Generators of Analytic Resolving Families for Distributed Order Equations and Perturbations

Abstract

Linear differential equations of a distributed order with an unbounded operator in a Banach space are studied in this paper. A theorem on the generation of analytic resolving families of operators for such equations is proved. It makes it possible to study the unique solvability of inhomogeneous equations. A perturbation theorem for the obtained class of generators is proved. The results of the work are illustrated by an example of an initial boundary value problem for the ultraslow diffusion equation with the lower-order terms with respect to the spatial variable.

1. Introduction

Equations with distributed order derivatives appear in various applied problems concerning certain physical or technical processes—for example, when processes are described by equations with fractional derivatives, the order of which depends on the process parameters: in the theory of viscoelasticity [1], in kinetic theory [2], for modelling diffusion with a logarithmic growth of the mean square displacement (ultraslow diffusion) [3] and so on (e.g., [4,5,6,7]). In the last several decades, numerical solutions searching for such equations were performed in [8,9] and many other papers. At the same time, in many scientific works equations with distributed fractional derivatives began to be investigated from the mathematical point of view: unique solvability, a qualitative behaviour of solutions. In this sense, I note the works of Nakhushev [4,10] on properties of distributed order differential operators; of Pskhu [11,12] on the solvability and qualitative properties of both ordinary differential equations of distributed order, and the diffusion equation of a distributed order in time; of Umarov and Gorenflo [13] on the unique solvability of multipoint problems, including the Cauchy problem, to the equation with a distributed Caputo derivative in time and with pseudodifferential operators with respect to the space variables; of Atanacković, Oparnica and Pilipović (see [14] and others) on the existence and the uniqueness of mild and classical solutions for a class of nonlinear distributed order equations, which arise in distributed derivatives models of viscoelasticity and system identification theory; and of Kochubei [3] on the properties of the ultraslow diffusion equation and initial boundary value problems for it.

Consider the distributed order equation

$$\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }z\left(t\right)d\alpha =Az\left(t\right)+g\left(t\right),t\in (0,T],$$

(1)

with the Gerasimov–Caputo derivative $D_t^{\alpha }$ and with a linear operator A in a Banach space $\mathcal{Z}$, $m-1<b\le m\in \mathbb{N}$, $\omega :[0,b]\to \mathbb{C}$, $T>0$, $g:[0,T]\to \mathcal{Z}$. In [15,16], this equation was studied in the case of a bounded operator A. Initial value problems for Equation (1) and for an analogous equation with a linear degenerate operator at the distributed order derivative are researched in [17,18] in the cases of sectorial operators and of the Gerasimov-Caputo or the Riemann-Liouville fractional derivatives under the integral in the left-hand side of the equation.

In the case of $b\in (0,1]$ necessary and sufficient conditions on, generally speaking, an unbounded operator A for the existence of an analytic resolving family of operators for the homogeneous equation

$$\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }z\left(t\right)d\alpha =Az\left(t\right),t>0,$$

(2)

were obtained in [19]. Under these conditions, a unique solvability theorem for the inhomogeneous Equation (1) were proved, and the obtained results were applied to the study of initial boundary value problems for a class of partial differential equations of distributed order with respect to time. In the present paper we generalised the results of [19] on generators of analytic resolving families of operators and the unique solvability of inhomogeneous Equation (1) for the case of arbitrary $b>0$. Moreover, a theorem on perturbations of generators for distributed order Equation (2) is proved. The obtained results are an extension of the analytic semigroup of operators theory to the case of distributed order equations.

In the second section, the generation theorem for analytic resolving families of operators for the distributed order Equation (2) is proved. It is applied to the study of the inhomogeneous Equation (1) in the third section. In the fourth section the perturbation theorem for generators of the resolving families of operators for Equation (2) is obtained. The abstract results are illustrated by an example of an initial boundary value problem for the ultraslow diffusion equation with the lower-order terms with respect to the spatial variable.

2. Generators of Resolving Families of Operators

Denote for $\beta >0$, $t>0$, ${\overline{\mathbb{R}}}_{+}:={\mathbb{R}}_{+}\cup \left\{0\right\}$, $h:{\overline{\mathbb{R}}}_{+}\to \mathcal{Z}$,

$$J_t^{\beta }h\left(t\right):=\frac{1}{\Gamma \left(\beta \right)}\underset{0}{\overset{t}{\int }}{(t-s)}^{\beta -1}h\left(s\right)ds.$$

Let $m-1<\alpha \le m\in \mathbb{N}$, $D_t^m$ be the usual derivative of the m-th order, $D_t^{\alpha }$ be the Gerasimov–Caputo derivative [20,21,22]:

$$D_t^{\alpha }h\left(t\right):=D_t^m{J}_t^{m-\alpha }\left(h\left(t\right)-\sum _{k=0}^{m-1}{h}^{\left(k\right)}\left(0\right)\frac{t^k}{k!}\right).$$

Let $\mathcal{Z}$ be a Banach space. The Laplace transform of a function $h:{\mathbb{R}}_{+}\to \mathcal{Z}$ will be denoted by $\mathrm{Lap}\left[h\right]$. By $\mathrm{Lap}\left(\mathcal{Z}\right)$ denote the set of functions $h:{\mathbb{R}}_{+}\to \mathcal{Z}$, such that the Laplace transform $\mathrm{Lap}\left[h\right]$ is defined. The Laplace transform of the Gerasimov–Caputo derivative of the order $\alpha >0$ satisfies the equality (see [23])

$$\mathrm{Lap}\left[D_t^{\alpha }h\right]\left(\lambda \right)={\lambda }^{\alpha }\mathrm{Lap}\left[h\right]\left(\lambda \right)-\sum _{k=0}^{m-1}{h}^{\left(k\right)}\left(0\right){\lambda }^{\alpha -1-k}.$$

(3)

We introduce also the notations $S_{\theta ,a}:=\{\mu \in \mathbb{C}:|arg(\mu -a)|<\theta ,\mu \ne a\}$ at $\theta \in (\pi /2,\pi ]$, $a\in \mathbb{R}$, ${\Sigma }_{\psi }:=\{t\in \mathbb{C}:|argt|<\psi ,t\ne 0\}$ at $\psi \in (0,\pi /2]$.

Theorem 1.

([24], Theorem 0.1, p. 5), ([25], Theorem 2.6.1, p. 84). Let ${\theta }_0\in (\pi /2,\pi ]$, $a\in \mathbb{R}$, $\mathcal{Z}$ be a Banach space, a map $H:(a,\infty )\to \mathcal{Z}$ be set. The next assertions are equivalent.

(i) 

There exists an analytic function $F:{\Sigma }_{{\theta }_0-\pi /2}\to \mathcal{Z}$, for every $\theta \in (\pi /2,{\theta }_0)$ there exists such $C\left(\theta \right)>0$, that for all $t\in {\Sigma }_{\theta -\pi /2}$ the inequality ${\parallel F\left(t\right)\parallel }_{\mathcal{Z}}\le C\left(\theta \right)e^{a\mathrm{Re}t}$ is satisfied; $\mathrm{Lap}\left[F\right]\left(\lambda \right)=H\left(\lambda \right)$ at $\lambda >a$.

(ii) 

The map H is analytically continued on $S_{{\theta }_0,a}$, for every $\theta \in (\pi /2,{\theta }_0)$ there exists such $K\left(\theta \right)>0$, such that for all $\lambda \in S_{\theta ,a}$

$${\parallel H\left(\lambda \right)\parallel }_{\mathcal{Z}}\le \frac{K\left(\theta \right)}{|\lambda -a|}.$$

Denote by $\mathcal{L}\left(\mathcal{Z}\right)$ the Banach space of all linear continuous operators from $\mathcal{Z}$ to $\mathcal{Z}$ denote by $\mathcal{C}l\left(\mathcal{Z}\right)$ the set of all linear closed operators, densely defined in $\mathcal{Z}$, acting in the space $\mathcal{Z}$. We supply the domain $D_A$ of an operator $A\in \mathcal{C}l\left(\mathcal{Z}\right)$ by the norm of its graph and, thus, we get the Banach space. Consider the Cauchy problem

$$z\left(0\right)=z_0,z^{\left(k\right)}\left(0\right)=0,k=1,2,\dots ,m-1,$$

(4)

for a distributed order equation

$$\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }z\left(t\right)d\alpha =Az\left(t\right),t>0,$$

(5)

where $0\le m-1<b\le m\in \mathbb{N}$, $\omega :[0,b]\to \mathbb{C}$. By solution of problem Equations (4) and (5) we call such function $z\in C^{m-1}({\overline{\mathbb{R}}}_{+};\mathcal{Z})\cap C({\mathbb{R}}_{+};D_A)$, that there exists $\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }z\left(t\right)d\alpha \in C({\mathbb{R}}_{+};\mathcal{Z})$ and equalities (4) and (5) are fulfilled.

If in the original problem the integral is taken from some $c\in (0,b)$ until b, then the corresponding equation can also be written as (5), putting $\omega \left(\alpha \right)\equiv 0$ for $\alpha \in (0,c)$.

Denote

$$W\left(\lambda \right):=\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right){\lambda }^{\alpha }d\alpha .$$

Lemma 1.

[19]. Let $b>0$, $\omega \in L_1(0,b)$. Then, W is analytic on the set $\mathbb{C}\backslash \{\lambda \in \mathbb{C}:\mathrm{Re}\lambda \le 0\}$.

Lemma 2.

Let $0\le m-1<b\le m\in \mathbb{N}$, ω be a piecewise continuous function on $(0,b)$, which be continuous from the left at the point b, $\omega \left(b\right)\ne 0$. Then there exist ${\theta }_0\in (\pi /2,\pi ]$, $a_0\ge 0$, such that

$$\exists C>0\exists \epsilon \in (0,1)\exists \varrho >0\forall \lambda \in S_{{\theta }_0,a_0}\backslash \{\lambda \in \mathbb{C}:|\lambda |<\varrho \}\left|W\left(\lambda \right)\right|\ge {C\left|\lambda \right|}^{m-1+\epsilon }.$$

(6)

Proof. 

At $c>0$, which is close enough to b, we have for some $C_1=C_1\left({\epsilon }_1\right)>0$ and for all sufficiently large $\left|\lambda \right|$

$$\left|\underset{c}{\overset{b}{\int }}\omega \left(\alpha \right){\lambda }^{\alpha }d\alpha \right|=\left|\omega \left(\xi \right)\right|\left|\underset{c}{\overset{b}{\int }}{\lambda }^{\alpha }d\alpha \right|=\left|\omega \left(\xi \right)\right|\left|\frac{{\lambda }^b-{\lambda }^c}{ln\lambda }\right|\ge C_1{\left|\lambda \right|}^{{\epsilon }_1}$$

for some $\xi \in (c,b)$ and every ${\epsilon }_1\in (c,b)$. Thus, at $\epsilon \in (c,{\epsilon }_1)$ for some $C>0$ and sufficiently large $\left|\lambda \right|$

$$\left|\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right){\lambda }^{\alpha }d\alpha \right|\ge C_1{\left|\lambda \right|}^{{\epsilon }_1}-{\left|\lambda \right|}^c\underset{0}{\overset{c}{\int }}\left|\omega \left(\alpha \right)\right|d\alpha \ge {C\left|\lambda \right|}^{\epsilon }.$$

 □

A family of operators $\left\{S\right(t)\in \mathcal{L}(\mathcal{Z}):t\ge 0\}$ is called resolving for Equation (5), if the next conditions are satisfied:

(i)

$S\left(t\right)$ is strongly continuous at $t\ge 0$, $S\left(0\right)=I$;

(ii)

$S\left(t\right)\left[D_A\right]\subset D_A$, $S\left(t\right)Ax=AS\left(t\right)x$ for all $x\in D_A$, $t\ge 0$;

(iii)

$S\left(t\right)z_0$ is a solution of Cauchy problem Equations (4) and (5) for all $z_0\in D_A$.

A resolving family of operators is called analytic if it has the analytic continuation to a sector ${\Sigma }_{{\psi }_0}$ at some ${\psi }_0\in (0,\pi /2]$. An analytic resolving family of operators $\left\{S\right(t)\in \mathcal{L}(\mathcal{Z}):t\ge 0\}$ has a type $({\psi }_0,a_0)$ at some ${\psi }_0\in (0,\pi /2]$, $a_0\in \mathbb{R}$, if for all $\psi \in (0,{\psi }_0)$, $a>a_0$ there exists such $C(\psi ,a)$, that for all $t\in {\Sigma }_{\psi }$ the inequality ${\parallel S\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le C(\psi ,a)e^{a\mathrm{Re}t}$ is satisfied.

Remark 1.

Similar concepts of the resolving family of operators, the analytic resolving family of operators, are used in the study of integral evolution equations [24] and fractional differential equations [23]. If we consider the first order derivative instead of the distributed order derivative, then it will be an analytic semigroup of operators for the equation $D_t^{1}z\left(t\right)=Az\left(t\right)$ [26,27,28].

Denote by $\rho \left(A\right)$ the resolvent set of an operator A. Let an operator $A\in \mathcal{C}l\left(\mathcal{Z}\right)$ satisfy the following conditions:

(1)

There exist such ${\theta }_0\in (\pi /2,\pi ]$, $a_0\ge 0$, that for $\lambda \in S_{{\theta }_0,a_0}$ we have $W\left(\lambda \right)\in \rho \left(A\right)$;

(2)

For every $\theta \in (\pi /2,{\theta }_0)$, $a>a_0$ there exists such $K(\theta ,a)>0$, that for all $\lambda \in S_{{\theta }_0,a_0}$

$${\parallel (W\left(\lambda \right)I-A)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le \frac{\left|\lambda \right|K(\theta ,a)}{\left|W\right(\lambda \left)\right||\lambda -a|}.$$

Then, we can say that the operator A belongs to the class ${\mathcal{A}}_W({\theta }_0,a_0)$.

If $A\in {\mathcal{A}}_W({\theta }_0,a_0)$, the operators

$$Z_0\left(t\right):=\frac{1}{2\pi i}\underset{\Gamma }{\int }\frac{e^{\lambda t}}{\lambda }W\left(\lambda \right){\left(W\left(\lambda \right)I-A\right)}^{-1}d\lambda$$

(7)

are defined at $t>0$. Here $\Gamma ={\Gamma }_{+}\cup {\Gamma }_{-}\cup {\Gamma }_0$, ${\Gamma }_{\pm }=\{\mu \in \mathbb{C}:\mu =a+r{e}^{\pm i\theta },r\in (\delta ,\infty )\}$, ${\Gamma }_0=\{\mu \in \mathbb{C}:\mu =a+\delta e^{i\phi },\phi \in (-\theta ,\theta )\}$ for some $\delta >0$, $a>a_0$, $\theta \in (\pi /2,{\theta }_0)$.

Theorem 2.

Let $m-1<b\le m\in \mathbb{N}$, ${\theta }_0\in (\pi /2,\pi ]$, $a_0\ge 0$, $\omega \in L_1(0,b)$, W satisfy condition (6). Then, there exists an analytic resolving family of operators of the type $({\theta }_0-\pi /2,a_0)$ for Equation (5), if and only if $A\in {\mathcal{A}}_W({\theta }_0,a_0)$. In this case, the resolving family of operators is unique, it has the form (7) and at $z_0\in D_A$ the function $z\left(t\right)=Z_0\left(t\right)z_0$ is a unique solution of problem (4), (5) in the space $\mathrm{Lap}\left(\mathcal{Z}\right)$.

Proof. 

Let $A\in {\mathcal{A}}_W({\theta }_0,a_0)$, $R>\delta$,

$${\Gamma }_R=\bigcup _{k=1}^4{\Gamma }_{k,R},{\Gamma }_{1,R}={\Gamma }_0,{\Gamma }_{2,R}=\{\lambda \in \mathbb{C}:\lambda =a+R{e}^{i\phi },\phi \in (-\theta ,\theta )\},$$

$${\Gamma }_{3,R}=\{\lambda \in \mathbb{C}:\lambda =a+r{e}^{i\theta },r\in [\delta ,R]\},{\Gamma }_{4,R}=\{\lambda \in \mathbb{C}:\lambda =a+r{e}^{-i\theta },r\in [\delta ,R]\},$$

${\Gamma }_R$ is the positively oriented closed loop. Let us also consider the contours

$${\Gamma }_{5,R}=\{\lambda \in \mathbb{C}:\lambda =a+r{e}^{i\theta },r\in [R,\infty )\},{\Gamma }_{6,R}=\{\lambda \in \mathbb{C}:\lambda =a+r{e}^{-i\theta },r\in [R,\infty )\},$$

then $\Gamma ={\Gamma }_{5,R}\cup {\Gamma }_{6,R}\cup {\Gamma }_R\backslash {\Gamma }_{2,R}$.

At $t>0$, $z_0\in D_A$ we have

$$Z_0\left(t\right)z_0=\frac{1}{2\pi i}\underset{\Gamma }{\int }\frac{e^{\lambda t}}{\lambda }W\left(\lambda \right){\left(W\left(\lambda \right)I-A\right)}^{-1}{z}_{0}d\lambda =$$

$$=\frac{1}{2\pi i}\underset{\Gamma }{\int }\frac{e^{\lambda t}}{\lambda }d\lambda z_0+\frac{1}{2\pi i}\underset{\Gamma }{\int }\frac{e^{\lambda t}}{\lambda }{\left(W\left(\lambda \right)I-A\right)}^{-1}A{z}_{0}d\lambda .$$

For $t\in [0,1]$, $\lambda \in \Gamma \backslash \{\lambda \in \mathbb{C}:|\lambda |\le \varrho \}$

$${∥\frac{e^{\lambda t}}{\lambda }{\left(W\left(\lambda \right)I-A\right)}^{-1}A{z}_0∥}_{\mathcal{Z}}\le \frac{e^{a+\delta }K(\theta ,a){\parallel A{z}_0\parallel }_{\mathcal{Z}}}{\left|W\right(\lambda \left)\right||\lambda -a|}\le \frac{C_1}{{\left|\lambda \right|}^{m+\epsilon }}.$$

Hence, the integral $Z_0\left(t\right)$ converges uniformly on $t\in [0,1]$ and by the continuity

$$Z_0\left(0\right)z_0=z_0+\frac{1}{2\pi i}\underset{\Gamma }{\int }\frac{1}{\lambda }{\left(W\left(\lambda \right)I-A\right)}^{-1}A{z}_{0}d\lambda =$$

$$=z_0+\underset{R\to \infty }{lim}\frac{1}{2\pi i}\left(\underset{{\Gamma }_R}{\int }-\underset{{\Gamma }_{2,R}}{\int }+\underset{{\Gamma }_{5,R}}{\int }+\underset{{\Gamma }_{6,R}}{\int }\right)\frac{1}{\lambda }{\left(W\left(\lambda \right)I-A\right)}^{-1}A{z}_{0}d\lambda =z_0,$$

since by Cauchy Theorem

$$\underset{{\Gamma }_R}{\int }\frac{1}{\lambda }{\left(W\left(\lambda \right)I-A\right)}^{-1}A{z}_{0}d\lambda =0,$$

at the same time for $t\in [0,1]$, $\lambda \in \Gamma \backslash \{\lambda \in \mathbb{C}:|\lambda |\le \varrho \}$

$${∥\underset{{\Gamma }_{s,R}}{\int }\frac{1}{\lambda }{\left(W\left(\lambda \right)I-A\right)}^{-1}A{z}_{0}d\lambda ∥}_{\mathcal{Z}}\le \frac{C_2}{R^{m-1+\epsilon }},s=2,5,6.$$

Similarly, at $t>0$, $z_0\in D_A$, $k\in \{1,2,\dots ,m-1\}$

$$\frac{1}{2\pi i}\underset{\Gamma }{\int }{\lambda }^{k-1}{e}^{\lambda t}W\left(\lambda \right){\left(W\left(\lambda \right)I-A\right)}^{-1}{z}_{0}d\lambda =$$

$$=\frac{1}{2\pi i}\underset{\Gamma }{\int }{\lambda }^{k-1}{e}^{\lambda t}d\lambda z_0+\frac{1}{2\pi i}\underset{\Gamma }{\int }{\lambda }^{k-1}{e}^{\lambda t}{\left(W\left(\lambda \right)I-A\right)}^{-1}A{z}_{0}d\lambda ,$$

at $t\in [0,1]$, $\lambda \in \Gamma \backslash \{\lambda \in \mathbb{C}:|\lambda |\le \varrho \}$

$${∥{\lambda }^{k-1}{e}^{\lambda t}{\left(W\left(\lambda \right)I-A\right)}^{-1}A{z}_0∥}_{\mathcal{Z}}\le \frac{{\left|\lambda \right|}^{k-1}{e}^{a+\delta }K(\theta ,a){\parallel A{z}_0\parallel }_{\mathcal{Z}}}{\left|W\right(\lambda \left)\right||\lambda -a|}\le \frac{C_1}{{\left|\lambda \right|}^{m-k+\epsilon }},$$

therefore,

$$Z_0^{\left(k\right)}\left(0\right)z_0=0+\frac{1}{2\pi i}\underset{\Gamma }{\int }{\lambda }^{k-1}{\left(W\left(\lambda \right)I-A\right)}^{-1}A{z}_{0}d\lambda =0.$$

Consequently, $Z_0(·)z_0\in C^{m-1}({\overline{\mathbb{R}}}_{+};\mathcal{Z})$, the function $z\left(t\right):=Z_0\left(t\right)z_0$ satisfies Cauchy conditions Equation (4). Since the operator A is closed and commutes with the operators ${(W\left(\lambda \right)I-A)}^{-1}$ on $D_A$, at $z_0\in D_A$ the inclusion $A{Z}_0(·)z_0\in C({\mathbb{R}}_{+};\mathcal{Z})$ is fulfilled also, i.e., $z(·):=Z_0(·)z_0\in C({\mathbb{R}}_{+};D_A)$.

At $\mathrm{Re}\mu >a$ we have

$$\mathrm{Lap}\left[Z_0\right]\left(\mu \right)=\frac{1}{2\pi i}\underset{\Gamma }{\int }\frac{W\left(\lambda \right)}{\lambda (\mu -\lambda )}{\left(W\left(\lambda \right)I-A\right)}^{-1}d\lambda .$$

Since $A\in {\mathcal{A}}_W({\theta }_0,a_0)$, then at $\lambda \in S_{\theta ,a}$

$${∥\frac{W\left(\lambda \right)}{\lambda (\mu -\lambda )}{\left(W\left(\lambda \right)I-A\right)}^{-1}∥}_{\mathcal{L}\left(\mathcal{Z}\right)}\le \frac{K(\theta ,a)}{|\lambda -a||\lambda -\mu |},$$

therefore,

$$\underset{R\to \infty }{lim}\frac{1}{2\pi i}\underset{{\Gamma }_{s,R}}{\int }\frac{W\left(\lambda \right)}{\lambda (\mu -\lambda )}{\left(W\left(\lambda \right)I-A\right)}^{-1}d\lambda =0,s=2,5,6,$$

and by the Cauchy integral formula

$$\mathrm{Lap}\left[Z_0\right]\left(\mu \right)=\underset{R\to \infty }{lim}\frac{1}{2\pi i}\underset{{\Gamma }_R}{\int }\frac{W\left(\lambda \right)}{\lambda (\mu -\lambda )}{\left(W\left(\lambda \right)I-A\right)}^{-1}d\lambda =\frac{W\left(\mu \right)}{\mu }{\left(W\left(\mu \right)I-A\right)}^{-1}.$$

Take in Theorem 1

$$H\left(\lambda \right)=\frac{W\left(\lambda \right)}{\lambda }{(W\left(\lambda \right)I-A)}^{-1},F=Z_0,$$

then it follows that the mapping $Z_0:{\Sigma }_{{\theta }_0-\pi /2}\to \mathcal{L}\left(\mathcal{Z}\right)$ is analytic and for every $\theta \in (\pi /2,{\theta }_0)$, $a>a_0$ there exists such $C(\theta ,a)>0$ that for all $t\in {\Sigma }_{\theta -\pi /2}$ $\parallel Z_0{\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le C(\theta ,a)e^{a\mathrm{Re}t}$. Thus, $z\in \mathrm{Lap}\left(\mathcal{Z}\right)$.

Next for $z\left(t\right):=Z_0\left(t\right)z_0$ at $z_0\in D_A$

$$\mathrm{Lap}\left[Az\right]\left(\mu \right)=\frac{W\left(\mu \right)}{\mu }{\left(W\left(\mu \right)I-A\right)}^{-1}A{z}_0.$$

Consequently, $\mathrm{Lap}\left[z\right]\left(\mu \right)\in D_A$, $A\mathrm{Lap}\left[z\right]\left(\mu \right)=\mathrm{Lap}\left[Az\right]\left(\mu \right)$, $\mathrm{Lap}\left[z\right]\left(\mu \right)$ and $\mathrm{Lap}\left[Az\right]\left(\mu \right)$ have analytic extensions on $S_{{\theta }_0,a_0}$, since $A\in {\mathcal{A}}_W({\theta }_0,a_0)$.

Using Formula (3) of the Laplace transform, we obtain

$$\mathrm{Lap}\left[\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }z\left(t\right)d\alpha \right]\left(\mu \right)=\frac{{\left(W\left(\mu \right)\right)}^2}{\mu }{\left(W\left(\mu \right)I-A\right)}^{-1}{z}_0-\frac{W\left(\mu \right)}{\mu }{z}_0=$$

$$=\frac{W\left(\mu \right)}{\mu }{\left(W\left(\mu \right)I-A\right)}^{-1}A{z}_0=\mathrm{Lap}\left[Az\right]\left(\mu \right).$$

We apply the inverse Laplace transform to both sides of the obtained equality and get equality (5) at all points of the function $Az$ continuity, i.e., for all $t>0$. Hence, z is a solution of problem Equations (4) and (5) and $\{Z_0\left(t\right)\in \mathcal{L}\left(\mathcal{Z}\right):t\ge 0\}$ is an analytic resolving family of operators of the type $({\theta }_0-\pi /2,a_0)$ for Equation (5).

Let there exist an analytic resolving family of operators $\left\{S\right(t)\in \mathcal{L}(\mathcal{Z}):t\ge 0\}$ of the type $({\theta }_0-\pi /2,a_0)$ for Equation (5), denote $\mathrm{Lap}\left[S\right]\left(\lambda \right):=H\left(\lambda \right)$, $\lambda >a_0$. From Equation (5) due to paragraph (ii) of the resolving family definition we obtain at $z_0\in D_A$ equalities

$$\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }S\left(t\right)z_{0}d\alpha =AS\left(t\right)z_0=S\left(t\right)A{z}_0,$$

hence, due to the closedness of the operator A at $\lambda >a_0$ $H\left(\lambda \right)\left[D_A\right]\subset D_A$,

$$\mathrm{Lap}\left[\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }S\left(t\right)z_{0}d\alpha \right]\left(\lambda \right)=W\left(\lambda \right)H\left(\lambda \right)z_0-\frac{W\left(\lambda \right)}{\lambda }{z}_0=H\left(\lambda \right)A{z}_0=AH\left(\lambda \right)z_0.$$

Therefore, the operator $W\left(\lambda \right)I-A:D_A\to \mathcal{Z}$ is bijective and

$$H\left(\lambda \right)=\frac{W\left(\lambda \right)}{\lambda }{(W\left(\lambda \right)I-A)}^{-1},\lambda >a_0.$$

By Theorem 1, it follows that $A\in {\mathcal{A}}_W({\theta }_0,a_0)$, $S\left(t\right)\equiv Z_0\left(t\right)$ by virtue of the uniqueness of the inverse Laplace transform.

If there exist two solutions $z_1$, $z_2$ of problem (4) and (5) from the class $\mathrm{Lap}\left(\mathcal{Z}\right)$, then their difference $y=z_1-z_2\in \mathrm{Lap}\left(\mathcal{Z}\right)$ is a solution of Equation (5) and satisfies the initial conditions $y^{\left(k\right)}\left(0\right)=0$, $k=0,1,\dots ,m-1$. Acting by the Laplace transform on both parts of Equation (5) and due to the initial conditions, we get the equality $W\left(\lambda \right)\mathrm{Lap}\left[y\right]\left(\lambda \right)=A\mathrm{Lap}\left[y\right]\left(\lambda \right).$ Since $A\in {\mathcal{A}}_W({\theta }_0,a_0)$, at $\lambda \in S_{{\theta }_0,a_0}$ we get the identity $\mathrm{Lap}\left[y\right]\left(\lambda \right)\equiv 0$. This means that $y\equiv 0$. Therefore, $z\left(t\right)=Z_0\left(t\right)z_0$ is a unique solution of problem (4) and (5) at $z_0\in D_A$ in the space $\mathrm{Lap}\left(\mathcal{Z}\right)$. □

Remark 2.

If we consider problem Equations (4) and (5) on the segment $[0,T]$, then we will continue the function y on $[T,\infty )$ by a continuous bounded way and, reasoning in the same way, we get the uniqueness of the solution on the segment.

Remark 3.

It is easy to show that under the conditions of Theorem 2 at $z_0\in D_{A^2}$ the inclusion $Z_0(·)z_0\in C({\overline{\mathbb{R}}}_{+};D_A)$ is fulfilled, i.e., $Z_0(·)z_0$ is continuous in the norm of $D_A$ at zero and satisfies Equation (5) at $t=0$.

Theorem 3.

Let $m-1<b\le m\in \mathbb{N}$, ${\theta }_0\in (\pi /2,\pi ]$, $a_0\ge 0$, $\omega \in L_1(0,b)$, W satisfy condition (6), $A\in {\mathcal{A}}_W({\theta }_0,a_0)$, $z_0,z_1,\dots ,z_{m-1}\in D_A$. Then there exists a unique in the space $\mathrm{Lap}\left(\mathcal{Z}\right)$ solution of the problem

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

(8)

for Equation (5). In this case, the solution is analytic in the sector ${\Sigma }_{{\theta }_0-\pi /2}$ and has the form $z\left(t\right)={\displaystyle \sum _{k=0}^{m-1}}{Z}_k\left(t\right)z_k$, where

$$Z_k\left(t\right)=\frac{1}{2\pi i}\underset{\Gamma }{\int }\frac{e^{\lambda t}}{{\lambda }^{k+1}}W\left(\lambda \right){\left(W\left(\lambda \right)I-A\right)}^{-1}d\lambda ,k=0,1,\dots ,m-1.$$

Proof. 

Reasoning as in the proof of the previous theorem, it is not difficult to show that by virtue of the conditions of the theorem $Z_k^{\left(k\right)}\left(t\right)=Z_0\left(t\right)$, therefore, maps $Z_k:{\Sigma }_{{\theta }_0-\pi /2}\to \mathcal{L}\left(\mathcal{Z}\right)$ are analytic, moreover, $Z_k^{\left(k\right)}\left(0\right)z_k=z_k$, $Z_k^{\left(l\right)}\left(0\right)z_k=0$, $k\in \{0,1,\dots ,m-1\}$, $l\in \{0,1,\dots ,m-1\}\backslash \left\{k\right\}$. □

3. Inhomogeneous Equation

A solution of Cauchy problem (8) for the inhomogeneous equation

$$\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }z\left(t\right)d\alpha =Az\left(t\right)+g\left(t\right),t\in (0,T),$$

(9)

where $m-1<b\le m\in \mathbb{N}$, $\omega :[0,b]\to \mathbb{C}$, $T>0$, $g\in C\left(\right[0,T];\mathcal{Z})$, is a such function $z\in C^{m-1}([0,T);\mathcal{Z})\cap C((0,T);D_A)$ that there exists $\underset{a}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }z\left(t\right)d\alpha \in C((0,T);\mathcal{Z})$ and equalities (8) and (9) are fulfilled.

Denote

$$Z\left(t\right):=\frac{1}{2\pi i}\underset{\Gamma }{\int }{e}^{\lambda t}{\left(W\left(\lambda \right)I-A\right)}^{-1}d\lambda .$$

First, we will consider the case of the increased smoothness of the function g in spatial variables: $g\in C([0,T];D_A)$.

Lemma 3.

Let $m-1<b\le m\in \mathbb{N}$, $\omega \in L_1(0,b)$, W satisfy condition (6), ${\theta }_0\in (\pi /2,\pi ]$, $a_0\ge 0$, $A\in {\mathcal{A}}_W({\theta }_0,a_0)$, $g\in C([0,T];D_A)$. Then, the function

$$z_g\left(t\right)=\underset{0}{\overset{t}{\int }}Z(t-s)g\left(s\right)ds$$

(10)

is a unique solution of problem (4) and (9) with $z_0=0$.

Proof. 

It is easy to show that $Z\left(t\right)$ has the analytic extension on ${\Sigma }_{{\theta }_0-\pi /2}$. We investigate the behavior of this function and its derivatives in the right vicinity of zero. Due to condition (6) and since $A\in {\mathcal{A}}_W({\theta }_0,a_0)$, we have

$$\parallel Z^{\left(k\right)}{\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le C_1\underset{\Gamma }{\int }\frac{e^{t\mathrm{Re}\lambda }ds}{{\left|\lambda \right|}^{m-1-k+\epsilon }},k=0,1,\dots ,m-1.$$

At $t\in (0,1]$ we have

$$\underset{{\Gamma }_0}{\int }\frac{e^{t\mathrm{Re}\lambda }ds}{{\left|\lambda \right|}^{m-1-k+\epsilon }}\le 2C_1\pi {\delta }^{2-m+k-\epsilon }{e}^{a+\delta },$$

$$\underset{{\Gamma }_{\pm }}{\int }\frac{e^{t\mathrm{Re}\lambda }}{{\left|\lambda \right|}^{m-1-k+\epsilon }}ds\le C_1\underset{\delta }{\overset{\infty }{\int }}\frac{e^{rtcos\theta }dr}{r^{m-1-k+\epsilon }}=C_1{(-tcos\theta )}^{m-2-k+\epsilon }\Gamma (2-m+k-\epsilon ),$$

at $k=0,1,\dots ,m-1$. Therefore, $Z^{\left(k\right)}\left(0\right)=0$ for $k=0,1,\dots ,m-2$, $\parallel Z^{(m-1)}{\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}=O\left(t^{\epsilon -1}\right)$ as $t\to 0+$. Consequently,

$$z_g^{\left(k\right)}\left(t\right)=\underset{0}{\overset{t}{\int }}{Z}^{\left(k\right)}(t-s)g\left(s\right)ds,$$

$\parallel z_g^{\left(k\right)}\left(t\right)\parallel \le C_2{t}^{m-1-k+\epsilon }\to 0$ as $t\to 0+$, $k=0,1,\dots ,m-2$. Therefore, conditions (4) with $z_0=0$ are fulfilled.

Define $g\left(t\right)=0$ at $t\ge T$, then $z_g=Z\ast g$, $\mathrm{Lap}\left[z_g\right]=\mathrm{Lap}\left[Z\right]\mathrm{Lap}\left[g\right].$ Reasoning as in the proof of Theorem 2, we get the equality $\mathrm{Lap}\left[Z\right]\left(\lambda \right)={\left(W\left(\lambda \right)I-A\right)}^{-1}$, since due to conditions (6) and $A\in {\mathcal{A}}_W({\theta }_0,a_0)$ at $\lambda \in S_{\theta ,a}\backslash \{\nu \in \mathbb{C}:|\nu |<\varrho \}$

$$∥\frac{1}{\mu -\lambda }{\left(W\left(\lambda \right)I-A\right)}^{-1}∥\le \frac{C_3}{{\left|\lambda \right|}^{m+\epsilon }}.$$

Hence,

$$\mathrm{Lap}\left[\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }{z}_{g}d\alpha \right]\left(\mu \right)=W\left(\mu \right){\left(W\left(\mu \right)I-A\right)}^{-1}\mathrm{Lap}\left[g\right]\left(\mu \right)=$$

$$=\mathrm{Lap}\left[g\right]\left(\mu \right)+A{\left(W\left(\mu \right)I-A\right)}^{-1}\mathrm{Lap}\left[g\right]\left(\mu \right).$$

Acting by the inverse Laplace transform on both sides of the obtained equality, we get

$$\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }{z}_g\left(t\right)d\alpha =g\left(t\right)+A(Z\ast g)\left(t\right)=g\left(t\right)+A{z}_g\left(t\right),$$

since $g\in C([0,T];D_A)$ and due to the closedness of the operator A the value $A(Z\ast g)\left(t\right)=Z\ast Ag\left(t\right)$ is finite.

The proof of the uniqueness of the problem solution is reduced in the standard way to the proof of the uniqueness for the homogeneous problem. By virtue of Remark 2, we get the required statement. □

Now let us consider the case of the increased smoothness of the function g with respect to the time variable.

By $C^{\gamma }([0,T];\mathcal{Z})$ at $\gamma \in (0,1]$ we denote the class of Hölderian functions, i.e., functions $f:[0,T]\to \mathcal{Z}$, such that there exists $C>0$, for all $t,s\in [0,T]$ the inequality ${\parallel f\left(t\right)-f\left(s\right)\parallel }_{\mathcal{Z}}\le C{|t-s|}^{\gamma }$ is satisfied.

Lemma 4.

Let $m-1<b\le m\in \mathbb{N}$, $\omega \in L_1(0,b)$, W satisfy condition (6), ${\theta }_0\in (\pi /2,\pi ]$, $a_0\ge 0$, $A\in {\mathcal{A}}_W({\theta }_0,a_0)$, $g\in C^{\gamma }([0,T];\mathcal{Z})$, $\gamma \in (0,1]$. Then, the function (10) is a unique solution of problem (4), (9) with $z_0=0$.

Proof. 

We have at $t\to 0+$, as in the proof of the previous lemma, ${\parallel Z\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}=O\left(t^{\epsilon -1}\right)$, $\parallel z_g{\left(t\right)\parallel }_{\mathcal{Z}}\le C_2{t}^{\epsilon }$. Moreover, $\mathrm{im}Z\left(t\right)\subset D_A$ at $t>0$, since due to the analyticity of $Z_1\left(t\right)$

$$AZ\left(t\right)=\frac{1}{2\pi i}\underset{\Gamma }{\int }A{(W\left(\lambda \right)I-A)}^{-1}{e}^{\lambda t}d\lambda =\frac{1}{2\pi i}\underset{\Gamma }{\int }W\left(\lambda \right){(W\left(\lambda \right)I-A)}^{-1}{e}^{\lambda t}d\lambda =Z_1\left(t\right).$$

Note that

$$\underset{{\Gamma }_{\pm }}{\int }\frac{\left|\lambda \right|e^{t\mathrm{Re}\lambda }}{|\lambda -a|}ds\le \underset{\delta }{\overset{\infty }{\int }}\frac{|r{e}^{i\theta }+a|e^{rtcos\theta }dr}{r}\le \frac{C_1}{-tcos\theta },$$

hence, ${\parallel AZ\left(t\right)\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}=O\left(t^{-1}\right)$ as $t\to 0+$,

$${\parallel AZ(t-s)(g\left(s\right)-g\left(t\right))\parallel }_{\mathcal{Z}}\le C_2{|t-s|}^{\gamma -1}.$$

Consequently, the integral

$$\underset{0}{\overset{t}{\int }}AZ(t-s)g\left(s\right)ds=\underset{0}{\overset{t}{\int }}AZ(t-s)(g\left(s\right)-g\left(t\right))ds+(Z_0\left(t\right)-I)g\left(t\right)$$

converges, therefore, $z_g\left(t\right)\in D_A$, $z_g\in C((0,T);D_A)$.

The rest of the proof is the same as for Lemma 3. □

From Theorem 2, Lemmas 3 and 4, we get the following result.

Theorem 4.

Let $m-1<b\le m\in \mathbb{N}$, $\omega \in L_1(0,b)$, W satisfy condition (6), ${\theta }_0\in (\pi /2,\pi ]$, $a_0\ge 0$, $A\in {\mathcal{A}}_W({\theta }_0,a_0)$, $g\in C([0,T];D_A)\cup C^{\gamma }([0,T];\mathcal{Z})$, $\gamma \in (0,1]$, $z_k\in D_A$, $k=0,1,\dots ,m-1$. Then the function

$$z\left(t\right)=\sum _{k=0}^{m-1}{Z}_k\left(t\right)z_k+\underset{0}{\overset{t}{\int }}Z(t-s)g\left(s\right)ds$$

is a unique solution of problem (8), (9).

Remark 4.

Note that

$$Z_0\left(t\right)=\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha -1}Z\left(t\right)d\alpha .$$

Indeed, Formula (3) and the values of the derivatives of Z in zero implies that

$$\mathrm{Lap}\left[\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha -1}Z\left(t\right)d\alpha \right]\left(\lambda \right)=\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right){\lambda }^{\alpha -1}{(W\left(\lambda \right)I-A)}^{-1}d\alpha =$$

$$=\frac{W\left(\lambda \right)}{\lambda }{\left(W\left(\lambda \right)I-A\right)}^{-1}=\mathrm{Lap}\left[Z_0\right]\left(\lambda \right).$$

Due to the uniqueness of the inverse Laplace transform, we get the required assertion.

4. Perturbations of Operators of the Class ${\mathcal{A}}_W({\theta }_0,a_0)$

The result of this section is analogous to the perturbation theorem for analytic semigroups of operators [28].

Theorem 5.

Let $m-1<b\le m\in \mathbb{N}$, $\omega \in L_1(0,b)$, W satisfy condition (6), ${\theta }_0\in (\pi /2,\pi )$, $a_0>0$, $A\in {\mathcal{A}}_W({\theta }_0,a_0)$, $B\in \mathcal{C}l\left(\mathcal{Z}\right)$, for all $x\in D_A\subset D_B$

$${\parallel Bx\parallel }_{\mathcal{Z}}\le {b\parallel Ax\parallel }_{\mathcal{Z}}+c{\parallel x\parallel }_{\mathcal{Z}},$$

(11)

where $b,c\ge 0$, there exists $q\in (0,1)$ such that $b(1+K(\theta ,a\left)\right)<q$ for every $\theta \in (\pi /2,{\theta }_0)$, $a>a_0$. Then $A+B\in {\mathcal{A}}_W({\theta }_0,a_1)$ at sufficiently large $a_1>a_0$.

Proof. 

Take $k\in \mathbb{N}\backslash \left\{1\right\}$, $\lambda \in S_{\theta ,ka}\subset S_{\theta ,a}$ at some $\theta \in (\pi /2,{\theta }_0)$, $a>a_0$, then due to (11)

$${\parallel B(W\left(\lambda \right)I-A)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}{\le b\parallel A(W\left(\lambda \right)I-A)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}+c{\parallel {(W\left(\lambda \right)I-A)}^{-1}\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le$$

$$\le b\left(1+\frac{\left|\lambda \right|K(\theta ,a)}{|\lambda -a|}\right)+\frac{c\left|\lambda \right|K(\theta ,a)}{|\lambda -a|\left|W\right(\lambda \left)\right|}.$$

In this case, the value

$$\frac{\left|\lambda \right|}{|\lambda -a|}\le \frac{1}{1-\frac{a}{\left|\lambda \right|}}\le \frac{1}{1-\frac{1}{ksin{\theta }_0}}$$

is close to 1 and

$$\frac{\left|\lambda \right|}{|\lambda -a|\left|W\right(\lambda \left)\right|}\le \frac{1}{\left(1-\frac{1}{ksin{\theta }_0}\right)C{[k{a}_{0}sin{\theta }_0]}^{m-1+\epsilon }}$$

is close to 0 for a large enough number k. Here C is the constant from inequality (6). Therefore, for such k the inequality

$${\parallel B(W\left(\lambda \right)I-A)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le b\left(1+\frac{K(\theta ,a)}{1-\frac{1}{ksin{\theta }_0}}\right)+\frac{cK(\theta ,a)}{\left(1-\frac{1}{ksin{\theta }_0}\right)C{[k{a}_{0}sin{\theta }_0]}^{m-1+\epsilon }}\le q<1$$

is fulfilled. Next we have

$${(W\left(\lambda \right)I-A-B)}^{-1}\le {(W\left(\lambda \right)I-A)}^{-1}{(I-B{(W\left(\lambda \right)I-A)}^{-1})}^{-1}=$$

$$={(W\left(\lambda \right)I-A)}^{-1}\sum _{n=0}^{\infty }{\left[B{(W\left(\lambda \right)I-A)}^{-1}\right]}^n,$$

$$\frac{|\lambda -ka|}{|\lambda -a|}=\left|1-\frac{(k-1)a}{\lambda -a}\right|\le 1+\frac{(k-1)a}{|\lambda -a|}\le 1+\frac{1}{sin{\theta }_0}.$$

Hence,

$${\parallel (W\left(\lambda \right)I-A-B)}^{-1}{\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}\le \frac{\left|\lambda \right|K(\theta ,a)}{(1-q)|\lambda -a|\left|W\right(\lambda \left)\right|}\le \frac{\left|\lambda \right|K(\theta ,a)\left(1+\frac{1}{sin{\theta }_0}\right)}{(1-q)|\lambda -ka|\left|W\right(\lambda \left)\right|}.$$

Consequently, $A+B\in {\mathcal{A}}_W({\theta }_0,k{a}_0)$, and for all $\theta \in (\pi /2,{\theta }_0)$, $a>a_0$

$$K(\theta ,ka)=\frac{K(\theta ,a)}{1-q}\left(1+\frac{1}{sin{\theta }_0}\right).$$

 □

Remark 5.

A bounded operator $B\in \mathcal{L}\left(\mathcal{Z}\right)$ satisfies condition (11) with $b=0$, $c={\parallel B\parallel }_{\mathcal{L}\left(\mathcal{Z}\right)}$.

Corollary 1.

Let $m-1<b\le m\in \mathbb{N}$, $\omega \in L_1(0,b)$, W satisfy condition (6), ${\theta }_0\in (\pi /2,\pi )$. Then the set

$$\bigcup _{a_0>0}{\mathcal{A}}_W({\theta }_0,a_0)$$

is open in the normed space $\mathcal{L}(D_A;\mathcal{Z})$.

Proof. 

An operator $B\in \mathcal{L}(D_A;\mathcal{Z})$ with a sufficiently small norm ${\parallel B\parallel }_{\mathcal{L}(D_A;\mathcal{Z})}=b$ satisfies condition (11) with $c=0$. □

5. Application to an Initial-Boundary Value Problem

Consider the initial-boundary value problem

$$u^{\left(k\right)}(x,0)=u_k\left(x\right),x\in (0,\pi ),k=0,1,\dots ,m-1,$$

(12)

$$u(0,t)=u(\pi ,t)=0,t\in (0,T),$$

(13)

for the equation of the ultraslow diffusion [3]

$$\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }u(x,t)d\alpha =u_{xx}(x,t)+r{u}_x(x,t)+su(x,t)+f(x,t),(x,t)\in (0,\pi )\times (0,T),$$

(14)

where $m-1<b\le m\in \mathbb{N}$, $\omega :[0,b]\to \mathbb{R}$, $r,s\in \mathbb{R}$, $f\in (0,\pi )\times [0,T]\to \mathbb{R}$. Set $\mathcal{Z}=L_2(0,\pi )$, $A=\frac{{\partial }^2}{\partial x^2}$, $D_A=H_0^2(0,\pi ):=\{v\in H^2(0,\pi ):v\left(0\right)=v\left(\pi \right)=0\},$$B=r\frac{\partial }{\partial x}+sI$, $D_B=H^1(0,\pi ).$ Then $A,B\in \mathcal{C}l\left(\mathcal{Z}\right)$.

Let $0\le b_0<b$, $\omega \left(\alpha \right)\equiv 0$ at $\alpha \in [0,b_0)$, $\omega \in C([b_0,b];\mathbb{R})$, $\omega \left(\alpha \right)\ge 0$ at $\alpha \in [b_0,b)$, $\omega \left(b\right)>0.$ Lemma 1 implies that $W\left(\lambda \right):=\underset{b_0}{\overset{b}{\int }}\omega \left(\alpha \right){\lambda }^{\alpha }d\alpha$ is analytic on the set $\mathbb{C}\backslash \{\lambda \in \mathbb{C}:\mathrm{Re}\lambda \le 0\}$, by Lemma 2 condition (6) is satisfied. By the mean value theorem $W\left(\lambda \right)=(b-b_0)\omega \left(\xi \right){\left|\lambda \right|}^{\xi }{e}^{i\xi arg\lambda }$, where $0\le b_0<\xi <b\le m$. Therefore, $argW\left(\lambda \right)=\xi arg\lambda$.

It is important for us to avoid the equalities $W\left(\lambda \right)=-k^2$, $k\in \mathbb{N}$, at $\lambda$ from the selected sector $S_{{\theta }_0,a_0}$, i.e., the equalities

$$arg\lambda =\pi /\xi ,\left|\lambda \right|={\left(\frac{k^2}{(b-b_0)\omega \left(\xi \right)}\right)}^{1/\xi },k\in \mathbb{N}.$$

Obviously, there exists a sector of the form $S_{{\theta }_0,a_0}$ at some ${\theta }_0\in (\pi /2,\pi )$, $a_0\ge 0$, only if

$$\xi \in \left(\frac{2}{4k_0+3},\frac{2}{4k_0+1}\right)$$

(15)

for some $k_0\in {\mathbb{N}}_0:=\mathbb{N}\cup \left\{0\right\}$. We can take any $a_0\ge 0$ and ${\theta }_0$ may be taken as arbitrary from $\left(\frac{\pi }{2},\frac{\pi }{\xi }-2\pi k_0\right)$, if $\xi \in \left[\frac{2}{4k_0+2},\frac{2}{4k_0+1}\right)$, ${\theta }_0$ is arbitrary from $\left(\frac{\pi }{2},2\pi (1-k_0)-\frac{\pi }{\xi }\right)$, if $\xi \in \left(\frac{2}{4k_0+3},\frac{2}{4k_0+2}\right).$

Take $a_0\ge 0$, ${\theta }_0$ according to the choice described above. We have at $v\in L_2(0,\pi )$, $\theta \in (\pi /2,{\theta }_0)$, $a>a_0$, $\lambda \in S_{\theta ,a}$

$${\parallel (W\left(\lambda \right)I-A)}^{-1}{v\parallel }_{L_2(0,\pi )}^2=\sum _{k=1}^{\infty }\frac{{\left|\langle v\left(x\right),sinkx\rangle \right|}^2}{|W\left(\lambda \right)+k^2{|}^2}=$$

$$=\frac{{\left|\lambda \right|}^2{\left|1-\frac{a}{\lambda }\right|}^2}{{\left|W\left(\lambda \right)\right|}^2{|\lambda -a|}^2\underset{l\in \mathbb{N},\lambda \in S_{\theta ,a}}{min}{|W\left(\lambda \right)+l^2|}^2}\sum _{k=1}^{\infty }{\left|\langle v\left(x\right),sinkx\rangle \right|}^2\le$$

$$\le \frac{{\left|\lambda \right|}^2{\left(1+\frac{1}{sin\theta }\right)}^2{\parallel v\parallel }_{L_2(0,\pi )}^2}{{\left|W\left(\lambda \right)\right|}^2{|\lambda -a|}^2\underset{l\in \mathbb{N},\lambda \in S_{\theta ,a}}{min}|(b-b_0)\omega \left(\xi \right){\left|\lambda \right|}^{\xi }{e}^{i\xi arg\lambda }+l^2{|}^2}.$$

Note that at $\lambda \in S_{\theta ,a}$ $cos(\xi arg\lambda )\ge cos\xi \theta$, where $0\le \xi \theta \le \xi {\theta }_0\in (0,\pi )$ by the construction of ${\theta }_0$.

Therefore,

$$|c{e}^{i\xi arg\lambda }+l^2{|}^2={(ccos(\xi arg\lambda )+l^2)}^2+c^2{sin}^2(\xi arg\lambda )=$$

$$=c^2+l^4+2c{l}^{2}cos(\xi arg\lambda )\ge c^2+l^4+2c{l}^{2}cos\xi \theta ={|c{e}^{i\xi \theta }+l^2|}^2,$$

$${\parallel (W\left(\lambda \right)I-A)}^{-1}{\parallel }_{\mathcal{L}\left(L_2(0,\pi )\right)}\le \frac{\left|\lambda \right|\left(1+\frac{1}{sin\theta }\right)}{\left|W\right(\lambda \left)\right||\lambda -a|sin\xi \theta }.$$

Thus, the following result is obtained.

Lemma 5.

Let $m-1<b\le m\in \mathbb{N}$, $0\le b_0<b$, $\omega \left(\alpha \right)\equiv 0$ at $\alpha \in [0,b_0)$, $\omega \in C([b_0,b];\mathbb{R})$, $\omega \left(\alpha \right)\ge 0$ at $\alpha \in [b_0,b)$, $\omega \left(b\right)>0$ and condition (15) be satisfied. Then the operator $A=\frac{{\partial }^2}{\partial x^2}$, $D_A=H_0^2(0,\pi ),$ acting in the space $L_2(0,\pi )$, belongs to the class ${\mathcal{A}}_W({\theta }_0,a_0)$ at some ${\theta }_0\in (\pi /2,\pi )$ and every $a_0\ge 0$.

Remark 6.

From the above arguments, it can be seen that for all $\theta \in (\pi /2,{\theta }_0)$, $a>a_0$

$$K(\theta ,a)=\frac{\left(1+\frac{1}{sin\theta }\right)}{sin\xi \theta }\le \frac{\left(1+\frac{1}{sin{\theta }_0}\right)}{sin\xi {\theta }_0}.$$

Remark 7.

It is clear that condition (15) is not constructive.However, if outside the interval of form (15) at some $k_0$ the function ω is equal to zero (e.g., outside the interval $(2/3,2)$), then condition (15) is obviously satisfied. Conversely, if the function ω is equal to zero outside of an interval $\left(\frac{2}{4k_0+5},\frac{2}{4k_0+3}\right)$ at some $k_0\in {\mathbb{N}}_0$, then $A\notin {\mathcal{A}}_W({\theta }_0,a_0)$ for every ${\theta }_0\in (\pi /2,\pi )$, $a_0\ge 0$.

Lemma 6.

Let $m-1<b\le m\in \mathbb{N}$, $0\le b_0<b$, $\omega \left(\alpha \right)\equiv 0$ at $\alpha \in [0,b_0)$, $\omega \in C([b_0,b];\mathbb{R})$, $\omega \left(\alpha \right)\ge 0$ at $\alpha \in [b_0,b)$, $\omega \left(b\right)>0$ and condition (15) be fulfilled. The operators $A=\frac{{\partial }^2}{\partial x^2}$, $D_A=H_0^2(0,\pi ),$$B=r\frac{\partial }{\partial x}+sI$, $D_B=H^1(0,\pi )$, act in the space $L_2(0,\pi )$. Then, for all $v\in D_A$

$${\parallel Bv\parallel }_{L_2(\Omega )}\le \frac{\pi \left|r\right|}{\sqrt{2}}{\parallel Av\parallel }_{L_2(\Omega )}+{\left|s\right|\parallel v\parallel }_{L_2(0,\pi )}.$$

(16)

Therefore, if

$$\left|r\right|<\frac{\sqrt{2}sin{\theta }_{0}sin\xi {\theta }_0}{\pi (1+sin{\theta }_0+sin{\theta }_{0}sin\xi {\theta }_0)},$$

then the operator $A+B$ belongs to the class ${\mathcal{A}}_W({\theta }_0,a_1)$ at sufficiently large $a_1>a_0$.

Proof. 

For $v\in D_A$, $x_1,x_2\in (0,\pi )$ due to the Hölder inequality

$${(v^{\prime }\left(x_2\right)-v^{\prime }\left(x_1\right))}^2={\left(\underset{x_1}{\overset{x_2}{\int }}{v}^{″}\left(x\right)dx\right)}^2\le |x_2-x_1|\underset{x_1}{\overset{x_2}{\int }}{\left(v^{″}\left(x\right)\right)}^{2}dx,$$

hence,

$${\left(v^{\prime }\left(x_2\right)\right)}^2-2v^{\prime }\left(x_1\right)v^{\prime }\left(x_2\right)+{\left(v^{\prime }\left(x_1\right)\right)}^2\le \pi \underset{0}{\overset{\pi }{\int }}{\left(v^{″}\left(x\right)\right)}^{2}dx.$$

By integrating both parts of this inequality twice, with respect to $x_1$ and $x_2$, we get

$$2\pi \underset{0}{\overset{\pi }{\int }}{\left(v^{\prime }\left(x\right)\right)}^{2}dx-2{\left(\underset{0}{\overset{\pi }{\int }}{v}^{\prime }\left(x\right)dx\right)}^2=2\pi \underset{0}{\overset{\pi }{\int }}{\left(v^{\prime }\left(x\right)\right)}^{2}dx\le {\pi }^3\underset{0}{\overset{\pi }{\int }}{\left(v^{″}\left(x\right)\right)}^{2}dx.$$

Therefore, inequality (16) holds. It remains to refer to Theorem 5 and Remark 5. □

Theorem 6.

Let $0\le b_0<b\le 1$, $\omega \left(\alpha \right)\equiv 0$ at $\alpha \in [0,b_0)$, $\omega \in C([b_0,b];\mathbb{R})$, $\omega \left(\alpha \right)\ge 0$ at $\alpha \in [b_0,b)$, $\omega \left(b\right)>0$ and condition (15) hold, $f\in C([0,T];H_0^2(0,\pi ))\cup C^{\gamma }([0,T];L_2(0,\pi ))$, $\gamma \in (0,1]$. Then there exists a unique solution of problem (12)–(14).

Proof. 

Problem (12)–(14) has the form of Cauchy problem (8) to the equation

$$\underset{0}{\overset{b}{\int }}\omega \left(\alpha \right)D_t^{\alpha }z\left(t\right)d\alpha =(A+B)z\left(t\right)+g\left(t\right),t\in (0,T),$$

with operators A and B, which are defined in this section. By Lemma 6 $A+B\in {\mathcal{A}}_W({\theta }_0,a_1)$, therefore, by Theorem 4, we get the required statement, noting that in this case $D_A=H_0^2(0,\pi )$. □

6. Conclusions

A generation theorem for analytic resolving families of operators to the linear distributed order equation is proved, the inhomogeneous equation unique solvability is studied and a perturbation theorem for the obtained class of generators is obtained. Results of the work are applied to the study of an initial boundary value problem for the ultraslow diffusion equation with the lower-order terms with respect to the spatial variable.