Integrated Resolving Functions for Equations with Gerasimov–Caputo Derivatives

Abstract

The concept of a $\beta$-integrated resolving function for a linear equation with a Gerasimov–Caputo fractional derivative is introduced into consideration. A number of properties of such functions are proved, and conditions for the solvability of the Cauchy problem to linear homogeneous and inhomogeneous equations are found in the case of the existence of a $\beta$-integrated resolving function. The necessary and sufficient conditions for the existence of such a function in terms of estimates on the resolvent of its generator are obtained. The example of a $\beta$-integrated resolving function for the Schrödinger equation is given. Thus, the paper discusses some aspects of the symmetry of the concepts of integrability and differentiability. Namely, it is shown that, in the absence of a sufficiently differentiable resolving function for a fractional differential equation, the problem of the existence of a solution can be solved by an integrated resolving function of the equation.

1. Introduction

One of the traditional approaches for solving broad classes of initial boundary value problems involving partial differential equations is using, from the theory of differential equations in Banach spaces, in particular, the theory of semigroups of operators [1,2,3,4]. As it is known, the resolving function $S:{\overline{\mathbb{R}}}_{+}\to \mathcal{L}\left(\mathcal{Z}\right)$ that defines the solution $z\left(t\right)=S\left(t\right)z_0$ of the Cauchy problem $z\left(0\right)=z_0$ to a first-order equation

$$D^{1}z\left(t\right)=Az\left(t\right)$$

(1)

($D^1$ is the first order differentiation operator) in a Banach space $\mathcal{Z}$ with, generally speaking, an unbounded linear operator A form a semigroup, i.e., $S\left(s\right)S\left(t\right)=S(s+t)$ for all $s,t\ge 0$, since the formal solution of such an equation is given by the exponential function. For fractional order equations

$$D^{\alpha }z\left(t\right)=Az\left(t\right),$$

($D^{\alpha }z$ is the Gerasimov–Caputo derivative of the order $\alpha >0$) the semigroup property of resolving function is absent, and there is no generalization of this property in the general case, since the class of Mittag–Leffler functions that are the formal solutions of such equations is too wide. However, in other respects, the advantages of these functional-analytical approaches remain valid in the study of fractional differential equations. Such advantages include the possibility of studying wide classes of initial boundary value problems for partial differential equations and systems of equations within one class of initial problems for an equation in a Banach space: proving the existence and uniqueness of the solution, obtaining a representation of a solution in the linear case and an approximate solution in the nonlinear one, etc. The technique of resolving families of operators (resolving functions) is used successfully when studying integro-differential equations [5,6], integral evolution equations [7,8], various fractional differential equations [9,10,11,12,13,14].

An important generalization of the notion of the operator semigroup is the k-times integrated operator semigroup [15,16]. The theory of the k-times integrated operator semigroups (see [15,16,17,18,19]) enables one to investigate the solvability of the Cauchy problem $z\left(0\right)=z_0$ for a first-order Equation (1) in the case where the operator A does not generate an operator semigroup S, but generates a k-times integrated semigroup of operators $S_k$. In the case of the existence of the semigroup S, $S_k$ is the k-th order primitive of S.

The Cauchy problem

$$D^{k}z\left(0\right)=z_k,k=0,1,\dots ,m-1,$$

(2)

for a linear homogeneous equation

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

(3)

was studied in terms of solution operators (resolving functions) in [9]. Here, $D^k$ are derivatives of k-th order, $k=0,1,\dots ,m-1$, $D^{\alpha }$ is the Gerasimov–Caputo derivative of the order $\alpha \in (m-1,m]$, $m\in \mathbb{N}$ (see the definition in the next section). In [9], the necessary and sufficient conditions in terms of the operator A resolvent are obtained for the existence of the solution operator (resolving function) of Equation (3). The properties of strongly continuous and analytic, exponentially bounded resolving functions were studied.

The aim of the present work is to extend the concept of a k-times integrated semigroup to the resolving functions of the fractional differential Equation (3). In this case, $\beta$-integrated resolving functions will be considered not only for $\beta \in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$, as for operator semigroups, but also for a fractional order $\beta \in {\mathbb{R}}_{+}$ of integration. This will allow us to assert the existence of a solution to the Cauchy problem (2) for Equation (3) or for the corresponding inhomogeneous equation in the case where there is no (sufficiently differentiable) resolving function, but there is a $\beta$-integrated resolving function (its existence conditions are less stringent). In partial cases, we obtain k-times integrated semigroups [15,16] ($\alpha =1$, $\beta =k\in \mathbb{N}$) and the resolving functions of the fractional order equation [9] ($\alpha >0$ is arbitrary, $\beta =0$). Thus, the symmetry properties (in some sense) of the concepts of integrability and differentiability are studied.

In the Section 2, some necessary definitions are given, including the new notion of a $\beta$-integrated resolving function. Some properties of this function are proved. Necessary and (separately) sufficient conditions for the existence of a $\beta$-integrated resolving function are obtained in terms of estimates for the resolvent of the generator A in some right complex half-plane. Section 3 contains the issues of the existence of a unique solution for the Cauchy problem to Equation (3), the corresponding general inhomogeneous equation and some special inhomogeneous equations. Mild solutions and classical solutions are considered. In the Section 4, two theorems on the necessary and sufficient conditions for the existence of a $\beta$-integrated resolving function are obtained. These conditions are formulated in terms of the resolvent of generator A on the semi-axis. The last section concerns the $\beta$-integrated resolving function for the linear time-fractional Schrödinger equation with the Dirichlet boundary condition.

2. $\mathbf{\beta }$-Integrated Resolving Functions and Some of Their Properties

Introduce the denotation ${\overline{\mathbb{R}}}_{+}:={\mathbb{R}}_{+}\cup \left\{0\right\}$, for $h:{\mathbb{R}}_{+}\to \mathcal{Z}$, where $\mathcal{Z}$ is a Banach space, the Riemann–Liouville integral of the order $\beta >0$ is

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

$J^{0}h\left(t\right):=h\left(t\right)$, $D^m$ is the derivative of the m-th order, $m\in \mathbb{N}$. The Gerasimov–Caputo derivative of the order $\alpha \in (m-1,m]$ for $h:{\overline{\mathbb{R}}}_{+}\to \mathcal{Z}$ is defined as [9] (p. 11, Formula (1.20)).

$$D^{\alpha }h\left(t\right):=D^m{J}^{m-\alpha }\left(h\left(t\right)-\sum _{k=0}^{m-1}{D}^{k}h\left(0\right)g_{k+1}\left(t\right)\right).$$

Here and further $D^{\beta }h\left(0\right):=\underset{t\to 0+}{lim}{D}^{\beta }h\left(t\right)$, $\beta \in \mathbb{R}.$ For $\alpha \le 0$, the denotation $D^{\alpha }h\left(t\right):=J^{-\alpha }h\left(t\right)$ will also be used.

Remark 1. 

A.N.Gerasimov [20] and M. Caputo [21] introduced the concept of a fractional derivative, named here by their names, independently of each other. A discussion of these issues can be found in [22].

The Laplace transform of a function $h:{\mathbb{R}}_{+}\to \mathcal{Z}$ will be denoted by $\widehat{h}$. The Laplace transform of the Riemann–Liouville integral and the Gerasimov–Caputo derivative satisfies the equalities (see, e. g., [9,23]).

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

Denote by $\mathcal{L}\left(\mathcal{Z}\right)$ the Banach algebra of all linear bounded operators from $\mathcal{Z}$ to $\mathcal{Z}$, the set of all linear closed operators, densely defined in $\mathcal{Z}$, acting to the space $\mathcal{Z}$ will be denoted by $\mathcal{C}ld\left(\mathcal{Z}\right)$. The denotation for the set of all linear closed operators, defined in $\mathcal{Z}$, acting on the space $\mathcal{Z}$ is $\mathcal{C}l\left(\mathcal{Z}\right)$. Endow the domain $D_A$ of an operator $A\in \mathcal{C}l\left(\mathcal{Z}\right)$ by the norm of its graph ${\parallel ·\parallel }_{D_A}={\parallel ·\parallel }_{\mathcal{Z}}+{\parallel A·\parallel }_{\mathcal{Z}}$ and obtain the Banach space $D_A$.

Consider the Cauchy problem

$$D^{k}z\left(0\right)=z_k,k=0,1,\dots ,m-1,$$

(4)

for a linear homogeneous equation

$$D^{\alpha }z\left(t\right)=Az\left(t\right),t\ge 0,$$

(5)

where $m-1<\alpha \le m\in \mathbb{N}$, $A\in \mathcal{C}l\left(\mathcal{Z}\right)$. As a solution to problem (4), (5) is a function $z\in C({\overline{\mathbb{R}}}_{+};D_A)\cap C^{m-1}({\overline{\mathbb{R}}}_{+};\mathcal{Z})$, such that $D^{\alpha }z\in C({\overline{\mathbb{R}}}_{+};\mathcal{Z})$ exists and conditions (4) and equality (5) for $t\in {\overline{\mathbb{R}}}_{+}$ hold.

Definition 1 

([9] (p. 20, Definition 2.3)) A function $S:{\overline{\mathbb{R}}}_{+}\to \mathcal{L}\left(\mathcal{Z}\right)$ is called resolving for Equation (5) if the following conditions are satisfied:

(i)

$S\left(t\right)$ is strongly continuous on ${\overline{\mathbb{R}}}_{+}$, $S\left(0\right)=I$;

(ii)

$S\left(t\right)\left[D_A\right]\subset D_A$, $S\left(t\right)A{z}_0=AS\left(t\right)z_0$ for all $z_0\in D_A$, $t\ge 0$;

(iii)

$S\left(t\right)z_0$ is a solution to the Cauchy problem $z\left(0\right)=z_0$, $D^{k}z\left(0\right)=0$, $k=1,\dots ,m-1$, to Equation (5) for every $z_0\in D_A$.

Remark 2. 

Often, the family of operators $\left\{S\right(t)\in \mathcal{L}(\mathcal{Z}):t\ge 0\}$ from Definition 1 is called a solution operator [9] (p. 20, Definition 2.3) or a resolving family of operators [11,12,13,14]. The second option seems more convenient to us. In addition, in this paper, it is more natural to use the corresponding mapping $S:{\overline{\mathbb{R}}}_{+}\to \mathcal{L}\left(\mathcal{Z}\right)$ not a family of operators, so we use the shorter term “resolving function”.

Remark 3. 

Concepts of the resolving function or resolving family of operators (resolving functions) are used in the study of first-order equations [1,2,3] ($C_0$-continuous resolving semigroup of operators), integro-differential equations [5], integral evolution equations [7], fractional differential equations [9].

Let $\mathcal{X}$ be a Banach space. For $F:{\overline{\mathbb{R}}}_{+}\to \mathcal{X}$, define the exponential growth bound as $\omega \left(F\right):=inf\{\omega \in \mathbb{R}:\underset{t\ge 0}{sup}\parallel e^{-\omega t}F\left(t\right){\parallel }_{\mathcal{X}}<\infty \}$.

Let S be a resolving function of Equation (5) with $\omega \left(S\right)<\infty$. It is known that $\widehat{S}\left(\lambda \right)={\lambda }^{\alpha -1}{({\lambda }^{\alpha }-A)}^{-1}$ for $\lambda >\omega \left(S\right)$ [9] (p. 21, Formula (2.6)). For $\beta \ge 0$, consider the function $S_{\beta }:{\overline{\mathbb{R}}}_{+}\to \mathcal{L}\left(\mathcal{Z}\right)$, where for $z_0\in \mathcal{Z}$

$$S_{\beta }\left(t\right)z_0:=\underset{0}{\overset{t}{\int }}\frac{{(t-s)}^{\beta -1}}{\Gamma \left(\beta \right)}S\left(s\right)z_{0}ds=J^{\beta }S\left(t\right)z_0.$$

Then, for $\mathrm{Re}\lambda >\omega \left(S\right)$, there exists the Laplace transform

$${\widehat{S}}_{\beta }\left(\lambda \right)={\lambda }^{-\beta }\widehat{S}\left(\lambda \right)={\lambda }^{\alpha -1-\beta }{({\lambda }^{\alpha }-A)}^{-1},$$

consequently, ${\lambda }^{\alpha -1}{({\lambda }^{\alpha }-A)}^{-1}={\lambda }^{\beta }{\widehat{S}}_{\beta }\left(\lambda \right).$

Definition 2. 

Let $\alpha >0$, $A\in \mathcal{C}l\left(\mathcal{Z}\right)$, $\beta \ge 0$. We call A a generator of a β-integrated resolving function, if $\omega \ge 0$ exists and a strongly continuous function $S_{\beta }:{\overline{\mathbb{R}}}_{+}\to \mathcal{L}\left(\mathcal{Z}\right)$ such that $\omega \left(S_{\beta }\right)\le \omega$, $({\omega }^{\alpha },\infty )\subset \rho \left(A\right)$ and equality ${\lambda }^{\alpha -1}{({\lambda }^{\alpha }-A)}^{-1}={\lambda }^{\beta }{\widehat{S}}_{\beta }\left(\lambda \right)$ is valid for all $\lambda >\omega$. In this case, $S_{\beta }$ is called the β-integrated resolving the function generated by A.

Remark 4. 

Due to [9] (p. 23, Theorem 2.9), the 0-integrated resolving function is a resolving function for Equation (5).

Remark 5. 

From Definition 2, it follows the one-to-one correspondence of generators and β-integrated resolving functions for a fixed $\beta \ge 0$.

Example 1. 

Consider $A\in \mathcal{L}\left(\mathcal{Z}\right)$, then the β-resolving function of Equation (5) is $S_{\beta }\left(t\right)=t^{\beta }{E}_{\alpha ,\beta +1}\left(A{t}^{\alpha }\right),$ where $E_{a,b}\left(z\right)$ is the Mittag–Leffler function. Indeed, for

$$\Gamma ={\{\mu \in \mathbb{C}:|arg(\mu -2\parallel A\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)})|=\pi /4\}$$

we have

$$S_{\beta }\left(t\right)=\frac{1}{2\pi i}\underset{\Gamma }{\int }{\mu }^{\alpha -1-\beta }{({\mu }^{\alpha }-A)}^{-1}{e}^{\mu t}d\mu =\frac{1}{2\pi i}\sum _{k=0}^{\infty }{A}^k\underset{\Gamma }{\int }{\mu }^{-1-\beta -k\alpha }{e}^{\mu t}d\mu =$$

$$=\frac{1}{2\pi i}\sum _{k=0}^{\infty }{t}^{\beta +k\alpha }{A}^k\underset{{\Gamma }_t}{\int }{\nu }^{-1-\beta -k\alpha }{e}^{\nu }d\nu =\sum _{k=0}^{\infty }\frac{t^{\beta +k\alpha }{A}^k}{\Gamma (\beta +k\alpha +1)}=t^{\beta }{E}_{\alpha ,\beta +1}\left(A{t}^{\alpha }\right),$$

where ${\Gamma }_t:=t\Gamma$.

Lemma 1. 

Let $\alpha >0$, $0\le {\beta }_1<{\beta }_2$, A be a generator of a ${\beta }_1$-integrated resolving function. Then, A is the generator of a ${\beta }_2$-integrated resolving function.

Proof. 

Let for $\lambda >\omega$

$${\lambda }^{\alpha -1}{({\lambda }^{\alpha }-A)}^{-1}={\lambda }^{{\beta }_1}{\widehat{S}}_{{\beta }_1}\left(\lambda \right)={\lambda }^{{\beta }_2}{\widehat{J^{{\beta }_2-{\beta }_1}S}}_{{\beta }_1}\left(\lambda \right)={\lambda }^{{\beta }_2}{\widehat{S}}_{{\beta }_2}\left(\lambda \right),$$

where $S_{{\beta }_2}:=J^{{\beta }_2-{\beta }_1}{S}_{{\beta }_1}:{\overline{\mathbb{R}}}_{+}\to \mathcal{L}\left(\mathcal{Z}\right)$ is a strongly continuous function. □

Proposition 1 

([16] (p. 110, Proposition 3.1.5)). Let $S:{\mathbb{R}}_{+}\to \mathcal{L}\left(\mathcal{Z}\right)$ be strongly continuous such that $\omega \left(S\right)<\omega$. Then, the following assertions hold:

(i) If $B\in \mathcal{L}\left(\mathcal{Z}\right)$ such that $B\widehat{S}\left(\lambda \right)=\widehat{S}\left(\lambda \right)B$ for all $\lambda >\omega$, then $BS\left(t\right)=S\left(t\right)B$ for all $t\ge 0$.

(ii) In particular, if $\widehat{S}\left(\mu \right)\widehat{S}\left(\lambda \right)=\widehat{S}\left(\lambda \right)\widehat{S}\left(\mu \right)$ for all $\lambda ,\mu >\omega$, then $S\left(t\right)S\left(s\right)=S\left(s\right)S\left(t\right)$ for all $t,s\ge 0$.

Lemma 2. 

Let $\alpha >0$, $\beta \ge 0$, A be the generator of a β-integrated resolving function $\{S_{\beta }\left(t\right)\in \mathcal{L}\left(\mathcal{Z}\right):t\ge 0\}$. Then, the following holds.

(i)

${(\mu -A)}^{-1}{S}_{\beta }\left(t\right)=S_{\beta }\left(t\right){(\mu -A)}^{-1}$, $t\ge 0$, $\mu \in \rho \left(A\right)$.

(ii)

$S_{\beta }\left(s\right)S_{\beta }\left(t\right)=S_{\beta }\left(t\right)S_{\beta }\left(s\right)$, $s\ge 0$, $t\ge 0$.

(iii)

If $z_0\in D_A$, then $S_{\beta }\left(t\right)z_0\in D_A$ and $A{S}_{\beta }\left(t\right)z_0=S_{\beta }\left(t\right)A{z}_0$ for all $t\ge 0$.

(iv)

Let $z_0\in D_A$ and $t\ge 0$. Then

$$J^{\alpha }{S}_{\beta }\left(t\right)A{z}_0=S_{\beta }\left(t\right)z_0-g_{\beta +1}\left(t\right)z_0.$$

(6)

In particular, $D^{\alpha }{S}_{\beta }\left(t\right)z_0=S_{\beta }\left(t\right)A{z}_0$, $\beta \in \{0,1,\dots ,m-1\},$

$$D^{\alpha }{S}_{\beta }\left(t\right)z_0=S_{\beta }\left(t\right)A{z}_0+g_{\beta -\alpha +1}\left(t\right)z_0,\beta \notin \{0,1,\dots ,m-1\},$$

$D^k{|}_{t=0}{S}_{\beta }\left(t\right)z_0=0$ for $k\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$ such that $k<\alpha$, $\beta \in \{0,1,\dots ,k-1\}\cup (k,\infty )$.

(v) Let $z_0\in \mathcal{Z}$ and $t\ge 0$. Then, $J^{\alpha }{S}_{\beta }\left(t\right)z_0\in D_A$ and

$$A{J}^{\alpha }{S}_{\beta }\left(t\right)z_0=S_{\beta }\left(t\right)z_0-g_{\beta +1}\left(t\right)z_0,$$

(7)

$S_{\beta }\left(0\right)=0$ for $\beta >0$.

(vi) Let $x,y\in \mathcal{Z}$ such that, for all, $t\ge 0$

$$J^{\alpha }{S}_{\beta }\left(t\right)y=S_{\beta }\left(t\right)x-g_{\beta +1}\left(t\right)x.$$

(8)

Then, $x\in D_A$ and $Ax=y$.

Proof. 

We have for $\lambda >\omega$ ${\lambda }^{\alpha -1}{({\lambda }^{\alpha }-A)}^{-1}={\lambda }^{\beta }{\widehat{S}}_{\beta }\left(\lambda \right)$, therefore, due to the resolvent identity, we have ${(\mu -A)}^{-1}{\widehat{S}}_{\beta }\left(\lambda \right)={\widehat{S}}_{\beta }\left(\lambda \right){(\mu -A)}^{-1}$, consequently, by Proposition 1, (i) we obtain assertion (i) of this lemma. Proposition 1 (ii) implies assertion (ii) of the lemma.

Take $z_0={(\mu -A)}^{-1}{y}_0\in D_A$ for some $\mu \in \rho \left(A\right)$, $y_0\in \mathcal{Z}$, and then, for all $t\ge 0$ $S_{\beta }\left(t\right)z_0=S_{\beta }\left(t\right){(\mu -A)}^{-1}{y}_0={(\mu -A)}^{-1}{S}_{\beta }\left(t\right)y_0\in D_A$,

$$A{S}_{\beta }\left(t\right)z_0=A{(\mu -A)}^{-1}{S}_{\beta }\left(t\right)y_0=\mu {(\mu -A)}^{-1}{S}_{\beta }\left(t\right)y_0-S_{\beta }\left(t\right)y_0=S_{\beta }A{(\mu -A)}^{-1}{y}_0.$$

Therefore, statement (iii) holds.

For $z_0\in D_A$, $t\ge 0$, $\lambda >\omega$, we have

$$z_0={\lambda }^{\beta +1}{\widehat{g}}_{\beta +1}\left(\lambda \right)z_0={({\lambda }^{\alpha }-A)}^{-1}({\lambda }^{\alpha }-A)z_0={\lambda }^{\beta +1}{\widehat{S}}_{\beta }\left(\lambda \right)z_0-{\lambda }^{\beta +1-\alpha }{\widehat{S}}_{\beta }\left(\lambda \right)A{z}_0=$$

$$={\lambda }^{\beta +1}{\widehat{S}}_{\beta }\left(\lambda \right)z_0-{\lambda }^{\beta +1}{\widehat{J^{\alpha }S}}_{\beta }\left(\lambda \right)A{z}_0.$$

By the inverse Laplace transform, we obtain equality (6). Hence, for $k\in {\mathbb{N}}_0$, $k<\alpha$, we have $D^k{S}_{\beta }\left(t\right)z_0=D^k{J}^{\alpha }{S}_{\beta }\left(s\right)A{z}_0=J^{\alpha -k}{S}_{\beta }\left(s\right)A{z}_0,$ if $\beta \in \{0,1,\dots ,k-1\}$, and

$$D^k{S}_{\beta }\left(t\right)z_0=D^k{J}^{\alpha }{S}_{\beta }\left(s\right)A{z}_0+g_{\beta -k+1}\left(t\right)z_0=J^{\alpha -k}{S}_{\beta }\left(s\right)A{z}_0+g_{\beta -k+1}\left(t\right)z_0,$$

if $\beta \in (k,\infty )$.

For $\mu \in \rho \left(A\right)$ and $z_0\in \mathcal{Z}$, assertions (i)–(iv) imply that

$$J^{\alpha }{S}_{\beta }\left(s\right)z_0=\mu {(\mu -A)}^{-1}{J}^{\alpha }{S}_{\beta }\left(s\right)z_0-A{(\mu -A)}^{-1}{J}^{\alpha }{S}_{\beta }\left(t\right)z_0=$$

$$=\mu {(\mu -A)}^{-1}{J}^{\alpha }{S}_{\beta }\left(s\right)z_0-J^{\alpha }{S}_{\beta }\left(t\right)A{(\mu -A)}^{-1}{z}_0=$$

$$=\mu {(\mu -A)}^{-1}{J}^{\alpha }{S}_{\beta }\left(s\right)z_0-{(\mu -A)}^{-1}{S}_{\beta }\left(t\right)z_0+g_{\beta +1}\left(t\right){(\mu -A)}^{-1}{z}_0.$$

(9)

Hence, ${(\mu -A)}^{-1}{S}_{\beta }\left(0\right)z_0=0$ for $\beta >0$, $S_{\beta }\left(0\right)z_0=0$. Additionally, it follows from (9) that $J^{\alpha }{S}_{\beta }\left(s\right)z_0\in D_A$ and

$$(\mu -A)J^{\alpha }{S}_{\beta }\left(s\right)z_0=\mu J^{\alpha }{S}_{\beta }\left(s\right)z_0-S_{\beta }\left(t\right)z_0+g_{\beta +1}\left(t\right)z_0.$$

Therefore, equality (7) is true.

If $x,y\in \mathcal{Z}$ such that (8) is valid, then

$${({\lambda }^{\alpha }-A)}^{-1}({\lambda }^{\alpha }x-y)={\lambda }^{\beta +1}{\widehat{S}}_{\beta }\left(\lambda \right)x-{\lambda }^{\beta -\alpha +1}{\widehat{S}}_{\beta }\left(\lambda \right)y=$$

$${\lambda }^{\beta +1}{\widehat{S}}_{\beta }\left(\lambda \right)x-{\lambda }^{\beta +1}{\widehat{J^{\alpha }S}}_{\beta }\left(\lambda \right)y={\lambda }^{\beta +1}{\widehat{g}}_{\beta +1}\left(\lambda \right)x=x.$$

Consequently, $x\in D_A$, ${\lambda }^{\alpha }x-y=({\lambda }^{\alpha }-A)x$ and $y=Ax$. □

Remark 6. 

It is known that the generators of β-integrated resolving functions may be not densely defined (see Remark 3.2.3 in [16] for $\alpha =1$, $\beta \in \mathbb{N}$), but due to assertion (v) of Lemma 2 $S_{\beta }\left(t\right)z_0=D^{\alpha }{J}^{\alpha }{S}_{\beta }\left(t\right)z_0\in {\overline{D}}_A$ for all $t\ge 0$, $z_0\in \mathcal{Z}$.

Remark 7. 

Apparently, there is no generalization of the semigroup property for the resolving functions in the case of $\alpha \ne 1$, $\beta =0$. Hence, there is no analogue of functional relation (3.9) ([16] p. 124) characterizing k-times integrated semigroups (the case of $\alpha =1$, $\beta =k\in \mathbb{N}$), and there is no generalization of Proposition 3.2.4 for $\alpha \ne 1$.

Theorem 1 

([16] (p. 81, Theorem 2.5.1)). Let $\mathcal{X}$ be a Banach space, $\omega \ge 0$, $q:\{\lambda \in \mathbb{C}:\mathrm{Re}\lambda >\omega \}\to \mathcal{X}$ be an analytic function with $\underset{\mathrm{Re}\lambda >\omega }{sup}{\parallel \lambda q\left(\lambda \right)\parallel }_{\mathcal{X}}<\infty$ and let $b>0$. Then, there exists $F\in C(\overline{\mathbb{R}};\mathcal{X})$ with $\underset{t>0}{sup}{\parallel e^{-\omega t}{t}^{-b}F\left(t\right)\parallel }_{\mathcal{X}}<\infty$ such that $q\left(\lambda \right)={\lambda }^b\widehat{F}\left(\lambda \right)$ for $\mathrm{Re}\lambda >\omega$.

Theorem 2. 

Let $\alpha >0$, $A\in \mathcal{C}l\left(\mathcal{Z}\right)$ and $\beta \ge 0$.

(i)

If $\omega \ge 0$, $K>0$, $b>0$ exists such that ${\lambda }^{\alpha }\in \rho \left(A\right)$ and the inequality $\parallel {\lambda }^{\alpha -1}{({\lambda }^{\alpha }-A)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K{|\lambda |}^{\beta -1-b}$ holds, whenever $\mathrm{Re}\lambda >\omega$, then A generates a β-integrated resolving function $S_{\beta }$ satisfying $\omega \left(S_{\beta }\right)\le \omega$.

(ii)

If A generates a β-integrated resolving function $S_{\beta }$ such that $\omega \left(S_{\beta }\right)<\infty$, then for $\omega >max\{\omega \left(S_{\beta }\right),0\}$, there exists K such that the inclusion ${\lambda }^{\alpha }\in \rho \left(A\right)$ and the inequality $\parallel {\lambda }^{\alpha -1}{({\lambda }^{\alpha }-A)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K{|\lambda |}^{\beta }$ are true whenever $\mathrm{Re}\lambda >\omega$.

Proof. 

Apply Theorem 1 for $q\left(\lambda \right)={\lambda }^{\alpha -1-\beta +b}{({\lambda }^{\alpha }-A)}^{-1}$. Then, a strongly continuous function $S_{\beta }:{\overline{\mathbb{R}}}_{+}\to \mathcal{L}\left(\mathcal{Z}\right)$ exists such that the equality ${\lambda }^{\alpha -1-\beta }{({\lambda }^{\alpha }-A)}^{-1}={\widehat{S}}_{\beta }\left(\lambda \right)$ is valid and assertion (i) is true. Moreover, there exists $K_1>0$ such that, for all $t\ge 0$$\parallel S_{\beta }{\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K_1{t}^b{e}^{\omega t}$.

Let $max\{\omega \left(S_{\beta }\right),0\}<{\omega }_1<\omega$. Then, there exists $K_1>0$ such that for all $t\ge 0$$\parallel S_{\beta }{\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K_1{e}^{{\omega }_{1}t}$. Therefore,

$$\parallel {\lambda }^{\alpha -1}{({\lambda }^{\alpha }-A)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}={\parallel {\lambda }^{\beta }{\widehat{S}}_{\beta }\left(\lambda \right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le \frac{K_1{|\lambda |}^{\beta }}{\mathrm{Re}\lambda -{\omega }_1}\le \frac{K_1{|\lambda |}^{\beta }}{\omega -{\omega }_1}$$

whenever $\mathrm{Re}\lambda >\omega$. □

3. Cauchy Problem for Equations with a Generator of $\mathbf{\beta }$-Integrated Function

Consider the Cauchy problem

$$D^{k}z\left(0\right)=z_k,k=0,1,\dots ,m-1,$$

(10)

$$D^{\alpha }z\left(t\right)=Az\left(t\right)+f\left(t\right),t\in [0,T],$$

(11)

where $T>0$, $z_k\in \mathcal{Z}$, $k=0,1,\dots ,m-1$, $f\in L_1(0,T;\mathcal{Z})$ and an operator A generates a $\beta$-integrated resolving function for $\beta \ge 0$.

By a mild solution of problem (10) and (11), we mean a function $z\in C\left(\right[0,T];\mathcal{Z})$ such that for all $t\in [0,T]$, $J^{\alpha }z\left(t\right)\in D_A$ and

$$z\left(t\right)=A{J}^{\alpha }z\left(t\right)+\sum _{k=0}^{m-1}{g}_{k+1}\left(t\right)z_k+J^{\alpha }f\left(t\right).$$

By a classical solution of (10) and (11), we understand a function $z\in C^{m-1}([0,T];\mathcal{Z})\cap C([0,T];D_A)$ such that $D^{\alpha }z\in C([0,T];\mathcal{Z})$, conditions (10) hold and equality (11) is valid for all $t\in [0,T]$.

Let $\lceil p\rceil$ be the minimal integer, which is equal to or greater than $p\in \mathbb{R}$.

Theorem 3. 

Let $m-1<\alpha \le m\in \mathbb{N}$, $n-1<\beta \le n\in \mathbb{N}$, $l=\lceil \frac{\beta }{\alpha }\rceil$, A generate a β-integrated resolving function $S_{\beta }$ such that $\omega \left(S_{\beta }\right)<\infty$, then the following assertions are valid.

(i)

If $z_k\in D_{A^l}$, $k=0,1,\dots ,m-1$, $f\equiv 0$, then there exists a unique mild solution to problem (10) and (11).

(ii)

If $z_k\in D_{A^{l+1}}$, $k=0,1,\dots ,m-1$, $f\equiv 0$, then there exists a unique classical solution to problem (10) and (11).

(iii)

If $\alpha \ge 1$, $z_k\in D_{A^l}$, $k=0,1,\dots ,m-1$, $f\in C([0,T];\mathcal{Z})\cap L_1(0,T;D_{A^l})$, then there exists a unique mild solution to problem (10) and (11).

(iv)

If $\alpha \ge 1$, $z_k\in D_{A^{l+1}}$, $k=0,1,\dots ,m-1$, $f\in C([0,T];\mathcal{Z})\cap L_1(0,T;D_{A^{l+1}})$, then there exists a unique classical solution to problem (10) and (11).

Proof. 

Due to Lemma 2 (iv) for $z_k\in D_{A^l}$

$$S_{\beta }\left(t\right)z_k=J^{\alpha }{S}_{\beta }\left(t\right)A{z}_k+g_{\beta +1}\left(t\right)z_k=J^{2\alpha }{S}_{\beta }\left(t\right)A^2{z}_k+g_{\beta +1}\left(t\right)z_k+g_{\alpha +\beta +1}\left(t\right)A{z}_k=$$

$$=\dots =J^{(l-1)\alpha }{S}_{\beta }\left(t\right)A^{l-1}{z}_k+\sum _{j=0}^{l-2}{g}_{j\alpha +\beta +1}\left(t\right)A^j{z}_k=J^{l\alpha }{S}_{\beta }\left(t\right)A^l{z}_k+\sum _{j=0}^{l-1}{g}_{j\alpha +\beta +1}\left(t\right)A^j{z}_k.$$

Hence, for $k=0,1,\dots ,m-1$

$$D^{\beta }{J}^k{S}_{\beta }\left(t\right)z_k=J^{(l-1)\alpha -\beta +k}{S}_{\beta }\left(t\right)A^{l-1}{z}_k+\sum _{j=0}^{l-2}{g}_{j\alpha +k+1}\left(t\right)A^j{z}_k=$$

$$=J^{l\alpha -\beta +k}{S}_{\beta }\left(t\right)A^l{z}_k+\sum _{j=0}^{l-1}{g}_{j\alpha +k+1}\left(t\right)A^j{z}_k=$$

$$=A{J}^{\alpha }\left(J^{(l-1)\alpha -\beta +k}{S}_{\beta }\left(t\right)A^{l-1}{z}_k+\sum _{j=0}^{l-2}{g}_{j\alpha +k+1}\left(t\right)A^j{z}_k\right)+g_{k+1}\left(t\right)z_k=$$

$$=A{J}^{\alpha }{D}^{\beta }{J}^k{S}_{\beta }\left(t\right)z_k+g_{k+1}\left(t\right)z_k.$$

Therefore, $z\left(t\right)=\sum _{k=0}^{m-1}{D}^{\beta }{J}^k{S}_{\beta }\left(t\right)z_k\in C([0,T];\mathcal{Z})$ is a mild solution to (10) and (11) with $f\equiv 0$. Let x be another mild solution of this problem. Then, $y=z-x$ satisfies the equality $y\left(t\right)=A{J}^{\alpha }y\left(t\right)$ for $t\in [0,T]$. Define $y\left(t\right)=0$ for $t>T$ and obtain $\widehat{y}\left(\lambda \right)={\lambda }^{-\alpha }A\widehat{y}\left(\lambda \right)$. Hence, for $\lambda >\omega$$\widehat{y}\left(\lambda \right)={({\lambda }^{\alpha }-A)}^{-1}0\equiv 0$ and $x\equiv z$.

For $z_k\in D_{A^{l+1}}$, we similarly have that

$$D^{\beta }{J}^k{S}_{\beta }\left(t\right)z_k=J^{l\alpha -\beta +k}{S}_{\beta }\left(t\right)A^l{z}_k+\sum _{j=0}^{l-1}{g}_{j\alpha +k+1}\left(t\right)A^j{z}_k=$$

$$=J^{(l+1)\alpha -\beta +k}{S}_{\beta }\left(t\right)A^{l+1}{z}_k+\sum _{j=0}^l{g}_{j\alpha +k+1}\left(t\right)A^j{z}_k,$$

$$D^{\alpha }{D}^{\beta }{J}^k{S}_{\beta }\left(t\right)z_k=J^{l\alpha -\beta +k}{S}_{\beta }\left(t\right)A^{l+1}{z}_k+\sum _{j=1}^l{g}_{(j-1)\alpha +k+1}\left(t\right)A^j{z}_k=$$

$$=A{J}^{l\alpha -\beta +k}{S}_{\beta }\left(t\right)A^l{z}_k+A\sum _{j=0}^{l-1}{g}_{j\alpha +k+1}\left(t\right)A^j{z}_k=A{D}^{\beta }{J}^k{S}_{\beta }\left(t\right)z_k,$$

$$D^k{|}_{t=0}{D}^{\beta }{J}^k{S}_{\beta }\left(t\right)z_k=D^k{{|}_{t=0}{J}^{l\alpha -\beta +k}{S}_{\beta }\left(t\right)A^l{z}_k+D^k|}_{t=0}\sum _{j=0}^{l-1}{g}_{j\alpha +k+1}\left(t\right)A^j{z}_k=$$

$$=J^{l\alpha -\beta }{S}_{\beta }\left(t\right)A^l{z}_k{|}_{t=0}+z_k=z_k,k=0,1,\dots ,m-1,$$

$$D^r{|}_{t=0}{D}^{\beta }{J}^k{S}_{\beta }\left(t\right)z_k=D^r{{|}_{t=0}{J}^{(l+1)\alpha -\beta +k}{S}_{\beta }\left(t\right)A^{l+1}{z}_k+D^r|}_{t=0}\sum _{j=0}^l{g}_{j\alpha +k+1}\left(t\right)A^j{z}_k=$$

$$=J^{(l+1)\alpha -\beta +k-r}{S}_{\beta }\left(t\right)A^{l+1}{z}_k{|}_{t=0}+0=0,r\in \{0,1,\dots ,m-1\}\setminus \left\{k\right\}.$$

Hence, $z\left(t\right)=\sum _{k=0}^{m-1}{D}^{\beta }{J}^k{S}_{\beta }\left(t\right)z_k\in C^{m-1}([0,T];\mathcal{Z})\cap C([0,T];D_A)$ is a classical solution to (10) and (11) with $f\equiv 0$. If two classical solutions exist, then there are two mild solutions. The previous assertion implies the uniqueness of a solution.

Let $f\in C([0,T];\mathcal{Z})\cap L_1(0,T;D_{A^l})$, then for $\alpha \ge 1$, $k=0,1,\dots ,n-1$

$$D^k{|}_{t=0}\underset{0}{\overset{t}{\int }}{J}^{\alpha -1}{S}_{\beta }(t-s)f\left(s\right)ds=$$

$$=D^k{{|}_{t=0}\underset{0}{\overset{t}{\int }}{J}^{(l+1)\alpha -1}{S}_{\beta }(t-s)A^{l}f\left(s\right)ds+D^k|}_{t=0}\underset{0}{\overset{t}{\int }}\sum _{j=0}^{l-1}{g}_{(j+1)\alpha +\beta }(t-s)A^{j}f\left(s\right)ds=$$

$$=\underset{0}{\overset{t}{\int }}{J}^{(l+1)\alpha -k-1}{S}_{\beta }(t-s)A^{l}f\left(s\right)ds{{|}_{t=0}+\underset{0}{\overset{t}{\int }}\sum _{j=0}^{l-1}{g}_{(j+1)\alpha +\beta -k}(t-s)A^{j}f\left(s\right)ds|}_{t=0}=0,$$

since

$${∥\underset{0}{\overset{t}{\int }}{g}_{\alpha +\beta -k}(t-s)f\left(s\right)ds∥}_{\mathcal{Z}}\le g_{\alpha +\beta -k+1}\left(t\right){\parallel f\parallel }_{C\left(\right[0,T];\mathcal{Z})}\to 0,t\to 0+.$$

(12)

Thus, due to Lemma 2 (iv)

$$D^{\beta }\underset{0}{\overset{t}{\int }}{J}^{\alpha -1}{S}_{\beta }(t-s)f\left(s\right)ds=D^n\underset{0}{\overset{t}{\int }}{J}^{\alpha -1+n-\beta }{S}_{\beta }(t-s)f\left(s\right)ds=$$

$$=D^n\underset{0}{\overset{t}{\int }}{J}^{l\alpha -\beta +n-1}{S}_{\beta }(t-s)A^{l-1}f\left(s\right)ds+D^n\underset{0}{\overset{t}{\int }}\sum _{j=0}^{l-2}{g}_{(j+1)\alpha +n}(t-s)A^{j}f\left(s\right)ds=$$

$$=\underset{0}{\overset{t}{\int }}{J}^{l\alpha -\beta -1}{S}_{\beta }(t-s)A^{l-1}f\left(s\right)ds+\underset{0}{\overset{t}{\int }}\sum _{j=0}^{l-2}{g}_{(j+1)\alpha }(t-s)A^{j}f\left(s\right)ds=$$

$$=\underset{0}{\overset{t}{\int }}{J}^{(l+1)\alpha -\beta -1}{S}_{\beta }(t-s)A^{l}f\left(s\right)ds+\underset{0}{\overset{t}{\int }}\sum _{j=0}^{l-1}{g}_{(j+1)\alpha }(t-s)A^{j}f\left(s\right)ds=$$

$$=A{J}^{\alpha }\left(\underset{0}{\overset{t}{\int }}{J}^{l\alpha -\beta -1}{S}_{\beta }(t-s)A^{l-1}f\left(s\right)ds+\underset{0}{\overset{t}{\int }}\sum _{j=0}^{l-2}{g}_{(j+1)\alpha }(t-s)A^{j}f\left(s\right)ds\right)+J^{\alpha }f\left(t\right),$$

consequently, $D^{\beta }\underset{0}{\overset{t}{\int }}{J}^{\alpha -1}{S}_{\beta }(t-s)f\left(s\right)ds\in C([0,T];\mathcal{Z})$ and

$$\sum _{k=0}^{m-1}{D}^{\beta }{J}^k{S}_{\beta }\left(t\right)z_k+D^{\beta }\underset{0}{\overset{t}{\int }}{J}^{\alpha -1}{S}_{\beta }(t-s)f\left(s\right)ds$$

is a mild solution to problem (10) and (11). Its uniqueness can be proven analogously to the homogeneous case.

If $f\in C([0,T];\mathcal{Z})\cap L_1(0,T;D_{A^{l+1}})$, then for $k=0,1,\dots ,m-1$

$$D^k{D}^{\beta }\underset{0}{\overset{t}{\int }}{J}^{\alpha -1}{S}_{\beta }(t-s)f\left(s\right)ds=$$

$$=D^k\underset{0}{\overset{t}{\int }}{J}^{(l+2)\alpha -\beta -1}{S}_{\beta }(t-s)A^{l+1}f\left(s\right)ds+D^k\underset{0}{\overset{t}{\int }}\sum _{j=0}^l{g}_{(j+1)\alpha }(t-s)A^{j}f\left(s\right)ds=$$

$$=\underset{0}{\overset{t}{\int }}{J}^{(l+2)\alpha -\beta -k-1}{S}_{\beta }(t-s)A^{l+1}f\left(s\right)ds+\underset{0}{\overset{t}{\int }}\sum _{j=0}^l{g}_{(j+1)\alpha -k}(t-s)A^{j}f\left(s\right)ds\in C([0,T];\mathcal{Z}),$$

consequently, taking into account (12), we have

$$D^k{|}_{t=0}{D}^{\beta }\underset{0}{\overset{t}{\int }}{J}^{\alpha -1}{S}_{\beta }(t-s)f\left(s\right)ds=0,$$

$$D^{\alpha }{D}^{\beta }\underset{0}{\overset{t}{\int }}{J}^{\alpha -1}{S}_{\beta }(t-s)f\left(s\right)ds=D^m{J}^{m-\alpha }{D}^{\beta }\underset{0}{\overset{t}{\int }}{J}^{\alpha -1}{S}_{\beta }(t-s)f\left(s\right)ds=$$

$$=D^m\underset{0}{\overset{t}{\int }}{J}^{(l+1)\alpha -\beta +m-1}{S}_{\beta }(t-s)A^{l+1}f\left(s\right)ds+D^m\underset{0}{\overset{t}{\int }}\sum _{j=0}^l{g}_{j\alpha +m}(t-s)A^{j}f\left(s\right)ds=$$

$$=\underset{0}{\overset{t}{\int }}{J}^{(l+1)\alpha -\beta -1}{S}_{\beta }(t-s)A^{l+1}f\left(s\right)ds+\underset{0}{\overset{t}{\int }}\sum _{j=1}^l{g}_{j\alpha }(t-s)A^{j}f\left(s\right)ds=$$

$$=A\left(\underset{0}{\overset{t}{\int }}{J}^{(l+1)\alpha -\beta -1}{S}_{\beta }(t-s)A^{l}f\left(s\right)ds+\underset{0}{\overset{t}{\int }}\sum _{j=0}^{l-1}{g}_{(j+1)\alpha }(t-s)A^{j}f\left(s\right)ds\right)=$$

$$=A{D}^{\beta }\underset{0}{\overset{t}{\int }}{J}^{\alpha -1}{S}_{\beta }(t-s)f\left(s\right)ds\in C([0,T];\mathcal{Z}).$$

Therefore,

$$\sum _{k=0}^{m-1}{D}^{\beta }{J}^k{S}_{\beta }\left(t\right)z_k+D^{\beta }\underset{0}{\overset{t}{\int }}{J}^{\alpha -1}{S}_{\beta }(t-s)f\left(s\right)ds$$

is a unique classical solution of problem (10) and (11). □

Lemma 3 

([16] (p. 130, Lemma 3.2.14). Let $\mathcal{X},\mathcal{Y}$ be Banach spaces and $V:{\overline{\mathbb{R}}}_{+}\to \mathcal{L}(\mathcal{X};\mathcal{Y})$. Assume that $V(·)x$ is exponentially bounded for all $x\in \mathcal{X}$. Then, there exist constants $M\ge 0$, $\omega \in \mathbb{R}$ such that ${\parallel V\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le M{e}^{\omega t}$ for all $t\ge 0$.

Theorem 4. 

Let $m-1<\alpha \le m\in \mathbb{N}$, $n-1<\beta \le n\in \mathbb{N}$, $A\in \mathcal{C}l\left(\mathcal{Z}\right)$. The following statements are equivalent:

(i)

A generates a β-integrated resolving function $S_{\beta }$ with $\omega \left(S_{\beta }\right)<\infty$.

(ii)

For all $z_0,z_1,\dots ,z_{m-1}\in \mathcal{Z}$, there exists a unique classical solution of problem

$$D^{k}y\left(0\right)=0,k=0,1,\dots ,m-1,$$

(13)

$$D^{\alpha }y\left(t\right)=Ay\left(t\right)+\sum _{k=0}^{m-1}{g}_{\beta +k+1}\left(t\right)z_k,t\in {\overline{\mathbb{R}}}_{+},$$

(14)

and it is exponentially bounded.

Proof. 

If A generates a $\beta$-integrated resolving function $S_{\beta }$, take

$$y\left(t\right)=J^{\alpha }\sum _{k=0}^{m-1}{J}^k{S}_{\beta }\left(t\right)z_k.$$

We have $K>0$, $\omega >max\{\omega \left(S_{\beta }\right),0\}$, such that for all $t\ge 0$$\parallel S_{\beta }{\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K{e}^{\omega t}$, hence, $\parallel J^{\alpha +k}{S}_{\beta }{\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K{e}^{\omega t}\frac{t^{\alpha +k}}{\Gamma (\alpha +k+1)}\le K_1{e}^{{\omega }_{1}t}$ for every ${\omega }_1>\omega$ and some $K_1>0$. By Lemma 2 (v)

$$D^{\alpha }y\left(t\right)=\sum _{k=0}^{m-1}{J}^k{S}_{\beta }\left(t\right)z_k=A\sum _{k=0}^{m-1}{J}^{k+\alpha }{S}_{\beta }\left(t\right)z_k+\sum _{k=0}^{m-1}{g}_{\beta +k+1}\left(t\right)z_k=$$

$$=Ay\left(t\right)+\sum _{k=0}^{m-1}{g}_{\beta +k+1}\left(t\right)z_k\in C([0,T];\mathcal{Z}),$$

conditions (13) obviously hold; hence, y is a classical solution of problem (13) and (14). Its uniqueness on every segment $[0,T]$ can be shown as in the proof of assertion (i) in Theorem 3; hence, the solution is unique on ${\overline{\mathbb{R}}}_{+}$. Consequently, statement (i) implies assertion (ii).

Let statement (ii) hold. Denote by $V(·)z_0$ the solution to problem (13) and (14) with an arbitrary $z_0\in \mathcal{Z}$ and $z_1=z_2=\dots =z_{m-1}=0$. The mapping $V\left(t\right):\mathcal{Z}\to D_A$ is evidently linear. Moreover, we have that $V\left(t\right)\in \mathcal{L}(\mathcal{Z};D_A)$. Indeed, the space $C({\mathbb{R}}_{+};D_A)$ is a Fréchet space with the seminorms $p_m\left(f\right):={sup\{\parallel f\left(t\right)\parallel }_{D_A}:\frac{1}{m}\le t\le m\}$. Define a mapping $\Phi :\mathcal{Z}\to C({\mathbb{R}}_{+};D_A)$ by $\Phi \left(z_0\right)=V(·)z_0$. Let $z_{0n}\to z_0$ in $\mathcal{Z}$ and $V(·)z_{0n}\to y$ in $C({\mathbb{R}}_{+};D_A)$ as $n\to \infty$. Hence, for $t>0$ $J^{\alpha }V\left(t\right)z_{0n}\to J^{\alpha }V\left(t\right)z_0$ as $n\to \infty$. Since $A{J}^{\alpha }V\left(t\right)z_{0n}=V\left(t\right)z_{0n}-g_{\alpha +\beta +1}\left(t\right)z_{0n}$ and A is closed, it follows that $J^{\alpha }V\left(t\right)z_0\in D_A$ and $A{J}^{\alpha }V\left(t\right)z_0=\underset{n\to \infty }{lim}(V\left(t\right)z_{0n}-g_{\alpha +\beta +1}\left(t\right)z_{0n})=V\left(t\right)z_0-g_{\alpha +\beta +1}\left(t\right)z_0$. Thus, $V\left(t\right)z_0$ is a solution of (13) and (14) and $V(·)z_0=\Phi \left(z_0\right)$. Then, $\Phi$ is closed and due to the closed graph theorem $\Phi$ is continuous. In particular, the mapping $\mathcal{Z}\to D_A$, $z_0\to V\left(t\right)z_0$ is continuous for $t>0$. The assumption together with Lemma 3 imply that, for suitable constants $K>0$, $\omega \ge 0$ ${\parallel V\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K{e}^{\omega t}$ for all $t\ge 0$. Therefore, $Q\left(\lambda \right)z_0:={\lambda }^{\beta +1}\widehat{V(·)z_0}\left(\lambda \right)$ is well defined for $\lambda >\omega$, since

$$\parallel J^{\alpha }AV\left(t\right)z_0{\parallel }_{\mathcal{Z}}=∥V\left(t\right)z_0-g_{\alpha +\beta +1}\left(t\right)z_0∥\le \left(K{e}^{\omega t}+g_{\alpha +\beta +1}\left(t\right)\right){\parallel z_0\parallel }_{\mathcal{Z}}.$$

There exists $\widehat{J^{\alpha }AV(·)z_0}\left(\lambda \right)$ and $\widehat{D^{\alpha }{J}^{\alpha }AV(·)z_0}\left(\lambda \right)=\widehat{AV(·)z_0}\left(\lambda \right)$, consequently, $Q\left(\lambda \right)z_0\in D_A$ for all $z_0\in \mathcal{Z}$, all $\lambda >\omega$ and

$$({\lambda }^{\alpha }-A)Q\left(\lambda \right)z_0={\lambda }^{\alpha +\beta +1}\widehat{V(·)z_0}\left(\lambda \right)-{\lambda }^{\beta +1}\widehat{AV(·)z_0}\left(\lambda \right)=$$

$$={\lambda }^{\alpha +\beta +1}\widehat{V(·)z_0}\left(\lambda \right)-{\lambda }^{\beta +1}\widehat{D^{\alpha }V(·)z_0}+{\lambda }^{\beta +1}\widehat{g_{\beta +1}}\left(\lambda \right)z_0=z_0$$

(15)

for $\lambda >\omega$. In order to show that ${\lambda }^{\alpha }-A$ is injective for $\lambda >\omega$, assume that $({\lambda }^{\alpha }-A)z_0=0$ for some $z_0\in D_A$ and $\lambda >\omega$. Then, the solution $V\left(t\right)z_0$ of (13), (14) is given by

$$V\left(t\right)z_0=\underset{0}{\overset{t}{\int }}{E}_{\alpha ,\alpha }\left({\lambda }^{\alpha }{s}^{\alpha }\right)g_{\beta +1}(t-s)z_{0}ds=\sum _{k=0}^{\infty }\frac{\Gamma (\alpha k+1){\lambda }^{\alpha k}{t}^{\alpha k+\beta +1}}{\Gamma (\alpha k+\alpha )\Gamma (\alpha k+\beta +2)}.$$

Hence, for some $c>0$ and for all $t\ge 0$, $\lambda >\omega$

$$c{t}^{\beta +1}{E}_{\alpha ,\beta +2}\left({\lambda }^{\alpha }{t}^{\alpha }\right)\parallel z_0{\parallel }_{\mathcal{Z}}\le \parallel V\left(t\right)z_0{\parallel }_{\mathcal{Z}}\le K{e}^{\omega t}{\parallel z_0\parallel }_{\mathcal{Z}},$$

$$c{t}^{\beta +1}{E}_{\alpha ,\beta +2}\left({\lambda }^{\alpha }{t}^{\alpha }\right)e^{-\omega t}\parallel z_0{\parallel }_{\mathcal{Z}}=\frac{c}{\alpha }{\lambda }^{-\beta -1}{e}^{(\lambda -\omega )t}{\parallel z_0\parallel }_{\mathcal{Z}}+$$

$$+e^{-\omega t}O\left({\left(\lambda t\right)}^{-\alpha }\right)\parallel z_0{\parallel }_{\mathcal{Z}}\le K{\parallel z_0\parallel }_{\mathcal{Z}}.$$

Here, we use asymptotic formula [9] for the Mittag–Leffler function. The last inequality is possible, if $z_0=0$ only. Hence, due to (15) ${({\lambda }^{\alpha }-A)}^{-1}=Q\left(\lambda \right)$ for $\lambda >\omega$ and $D^{\alpha }V$ is a $\beta$-integrated resolving function generated by A. Indeed, $S_{\beta }\left(t\right)z_0:=D^{\alpha }V\left(t\right)z_0=AV\left(t\right)z_0+\frac{t^{\beta }{z}_0}{\Gamma (\beta +1)}$ exists for all $t>0$ and all $z_0\in \mathcal{Z}$ and $D^{k}V\left(0\right)=0$, $k=0,1,\dots ,m-1$. Therefore, ${\widehat{S}}_{\beta }\left(\lambda \right)={\lambda }^{\alpha -\beta -1}Q\left(\lambda \right)={\lambda }^{\alpha -\beta -1}{({\lambda }^{\alpha }-A)}^{-1}$, so $S_{\beta }$ is a $\beta$-integrated resolving function generated by A. Consequently, assertion (ii) implies (i). □

4. Criterion of Existence for $\mathbf{\beta }$-Integrated Resolving Function

Herein, we will use the next two statements.

Theorem 5 

([16] (p. 78, Theorem 2.4.1)). Let $K>0$, $\omega \in \mathbb{R}$. If $\mathcal{X}$ has the Radon–Nikodym property, then for any $r\in C^{\infty }((\omega ,\infty );\mathcal{X})$ the following assertions are equivalent:

(i)

${\parallel (\lambda -\omega )}^{k+1}\frac{1}{k!}{r}^{\left(k\right)}{\left(\lambda \right)\parallel }_{\mathcal{X}}\le K$ for all $\lambda >\omega ,k\in {\mathbb{N}}_0.$

(ii)

There exists $G:{\overline{\mathbb{R}}}_{+}\to \mathcal{X}$ such that $G\left(0\right)=0$, ${\parallel G\left(t\right)-G\left(s\right)\parallel }_{\mathcal{X}}\le K\underset{s}{\overset{t}{\int }}{e}^{\omega \tau }d\tau$, $0\le s\le t$, and $r\left(\lambda \right)=\widehat{dG}\left(\lambda \right)$ for all $\lambda >\omega$.

Theorem 6 

([16] (p. 78, Theorem 2.4.2)). Let $K>0$, $\omega \in \mathbb{R}$. For any $r\in C^{\infty }((\omega ,\infty );\mathcal{X})$, the following assertions are equivalent:

(i)

${\parallel (\lambda -\omega )}^{k+1}\frac{1}{k!}{r}^{\left(k\right)}{\left(\lambda \right)\parallel }_{\mathcal{X}}\le K$ for all $\lambda >\omega ,k\in {\mathbb{N}}_0.$

(ii)

There exists $g\in L_{1,\mathrm{loc}}({\mathbb{R}}_{+};\mathcal{X})$ such that ${\parallel g\left(t\right)\parallel }_{\mathcal{X}}\le K{e}^{\omega t}$ for almost all $t\ge 0$ and $r\left(\lambda \right)=\widehat{g}\left(\lambda \right)$ for all $\lambda >\omega$.

Theorem 7. 

Let $A\in \mathcal{C}l\left(\mathcal{Z}\right)$, $\alpha >0$, $\beta ,\omega \ge 0$, $K>0$. Then, the following statements are equivalent:

(i)

$({\omega }^{\alpha },\infty )\subset \rho \left(A\right)$ and for all $\lambda >\omega$, $k\in {\mathbb{N}}_0$

$${∥\frac{{(\lambda -\omega )}^{k+1}}{k!}{\left[{\lambda }^{\alpha -\beta -1}{({\lambda }^{\alpha }-A)}^{-1}\right]}^{\left(k\right)}∥}_{\mathcal{L}\left(\mathcal{Z}\right)}\le K.$$

(ii)

A generates a $(\beta +1)$-integrated resolving function $S_{\beta +1}$ on $\mathcal{Z}$ satisfying

$$\parallel S_{\beta +1}\left(t\right)-S_{\beta +1}{\left(s\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K\underset{s}{\overset{t}{\int }}{e}^{\omega \tau }d\tau ,0\le s\le t.$$

(16)

Proof. 

Let assertion (i) be true. From Theorem 6, it follows that there is $S_{\beta }\in L_{1,\mathrm{loc}}({\mathbb{R}}_{+};\mathcal{Z})$ with $\parallel S_{\beta }{\left(t\right)\parallel }_{\mathcal{Z}}\le K{e}^{\omega t}$ for almost all $t\ge 0$ such that ${\lambda }^{\alpha -1}{({\lambda }^{\alpha }-A)}^{-1}={\lambda }^{\beta }{\widehat{S}}_{\beta }\left(\lambda \right)$ for all $\lambda >\omega$. Then, $S_{\beta +1}=J^1{S}_{\beta }$ is a strongly continuous function, which is generated by the A$(\beta +1)$-integrated resolving function satisfying (16).

Conversely, assume that (ii) holds. By the definition of the $(\beta +1)$-integrated resolving function $S_{\beta +1}$, there exists ${\omega }_1\ge \omega$ such that ${\lambda }^{\alpha -1}{({\lambda }^{\alpha }-A)}^{-1}={\lambda }^{\beta +1}{\widehat{S}}_{\beta +1}\left(\lambda \right)$ for all $\lambda >{\omega }_1$. Then, inequality (16) also holds with a constant $\omega ={\omega }_1$. Without limiting generality, we can assume that $\omega ={\omega }_1>0$. Take $G\left(t\right)=S_{\beta +1}\left(t\right)$, $t\ge 0$; due to Lemma 2 (v) $S_{\beta +1}\left(0\right)=0$, hence,

$$\parallel S_{\beta +1}{\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K\underset{0}{\overset{t}{\int }}{e}^{\omega \tau }d\tau =\frac{K}{\omega }(e^{\omega t}-1)\le K_1{e}^{\omega t},K_1>0,$$

and $\omega \left(S_{\beta +1}\right)\le \omega$. Due to the equality

$${\widehat{dS}}_{\beta +1}\left(\lambda \right):=\underset{0}{\overset{\infty }{\int }}{e}^{-\lambda s}d{S}_{\beta +1}\left(s\right)=-S_{\beta +1}\left(0\right)+\lambda {\widehat{S}}_{\beta +1}\left(\lambda \right),\mathrm{Re}\lambda >\omega \left(S_{\beta +1}\right),$$

we have ${\widehat{dS}}_{\beta +1}\left(\lambda \right)=\lambda {\widehat{S}}_{\beta +1}\left(\lambda \right)={\lambda }^{\alpha -\beta -1}{({\lambda }^{\alpha }-A)}^{-1}$ and assertion (i) follows from Theorem 5. □

Define

$${\mathrm{Lip}}_{\omega }({\overline{\mathbb{R}}}_{+};\mathcal{Z}):=\left\{G:{\overline{\mathbb{R}}}_{+}\to \mathcal{Z}:G\left(0\right)=0,{\parallel G\parallel }_{{\mathrm{Lip}}_{\omega }({\overline{\mathbb{R}}}_{+};\mathcal{Z})}<\infty \right\},$$

$${\parallel G\parallel }_{{\mathrm{Lip}}_{\omega }({\overline{\mathbb{R}}}_{+};\mathcal{Z})}:=\underset{t>s\ge 0}{sup}\frac{{\parallel G\left(t\right)-G\left(s\right)\parallel }_{\mathcal{Z}}}{\underset{s}{\overset{t}{\int }}{e}^{\omega r}dr},C_{\omega }^1({\overline{\mathbb{R}}}_{+};\mathcal{Z})=C^1({\overline{\mathbb{R}}}_{+};\mathcal{Z})\cap {\mathrm{Lip}}_{\omega }({\overline{\mathbb{R}}}_{+};\mathcal{Z}).$$

Lemma 4 

([16] (p. 133, Lemma 3.3.3)). For $\omega \in \mathbb{R}$, the space $C_{\omega }^1({\overline{\mathbb{R}}}_{+};\mathcal{Z})$ is a closed subspace of ${\mathrm{Lip}}_{\omega }({\overline{\mathbb{R}}}_{+};\mathcal{Z})$. In particular, if $S\in {\mathrm{Lip}}_{\omega }({\overline{\mathbb{R}}}_{+};\mathcal{L}\left(\mathcal{Z}\right))$, then $\{z_0\in \mathcal{Z}:S(·)z_0\in C^1({\overline{\mathbb{R}}}_{+};\mathcal{Z})\}$ is a closed subspace of $\mathcal{Z}$.

Theorem 8. 

Let $A\in \mathcal{C}ld\left(\mathcal{Z}\right)$, $\alpha \ge 1$, $\beta ,\omega \ge 0$, $K>0$. Then, the following assertions are equivalent:

(i)

$({\omega }^{\alpha },\infty )\subset \rho \left(A\right)$ and for all $\lambda >\omega$, $k\in {\mathbb{N}}_0$

$${∥\frac{{(\lambda -\omega )}^{k+1}}{k!}{\left[{\lambda }^{\alpha -\beta -1}{({\lambda }^{\alpha }-A)}^{-1}\right]}^{\left(k\right)}∥}_{\mathcal{L}\left(\mathcal{Z}\right)}\le K.$$

(ii)

A generates a β-integrated resolving function $S_{\beta }$ on $\mathcal{Z}$ satisfying

$$\parallel S_{\beta }{\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le K{e}^{\omega t},t\ge 0.$$

Proof. 

Assume that statement (ii) holds. Then, A also generates a $(\beta +1)$-integrated resolving function $S_{\beta +1}=J^1{S}_{\beta }$ on $\mathcal{Z}$ which satisfies the assertion (ii) of Theorem 7. Hence, statement (i) follows from that theorem.

Conversely, assume that (i) holds. By Theorem 7, A generates a $(\beta +1)$-integrated resolving function $S_{\beta +1}$ on $\mathcal{Z}$ such that $S_{\beta +1}\in {\mathrm{Lip}}_{\omega }({\overline{\mathbb{R}}}_{+};\mathcal{L}\left(\mathcal{Z}\right))$ with $\parallel S_{\beta +1}{\parallel }_{{\mathrm{Lip}}_{\omega }({\overline{\mathbb{R}}}_{+};\mathcal{L}\left(\mathcal{Z}\right))}\le K$. By Lemma 2 (iv), under condition $\alpha \ge 1$ $S_{\beta }\left(t\right)z_0:=D^1{S}_{\beta +1}\left(t\right)z_0=J^{\alpha -1}{S}_{\beta +1}\left(t\right)A{z}_0+g_{\beta +1}\left(t\right)$ exists for all $z_0\in D_A$. Thus, $t\to S_{\beta }\left(t\right)z_0$ is continuous on ${\overline{\mathbb{R}}}_{+}$ for $\alpha \ge 1$, $\beta \ge 0$. By Lemma 4, the definition of $S_{\beta }\left(t\right)z_0$ given by this way is also meaningful for $z_0\in {\overline{D}}_A$ and $t\to S_{\beta }\left(t\right)z_0$ is also continuous for $z_0\in {\overline{D}}_A$. By assumption ${\overline{D}}_A=\mathcal{Z}$, therefore A is the generator of the $\beta$-integrated resolving function $S_{\beta }$ on $\mathcal{Z}$ which is exponentially bounded due to inequality (16) for $S_{\beta +1}$. □

Remark 8. 

The implication from (ii) to (i) is valid for arbitrary $\alpha >0$ and for $A\in \mathcal{C}l\left(\mathcal{Z}\right)$.

5. $\mathbf{\beta }$-Integrated Resolving Function for the Time-Fractional Schrödinger Equation

Let $d\in \mathbb{N}$, $\Omega \subset {\mathbb{R}}^d$ be a bounded region with a smooth boundary $\partial \Omega$. The initial boundary value problem

$$w(\xi ,0)=w_0\left(\xi \right),D_t^{1}w(\xi ,0)=w_1\left(\xi \right),\xi \in \Omega ,$$

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

(17)

for the linear time-fractional Schrödinger equation

$$D^{\alpha }w(\xi ,t)=i\Delta w(\xi ,t),(\xi ,t)\in \Omega \times {\overline{\mathbb{R}}}_{+},$$

(18)

with $1<\alpha \le 2$ can be reduced to problem (4), (5), if we take $\mathcal{Z}=L_2(\Omega )$, the unbounded operator $A\in \mathcal{C}ld\left(\mathcal{Z}\right)$ is defined as $Av=i\Delta v$, $D_A:=\{v\in H^2(\Omega ):v\left(\xi \right)=0,\xi \in \partial \Omega \}$. It is known that $\sigma \left(A\right)=\{i{\lambda }_k:k\in \mathbb{N}\}$, where ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_k\ge \dots$ are the real non-positive eigenvalues of the corresponding Laplace operator numbered in ascending order, taking into account their multiplicities. Let $\left\{{\phi }_k\right\}$ be the corresponding eigenfunctions of A, which form an orthonormal basis in $L_2(\Omega )$.

The $\beta$-resolving function of Equation (18) with boundary condition (17) for $\beta \ge 0$ has the form

$$S_{\beta }\left(t\right)z_0=\sum _{k=1}^{\infty }{t}^{\beta }{E}_{\alpha ,\beta +1}\left(t^{\alpha }i{\lambda }_k\right)\langle z_0,{\phi }_k\rangle {\phi }_k,z_0\in L_2(\Omega ),$$

where $\langle ·,·\rangle$ is the inner product in $L_2(\Omega )$. Indeed, for $\lambda >0$ using the Mittag–Leffler function definition $E_{\alpha ,\beta +1}\left(z\right):=\sum _{l=0}^{\infty }\frac{z^l}{\Gamma (\alpha l+\beta +1)}$ and the equality $\widehat{t^{\alpha l+\beta }}=\Gamma (\alpha l+\beta +1){\lambda }^{-\alpha l-\beta -1}$ we have

$$\widehat{S_{\beta }(·)z_0}\left(\lambda \right)=\sum _{k=1}^{\infty }\sum _{l=0}^{\infty }\frac{i^l{\lambda }_k^l}{{\lambda }^{\alpha l+\beta +1}}\langle z_0,{\phi }_k\rangle {\phi }_k={\lambda }^{-\beta -1}\sum _{k=1}^{\infty }\frac{\langle z_0,{\phi }_k\rangle {\phi }_k}{1-{\lambda }^{-\alpha }i{\lambda }_k}=$$

$$={\lambda }^{\alpha -\beta -1}\sum _{k=1}^{\infty }\frac{\langle z_0,{\phi }_k\rangle {\phi }_k}{{\lambda }^{\alpha }-i{\lambda }_k}={\lambda }^{\alpha -\beta -1}{({\lambda }^{\alpha }-A)}^{-1}.$$

Thus, the conditions of Definition 2 are satisfied.

6. Conclusions

The case is investigated when an unbounded operator at the unknown function in a linear equation resolved with respect to the Gerasimov–Caputo derivative does not generate a resolving function (a resolving family of operators), but satisfies weaker conditions sufficient to generate a $\beta$-integrated resolving function. Conditions for the unique solvability of the equation in the sense of classical and mild solutions are obtained. Necessary and sufficient conditions for the existence of a $\beta$-integrated function is found in terms of the resolvent of an operator A. These results are planned to be extended to multi-term equations with Gerasimov–Caputo derivatives, equations with Riemann–Liouville, Hilfer, and Dzhrbashyan–Nersesyan fractional derivatives and distributed derivatives.