General Fractional Calculus Operators of Distributed Order

Abstract

In this paper, two types of general fractional derivatives of distributed order and a corresponding fractional integral of distributed type are defined, and their basic properties are investigated. The general fractional derivatives of distributed order are constructed for a special class of one-parametric Sonin kernels with power law singularities at the origin. The conventional fractional derivatives of distributed order based on the Riemann–Liouville and Caputo fractional derivatives are particular cases of the general fractional derivatives of distributed order introduced in this paper.

1. Introduction

The formal constructions of the operators nowadays referred to as the general fractional integrals (GFIs) and the general fractional derivatives (GFDs) were suggested for the first time by Sonin in [1]. In this paper, Sonin extends Abel’s method for solving the Abel integral equation presented in [2,3] to a class of integral equations with so-called Sonin kernels. He recognized that the basic ingredient of Abel’s method for solving the Abel integral equation (in modern notations),

$$f\left(t\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_0^t{(t-\tau )}^{\alpha -1}u\left(\tau \right)d\tau ,t>0,$$

(1)

is nothing more than a simple formula for the power law kernels,

$$(h_{\alpha }\ast h_{1-\alpha })\left(t\right)\equiv 1,t>0,h_{\alpha }\left(t\right):=\frac{t^{\alpha -1}}{\Gamma \left(\alpha \right)},\alpha >0,$$

(2)

and suggested its generalization in the form

$$(\kappa \ast k)\left(t\right)\equiv 1,t>0,$$

(3)

where ∗ stands for the Laplace convolution. Nowadays, Condition (3) is referred to as the Sonin condition, and the functions that satisfy this condition are called Sonin kernels.

The simplest pair of Sonin kernels are the power law kernels $h_{\alpha }$ and $h_{1-\alpha }$, which were known already to Abel; see Relation (2). In [1], Sonin introduced an important class of Sonin kernels that can be represented in the following form:

$$\kappa \left(t\right)=h_{\alpha }\left(t\right)·{\kappa }_1\left(t\right),{\kappa }_1\left(t\right)=\sum _{k=0}^{+\infty }{a}_k{t}^k,a_0\ne 0,0<\alpha <1,$$

(4)

$$k\left(t\right)=h_{1-\alpha }\left(t\right)·k_1\left(t\right),k_1\left(t\right)=\sum _{k=0}^{+\infty }{b}_k{t}^k,$$

(5)

where the functions ${\kappa }_1={\kappa }_1\left(t\right)$ and $k_1=k_1\left(t\right)$ are analytical on $\mathbb{R}$ and their coefficients satisfy the following infinite system of linear equations with a triangular matrix:

$$a_0{b}_0=1,\sum _{k=0}^n\Gamma (k+1-\alpha )\Gamma (\alpha +n-k)a_{n-k}{b}_k=0,n=1,2,3,\dots .$$

In particular, he derived the famous pair of Sonin kernels

$$\kappa \left(t\right)={\left(\sqrt{t}\right)}^{\alpha -1}{J}_{\alpha -1}\left(2\sqrt{t}\right),k\left(t\right)={\left(\sqrt{t}\right)}^{-\alpha }{I}_{-\alpha }\left(2\sqrt{t}\right),0<\alpha <1$$

in terms of the Bessel function $J_{\nu }$ and the modified Bessel function $I_{\nu }$.

Following Abel, Sonin formally solved the integral equation with Sonin kernel $\kappa$

$$f\left(t\right)=(\kappa \ast u)\left(t\right)={\int }_0^t\kappa (t-\tau )u\left(\tau \right)d\tau$$

and represented its solution in the form

$$u\left(t\right)=\frac{d}{dt}(k\ast f)\left(t\right)=\frac{d}{dt}{\int }_0^{t}k(t-\tau )u\left(\tau \right)d\tau ,$$

where the kernel k is the Sonin kernel associated with the kernel $\kappa$ through the Sonin condition (3). In particular, for the Abel integral equation, Equation (1), the solution takes the well-known form:

$$u\left(t\right)=\frac{d}{dt}(h_{1-\alpha }\ast f)\left(t\right)=\frac{d}{dt}\frac{1}{\Gamma (1-\alpha )}{\int }_0^t{(t-\tau )}^{-\alpha }f\left(\tau \right)d\tau ,t>0.$$

(6)

However, Sonin did not interpret his results in terms of fractional calculus. This was performed by Kochubei much later in his paper [4], where, among other things, he introduced and investigated the regularized GFD or the GFD of the Caputo type in the form

$${(}_{\ast }{\mathbb{D}}_{\left(k\right)}u)\left(t\right)=\frac{d}{dt}(k\ast u)\left(t\right)-u\left(0\right)k\left(t\right),t>0$$

(7)

with Sonin kernel k from a special class $\mathcal{K}$ of kernels described in terms of their Laplace transforms (see [4] for details). Kochubei also considered the GFD of the Riemann–Liouville type

$$\left({\mathbb{D}}_{\left(k\right)}u\right)\left(t\right)=\frac{d}{dt}(k\ast u)\left(t\right),t>0.$$

(8)

Evidently, the regularized GFD (7) and the GFD (8) are connected to each other by the relation

$${(}_{\ast }{\mathbb{D}}_{\left(k\right)}u)\left(t\right)=\left({\mathbb{D}}_{\left(k\right)}[u(·)-u\left(0\right)]\right)\left(t\right)=\left({\mathbb{D}}_{\left(k\right)}u\right)\left(t\right)-u\left(0\right)k\left(t\right),t>0.$$

(9)

On the space of absolutely continuous functions, another useful representation of the regularized GFD is valid in the following form:

$${(}_{\ast }{\mathbb{D}}_{\left(k\right)}u)\left(t\right)=(k\ast u^{\prime })\left(t\right),t>0.$$

(10)

For Sonin kernels k from class $\mathcal{K}$, Kochubei showed existence of the associated completely monotonic Sonin kernels $\kappa$ and introduced the corresponding GFI as follows:

$$\left({\mathbb{I}}_{\left(\kappa \right)}u\right)\left(t\right)=(\kappa \ast u)\left(t\right)={\int }_0^t\kappa (t-\tau )u\left(\tau \right)d\tau .$$

(11)

It is worth mentioning that the construction of the GFI and GFDs with Sonin kernels was first formally introduced by Sonin in [1]. However, their theory essentially depends on the kernels, or on the classes of kernels, and on the spaces of functions where these operators are studied. Even in the case of the power law Sonin kernels $h_{\alpha }$ and $h_{1-\alpha }$, which generate the Riemann–Liouville fractional integral and the Riemann–Liouville and Caputo fractional derivatives, the properties of these fractional calculus (FC) operators are very different for different spaces of functions. In [4], Kochubei introduced and investigated a very important special case of the GFI and GFDs with Sonin kernels from the class $\mathcal{K}$. However, one of the conditions posed on the kernels from $\mathcal{K}$ (the Laplace transform of the kernel k has to be a Stieltjes function) is very restrictive. As a consequence, not all of the known Sonin kernels belong to this class.

In the subsequent publications devoted to the GFI and GFDs, essentially larger classes of Sonin kernels compared to those suggested in [1,4] were introduced and investigated. In [5], the GFI and GFDs with Sonin kernels from class ${\mathcal{S}}_{-1}$ were studied. The Sonin kernels from ${\mathcal{S}}_{-1}$ are the functions continuous on ${\mathbb{R}}_{+}$ that have an integrable singularity of the power-law type at origin (see [5] for details). The class ${\mathcal{S}}_{-1}$ is a very general one and contains both the kernels (4) and (5) introduced by Sonin and the Kochubei class $\mathcal{K}$ of Sonin kernels.

In [6], the Sonin condition (3) was generalized, and the GFI and GFDs of arbitrary order were introduced (Operators (7), (8), and (11) with Sonin kernels have a “generalized order” less than one). In [7], the so-called 1st level GFD was suggested. This derivative is constructed as a composition of two GFIs with different kernels and the first-order derivative and contains both the GFD (8) and the regularized GFD (10) as its particular cases. In the recent publications by Tarasov, the multi-kernel approach for the definition of the GFI and GFDs, the general vector calculus based on the GFI and GFDs, and the Riesz form of the general FC operators in the multi-dimensional space were suggested; see [8,9,10], respectively. We also refer readers to the paper [11], where the left- and right-sided GFIs and GFDs on a finite interval were introduced and investigated. For an overview of the recent publications devoted to fractional differential equations, both ordinary and partial, with GFDs, we refer readers to the recent survey paper [12].

It is worth mentioning that the GFI and GFDs introduced in the publications mentioned above have already had important applications. In the papers [13,14,15,16,17], Tarasov suggested several non-local physical theories based on the GFI and GFDs with Sonin kernels, including general fractional dynamics, general non-Markovian quantum dynamics, general non-local electrodynamics, non-local classical theory of gravity, and non-local statistical mechanics. Furthermore, the GFI and GFDs with Sonin kernels were used in some mathematical models for anomalous diffusion and in linear viscoelasticity; see, e.g., [18,19,20,21,22].

Another active direction of research in modern FC concerns an object that does not possess a counterpart in the world of integer order derivatives, namely, the so-called fractional derivatives of distributed order. This interest has a clear physical background. Recently, a lot of attention in the modeling of anomalous diffusion processes has been attracted by the so-called ultra-slow diffusion, which is characterized by the logarithmic behavior of the mean squared displacement of the diffusing particles; see, e.g., [23,24,25,26] and the references therein. One of the most promising approaches for describing such processes is by means of time-fractional diffusion equations with fractional derivatives of distributed order. From a mathematical viewpoint, the fractional derivatives of distributed order and the fractional differential equations with these derivatives were studied, e.g., in [27,28,29,30,31].

Until now, the definitions of fractional derivatives of distributed order were based on conventional fractional derivatives and especially on the Riemann–Liouville and Caputo derivatives. The main subject of this paper is the introduction of the concept of GFDs of distributed order and the investigation of their basic properties. These operators are a generalization of both GFDs and the fractional derivatives of distributed order introduced so far. Moreover, in this paper, we also define the corresponding fractional integrals of distributed type and prove the fundamental theorems of FC for these integrals and the GFDs of distributed order.

The rest of the paper is organized as follows: In Section 2, a one-parametric class of Sonin kernels is introduced, and GFDs of distributed order with the kernels from this class are defined and investigated. Section 3 is devoted to fractional integrals of distributed type and some connections between GFDs and the fractional integral of distributed type in the form of two fundamental theorems of FC. In the final section, Section 4, some examples of GFDs of distributed order and the corresponding fractional integrals of distributed type are presented.

2. General Fractional Derivatives of Distributed Order

First, we remind the readers of the definitions of the Riemann–Liouville and Caputo fractional derivatives of distributed order on the interval $(a,b),0\le a<b\le 1$, respectively:

$$\left({\mathbb{D}}^{w}u\right)\left(t\right)={\int }_a^b\left(D_{0+}^{\alpha }u\right)\left(t\right)w\left(\alpha \right)d\alpha ,$$

(12)

$${(}_{\ast }{\mathbb{D}}^{w}u)\left(t\right)={\int }_a^b{(}_{\ast }{D}_{0+}^{\alpha }u)\left(t\right)w\left(\alpha \right)d\alpha ,$$

(13)

where the weight function w satisfies the properties $w\in C\left(\right[a,b\left]\right)$, $w\left(\alpha \right)\ge 0$$\forall \alpha \in [a,b]$ and $w\left(\alpha \right)\not\equiv 0$, the Riemann–Liouville fractional derivative of the order $\alpha ,0<\alpha <1$, is defined by

$$\left(D_{0+}^{\alpha }u\right)\left(t\right)=\frac{d}{dt}(h_{1-\alpha }\ast u)\left(t\right),t>0,$$

(14)

and the Caputo fractional derivative of the order $\alpha ,0<\alpha <1$, is given by

$${(}_{\ast }{D}_{0+}^{\alpha }u)\left(t\right)=(h_{1-\alpha }\ast u^{\prime })\left(t\right),t>0.$$

(15)

Please note that in some FC publications, the distributed order derivatives (12) and (13) are defined on the interval $(a,b)$ with $0\le a<b\le 2$. However, in this paper, we restrict ourselves to the case $0\le a<b\le 1$.

For a generalization of the distributed order fractional derivatives (12) and (13) to the case of GFDs of distributed order, we need a class of Sonin kernels that explicitly depend on a parameter that can be interpreted as a “generalized order” of the corresponding GFDs.

In what follows, we deal with Sonin kernels $k_{\alpha }\left(t\right)$ and their associated kernels ${\kappa }_{\alpha }\left(t\right)$ that satisfy the following two constraints:

• (C1) The Sonin condition

  $$(k_{\alpha }\ast {\kappa }_{\alpha })\left(t\right)\equiv 1,t>0$$

  (16)

  holds valid for all $\alpha \in (a,b)$, where $0\le a<b\le 1.$
• (C2) The kernels k and $\kappa$ can be represented as follows:

  $$k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)k_1(t,\alpha ),{\kappa }_{\alpha }\left(t\right)=h_{\alpha }\left(t\right){\kappa }_1(t,\alpha ),$$

  (17)

  where $k_1,{\kappa }_1\in C([0,+\infty )\times [a,b])$ and $k_1(0,\alpha )\ne 0,{\kappa }_1(0,\alpha )\ne 0,t>0,\alpha \in (a,b)$.

The class of Sonin kernels that satisfy conditions (C1) and (C2) will be denoted by ${}_{\ast }{S}_{\alpha }$. Evidently, any associated kernel ${\kappa }_{\alpha }$ to a kernel $k_{\alpha }\in {}_{\ast }{S}_{\alpha }$ also belongs to the class ${}_{\ast }{S}_{\alpha }$, and $k_{\alpha }$ is its associated kernel.

Please note that the kernels from the class ${}_{\ast }{S}_{\alpha }$ are functions of two variables: $t(t\in {\mathbb{R}}_{+})$ and $\alpha (\alpha \in (a,b\left)\right)$. The simplest and very important examples of this kind are the Sonin kernels $k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right),\alpha \in (0,1)$, and ${\kappa }_{\alpha }\left(t\right)=h_{\alpha }\left(t\right),\alpha \in (0,1)$ of the Riemann–Liouville fractional derivative and the Riemann–Liouville fractional integral, respectively.

In general, Sonin kernels from the class ${}_{\ast }{S}_{\alpha }$ and their associated kernels are the power law functions $h_{1-\alpha }\left(t\right)$ and $h_{\alpha }\left(t\right)$, which are disturbed by (multiplied with) some continuous functions (compare to the kernels (4) and (5) introduced by Sonin). These continuous functions can depend on $\alpha$ and/or other parameters or not. Most of the known Sonin kernels (see, e.g., [5]) belong to the class ${}_{\ast }{S}_{\alpha }$, and thus our theory will cover many known particular cases, including those presented in the examples from our paper; see Section 4. However, as mentioned in [32], Sonin kernels can also possess other kinds of singularities at the origin, say, the ones of the power-logarithmic type. Such kernels are not covered by our theory.

It is also worth mentioning that, for any fixed $\alpha \in (a,b)$, Sonin kernels from class ${}_{\ast }{S}_{\alpha }$ are functions of the t-variable that belong to the space $C_{-1}(0,+\infty )$ defined as follows:

$$C_{-1}(0,+\infty )=\{u:u=t^{p}v\left(t\right),t>0,p>-1,v\in C[0,\infty )\}.$$

(18)

The space $C_{-1}(0,+\infty )$ was employed in several publications devoted to GFIs and GFDs with Sonin kernels from class ${\mathcal{S}}_{-1}$ (see, e.g., [5]):

$$(\kappa ,k\in {\mathcal{S}}_{-1})\iff (\kappa ,k\in C_{-1}(0,+\infty ))\wedge ((\kappa \ast k)\left(t\right)=1,t>0).$$

(19)

For a fixed $\alpha \in (a,b)$, the Sonin kernels from class ${}_{\ast }{S}_{\alpha }$ introduced above evidently belong to class ${\mathcal{S}}_{-1}$. Thus, we can use the results derived in these publications for the analysis of GFDs and GFIs with the kernels from class ${}_{\ast }{S}_{\alpha }$.

In what follows, we also employ another useful subspace of the space $C_{-1}(0,+\infty )$, defined by

$$C_{-1}^n(0,+\infty )=\{u:u^{\left(n\right)}\in C_{-1}(0,+\infty )\},n\in \mathbb{N}.$$

(20)

For a kernel $k_{\alpha }\in {}_{\ast }{S}_{\alpha }$, the GFD (8) and the regularized GFD (10) are defined for all values of $\alpha$ from the interval $(a,b)$ as follows:

$$\left({\mathbb{D}}_{k_{\alpha }}u\right)\left(t\right)=\frac{d}{dt}{\int }_0^t{k}_{\alpha }(t-\tau )u\left(\tau \right)d\tau ,t>0,$$

(21)

$${(}_{\ast }{\mathbb{D}}_{k_{\alpha }}u)\left(t\right)={\int }_0^t{k}_{\alpha }(t-\tau )u^{\prime }\left(\tau \right)d\tau ,t>0.$$

(22)

Now, we proceed with the definitions of the corresponding general fractional derivatives of distributed order.

Definition 1.

Let a kernel $k_{\alpha }$ belong to the class ${}_{\ast }{S}_{\alpha }$ and a weight function w satisfy conditions $w\in C\left(\right[a,b\left]\right)$, $w\left(\alpha \right)\ge 0,\alpha \in [a,b]$, and $w\left(\alpha \right)\not\equiv 0,\alpha \in [a,b]$.

The general distributed order fractional derivative (GDOFD) of the Riemann–Liouville type and the regularized or Caputo-type GDOFD are defined as follows, respectively:

$$\left({\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(t\right)={\int }_a^b\left({\mathbb{D}}_{k_{\alpha }}u\right)\left(t\right)w\left(\alpha \right)d\alpha ,$$

(23)

$$\left({}_{\ast }{\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(t\right)={\int }_a^b\left({}_{\ast }{\mathbb{D}}_{k_{\alpha }}u\right)\left(t\right)w\left(\alpha \right)d\alpha .$$

(24)

Remark 1.

The GDOFDs (23) and (24) are well-defined, in particular for the functions from the space $C_{-1}^1(0,+\infty )$. Indeed, let the inclusion $u\in C_{-1}^1(0,+\infty )$ hold true. Then,

$$u^{\prime }=t^{\beta }{u}_1\left(t\right),t>0,\beta >-1,u_1\in C[0,\infty ).$$

The regularized GFD with a kernel $k_{\alpha }\in {}_{\ast }{S}_{\alpha }$ takes the form

$$\begin{array}{ccc}\hfill {(}_{\ast }{\mathbb{D}}_{k_{\alpha }}u)\left(t\right)& =& {\int }_0^t{h}_{1-\alpha }(t-\tau )k_1(t-\tau ,\alpha ){\tau }^{\beta }{u}_1\left(\tau \right)d\tau .\hfill \end{array}$$

Because $h_{1-\alpha }(t-\tau ){\tau }^{\beta }$ is integrable and of one sign and $k_1(t-\tau ,\alpha )u_1\left(\tau \right)$ is continuous, applying the mean value theorem for the last integral yields

$$\begin{array}{ccc}\hfill {(}_{\ast }{\mathbb{D}}_{k_{\alpha }}u)\left(t\right)& =& u_1\left({\tau }_0\right)k_1(t-{\tau }_0,\alpha ){\int }_0^t{h}_{1-\alpha }(t-\tau ){\tau }^{\beta }d\tau \hfill \\ & =& u_1(t-{\tau }_0)k_1({\tau }_0,\alpha )\frac{\Gamma (\beta +1)}{\Gamma (\beta +2-\alpha )}{t}^{\beta +1-\alpha }.\hfill \end{array}$$

(25)

Due to the inequalities $\beta >-1$ and $0<\alpha <1$, we thus arrive at the inclusion

$${(}_{\ast }{\mathbb{D}}_{k_{\alpha }}u)\left(t\right)=u_1(t-{\tau }_0)k_1({\tau }_0,\alpha )\frac{\Gamma (\beta +1)}{\Gamma (\beta +2-\alpha )}{t}^{\beta +1-\alpha }\in C((0,\infty )\times [a,b]),$$

which proves that the regularized GDOFD (24) is well-defined in the space $C_{-1}^1(0,+\infty ).$

To prove the same statement for the GDOFD (23), we employ the result proved above and the relation (9) between the regularized GFD (7) and the GFD (8) that is valid in the space $C_{-1}^1(0,+\infty ).$

Now, we derive a useful representation of the regularized GDOFD (24) that, by definition, is the following iterated integral

$$\begin{array}{ccc}\hfill \left({}_{\ast }{\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(t\right)& =& {\int }_a^b\left({}_{\ast }{\mathbb{D}}_{k_{\alpha }}u\right)\left(t\right)w\left(\alpha \right)d\alpha \hfill \\ & =& {\int }_a^b{\int }_0^t{k}_{\alpha }(t-\tau )u^{\prime }\left(\tau \right)d\tau w\left(\alpha \right)d\alpha .\hfill \end{array}$$

Representation (25) ensures that the corresponding double integral

$${\int }_a^b{\int }_0^t{k}_{\alpha }(t-\tau )u^{\prime }\left(\tau \right)w\left(\alpha \right)d\tau d\alpha$$

is absolutely integrable. Thus, by using Fubini’s theorem, we can interchange the order of integration in the iterated integral and obtain the following formula:

$$\begin{array}{ccc}\hfill \left({}_{\ast }{\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(t\right)& =& {\int }_0^t{u}^{\prime }\left(\tau \right){\int }_a^b{k}_{\alpha }(t-\tau )w\left(\alpha \right)d\alpha d\tau \hfill \end{array}$$

$$\begin{array}{ccc}& =& {\int }_0^t{k}^w(t-\tau )u^{\prime }\left(\tau \right)d\tau ,\hfill \end{array}$$

(26)

$$\begin{array}{ccc}& =& (k^w\ast u^{\prime })\left(t\right),\hfill \end{array}$$

(27)

where

$$k^w\left(t\right)={\int }_a^b{k}_{\alpha }\left(t\right)w\left(\alpha \right)d\alpha ,$$

(28)

and ∗ denotes the Laplace convolution.

Remark 2.

The last formula looks like Representation (10) of the regularized GFD with the kernel $k^w$. Because kernel $k_{\alpha }$ belongs to the space $C_{-1}(0,+\infty )$ for all $\alpha \in (a,b)$, we have the inclusion $k^w\left(t\right)={\int }_a^b{k}_{\alpha }\left(t\right)w\left(\alpha \right)d\alpha \in C_{-1}(0,+\infty )$. Moreover, as we see in the next section, under some additional conditions, the function $k^w$ is a Sonin kernel from the class ${\mathcal{S}}_{-1}$, and thus the GDOFD (24) can be interpreted as a regularized GFD (10) with the kernel $k^w\in {\mathcal{S}}_{-1}$. In its turn, this means that the GDOFDs with such kernels and weight functions are a special subclass of the regularized GFDs with Sonin kernels from the class ${\mathcal{S}}_{-1}$. Thus, one can employ the results derived in the publications devoted to GFDs (see, e.g., [5,6,11]) for investigation of the GDOFDs.

Applying the same procedure to the GDOFD (23), we arrive at the analogous representation

$$\left({\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(t\right)=\frac{d}{dt}(k^w\ast u)\left(t\right),$$

(29)

where the kernel $k^w$ is defined as in (28).

As in the case of the GDOFD (24), under some additional conditions (see the next section), the GDOFD (23) can be interpreted as a GFD of the Riemann–Liouville type with the kernel $k^w\in {\mathcal{S}}_{-1}$.

Now, we discuss an important relation between the GDOFD (23) and the regularized GDOFD (24). To derive it, we employ the relation

$$\left({\mathbb{D}}_{k_{\alpha }}u\right)\left(t\right)=\left({}_{\ast }{\mathbb{D}}_{k_{\alpha }}u\right)\left(t\right)+u\left(0\right)k_{\alpha }\left(t\right),\alpha \in (a,b),$$

(30)

which holds true for any $u\in C_{-1}^1(0,\infty )$ and for any kernel from the class ${\mathcal{S}}_{-1}$; see [5]. As already mentioned, for a fixed $\alpha \in (a,b)$, the kernels from the class ${}_{\ast }{S}_{\alpha }$ belong to the class ${\mathcal{S}}_{-1}$. Using Formula (30), we thus arrive at the relation

$$\begin{array}{ccc}\hfill \left({\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(t\right)& =& {\int }_a^b\left({\mathbb{D}}_{k_{\alpha }}u\right)\left(t\right)w\left(\alpha \right)d\alpha \hfill \\ & =& {\int }_a^b\left(\left({}_{\ast }{\mathbb{D}}_{k_{\alpha }}u\right)\left(t\right)+u\left(0\right)k_{\alpha }\left(t\right)\right)w\left(\alpha \right)d\alpha \hfill \\ & =& \left({}_{\ast }{\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(t\right)+u\left(0\right){\int }_a^b{k}_{\alpha }\left(t\right)w\left(\alpha \right)d\alpha \hfill \\ & =& \left({}_{\ast }{\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(t\right)+u\left(0\right)k^w\left(t\right).\hfill \end{array}$$

(31)

Some examples of the GDOFDs introduced above are provided in Section 4.

3. General Fractional Integrals of Distributed Type

To introduce the general fractional integrals of distributed type (GFIDs), in what follows, we impose some additional conditions on the kernels $k_{\alpha }\left(t\right)$ from the class ${}_{\ast }{S}_{\alpha }$:

• (C3) The Laplace transform

  $$K_{\alpha }\left(\rho \right)=\left(\mathcal{L}{k}_{\alpha }\right)\left(\rho \right):={\int }_0^{\infty }{e}^{-\rho t}{k}_{\alpha }\left(t\right)dt$$

  exists for all $\rho >0$ and $\alpha \in (a,b)$.
• (C4) The Laplace transform $K_{\alpha }\left(\rho \right)$ satisfies the following standard conditions:

  $$\begin{array}{ccc}\hfill & & \rho K_{\alpha }\left(\rho \right)\to \infty ,\mathrm{as}\rho \to \infty ,\hfill \end{array}$$

  (32)

  $$\begin{array}{ccc}\hfill & & \rho K_{\alpha }\left(\rho \right)\to 0,\mathrm{as}\rho \to 0.\hfill \end{array}$$

  (33)

The class of the kernels from ${}_{\ast }{S}_{\alpha }$ that satisfy conditions (C3) and (C4) will be denoted by ${}_{\ast }{\widehat{S}}_{\alpha }$. In what follows, we always consider the operators with the kernels from the class ${}_{\ast }{\widehat{S}}_{\alpha }$, i.e., the kernels that satisfy conditions (C1)–(C4).

For a definition of the GFIDs, we need some auxiliary results. First, we mention an evident relation

$$K^w\left(\rho \right):=\left(\mathcal{L}{k}^w\right)\left(\rho \right)={\int }_a^b{K}_{\alpha }\left(\rho \right)w\left(\alpha \right)d\alpha$$

(34)

between the Laplace transform $K^w\left(\rho \right)$ of the kernel $k_w$ given by (28) and the Laplace transform $K_{\alpha }\left(\rho \right)$ of the kernel $k_{\alpha }$. Moreover, for $k_{\alpha }\in {}_{\ast }{\widehat{S}}_{\alpha }$, the Laplace transform $K^w\left(\rho \right)$ satisfies Conditions (32) and (33).

The last formula, the Laplace convolution theorem, and the known Laplace transform formula for the first-order derivative result in the following useful representation of the Laplace transform for the regularized GDOFD (24):

$$\left(\mathcal{L}{}_{\ast }{\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(\rho \right)=K^w\left(\rho \right)(\rho \left(\mathcal{L}u\right)\left(\rho \right)-u\left(0\right)).$$

(35)

Now, let us consider the following initial-value problem for the fractional differential equation with the regularized GDOFD (24):

$$\left({}_{\ast }{\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(t\right)=g\left(t\right),u\left(0\right)=0.$$

(36)

Assuming the existence of the Laplace transform of the function g for all $\rho >0$, we apply the Laplace transform to the above equation and, using Formula (35), we obtain the relation

$$K^w\left(\rho \right)\rho \left(\mathcal{L}u\right)\left(\rho \right)=\left(\mathcal{L}g\right)\left(\rho \right).$$

The solution u to Problem (36) can be formally represented as follows:

$$u\left(t\right)=\left({\mathcal{L}}^{-1}\frac{\left(\mathcal{L}g\right)\left(\rho \right)}{\rho K^w\left(\rho \right)}\right)\left(t\right).$$

The convolution theorem for the Laplace transform leads then to the representation

$$u\left(t\right)=({\psi }_w\ast g)\left(t\right)$$

(37)

of the solution u in the time-domain, where the kernel ${\psi }_w$ is defined in terms of the inverse Laplace transform:

$${\psi }_w\left(t\right)=\left({\mathcal{L}}^{-1}\frac{1}{\rho K^w\left(\rho \right)}\right)\left(t\right).$$

(38)

The function ${\psi }_w$ will play the role of a kernel of the GFID. In the following theorem, we provide some important characteristics of this function.

Theorem 1.

Let the kernel $k_{\alpha }$ be from the class ${}_{\ast }{\widehat{S}}_{\alpha }$.

Then, the function ${\psi }_w$ defined by (38) belongs to the space $C_{-1}(0,\infty )$. Moreover, the functions $k^w$ and ${\psi }_w$ form a pair of Sonin kernels from the class ${\mathcal{S}}_{-1}$.

Proof. 

By definition, any kernel $k_{\alpha }\in {}_{\ast }{\widehat{S}}_{\alpha }$ can be represented in the form

$$k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)k_1(t,\alpha ),k_1(t,\alpha )\in C([0,\infty )\times [a,b]),k_1(0,\alpha )\ne 0,a<\alpha <b.$$

For the Laplace transform $K_{\alpha }\left(\rho \right)$, we have the expression

$$K_{\alpha }\left(\rho \right)=\left(\mathcal{L}{k}_{\alpha }\right)\left(\rho \right)={\int }_0^{\infty }{e}^{-\rho t}{k}_{\alpha }\left(t\right)dt={\int }_0^{\infty }{e}^{-\rho t}{h}_{1-\alpha }\left(t\right)k_1(t,\alpha )dt.$$

Because $k_1(t,\alpha )\in C([0,\infty )\times [a,b])$ and $e^{-\rho t}{h}_{1-\alpha }\left(t\right)\in L^1(0,\infty ),$ applying the mean value theorem to the last integral yields the representation

$$K_{\alpha }\left(\rho \right)=k_1(t_0,\alpha ){\int }_0^{\infty }{e}^{-\rho t}{h}_{1-\alpha }\left(t\right)dt=k_1(t_0,\alpha ){\int }_0^{\infty }\frac{e^{-\rho t}{t}^{-\alpha }}{\Gamma (1-\alpha )}dt=\frac{k_1(t_0,\alpha )}{{\rho }^{1-\alpha }},$$

where $t_0>0$ and $k_1(t_0,\alpha )\ne 0$.

Taking into account the last formula and Relation (34), the Laplace transform $K^w\left(\rho \right)$ of the kernel $k_w$ takes the form

$$K^w\left(\rho \right)={\int }_a^b{K}_{\alpha }\left(\rho \right)w\left(\alpha \right)d\alpha ={\int }_a^b\frac{k_1(t_0,\alpha )}{{\rho }^{1-\alpha }}w\left(\alpha \right)d\alpha =\frac{1}{\rho }{\int }_a^b{k}_1(t_0,\alpha ){\rho }^{\alpha }w\left(\alpha \right)d\alpha .$$

Applying the mean value theorem to the last integral, we arrive at the representation

$$K^w\left(\rho \right)={\rho }^{{\alpha }_0-1}W{k}_1(t_0,{\alpha }_0)$$

with ${\alpha }_0\in (a,b)$ and $W={\int }_a^{b}w\left(\alpha \right)d\alpha >0.$ Thus, the relation

$$\frac{1}{\rho K^w\left(\rho \right)}=\frac{1}{k_1(t_0,{\alpha }_0)W}\frac{1}{{\rho }^{{\alpha }_0}}$$

holds true. The inverse Laplace transform of the right-hand side of the last formula is well-known, and we arrive at the representation

$${\psi }_w\left(t\right)=\left({\mathcal{L}}^{-1}\frac{1}{\rho K^w\left(\rho \right)}\right)\left(t\right)=\frac{1}{k_1(t_0,{\alpha }_0)W}\frac{t^{{\alpha }_0-1}}{\Gamma \left({\alpha }_0\right)},$$

which immediately implicates the inclusion ${\psi }_w\in C_{-1}(0,\infty )$.

By definition, $\left(\mathcal{L}{\psi }_w\right)\left(\rho \right)=\frac{1}{\rho K^w\left(\rho \right)}$. Then, we obtain the relation

$$\left(\mathcal{L}{\psi }_w\right)\left(\rho \right)\left(\mathcal{L}{k}^w\right)\left(\rho \right)=\frac{1}{\rho },$$

which in time-domain can be rewritten as

$$(k^w\ast {\psi }_w)\left(t\right)=1,t>0.$$

Thus, the functions ${\psi }_w$ and $k^w$ belong to the space $C_{-1}(0,\infty )$ and form a pair of Sonin kernels, i.e., ${\psi }_w,k^w\in {\mathcal{S}}_{-1}.$ □

Motivated by the form (37) of the solution to the fractional differential Equation (36) and by Theorem 1, we now proceed with defining the GFIDs.

Definition 2.

Let kernel $k_{\alpha }$ be from class ${}_{\ast }{\widehat{S}}_{\alpha }$.

The general fractional integral operator of distributed type (GFID) is defined by

$$\left({\mathbb{I}}_{{\psi }_w}u\right)\left(t\right)=({\psi }_w\ast u)\left(t\right)={\int }_0^t{\psi }_w(t-\tau )u\left(\tau \right)d\tau ,$$

(39)

where the function ${\psi }_w$ is as in (38).

Remark 3.

For a kernel $k_{\alpha }\in {}_{\ast }{\widehat{S}}_{\alpha }$, let ${\kappa }_{\alpha }$ be its associated Sonin kernel. The Sonin condition in Laplace domain takes the form

$$\left(\mathcal{L}{k}_{\alpha }\right)\left(\rho \right)·\left(\mathcal{L}{\kappa }_{\alpha }\right)\left(\rho \right)=\frac{1}{\rho }.$$

Then, we obtain the formulas

$$K^w\left(\rho \right)={\int }_a^b{K}_{\alpha }\left(\rho \right)w\left(\alpha \right)d\alpha ={\int }_a^b\frac{1}{\rho \left(\mathcal{L}{\kappa }_{\alpha }\right)\left(\rho \right)}w\left(\alpha \right)d\alpha$$

and

$$\frac{1}{\rho K^w\left(\rho \right)}={\left({\int }_a^b\frac{1}{\left(\mathcal{L}{\kappa }_{\alpha }\right)\left(\rho \right)}w\left(\alpha \right)d\alpha \right)}^{-1}.$$

Thus, we arrive at another representation of the kernel ${\psi }_w$ in terms of the kernel ${\kappa }_{\alpha }$ of the corresponding GFI:

$${\psi }_w\left(t\right)=\left({\mathcal{L}}^{-1}{\left({\int }_a^b\frac{1}{\left(\mathcal{L}{\kappa }_{\alpha }\right)\left(\rho \right)}w\left(\alpha \right)d\alpha \right)}^{-1}\right)\left(t\right).$$

As shown in Theorem 1, kernel ${\psi }_w$ of the GFID (39) is from the class ${\mathcal{S}}_{-1}$ of Sonin kernels. Thus, Operator (39) is a special case of GFIs with the kernels from ${\mathcal{S}}_{-1}$, and we can employ the results already derived for GFIs in the space $C_{-1}(0,\infty )$ (see, e.g., [5,6] and subsequent publications). In particular, the following properties are worth mentioning:

$$\begin{array}{ccc}\hfill {\mathbb{I}}_{{\psi }_w}& :& C_{-1}(0,\infty )\to C_{-1}(0,\infty )\left(\text{mapping property}\right),\hfill \\ \hfill {\mathbb{I}}_{{\psi }_{w_1}}{\mathbb{I}}_{{\psi }_{w_2}}& =& {\mathbb{I}}_{{\psi }_{w_2}}{\mathbb{I}}_{{\psi }_{w_1}}\left(\text{commutativity law}\right)\hfill \\ \hfill {\mathbb{I}}_{{\psi }_{w_1}}{\mathbb{I}}_{{\psi }_{w_2}}& =& {\mathbb{I}}_{{\psi }_{w_1}\ast {\psi }_{w_2}}\left(\text{index law}\right).\hfill \end{array}$$

(40)

According to Theorem 1, the kernel $k^w$ of the GDOFD (23) and of the regularized GDOFD (24) is a Sonin kernel associated with the kernel ${\psi }_w$ of the GFID (39). Thus, we can apply the first and second fundamental theorems for GFDs and GFIs with Sonin kernels from class ${\mathcal{S}}_{-1}$ derived in [5] and arrive at the following important results:

Theorem 2

(First Fundamental Theorem for Distributed Order Fractional Operators). Let kernel $k_{\alpha }$ be from class ${}_{\ast }{\widehat{S}}_{\alpha }$.

Then, the GDOFD (23) and the regularized GDOFD (24) are the left-inverse operators to the GFID (39):

$$\begin{array}{ccc}\hfill \left({\mathbb{D}}_{k_{\alpha }}^w{\mathbb{I}}_{{\psi }_w}u\right)\left(t\right)& =& u\left(t\right),u\in C_{-1}(0,\infty ),t>0\hfill \end{array}$$

(41)

$$\begin{array}{ccc}\hfill \left({}_{\ast }{\mathbb{D}}_{k_{\alpha }}^w{\mathbb{I}}_{{\psi }_w}u\right)\left(t\right)& =& u\left(t\right),u\in C_{-1}^1(0,\infty ),t>0.\hfill \end{array}$$

(42)

Theorem 3

(Second Fundamental Theorem for Distributed Order Fractional Operators). Let kernel $k_{\alpha }$ be from class ${}_{\ast }{\widehat{S}}_{\alpha }$ and $u\in C_{-1}^1(0,\infty )$.

Then, the relations

$$\begin{array}{ccc}\hfill \left({\mathbb{I}}_{{\psi }_w}{}_{\ast }{\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(t\right)& =& u\left(t\right)-u\left(0\right),t>0\hfill \end{array}$$

(43)

$$\begin{array}{ccc}\hfill (\left({\mathbb{I}}_{{\psi }_w}{\mathbb{D}}_{k_{\alpha }}^{w}u\right)\left(t\right)& =& u\left(t\right),t>0\hfill \end{array}$$

(44)

hold valid.

For the proofs of the fundamental theorems for GFDs and GFIs with Sonin kernels from class ${\mathcal{S}}_{-1}$, we refer interested readers to [5].

4. Examples of the General Fractional Operators of Distributed Order

In this section, we discuss three particular examples of Sonin kernels from class ${}_{\ast }{\widehat{S}}_{\alpha }$ and the corresponding general fractional operators of distributed order.

First example: We start with the power law kernel $k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)$ of the Riemann–Liouville and Caputo fractional derivatives and the associated kernel ${\kappa }_{\alpha }\left(t\right)=h_{\alpha }\left(t\right)$ of the Riemann–Liouville fractional integral.

In this case, the GDOFDs introduced in this paper are nothing more than the Riemann–Liouville and Caputo fractional derivatives of distributed order on the interval $(0,1)$, defined as in (12) and (13), respectively. As mentioned in the introduction, the distributed order fractional derivatives of the Riemann–Liouville and Caputo types are well-studied (see, e.g., [27,28,29,30,31]) and have many applications. In this example, we look at these operators from the viewpoint of our general theory.

For kernels $k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)$ and ${\kappa }_{\alpha }\left(t\right)=h_{\alpha }\left(t\right)$, we set $k_1(t,\alpha )={\kappa }_1(t,\alpha )=1$ and $a=0,b=1$, and show that they belong to class ${}_{\ast }{\widehat{S}}_{\alpha }$ of Sonin kernels, introduced in Section 2. Indeed, they evidently satisfy conditions (C1) and (C2).

Moreover, the Laplace transform $K_{\alpha }\left(\rho \right)$ of the kernel $k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)$ does exist for $\rho >0$, and can be written down in an explicit form:

$$K_{\alpha }\left(\rho \right)=\left(\mathcal{L}{k}_{\alpha }\right)\left(\rho \right)={\rho }^{\alpha -1},\rho >0.$$

For the function $K_{\alpha }\left(\rho \right)={\rho }^{\alpha -1}$, Conditions (32) and (33) are evidently satisfied, and thus kernel $k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)$ belongs to class ${}_{\ast }{\widehat{S}}_{\alpha }$ of Sonin kernels.

This means that all of the results that were presented in the previous sections, including the properties of the corresponding GFID and the fundamental theorems of FC for the GDOFDs and the GFID, hold true. However, we found it instructive to perform some independent calculations and derivations and to establish some explicit formulas that are not possible in the general case.

For the power law kernels, Formula (34) takes the form

$$\begin{array}{c}\hfill K^w\left(\rho \right)=\left(\mathcal{L}{k}^w\right)\left(\rho \right)={\int }_0^1{\rho }^{\alpha -1}w\left(\alpha \right)d\alpha .\end{array}$$

Because $w\in C[0,1]$ and ${\rho }^{\alpha -1}$ is integrable and of one sign for $0<\alpha <1$, applying the mean value theorem for the last integral yields the relation

$$\begin{array}{c}\hfill K^w\left(\rho \right)=w\left({\alpha }_0\right){\int }_0^1{\rho }^{\alpha -1}d\alpha =w\left({\alpha }_0\right)\frac{\rho -1}{\rho ln\left(\rho \right)},\end{array}$$

(45)

for some ${\alpha }_0$ ($0<{\alpha }_0<1)$. We also mention that $w\left({\alpha }_0\right)\ne 0$ because of the evident inequality $K^w\left(\rho \right)>0$ for any $\rho >0$.

Now, we obtain the representation

$$\frac{1}{\rho K^w\left(\rho \right)}=\frac{1}{w\left({\alpha }_0\right)}\frac{ln\left(\rho \right)}{\rho -1}.$$

Because the function on the right-hand side of the last formula has a finite many singular points and tends to 0 as $\rho \to \infty$, its inverse Laplace transform

$${\psi }_w\left(t\right)=\frac{1}{w\left({\alpha }_0\right)}\left({\mathcal{L}}^{-1}\frac{ln\left(\rho \right)}{\rho -1}\right)\left(t\right)$$

is well-defined and can be represented in explicit form ([33], p. 1027):

$${\psi }_w\left(t\right)=\frac{1}{w\left({\alpha }_0\right)}\left({\mathcal{L}}^{-1}\frac{ln\left(\rho \right)}{\rho -1}\right)\left(t\right)=\frac{1}{w\left({\alpha }_0\right)}{e}^t{E}_1\left[t\right],$$

where $E_1$ is the exponential integral $E_1\left[t\right]:={\int }_1^{\infty }\frac{e^{-tx}}{x}dx$.

Second example: In this example, we consider the kernels

$$k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)exp(-\mu t),{\kappa }_{\alpha }\left(t\right)=h_{\alpha }\left(t\right)exp(-\mu t)+\mu {\int }_0^t{h}_{\alpha }\left(s\right)exp(-\mu s)ds$$

with $\mu \ge 0$. It is well known (see, e.g., [5]) that, for any $\alpha \in (0,1)$, the function $k_{\alpha }\left(t\right)$ is a Sonin kernel and ${\kappa }_{\alpha }\left(t\right)$ is its associated Sonin kernel. Furthermore, direct calculations show that these kernels are from the class ${}_{\ast }{S}_{\alpha }$ of Sonin kernels with $a=b=1$.

The Laplace transform $K_{\alpha }\left(\rho \right)$ of the kernel $k_{\alpha }$ can be explicitly evaluated:

$$K_{\alpha }\left(\rho \right)=\left(\mathcal{L}{k}_{\alpha }\right)\left(\rho \right)=\frac{1}{{(\rho +\mu )}^{1-\alpha }},\rho +\mu >0.$$

For this function, condition (C4) is also satisfied, and thus kernels $k_{\alpha }$ and ${\kappa }_{\alpha }$ belong to the class ${}_{\ast }{\widehat{S}}_{\alpha }$ of Sonin kernels.

Then, we proceed with Formula (34), which takes the form

$$\begin{array}{c}\hfill K^w\left(\rho \right)=\left(\mathcal{L}{k}^w\right)\left(\rho \right)={\int }_0^1\frac{1}{{(\rho +\mu )}^{1-\alpha }}w\left(\alpha \right)d\alpha =\frac{1}{(\rho +\mu )}{\int }_0^1{(\rho +\mu )}^{\alpha }w\left(\alpha \right)d\alpha .\end{array}$$

Because ${(\rho +\mu )}^{\alpha }\in C[0,1]$ and the weight function w is integrable and of one sign for $0<\alpha <1,$ applying the mean value theorem for the last integral yields the relation

$$\begin{array}{c}\hfill K^w\left(\rho \right)=\frac{1}{(\rho +\mu )}{(\rho +\mu )}^{{\alpha }_0}W={(\rho +\mu )}^{{\alpha }_0-1}W,\end{array}$$

(46)

for some ${\alpha }_0(0<{\alpha }_0<1)$ and with $W={\int }_0^{1}w\left(\alpha \right)d\alpha .$

Then, we obtain the formula

$$\frac{1}{\rho K^w\left(\rho \right)}=\frac{1}{W}\frac{1}{\rho }\frac{\rho +\mu }{{(\rho +\mu )}^{{\alpha }_0}}=\frac{1}{W}\left(\frac{1}{{(\rho +\mu )}^{{\alpha }_0}}+\frac{\mu }{\rho {(\rho +\mu )}^{{\alpha }_0}}\right).$$

By definition, the kernel function ${\psi }_w\left(t\right)$ of the GFID is the inverse Laplace transform of the last expression. Thus, we arrive at the representation

$${\psi }_w\left(t\right)=\frac{1}{W}\left(\frac{t^{{\alpha }_0-1}}{\Gamma \left({\alpha }_0\right)}exp(-\mu t)+\mu {\int }_0^t\frac{s^{{\alpha }_0-1}}{\Gamma \left({\alpha }_0\right)}exp(-\mu s)ds)\right).$$

As we see, kernel ${\psi }_w$ has the form of kernel ${\kappa }_{\alpha }$ with a certain ${\alpha }_0\in (0,1)$ that depends on $\mu$ and the weight function w. Thus, it is well-defined and belongs to the class ${\mathcal{S}}_{-1}$ of Sonin kernels, as stated in Theorem 1.

Third example: In the last example, we consider Sonin kernels (see, e.g., [5])

$$k_{\alpha }\left(t\right)=t^{-\alpha }{E}_{\gamma ,1-\alpha }(-t^{\gamma }),{\kappa }_{\alpha }\left(t\right)=h_{\alpha }\left(t\right)+h_{\alpha +\gamma }\left(t\right),t>0,0<\alpha ,\gamma <1.$$

First, we represent kernel $k_{\alpha }$ in the form

$$k_{\alpha }\left(t\right)=h_{1-\alpha }\left(t\right)\Gamma (1-\alpha )E_{\gamma ,1-\alpha }(-t^{\gamma }).$$

It is easy to verify that $k_{\alpha }\left(t\right)\in {}_{\ast }{S}_{\alpha }$, provided that $0\le a<b<1.$ Similarly, ${\kappa }_{\alpha }\left(t\right)\in {}_{\ast }{S}_{\alpha },$ provided that $0<a<b\le 1.$ Thus, $k_{\alpha },{\kappa }_{\alpha }\in {}_{\ast }{S}_{\alpha }$, provided that $0<a<b<1$ (in the previous examples, we had the case $a=0$ and $b=1$). In the further derivations, we assume that this condition holds valid.

The Laplace transform $K_{\alpha }\left(\rho \right)$ of the kernel $k_{\alpha }$ can be explicitly evaluated:

$$K_{\alpha }\left(\rho \right)=\left(\mathcal{L}{k}_{\alpha }\right)\left(\rho \right)=\frac{{\rho }^{\gamma +\alpha -1}}{{\rho }^{\gamma }+1},\rho >0.$$

For $0<\alpha ,\gamma <1$, this function satisfies condition (C4), and thus kernels $k_{\alpha }$ and ${\kappa }_{\alpha }$ belong to class ${}_{\ast }{\widehat{S}}_{\alpha }$ of Sonin kernels.

Formula (34) now takes the form

$$\begin{array}{c}\hfill K^w\left(\rho \right)=\left(\mathcal{L}{k}^w\right)\left(\rho \right)={\int }_a^b\frac{{\rho }^{\gamma +\alpha -1}}{{\rho }^{\gamma }+1}w\left(\alpha \right)d\alpha =\frac{{\rho }^{\gamma -1}}{{\rho }^{\gamma }+1}{\int }_a^b{\rho }^{\alpha }w\left(\alpha \right)d\alpha .\end{array}$$

Because ${\rho }^{\alpha }\in C[a,b]$ and w is integrable and of one sign for $a<\alpha <b,$ applying the mean value theorem for the last integral yields the relation

$$\begin{array}{c}\hfill K^w\left(\rho \right)=\frac{{\rho }^{\gamma -1}}{{\rho }^{\gamma }+1}{\rho }^{{\alpha }_0}{\int }_a^{b}w\left(\alpha \right)d\alpha =\frac{{\rho }^{\gamma +{\alpha }_0-1}}{{\rho }^{\gamma }+1}W\end{array}$$

(47)

for some ${\alpha }_0(a<{\alpha }_0<b)$ and with $W={\int }_a^{b}w\left(\alpha \right)d\alpha .$

From the last formula, we obtain

$$\frac{1}{\rho K^w\left(\rho \right)}=\frac{1}{W}\frac{{\rho }^{\gamma }+1}{{\rho }^{\gamma +{\alpha }_0}}=\frac{1}{W}({\rho }^{-{\alpha }_0}+{\rho }^{-\gamma -{\alpha }_0}).$$

Applying the inverse Laplace transform to the right-hand side of the last formula, we arrive at the following representation for the kernel ${\psi }_w$ of the corresponding GFID:

$${\psi }_w\left(t\right)=\frac{1}{W}\left(\frac{t^{{\alpha }_0-1}}{\Gamma \left({\alpha }_0\right)}+\frac{t^{\gamma +{\alpha }_0-1}}{\Gamma (\gamma +{\alpha }_0)}\right).$$

This function is well-defined for $t>0$, belongs to the space $C_{-1}(0,\infty )$, and is a Sonin kernel from the class ${\mathcal{S}}_{-1}$, as predicted by Theorem 1.