Scale-Invariant General Fractional Calculus: Mellin Convolution Operators

Abstract

General fractional calculus (GFC) of operators that is defined through the Mellin convolution instead of Laplace convolution is proposed. This calculus of Mellin convolution operators can be considered as an analogue of the Luchko GFC for the Laplace convolution operators. The proposed general fractional differential operators are generalizations of scaling (dilation) differential operator for the case of general form of nonlocality. Semi-group and scale-invariant properties of these operators are proven. The Hadamard and Hadamard-type fractional operators are special case of the proposed operators. The fundamental theorems for the scale-invariant general fractional operators are proven. The proposed GFC can be applied in the study of dynamics, which is characterized by nonlocality and scale invariance.

1. Introduction

Scale invariance is an important property in various processes and systems. For example, the scale invariance is studied in transport and relaxation processes [1], in the trapped ion system [2]; in slow neutron scattering [3]; in different economic phenomena [4]; in stock markets before crashes [5]; in many diverse behaviors of humans and animals [6]; in the behavior of brain neurons [7]; in processes of self-organization of collective behavior, observed in a variety of artificial and natural systems [8]; and in modern physics, the group of conformal transformations of the space–time is used in the description of the symmetry properties of the equations of electromagnetic fields in a vacuum [9], including scaling [10] (p. 409).

Properties of scaling are explored by using the transformation of a function $f\left(x\right)$ under the change of the variable x by some scale factor $\rho$: $x\to \rho x$. For ${\mathbb{R}}^1$, the dilation operator [11] (p. 95), [12] (p. 11), is:

$${\Pi }_{\rho }f\left(t\right)=f\left(\rho t\right),$$

(1)

where $t\in \mathbb{R}$ and $\rho >0$. The differential operator of dilation (scaling) of integer order [11] (p. 95), that has the form

$${\mathcal{D}}_t^m={\left(t\frac{d}{dt}\right)}^m$$

(2)

is scale-invariant since:

$${\Pi }_{\rho }{\mathcal{D}}_t^{m}f\left(t\right)={\mathcal{D}}_t^m{\Pi }_{\rho }f\left(t\right),$$

(3)

where $m\in \mathbb{N}$. One can also consider the differential operator of integer order:

$${\mathcal{D}}_t^{m,\beta }=t^{-\beta }{\mathcal{D}}_t^m{t}^{\beta },$$

(4)

which is a scalar-invariant operator since:

$${\Pi }_{\rho }{\mathcal{D}}_t^{m,\beta }f\left(t\right)={\mathcal{D}}_t^{m,\beta }{\Pi }_{\rho }f\left(t\right),$$

(5)

where $\rho >0$ and $t>0$.

Equations (2) and (4) are integer-order differential operators. In mathematics, the differential operators of arbitrary and non-integer orders are well-known fractional operators [11,12,13,14,15,16,17], which are actively applied to describe nonlocality of the power-law type for various systems and processes [18,19,20,21,22,23,24,25,26] (see also the hanbooks containing 35 reviews in different areas of physics [27,28]). For some fractional differential and integral operators, the scaling properties are considered. As an example, the scaling property of the Riemann–Liouville fractional integral $I_t^{\alpha }$ of order $\alpha >0$ [11] (p. 96), is described as:

$${\Pi }_{\rho }{I}_t^{\alpha }\left[\tau \right]f\left(\tau \right)={\rho }^{\alpha }{I}_t^{\alpha }\left[\tau \right]{\Pi }_{\rho }f\left(\tau \right).$$

(6)

For the Marchaud fractional derivative ${}^M{D}_t^{\alpha }\left[\tau \right]$ of arbitrary order $\alpha >0$, the property of scaling is described as:

$${\Pi }_{\rho }{}^M{D}_t^{\alpha }\left[\tau \right]f\left(\tau \right)={\rho }^{-\alpha }{}^M{D}_t^{\alpha }\left[\tau \right]{\Pi }_{\rho }f\left(\tau \right),$$

(7)

where $\alpha >0$ (see Equation (5).62 in [11] (p. 111)). Note that these fractional operators of arbitrary order $\alpha >0$ cannot be considered as scale-invariant operators.

For applied sciences, it is important to generalize the differential operator of dilation ${\mathcal{D}}_t^m$ from orders $m\in \mathbb{N}$ to arbitrary $\alpha \in {\mathbb{R}}_{+}$. For the first time, fractional integral operators of arbitrary order, which are scale-invariant operators, were proposed by Jacques S. Hadamard [29] in 1892. These operators are called the Hadamard fractional integrals and derivatives. Their properties are considered in Sections 18.3, 23.1 of [11], in Section 2.7 of [12] (pp. 110–120), and in the book [30]. The calculus of fractional Hadamard operators is currently being actively developed [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51], including the so-called Hadamard-type fractional calculus [31,32,33,34,35,36,37], its Caputo form [41,42,49], and other generalizations [45,47,48]. The Hadamard-type and Hadamard fractional operators are researched and applied in various works by Butzer, Kilbas, and Trujillo in [31,34,35]; by Kilbas in [32]; by Kilbas and Titjura in [33]; by Kilbas, Marzan, and Tityura in [36,37]; by Luchko in [38]; by Luchko, and Kiryakova in [39]; by Klimek in [40]; by Jarad, Abdeljawad, and Baleanu in [41]; by Gambo, Jarad, Baleanu, and Abdeljawad in [42]; by Kamocki in [43]; by Almeida in [44]; by Li Ma and Changpin Li [45,46]; by Garra and Polito in [47]; by Garra, Orsingher and Polito in [48]; by Zafar, Rehman, and Shams in [49]; by Fahad, Fernandez, Rehman, and Siddiqi in [50]; and by Weiwei Liu and Lishan Liu [51]. It should be noted that the need to generalize the Hadamard-type fractional calculus for the general form of operator kernels is justified in work [52]. A time-scale invariant fractional dynamics is described in [53], where it is proven the behavior of the system is independent of the kick period for the case of zero initial conditions. Fractional differential and integral operators with continuously distributed scaling (dilation) are suggested in [54] (pp. 160–167), (see Section 9 thereof), which generalize the Erdelyi–Kober operators by using a random scale factor $\rho$. An entropy interpretation of Hadamard-type fractional operators is proposed in [55]. Some fractional linear scale invariant systems are described by Ortigueira in [56]. Note that a special form of fractional scale calculus is proposed by Ortigueira and Bohannan in [57].

Unfortunately, the Hadamard-type fractional operators describe only one type of nonlocality. Therefore, it is important to generalize the Hadamard-type fractional differential and integral operators from the Hadamard-type kernels to a wide class of operator kernels to describe different types of processes and systems with nonlocal scaling in time and space.

The motivation for the proposed work is the necessity and importance of a scale-invariant calculus, which makes it possible to describe a wider class of non-local phenomena, systems, and processes with scale-invariance. It is important to generalize the Hadamard-type fractional calculus and have generalizations of the Hadamard-type fractional differential and integral operators. For example, one can consider the following form of such operators:

$$D_{a,\left(K\right)}^{t,m}\left[\tau \right]f\left(\tau \right)={\int }_a^{t}K(t,\tau ){\mathcal{D}}_{\tau }^{m}f\left(\tau \right)\frac{d\tau }{\tau },$$

(8)

where $m\in \mathbb{N}$, $t\in (a,b)$ with $0\le a<b\le \infty$, and $K(t,\tau )$ is the kernel that is designed to describe the nonlocality. To have scale-invariant operators, one should impose the condition:

$${\Pi }_{\rho }{D}_{a,\left(K\right)}^{t,m}\left[\tau \right]f\left(\tau \right)=D_{\left(K\right)}^{t,m}\left[\tau \right]{\Pi }_{\rho }f\left(\tau \right).$$

(9)

To have a general fractional calculus (GFC), one should define the integral operator:

$$J_{a,\left(M\right)}^t\left[\tau \right]f\left(\tau \right)={\int }_a^{t}M(t,\tau )f\left(\tau \right)\frac{d\tau }{\tau },$$

(10)

such that:

$${\Pi }_{\rho }{J}_{a,\left(K\right)}^t\left[\tau \right]f\left(\tau \right)=J_{\left(K\right)}^t\left[\tau \right]{\Pi }_{\rho }f\left(\tau \right).$$

(11)

In order to obtain the general fractional operators Equations (8) and (10) form a general fractional calculus (GFC), the fundamental theorems should be satisfied [58,59]. For example, the first fundamental theorem should state the equation:

$$D_{a,\left(K\right)}^{t,m}\left[\tau \right]J_{a,\left(K\right)}^{\tau }\left[s\right]f\left(s\right)=f\left(t\right)$$

(12)

for some function spaces and some sets of pairs of operator kernels $(M,K)$.

The problem of finding the types of kernels of the operators satisfying these conditions was first posed and described in [52] (see also [53]). To solve this problem, one can use the analogy with fractional calculus, in which fractional operators are expressed in terms of the Laplace convolution. There is a well-known approach to describing the Laplace convolution type of fractional operators. One of the most interesting and promising fractional calculus of the Laplace convolution operators is the general fractional calculus (GFC). This calculus is based on the Laplace convolution and concepts of kernel pairs that were proposed by Sonin [60] (see also [61]) in 1884, and then has been generalized by Luchko in 2021. The terms “general fractional calculus” (GFC), “general fractional integral” (GFI) operators, and “general fractional differential” (GFD) operators appeared much later, in the Kochubei paper [62] in 2011 (see also [63,64]). To form a fractional calculus, these general fractional operators should have analogues of the fundamental theorems of the standard calculus. The GFC of the Laplace convolution operators was developed in various works [65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82]. In the last few years, the Luchko GFC has been intensively developed in the Luchko papers [83,84,85,86,87,88,89,90,91,92] and in works of other scientists.

To apply this analogy with the Luchko GFC, it is necessary to use the Mellin convolution or its modification instead of the Laplace convolution, which is used in the GFC in the Luchko form. The proposed generalization of the Mellin transform is the simplest generalization for which this transformation has the associativity property. Note that various generalizations of the Mellin transform are known (for example, see [93,94]). The proposed GFC, which is suggested in this paper, can be called the scale-invariant GFC. Obviously, the scale-invariant GFC must contain the Hadamard-type fractional operators [31,32] as a special case. This requirement is in fact analogous to the fact that the Luchko GFC contains the well-known fractional operators of the Riemann–Liouville and Caputo types as a special case.

In Section 2, scale-invariant general fractional integral operators and fractional differential operators are proposed. Section 2.1 explains the need to modify the standard Mellin convolution. A modification (generalization) of the standard Mellin convolution is proposed and some properties of this convolution are described. In Section 2.2, some sets of functions and sets of kernel pairs are proposed. Some properties of these kernels are proven. In Section 2.3, the scale-invariant general fractional integral operator is defined and properties are described. In Section 2.4, scale-invariant general fractional differential operators are defined and the scale property is considered. In Section 3, the fundamental theorems of scale-invariant general fractional operators are proven. In Section 3.1, the well-known fundamental theorems for Hadamard-type fractional calculus are described. In Section 3.2, the first fundamental theorems for scale-invariant general fractional integral and differential operators are proven. In Section 3.3, the second fundamental theorems for scale-invariant general fractional differential and integral operators are proven. In Section 4, a short conclusion is given.

2. Scale-Invariant General Fractional Operators

This section proposes the following: a generalization of the Mellin convolution, sets of functions and kernel pairs, scale-invariant GF integral (GFI) operators, and scale-invariant GF differential (GFD) operators.

2.1. Generalization of the Mellin Convolution

To define scale-invariant general fractional operators, one can use a generalization of the Mellin convolution instead of the Laplace convolution, which is used in the Luchko GFC.

The standard Mellin convolution is defined in the following form (see [12] (p. 22), and [13] (p. 253)).

Definition 1. 

The Mellin convolution of the two functions $f\left(x\right)$ and $g\left(x\right)$, which are defined on ${\mathbb{R}}_{+}=(0,\infty )$, is defined by the equation:

$$(f\circ g)\left(x\right)={\int }_0^{x}f\left(\frac{x}{u}\right)g\left(u\right)\frac{du}{u},$$

(13)

for all $x\in {\mathbb{R}}_{+}$.

The Mellin convolution Equation (13) has the commutativity property [12] (p. 22), as:

$$(f\circ g)\left(x\right)=(g\circ f)\left(x\right),$$

(14)

where $x>0$.

Remark 1. 

Unfortunately, the Mellin convolution Equation (13) does not have the property of associativity for a wide class of functions. In general, the rule:

$$\left(\right(f\circ g)\circ h)\left(x\right)\ne (f\circ (g\circ h\left)\right)\left(x\right)$$

(15)

is violated for the Mellin convolution Equation (13).

• Let us prove the violation of the associativity of the Mellin convolution Equation (13).
• The standard Mellin convolution Equation (13) of $f\left(x\right)$ and $k\left(x\right)$ is written as:

$$(f\circ k)\left(x\right)={\int }_0^{x}f\left(\frac{x}{u}\right)k\left(u\right)\frac{du}{u},$$

(16)

where $x>u>0$. Let the function $k\left(u\right)$ be the Mellin convolution in the form

$$k\left(u\right)=(g\circ h)\left(u\right)={\int }_0^{u}g\left(\frac{u}{t}\right)h\left(t\right)\frac{dt}{t},$$

(17)

where $u>t>0$. Therefore, substitution of Equation (17) into Equation (16) gives:

$$\begin{array}{c}(f\circ (g\circ h))\left(x\right)={\int }_0^{x}f\left(\frac{x}{u}\right)(g\circ h)\left(u\right)\frac{du}{u}=\\ {\int }_c^{x}f\left(\frac{x}{u}\right)\left({\int }_0^{u}g\left(\frac{u}{t}\right)h\left(t\right)\frac{dt}{t}\right)\frac{du}{u}=\\ {\int }_0^x\left({\int }_t^{x}f\left(\frac{x}{u}\right)g\left(\frac{u}{t}\right)\frac{du}{u}\right)h\left(t\right)\frac{dt}{t}\ne ((f\circ g)\circ h)\left(x\right).\end{array}$$

(18)

Here, Dirichlet’s equation is used (see Equation (1).32 in [11] (p. 9)) in the form

$${\int }_c^{x}du{\int }_c^{u}F(u,t)dt={\int }_c^{x}dt{\int }_t^{x}F(u,t)du,$$

(19)

for the function

$$F(u,t)=\frac{1}{ut}f\left(\frac{x}{u}\right)g\left(\frac{u}{t}\right)h\left(t\right),$$

(20)

where $x>u>t>0$.

• One can see that Equation (18) contains the expression:

$${\int }_t^{x}f\left(\frac{x}{u}\right)g\left(\frac{u}{t}\right)\frac{du}{u}\ne {\int }_0^{t}f\left(\frac{t}{u}\right)g\left(u\right)\frac{du}{u}=(f\circ g)\left(t\right).$$

(21)

As a result, the associativity of the Mellin convolution Equation (13) is violated, in the general case.

The property of associativity is necessary to generalize both the semi-group property of GFI operators, and for the fulfillment of the fundamental theorems of the suggested GFC.

Remark 2. 

In additional to the violation of associativity of the Mellin convolution Equation (13), the Hadamard fractional integrals on finite intervals $[a,b]$ cannot be rewritten by using the standard Mellin convolution.

• The standard Mellin convolution cannot be used to represent the Hadamard-type fractional operators on finite intervals $[a,b]$ as Mellin convolution operators.

Let us define a generalization of the Mellin convolution Equation (13).

Definition 2. 

Let $0<a<b<\infty$.

• The generalized Mellin convolution (GM-convolution) of the two functions $f\left(x\right)$ and $g\left(x\right)$, which are defined on $(a,b)$, is:

$$\left(f{\ast }_{\mathcal{M}}g\right)\left(\frac{x}{t}\right)={\int }_t^{x}f\left(\frac{x}{u}\right)g\left(\frac{u}{t}\right)\frac{du}{u},$$

(22)

for all $t,x\in [a,b]$ such that $0<a<t<x<b$.

Remark 3. 

Let us explain the reason why the notation $\left(f{\ast }_{\mathcal{M}}g\right)(x/t)$ is used instead of the notation $\left(f{\ast }_{\mathcal{M}}g\right)(x,t)$. In other words, let us demonstrate that the right side of Equation (22) depends on the ratio of the variables x and t.

• Using the variable $z=u/t$ instead of u, the integral of Equation (22) becomes:

$${\int }_t^{x}f\left(\frac{x}{u}\right)g\left(\frac{u}{t}\right)\frac{du}{u}={\int }_1^{x/t}f\left(\frac{(x/t)}{u}\right)g\left(z\right)\frac{dz}{z},$$

(23)

where $x>t$.

• Note that the generalized Mellin convolution Equation (22) can be written by an equation similar to Equation (13) of the standard Mellin convolution:

$$\left(f{\ast }_{\mathcal{M}}g\right)\left(x\right)={\int }_1^{x}f\left(\frac{x}{u}\right)g\left(u\right)\frac{du}{u},$$

(24)

for all $x>1$. However, in the generalized Mellin convolution, the lower limit of the integral is equal to one and not to zero, which is used in Equation (13).

Remark 4. 

Let us give an additional explanation of the reasons for considering the GM-convolution instead of the standard Mellin convolution. These comments and explanations will be given for Laplace convolution, which is actively used in the general fractional calculus.

• Let us consider fractional integrals expressed in terms of the Laplace convolution:

$$\left(f{\ast }_{L}g\right)\left(x\right)={\int }_0^{x}f(x-u)g\left(u\right)du,$$

(25)

where $x>u>0$. Let us first give some comments about the fractional integral for finite intervals $(a,b)$. The Riemann–Liouville fractional integrals on the finite interval $(a,b)$ are defined [11,12] by the equation:

$$\left(I_{a+}^{\alpha }g\right)\left(x\right)={\int }_a^x{h}_{\alpha }(x-u)g\left(u\right)du,$$

(26)

where:

$$h_{\alpha }(x-u)=\frac{{(x-u)}^{\alpha -1}}{\Gamma \left(\alpha \right)}$$

with $a<x<b$, $\alpha >0$ and $g\left(x\right)\in L_1(a,b)$.

• It should be noted that the integrals in Equation (26) on the finite interval $(a,b)$ usually consider for functions:

$$g\left(x\right)=f(x-a),$$

(27)

where $f\left(x\right)\in L_1(0,b-a)$ (see Table 9.1 of [11] (p. 173)). Almost all equations of Table 9.1 in [11] are presented as:

$${\int }_a^x{h}_{\alpha }(x-u)f(u-a)du=G(x-a).$$

(28)

For example, Equation (23) of Table 9.1 in [11] (p. 173) has the form

$${\int }_a^x{h}_{\alpha }(x-u)e_{\mu ,\beta }(u-a)du=e_{\mu ,\alpha +\beta }(x-a),$$

(29)

where $\alpha >0$, $\beta >0$, $\mu >0$ and

$$e_{\mu ,\beta }(u-a)={(u-a)}^{\beta -1}{E}_{\mu ,\beta }\left[{(u-a)}^{\mu }\right].$$

As a result, the operators in Equation (26) on the finite interval $(a,b)$ can be defined as:

$$\left(I_{a+}^{\alpha }f\right)\left(x\right)={\int }_a^x{h}_{\alpha }(x-u)f(u-a)du.$$

(30)

Representations of Equation (30) are used to extend the GFC on finite intervals $(a,b)$ [80,92]. In this extension of GFC, one can use $-\infty <a<b<\infty$, or $-\infty <a<b=\infty$.

• The general fractional $I_{\left(M\right),a+}^x$ on the interval $(a,b)$, where $-\infty <a<b\le \infty$, can be defined by the equation:

$$I_{\left(M\right),a+}^x\left[t\right]f(t-a):={\int }_a^{x}M(x-t)f(t-a)dt,$$

(31)

where $x\ge t>a$.

• Therefore the following convolution should be used

$$(f\ast g)(t-a)={\int }_a^{x}f(x-u)g(u-a)du,$$

(32)

where $x>u>a$. Using the variable $z=u-a$, one can obtain:

$${\int }_a^{x}f(x-u)g(u-a)du={\int }_0^{x-a}f((x-a)-z)g\left(z\right)dz=\left(f{\ast }_{L}g\right)(x-a).$$

(33)

As a result, convolution Equation (32) can be expressed through the standard Laplace convolution Equation (25).

• When considering convolutions, in which the difference $x-a$ is replaced by the ratio $x/a$, this relationship with standard convolution is violated. The generalized Mellin convolution Equation (22) cannot be expressed through the standard Mellin convolution Equation (13) in the general case (see Remark 3). Therefore, in the general case, we have:

$$(f\circ g)\left(\frac{x}{a}\right)\ne \left(f{\ast }_{\mathcal{M}}g\right)\left(\frac{x}{a}\right).$$

(34)

The commutativity property can be proven for the generalized Mellin convolution in Equation (22).

Theorem 1. 

The generalized Mellin convolution has the commutativity property in the form

$$\left(f{\ast }_{\mathcal{M}}g\right)\left(\frac{x}{t}\right)=\left(g{\ast }_{\mathcal{M}}f\right)\left(\frac{x}{t}\right)$$

(35)

for $x>t>0$.

Proof. 

The generalized Mellin convolution of the two functions $f\left(x\right)$ and $g\left(x\right)$ is written as:

$$\left(f{\ast }_{\mathcal{M}}g\right)\left(\frac{x}{t}\right)={\int }_t^{x}f\left(\frac{x}{u}\right)g\left(\frac{u}{t}\right)\frac{du}{u},$$

(36)

where $x>u>t>0$. Using the new variable $z=\left(tx\right)/u$, then:

$$\frac{du}{u}=-\frac{dz}{z},\frac{x}{u}=\frac{z}{t},\frac{u}{t}=\frac{x}{z},$$

and Equation (36) becomes:

$$\left(f{\ast }_{\mathcal{M}}g\right)\left(\frac{x}{t}\right)=-{\int }_x^{t}f\left(\frac{z}{t}\right)g\left(\frac{x}{z}\right)\frac{dz}{z}=$$

$${\int }_t^{x}g\left(\frac{x}{z}\right)f\left(\frac{z}{t}\right)\frac{dz}{z}=\left(g{\ast }_{\mathcal{M}}f\right)\left(\frac{x}{t}\right),$$

(37)

where $x>z>t>0$. □

The associativity property can be proven for the generalized Mellin convolution Equation (22).

Theorem 2. 

The generalized Mellin convolution has the associativity property in the form

$$\left(f{\ast }_{\mathcal{M}}\left(g{\ast }_{\mathcal{M}}h\right)\right)\left(\frac{x}{c}\right)=\left(\left(f{\ast }_{\mathcal{M}}g\right){\ast }_{\mathcal{M}}h\right))\left(\frac{x}{c}\right),$$

(38)

for $x>c>0$.

Proof. 

The generalized Mellin convolution of the two functions $f\left(x\right)$ and $k\left(x\right)$ can be written as:

$$\left(f{\ast }_{\mathcal{M}}k\right)\left(\frac{x}{c}\right)={\int }_c^{x}f\left(\frac{x}{u}\right)k\left(\frac{u}{c}\right)\frac{du}{u},$$

(39)

where $x>u>c>0$. Let the function $k\left(u/c\right)$ be the generalized Mellin convolution of the two functions g and h in the form

$$k\left(\frac{u}{c}\right)=\left(g{\ast }_{\mathcal{M}}h\right)\left(\frac{u}{c}\right)={\int }_c^{u}g\left(\frac{u}{t}\right)h\left(\frac{t}{c}\right)\frac{dt}{t},$$

(40)

where $u>t>c>0$. Therefore, substitution of the expression in Equation (40) into Equation (39) gives:

$$\begin{array}{c}\left(f{\ast }_{\mathcal{M}}\left(g{\ast }_{\mathcal{M}}h\right)\right)\left(\frac{x}{c}\right)={\int }_c^{x}f\left(\frac{x}{u}\right)\left(g{\ast }_{\mathcal{M}}h\right)\left(\frac{u}{c}\right)\frac{du}{u}=\\ {\int }_c^{x}f\left(\frac{x}{u}\right)\left({\int }_c^{u}g\left(\frac{u}{t}\right)h\left(\frac{t}{c}\right)\frac{dt}{t}\right)\frac{du}{u}=\\ {\int }_c^x\left({\int }_t^{x}f\left(\frac{x}{u}\right)g\left(\frac{u}{t}\right)\frac{du}{u}\right)h\left(\frac{t}{c}\right)\frac{dt}{t}=\left(\left(f{\ast }_{\mathcal{M}}g\right){\ast }_{\mathcal{M}}h\right)\left(\frac{x}{c}\right).\end{array}$$

(41)

Here, the Dirichlet equation is used (see Equation (1).32 in [11] (p. 9)) in the form

$${\int }_c^{x}du{\int }_c^{u}F(u,t)dt={\int }_c^{x}dt{\int }_t^{x}F(u,t)du,$$

(42)

for the function:

$$F(u,t)=f\left(\frac{x}{u}\right)g\left(\frac{u}{t}\right)h\left(\frac{t}{c}\right),$$

(43)

where $x>u>t>c>0$. □

As a result, in this subsection the commutativity and associativity of the GM-convolution are proven.

2.2. Sets of Functions and Kernel Pairs

In order to define general integral (and differential) operators through the generalized Mellin convolution by the equations:

$${\mathcal{I}}_{a,\left(M\right)}^x\left[u\right]f\left(u\right)=\left(M{\ast }_{\mathcal{M}}f\right)\left(\frac{x}{a}\right),$$

one should consider sets of operator kernels $M\left(x\right)$ and functions $f\left(x\right)$ for the interval $(a,b)\subset (0,\infty )$.

Let us define a set of kernels given on the interval $(a,b)$, where $0<a<b\le \infty$.

Definition 3. 

Let $\beta ,\mu \in \mathbb{R}$, and let a function $M\left(z\right)$ be represented in the form

$$M\left(z\right)=z^{-\beta }G\left(z\right),$$

(44)

for all $z>0$, where $\beta >\mu$ and $G\left(z\right)\in C(0,\infty )$.

• Then, the set of such kernels $M\left(z\right)$ is denoted as $C_{-\mu }(0,\infty )$.

Remark 5. 

If $0<a<t<x<b<\infty$, one can consider kernel Equation (44) in the form

$$M\left(\frac{x}{t}\right)={\left(\frac{x}{t}\right)}^{-\beta }G\left(\frac{x}{t}\right),$$

(45)

where $G\left(x\right)\in C(1,b/a)$ instead of $G\left(z\right)\in C(0,\infty )$.

Example 1. 

As an example of the functions $M\left(z\right)$, which belong to the set $C_{-\mu }(0,\infty )$, one can consider the kernel

$$M\left(z\right)=M_{\alpha ,\beta }\left(z\right)=\frac{1}{\Gamma \left(\alpha \right)}{z}^{-\beta }{\left(\ln \left(z\right)\right)}^{\alpha -1},(\beta >\mu ,\alpha >0),$$

(46)

where $z=x/t$ and $0<t<x<\infty$, which can be written as:

$$M_{\alpha ,\beta }\left(\frac{x}{t}\right)=\frac{1}{\Gamma \left(\alpha \right)}{\left(\frac{t}{x}\right)}^{\beta }{\left(\ln \frac{x}{t}\right)}^{\alpha -1}.$$

(47)

For the GM-convolution Equation (22) of kernels Equation (46), one can prove the following theorem.

Theorem 3. 

Let $\alpha ,\sigma \in {\mathbb{R}}_{+}$ and $\beta \in \mathbb{R}$, and let kernels $M_{\alpha ,\beta }\left(x\right)$, $M_{\sigma ,\beta }\left(x\right)$ be defined by Equation (46).

• Then, kernels $M_{\alpha ,\beta }\left(x\right)$ and $M_{\sigma ,\beta }\left(x\right)$ satisfy the following property:

$$\left(M_{\alpha ,\beta }{\ast }_{\mathcal{M}}{M}_{\sigma ,\beta }\right)\left(\frac{x}{t}\right)=M_{\alpha +\sigma ,\beta }\left(\frac{x}{t}\right).$$

(48)

In a particular case, one can obtain:

$$\left(M_{m,\beta }{\ast }_{\mathcal{M}}{M}_{n,\beta }\right)\left(\frac{x}{t}\right)=M_{m+n,\beta }\left(\frac{x}{t}\right),$$

(49)

where $n,m\in \mathbb{N}$.

Proof. 

Let us consider the kernels

$$M_{\alpha ,\beta }\left(z\right)=\frac{1}{\Gamma \left(\alpha \right)}{z}^{-\beta }{\left(\ln \left(z\right)\right)}^{\alpha -1},(\beta >\mu \alpha >0),$$

(50)

$$M_{\sigma ,\beta }\left(z\right)=\frac{1}{\Gamma \left(\sigma \right)}{z}^{-\beta }{\left(\ln \left(z\right)\right)}^{\sigma -1},(\beta >\mu \sigma >0).$$

(51)

The GM-convolution of these functions has the form

$$\begin{array}{c}\left(M_{\alpha ,\beta }{\ast }_{\mathcal{M}}{M}_{\sigma ,\beta }\right)\left(\frac{x}{t}\right)={\int }_t^x{M}_{\alpha ,\beta }\left(\frac{x}{u}\right)M_{\sigma ,\beta }\left(\frac{u}{t}\right)\frac{du}{u}=\\ {\int }_t^x\frac{du}{u}\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\sigma \right)}{\left(\frac{x}{u}\right)}^{-\beta }{\left(\frac{u}{t}\right)}^{-\beta }{\left(\ln \left(\frac{x}{u}\right)\right)}^{\alpha -1}{\left(\ln \left(\frac{u}{t}\right)\right)}^{\sigma -1}=\\ {\left(\frac{x}{t}\right)}^{-\beta }\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\sigma \right)}{\int }_t^x\frac{du}{u}{\left(\ln \left(\frac{x}{u}\right)\right)}^{\alpha -1}{\left(\ln \left(\frac{u}{t}\right)\right)}^{\sigma -1}.\end{array}$$

(52)

Let us use the variable

$$y=\frac{\ln (u/t)}{\ln (x/t)};$$

the integral of Equation (52) becomes:

$$\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\sigma \right)}{\int }_t^x\frac{du}{u}{\left(\ln \left(\frac{x}{u}\right)\right)}^{\alpha -1}{\left(\ln \left(\frac{u}{t}\right)\right)}^{\sigma -1}=$$

$$\frac{1}{\Gamma \left(\alpha \right)\Gamma \left(\sigma \right)}{\left(\ln \left(\frac{x}{t}\right)\right)}^{\alpha +\sigma -1}{\int }_0^{1}dy{(1-y)}^{\alpha -1}{y}^{\sigma -1}=$$

$$\frac{B(\sigma ,\alpha )}{\Gamma \left(\alpha \right)\Gamma \left(\sigma \right)}{\left(\ln \left(\frac{x}{t}\right)\right)}^{\alpha +\sigma -1}=\frac{1}{\Gamma (\alpha +\sigma )}{\left(\ln \left(\frac{x}{t}\right)\right)}^{\alpha +\sigma -1}.$$

As a result, we obtain:

$$\left(M_{\alpha ,\beta }{\ast }_{\mathcal{M}}{M}_{\sigma ,\beta }\right)\left(\frac{x}{t}\right)=\frac{1}{\Gamma (\alpha +\sigma )}{\left(\frac{x}{t}\right)}^{-\beta }{\left(\ln \left(\frac{x}{t}\right)\right)}^{\alpha +\sigma -1}=M_{\alpha +\sigma ,\beta }\left(\frac{x}{t}\right),$$

(53)

where $\alpha >0$, $\sigma >0$, $\beta >\mu$. □

Let us define the scale-invariant analog of the Sonin and Luchko conditions for the kernel pairs.

Definition 4. 

Let $M\left(x\right)$ and $K\left(x\right)$ be functions that satisfy the following conditions.

(1) 

The function $M\left(z\right)$ and $K\left(z\right)$ belong to the set $C_{-\mu }(0,\infty )$.

(2) 

The generalized Mellin convolution of the functions $M\left(x\right)$ and $K\left(x\right)$ has the form

$$\left(M{\ast }_{\mathcal{M}}K\right)\left(x\right)=M_{m,\beta }\left(x\right),$$

(54)

where:

$$M_{m,\beta }\left(x\right)=\frac{1}{\Gamma \left(m\right)}{x}^{-\beta }{\left(\ln x\right)}^{m-1},$$

(55)

with $m\in \mathbb{N}$, $\beta \in \mathbb{R}$, and $\beta >\mu$.

• Then, the set of such kernel pairs is denoted as ${\mathcal{M}}_{m,\beta }$.

Remark 6. 

Equation (54) can be written as:

$$\left(M{\ast }_{\mathcal{M}}K\right)\left(\frac{x}{t}\right)={\int }_t^{x}M\left(\frac{x}{u}\right)K\left(\frac{u}{t}\right)\frac{du}{u}=M_{m,\beta }\left(\frac{x}{t}\right),$$

(56)

where $0<a<t<u<x<b<\infty$.

Example 2. 

Let $\alpha \in {\mathbb{R}}_{+}$, $m-1<\alpha <m$, and $\beta \in \mathbb{R}$, $m\in \mathbb{N}$.

• Let us consider the kernel pair $(M,K)$ that belongs to the set ${\mathcal{M}}_{m,\beta }$ with the kernels

$$M\left(z\right)=M_{\alpha ,\beta }\left(z\right),K\left(z\right)=M_{m-\alpha ,\beta }\left(z\right),$$

(57)

where $\beta >\mu$.

• Then, using Theorem 3 with $\sigma =m-\alpha$, one can obtain:

$$\left(M_{\alpha ,\beta }{\ast }_{\mathcal{M}}{M}_{m-\alpha ,\beta }\right)\left(x\right)=M_{m,\beta }\left(x\right).$$

(58)

Let us define a set of functions given on interval $(a,b)$, where using Definition 3 and Remark 5, let us introduce notation for the set of functions that are acted upon by operators. As a set of functions given on interval $(a,b)$, where $0<a<b\le \infty$, one can consider the set $C_{-\eta }(0,\infty )$. In this case, the function $f\left(z\right)$ can be represented as:

$$f\left(z\right)=z^{-q}g\left(z\right)$$

(59)

for all $z>0$, where $q<\eta$ and $g\left(z\right)\in C(0,\infty )$, where $q,\eta \in \mathbb{R}$. If $0<a<t<x<b<\infty$, one can consider Equation (59) in the form

$$f\left(\frac{x}{t}\right)={\left(\frac{x}{t}\right)}^{-q}g\left(\frac{x}{t}\right),$$

(60)

where $g\left(z\right)\in C(1,b/a)$ instead of $g\left(z\right)\in C(0,\infty )$. These notations will be used later.

Remark 7. 

For the scale-invariant calculus, in the sets $C_{-\eta }(0,\infty )$ and $C_{-\mu }(0,\infty )$, there will be used the assumption $\mu >\eta$, and:

$$\beta >\mu >\eta >q.$$

(61)

In some cases, μ can be considered as the infimum of β, and η as the supremum of q, i.e., $\mu =\sup \left(\beta \right)$ and $\eta =\inf \left(q\right)$.

Remark 8. 

Let kernel $M\left(z\right)$ belong to the set $C_{-\mu }(0,\infty )$, and let $f\left(t\right)\in C_{-\eta }(0,\infty )$ with $\eta <\mu <\beta$.

• Then, the generalized Mellin convolution of the kernel $M\left(x\right)$ and the function $f\left(x\right)$ can be represented as:

$$\begin{array}{c}\left(M{\ast }_{\mathcal{M}}f\right)\left(\frac{x}{t}\right)={\int }_t^{x}M\left(\frac{x}{u}\right)f\left(\frac{u}{t}\right)\frac{du}{u}=\\ x^{-\beta }{t}^q{\int }_t^{x}G\left(\frac{x}{u}\right)g\left(\frac{u}{t}\right)u^{\beta -q}\frac{du}{u}=x^{-\beta }{t}^q{\int }_t^{x}G\left(\frac{x}{u}\right)g\left(\frac{u}{t}\right)u^{\beta -q-1}du,\end{array}$$

(62)

where $\beta >q$ such that $\beta -q-1>-1$, and the product $G\left(x/u\right)g(u/t)$ is a continuous function as a function of the variable u for all values of u satisfying the inequality

$$0<a<t<u<x<\infty .$$

(63)

Using the first Weierstrass theorem (the extreme value theorem), which states that a continuous function on the closed interval $[a,b]$ is bounded on this interval, one can prove that the GM-convolution $\left(M{\ast }_{\mathcal{M}}f\right)(x/t)$ is bounded on the closed interval $[a,b]$, where $0<a<t<x<b<\infty$.

Let us give a definition of the set $X_q^p(a,b)$, [32] (p. 1193).

Definition 5. 

Let $0\le a<b\le \infty$.

• The space $X_q^p(a,b)$ with $q\in \mathbb{R}$ and $p\ge 1$ is an $L_p$-space with the power weight, which consists of those real-valued Lebesgue measurable functions $f\left(t\right)$ on $(a,b)$ for which:

$${\left|\right|f\left|\right|}_{X_q^p}={\left({\int }_a^b\frac{dt}{t}{\left|t^{q}f\left(t\right)\right|}^p\right)}^{1/p}<\infty .$$

(64)

In particular, when $q=1/p$, the space $X_q^p(a,b)$ coincides with the space $L_p(a,b)$, i.e., $X_{1/p}^p(a,b)=L_p(a,b)$.

Theorem 4. 

Let $\alpha ,\sigma \in {\mathbb{R}}_{+}$, $\beta \in \mathbb{R}$, and let $f\left(x\right)\in X_q^p(a,b)$ with $\beta >q$, $1\le p\le \infty$.

• Let kernels $M_{\alpha ,\beta }\left(x\right)$, $M_{\sigma ,\beta }\left(x\right)$ be defined by Equation (46).
• Then, the property

$$\left(M_{\alpha ,\beta }{\ast }_{\mathcal{M}}\left(M_{\sigma ,\beta }{\ast }_{\mathcal{M}}f\right)\right)\left(\frac{x}{t}\right)=\left(M_{\alpha +\sigma ,\beta }{\ast }_{\mathcal{M}}f\right)\left(\frac{x}{t}\right)$$

(65)

holds, where $b>x>t\ge a>0$.

Proof. 

Theorem 4 directly follows from the associativity of the GM-convolution, which is proven in Theorem 2 and Theorem 3 that describe the GM-convolution of the kernels $M_{\alpha ,\beta }\left(x\right)$, $M_{\sigma ,\beta }\left(x\right)$. □

Remark 9. 

For the case $t=a$, Theorem 4 was proven as Theorem 4.1 in [32] (pp. 1200–1201), f in the form

$$\left(M_{\alpha ,\beta }{\ast }_{\mathcal{M}}\left(M_{\sigma ,\beta }{\ast }_{\mathcal{M}}f\right)\right)(x/a)=\left(M_{\alpha +\sigma ,\beta }{\ast }_{\mathcal{M}}f\right)(x/a),$$

(66)

where the function $f(x/a)$ should be used instead of $f\left(x\right)$ in the proof.

2.3. Scale-Invariant GF Integral Operators

The GFI operator with the kernel $M\left(x\right)$ is defined by the following definition.

Definition 6. 

Let kernel pair $(M,K)$ belong to the set ${\mathcal{M}}_{m,\beta }$.

• The scale-invariant general fractional integral operator with the kernel $M\left(x\right)$ is defined by the equation

$${\mathcal{J}}_{a,\left(M\right)}^t\left[\tau \right]f\left(\tau \right):=\left(M{\ast }_{\mathcal{M}}f\right)\left(\frac{t}{a}\right)={\int }_a^{t}M\left(\frac{t}{\tau }\right)f\left(\frac{\tau }{a}\right)\frac{d\tau }{\tau },$$

(67)

where $0<a<\tau <t<b<\infty$.

Note that for $a>0$, in many cases, it is possible to define and use the function $f^{\ast }\left(\tau \right)=f\left(a\tau \right)$ instead of $f\left(x\right)$, since $f^{\ast }(\tau /a)=f\left(\tau \right)$.

Example 3. 

For example, one can consider the kernel

$$M\left(\frac{t}{\tau }\right)=M_{\alpha ,\beta }\left(\frac{t}{\tau }\right)=\frac{1}{\Gamma \left(\alpha \right)}{\left(\frac{t}{\tau }\right)}^{-\beta }{\left(\ln \frac{t}{\tau }\right)}^{\alpha -1},$$

(68)

where $t>\tau >a>0$.

• General fractional integral Equation (67) with kernel Equation (68) is called the Hadamard-type fractional integral of the order $\alpha >0$ (see Equation (1.3) in [32] for $a>0$, where $f\left(x\right)$ is used instead of $f(x/a)$). The general fractional integral with kernel Equation (68) is written as:

$${\mathcal{J}}_{a,\left(M_{\alpha ,\beta }\right)}^t\left[\tau \right]f\left(\tau \right)={\mathcal{J}}_{a,\beta }^{t,\alpha }\left[\tau \right]f\left(\frac{\tau }{a}\right)=\frac{1}{\Gamma \left(\alpha \right)}{\int }_a^t{\left(\frac{t}{\tau }\right)}^{-\beta }{\left(\ln \frac{t}{\tau }\right)}^{\alpha -1}f\left(\frac{\tau }{a}\right)\frac{d\tau }{\tau },$$

(69)

where $\beta \in \mathbb{R}$ and $\beta >\eta$.

• The Hadamard-type fractional integral operators with standard Mellin convolution were first proposed by Paul L. Butzer, Anatoly A. Kilbas, and Juan J. Trujillo in 2002 [31,32]. The properties of the Hadamard-type fractional integral operators are described in [31,32,33,34,35,36,37]. Integral operator Equation (69) with $\beta =0$ is called the Hadamard fractional integral operators. These operators were proposed by Jacques S. Hadamard [29] in 1892. The properties of these operators are described in [12] (pp. 110–120) (see also Sections 18.3, 23.1 of [11] and [30], respectively).

The Hadamard-type fractional integral operator ${\mathcal{J}}_{\left(M_{\alpha ,\beta }\right)}^t$ is bounded in the space $X_q^p(a,b)$, where $p\ge 1$. This statement is proven as Theorem 2.1 in [32] (pp.1194–1195)).

Example 4. 

For example, one can consider the kernel Equation (68) with $\alpha =m\in \mathbb{N}$ in the form

$$M\left(\frac{t}{\tau }\right)=M_{m,\beta }\left(\frac{t}{\tau }\right)=\frac{1}{\Gamma \left(m\right)}{\left(\frac{t}{\tau }\right)}^{-\beta }{\left(\ln \frac{t}{\tau }\right)}^{m-1},$$

(70)

where $t>\tau >a>0$. General fractional integral Equation (67) with kernel Equation (70) is the standard integral operator of the integer order $m\in \mathbb{N}$ (see Equation (1).1 in [31,32] for $a>0$ and [32] for $a=0$), that has the form

$$\begin{array}{c}{\mathcal{J}}_{a,\beta }^{t,m}\left[\tau \right]f\left(\tau \right)=t^{-\beta }{\int }_a^t\frac{d{t}_1}{t_1}{\int }_a^{t_1}\frac{d{t}_2}{t_2}\dots {\int }_a^{t_{m-1}}{t}_m^{\beta }f\left(t_m\right)\frac{d{t}_m}{t_m}=\\ \frac{1}{\Gamma \left(m\right)}{\int }_a^t{\left(\frac{\tau }{t}\right)}^{\beta }{\left(\ln \frac{t}{\tau }\right)}^{m-1}f\left(\tau \right)\frac{d\tau }{\tau },\end{array}$$

(71)

where $\Gamma \left(m\right)=(m-1)!$ for $m\in \mathbb{N}$.

• The operator Equation (69) with $\alpha =m\in \mathbb{N}$ has the form

$${\mathcal{J}}_{a,\left(M_{\alpha ,\beta }\right)}^t\left[\tau \right]f\left(\tau \right)={\mathcal{J}}_{a,\beta }^{t,m}\left[\tau \right]f\left(\frac{\tau }{a}\right),$$

(72)

where $\beta \in \mathbb{R}$ and $\beta >\eta$.

The boundedness of the GF operator is given, for example, by the following statement.

Theorem 5. 

Let kernel pair $(M,K)$ belong to the set ${\mathcal{M}}_{m,\beta }$, let kernel $M\left(z\right)$ belong to the set $C_{-\mu }(0,\infty )$, and let $f\left(t\right)\in C_{-\eta }(0,\infty )$ with $\eta <\mu <\beta$.

• Then, the scale-invariant general fractional integral operator with the kernel $M\left(x\right)$ of the function $f\left(x\right)$ is bounded on the the closed interval $[a,b]$, where $0<a<t<x<b<\infty$.

Proof. 

The proof directly follows from the definitions of the sets $C_{-\mu }(0,\infty )$ and $C_{-\eta }(0,\infty )$, properties of continuous functions, the first Weierstrass theorem (the extreme value theorem), and Remark 5. □

The scaling property of the general fractional integral is described by the following theorem.

Theorem 6 

(Scale-invariance of GFI operators). Let kernels $M_1\left(x\right)$, $M_2\left(x\right)$ belong to the set ${\mathcal{M}}_{\alpha ,\beta }$.

• Then, general fractional integral operators Equation (67) satisfy the following scaling property:

$${\Pi }_{\rho }{\mathcal{J}}_{a,\left(M\right)}^x\left[\tau \right]f\left(u\right)={\mathcal{J}}_{a,\left(M\right)}^x\left[u\right]{\Pi }_{\rho }f\left(u\right),$$

(73)

where $\rho >0$, $b>t>a$ and:

$${\Pi }_{\rho }f\left(t\right)=f\left(\rho t\right)$$

(74)

with $\rho >0$.

Proof. 

Using the definition of the operator ${\Pi }_{\rho }$ one can obtain:

$$\begin{array}{c}{\Pi }_{\rho }{\mathcal{J}}_{a,\left(M\right)}^x\left[u\right]f\left(u\right)={\Pi }_{\rho }{\int }_a^{x}M\left(\frac{x}{u}\right)f\left(\frac{u}{a}\right)\frac{du}{u}=\\ {\int }_a^{\rho x}M\left(\frac{\rho x}{u}\right)f\left(\frac{u}{a}\right)\frac{du}{u}={\int }_a^{x}M\left(\frac{\rho x}{\rho z}\right)f\left(\frac{\rho z}{a}\right)\frac{dz}{z}=\\ {\int }_a^{x}M\left(\frac{x}{z}\right)f\left(\rho \frac{z}{a}\right)\frac{dz}{z}={\mathcal{J}}_{a,\left(M\right)}^x\left[z\right]{\Pi }_{\rho }f\left(z\right),\end{array}$$

(75)

where the variable $z=u/\rho$ is used. □

Remark 10. 

Note that as a corollary of the property, which states the dependence of the GM-convolution on the ratio $x/a$ (see Remark 3), one can state that the GFI operator does not change with the simultaneous transformation $x\to \rho x$ and $a\to \rho a$, $b\to \rho b$.

The following theorem shows the semi-group property of the scale-invariant GF integral operators.

Theorem 7 

(Semi-group property of GFI operators). Let kernels $M_1\left(x\right)$, $M_2\left(x\right)$ belong to the set ${\mathcal{M}}_{\alpha ,\beta }$.

• Then, the property

$${\mathcal{J}}_{a,\left(M_1\right)}^x\left[u\right]{\mathcal{J}}_{a,\left(M_2\right)}^u\left[t\right]f\left(t\right)={\mathcal{J}}_{a,\left(M_1{\ast }_{\mathcal{M}}{M}_2\right)}^x\left[t\right]f\left(t\right),$$

(76)

holds, where $b>x>t>a>0$.

Proof. 

Using Definition 6 and Equation (66) of Theorem 4 or the associativity of the GM-convolution, one can obtain:

$$\begin{array}{c}{\mathcal{J}}_{a,\left(M_1\right)}^x\left[u\right]{\mathcal{J}}_{a,\left(M_2\right)}^u\left[t\right]f\left(t\right)=\left(M_1{\ast }_{\mathcal{M}}\left(M_2{\ast }_{\mathcal{M}}f\right)\right)\left(\frac{x}{a}\right)=\\ \left(\left(M_1{\ast }_{\mathcal{M}}{M}_2\right){\ast }_{\mathcal{M}}f\right)\left(\frac{x}{a}\right)={\mathcal{J}}_{a,\left(M_1{\ast }_{\mathcal{M}}{M}_2\right)}^x\left[t\right]f\left(t\right).\end{array}$$

(77)

□

2.4. Scale-Invariant GF Differential Operators

Let us present the definition of the scale-invariant general fractional differential operators.

Definition 7. 

Let kernel pair $(M_m,K_m)$ belong to the set ${\mathcal{M}}_{m,\beta }$, and $0<a<\tau <t<b<\infty$.

• The scale-invariant general fractional differential operator of the Riemann–Liouville type with the kernel $K_m\left(x\right)$ is defined by the equation:

$$\begin{array}{c}{\mathcal{D}}_{a,\left(K\right)}^t\left[\tau \right]f\left(\tau \right)={\mathcal{D}}_t^{m,\beta }{\mathcal{J}}_{a,\left(K\right)}^t\left[\tau \right]f\left(\tau \right)=\\ {\mathcal{D}}_t^{m,\beta }\left(K_m{\ast }_{\mathcal{M}}f\right)\left(\frac{t}{a}\right)={\mathcal{D}}_t^{m,\beta }{\int }_a^t{K}_m\left(\frac{t}{\tau }\right)f\left(\frac{\tau }{a}\right)\frac{d\tau }{\tau }.\end{array}$$

(78)

The scale-invariant general fractional differential operator of the Caputo type with the kernel $K_m\left(x\right)$ is defined by the equation:

$$\begin{array}{c}{\mathcal{D}}_{a,\left(K\right)}^{t,\ast }\left[\tau \right]f\left(\tau \right)={\mathcal{J}}_{a,\left(K\right)}^t\left[\tau \right]{\mathcal{D}}_{\tau }^{m,\beta }f\left(\tau \right)=\\ \left(K_m{\ast }_{\mathcal{M}}{\mathcal{D}}_{\tau }^{m,\beta }f\right)\left(\frac{t}{a}\right)={\int }_a^t\frac{d\tau }{\tau }{K}_m\left(\frac{t}{\tau }\right)\left({\mathcal{D}}_{\tau }^{m,\beta }f\right)\left(\frac{\tau }{a}\right).\end{array}$$

(79)

Here, ${\mathcal{J}}_{a,\left(K_m\right)}^t$ is the GFI operator with the kernel $K_m$, $t>a$, and the operator

$${\mathcal{D}}_t^{m,\beta }f\left(t\right)=t^{-\beta }{\left(t\frac{d}{dt}\right)}^m{t}^{\beta }f\left(t\right)$$

(80)

is a differential operator of the integer order $m\in \mathbb{N}$.

The general fractional differential operator Equation (78) can be considered as a generalization of the Hadamard-type fractional derivatives (see Equation (1.4) in [32] (p. 1192)).

Example 5. 

For example, one can consider the GFD kernel $K_m\left(x\right)$ in the form

$$K_m\left(\frac{t}{\tau }\right)=M_{m-\alpha ,\beta }\left(\frac{t}{\tau }\right)=\frac{1}{\Gamma (m-\alpha )}{\left(\frac{t}{\tau }\right)}^{-\beta }{\left(\ln \frac{t}{\tau }\right)}^{m-\alpha -1},$$

(81)

where $t>\tau >a$.

• General fractional differential operator Equation (78) with kernel Equation (81) can be represented by the equation

$$\begin{array}{c}{\mathcal{D}}_{a,\left(K_m\right)}^t\left[\tau \right]f\left(\tau \right)={\mathcal{D}}_t^{m,\beta }{\mathcal{J}}_{a,\left(M_{m-\alpha ,\beta }\right)}^t\left[\tau \right]f\left(\tau \right)=\\ \frac{1}{\Gamma (m-\alpha )}{t}^{-\beta }{\left(t\frac{d}{dt}\right)}^m{t}^{\beta }{\int }_a^t\frac{d\tau }{\tau }{\left(\frac{\tau }{t}\right)}^{\beta }{\left(\ln \frac{t}{\tau }\right)}^{m-\alpha +1}f\left(\frac{\tau }{a}\right),\end{array}$$

(82)

where ${\mathcal{D}}_{a,\beta }^{t,\alpha }$ is the Hadamard-type fractional integral operator, $t>a>0$.

• The operator ${\mathcal{D}}_{a,\beta }^{t,\alpha }$ that is defined by equation

$${\mathcal{D}}_{a,\beta }^{t,\alpha }\left[\tau \right]f\left(\frac{\tau }{a}\right)={\mathcal{D}}_{a,\left(K_m\right)}^t\left[\tau \right]f\left(\tau \right)$$

(83)

is called the Hadamard-type fractional differential operator of the order $\alpha \in (m-1,m)$, $m\in \mathbb{N}$, [31,32]. Equation (83) with $\beta =0$ is called the Hadamard fractional differential operator (see in [12] (pp. 110–120), [30] and [11] (p. 332)).

• Equation (82) exists almost everywhere on the space $A{C}_{\mathcal{D},\beta }^m(a,b)$. This statement is proven as Theorem 3.2 in [32] (p. 1198). The space $A{C}_{\mathcal{D},\beta }^m[a,b]$ consists of functions $f\left(t\right)$ on $[a,b]$ that have ${\mathcal{D}}_t^k\left(t^{\beta }f\left(t\right)\right)$ for $k=1,...m-1$, and ${\mathcal{D}}_t^{m-1}\left(t^{\beta }f\left(t\right)\right)$ is absolutely continuous on $[a,b]$, where $\mathcal{D}$ denotes the operator ${\mathcal{D}}_t=td/dt$ (see [32] (p. 1193)).
• The GFD operator of Caputo-type Equation (79) with kernel Equation (81) can be represented by the equation

$$\begin{array}{c}{\mathcal{D}}_{a,\left(K_m\right)}^{t,\ast }\left[\tau \right]f\left(\tau \right)={\mathcal{J}}_{a,\left(M_{m-\alpha }\right)}^{t,\ast }\left[\tau \right]{\mathcal{D}}_{\tau }^{m,\beta }f\left(\tau \right)=\\ \frac{1}{\Gamma (m-\alpha )}{\int }_a^t\frac{d\tau }{\tau }{\left(\frac{\tau }{t}\right)}^{\beta }{\left(\ln \frac{t}{\tau }\right)}^{m-\alpha +1}\left({\mathcal{D}}_{\tau }^{m,\beta }\right)f\left(\frac{t}{a}\right),\end{array}$$

(84)

where $t>a>0$.

• The operator ${\mathcal{D}}_{a,\beta }^{t,\alpha ,\ast }$ that is defined by equation:

$${\mathcal{D}}_{a,\beta }^{t,\alpha ,\ast }\left[\tau \right]f\left(\frac{\tau }{a}\right)={\mathcal{D}}_{a,\left(K_m\right)}^{t,\ast }\left[\tau \right]f\left(\tau \right),$$

(85)

which is called the Caputo modification of the Hadamard-type fractional differential operator [41,42].

Remark 11. 

For $\alpha =m\in \mathbb{N}$, the Hadamard-type fractional differential operator is the integer-order differential operator [12] (p. 112), in the form

$${\mathcal{D}}_{a,\beta }^{t,m}\left[\tau \right]f\left(\tau \right)={\mathcal{D}}_t^{m,\beta }f\left(t\right)=t^{-\beta }{\left(t\frac{d}{dt}\right)}^m\left(t^{\beta }f\left(t\right)\right).$$

(86)

For $\beta =0$, and $\alpha =m\in \mathbb{N}$, Equation (82) is the differential operator of dilation (scaling) of the integer order

$${\mathcal{D}}_{a,0}^{t,m}f\left(t\right)={\mathcal{D}}_t^{m}f\left(t\right)={\left(t\frac{d}{dt}\right)}^{m}f\left(t\right).$$

(87)

Therefore, the GFD operators in Equation (83) with $\alpha =m\in \mathbb{N}$ are differential operators of the integer order $m\in \mathbb{N}$ that do not depend on the parameter $a>0$.

• As a result, the GFD operator is:

$${\mathcal{D}}_{a,\left(M_{m-\alpha }\right)}^t\left[\tau \right]f\left(\tau \right)={\mathcal{D}}_{a,\beta }^{t,\alpha }\left[\tau \right]f\left(\frac{\tau }{a}\right)={\mathcal{D}}_t^{\alpha ,\beta }f\left(\frac{t}{a}\right),$$

(88)

Equations (86) and (87) do not depend on the parameter $a>0$ and are operators of integer order $m\in \mathbb{N}$.

The scaling property of the general fractional differential operator is described by the following theorem.

Theorem 8 

(Scale-invariance of GFD operators). Let kernel pair $(M_m,K_m)$ belong to the set ${\mathcal{M}}_{m,\beta }$, and $0<a<\tau <t<\infty$.

• Then, the GF differential Equations (78) and (79) satisfy the following scaling properties:

$${\Pi }_{\rho }{\mathcal{D}}_{a,\left(K_m\right)}^t\left[\tau \right]f\left(\tau \right)={\mathcal{D}}_{a,\left(K_m\right)}^t\left[\tau \right]{\Pi }_{\rho }f\left(\tau \right),$$

(89)

$${\Pi }_{\rho }{\mathcal{D}}_{a,\left(K_m\right)}^{t,\ast }\left[\tau \right]f\left(\tau \right)={\mathcal{D}}_{a,\left(K_m\right)}^{t,\ast }\left[\tau \right]{\Pi }_{\rho }f\left(\tau \right),$$

(90)

where $\rho >0$, $a>0$, $t\in (a,b)$.

Proof. 

The proofs of Equations (89) and (90) are similar to the proof of the scale property of the GF integral operator in Theorem 6.

• Let us consider the scaling properties of the GFD operator of the Riemann–Liouville type. Using the definition of this operator, one can obtain:

$${\Pi }_{\rho }{\mathcal{D}}_{a,\left(K\right)}^t\left[\tau \right]f\left(\tau \right)={\Pi }_{\rho }{\mathcal{D}}_t^{m,\beta }{\mathcal{J}}_{a,\left(K\right)}^t\left[\tau \right]f\left(\tau \right).$$

(91)

Using the scaling property

$${\Pi }_{\rho }{\mathcal{D}}_t^{m,\beta }f\left(t\right)={\mathcal{D}}_t^{m,\beta }{\Pi }_{\rho }f\left(t\right),$$

(92)

Equation (91) becomes:

$${\Pi }_{\rho }{\mathcal{D}}_{a,\left(K\right)}^t\left[\tau \right]f\left(\tau \right)={\mathcal{D}}_t^{m,\beta }{\Pi }_{\rho }{\mathcal{J}}_{a,\left(K\right)}^t\left[\tau \right]f\left(\tau \right).$$

(93)

Then, using the scaling property of the GFI operators (see Theorem 6) in the form

$${\Pi }_{\rho }{\mathcal{J}}_{a,\left(M\right)}^x\left[\tau \right]f\left(u\right)={\mathcal{J}}_{a,\left(M\right)}^x\left[u\right]{\Pi }_{\rho }f\left(u\right),$$

(94)

Equation (93) becomes:

$${\Pi }_{\rho }{\mathcal{D}}_{a,\left(K\right)}^t\left[\tau \right]f\left(\tau \right)={\mathcal{D}}_t^{m,\beta }{\mathcal{J}}_{a,\left(K\right)}^t\left[\tau \right]{\Pi }_{\rho }f\left(\tau \right)={\mathcal{D}}_{a,\left(K\right)}^t\left[\tau \right]{\Pi }_{\rho }f\left(\tau \right),$$

(95)

which proves the scale-invariant property of the GFD operator of the Riemann–Liouville type.

• Similar transformations are made for the GFD operator of the Caputo. Using the definition of this operator, the scaling property of the GFI operators and the scaling property of the operator ${\mathcal{D}}_t^{m,\beta }$, one can obtain the following:

$$\begin{array}{c}{\Pi }_{\rho }{\mathcal{D}}_{a,\left(K\right)}^{t,\ast }\left[\tau \right]f\left(\tau \right)={\Pi }_{\rho }{\mathcal{J}}_{a,\left(K\right)}^t\left[\tau \right]{\mathcal{D}}_{\tau }^{m,\beta }f\left(\tau \right)=\\ {\mathcal{J}}_{a,\left(K\right)}^t\left[\tau \right]{\Pi }_{\rho }{\mathcal{D}}_{\tau }^{m,\beta }f\left(\tau \right).={\mathcal{J}}_{a,\left(K\right)}^t\left[\tau \right]{\mathcal{D}}_{\tau }^{m,\beta }{\Pi }_{\rho }f\left(\tau \right)={\mathcal{D}}_{a,\left(K\right)}^{t,\ast }\left[\tau \right]{\Pi }_{\rho }f\left(\tau \right),\end{array}$$

(96)

which proves the scale-invariant property of the GFD operator of the Caputo type. □

3. Fundamental Theorems of Scale-Invariant GF Operators

3.1. Fundamental Theorems for Hadamard-Type Fractional Operators

Let us first present fundamental theorems for the well-known fractional operators.

The Hadamard fractional operators can be given as:

$${\mathcal{J}}_{t,a+}^{\alpha }\left[\tau \right]f\left(\frac{\tau }{a}\right)={\mathcal{D}}_{a,\left(M_{\alpha ,0}\right)}^t\left[\tau \right]f\left(\tau \right),$$

(97)

$${\mathcal{D}}_{t,a+}^{\alpha }\left[\tau \right]f\left(\frac{\tau }{a}\right)={\mathcal{D}}_{a,\left(M_{m-\alpha ,0}\right)}^t\left[\tau \right]f\left(\tau \right).$$

(98)

The Hadamard-type fractional operators can be given as:

$${\mathcal{J}}_{a,\beta }^{t,\alpha }\left[\tau \right]f\left(\frac{\tau }{a}\right)={\mathcal{D}}_{a,\left(M_{\alpha ,\beta }\right)}^t\left[\tau \right]f\left(\tau \right),$$

(99)

$${\mathcal{D}}_{a,\beta }^{t,\alpha }\left[\tau \right]f\left(\frac{\tau }{a}\right)={\mathcal{D}}_{a,\left(M_{m-\alpha ,\beta }\right)}^t\left[\tau \right]f\left(\tau \right).$$

(100)

The first fundamental theorem for the Hadamard and Hadamard-type fractional operators has the following form.

Theorem 9. 

(a) Let $f\left(t\right)\in L_p(a,b)$, where $0<a<b\le \infty$ and $1\le p\le \infty$.

Then, the equation

$${\mathcal{D}}_{t,a+}^{\alpha }\left[\tau \right]{\mathcal{J}}_{\tau ,a+}^{\alpha }\left[s\right]f\left(s\right)=f\left(t\right)$$

(101)

holds for all $t\in (a,b)$ and $\alpha >0$.

(b) 

Let $f\left(t\right)\in X_q^p(a,b)$, where $0<a<b\le \infty$, $1\le p\le \infty$.

Then, the equation

$${\mathcal{D}}_{a,\beta }^{t,\alpha }\left[\tau \right]{\mathcal{J}}_{a,\beta }^{\tau ,\alpha }\left[s\right]f\left(s\right)=f\left(t\right)$$

(102)

holds for all $t\in (a,b)$ and $\alpha >0$, $\beta >q$.

Theorem 9 is proven as Property 2.28 in [12] (p. 116).

Definition 8. 

Function $f\left(x\right)$ belongs to the set ${\mathcal{J}}_{a,\beta }^{t,\alpha }\left(X_q^p(a,b)\right)$, if it can be represented as:

$$f\left(t\right)={\mathcal{J}}_{a,\beta }^{t,\alpha }\left[\tau \right]g\left(\tau \right),$$

where $g\left(x\right)$ belongs to the set $X_q^p(a,b)$, $\alpha >0$, $\beta >q$ with $0<a<b\le \infty$.

The second fundamental theorem for the Hadamard and Hadamard-type fractional operators has the following form.

Theorem 10. 

(a) Let $f\left(t\right)\in {\mathcal{J}}_{a+}^{t,\alpha }\left(L_p(a,b)\right)$, where $0<a<b\le \infty$ and $1\le p\le \infty$.

Then, the equation

$${\mathcal{J}}_{a+}^{t,\alpha }\left[\tau \right]{\mathcal{D}}_{a+}^{\tau ,\alpha }\left[s\right]f\left(s\right)=f\left(t\right)$$

(103)

is satisfied for all $x\in (a,b)$ and $\alpha >0$.

(b) 

Let $f\left(t\right)\in {\mathcal{J}}_{a,\beta }^{t,\alpha }\left(X_q^p(a,b)\right)$, where $1\le p\le \infty$, and $\beta >q$.

Then, the equation

$${\mathcal{J}}_{a,\beta }^{t,\alpha }\left[\tau \right]{\mathcal{D}}_{a,\beta }^{\tau ,\alpha }\left[s\right]f\left(s\right)=f\left(t\right)$$

(104)

holds for all $t\in (a,b)$ and $\alpha >0$.

Theorem 10 is proven as Lemma 2.35 in [12] (p. 117), where Theorem 10 (b) is proven for $a=0$ and $b=\infty$.

Theorem 11. 

Let $0<a<b\le \infty$ and let function $f\left(t\right)$ satisfy the condition

$$g\left(t\right)={\mathcal{J}}_{a,\beta }^{t,m-\alpha }\left[\tau \right]f\left(\tau \right)\in A{C}_{\delta ,\beta }^m[a,b],$$

(105)

where $m=\left[\alpha \right]+1$.

• Then, the equation

$${\mathcal{J}}_{a,\beta }^{t,\alpha }\left[\tau \right]{\mathcal{D}}_{a,\beta }^{\tau ,\alpha }\left[s\right]f\left(s\right)=f\left(t\right)-{\left(\frac{a}{t}\right)}^{\beta }\sum _{k=1}^m\frac{1}{\Gamma (\alpha -k+1)}{\left(\ln \frac{t}{a}\right)}^{\alpha -k}\underset{t\to a+}{\lim }{\mathcal{D}}_{a,\beta }^{t,\alpha -k}f\left(t\right)$$

(106)

is satisfied for $m\ge 2$.

• For $m=1$,

$${\mathcal{J}}_{a,\beta }^{t,\alpha }\left[\tau \right]{\mathcal{D}}_{a,\beta }^{\tau ,\alpha }\left[s\right]f\left(s\right)=f\left(t\right)-\frac{1}{\Gamma \left(\alpha \right)}{\left(\frac{a}{t}\right)}^{\beta }{\left(\ln \frac{t}{a}\right)}^{\alpha -1}\underset{t\to a+}{\lim }{\mathcal{J}}_{a,\beta }^{t,\alpha -k}f\left(t\right)$$

(107)

for all $t\in (a,b)$.

For $\beta =0$, Theorem 11 is proven in [33] in 2002 (see also Theorem 2.3 in [12] (p. 116)). Theorem 11 is also proven as Theorem 3.15 in [50] (p. 12).

In particular, Theorem 11 is satisfied for $\alpha =m\in \mathbb{N}$ in the form

$${\mathcal{J}}_{a,\beta }^{t,m}\left[\tau \right]{\mathcal{D}}_{a,\beta }^{\tau ,m}\left[s\right]f\left(\frac{s}{a}\right)=f\left(\frac{t}{a}\right)-{\left(\frac{a}{t}\right)}^{\beta }\sum _{k=0}^{m-1}\frac{1}{\Gamma (k+1)}{\left(\ln \frac{t}{a}\right)}^k\underset{t\to a+}{\lim }{\mathcal{D}}_t^{k,\beta }f\left(\frac{t}{a}\right),$$

(108)

if $f\left(\tau \right)\in A{C}_{\delta ,\beta }^m[a,b]$. For $\beta =0$, see Equation (2).7.49 in [12] (p. 116), and in [33].

Remark 12. 

Theorems 9–11 for finite interval $[a,b]$ can be represented through the generalized Mellin convolution by replacing the functions $f\left(x\right)$ with function $f(x/a)$ and:

$${\mathcal{J}}_{a,\beta }^{t,\alpha }\left[\tau \right]f\left(\frac{\tau }{a}\right)={\mathcal{J}}_{a,\left(M_{\alpha ,\beta }\right)}^t\left[\tau \right]f\left(\tau \right),{\mathcal{D}}_{a,\beta }^{t,\alpha }\left[\tau \right]f\left(\frac{\tau }{a}\right)={\mathcal{D}}_{a,\left(M_{\alpha ,\beta }\right)}^t\left[\tau \right]f\left(\tau \right),$$

(109)

where $0<a<\tau <t<b<\infty$.

3.2. First Fundamental Theorems for General Fractional Operators

Let us prove fundamental theorem of the proposed scale-invariant general fractional calculus. In these theorems, the following conditions are assumed:

$$\beta >\mu >\eta >q,$$

where the parameters $\beta$, $\mu$ characterize the set of kernels $M\left(z\right)$ and the parameters $\eta$, q characterize the set of functions $f\left(t\right)$.

The first fundamental theorem for scale-invariant general fractional operators has the following form.

Theorem 12 

(First FT for GFD operator of Riemann–Liouville type). Let kernel pair $(M,K)$ belong to the set ${\mathcal{M}}_{m,\beta }$.

• Let $f\left(t\right)\in C_{-\eta }(0,\infty )$ with $\eta <\beta$ and $f\left(t\right)\in X_q^p(a,b)$ with $1\le p\le \infty$.
• Then, the equation

$${\mathcal{D}}_{a,\left(K\right)}^t\left[\tau \right]{\mathcal{J}}_{a,\left(M\right)}^{\tau }\left[s\right]f\left(s\right)=f\left(\frac{t}{a}\right)$$

(110)

• holds for all $t\in (a,b)$.

Proof. 

Using Definitions 6 and 7 in the form

$${\mathcal{D}}_{a,\left(K\right)}^t\left[\tau \right]g\left(\tau \right)={\mathcal{D}}_t^{m,\beta }{\mathcal{J}}_{a,\left(K\right)}^t\left[\tau \right]g\left(\tau \right)={\mathcal{D}}_t^{m,\beta }\left(K{\ast }_{\mathcal{M}}g\right)\left(\frac{t}{a}\right),$$

(111)

$$g\left(\frac{\tau }{a}\right)={\mathcal{J}}_{a,\left(M\right)}^{\tau }\left[s\right]f\left(s\right)=\left(M{\ast }_{\mathcal{M}}f\right)\left(\frac{\tau }{a}\right),$$

(112)

where:

$${\mathcal{D}}_t^{m,\beta }=t^{-\beta }{\left(t\frac{d}{dt}\right)}^m{t}^{\beta },$$

(113)

one can obtain:

$${\mathcal{D}}_{a,\left(K\right)}^t\left[\tau \right]{\mathcal{J}}_{a,\left(M\right)}^{\tau }\left[s\right]f\left(s\right)={\mathcal{D}}_t^{m,\beta }\left(K{\ast }_{\mathcal{M}}\left(M{\ast }_{\mathcal{M}}f\right)\right)\left(\frac{t}{a}\right).$$

(114)

Using the associativity property of the GM-convolution, Equation (114) becomes:

$${\mathcal{D}}_t^{m,\beta }\left(K{\ast }_{\mathcal{M}}\left(M{\ast }_{\mathcal{M}}f\right)\right)\left(\frac{t}{a}\right)={\mathcal{D}}_t^{m,\beta }\left(\left(K{\ast }_{\mathcal{M}}M\right){\ast }_{\mathcal{M}}f\right)\left(\frac{t}{a}\right)={\mathcal{D}}_t^{m,\beta }\left(M_{m,\beta }{\ast }_{\mathcal{M}}f\right)\left(\frac{t}{a}\right),$$

(115)

where the fact that $(M,K)\in {\mathcal{M}}_{m,\beta }$ is taken into account by using Equation (54) of Definition 4.

• Then, using Equation (102) of Theorem 9 for $\alpha =m\in \mathbb{N}$ and ${\mathcal{D}}_{a,\beta }^{t,m}={\mathcal{D}}_t^{m,\beta }$, one can obtain:

$${\mathcal{D}}_t^{m,\beta }\left(M_{m,\beta }{\ast }_{\mathcal{M}}f\right)\left(\frac{t}{a}\right)={\mathcal{D}}_{a,\beta }^{t,m}{\int }_a^t{M}_{m,\beta }\left(\frac{t}{\tau }\right)f\left(\frac{\tau }{a}\right)\frac{d\tau }{\tau }=$$

$${\mathcal{D}}_{a,\beta }^{t,m}{\mathcal{J}}_{a,\beta }^{t,m}\left[\tau \right]f\left(\frac{\tau }{a}\right)=\left(\frac{t}{a}\right),$$

(116)

if $f\left(x\right)\in X_q^p(a,b)$, where $1\le p\le \infty$ and $\beta >q$. □

Remark 13. 

Note that Equation (115) cannot be realized for the standard Mellin convolution, since the associative property for this convolution is violated, in general.

Theorem 13 

(First FT for GFD operator of Caputo type). Let kernel pair $(M,K)$ belong to the set ${\mathcal{M}}_{m,\beta }$

• Let $f\left(x\right)\in {\mathcal{J}}_{a,\left(K\right)}^t\left(X_q^p(a,b)\right)$, where $1\le p\le \infty$, and let $q<\beta$.
• Then, the equation

$${\mathcal{D}}_{a,\left(K\right)}^{t,\ast }\left[\tau \right]{\mathcal{J}}_{a,\left(M\right)}^{\tau }\left[s\right]f\left(s\right)=f\left(\frac{t}{a}\right)$$

(117)

holds for all $t\in (a,b)$.

Proof. 

Using Definitions 6 and 7 in the form

$${\mathcal{D}}_{a,\left(K\right)}^{t,\ast }\left[\tau \right]g\left(\tau \right)={\mathcal{J}}_{a,\left(K\right)}^t\left[\tau \right]{\mathcal{D}}_{\tau }^{m,\beta }g\left(\tau \right)=\left(K{\ast }_{\mathcal{M}}{\mathcal{D}}_t^{m,\beta }g\right)\left(\frac{t}{a}\right),$$

(118)

$$g\left(\frac{\tau }{a}\right)={\mathcal{J}}_{a,\left(M\right)}^{\tau }\left[s\right]f\left(s\right)=\left(M{\ast }_{\mathcal{M}}f\right)\left(\frac{\tau }{a}\right),$$

(119)

where:

$${\mathcal{D}}_t^{m,\beta }=t^{-\beta }{\left(t\frac{d}{dt}\right)}^m{t}^{\beta },$$

(120)

one can obtain:

$${\mathcal{D}}_{a,\left(K\right)}^{t,\ast }\left[\tau \right]{\mathcal{J}}_{a,\left(M\right)}^{\tau }\left[s\right]f\left(s\right)=\left(K{\ast }_{\mathcal{M}}{\mathcal{D}}_{\tau }^{m,\beta }\left(M{\ast }_{\mathcal{M}}f\right)\right)\left(\frac{t}{a}\right).$$

(121)

Using that $f\left(t\right)\in {\mathcal{J}}_{a,\left(K\right)}^t\left(X_q^p(a,b)\right)$, the function $f\left(t\right)$ can be represented as:

$$f\left(\frac{s}{a}\right)={\mathcal{J}}_{a,\left(K\right)}^s\left[w\right]\phi \left(w\right)=\left(K{\ast }_{\mathcal{M}}\phi \right)\left(\frac{s}{a}\right),$$

(122)

where $\phi \left(t\right)\in X_q^p(a,b)$, which gives:

$${\mathcal{D}}_t^{m,\beta }\left(M{\ast }_{\mathcal{M}}f\right)\left(\frac{t}{a}\right)={\mathcal{D}}_t^{m,\beta }\left(M{\ast }_{\mathcal{M}}\left(K{\ast }_{\mathcal{M}}\phi \right)\right)\left(\frac{t}{a}\right).$$

(123)

Using the associativity property of the GM-convolution, Equation (123) becomes:

$${\mathcal{D}}_t^{m,\beta }\left(M{\ast }_{\mathcal{M}}\left(K{\ast }_{\mathcal{M}}\phi \right)\right)\left(\frac{t}{a}\right)={\mathcal{D}}_t^{m,\beta }\left(\left(M{\ast }_{\mathcal{M}}K\right){\ast }_{\mathcal{M}}\phi \right)\left(\frac{t}{a}\right)={\mathcal{D}}_t^{m,\beta }\left(M_{m,\beta }{\ast }_{\mathcal{M}}\phi \right)\left(\frac{t}{a}\right),$$

(124)

where the fact that $(M,K)\in {\mathcal{M}}_{m,\beta }$ is taken into account by using Equation (54) of Definition 4.

• Then, using Equation (102) of Theorem 9 for $\alpha =m\in \mathbb{N}$, and equation

$${\mathcal{D}}_t^{m,\beta }\left(M_{m,\beta }{\ast }_{\mathcal{M}}\phi \right)\left(\frac{t}{a}\right)={\mathcal{D}}_{a,\beta }^{t,m}{\int }_a^t{M}_{m,\beta }\left(\frac{t}{\tau }\right)\phi \left(\frac{\tau }{a}\right)\frac{d\tau }{\tau }=\phi \left(\frac{t}{a}\right),$$

(125)

one can obtain:

$${\mathcal{D}}_t^{m,\beta }\left(M_{m,\beta }{\ast }_{\mathcal{M}}\phi \right)\left(\frac{t}{a}\right)=\phi \left(\frac{t}{a}\right).$$

(126)

As a result, using Equations (121)–(123), one can obtain:

$${\mathcal{D}}_{a,\left(K\right)}^{t,\ast }\left[\tau \right]{\mathcal{J}}_{a,\left(M\right)}^{\tau }\left[s\right]f\left(s\right)=\left(K{\ast }_{\mathcal{M}}\phi \right)\left(\frac{t}{a}\right)=f\left(\frac{t}{a}\right),$$

(127)

where $0<a<\tau <t<b<\infty$. □

Remark 14. 

Note that in the proven forms of the first fundamental theorems of the scale-invariant CFC, the property of associativity of the generalized Mellin convolution was used.

3.3. Second Fundamental Theorems for General Fractional Operators

Theorem 14 

(Second FT for GFD operator of Caputo type). Let kernel pair $(M,K)$ belong to the set ${\mathcal{M}}_{m,\beta }$:

(a) 

Let $f\left(t\right)\in {\mathcal{J}}_{a,\beta }^{t,m}\left(X_q^p(0,\infty )\right)$, where $1\le p\le \infty$, and let $\beta >q$.

Then, the equation

$${\mathcal{J}}_{a,\left(M\right)}^t\left[\tau \right]{\mathcal{D}}_{a,\left(K\right)}^{\tau ,\ast }\left[s\right]f\left(s\right)=f\left(\frac{t}{a}\right)$$

(128)

holds for all $t\in (a,b)$.

(b) 

Let $f\left(t\right)\in A{C}_{\delta ,\beta }^m(a,b)$, where $\beta >q$.

Then, the equation

$${\mathcal{J}}_{a,\left(M\right)}^t\left[\tau \right]{\mathcal{D}}_{a,\left(K\right)}^{\tau ,\ast }\left[s\right]f\left(s\right)=f\left(\frac{t}{a}\right)-{\left(\frac{a}{t}\right)}^{\beta }\sum _{k=0}^{m-1}\frac{1}{\Gamma (k+1)}{\left(\ln \frac{t}{a}\right)}^k\underset{t\to a+}{\lim }{\mathcal{D}}_t^{k,\beta }f\left(\frac{t}{a}\right)$$

(129)

holds for all $t\in (a,b)$.

Proof. 

Using Definitions 6 and 7 in the form

$${\mathcal{J}}_{a,\left(M\right)}^t\left[\tau \right]g\left(\tau \right)=\left(M{\ast }_{\mathcal{M}}g\right)\left(\frac{t}{a}\right),$$

(130)

$$g\left(\frac{\tau }{a}\right)={\mathcal{D}}_{a,\left(K\right)}^{\tau ,\ast }\left[s\right]f\left(s\right)={\mathcal{J}}_{a,\left(K\right)}^{\tau }\left[s\right]{\mathcal{D}}_s^{m,\beta }f\left(s\right)=\left(K{\ast }_{\mathcal{M}}{\mathcal{D}}_s^{m,\beta }f\right)\left(\frac{\tau }{a}\right),$$

(131)

one can obtain:

$${\mathcal{J}}_{a,\left(M\right)}^t\left[\tau \right]{\mathcal{D}}_{a,\left(K\right)}^{\tau ,\ast }\left[s\right]f\left(s\right)=\left(M{\ast }_{\mathcal{M}}\left(K{\ast }_{\mathcal{M}}{\mathcal{D}}_s^{m,\beta }f\right)\right)\left(\frac{t}{a}\right).$$

(132)

Using the associativity property of the GM-convolution, Equation (132) becomes:

$$\left(M{\ast }_{\mathcal{M}}\left(K{\ast }_{\mathcal{M}}{\mathcal{D}}_s^{m,\beta }f\right)\right)\left(\frac{t}{a}\right)=\left(M{\ast }_{\mathcal{M}}K\right){\ast }_{\mathcal{M}}{\mathcal{D}}_s^{m,\beta }f)\left(\frac{t}{a}\right)=\left(M_{m,\beta }{\ast }_{\mathcal{M}}{\mathcal{D}}_s^{m,\beta }f\right)\left(\frac{t}{a}\right),$$

(133)

where the fact that $(M,K)\in {\mathcal{M}}_{m,\beta }$ is taken into account by using Equation (54) of Definition 4.

If $f\left(x\right)\in {\mathcal{J}}_{a,\beta }^{t,m}\left(X_q^p(a,b)\right)$, then using the second fundamental theorem for Hadamard-type fractional operators in the form of Theorem 10 with $\alpha =m\in \mathbb{N}$, one can obtain:

$$\left(M_{m,\beta }{\ast }_{\mathcal{M}}{\mathcal{D}}_s^{m,\beta }f\right)\left(\frac{t}{a}\right)={\mathcal{J}}_{a,\beta }^{t,m}\left[\tau \right]{\mathcal{D}}_{a,\beta }^{\tau ,m}\left[s\right]f\left(\frac{s}{a}\right)=f\left(\frac{t}{a}\right),$$

(134)

if $f\left(x\right)\in {\mathcal{J}}_{a,\beta }^{t,m}\left(X_q^p(a,b)\right)$, where $1\le p\le \infty$, and $\beta >q$.

• If $f\left(t\right)\in A{C}_{\delta ,\beta }^m(a,b)$, then using Equation (108), Equation (133) can be represented as:

$$\begin{array}{c}\left(M{\ast }_{\mathcal{M}}\left(K{\ast }_{\mathcal{M}}{\mathcal{D}}_s^{m,\beta }f\right)\right)\left(\frac{t}{a}\right)=\left(M_{m,\beta }{\ast }_{\mathcal{M}}{\mathcal{D}}_s^{m,\beta }f\right)\left(\frac{t}{a}\right)={\mathcal{J}}_{a,\beta }^{t,m}\left[\tau \right]{\mathcal{D}}_{a,\beta }^{\tau ,m}\left[s\right]f\left(\frac{s}{a}\right)=\\ f\left(\frac{t}{a}\right)-{\left(\frac{a}{t}\right)}^{\beta }\sum _{k=0}^{m-1}\frac{1}{\Gamma (k+1)}{\left(\ln \frac{t}{a}\right)}^k\underset{t\to a+}{\lim }{\mathcal{D}}_t^{k,\beta }f\left(\frac{t}{a}\right),\end{array}$$

(135)

for all $t>a>0$. □

Definition 9. 

Let kernel pair $(M,K)$ belongs to the set ${\mathcal{M}}_{m,\beta }$, and let function $f\left(t\right)$ be represented by the equation

$$f\left(\frac{t}{a}\right)=\left(M{\ast }_{\mathcal{M}}\phi \right)\left(\frac{t}{a}\right)={\mathcal{J}}_{\left(M\right)}^t\left[\tau \right]\phi \left(\tau \right),$$

(136)

where $\phi \left(t\right)\in X_q^p(a,b)$.

• Then, the set of such function $f\left(t\right)$ is denoted as ${\mathcal{J}}_{a,\left(M\right)}^t\left(X_q^p(a,b)\right)$.

Theorem 15 

(Second FT for GFD operator of Riemann–Liouville type). Let kernel pair $(M,K)$ belong to the set ${\mathcal{M}}_{m,\beta }$.

• Let $f\left(t\right)\in {\mathcal{J}}_{a,\left(M\right)}^t\left(X_q^p(a,b)\right)$, where $1\le p\le \infty$, and let $q<\beta$.
• Then, the equation

$${\mathcal{J}}_{a,\left(M\right)}^t\left[\tau \right]{\mathcal{D}}_{a,\left(K\right)}^{\tau }\left[s\right]f\left(s\right)=f\left(\frac{t}{a}\right)$$

(137)

holds for all $t\in (a,b)$.

Proof. 

Using Definitions 6 and 7 in the form

$${\mathcal{J}}_{a,\left(M\right)}^t\left[\tau \right]g\left(\tau \right)=\left(M{\ast }_{\mathcal{M}}g\right)\left(\frac{\tau }{a}\right),$$

(138)

$$g\left(\frac{\tau }{a}\right)={\mathcal{D}}_{a,\left(K\right)}^{\tau }\left[s\right]f\left(s\right)={\mathcal{D}}_t^{m,\beta }\left(K{\ast }_{\mathcal{M}}f\right)\left(\frac{t}{a}\right),$$

(139)

one can obtain:

$${\mathcal{J}}_{a,\left(M\right)}^t\left[\tau \right]{\mathcal{D}}_{a,\left(K\right)}^{\tau }\left[s\right]f\left(s\right)=\left(M{\ast }_{\mathcal{M}}{\mathcal{D}}_t^{m,\beta }\left(K{\ast }_{\mathcal{M}}f\right)\right)\left(\frac{t}{a}\right).$$

(140)

Using that $f\left(t\right)\in {\mathcal{J}}_{a,\left(M\right)}^t\left(X_q^p(a,b)\right)$ or $f\left(t\right)\in {\mathcal{J}}_{a,\left(M\right)}^t\left(A{C}_{\delta ,\beta }^m(a,b)\right)$, one can use the representation

$$f\left(\frac{s}{a}\right)={\mathcal{J}}_{\left(M\right)}^s\left[w\right]\phi \left(w\right)=\left(M{\ast }_{\mathcal{M}}\phi \right)\left(\frac{s}{a}\right),$$

(141)

where $\phi \left(t\right)\in X_q^p(a,b)$, which gives:

$${\mathcal{D}}_t^{m,\beta }\left(K{\ast }_{\mathcal{M}}f\right)\left(\frac{t}{a}\right)={\mathcal{D}}_t^{m,\beta }\left(K{\ast }_{\mathcal{M}}\left(M{\ast }_{\mathcal{M}}\phi \right)\right)\left(\frac{t}{a}\right).$$

(142)

Using the associativity property of the GM-convolution, Equation (142) becomes:

$${\mathcal{D}}_t^{m,\beta }\left(K{\ast }_{\mathcal{M}}\left(M{\ast }_{\mathcal{M}}\phi \right)\right)\left(\frac{t}{a}\right)={\mathcal{D}}_t^{m,\beta }\left(\left(K{\ast }_{\mathcal{M}}M\right){\ast }_{\mathcal{M}}\phi \right)\left(\frac{t}{a}\right).$$

(143)

Using the associativity property of the GM-convolution, Equation (123) becomes:

$${\mathcal{D}}_t^{m,\beta }\left(\left(K{\ast }_{\mathcal{M}}M\right){\ast }_{\mathcal{M}}\phi \right)\left(\frac{t}{a}\right)={\mathcal{D}}_t^{m,\beta }\left(M_{m,\beta }{\ast }_{\mathcal{M}}\phi \right)\left(\frac{t}{a}\right)={\mathcal{D}}_t^{m,\beta }{\mathcal{J}}_{a,\beta }^{t,m}\phi \left(\frac{t}{a}\right),$$

(144)

where the fact that $(M,K)\in {\mathcal{M}}_{m,\beta }$ is taken into account by using Equation (54) of Definition 4.

Using the first fundamental theorem for the Hadamard-type fractional operators with $\alpha =m$, one can obtain:

$${\mathcal{D}}_t^{m,\beta }\left(M_{m,\beta }{\ast }_{\mathcal{M}}\phi \right)\left(\frac{t}{a}\right)={\mathcal{D}}_t^{m,\beta }{\mathcal{J}}_{a,\beta }^{t,m}\left[\tau \right]\phi \left(\frac{\tau }{a}\right)=\phi \left(\frac{t}{a}\right),$$

(145)

if $\phi \left(t\right)\in X_q^p(a,b)$.

• As a result, using Equation (141), Equation (144) can be written as:

$$\left(M{\ast }_{\mathcal{M}}{\mathcal{D}}_t^{m,\beta }\left(K{\ast }_{\mathcal{M}}f\right)\right)\left(\frac{t}{a}\right)=\left(M{\ast }_{\mathcal{M}}\phi \right)\left(\frac{t}{a}\right)=f\left(\frac{t}{a}\right),$$

(146)

if $f\left(t\right)\in {\mathcal{J}}_{a,\left(M\right)}^t\left(X_q^p(a,b)\right)$ and $t>a>0$. □

Remark 15. 

It should be also noted that in the proven forms of the second fundamental theorems of the scale-invariant GFC, the property of associativity of the generalized Mellin convolution was used.

4. Conclusions

In this article, a GFC of scale-invariant operators is considered. The standard Hadamard-type fractional operators describe only one type of nonlocality. It is also proposed to expand the set of admissible kernels of fractional operators that make it possible to describe scale-invariant nonlocalities. Scale-invariant general fractional integral (GFI) operators and scale-invariant general fractional differential (GFD) operators are suggested and properties of these operators are proven.

The following results are proposed in this paper.

• A modification of the standard Mellin convolution is proposed and some properties of this convolution are proven.
• A set of kernel pairs, which can be considered as an analog of the Sonin and Luchko sets of operator kernel pairs for the Laplace convolution operators, are defined. Some properties of these kernels are proven.
• The scale-invariant GFI operator is defined and properties are described.
• The scale-invariant GFD operators are defined and scale-invariance property is proven.
• The fundamental theorems of scale-invariant general fractional operators are proven.

The proposed scale-invariant operators of GFC are intended to serve as a tool for describing a wide set of processes and systems with nonlocal scaling in time and space.

As further research and development of the scale-invariant GFC, the following can be noted as examples: (a) GFC of scale-invariant n-fold sequential GFD operators [86,87]; (b) generalized Mellin convolutional Taylor formula for scale-invariant case [87]; (c) multi-kernel extension of the scale-invariant general fractional calculus [80].