{\alpha }\widehat{h}\left(\lambda \right)-\sum _{k=0}^{m-1}{D}^{\alpha -m+k}h\left(0\right){\lambda }^{m-1-k}.$$

$$\mathfrak{L}\left[D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}h\right]\left(\lambda \right)={\lambda }^{{\sigma }_n}\widehat{h}\left(\lambda \right)-\sum _{k=0}^{n-1}{\lambda }^{{\sigma }_n-{\sigma }_k-1}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}h\left(0\right).$$

(4)

Hereafter, ${\sigma }_k:={\alpha }_0+{\alpha }_1+\dots +{\alpha }_k-1$, $k=0,1,\dots ,n$, $D^{\alpha -m+k}h\left(0\right):=\underset{t\to 0+}{lim}{D}^{\alpha -m+k}h\left(t\right)$, $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}h\left(0\right):=\underset{t\to 0+}{lim}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}h\left(t\right)$.

Let $\mathcal{L}\left(\mathcal{Z}\right)$ be the Banach space of all linear bounded operators from $\mathcal{Z}$ to $\mathcal{Z}$, and $\mathcal{C}l\left(\mathcal{Z}\right)$ be the set of all linear closed operators, densely defined in $\mathcal{Z}$, acting onto the space $\mathcal{Z}$. Endow the domain $D_A$ of an operator $A\in \mathcal{C}l\left(\mathcal{Z}\right)$ by the norm of its graph and obtain the Banach space $D_A$.

Consider the problem

$$D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}z\left(0\right)=z_k,k=0,1,\dots ,n-1,$$

(5)

for the linear homogeneous equation

$$D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}z\left(t\right)=Az\left(t\right),t\in {\mathbb{R}}_{+},$$

(6)

where the Dzhrbashyan–Nersesyan derivatives $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}z$, $k=0,1,\dots ,n$, associated with the sequence ${\left\{{\alpha }_k\right\}}_0^n$, are defined by (2), (3), and $A\in \mathcal{C}l\left(\mathcal{Z}\right)$. By solving problems (5) and (6), we call the function $z\in C({\mathbb{R}}_{+};D_A)$, such that $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}z\in AC({\overline{\mathbb{R}}}_{+};\mathcal{Z})$, $k=0,1,\dots ,n-1$, $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}z\in C({\mathbb{R}}_{+};\mathcal{Z})$, and conditions (5) and equality (6) for $t\in {\mathbb{R}}_{+}$ are fulfilled. Hereafter, ${\overline{\mathbb{R}}}_{+}={\mathbb{R}}_{+}\cup \left\{0\right\}$.

Note that in the case of the Gerasimov–Caputo derivative, conditions (5) have the form of the Cauchy conditions, since ${\alpha }_k=1$, $k=0,1,\dots ,n-1$, $\alpha -n+1\in (0,1)$; hence, $D^{\left\{{\alpha }_0\right\}}z\left(0\right)=D^{0}z\left(0\right)=z\left(0\right)$, $D^{\{{\alpha }_0,{\alpha }_1\}}z\left(0\right)=D^0{D}^{1}z\left(0\right)=D^{1}z\left(0\right)$, …, $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}z\left(0\right)=D^0{D}^{k}z\left(0\right)=D^{k}z\left(0\right)$, $k=0,1,\dots ,n-1$. If we consider the equation with the Riemann–Liouville derivative, then problem (5) will be a Cauchy-type problem in terms of the classical monograph [30]: $D^{\left\{{\alpha }_0\right\}}z\left(0\right)=D^{\alpha -n}z\left(0\right)$, $D^{\{{\alpha }_0,{\alpha }_1\}}z\left(0\right)=D^0{D}^{\alpha -n+1}z\left(0\right)$, …, $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}z\left(0\right)=D^0{D}^{\alpha -n+k}z\left(0\right)=D^{\alpha -n+k}z\left(0\right)$, $k=0,1,\dots ,n-1$, since ${\alpha }_0=\alpha -n+1\in (0,1)$, ${\alpha }_k=1$, $k=1,2,\dots ,n$.

Define the resolving family of operators for Equation (6).

Definition 1.

A family of operators $\{S\left(t\right)\in \mathcal{L}\left(\mathcal{Z}\right):t\in {\mathbb{R}}_{+}\}$ is considered to be resolving for Equation (6), if the following conditions are satisfied:

(i) There exist $K\in {\mathbb{R}}_{+}$ and $\omega \in {\overline{\mathbb{R}}}_{+}$ such that ${\parallel S\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K{\mathrm{e}}^{\omega t}{t}^{{\alpha }_0-1}$ for all $t\in {\mathbb{R}}_{+}$;

(ii) The family $\{S\left(t\right):t\in {\mathbb{R}}_{+}\}$ is strongly continuous on ${\mathbb{R}}_{+}$, $s\text{-}\underset{t\to 0+}{lim}{J}^{1-{\alpha }_0}S\left(t\right)=I$;

(iii) $S\left(t\right)\left[D_A\right]\subset D_A$, $S\left(t\right)A{z}_0=AS\left(t\right)z_0$ for all $z_0\in D_A$, $t\in {\mathbb{R}}_{+}$;

(iv) For every $z_0\in D_A$, $S\left(t\right)z_0$ is a solution of the Cauchy-type problem $D^{\left\{{\alpha }_0\right\}}z\left(0\right)=z_0$, $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}z\left(0\right)=0$, $k=1,2,\dots ,n-1$, to Equation (6).

We will write $A\in C^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$, if there exists a resolving family of operators for Equation (6) with a constant $\omega \in {\overline{\mathbb{R}}}_{+}$ in condition (i).

Remark 1.

Note that for $A\in \mathcal{L}\left(\mathcal{Z}\right)$, the operators of the resolving family of operators for Equation (6) are defined by the Mittag–Leffler function: $S\left(t\right)=t^{{\alpha }_0-1}{E}_{{\sigma }_n,{\alpha }_0}\left(t^{{\sigma }_n}A\right)$, $t\in {\mathbb{R}}_{+}.$

Definition 2.

An operator $A\in \mathcal{C}l\left(\mathcal{Z}\right)$ is called an operator of the class ${\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$ for some constant $\omega \in {\overline{\mathbb{R}}}_{+}$, if the next two conditions hold:

(i) If $\mathrm{Re}\lambda >\omega$, then ${\lambda }^{{\sigma }_n}\in \rho \left(A\right):=\{\mu \in \mathbb{C}:{(\mu -A)}^{-1}\in \mathcal{L}\left(\mathcal{Z}\right)\};$

(ii) There exists a constant $K\in {\mathbb{R}}_{+}$, such that for $\mathrm{Re}\lambda >\omega$ and for all $m\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$,

$${\displaystyle {∥\frac{d^m}{d{\lambda }^m}\left({\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}-A)}^{-1}\right)∥}_{\mathcal{L}\left(\mathcal{Z}\right)}\le \frac{K\mathrm{\Gamma }({\alpha }_0+m)}{{(\mathrm{Re}\lambda -\omega )}^{{\alpha }_0+m}}.}$$

Remark 2.

Define the Gerasimov–Caputo derivative of an order $\alpha \in (n-1,n)$, $n\in \mathbb{N}$, as

$${}^G\mathit{D}^{\alpha }h\left(t\right):=D^{\alpha }\left(h\left(t\right)-\sum _{k=0}^{m-1}{D}^{k}h\left(0\right){\displaystyle \frac{t^k}{k!}}\right).$$

In Theorem 2.8 [10], it was proved that there exists a resolving family of the equation ${}^G\mathit{D}^{\alpha }z\left(t\right)=Az\left(t\right)$ if and only if the operator $A\in \mathcal{C}l\left(\mathcal{Z}\right)$ satisfies conditions of Definition 2 with ${\sigma }_n=\alpha$, ${\alpha }_0=1$. So, in this case, we have the class of operators ${\mathcal{C}}_{\{1,1,\dots ,1,\alpha -n+1\}}\left(\omega \right).$

Remark 3.

Theorem 3.2 in [13] states that there exists a resolving family of the equation $D^{\alpha }z\left(t\right)=Az\left(t\right)$, $n-1<\alpha \le n\in \mathbb{N}$, if and only if the operator $A\in \mathcal{C}l\left(\mathcal{Z}\right)$ satisfies the conditions of Definition 2 with ${\sigma }_n=\alpha$, ${\alpha }_0=\alpha -n+1$, i.e., for operators from the class $\left\}\right\}{C}_{\{\alpha -n+1,1,1,\dots ,1\}}\left(\omega \right).$

Remark 4.

In Theorem 3 in [16], it was proved that the necessary and sufficient conditions of the existence of a resolving family for the equation $D^{\alpha ,\beta }z\left(t\right)=Az\left(t\right)$ with the Hilfer derivative $D^{\alpha ,\beta }$, $n-1<\alpha \le n\in \mathbb{N}$, $\beta \in [0,1]$, are the conditions of Definition 2 with ${\sigma }_n=\alpha$, ${\alpha }_0=(1-\beta )(\alpha -n)+1$. Here, we see the class of operators ${\mathcal{C}}_{\left\{\right(1-\beta \left)\right(\alpha -n)+1,1,1,\dots ,1,\beta (\alpha -n)+1\}}\left(\omega \right).$

Lemma 1.

Let $m-1<\alpha \le m\in \mathbb{N}$. Then, ${\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)\subset {\mathcal{C}}_{{\sigma }_n,G}\left(\omega \right)$.

Proof. 

Denote $H\left(\lambda \right):={\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}-A)}^{-1}$ for $\lambda >\omega$. For ${\sigma }_n>0$, $m\in \mathbb{N}$, and $\mathrm{Re}\lambda >\omega$, we have

$${\displaystyle \frac{d^m}{d{\lambda }^m}}\left({\lambda }^{{\sigma }_n-1}{({\lambda }^{{\sigma }_n}I-A)}^{-1}\right)={\displaystyle \frac{d^m}{d{\lambda }^m}}\left({\lambda }^{{\alpha }_0-1}H\left(\lambda \right)\right)=\sum _{k=0}^m{C}_m^k{\displaystyle \frac{d^{m-k}}{d{\lambda }^{m-k}}}{\lambda }^{{\alpha }_0-1}{\displaystyle \frac{d^k}{d{\lambda }^k}}H\left(\lambda \right)=$$

$$=\sum _{k=0}^m{C}_m^k({\alpha }_0-1)({\alpha }_0-2)\dots ({\alpha }_0-m+k){\lambda }^{{\alpha }_0-1-m+k}{\displaystyle \frac{d^k}{d{\lambda }^k}}H\left(\lambda \right),$$

$${∥{\displaystyle \frac{d^m}{d{\lambda }^m}}\left({\lambda }^{{\sigma }_n-1}{({\lambda }^{{\sigma }_n}I-A)}^{-1}\right)∥}_{\mathcal{L}\left(\mathcal{Z}\right)}\le$$

$$\le K\sum _{k=0}^m{C}_m^k(1-{\alpha }_0)(2-{\alpha }_0)\dots (m-k-{\alpha }_0){\left|\lambda \right|}^{{\alpha }_0-1-m+k}{\displaystyle \frac{\mathrm{\Gamma }({\alpha }_0+k)}{{(\mathrm{Re}\lambda -\omega )}^{{\alpha }_0+k}}}\le$$

$$\le {\displaystyle \frac{K\mathrm{\Gamma }\left({\alpha }_0\right)}{{(\mathrm{Re}\lambda -\omega )}^{m+1}}}\sum _{k=0}^m{C}_m^k(1-{\alpha }_0)(2-{\alpha }_0)\dots (m-k-{\alpha }_0)\times$$

$$\times ({\alpha }_0+k-1)({\alpha }_0+k-2)\dots {\alpha }_0={\displaystyle \frac{K\mathrm{\Gamma }\left({\alpha }_0\right)m!}{{(\mathrm{Re}\lambda -\omega )}^{m+1}}}$$

due to the equality

$$\sum _{k=0}^m{C}_m^k(1-{\alpha }_0)(2-{\alpha }_0)\dots (m-k-{\alpha }_0)({\alpha }_0+k-1)({\alpha }_0+k-2)\dots {\alpha }_0=m!,$$

which is proven in [12] (see Formula (10)). □

Lemma 2.

Let ${\alpha }_k\in (0,1]$, $k=0,1,\dots ,n$, and consider that there exists a resolving family of operators $\{S\left(t\right)\in \mathcal{L}\left(\mathcal{Z}\right):t\in {\mathbb{R}}_{+}\}$ for Equation (6). Then,

(i) For an arbitrary $z_k\in D_A$, the function $J^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}S\left(t\right)z_k$ is a solution of the problem

$$D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_l\}}z\left(0\right)=0,l\in \{0,1,\dots ,m-1\}\setminus \left\{k\right\},D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}z\left(0\right)=z_k$$

for Equation (6);

(ii) For an arbitrary $z_0,z_1,\dots ,z_{n-1}\in D_A$, the function $S\left(t\right)z_0+{\displaystyle \sum _{k=1}^{n-1}}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}S\left(t\right)z_k$ is a solution of the problem $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}z\left(0\right)=z_k$, $k=0,1,\dots ,n-1$, for Equation (6).

Proof. 

(i) For $z_k\in D_A$, $k=1,2,\dots ,n-1$, we have

$$\underset{t\to 0+}{lim}{D}^{\left\{{\alpha }_0\right\}}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}S\left(t\right)z_k=\underset{t\to 0+}{lim}{J}^{{\alpha }_1+\dots +{\alpha }_k+1-{\alpha }_0}S\left(t\right)z_k=0,$$

since ${\alpha }_1+\dots +{\alpha }_k+1-{\alpha }_0>0$. For $k>1$, using the equality

$$D^1{J}^{\beta }h\left(t\right)={\displaystyle \frac{t^{\beta -1}}{\mathrm{\Gamma }\left(\beta \right)}}h\left(0\right)+J^{\beta }{D}^{1}h\left(t\right),$$

we get

$$\underset{t\to 0+}{lim}{D}^{\{{\alpha }_0,{\alpha }_1\}}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}S\left(t\right)z_k=\underset{t\to 0+}{lim}{J}^{1-{\alpha }_1}{D}^1{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}{J}^{1-{\alpha }_0}S\left(t\right)z_k=$$

$$=\underset{t\to 0+}{lim}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}{J}^{1-{\alpha }_1}{D}^1{J}^{1-{\alpha }_0}S\left(t\right)z_k+\underset{t\to 0+}{lim}{J}^{1-{\alpha }_1}{\displaystyle \frac{t^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k-1}}{\mathrm{\Gamma }({\alpha }_1+{\alpha }_2+\dots +{\alpha }_k)}}{z}_k=0,$$

for $k>2$

$$\underset{t\to 0+}{lim}{D}^{\{{\alpha }_0,{\alpha }_1,{\alpha }_2\}}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}S\left(t\right)z_k=\underset{t\to 0+}{lim}{J}^{1-{\alpha }_2}{D}^1{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}{D}^{\{{\alpha }_0,{\alpha }_1\}}S\left(t\right)z_k+$$

$$+\underset{t\to 0+}{lim}{J}^{1-{\alpha }_2}{D}^1{\displaystyle \frac{t^{{\alpha }_2+{\alpha }_3+\dots +{\alpha }_k}}{\mathrm{\Gamma }({\alpha }_2+{\alpha }_3+\dots +{\alpha }_k+1)}}{z}_k=$$

$$=\underset{t\to 0+}{lim}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}{D}^{\{{\alpha }_0,{\alpha }_1,{\alpha }_2\}}S\left(t\right)z_k+0+\underset{t\to 0+}{lim}{\displaystyle \frac{t^{{\alpha }_3+{\alpha }_4+\dots +{\alpha }_k}{z}_k}{\mathrm{\Gamma }({\alpha }_3+{\alpha }_4+\dots +{\alpha }_k+1)}}=0.$$

Further, we get

$$\underset{t\to 0+}{lim}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{k-1}\}}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}S\left(t\right)z_k=$$

$$=\underset{t\to 0+}{lim}{J}^{1-{\alpha }_{k-1}}{D}^1{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{k-2}\}}S\left(t\right)z_k+\underset{t\to 0+}{lim}{\displaystyle \frac{t^{{\alpha }_k}{z}_k}{\mathrm{\Gamma }({\alpha }_k+1)}}=$$

$$=\underset{t\to 0+}{lim}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{k-1}\}}S\left(t\right)z_k=0,$$

$$\underset{t\to 0+}{lim}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}S\left(t\right)z_k=$$

$$=\underset{t\to 0+}{lim}{J}^{1-{\alpha }_k}{D}^1{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{k-1}\}}S\left(t\right)z_k+z_k=$$

$$=\underset{t\to 0+}{lim}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}S\left(t\right)z_k+z_k=z_k,$$

and for $l=k+1,k+2,\dots ,n-1$

$$\underset{t\to 0+}{lim}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_l\}}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}S\left(t\right)z_k=\underset{t\to 0+}{lim}{J}^{1-{\alpha }_l}{D}^1{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_{l-1}\}}S\left(t\right)z_k=$$

$$=\underset{t\to 0+}{lim}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_l\}}S\left(t\right)z_k=0.$$

Here we consider that $\underset{t\to 0+}{lim}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_l\}}S\left(t\right)z_k=0$, $l=1,2,\dots ,n-1$ due to the definition of the resolving family of operators.

Assertion (i) implies statement (ii). □

Lemma 3.

Let ${\alpha }_k\in (0,1]$, $k=0,1,\dots ,n$, $A\in C^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$. Then, we have that $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$, and for the resolving family of operators $\{S\left(t\right)\in \mathcal{L}\left(\mathcal{Z}\right):t\in {\mathbb{R}}_{+}\}$ for Equation (6),

$$\widehat{S}\left(\lambda \right)={\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1},\mathrm{Re}\lambda >\omega .$$

(7)

Proof. 

For $\mathrm{Re}\lambda >\omega$, $z_0\in D_A$, from (4) and (6), we obtain that

$$\mathfrak{L}\left[D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}S\left(t\right)z_0\right]\left(\lambda \right)z_0={\lambda }^{{\sigma }_n}\widehat{S}\left(\lambda \right)z_0-{\lambda }^{{\sigma }_n-{\alpha }_0}{z}_0=A\widehat{S}\left(\lambda \right)z_0=\widehat{S}\left(\lambda \right)A{z}_0,$$

due to the closedness of the operator A. Conditions (iii) and (iv) of the definition of the resolving family are used here. Therefore, there exists the inverse operator for ${\lambda }^{{\sigma }_n}I-A:D_A\to \mathcal{Z}$ and equality (7) holds. Since $\widehat{S}\left(\lambda \right)$ is bounded, we have the inclusion ${\lambda }^{{\sigma }_n}\in \rho \left(A\right)$. Differentiate the equality

$$\underset{0}{\overset{\infty }{\int }}S\left(t\right){\mathrm{e}}^{-\lambda t}dt={\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}$$

with respect to $\lambda$ and obtain for $\mathrm{Re}\lambda >\omega$, $m\in {\mathbb{N}}_0$

$${∥{\displaystyle \frac{d^m}{d{\lambda }^m}}\left({\lambda }^{{\sigma }_n-{\sigma }_0-1}{({\lambda }^{{\sigma }_n}I-A)}^{-1}\right)∥}_{\mathcal{L}\left(\mathcal{Z}\right)}\le K\underset{0}{\overset{\infty }{\int }}{t}^{{\alpha }_0-1+m}{\mathrm{e}}^{(\omega -\mathrm{Re}\lambda )t}dt=$$

$$=K\widehat{t^{m+{\alpha }_0-1}}(\mathrm{Re}\lambda -\omega )={\displaystyle \frac{K\mathrm{\Gamma }(m+{\alpha }_0)}{{(\mathrm{Re}\lambda -\omega )}^{m+{\alpha }_0}}};$$

hence, $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right).$ □

From the above proof, the next statement follows immediately.

Corollary 1.

Let ${\alpha }_k\in (0,1]$, $k=0,1,\dots ,n$. Then $C^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)\subset {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$.

Lemma 4.

Let ${\alpha }_k\in (0,1]$, $k=0,1,\dots ,n$, ${\sigma }_n>0$, $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$. Then, for every $r\in \mathbb{N}$, $z_0\in \mathcal{Z}$

$$\underset{m\to \infty }{lim}{\left(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right)}^r{z}_0=z_0.$$

Proof. 

We have $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$, hence, for $m>\omega$

$$\parallel {(m^{{\sigma }_n}I-A)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le {\displaystyle \frac{K}{m^{{\sigma }_n-{\alpha }_0}{(m-\omega )}^{{\alpha }_0}}},{\parallel m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le C_1.$$

For $z_0\in D_A$, $\underset{m\to \infty }{lim}{m}^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}{z}_0=z_0+\underset{m\to \infty }{lim}{(m^{{\sigma }_n}I-A)}^{-1}A{z}_0=z_0.$ Since $D_A$ is dense in $\mathcal{Z}$, we have $\underset{m\to \infty }{lim}{m}^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}{z}_0=z_0$ for every $z_0\in \mathcal{Z}$.

Furthermore, for $z_0\in \mathcal{Z}$

$$\underset{m\to \infty }{lim}{\left(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right)}^2{z}_0=\underset{m\to \infty }{lim}{\left(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right)}^2-m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1})z_0+$$

$$+\underset{m\to \infty }{lim}{m}^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}{z}_0=z_0,$$

and since $m\to \infty$,

$$\parallel ({(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1})}^2-m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1})z_0{\parallel }_{\mathcal{Z}}\le C_1{\parallel m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}{z}_0-z_0\parallel }_{\mathcal{Z}}\to 0.$$

Similarly, we can prove the statement of the lemma for arbitrary $r\in \mathbb{N}$. □

Corollary 2.

Let ${\alpha }_k\in (0,1]$, $k=0,1,\dots ,n$, ${\sigma }_n>0$, $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$. Then, for any $r\in \mathbb{N}$, $D_{A^r}$ is dense in $\mathcal{Z}$.

Proof. 

For every $z_0\in \mathcal{Z}$, ${\left(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right)}^r{z}_0\in D_{A^r}$, $\underset{m\to \infty }{lim}{\left(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right)}^r{z}_0=z_0.$ □

Corollary 3.

Let ${\alpha }_k\in (0,1]$, $k=0,1,\dots ,n$, ${\sigma }_n>0$, $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$. Then, for any $r\in \mathbb{N}$, $D_{A^r}$ is dense in $D_A$.

Proof. 

Due to Lemma 4, for every $z_0\in D_A$ $\underset{m\to \infty }{lim}{\left(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right)}^r{z}_0=z_0$,

$$\underset{m\to \infty }{lim}A{\left(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right)}^r{z}_0=\underset{m\to \infty }{lim}{\left(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right)}^{r}A{z}_0=A{z}_0.$$

Therefore,

$$\parallel {\left(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right)}^r{z}_0-z_0{\parallel }_{D_A}={\parallel {\left(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right)}^r{z}_0-z_0\parallel }_{\mathcal{Z}}+$$

$$+\parallel A{\left(m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right)}^r{z}_0-A{z}_0{\parallel }_{\mathcal{Z}}\to 0,m\to \infty .$$

□

Theorem 1.

Let ${\alpha }_k\in (0,1]$, $k=0,1,\dots ,n$, and consider that there exists a resolving family $\{S\left(t\right)\in \mathcal{L}\left(\mathcal{Z}\right):t\in {\mathbb{R}}_{+}\}$ for Equation (6). Then, the family $\{J^{1-{\alpha }_0}S\left(t\right)\in \mathcal{L}\left(\mathcal{Z}\right):t\in {\overline{\mathbb{R}}}_{+}\}$ is continuous in the norm of $\mathcal{L}\left(\mathcal{Z}\right)$ in the point $t=0$, if and only if $A\in \mathcal{L}\left(\mathcal{Z}\right)$.

Proof. 

Let $t\in {\mathbb{R}}_{+}$, then

$$\parallel J^{1-{\alpha }_0}{S\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K\underset{0}{\overset{t}{\int }}{\displaystyle \frac{{(t-s)}^{-{\alpha }_0}}{\mathrm{\Gamma }(1-{\alpha }_0)}}{s}^{{\alpha }_0-1}{\mathrm{e}}^{\omega s}ds\le K\mathrm{\Gamma }\left({\alpha }_0\right){\mathrm{e}}^{\omega t}.$$

Lemma 3 implies that for $\mathrm{Re}\lambda >\omega$$\widehat{J^{1-{\alpha }_0}S}\left(\lambda \right)={\lambda }^{{\sigma }_n-1}{({\lambda }^{{\sigma }_n}I-A)}^{-1}$, i.e.,

$${\lambda }^{{\sigma }_n-1}{({\lambda }^{{\sigma }_n}I-A)}^{-1}-{\displaystyle \frac{I}{\lambda }}=\underset{0}{\overset{\infty }{\int }}{\mathrm{e}}^{-\lambda t}(J^{1-{\alpha }_0}S\left(t\right)-I)dt.$$

Let $\eta \left(t\right)=\parallel J^{1-{\alpha }_0}{S\left(t\right)-I\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}$ be a continuous function on the segment $[0,1]$ and $\eta \left(0\right)=0$. Then, for $\epsilon >0$, take $\delta >0$ such that $\eta \left(t\right)\le \epsilon$ for all $t\in [0,\delta ]$ and obtain

$${∥{\lambda }^{{\sigma }_n-1}{({\lambda }^{{\sigma }_n}I-A)}^{-1}-{\displaystyle \frac{I}{\lambda }}∥}_{\mathcal{L}\left(\mathcal{Z}\right)}\le \underset{0}{\overset{\delta }{\int }}{\mathrm{e}}^{-\lambda t}\eta \left(t\right)dt+\underset{\delta }{\overset{\infty }{\int }}{\mathrm{e}}^{-\lambda t}\eta \left(t\right)dt\le {\displaystyle \frac{\epsilon }{\lambda }}+o\left({\displaystyle \frac{1}{\lambda }}\right)$$

as $\mathrm{Re}\lambda \to +\infty$, since $\eta \left(t\right)\le K\mathrm{\Gamma }\left({\alpha }_0\right){\mathrm{e}}^{\omega t}+1$ for $t\ge 0$. Therefore, for sufficiently large $\mathrm{Re}\lambda$, $\parallel {\lambda }^{{\sigma }_n}{({\lambda }^{{\sigma }_n}I-A)}^{-1}{-I\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}<1.$ Consequently, ${\lambda }^{{\sigma }_n}{({\lambda }^{{\sigma }_n}I-A)}^{-1}$ is a continuously invertible operator and ${\left[{\lambda }^{{\sigma }_n}{({\lambda }^{{\sigma }_n}I-A)}^{-1}\right]}^{-1}={\lambda }^{-{\sigma }_n}({\lambda }^{{\sigma }_n}I-A)\in \mathcal{L}\left(\mathcal{Z}\right)$, $A\in \mathcal{L}\left(\mathcal{Z}\right)$.

For $A\in \mathcal{L}\left(\mathcal{Z}\right)$, we will represent the resolving family of operators $\{S\left(t\right)\in \mathcal{L}\left(\mathcal{Z}\right):t\in {\mathbb{R}}_{+}\}$ as the inverse Laplace transform of ${\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}$, which is defined for ${\left|\lambda \right|>\parallel A\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}^{1/{\sigma }_n}$. For this aim, take ${\mathrm{\Gamma }}_R:=\{R{\mathrm{e}}^{i\phi }\in \mathbb{C}:\phi \in (-\pi ,\pi )\}\cup \{r{\mathrm{e}}^{i\pi }\in \mathbb{C}:r\in [R,\infty )\}\cup \{r{\mathrm{e}}^{-i\pi }\in \mathbb{C}:r\in [R,\infty )\}$ for ${R>\parallel A\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}^{1/{\sigma }_n}$; hence, for $t\in {\mathbb{R}}_{+}$

$$J^{1-{\alpha }_0}S\left(t\right)={\displaystyle \frac{1}{2\pi i}}\underset{{\mathrm{\Gamma }}_R}{\int }{\lambda }^{{\sigma }_n-1}{({\lambda }^{{\sigma }_n}I-A)}^{-1}{\mathrm{e}}^{\lambda t}d\lambda ={\displaystyle \frac{1}{2\pi i}}\underset{{\mathrm{\Gamma }}_R}{\int }\sum _{k=0}^{\infty }{\lambda }^{-{\sigma }_{n}k-1}{A}^k{\mathrm{e}}^{\lambda t}d\lambda =$$

$$=\sum _{k=0}^{\infty }{\displaystyle \frac{t^{{\sigma }_{n}k}{A}^k}{\mathrm{\Gamma }({\sigma }_{n}k+1)}}=E_{{\sigma }_n,1}\left(t^{{\sigma }_n}A\right),$$

where $E_{\alpha ,\beta }\left(z\right):={\displaystyle \sum _{n=0}^{\infty }}{\displaystyle \frac{z^n}{\mathrm{\Gamma }(\alpha n+\beta )}}$ is the Mittag–Leffler function. Then, as $t\to 0+$,

$$\parallel J^{1-{\alpha }_0}{S\left(t\right)-I\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le \sum _{k=1}^{\infty }{\displaystyle \frac{t^{{\sigma }_{n}k}{\parallel A\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}^k}{\mathrm{\Gamma }({\sigma }_{n}k+1)}}=t^{{\sigma }_n}{\parallel A\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}{E}_{{\sigma }_n,{\sigma }_n+1}(t^{{\sigma }_n}{\parallel A\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)})\to 0.$$

□

3. Existence of the Resolving Family of Operators

Theorem 2.

Let ${\alpha }_k\in (0,1]$, $k=0,1,\dots ,n$, ${\sigma }_n>2$, $A\in C_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$. Then, $A\in \mathcal{L}\left(\mathcal{Z}\right)$.

Proof. 

For $A\in C_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$, we have

$$F_{{\sigma }_n,\omega }:=\left\{{\lambda }^{{\sigma }_n}:\mathrm{Re}\lambda >\omega ,|arg\lambda |\le {\displaystyle \frac{\pi }{{\sigma }_n}}\right\}\subset \rho \left(A\right),$$

and hence, $\rho \left(A\right)$ occupies the entire complex plane, with the exception of some bounded sets. Therefore, for every $\mu \in \mathbb{C}$ with a large enough $\left|\mu \right|$, we have $\mu ={\lambda }^{{\sigma }_n}\in F_{{\sigma }_n,\omega }$. Then, Corollary 1 implies that

$$\parallel \mu R_{\mu }{\left(A\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le {\displaystyle \frac{K\mathrm{\Gamma }\left({\alpha }_0\right){\left|\lambda \right|}^{{\alpha }_0}}{{(\mathrm{Re}\lambda -\omega )}^{{\alpha }_0}}}\le {\displaystyle \frac{K\mathrm{\Gamma }\left({\alpha }_0\right){\left|\lambda \right|}^{{\alpha }_0}}{\left(\right|\lambda |cos\frac{\pi }{{\sigma }_n}{-\omega )}^{{\alpha }_0}}}\to {\displaystyle \frac{K\mathrm{\Gamma }\left({\alpha }_0\right)}{{cos}^{{\alpha }_0}\frac{\pi }{{\sigma }_n}}},\left|\mu \right|\to \infty .$$

Therefore, Lemma 5.2 [7] implies the boundedness of the operator A. □

For $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$, $\mathrm{Re}\lambda >\omega$, $H\left(\lambda \right):={\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}-A)}^{-1}$, $m\in \mathbb{N}$, $t\in {\mathbb{R}}_{+}$, define by the Phillips inversion formula ([1,32], Theorem 6.3.3)

$${\mathcal{S}}_m\left(t\right):={\mathrm{e}}^{-mt}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{\left(m(m+\omega )t\right)}^{k+1}}{k!(k+1)!}}{H}^{\left(k\right)}(m+\omega ).$$

This series converges uniformly on every segment $[1/T,T]$, $T>0$.

By the Stirling formula for $k\to \infty$,

$${\displaystyle \frac{\mathrm{\Gamma }(k+1-\delta )}{\mathrm{\Gamma }(k+1)}}\sim C_1{k}^{-\delta },{\displaystyle \frac{\mathrm{\Gamma }(k+2)}{\mathrm{\Gamma }(k+2+\delta )}}\sim C_2{k}^{-\delta };$$

therefore, the inequality

$${\displaystyle \frac{\mathrm{\Gamma }(k+1-\delta )}{k!(k+1)!}}={\displaystyle \frac{\mathrm{\Gamma }(k+1-\delta )}{\mathrm{\Gamma }(k+1)\mathrm{\Gamma }(k+2)}}\le {\displaystyle \frac{C_3}{\mathrm{\Gamma }(k+2+\delta )}},\delta >0$$

is valid and for $t\in {\mathbb{R}}_{+}$, $\delta =1-{\alpha }_0$,

$$\parallel {\mathcal{S}}_m{\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K{\mathrm{e}}^{-mt}{m}^{1-{\alpha }_0}\sum _{k=0}^{\infty }{\displaystyle \frac{\mathrm{\Gamma }({\alpha }_0+k){\left((m+\omega )t\right)}^{k+1}}{k!(k+1)!}}\le$$

$$\le C_4{\mathrm{e}}^{-mt}{m}^{1-{\alpha }_0}\sum _{k=0}^{\infty }{\displaystyle \frac{{\left((m+\omega )t\right)}^{k+1}}{\mathrm{\Gamma }(k+3-{\alpha }_0)}}\le C_4{\mathrm{e}}^{-mt}{m}^{1-{\alpha }_0}{E}_{1,2-{\alpha }_0}\left((m+\omega )t\right).$$

Due to the asymptotic expansion of the Mittag–Leffler function

$$E_{\beta ,\gamma }\left(s\right)={\displaystyle \frac{1}{\beta }}{s}^{{\displaystyle \frac{1-\gamma }{\beta }}}{\mathrm{e}}^{s^{1/\beta }}+O\left(s^{-1}\right),s\to +\infty ,$$

we have for $t>0$

$$\parallel {\mathcal{S}}_m{\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le C_5{\mathrm{e}}^{-mt}{m}^{1-{\alpha }_0}{\left((m+\omega )t\right)}^{{\alpha }_0-1}{\mathrm{e}}^{(m+\omega )t}\le C_5{t}^{{\alpha }_0-1}{\mathrm{e}}^{\omega t}.$$

(8)

It is easy to prove that ${\mathcal{S}}_m\left(t\right)$ is an infinitely differentiable family for $t\in {\mathbb{R}}_{+}$. Hereafter, a strong limit is denoted by $s\text{-}\underset{m\to \infty }{lim}$.

Lemma 5.

Let ${\alpha }_k\in (0,1]$, $k=0,1,\dots ,n$, $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$. Then, there exists the strong limit $s\text{-}\underset{m\to \infty }{lim}{\mathcal{S}}_m\left(t\right)$, which is uniform with respect to t on every segment $[1/T,T]$, $T>0$.

Proof. 

For $r\in \mathbb{N}$, such that $r-1\le 1/{\sigma }_n<r$, we have $(r-1){\sigma }_n\le 1<r{\sigma }_n$; then for $z_0\in D_{A^r}$,

$$H\left(\lambda \right)z_0={\lambda }^{-{\alpha }_0}{z}_0+{\lambda }^{-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}A{z}_0=\dots =$$

$$=\sum _{k=0}^{r-1}{\lambda }^{-{\alpha }_0-k{\sigma }_n}{A}^k{z}_0+{\lambda }^{-{\alpha }_0-(r-1){\sigma }_n}{({\lambda }^{{\sigma }_n}I-A)}^{-1}{A}^r{z}_0.$$

Consequently,

$${∥H\left(\lambda \right)z_0-\sum _{k=0}^{r-1}{\lambda }^{-{\alpha }_0-k{\sigma }_n}{A}^k{z}_0∥}_{\mathcal{Z}}\le {\displaystyle \frac{K\parallel A^r{z}_0{\parallel }_{\mathcal{Z}}}{{\left|\lambda \right|}^{r{\sigma }_n}{(\mathrm{Re}\lambda -\omega )}^{{\alpha }_0}}}$$

and the integral

$$\mathrm{v}·\mathrm{p}·{\displaystyle \frac{1}{2\pi i}}\underset{s-i\infty }{\overset{s+i\infty }{\int }}{\mathrm{e}}^{\lambda t}H\left(\lambda \right)z_{0}d\lambda =\sum _{k=0}^{r-1}{\displaystyle \frac{t^{{\alpha }_0+k{\sigma }_n-1}{A}^k{z}_0}{\mathrm{\Gamma }({\alpha }_0+k{\sigma }_n)}}+$$

$$+{\displaystyle \frac{1}{2\pi i}}\underset{s-i\infty }{\overset{s+i\infty }{\int }}{\mathrm{e}}^{\lambda t}\left(H\left(\lambda \right)z_0-\sum _{k=0}^{r-1}{\lambda }^{-{\alpha }_0-k{\sigma }_n}{A}^k{z}_0\right)d\lambda ,s>\omega$$

converges.

For $\mathrm{Re}\lambda >\omega$, we obtain

$$H_m\left(\lambda \right):={\widehat{\mathcal{S}}}_m\left(\lambda \right)=\underset{0}{\overset{\infty }{\int }}{\mathrm{e}}^{-(m+\lambda )t}\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{\left(m(m+\omega )t\right)}^{k+1}}{k!(k+1)!}}{H}^{\left(k\right)}(m+\omega )dt=$$

$$=\sum _{k=0}^{\infty }{\displaystyle \frac{{(-1)}^k{\left(m(m+\omega )\right)}^{k+1}{H}^{\left(k\right)}(m+\omega )}{k!{(m+\lambda )}^{k+2}}}=$$

$$={\displaystyle \frac{m(m+\omega )}{{(m+\lambda )}^2}}\sum _{k=0}^{\infty }{\left({\displaystyle \frac{\lambda (m+\omega )}{m+\lambda }}-(m+\omega )\right)}^k{\displaystyle \frac{H^{\left(k\right)}(m+\omega )}{k!}}={\displaystyle \frac{m(m+\omega )}{{(m+\lambda )}^2}}H\left({\displaystyle \frac{\lambda (m+\omega )}{m+\lambda }}\right),$$

$\underset{n\to \infty }{lim}{H}_m\left(\lambda \right)=H\left(\lambda \right)$ due to the analyticity of $H\left(\lambda \right)$ on $\{\lambda \in \mathbb{C}:\mathrm{Re}\lambda >\omega \}$.

For ${\sigma }_n\in (0,2]$, $\mathrm{Re}\lambda \in [\omega +a,\omega +b]$, $0<a<b$, and sufficiently large $m\in \mathbb{N}$,

$${∥H_m\left(\lambda \right)∥}_{\mathcal{L}\left(\mathcal{Z}\right)}\le {\displaystyle \frac{{(m+\omega )}^2}{{|m+\lambda |}^2}}{∥H\left({\displaystyle \frac{\lambda (m+\omega )}{m+\lambda }}\right)∥}_{\mathcal{L}\left(\mathcal{Z}\right)}\le {\displaystyle \frac{{(m+\omega )}^2}{{|m+\lambda |}^2}}{\displaystyle \frac{K}{{\left(\mathrm{Re}\frac{\lambda (m+\omega )}{m+\lambda }-\omega \right)}^{{\alpha }_0}}}\le$$

$$\le {\displaystyle \frac{C_1{(1+\frac{\omega }{m})}^2}{|\frac{1}{m}+\frac{1}{\lambda }{|}^2{\left|\lambda \right|}^2}}\le {\displaystyle \frac{C_1{(1+\omega )}^2}{{\left|\lambda \right|}^2}}.$$

Here, we use the above for $\lambda =x+iy$, $x\in [a,b]$, $y\in \mathbb{R}$

$$\mathrm{Re}{\displaystyle \frac{\lambda (m+\omega )}{m+\lambda }}-\omega ={\displaystyle \frac{x(m+x)+y^2}{{(m+x)}^2+y^2}}(m+\omega )-\omega =$$

$$={\displaystyle \frac{m(m+x)(x-\omega )+m{y}^2}{{(m+x)}^2+y^2}}\ge {\displaystyle \frac{m(m+x)(a-\omega )+m{y}^2}{{(m+x)}^2+y^2}}\ge {\displaystyle \frac{m(a-\omega )}{m+x}}\ge {\displaystyle \frac{a-\omega }{1+b}}.$$

Take $s>\omega$, $z_0\in D_{A^r}$ and pass to limit $m\to \infty$ in the equality

$${\mathcal{S}}_m\left(t\right)z_0=\mathrm{v}·\mathrm{p}·{\displaystyle \frac{1}{2\pi i}}\underset{s-i\infty }{\overset{s+i\infty }{\int }}{\mathrm{e}}^{\lambda t}{H}_m\left(\lambda \right)z_{0}d\lambda ,t\in {\mathbb{R}}_{+}.$$

The set $D_{A^r}$ is dense in $\mathcal{Z}$ by Corollary 2 and the family of operators $\{{\mathcal{S}}_m\left(t\right)\in \mathcal{L}\left(\mathcal{Z}\right):t\in {\mathbb{R}}_{+}\}$ is uniformly bounded on every segment $[1/T,T]$, $T>0$. Therefore, we have the existence of the strong limit s-$\underset{m\to \infty }{lim}{\mathcal{S}}_m\left(t\right)$, which is uniform with respect to t on every segment $[1/T,T]$. □

Denote $Z\left(t\right):=s\text{-}\underset{m\to \infty }{lim}{\mathcal{S}}_m\left(t\right)$, $t\in {\mathbb{R}}_{+}$; then, inequality (8) implies that

$${\parallel Z\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le C{t}^{{\alpha }_0-1}{\mathrm{e}}^{\omega t}.$$

(9)

Due to Lemma 5, $Z\left(t\right)$ is strongly continuous for $t\in {\mathbb{R}}_{+}$.

Remark 5.

In the proof of Lemma 5, it is shown that for $s>\omega$, $z_0\in D_{A^r}$,

$$Z\left(t\right)z_0=\mathrm{v}·\mathrm{p}·{\displaystyle \frac{1}{2\pi i}}\underset{s-i\infty }{\overset{s+i\infty }{\int }}{\mathrm{e}}^{\lambda t}H\left(\lambda \right)z_{0}d\lambda .$$

(10)

Let the integral from the right-hand side of equality (10) be convergent for some $z_0\in \mathcal{Z}$. Then, this equality is true for this $z_0$. Indeed, for some $m\in \mathbb{N}$, we have

$$Z\left(t\right){\left[m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right]}^r{z}_0=\mathrm{v}·\mathrm{p}·{\displaystyle \frac{1}{2\pi i}}\underset{s-i\infty }{\overset{s+i\infty }{\int }}{\mathrm{e}}^{\lambda t}H\left(\lambda \right){\left[m^{{\sigma }_n}{(m^{{\sigma }_n}I-A)}^{-1}\right]}^r{z}_{0}d\lambda .$$

Passing to $m\to \infty$, we obtain (10) for this $z_0$ due to Lemma 4.

In particular, for $z_0\in D_A$, $\mathrm{Re}\lambda >\omega$

$$\parallel J^{{\sigma }_n}Z\left(t\right)A{z}_0{\parallel }_{\mathcal{Z}}\le {\displaystyle \frac{K\parallel A{z}_0{\parallel }_{\mathcal{Z}}}{\mathrm{\Gamma }\left({\sigma }_n\right)}}\underset{0}{\overset{t}{\int }}{(t-s)}^{{\sigma }_n-1}{\mathrm{e}}^{\omega s}{s}^{{\alpha }_0-1}ds\le {\displaystyle \frac{K\parallel A{z}_0{\parallel }_{\mathcal{Z}}\mathrm{\Gamma }\left({\alpha }_0\right){\mathrm{e}}^{\omega t}{t}^{{\alpha }_0+{\sigma }_n-1}}{\mathrm{\Gamma }({\alpha }_0+{\sigma }_n)}},$$

$$\mathfrak{L}\left[J^{{\sigma }_n}Z\left(t\right)A{z}_0\right]\left(\lambda \right)={\lambda }^{-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}A{z}_0=-{\lambda }^{-{\alpha }_0}{z}_0+{\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}{z}_0.$$

Hence, ${\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}{z}_0=\mathfrak{L}\left[J^{{\sigma }_n}Z\left(t\right)A{z}_0+{\displaystyle \frac{t^{{\alpha }_0-1}}{\mathrm{\Gamma }\left({\alpha }_0\right)}}{z}_0\right]\left(\lambda \right).$ Thus, (10) is valid for $z_0\in D_A$.

Theorem 3.

Let ${\alpha }_k\in (0,1]$, $k=0,1,\dots ,n$, ${\alpha }_0+{\alpha }_n>1$. Then $C^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)={\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$.

Proof. 

The inclusion $C^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)\subset {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$ is shown in Corollary 1. Here, we will prove the inverse inclusion.

If $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$, then due to Lemma 5, there exists a strongly continuous family of operators $\{Z\left(t\right)=s\text{-}\underset{m\to \infty }{lim}{\mathcal{S}}_m\left(t\right)\in \mathcal{L}\left(\mathcal{Z}\right):t\in {\mathbb{R}}_{+}\}$.

For $\mathrm{Re}\lambda >\omega$, the derivatives $H^{\left(k\right)}\left(\lambda \right):={\displaystyle \frac{d^k}{d{\lambda }^k}}\left({\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}\right)$ are sums of natural powers of ${({\lambda }^{{\sigma }_n}I-A)}^{-1}$, multiplied by scalar functions; consequently,

$${\displaystyle \frac{d^k}{d{\lambda }^k}}\left({\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}\right)z_0\in D_A,z_0\in \mathcal{Z},$$

$$A{\displaystyle \frac{d^k}{d{\lambda }^k}}\left({\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}\right)z_0={\displaystyle \frac{d^k}{d{\lambda }^k}}\left({\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}\right)A{z}_0,z_0\in D_A.$$

Since A is closed, these relations imply that for $z_0\in D_A$${\mathcal{S}}_m\left(t\right)z_0\in D_A$, $A{\mathcal{S}}_m\left(t\right)z_0={\mathcal{S}}_m\left(t\right)A{z}_0$. Let $m\to \infty$, then $Z\left(t\right)z_0\in D_A$, $AZ\left(t\right)z_0=Z\left(t\right)A{z}_0$ for any $z_0\in D_A$.

Taking into account Remark 5, we get for $z_0\in D_A$

$$\mathfrak{L}\left[Z\left(t\right)A{z}_0\right]={\lambda }^{{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}A{z}_0={\lambda }^{2{\sigma }_n-{\alpha }_0}{({\lambda }^{{\sigma }_n}I-A)}^{-1}{z}_0-{\lambda }^{{\sigma }_n-{\alpha }_0}{z}_0=$$

$$={\lambda }^{{\sigma }_n}H\left(\lambda \right)z_0-{\lambda }^{{\sigma }_n-{\alpha }_0}{z}_0,{\lambda }^{-{\sigma }_n}\mathfrak{L}\left[Z\left(t\right)A{z}_0\right]=H\left(\lambda \right)z_0-{\lambda }^{-{\alpha }_0}{z}_0,$$

$$J^{{\sigma }_n}Z\left(t\right)A{z}_0={\mathfrak{L}}^{-1}\left[{\lambda }^{-{\sigma }_n}\mathfrak{L}\left[Z\left(t\right)A{z}_0\right]\right]={\mathfrak{L}}^{-1}[H\left(\lambda \right)z_0-{\lambda }^{-{\alpha }_0}{z}_0]=Z\left(t\right)z_0-{\displaystyle \frac{t^{{\alpha }_0-1}{z}_0}{\mathrm{\Gamma }\left({\alpha }_0\right)}},$$

$$J^{1-{\alpha }_0}{J}^{{\sigma }_n}Z\left(t\right)A{z}_0=J^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}Z\left(t\right)A{z}_0=J^{1-{\alpha }_0}Z\left(t\right)z_0-z_0,$$

(11)

$$D^{\{{\alpha }_0,{\alpha }_1\}}{J}^{{\sigma }_n}Z\left(t\right)A{z}_0={}^G\mathit{D}^{{\alpha }_1}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}Z\left(t\right)A{z}_0=$$

$$=J^{{\alpha }_2+\dots +{\alpha }_n}Z\left(t\right)A{z}_0=D^{\{{\alpha }_0,{\alpha }_1\}}Z\left(t\right)z_0,$$

(12)

for $l=2,3,\dots ,n-1$,

$$D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_l\}}{J}^{{\sigma }_n}Z\left(t\right)A{z}_0=J^{{\alpha }_{l+1}+\dots +{\alpha }_n}Z\left(t\right)A{z}_0=D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_l\}}Z\left(t\right)z_0,$$

(13)

$$D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}{J}^{{\sigma }_n}Z\left(t\right)A{z}_0=Z\left(t\right)A{z}_0=AZ\left(t\right)z_0=D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}Z\left(t\right)z_0.$$

(14)

Due to (9), we have for $l=0,1,\dots ,n-1$

$$\parallel J^{{\alpha }_{l+1}+{\alpha }_2+\dots +{\alpha }_n}Z\left(t\right)A{z}_0{\parallel }_{\mathcal{Z}}\le C{\mathrm{e}}^{\omega t}{t}^{{\alpha }_0+{\alpha }_{l+1}+{\alpha }_{l+2}+\dots +{\alpha }_n-1}{\parallel A{z}_0\parallel }_{\mathcal{Z}}\to 0,t\to 0+,$$

(15)

since ${\alpha }_0+{\alpha }_n>1$. Therefore, (11) implies that

$$0=\underset{t\to 0+}{lim}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}Z\left(t\right)A{z}_0=\underset{t\to 0+}{lim}{J}^{1-{\alpha }_0}Z\left(t\right)z_0-z_0.$$

Due to (9), we have the estimate $\parallel D^{\left\{{\alpha }_0\right\}}{Z\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le C\mathrm{\Gamma }\left({\alpha }_0\right){\mathrm{e}}^{\omega t}.$ Since $D_A$ is dense in $\mathcal{Z}$, we have that $D^{\left\{{\alpha }_0\right\}}Z\left(0\right)z_0=z_0$ for every $z_0\in \mathcal{Z}$.

From (12), (13), and (15), it follows that $\underset{t\to 0+}{lim}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_l\}}Z\left(t\right)z_0=0$ for every $z_0\in D_A$, $l=1,2,\dots ,n-1$.

Due to (14) for $z_0\in D_A$ we have $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}Z\left(t\right)z_0=Z\left(t\right)A{z}_0\in C({\mathbb{R}}_{+};\mathcal{Z})$ and for every $l=0,1,\dots ,n-1$ $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_l\}}Z\left(t\right)z_0\in AC({\overline{\mathbb{R}}}_{+};\mathcal{Z})$. Therefore, $\{Z\left(t\right)\in \mathcal{L}\left(\mathcal{Z}\right):t\in {\mathbb{R}}_{+}\}$ satisfies all the conditions of the definition of a resolving family of operators for Equation (6). □

Remark 6.

The uniqueness of the resolving family of operators for Equation (6) follows from Lemma 3.

Corollary 4.

Let ${\alpha }_k\in (0,1]$, $k=0,1,\dots ,n$, ${\alpha }_0+{\alpha }_n>1$, $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$. Then, for every $z_0,z_1,\dots ,z_{n-1}\in D_A$, there exists a unique solution for problem (5), (6). The solution has the form $z\left(t\right)=Z\left(t\right)z_0+\sum _{k=1}^{n-1}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}Z\left(t\right)z_k$.

Proof. 

Note that for $k=1,2,\dots ,n-1$, $z_k\in D_A$, $l=0,1,\dots ,k-1$,

$$\underset{t\to 0+}{lim}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_l\}}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}Z\left(t\right)z_k=\underset{t\to 0+}{lim}{J}^{{\alpha }_{l+1}+{\alpha }_{l+2}+\dots +{\alpha }_k}{J}^{1-{\alpha }_0}Z\left(t\right)z_k=0,$$

$$\underset{t\to 0+}{lim}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}Z\left(t\right)z_k=\underset{t\to 0+}{lim}{J}^{1-{\alpha }_0}Z\left(t\right)z_k=z_k.$$

Due to (11) for $z_k\in D_A$, $l=k+1,k+2,\dots ,n-1$

$$\underset{t\to 0+}{lim}{D}^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_l\}}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}Z\left(t\right)z_k=\underset{t\to 0+}{lim}{}^G\mathit{D}^{{\alpha }_l}{}^G\mathit{D}^{{\alpha }_{l-1}}\dots {}^G\mathit{D}^{{\alpha }_{k+1}}{J}^{1-{\alpha }_0}Z\left(t\right)z_k=$$

$$=\underset{t\to 0+}{lim}{}^G\mathit{D}^{{\alpha }_1}{}^G\mathit{D}^{{\alpha }_{l-1}}\dots {}^G\mathit{D}^{{\alpha }_{k+1}}(J^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}Z\left(t\right)A{z}_k-z_k)=$$

$$=\underset{t\to 0+}{lim}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k+{\alpha }_{l+1}+{\alpha }_{l+2}+\dots +{\alpha }_n}Z\left(t\right)A{z}_k=0.$$

Furthermore,

$$D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}Z\left(t\right)z_k={}^G\mathit{D}^{{\alpha }_n}{}^G\mathit{D}^{{\alpha }_{n-1}}\dots {}^G\mathit{D}^{{\alpha }_{k+1}}{J}^{1-{\alpha }_0}Z\left(t\right)z_k=$$

$$={}^G\mathit{D}^{{\alpha }_n}{}^G\mathit{D}^{{\alpha }_{n-1}}\dots {}^G\mathit{D}^{{\alpha }_{k+1}}(J^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}Z\left(t\right)A{z}_k-z_k)=$$

$$=J^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}Z\left(t\right)A{z}_k=A{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}Z\left(t\right)z_k.$$

Hence, for every $z_k\in D_A$, the function $J^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}Z\left(t\right)z_k$ is a solution of the problem $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}z\left(0\right)=z_k$, $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_l\}}z\left(0\right)=0$, $l\in \{0,1,\dots ,n-1\}\setminus \left\{k\right\}$ for Equation (6).

The uniqueness of the problem solution still needs to be shown. Using the integral operator $J^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}$ on the both sides of Equation (6), we obtain

$$g_{1-{\alpha }_0}\mathrm{\ast }z\left(t\right)=J^{1-{\alpha }_0}z\left(t\right)=J^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}Az\left(t\right)+z_0=g_{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}\mathrm{\ast }Az\left(t\right)+z_0,$$

(16)

where z satisfies the initial conditions $D^{\left\{{\alpha }_0\right\}}z\left(0\right)=z_0$, $D^{\{{\alpha }_0,\dots ,{\alpha }_k\}}z\left(0\right)=0$ for $k=1,2,\dots ,n-1$. Hereafter, $g_{\beta }\left(t\right):=t^{\beta -1}/\mathrm{\Gamma }\left(\beta \right)$; the Laplace convolution $\ast$ is defined by the equality

$$\left(f\mathrm{\ast }g\right)\left(t\right)=\underset{0}{\overset{t}{\int }}f(t-s)g\left(s\right)ds.$$

Here, by the definition of the solution, we have $D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}z\in AC({\overline{\mathbb{R}}}_{+};\mathcal{Z})$ for every $k=0,1,\dots ,n-1$.

Since (16) is valid for $Z\left(t\right)z_0$, we have $(g_{1-{\alpha }_0}\mathrm{\ast }Z\left(t\right)-g_{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}\mathrm{\ast }AZ\left(t\right))z_0\equiv z_0;$ therefore, for another solution z, due to the linearity and commutativity of the Laplace convolution,

$$1\mathrm{\ast }z=1\mathrm{\ast }(g_{1-{\alpha }_0}\mathrm{\ast }Z-g_{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}\mathrm{\ast }AZ)z=(g_{1-{\alpha }_0}\mathrm{\ast }Z-g_{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}\mathrm{\ast }AZ)\mathrm{\ast }z=$$

$$=Z\mathrm{\ast }{g}_{1-{\alpha }_0}\mathrm{\ast }z-Z\mathrm{\ast }{g}_{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}\mathrm{\ast }Az=$$

$$=Z\mathrm{\ast }(g_{1-{\alpha }_0}\mathrm{\ast }z-g_{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}\mathrm{\ast }Az)=Z\mathrm{\ast }{z}_0=1\mathrm{\ast }Z{z}_0.$$

Differentiating by t, we get $z\left(t\right)=Z\left(t\right)z_0$ for all $t\in {\mathbb{R}}_{+}$.

Let z be a solution of (5), (6). Then,

$$z\left(t\right)-\sum _{k=1}^{n-1}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}Z\left(t\right)z_k$$

is a solution of such problem with $z_1=z_2=\dots =z_{n-1}=0$. Therefore,

$$z\left(t\right)-\sum _{k=1}^{n-1}{J}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_k}Z\left(t\right)z_k=Z\left(t\right)z_0.$$

□

4. An Example and an Application

Take the Banach space $l_2$ of sequences $z={\left\{z_m\right\}}_{m=1}^{\infty }$, where $z_m\in \mathbb{C}$ for $m\in \mathbb{N}$, such that the series ${\parallel z\parallel }_{l_2}^2={\sum }_{m=1}^{\infty }{\left|z_m\right|}^2$ converges.

Consider the initial value problem

$$D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}{z}_m\left(0\right)=z_{km},m\in \mathbb{N},$$

(17)

$$D^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}{z}_m\left(t\right)={\mu }_m{\mathrm{e}}^{i\pi \beta /2}{z}_m\left(t\right),t\in {\mathbb{R}}_{+},m\in \mathbb{N},$$

(18)

where $0<{\alpha }_k\le 1$, $k=0,1,\dots ,n$, ${\sigma }_n={\alpha }_0+{\alpha }_1+\dots +{\alpha }_n-1>0$, $-2<\beta \le 2$, $z_k={\left\{z_{km}\right\}}_{m=1}^{\infty }\in l_2$, $k=0,1,\dots ,n-1$, ${\left\{{\mu }_m\right\}}_{m=1}^{\infty }$ is a real positive monotone sequence, such that $\underset{m\to \infty }{lim}{\mu }_m=+\infty$. Set $\mathcal{Z}=l_2$, $A_{\beta }z={\left\{{\mu }_m{\mathrm{e}}^{i\pi \beta /2}{z}_m\right\}}_{m=1}^{\infty }$ for $z={\left\{z_m\right\}}_{m=1}^{\infty }\in l_2$, such that ${\left\{{\mu }_m{z}_m\right\}}_{m=1}^{\infty }\in l_2$, which makes up the domain $D_{A_{\beta }}$. It is easy to show that for every $\beta \in (-2,2]$, $A_{\beta }\in \mathcal{C}l\left(l_2\right)$ and $\sigma \left(A_{\beta }\right)=\{{\mu }_m{\mathrm{e}}^{i\pi \beta /2}:m\in \mathbb{N}\}$. Denote ${\Pi }_{\omega }^{{\sigma }_n}:=\{{\mu }^{{\sigma }_n}\in \mathbb{C}:\mathrm{Re}\mu >\omega \}$, and then, for every $\beta \in (-{\sigma }_n,{\sigma }_n)$ and $\omega \ge 0$, we have $\sigma \left(A_{\beta }\right)\cap {\Pi }_{\omega }^{{\sigma }_n}\ne \varnothing$ and $A\notin {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(\omega \right)$. So, we will consider $\beta \in (-2,-{\sigma }_n]\cup [{\sigma }_n,2]$ only.

Solving problem (17), (18) with arbitrary ${\left\{z_{0m}\right\}}_{m=1}^{\infty }\in D_{A_{\beta }}$ and with $z_{km}=0$, $k=1,2,\dots ,n-1$, $m\in \mathbb{N}$, for every $m\in \mathbb{N}$, respectively, we can determine that the resolving family of equations must have the form $S_{\beta }\left(t\right)z_0={\left\{t^{{\alpha }_0-1}{E}_{{\sigma }_n,{\alpha }_0}\left(t^{{\sigma }_n}{\mu }_m{\mathrm{e}}^{i\pi \beta /2}\right)z_{0m}\right\}}_{m=1}^{\infty }$ (see [17] and Remark 1). Since $arg\left({\mu }_m{\mathrm{e}}^{i\pi \beta /2}\right)=\pi \beta /2$, we have for a fixed $t>0$, $\beta \in (-2,-{\sigma }_n)\cup ({\sigma }_n,2]$ due to the asymptotic expansion of the Mittag–Leffler function:

$$|t^{{\alpha }_0-1}{E}_{{\sigma }_n,{\alpha }_0}\left(t^{{\sigma }_n}{\mu }_m{\mathrm{e}}^{i\pi \beta /2}\right)|\le C_1{t}^{{\alpha }_0-1}{\left(t^{{\sigma }_n}{\mu }_m\right)}^{-1}={\displaystyle \frac{C_1}{{\mu }_m{t}^{{\alpha }_1+{\alpha }_2+\dots +{\alpha }_n}}},m\to \infty .$$

Therefore, $S_{\beta }\left(t\right)\in \mathcal{L}\left(l_2\right)$, and it can be checked that $A_{\beta }\in C_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(0\right)$ for every $\beta \in (-2,-{\sigma }_n)\cup ({\sigma }_n,2]$.

Take $\beta =\pm {\sigma }_n$ and $arg\left(n{\mathrm{e}}^{\pm i\pi {\sigma }_n/2}\right)=\pm \pi {\sigma }_n/2$, and for a fixed $t>0$, using the asymptotic expansion of the Mittag–Leffler function, we can obtain

$$|t^{{\alpha }_0-1}{E}_{{\sigma }_n,{\alpha }_0}\left(t^{{\sigma }_n}{\mu }_m{\mathrm{e}}^{i\pi {\sigma }_n/2}\right)|\le C_2{t}^{{\alpha }_0-1}{\left(t^{{\sigma }_n}{\mu }_m\right)}^{{\displaystyle \frac{1-{\alpha }_0}{{\sigma }_n}}}\left|{\mathrm{e}}^{i{\mu }_m^{1/{\sigma }_n}t}\right|=C_2{\mu }_m^{{\displaystyle \frac{1-{\alpha }_0}{{\sigma }_n}}},m\to \infty .$$

So, $S_{\pm {\sigma }_n}\left(t\right)\notin \mathcal{L}\left(l_2\right)$ for ${\alpha }_0<1$ and $S_{\pm {\sigma }_n}\left(t\right)\in \mathcal{L}\left(l_2\right)$ for ${\alpha }_0=1$. Hence, $A_{\pm {\sigma }_n}\in C_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(0\right)$ only for ${\alpha }_0=1$.

Let $\mathrm{\Omega }\subset {\mathbb{R}}^d$ be a bounded domain with a smooth boundary $\partial \mathrm{\Omega }$. Consider the problem for equation, which can be interpreted as an equation of ultraslow dynamics of non-relativistic systems:

$$D_t^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}v(\xi ,t)=i\mathrm{\Delta }v(\xi ,t),(\xi ,t)\in \mathrm{\Omega }\times {\mathbb{R}}_{+},$$

(19)

$$v(\xi ,t)=0,(\xi ,t)\in \partial \mathrm{\Omega }\times {\mathbb{R}}_{+},$$

(20)

$$D_t^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}v(\xi ,0)=v_k\left(\xi \right),\xi \in \mathrm{\Omega },k=0,1,\dots ,n-1.$$

(21)

Here, $D_t^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}$, $k=0,1,\dots ,n$, are the Dzhrbashyan–Nersesyan fractional derivatives with respect to the variable t, and $\mathrm{\Delta }$ is the Laplace operator. Take $\mathcal{Z}=L_2\left(\mathrm{\Omega }\right)$ with the inner product $\langle ·,·\rangle$, $Av=i\mathrm{\Delta }v$, $D_A:=H_0^2\left(\mathrm{\Omega }\right):=\{v\in H^2\left(\mathrm{\Omega }\right):v\left(\xi \right)=0,\xi \in \partial \mathrm{\Omega }\}$, $z_k=v_k(·)\in D_A$, $k=0,1,\dots ,n-1$, $z\left(t\right)=v(·,t)\in \mathcal{Z}$, $t\ge 0$. Let $\sigma \left(A\right)=\{i{\lambda }_m:m\in \mathbb{N}\}$, $0>{\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_m\ge \dots$ and ${\phi }_m$ be eigenfunctions, which correspond to eigenvalues $i{\lambda }_m$ and $m\in \mathbb{N}$. Consider ${\mu }_m=-{\lambda }_m$, $z_m\left(t\right):=\langle v(·,t),{\phi }_m\rangle$, $z_{km}\left(t\right):=\langle v_k(·),{\phi }_m\rangle$, $k=0,1,\dots ,n-1$, and $m\in \mathbb{N}$, and then $\left\{z_{km}\right\}$, $\left\{z_m\left(t\right)\right\}\in l_2$, $t\ge 0$. We have thus solved problem (17), (18) with $A=A_{-1}\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(0\right)$ if ${\sigma }_n\le 1$, and $A=A_{-1}\notin {\mathcal{C}}_{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n}\left(\omega \right)$ for any $\omega \ge 0$, if ${\sigma }_n\in (1,2]$.

Analogously, we can consider the problem

$$D_t^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}v(\xi ,t)=i\mathrm{\Delta }{v}^p(\xi ,t),(\xi ,t)\in \mathrm{\Omega }\times {\mathbb{R}}_{+},$$

(22)

$${\mathrm{\Delta }}^{l}v(\xi ,t)=0,l=0,1,\dots ,p-1,(\xi ,t)\in \partial \mathrm{\Omega }\times {\mathbb{R}}_{+},$$

(23)

$$D_t^{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_k\}}v(\xi ,0)=v_k\left(\xi \right),\xi \in \mathrm{\Omega },k=0,1,\dots ,n-1.$$

(24)

with $p\in \mathbb{N}$ and obtain that the operator $Av=i{\mathrm{\Delta }}^{p}v$, $D_A:=H_0^2\left(\mathrm{\Omega }\right):=\{v\in H^{2p}\left(\mathrm{\Omega }\right):{\mathrm{\Delta }}^{l}v\left(\xi \right)=0,,l=0,1,\dots ,p-1,\xi \in \partial \mathrm{\Omega }\}$ can be reduced to the operator $A_{-1}$ in problem (17), (18) for an odd p and to $A_1$ for an even p. Hence, $A\in {\mathcal{C}}_{\{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n\}}\left(0\right)$ if ${\sigma }_n\le 1$, and $A\notin {\mathcal{C}}_{{\alpha }_0,{\alpha }_1,\dots ,{\alpha }_n}\left(\omega \right)$ for any $\omega \ge 0$ if ${\sigma }_n\in (1,2]$.

5. Conclusions

The initial value problem for a linear homogeneous differential equation in a Banach space with the Dzhrbashyan–Nersesyan derivative and a closed operator A at the unknown function is studied in this paper. A Hille–Yosida-type theorem on generation by the operator A of a strongly continuous resolving family of operators for such an equation is proven using Phillips-type approximations. The obtained results are illustrated by examples of partial differential equations.

Further, based on the results of this work, we will investigate the corresponding linear inhomogeneous equation and obtain a representation of its solution using the resolving family of operators of the linear homogeneous equation. This will make it possible to move on to the study of quasi-linear equations with a linear part having a resolving family of operators. In addition, we plan to consider the principle of subordination for linear equations with a Dzhrbashyan–Nersesyan derivative.