On a Generic Fractional Derivative Associated with the Riemann–Liouville Fractional Integral

Abstract

In this paper, a generic fractional derivative is defined as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. Then, the theory of the left-invertible operators developed by Przeworska-Rolewicz is applied to deduce its properties. In particular, we characterize its domain, null-space, and projector operator; establish the interrelations between its different realizations; and present a generalized fractional Taylor formula involving the generic fractional derivative. Then, we consider the fractional relaxation equation containing the generic fractional derivative, derive a closed-form formula for its unique solution, and study its complete monotonicity.

1. Introduction

Presently, the literature devoted to fractional integrals and derivatives and fractional differential equations is huge, and every day, new contributions are published. However, some basic questions related to the nature of fractional derivatives, the properties that they are expected to possess, and the interrelations between different kinds of fractional derivatives are still not completely clarified, even in the simplest case of fractional derivatives acting on the functions of a single variable. Moreover, the availability of several unequal definitions of the fractional derivatives leads to the repeating of derivation of the same mathematical results formulated for different fractional derivatives and to problems in choosing an appropriate definition for the mathematical models involving fractional derivatives.

In this paper, a partial solution to the problems mentioned above is suggested. We focus on the case of time-fractional derivatives of the functions of a single variable, and refer to [1] for a discussion of the space-fractional derivatives of functions depending on several variables and to [2,3,4] for surveys of other types of fractional derivatives. It is worth mentioning that in the publications [5,6,7], some axioms or desiderata for the one-parameter fractional calculus (FC) operators of the functions of a single variable were suggested. However, several questions regarding their background, as well as possible realizations of these axioms systems, remained open.

In [8], an abstract schema for construction of the one-parameter families of fractional derivatives for the functions of a single variable was developed. In particular, this schema includes the Riemann–Liouville, Caputo, Hilfer, and Djrbashian–Nersessian, or the nth level time-fractional derivatives, that were defined in the available literature so far. In this paper, we follow and deepen the ideas suggested in [8], and embed them into the framework of the theory of left-invertible operators suggested by Przeworska-Rolewicz in [9]. It is worth mentioning that whereas many of publications by Przeworska-Rolewicz are devoted to the theory of right-invertible operators, the case of left-invertible ones was considered only in [9].

Under some very reasonable assumptions, the only family of one-parameter fractional integrals of the functions of one variable defined on a finite interval is the family of Riemann–Liouville fractional integrals (see [8,10] for details). The well-accepted definition of one-parameter fractional derivatives is in the form of left-inverse operators to fractional integrals. Thus, for the functions of a single variable, one should focus on the Riemann–Liouville fractional integral and its left-inverse operators as fractional derivatives. The class of such operators is not empty, and contains all reasonable time-fractional derivatives introduced so far, including the Riemann–Liouville, Caputo, and Hilfer fractional derivatives.

According to the terminology of Przeworska-Rolewicz, the Riemann–Liouville fractional integral belongs to the class of left-invertible operators. In this paper, we apply the theory of left-invertible operators developed in [9] to the Riemann–Liouville fractional integral and a generic fractional derivative associated with this integral. It is worth mentioning that many basic and advanced properties of this generic fractional derivative directly follow from its definition as a set of the linear operators left-inverse to the Riemann–Liouville fractional integral. By derivation of these properties, we do not use any explicit formulas for the fractional derivatives, and thus, the obtained results are generic and cover all kinds of the time-fractional derivatives introduced so far, including the Riemann–Liouville, Caputo, and Hilfer fractional derivatives. The problem of providing a constructive description of all realizations of the generic fractional derivative associated with the Riemann–Liouville fractional integral is still open. However, it is known that there exist infinitely many different families of fractional derivatives of this kind, in the form of nth-level fractional derivatives (see [8] for details).

The structure of the rest of this paper is as follows. In Section 2, we provide a definition and the basic properties of the generic fractional derivative of the functions of a single variable, including a characterization of its domain, null-space, and projector operator. In Section 3, some advanced results formulated for the generic fractional derivative of the functions of a single variable are presented. In particular, we discuss the interrelations between different realisations of the generic fractional derivative and formulate a generalized fractional Taylor formula involving the Riemann–Liouville fractional integral and the generic fractional derivative associated with this integral. Section 4 is devoted to analysis of the initial-value problems for the fractional relaxation equation with the generic fractional derivative introduced in the second section. For this problem, a closed-form formula for its unique solution is derived in explicit form, and complete monotonicity of the solution is studied. Finally, in Section 5, some conclusions and open problems for research are formulated.

2. Definition and Basic Properties of the Generic Fractional Derivative

Nowadays, several unequal definitions of the time-fractional derivatives of the functions of a single variable including the Riemann–Liouville, Caputo, Hilfer, and Djrbashian–Nersessian or the nth-level fractional derivatives are actively used in the FC literature.

In contrary to this situation, there exists only one family of the fractional integrals $I^{\alpha },\alpha >0$ that satisfies some natural conditions mentioned below. Let X be either $L_p(0,1),$ $1\le p<+\infty$ or $C\left(\right[0,1\left]\right)$ and the following axioms hold true:

A1.

$\left(I^{1}x\right)\left(t\right)={\int }_0^{t}x\left(\tau \right)d\tau ,x\in X$.

A2.

$\left(I^{\alpha }{I}^{\beta }x\right)\left(t\right)=\left(I^{\alpha +\beta }x\right)\left(t\right),\alpha ,\beta >0,x\in X$.

A3.

$\alpha \to I^{\alpha }$ is a continuous map of $(0,+\infty )$ into $\mathcal{L}\left(X\right)$ for some Hausdorff topology on $\mathcal{L}\left(X\right)$, weaker than the norm topology, where $\mathcal{L}\left(X\right)$ is the set of all linear operators acting from X to X.

A4.

$x\in C\left(\right[0,1\left]\right)$ and $x\left(t\right)\ge 0$ or $x\in L_p(0,1)$ and $x\left(t\right)\ge 0$ almost everywhere on $[0,1]$ ⇒$\left(I^{\alpha }x\right)\left(t\right)\ge 0$ or $\left(I^{\alpha }x\right)\left(t\right)\ge 0$ almost everywhere on $[0,1]$, respectively, for all $\alpha >0$.

Then, the operators $I^{\alpha },\alpha >0$ are the well-known Riemann–Liouville fractional integrals

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

(1)

where $\Gamma \left(\alpha \right)$ is the Euler gamma-function defined as the analytic continuation of the integral

$$\Gamma \left(\alpha \right)={\int }_0^{+\infty }{t}^{\alpha -1}{e}^{-t}dt,\Re \left(\alpha \right)>0$$

to a holomorphic function in the whole complex plane except zero and the negative integers, where the function has simple poles.

The actual meaning of this important result derived in [10] by Cartwright and McMullen is that the only “true” one-parameter fractional integrals $I^{\alpha },\alpha >0$ of the functions of a single variable defined on a finite interval are the family of the Riemann–Liouville fractional integrals. For other properties of the Riemann–Liouville fractional integral, we refer to the encyclopedic book [4].

On the other hand, in the FC literature, several different families of the operators known as the fractional derivatives of the functions of a single variable are employed. For the sake of simplicity of the formulas, we restrict ourselves to the case of the fractional derivatives with the order $\alpha \in (0,1)$. The case $\alpha \ge 1$ can be treated in the similar manner.

For a long time, the most used fractional derivative of the functions of a single variable was the Riemann–Liouville fractional derivative defined as follows:

$$\left(D_{RL}^{\alpha }x\right)\left(t\right):={\displaystyle \frac{d}{dt}}\left(I^{1-\alpha }x\right)\left(t\right)={\displaystyle \frac{d}{dt}}{\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{(t-\tau )}^{-\alpha }x\left(\tau \right)d\tau ,0<\alpha <1.$$

(2)

In the last few decades, the so-called Caputo fractional derivative, which was introduced already by Abel in [11,12], has started to be actively employed in FC, and especially in the theory of the fractional differential equations, both ordinary and partial ones. This derivative is defined by the formula

$$\left(D_C^{\alpha }x\right)\left(t\right):=\left(I^{1-\alpha }{x}^{\prime }\right)\left(t\right),0<\alpha <1.$$

(3)

Recently, the generalized Riemann–Liouville or the Hilfer fractional derivative of order $\alpha$ $(0<\alpha <1)$ and type $\beta$ $(0\le \beta \le 1)$ was introduced in [13] as follows:

$$(D_H^{\alpha ,\beta }x)\left(t\right):=(I^{\beta (1-\alpha )}{\displaystyle \frac{d}{dt}}{I}^{(1-\alpha )(1-\beta )}x)\left(t\right).$$

(4)

In this paper, we use another parametrization of the Hilfer fractional derivative that is obtained from Formula (4) by setting ${\gamma }_1=\beta (1-\alpha )$:

$$\left(D_H^{\alpha ,{\gamma }_1}x\right)\left(t\right)=(I^{{\gamma }_1}{\displaystyle \frac{d}{dt}}{I}^{1-\alpha -{\gamma }_1}x)\left(t\right),0<\alpha <1,0\le {\gamma }_1\le 1-\alpha .$$

(5)

Finally, we mention the Djrbashian–Nersessian operator introduced in [14]. Here, we employ its parametrization, that was suggested in [8], in the form of the nth-level fractional derivative of order $\alpha$ $(0<\alpha <1)$ and type $\gamma =({\gamma }_1,{\gamma }_2,\dots ,{\gamma }_n)$:

$$(D_{nL}^{\alpha ,\left(\gamma \right)}x)\left(t\right):=\left(\prod _{k=1}^n(I^{{\gamma }_k}{\displaystyle \frac{d}{dt}})\right)\left(I^{n-\alpha -s_n}x\right)\left(t\right),$$

(6)

where

$$s_k:=\sum _{i=1}^k{\gamma }_i,k=1,2,\dots ,n,$$

(7)

and the conditions

$$0\le {\gamma }_k\mathrm{and}\alpha +s_k\le k,k=1,2,\dots ,n$$

(8)

are satisfied. To avoid a reduction of an nth-level fractional derivative to a derivative of a lower level, we also suppose that the conditions

$$n-1<\alpha +s_n\mathrm{and}{\gamma }_k<1,k=2,\dots ,n$$

(9)

hold valid.

The main property of the fractional derivatives mentioned above is that, on certain spaces of functions, they are left-inverse operators to the Riemann-Liouville fractional integral (1). They satisfy the so-called first fundamental theorem of FC (see [8] for details). In particular, as shown in [8], it is the case for the linear space

$$X_{FT}:=\left\{x:I^{\alpha }x\in \mathrm{AC}\left([0,1]\right)\mathrm{and}\left(I^{\alpha }x\right)\left(0\right)=0\right\},$$

(10)

where $\mathrm{AC}\left(\right[0,1\left]\right)$ is the space of the functions that are absolutely continuous on the interval $[0,1]$. Any function f from the space $\mathrm{AC}\left(\right[0,1\left]\right)$ admits a representation in the form

$$f\left(t\right)=f\left(0\right)+{\int }_0^t\varphi \left(\tau \right)d\tau ,t\in [0,1],$$

(11)

with a function $\varphi \in L_1(0,1)$. This leads to the natural definition of its derivative in the weak sense:

$${\displaystyle \frac{df}{dt}}:=\varphi \left(t\right)\in L_1(0,1).$$

According to Theorem 2.3 from [4], the space $X_{FT}$ given by Formula (10) is also characterized as follows:

$$X_{FT}=I^{1-\alpha }\left(L_1(0,1)\right).$$

It is also worth mentioning that the first fundamental theorem of FC is valid for all fractional derivatives mentioned above on the space $X_{FT}$ defined by Equation (10) (see [8] for the proofs).

For some fractional derivatives, this space can be extended. Say the Riemann–Liouville fractional derivative (2) is a left-inverse operator to the Riemann–Liouville fractional integral (1) on the space $L_1(0,1)$.

The main objective of this paper is to suggest a unified approach to different kinds of the time-fractional derivatives of the functions of a single variable introduced so far. For this aim, the definitions and results presented in [9] for the left-invertible operators are adjusted to the case of the fractional integrals and derivatives of the functions of a single variable.

Let X be a linear space over $\mathbb{R}$ or $\mathbb{C}$. For an operator $A\in \mathcal{L}\left(X\right)$, its domain and null-space are denoted by $D_A$ and $N_A$, respectively.

Definition 1 

([9]). An operator $R\in \mathcal{L}\left(X\right)$ with $D_R=X$ is said to be left-invertible if there exists an operator $L\in \mathcal{L}\left(X\right)$ with $D_L\subset R\left(X\right)$ such that

$$LR=EonX,$$

(12)

where E denotes the identity operator.

As already mentioned, the Riemann–Liouville fractional integral defined by (1) possesses left-inverse operators (in particular, the Riemann–Liouville fractional derivative on the space $L_1(0,1)$), and thus, it belongs to the class of left-invertible operators. Following ref. [8] and using Definition 1, we now introduce a natural concept of a generic fractional derivative of the functions of a single variable.

Definition 2. 

Let X be a linear space over $\mathbb{R}$ or $\mathbb{C}$ and $D_{I^{\alpha }}=X$, where $I^{\alpha },\alpha >0$ is the Riemann–Liouville fractional integral defined by (1).

The set of all linear operators ${\mathbb{D}}^{\alpha },\alpha >0,$ left-inverse to the Riemann–Liouville fractional integral $I^{\alpha }$ in sense of Definition 1, is called a generic fractional derivative associated with the Riemann–Liouville fractional integral.

It is worth emphasizing that the generic fractional derivative is an infinite set of linear operators that in particular includes the Riemann–Liouville, Caputo, Hilfer, and nth-level fractional derivatives (see [8] for the proofs). By ${\mathbb{D}}^{\alpha }$, we denote a realization of the generic fractional derivative, i.e, a certain linear operator left-inverse to the Riemann–Liouville fractional integral $I^{\alpha }$. In what follows, we refer to ${\mathbb{D}}^{\alpha }$ as a fractional derivative associated with the Riemann–Liouville integral.

Remark 1. 

For the operators $R=I^{\alpha }$ and $L={\mathbb{D}}^{\alpha }$, Formula (12) from Definition 1 takes the form

$$\left({\mathbb{D}}^{\alpha }{I}^{\alpha }x\right)\left(t\right)=x\left(t\right),x\in X.$$

(13)

Formula (13) is called the first fundamental theorem of FC for the fractional derivative ${\mathbb{D}}^{\alpha }$ and the Riemann–Liouville fractional integral $I^{\alpha }$.

As mentioned in [8], for a concrete fractional derivative ${\mathbb{D}}^{\alpha }$, the space X, where the relation (13) is valid, can be narrower compared to the space where the Riemann–Liouville integral is defined. Thus, in general, these two spaces are diverse. However, to keep formulations and derivations of our results clearly arranged, in what follows, we use the same notation for both spaces if its meaning is clear from the context.

Remark 2. 

Axioms A1–A4 of the fractional integrals, along with Formula (13), can be interpreted as a system of axioms for the one-parameter families of fractional integrals and derivatives. In contrast to the axioms suggested in [5,6,7], most of these axioms describe the properties of fractional integrals, and the only axiom related to fractional derivatives is its definition by Formula (13). As a consequence, the family of the fractional integrals satisfying axioms A1–A4 is unique. As for the families of fractional derivatives satisfying the axiom (13), there are infinitely many such families, including the Riemann-Liouville, Caputo, Hilfer, and nth-level fractional derivatives mentioned at the beginning of this section.

For investigation of the generic fractional derivative associated with the Riemann–Liouville fractional integral, a concept of its projector operator plays a very important role.

Definition 3. 

Let ${\mathbb{D}}^{\alpha }$ be a fractional derivative associated with the Riemann-Liouville fractional integral $I^{\alpha }$.

The operator

$${\mathbb{P}}^{\alpha }=E-I^{\alpha }{\mathbb{D}}^{\alpha } on D_{{\mathbb{D}}^{\alpha }}$$

(14)

is called a projector operator of the fractional derivative ${\mathbb{D}}^{\alpha }$.

Remark 3. 

Formula (14) can be rewritten in the form of the so-called second fundamental theorem of FC for the fractional derivative ${\mathbb{D}}^{\alpha }$; see [8]:

$$\left(I^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)=x\left(t\right)-\left({\mathbb{P}}^{\alpha }x\right)\left(t\right),x\in D_{{\mathbb{D}}^{\alpha }}.$$

(15)

As soon as we have an explicit formula for the projector operator ${\mathbb{P}}^{\alpha }$ of a certain fractional derivative ${\mathbb{D}}^{\alpha }$, we immediately obtain an explicit form of the second fundamental theorem of FC for this fractional derivative.

For the Riemann–Liouville, Caputo, and Hilfer fractional derivatives, the explicit formulas for their projector operators are known (see, e.g., [8]). A closed-form formula for the projector operator of the nth-level fractional derivative was recently derived in [15]. For convenience, we use the notation

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

(16)

As already mentioned in this paper, we restrict ourselves to the case of fractional derivatives of order $\alpha \in (0,1)$, and then, the formulas of the projector operators for the Riemann–Liouville, Caputo, Hilfer, and the nth-level fractional derivatives take the following forms, respectively ([8,15]):

$$\left({\mathbb{P}}_{RL}^{\alpha }x\right)\left(t\right):=(E-I^{\alpha }{D}_{RL}^{\alpha }x)\left(t\right)=\left(I^{1-\alpha }x\right)\left(0\right)h_{\alpha }\left(t\right),$$

(17)

$$\left({\mathbb{P}}_C^{\alpha }x\right)\left(t\right):=(E-I^{\alpha }{D}_C^{\alpha }x)\left(t\right)=x\left(0\right)h_1\left(t\right),$$

(18)

$$\left({\mathbb{P}}_H^{\alpha }x\right)\left(t\right):=(E-I^{\alpha }{D}_H^{\alpha ,{\gamma }_1}x)\left(t\right)=\left(I^{1-\alpha -{\gamma }_1}x\right)\left(0\right)h_{\alpha +{\gamma }_1}\left(t\right),$$

(19)

$$\left(P_{nL}^{\alpha }x\right)\left(t\right):=(E-I^{\alpha }{D}_{nL}^{\alpha ,\left(\gamma \right)}x)\left(t\right)=\sum _{k=1}^n{p}_k{h}_{\alpha +s_k-k+1}\left(t\right),$$

(20)

$$s_k=\sum _{i=1}^k{\gamma }_i,p_k=\left(\prod _{i=k+1}^n\left(I^{{\gamma }_i}{\displaystyle \frac{d}{dt}}\right)I^{n-\alpha -s_n}x\right)\left(0\right),k=1,2,\dots ,n.$$

(21)

Remark 4. 

The Riemann–Liouville, Caputo, and Hilfer fractional derivatives are interpreted as the fractional derivatives of the first level because they contain just one derivative of the first order, and thus, their null-spaces are one-dimensional under the condition $\alpha \in (0,1)$. All of these derivatives are particular cases of the nth-level fractional derivative $D_{nL}^{\alpha ,\left(\gamma \right)}$ with $n=1$.

In the following theorem, some basic properties of the fractional derivatives associated with the Riemann–Liouville fractional integral $I^{\alpha }$ as well as their projector operators are given.

Theorem 1. 

Let ${\mathbb{D}}^{\alpha }$ be a fractional derivative associated with the Riemann–Liouville fractional integral $I^{\alpha }$ defined on the linear space X, and let ${\mathbb{P}}^{\alpha }$ be its projector operator. Then the following properties hold true:

(P1) The operator ${\mathbb{P}}^{\alpha }$ is a projector:

$$\left({\left({\mathbb{P}}^{\alpha }\right)}^{2}x\right)\left(t\right)=\left({\mathbb{P}}^{\alpha }x\right)\left(t\right),x\in D_{{\mathbb{D}}^{\alpha }}.$$

(22)

(P2) The image of ${\mathbb{P}}^{\alpha }$ belongs to the null-space of ${\mathbb{D}}^{\alpha }$:

$$\left({\mathbb{D}}^{\alpha }{\mathbb{P}}^{\alpha }x\right)\left(t\right)=0,x\in D_{{\mathbb{D}}^{\alpha }}.$$

(23)

(P3) The image of $I^{\alpha }$ belongs to the null-space of ${\mathbb{P}}^{\alpha }$:

$$\left({\mathbb{P}}^{\alpha }{I}^{\alpha }x\right)\left(t\right)=0,x\in X.$$

(24)

(P4) The null-space of ${\mathbb{D}}^{\alpha }$ is characterized in terms of its projector operator ${\mathbb{P}}^{\alpha }$ as follows:

$$N_{{\mathbb{D}}^{\alpha }}=\left\{x\in D_{{\mathbb{D}}^{\alpha }}:\left({\mathbb{P}}^{\alpha }x\right)\left(t\right)=x\left(t\right)\right\}.$$

(25)

(P5) The null-space of ${\mathbb{P}}^{\alpha }$ is characterized as follows:

$$N_{{\mathbb{P}}^{\alpha }}=\left\{x:x\left(t\right)=\left(I^{\alpha }\varphi \right)\left(t\right),\varphi \in X\right\}.$$

(26)

(P6) For any $n\in \mathbb{N}$, we have

$$x\left(t\right)=\left(I^{\alpha n}\varphi \right)\left(t\right),\varphi \in X\Rightarrow \left({\mathbb{P}}^{\alpha }{\left({\mathbb{D}}^{\alpha }\right)}^{k}x\right)\left(t\right)=0,k=0,1,\dots ,n-1.$$

(27)

Proof. 

Most of the properties formulated above are valid for any left-invertible operators, their left-inverse operators, and their projector operators; see [9]. Because the Riemann–Liouville fractional integral is left-invertible, we can apply these results for the generic fractional derivative in sense of Definition 2. However, for the reader’s convenience, we reproduce here the proofs presented in [9] and adjust them to the case of the fractional derivatives associated with the Riemann–Liouville fractional integral.

Proof of P1: For $x\in D_{{\mathbb{D}}^{\alpha }}$, the projector operator ${\mathbb{P}}^{\alpha }$ is well-defined. Employing the definitions of the fractional derivative ${\mathbb{D}}^{\alpha }$ and its projector operator, we arrive at the following chain of equations:

$$\left({\left({\mathbb{P}}^{\alpha }\right)}^{2}x\right)\left(t\right)=\left((E-I^{\alpha }{\mathbb{D}}^{\alpha })(E-I^{\alpha }{\mathbb{D}}^{\alpha })x\right)\left(t\right)=$$

$$x\left(t\right)-\left(I^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)-\left(I^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)+\left(I^{\alpha }{\mathbb{D}}^{\alpha }{I}^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)=$$

$$x\left(t\right)-\left(I^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)-\left(I^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)+\left(I^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)=$$

$$x\left(t\right)-\left(I^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)=\left({\mathbb{P}}^{\alpha }x\right)\left(t\right).$$

Proof of P2: For $x\in D_{{\mathbb{D}}^{\alpha }}$, we obtain

$$\left({\mathbb{D}}^{\alpha }{\mathbb{P}}^{\alpha }x\right)\left(t\right)=\left({\mathbb{D}}^{\alpha }(E-I^{\alpha }{\mathbb{D}}^{\alpha })x\right)\left(t\right)=\left({\mathbb{D}}^{\alpha }x\right)\left(t\right)-\left({\mathbb{D}}^{\alpha }{I}^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)=$$

$$\left({\mathbb{D}}^{\alpha }x\right)\left(t\right)-\left({\mathbb{D}}^{\alpha }x\right)\left(t\right)=0.$$

Proof of P3: For $x\in X$, the definitions of the fractional derivative ${\mathbb{D}}^{\alpha }$ and its projector operator lead to the formula

$$\left({\mathbb{P}}^{\alpha }{I}^{\alpha }x\right)\left(t\right)=\left((E-I^{\alpha }{\mathbb{D}}^{\alpha })I^{\alpha }x\right)\left(t\right)=\left(I^{\alpha }x\right)\left(t\right)-\left(I^{\alpha }{\mathbb{D}}^{\alpha }{I}^{\alpha }x\right)\left(t\right)=\left(I^{\alpha }x\right)\left(t\right)-\left(I^{\alpha }x\right)\left(t\right)=0.$$

Proof of P4: Let $x\in D_{{\mathbb{D}}^{\alpha }}$ and $\left({\mathbb{P}}^{\alpha }x\right)\left(t\right)=x\left(t\right)$. Then,

$$\left({\mathbb{D}}^{\alpha }x\right)\left(t\right)=\left({\mathbb{D}}^{\alpha }{\mathbb{P}}^{\alpha }x\right)\left(t\right)=0$$

according to property P2, and we have the inclusion $x\in N_{{\mathbb{D}}^{\alpha }}$.

Now let $x\in N_{{\mathbb{D}}^{\alpha }}$, i.e., $\left({\mathbb{D}}^{\alpha }x\right)\left(t\right)=0$. Then

$$\left({\mathbb{P}}^{\alpha }x\right)\left(t\right)=\left((E-I^{\alpha }{\mathbb{D}}^{\alpha })x\right)\left(t\right)=x\left(t\right)-\left(I^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)=x\left(t\right)-\left(I^{\alpha }0\right)\left(t\right)=x\left(t\right)$$

and the proof of P4 is completed.

Proof of P5: Let $x\left(t\right)=\left(I^{\alpha }\varphi \right)\left(t\right),\varphi \in X$. Then, property P3 implicates the formula

$$\left({\mathbb{P}}^{\alpha }x\right)\left(t\right)=\left({\mathbb{P}}^{\alpha }{I}^{\alpha }\varphi \right)\left(t\right)=0$$

and $x\in N_{{\mathbb{P}}^{\alpha }}$.

Now, let $x\in N_{{\mathbb{P}}^{\alpha }}$, i.e., $\left({\mathbb{P}}^{\alpha }x\right)\left(t\right)=0$. Then,

$$\left((E-I^{\alpha }{\mathbb{D}}^{\alpha })x\right)\left(t\right)=0\Rightarrow x\left(t\right)=\left(I^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)\Rightarrow x\left(t\right)=\left(I^{\alpha }\varphi \right)\left(t\right),\varphi \left(t\right)=\left({\mathbb{D}}^{\alpha }x\right)\left(t\right).$$

Proof of P6: Let $n\in \mathbb{N}$ and $x\left(t\right)=\left(I^{\alpha n}\varphi \right)\left(t\right),\varphi \in X$. Then, for $k=0,1,\dots ,n-1$, we obtain

$$\left({\mathbb{P}}^{\alpha }{\left({\mathbb{D}}^{\alpha }\right)}^{k}x\right)\left(t\right)=\left((E-I^{\alpha }{\mathbb{D}}^{\alpha }){\left({\mathbb{D}}^{\alpha }\right)}^{k}x\right)\left(t\right)=\left((E-I^{\alpha }{\mathbb{D}}^{\alpha }){\left({\mathbb{D}}^{\alpha }\right)}^k\left(I^{\alpha n}\varphi \right)\right)\left(t\right)=$$

$$\left((E-I^{\alpha }{\mathbb{D}}^{\alpha }){\left({\mathbb{D}}^{\alpha }\right)}^k{\left(I^{\alpha }\right)}^n\varphi \right)\left(t\right)=\left((E-I^{\alpha }{\mathbb{D}}^{\alpha }){\left(I^{\alpha }\right)}^{n-k}\varphi \right)\left(t\right)=$$

$$\left({\left(I^{\alpha }\right)}^{n-k}\varphi \right)\left(t\right)-\left(I^{\alpha }{\mathbb{D}}^{\alpha }{\left(I^{\alpha }\right)}^{n-k}\varphi \right)\left(t\right)=\left({\left(I^{\alpha }\right)}^{n-k}\varphi \right)\left(t\right)-\left(I^{\alpha }{\left(I^{\alpha }\right)}^{n-k-1}\varphi \right)\left(t\right)=$$

$$\left({\left(I^{\alpha }\right)}^{n-k}\varphi \right)\left(t\right)-\left({\left(I^{\alpha }\right)}^{n-k}\varphi \right)\left(t\right)=0.$$

The proof of the Theorem is completed. □

Remark 5. 

It is worth mentioning that in the proof of Theorem 1, we did not employ any explicit formulas for the fractional derivatives or for their projector operators. The main tool used in the proof was the first fundamental theorem of FC valid for all realizations of the generic fractional derivative by definition. Thus, properties P1-P6 are valid for all realizations of the generic fractional derivative, both for the known and for not-yet-introduced ones.

The results formulated in Theorem 1 can be used for a concrete fractional derivative that leads to some known or new formulas. In particular, property P4 applied to the Riemann–Liouville, Caputo, Hilfer, and nth-level fractional derivatives in combination with Formulas (17)–(20) for their projector operators leads to the following characterization of their null-spaces (see [8,15]):

$$N_{D_{RL}^{\alpha }}=\{c_1{h}_{\alpha }\left(t\right),c_1\in \mathbb{R}\},$$

(28)

$$N_{D_C^{\alpha }}=\{c_1{h}_1\left(t\right),c_1\in \mathbb{R}\},$$

(29)

$$N_{D_H^{\alpha ,{\gamma }_1}}=\{c_1{h}_{\alpha +{\gamma }_1}\left(t\right),c_1\in \mathbb{R}\},$$

(30)

$$N_{D_{nL}^{\alpha ,\left(\gamma \right)}}=\{\sum _{k=1}^n{c}_k{h}_{\alpha +s_k-k+1}\left(t\right),s_k=\sum _{i=1}^k{\gamma }_i,c_k\in \mathbb{R},k=1,2,\dots ,n\}.$$

(31)

3. Advanced Properties of the Generic Fractional Derivative

In this section, we derive some advanced properties of the generic fractional derivative introduced in the previous section, including a characterization of its domain, a formula that is connecting its different realizations, and the generalized fractional Taylor formula involving the generic fractional derivative.

We start with a simple but important result regarding the domain of the generic fractional derivative.

Theorem 2. 

Let ${\mathbb{D}}^{\alpha },\alpha \in (0,1)$ be a fractional derivative associated with the Riemann–Liouville fractional integral $I^{\alpha }$ defined on the linear space X.

Then, for any $x\in D_{{\mathbb{D}}^{\alpha }}$, there exist the functions $x_1\in X$ and $x_2\in N_{{\mathbb{D}}^{\alpha }}$, such that

$$x\left(t\right)=\left(I^{\alpha }{x}_1\right)\left(t\right)+x_2\left(t\right),x_1\in X,x_2\in N_{{\mathbb{D}}^{\alpha }}.$$

(32)

Proof. 

Let $x\in D_{{\mathbb{D}}^{\alpha }}$. By definition, ${\mathbb{D}}^{\alpha }\in \mathcal{L}\left(X\right)$, and then

$$\left({\mathbb{D}}^{\alpha }x\right)\left(t\right)=x_1\left(t\right),x_1\in X.$$

(33)

Now, we apply the Riemann–Liouville fractional integral $I^{\alpha }$ to both sides of the Formula (33), and obtain the relation

$$\left(I^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)=\left(I^{\alpha }{x}_1\right)\left(t\right),x_1\in X$$

(34)

which can be rewritten in terms of the projector operator of the fractional derivative ${\mathbb{D}}^{\alpha }$ as follows:

$$\left(I^{\alpha }{\mathbb{D}}^{\alpha }x\right)\left(t\right)=x\left(t\right)-\left({\mathbb{P}}^{\alpha }x\right)\left(t\right)=\left(I^{\alpha }{x}_1\right)\left(t\right),x_1\in X.$$

(35)

According to property P2 of Theorem 1, the image of the projector operator belongs to the null-space of ${\mathbb{D}}^{\alpha }$, and thus, we arrive at the representation

$$x\left(t\right)=\left(I^{\alpha }{x}_1\right)\left(t\right)+\left({\mathbb{P}}^{\alpha }x\right)\left(t\right)=\left(I^{\alpha }{x}_1\right)\left(t\right)+x_2\left(t\right),x_1\in X,x_2={\mathbb{P}}^{\alpha }x\in N_{{\mathbb{D}}^{\alpha }}$$

(36)

which completes the proof of the Theorem. □

Representation (32) and Formula (13) (the first fundamental theorem of FC) clarify the reason for discrepancy in the mapping properties of different realizations of the generic fractional derivative that is caused by their unequal domains and null-spaces.

Now, we derive a formula connecting different realizations of the generic fractional derivative under the assumption that their null-spaces are finite-dimensional as it is the case for the Riemann–Liouville, Caputo, Hilfer, and nth-level fractional derivatives (see Formulas (28)–(31)).

Theorem 3. 

Let ${\mathbb{D}}_1^{\alpha }$ and ${\mathbb{D}}_2^{\alpha }$ be two fractional derivatives associated with the Riemann–Liouville fractional integral and the null-space of ${\mathbb{D}}_1^{\alpha }$ be finite-dimensional, i.e.,

$$N_{{\mathbb{D}}_1^{\alpha }}=\{c_1{\varphi }_1\left(t\right)+\cdots +c_n{\varphi }_n\left(t\right),c_1,\dots ,c_n\in \mathbb{R}\},$$

(37)

where the functions ${\varphi }_1,\dots ,{\varphi }_n$ are linearly independent, and the inclusion

$$N_{{\mathbb{D}}_1^{\alpha }}\subset D_{{\mathbb{D}}_2^{\alpha }}$$

(38)

holds true.

Then, the fractional derivatives ${\mathbb{D}}_1^{\alpha }$ and ${\mathbb{D}}_2^{\alpha }$ are connected by the relation

$$\left({\mathbb{D}}_2^{\alpha }x\right)\left(t\right)=\left({\mathbb{D}}_1^{\alpha }x\right)\left(t\right)+\sum _{j=1}^n{a}_j\left({\mathbb{D}}_2^{\alpha }{\varphi }_j\right)\left(t\right),x\in D_{{\mathbb{D}}_1^{\alpha }},$$

(39)

where the coefficients $a_j,j=1,\dots ,n$ are as in the representation

$$\left({\mathbb{P}}_1^{\alpha }x\right)\left(t\right)=\sum _{j=1}^n{a}_j{\varphi }_j\left(t\right)$$

(40)

of the element ${\mathbb{P}}_1^{\alpha }x$ from the null-space $N_{{\mathbb{D}}_1^{\alpha }}$, where ${\mathbb{P}}_1^{\alpha }$ is the projector of the fractional derivative ${\mathbb{D}}_1^{\alpha }$.

Proof. 

According to Theorem 2, any $x\in D_{{\mathbb{D}}_1^{\alpha }}$ can be represented as follows:

$$x\left(t\right)=\left(I^{\alpha }{x}_1\right)\left(t\right)+x_2\left(t\right),x_1\in X,x_2\in N_{{\mathbb{D}}_1^{\alpha }}.$$

The null-space of ${\mathbb{D}}_1^{\alpha }$ is finite-dimensional and is given by (37). Thus, we obtain the relation

$$x\left(t\right)=\left(I^{\alpha }{x}_1\right)\left(t\right)+\sum _{j=1}^n{c}_j{\varphi }_j\left(t\right).$$

(41)

Because of inclusion (38) and Formula (13) (the first fundamental theorem of FC), we can apply the operator ${\mathbb{D}}_2^{\alpha }$ to the right-hand side of Formula (41), and thus also to its left-hand side, and arrive at the equation

$$\left({\mathbb{D}}_2^{\alpha }x\right)\left(t\right)=x_1\left(t\right)+\sum _{j=1}^n{c}_j\left({\mathbb{D}}_2^{\alpha }{\varphi }_j\right)\left(t\right).$$

(42)

Acting on representation (41) with the fractional derivative ${\mathbb{D}}_1^{\alpha }$ immediately leads to the relation

$$\left({\mathbb{D}}_1^{\alpha }x\right)\left(t\right)=x_1\left(t\right)$$

(43)

and thus,

$$\left({\mathbb{D}}_2^{\alpha }x\right)\left(t\right)=\left({\mathbb{D}}_1^{\alpha }x\right)\left(t\right)+\sum _{j=1}^n{c}_j\left({\mathbb{D}}_2^{\alpha }{\varphi }_j\right)\left(t\right).$$

(44)

Now, let us determine the coefficients $c_j$ in the last representation. According to property P2 from Theorem 1, the inclusion ${\mathbb{P}}_1^{\alpha }x\in N_{{\mathbb{D}}_1^{\alpha }}$ holds true, and that leads to representation (40), with certain coefficients $a_j,j=1,\dots ,n$ that depend on the function x.

On the other hand, applying the projector operator ${\mathbb{P}}_1^{\alpha }$ to representation (41), we obtain the equation

$$\left({\mathbb{P}}_1^{\alpha }x\right)\left(t\right)=\left({\mathbb{P}}_1^{\alpha }{I}^{\alpha }{x}_1\right)\left(t\right)+\sum _{j=1}^n{c}_j\left({\mathbb{P}}_1^{\alpha }{\varphi }_j\right)\left(t\right).$$

(45)

Property P3 of Theorem 1 implicates

$$\left({\mathbb{P}}_1^{\alpha }{I}^{\alpha }{x}_1\right)\left(t\right)=0,$$

(46)

whereas property P4 leads to the relations

$$\left({\mathbb{P}}_1^{\alpha }{\varphi }_j\right)\left(t\right)={\varphi }_j\left(t\right),j=1,\dots ,n.$$

(47)

Taking into account Formulas (46) and (47), we rewrite Equation (45) as follows:

$$\left({\mathbb{P}}_1^{\alpha }x\right)\left(t\right)=\sum _{j=1}^n{c}_j{\varphi }_j\left(t\right).$$

(48)

Comparing this equation with relation (40), we see that $c_j=a_j,j=1,\dots ,n$, where the coefficients $a_j$ are defined as in representation (40). The proof of the Theorem is completed. □

Remark 6. 

To employ Formula (39) for fractional derivatives whose null-spaces are linear combinations of some power law functions, we use the following formula:

$$\left({\mathbb{D}}^{\alpha }{h}_{\beta }\right)\left(t\right)=h_{\beta -\alpha }\left(t\right),\alpha >0,h_{\beta }\in D_{{\mathbb{D}}^{\alpha }}\setminus N_{{\mathbb{D}}^{\alpha }}.$$

(49)

Formula (49) easily follows from the first fundamental theorem of FC valid for any realization of the generic fractional derivative by definition and the formula for convolution of two power law functions ([4,16]):

$$\left(I^{\alpha }{h}_{\beta }\right)\left(t\right)=(h_{\alpha }\ast h_{\beta })\left(t\right)=h_{\alpha +\beta }\left(t\right),\alpha >0,\beta >0.$$

Example 1. 

We start with the derivation of the known relation between the Riemann–Liouville and Caputo fractional derivatives (see, e.g., [16]).

Setting ${\mathbb{D}}_1^{\alpha }=D_C^{\alpha }$ and ${\mathbb{D}}_2^{\alpha }=D_{RL}^{\alpha }$ in Theorem 3, we immediately see that the inclusion $N_{D_C^{\alpha }}=\{c_1{h}_1\left(t\right),c_1\in \mathbb{R}\}\subset D_{D_{RL}^{\alpha }}$ holds true, and thus, this theorem is applicable.

Relation (39) from Theorem 3 adjusted to the case of the Riemann–Liouville and Caputo fractional derivatives by means of Formula (18) takes the form

$$\left(D_{RL}^{\alpha }x\right)\left(t\right)=\left(D_C^{\alpha }x\right)\left(t\right)+x\left(0\right)\left(D_{RL}^{\alpha }{h}_1\right)\left(t\right)=\left(D_C^{\alpha }x\right)\left(t\right)+x\left(0\right)h_{1-\alpha }\left(t\right),x\in D_{D_C^{\alpha }}.$$

(50)

It is worth mentioning that we cannot apply Theorem 3 in the case of ${\mathbb{D}}_1^{\alpha }=D_{RL}^{\alpha }$ and ${\mathbb{D}}_2^{\alpha }=D_C^{\alpha }$ because the null-space $N_{D_{RL}^{\alpha }}=\{c_1{h}_{\alpha }\left(t\right),c_1\in \mathbb{R}\}$ does not belong to the domain $D_{D_C^{\alpha }}$ of the Caputo fractional derivative.

Example 2. 

Now, we derive a new relation between the Riemann–Liouville and the nth-level fractional derivatives.

In Theorem 3, we set ${\mathbb{D}}_1^{\alpha }=D_{nL}^{\alpha ,\left(\gamma \right)}$ and ${\mathbb{D}}_2^{\alpha }=D_{RL}^{\alpha }$. The null-space of the nth-level fractional derivative is given by Formula (31). It is n-dimensional, and the inclusion

$$N_{D_{nL}^{\alpha ,\left(\gamma \right)}}=\{\sum _{k=1}^n{c}_k{h}_{\alpha +s_k-k+1}\left(t\right),c_1,\dots ,c_n\in \mathbb{R}\}\subset D_{D_{RL}^{\alpha }}$$

holds true because all of the exponents

$${\beta }_k=\alpha +s_k-k,k=1,\dots ,n$$

of the power law functions $h_{\alpha +s_k-k+1}$ satisfy the inequality $-1<{\beta }_k$ that is ensured by conditions (8) and (9) posed on the parameters of the nth-level fractional derivative $D_{nL}^{\alpha }$.

Taking into consideration Formula (20) for the projector operator of the nth-level fractional derivative, as well as Formula (49), the relation (39) from Theorem 3 leads to the formula

$$\left(D_{RL}^{\alpha }x\right)\left(t\right)=\left(D_{nL}^{\alpha ,\left(\gamma \right)}x\right)\left(t\right)+\sum _{k=1}^n{p}_k{h}_{s_k-k+1}\left(t\right),x\in D_{D_{nL}^{\alpha ,\left(\gamma \right)}},$$

(51)

$$s_k=\sum _{i=1}^k{\gamma }_i,p_k=\left(\prod _{i=k+1}^n\left(I^{{\gamma }_i}{\displaystyle \frac{d}{dt}}\right)I^{n-\alpha -s_n}x\right)\left(0\right),k=1,2,\dots ,n.$$

In the rest of this section, we present a generalized Taylor formula for the generic fractional derivative associated with the Riemann–Liouville fractional integral. To derive this formula, we do not use any explicit expressions for the fractional derivatives, and just employ their definition as the left-inverse operators to the Riemann–Liouville fractional integral.

Theorem 4. 

Let ${\mathbb{D}}^{\alpha }$ be any fractional derivative associated with the Riemann–Liouville fractional integral $I^{\alpha }$ and $x\in D_{{\left({\mathbb{D}}^{\alpha }\right)}^n},n\in \mathbb{N}$.

Then, the generalized Taylor formula

$$x\left(t\right)=\sum _{k=0}^{n-1}\left(I^{\alpha k}\left({\mathbb{P}}^{\alpha }\left({\left({\mathbb{D}}^{\alpha }\right)}^{k}x\right)\right)\right)\left(t\right)+r_n\left(t\right),n\in \mathbb{N}$$

(52)

holds valid, where the remainder $r_n$ is given by the formula

$$r_n\left(t\right)=\left(I^{\alpha n}\left({\left({\mathbb{D}}^{\alpha }\right)}^{n}x\right)\right)\left(t\right).$$

(53)

Under the condition ${\left({\mathbb{D}}^{\alpha }\right)}^{n}x\in C\left([0,a]\right),a>0$, the remainder $r_n$ can be represented as follows:

$$r_n\left(t\right)=\left({\left({\mathbb{D}}^{\alpha }\right)}^{n}x\right)\left(c\right)h_{\alpha n+1}\left(t\right),t\in (0,a),$$

(54)

where c is a suitably chosen value from the interval $[0,t]$.

Proof. 

The basic element of the proof is the operator identity

$$E=\sum _{k=0}^{n-1}{I}^{\alpha k}{\mathbb{P}}^{\alpha }{\left({\mathbb{D}}^{\alpha }\right)}^k+I^{\alpha n}{\left({\mathbb{D}}^{\alpha }\right)}^n,n\in \mathbb{N}$$

(55)

on the space $D_{{\left({\mathbb{D}}^{\alpha }\right)}^n}$. A formula of type (55) holds true for any left-invertible operator (see [9]). For the reader’s convenience, we provide a proof of (55) by the method of mathematical induction.

For $n=1$, Formula (55) takes the form

$$E={\mathbb{P}}^{\alpha }+I^{\alpha }{\mathbb{D}}^{\alpha }$$

(56)

which is nothing else but the definition of the projector operator.

Let the identity (55) be true for $n=N$, i.e.,

$$E=\sum _{k=0}^{N-1}{I}^{\alpha k}{\mathbb{P}}^{\alpha }{\left({\mathbb{D}}^{\alpha }\right)}^k+I^{\alpha N}{\left({\mathbb{D}}^{\alpha }\right)}^N.$$

(57)

Using Formula (57) and the definition of the projector operator, for $n=N+1$, we obtain the following chain of equations:

$$\sum _{k=0}^N{I}^{\alpha k}{\mathbb{P}}^{\alpha }{\left({\mathbb{D}}^{\alpha }\right)}^k+I^{\alpha (N+1)}{\left({\mathbb{D}}^{\alpha }\right)}^{N+1}=$$

$$\sum _{k=0}^{N-1}{I}^{\alpha k}{\mathbb{P}}^{\alpha }{\left({\mathbb{D}}^{\alpha }\right)}^k+I^{\alpha N}{\mathbb{P}}^{\alpha }{\left({\mathbb{D}}^{\alpha }\right)}^N+I^{\alpha (N+1)}{\left({\mathbb{D}}^{\alpha }\right)}^{N+1}=$$

$$E-I^{\alpha N}{\left({\mathbb{D}}^{\alpha }\right)}^N+I^{\alpha N}{\mathbb{P}}^{\alpha }{\left({\mathbb{D}}^{\alpha }\right)}^N+I^{\alpha (N+1)}{\left({\mathbb{D}}^{\alpha }\right)}^{N+1}=$$

$$E-I^{\alpha N}{\left({\mathbb{D}}^{\alpha }\right)}^N+I^{\alpha N}(I-I^{\alpha }{\mathbb{D}}^{\alpha }){\left({\mathbb{D}}^{\alpha }\right)}^N+I^{\alpha (N+1)}{\left({\mathbb{D}}^{\alpha }\right)}^{N+1}=$$

$$E-I^{\alpha N}{\left({\mathbb{D}}^{\alpha }\right)}^N+I^{\alpha N}{\left({\mathbb{D}}^{\alpha }\right)}^N-I^{\alpha N}{I}^{\alpha }{\mathbb{D}}^{\alpha }{\left({\mathbb{D}}^{\alpha }\right)}^N+I^{\alpha (N+1)}{\left({\mathbb{D}}^{\alpha }\right)}^{N+1}=E$$

which completes the proof of identity (55). This identity can be immediately interpreted as the generalized fractional Taylor formula (52) with the remainder in the following form:

$$r_n\left(t\right)=\left(I^{\alpha n}\left({\left({\mathbb{D}}^{\alpha }\right)}^{n}x\right)\right)\left(t\right)=(h_{\alpha n}\ast \left({\left({\mathbb{D}}^{\alpha }\right)}^{n}x\right))\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha n\right)}}{\int }_0^t{(t-\tau )}^{\alpha n-1}\left({\left({\mathbb{D}}^{\alpha }\right)}^{n}x\right)\left(\tau \right)d\tau .$$

Under the condition ${\left({\mathbb{D}}^{\alpha }\right)}^{n}x\in C\left([0,a]\right),a>0$, we can apply the generalized mean value theorem for integrals to the integral at the right-hand side of the last formula because the function $h_{\alpha n}$ is integrable and positive on the interval $(0,t)$. Then, we obtain

$$r_n\left(t\right)={\displaystyle \frac{\left({\left({\mathbb{D}}^{\alpha }\right)}^{n}x\right)\left(c\right)}{\Gamma \left(\alpha n\right)}}{\int }_0^t{(t-\tau )}^{\alpha n-1}d\tau =\left({\left({\mathbb{D}}^{\alpha }\right)}^{n}x\right)\left(c\right)(h_{\alpha n}\ast h_1)\left(t\right)=\left({\left({\mathbb{D}}^{\alpha }\right)}^{n}x\right)\left(c\right)h_{\alpha n+1}\left(t\right),$$

where c is a suitably chosen value from the interval $[0,t]$. This completes the proof of the Theorem. □

It is worth mentioning that the remainder $r_n$ defined as in (54) is provided in the Lagrange form. Indeed, for $\alpha =1$ (the case of the Taylor formula with the integer-order derivatives) and $x^{\left(n\right)}\in C\left([0,a]\right),a>0$, we have the formula

$$r_n\left(t\right)=\left(I^n{x}^{\left(n\right)}\right)\left(t\right)=x^{\left(n\right)}\left(c\right)h_{n+1}\left(t\right)=x^{\left(n\right)}\left(c\right){\displaystyle \frac{t^n}{\Gamma (n+1)}}={\displaystyle \frac{x^{\left(n\right)}\left(c\right)}{n!}}{t}^n,$$

(58)

where c is a suitably chosen value from the interval $[0,t]$.

Let us now consider two known and one new particular cases of the generalized Taylor formula (52).

Example 3. 

For the Riemann–Liouville fractional derivative with the projector operator given by relation (17), the generalized Taylor formula (52) takes the form known from [17,18]:

$$x\left(t\right)=\sum _{k=0}^{n-1}\left(I^{1-\alpha }{\left(D_{RL}^{\alpha }\right)}^{k}x\right)\left(0\right)h_{\alpha k+\alpha }\left(t\right)+r_n\left(t\right),r_n\left(t\right)=\left(I^{\alpha n}\left({\left(D_{RL}^{\alpha }\right)}^{n}x\right)\right)\left(t\right),n\in \mathbb{N}.$$

(59)

If the inclusion ${\left(D_{RL}^{\alpha }\right)}^{n}x\in C\left([0,a]\right),a>0$ holds true, the remainder $r_n$ can be represented as in Formula (54):

$$r_n\left(t\right)=\left({\left(D_{RL}^{\alpha }\right)}^{n}x\right)\left(c\right)h_{\alpha n+1}\left(t\right),c\in [0,t].$$

(60)

Example 4. 

The projector operator for the Caputo fractional derivative is given by Formula (18). Employing this formula, we represent the generalized Taylor formula for the Caputo fractional derivative as follows ([18]):

$$x\left(t\right)=\sum _{k=0}^{n-1}\left({\left(D_C^{\alpha }\right)}^{k}x\right)\left(0\right)h_{\alpha k+1}\left(t\right)+r_n\left(t\right),r_n\left(t\right)=\left(I^{\alpha n}\left({\left(D_C^{\alpha }\right)}^{n}x\right)\right)\left(t\right),n\in \mathbb{N}.$$

(61)

For the functions satisfying the condition ${\left(D_C^{\alpha }\right)}^{n}x\in C\left([0,a]\right),a>0$, the remainder $r_n$ can be represented in the Lagrange form according to Formula (54):

$$r_n\left(t\right)=\left({\left({\mathbb{D}}_C^{\alpha }\right)}^{n}x\right)\left(c\right)h_{\alpha n+1}\left(t\right),c\in [0,t].$$

(62)

Example 5. 

In this example, for the first time, we present a generalized Taylor formula for the Hilfer fractional derivative. Its projector operator is given by relation (19). Then the generalized Taylor formula (52) takes the following form:

$$x\left(t\right)=\sum _{k=0}^{n-1}\left(I^{1-\alpha -{\gamma }_1}{\left(D_H^{\alpha ,{\gamma }_1}\right)}^{k}x\right)\left(0\right)h_{\alpha k+\alpha +{\gamma }_1}\left(t\right)+r_n\left(t\right),r_n\left(t\right)=\left(I^{\alpha n}\left({\left(D_H^{\alpha ,{\gamma }_1}\right)}^{n}x\right)\right)\left(t\right),n\in \mathbb{N}.$$

(63)

Under the condition ${\left(D_H^{\alpha ,{\gamma }_1}\right)}^{n}x\in C\left([0,a]\right),a>0$, the remainder $r_n$ can be represented in the Lagrange form (54) as follows:

$$r_n\left(t\right)=\left({\left({\mathbb{D}}_H^{\alpha ,{\gamma }_1}\right)}^{n}x\right)\left(c\right)h_{\alpha n+1}\left(t\right),c\in [0,t].$$

(64)

As expected, for ${\gamma }_1=0$ and ${\gamma }_1=1-\alpha$, the generalized Taylor formula (63) for the Hilfer fractional derivative is reduced to the generalized Taylor formula (59) for the Riemann–Liouville fractional derivative and the generalized Taylor Formula (61) for the Caputo fractional derivative, respectively.

4. Fractional Relaxation Equation with the Generic Fractional Derivative

In this section, we deal with an initial-value problem for the fractional relaxation equation with the generic fractional derivative associated with the Riemann–Liouville fractional integral. The particular cases of this problem with the Riemann–Liouville, Caputo, and Hilfer fractional derivatives are well known. Recently, the fractional relaxation equation with the nth-level fractional derivative was considered in [15]. In the particular cases mentioned above, the fractional relaxation equations were solved by employing the individual form of the corresponding fractional derivatives. In this paper, we derive an explicit formula for the solution of the fractional relaxation equation with the generic fractional derivative just by using its definition as a set of linear operators left-inverse to the Riemann–Liouville fractional integral.

Theorem 5. 

The initial-value problem for the fractional relaxation equation

$$\left\{\begin{array}{cc}\left({\mathbb{D}}^{\alpha }y\right)\left(t\right)=-\lambda y\left(t\right),\hfill & 0<\alpha <1,\lambda >0,t>0,\hfill \\ \left({\mathbb{P}}^{\alpha }y\right)\left(t\right)=y_0\left(t\right),\hfill & y_0\left(t\right)\in N_{{\mathbb{D}}^{\alpha }},\hfill \end{array}\right.$$

(65)

where ${\mathbb{D}}^{\alpha }$ is any fractional derivative associated with the Riemann–Liouville fractional integral $I^{\alpha }$ and ${\mathbb{P}}^{\alpha }$ is its projector operator, possesses a unique solution given by the formula

$$y\left(t\right)=y_0\left(t\right)-\lambda ({\tau }^{\alpha -1}{E}_{\alpha ,\alpha }(-\lambda {\tau }^{\alpha })\ast y_0\left(\tau \right))\left(t\right),$$

(66)

where $E_{\alpha ,\beta }$ is the two-parameter Mittag–Leffler function defined by the convergent series

$$E_{\alpha ,\beta }\left(z\right):=\sum _{k=0}^{+\infty }{\displaystyle \frac{z^k}{\Gamma (\alpha k+\beta )}},z,\beta \in \mathbb{C},\alpha >0.$$

(67)

Proof. 

To derive the solution Formula (66), we first act with the Riemann–Liouville fractional integral on the fractional relaxation equation (first line in (65)) and use the initial condition (second line in (65)) as well as the definition of the projector operator:

$$\left({\mathbb{D}}^{\alpha }y\right)\left(t\right)=-\lambda y\left(t\right)\iff \left(I^{\alpha }{\mathbb{D}}^{\alpha }y\right)\left(t\right)=-\lambda \left(I^{\alpha }y\right)\left(t\right)\iff y\left(t\right)-\left({\mathbb{P}}^{\alpha }y\right)\left(t\right)=-\lambda \left(I^{\alpha }y\right)\left(t\right)\iff$$

$$y\left(t\right)-y_0\left(t\right)=-\lambda \left(I^{\alpha }y\right)\left(t\right)\iff y\left(t\right)+\lambda \left(I^{\alpha }y\right)\left(t\right)=y_0\left(t\right).$$

The last equation in this chain of equations is the conventional Abel integral equation of the second kind, whose solution in explicit form is given by Formula (66) (see, e.g., [4,19,20]). This completes the proof of the Theorem. □

Remark 7. 

It is worth mentioning that the solution Formula (66) holds valid for any fractional derivative associated with the Riemann–Liouville fractional integral $I^{\alpha }$, including the Riemann–Liouville, Caputo, Hilfer, and the nth-level fractional derivatives. However, the form of the projector operator, and thus the form of the initial condition (second line of (65)), is not the same for different realizations of the generic fractional derivative. This leads to essentially different forms of the solution Formula (66) for the particular fractional derivatives introduced so far.

Remark 8. 

The initial condition (second line in (65)) for the fractional relaxation equation is provided in terms of the projector operator ${\mathbb{P}}^{\alpha }$. Using an individual form of the projector operator (say, given by Formulas (17), (18), (19), or (20)), this initial condition can be represented in the conventional form, i.e., in terms of the values of certain FC operators applied to the unknown function y evaluated at the point zero.

The projector operators for the Riemann-Liouville, Caputo, Hilfer, and the nth-level fractional derivatives given by Formulas (17)–(20), respectively, are expressed in terms of the linear combinations of the power law functions. To specify the solution Formula (66) for these fractional derivatives, we start with the relation

$$({\tau }^{\alpha -1}{E}_{\alpha ,\alpha }(-\lambda {\tau }^{\alpha })\ast h_{\beta }\left(\tau \right))\left(t\right)=t^{\alpha +\beta -1}{E}_{\alpha ,\alpha +\beta }(-\lambda t^{\alpha }),\alpha >0,\beta >0,$$

(68)

which easily follows from the convolution formula ([4,16])

$$(h_{\alpha }\left(\tau \right)\ast h_{\beta }\left(\tau \right))\left(t\right)=h_{\alpha +\beta }\left(t\right),\alpha >0,\beta >0,$$

(69)

and the term-by-term integration of the series for the Mittag–Leffler function from the left-hand side of Formula (68).

Using relation (68), the solution Formula (66) for the initial-value problem (65) with the initial condition $y_0\left(t\right)=h_{\beta }\left(t\right)$ takes the following form:

$$\begin{array}{c}y\left(t\right)=h_{\beta }\left(t\right)-\lambda ({\tau }^{\alpha -1}{E}_{\alpha ,\alpha }(-\lambda {\tau }^{\alpha })\ast h_{\beta }\left(\tau \right))\left(t\right)=h_{\beta }\left(t\right)-\lambda t^{\alpha +\beta -1}{E}_{\alpha ,\alpha +\beta }(-\lambda t^{\alpha })=\\ t^{\beta -1}{E}_{\alpha ,\beta }(-\lambda t^{\alpha }),\alpha >0,\beta >0.\end{array}$$

(70)

Representation (70) and Formulas (17)–(20) for the projector operators of the Riemann-Liouville, Caputo, Hilfer, and nth-level fractional derivatives immediately lead to the known solution formulas for the fractional relaxation equations with these fractional derivatives.

Example 6. 

For the nth-level fractional derivative, Formula (20) for its projector operator leads to the following formulation of the initial-value problem (65) for the nth-level fractional derivative ([15]):

$$\left\{\begin{array}{c}\left(D_{nL}^{\alpha ,\left(\gamma \right)}y\right)\left(t\right)=-\lambda y\left(t\right),0<\alpha <1,\lambda >0,t>0,\hfill \\ \left({\prod }_{i=k+1}^n\left(I^{{\gamma }_i}{\displaystyle \frac{d}{dt}}\right)I^{n-\alpha -s_n}y\right)\left(0\right)=y_k,k=1,\dots ,n,\hfill \end{array}\right.$$

(71)

where $s_k,k=1,\dots ,n$ are given by Formula (7).

Using Formula (20) for the projector operator and the representation (70), the solution Formula (66) for problem (71) is represented as follows:

$$y\left(t\right)=\sum _{k=1}^n{y}_k{t}^{\alpha +s_k-k}{E}_{\alpha ,\alpha +s_k-k+1}(-\lambda t^{\alpha }),$$

(72)

where $s_k,k=1,\dots ,n$ are given by Formula (7). The solution Formula (72) was derived for the first time in [15] using the Laplace transform method.

Probably the most important particular cases of the initial-value problem (71) for the fractional relaxation equation are those with first-level fractional derivatives ($n=1$ in (71)), i.e., with the Riemann–Liouville, Caputo, and Hilfer fractional derivatives:

(1)

Problem (71), with the Hilfer fractional derivative ($n=1$, $0\le {\gamma }_1\le 1-\alpha$), takes the form

$$\left\{\begin{array}{c}\left(D_H^{\alpha ,{\gamma }_1}y\right)\left(t\right)=-\lambda y\left(t\right),0<\alpha <1,\lambda >0,t>0,\hfill \\ \left(I^{1-\alpha -{\gamma }_1}y\right)\left(0\right)=y_1.\hfill \end{array}\right.$$

(73)

Its solution is given by the formula

$$y\left(t\right)=y_1{x}^{\alpha +{\gamma }_1-1}{E}_{\alpha ,\alpha +{\gamma }_1}(-\lambda t^{\alpha }).$$

(74)

(2)

Problem (71), with the Riemann–Liouville fractional derivative ($n=1$, ${\gamma }_1=0$), takes the form

$$\left\{\begin{array}{c}\left(D_{RL}^{\alpha }y\right)\left(t\right)=-\lambda y\left(t\right),0<\alpha <1,\lambda >0,t>0,\hfill \\ \left(I^{1-\alpha }y\right)\left(0\right)=y_1.\hfill \end{array}\right.$$

(75)

Its solution is given by the formula

$$y\left(t\right)=y_1{x}^{\alpha -1}{E}_{\alpha ,\alpha }(-\lambda t^{\alpha }).$$

(76)

(3)

Problem (71), with the Caputo fractional derivative ($n=1$, ${\gamma }_1=1-\alpha$), takes the form

$$\left\{\begin{array}{c}\left(D_C^{\alpha }y\right)\left(t\right)=-\lambda y\left(t\right),0<\alpha <1,\lambda >0,t>0,\hfill \\ y\left(0\right)=y_1.\hfill \end{array}\right.$$

(77)

Its solution is given by the formula

$$y\left(t\right)=y_1{E}_{\alpha ,1}(-\lambda t^{\alpha }).$$

(78)

Finally, let us discuss the complete monotonicity of the solution (66) to the initial-value problem (65) for the fractional relaxation equation in the case that the null-space of the fractional derivative ${\mathbb{D}}^{\alpha }$ is built by the linear combinations of some power law functions:

$$N_{{\mathbb{D}}^{\alpha }}=\{c_1{h}_{{\beta }_1}\left(t\right)+\cdots +c_m{h}_{{\beta }_m}\left(t\right),c_1,\dots ,c_m\in \mathbb{R}\}.$$

(79)

Theorem 6. 

Let a fractional derivative ${\mathbb{D}}^{\alpha }$ associated with the Riemann–Liouville fractional integral satisfy the condition (79), and the exponents ${\beta }_k$ fulfill the restrictions $\alpha \le {\beta }_k\le 1,k=1,\dots ,m$, and $y_k\ge 0,k=1,\dots ,m$.

Then, the unique solution to the initial-value problem

$$\left\{\begin{array}{c}\left({\mathbb{D}}^{\alpha }y\right)\left(t\right)=-\lambda y\left(t\right),0<\alpha <1,\lambda >0,t>0,\hfill \\ \left({\mathbb{P}}^{\alpha }y\right)\left(t\right)=y_1{h}_{{\beta }_1}\left(t\right)+\cdots +y_m{h}_{{\beta }_m}\left(t\right)\hfill \end{array}\right.$$

(80)

is completely monotonic.

Proof. 

According to Theorem 5, the unique solution of the initial-value problem (80) is given by Formula (66). For the fractional derivatives with the null-space given by (79), we employ Formula (70) and obtain the solution in the form

$$y\left(t\right)=\sum _{k=1}^m{y}_k{t}^{{\beta }_k-1}{E}_{\alpha ,{\beta }_k}(-\lambda t^{\alpha }).$$

(81)

Now, we use the result derived in [15], that the function

$$h_{\alpha ,\beta ,\gamma ,\lambda }\left(t\right):=t^{\gamma -1}{E}_{\alpha ,\beta }(-\lambda t^{\alpha })$$

(82)

is completely monotonic under the conditions

$$0<\alpha \le 1,\alpha \le \beta ,0<\gamma \le 1,\lambda >0.$$

(83)

In our case, the parameters of problem (80) satisfy the conditions $\alpha \le {\beta }_k\le 1,$ for $k=1,\dots ,m$ and $\lambda >0$, and thus, all of the functions $t^{{\beta }_k-1}{E}_{\alpha ,{\beta }_k}(-\lambda t^{\alpha }),k=1,\dots ,m$ from the sum at the right-hand side of the solution Formula (81) are completely monotonic. Because $y_k\ge 0,$ $k=1,\dots ,m$ and any linear combination of completely monotonic functions with non-negative coefficients is completely monotonic, the proof of the Theorem is completed. □

Example 7. 

Let us consider the initial-value problem (71) for the fractional relaxation equation with the nth-level fractional derivative. The null-space of the nth-level fractional derivative has the form (79) (see Formula (31)). Thus, we apply Theorem (6), which implies that the unique solution to the initial-value problem (71) is completely monotonic under the conditions

$$k-1\le s_k,k=1,\dots ,n,$$

(84)

where $s_k,k=1,\dots ,n$ are given by Formula (7).

For $n=1$ and ${\gamma }_1\ge 0$, the condition (84) is satisfied because $s_1={\gamma }_1\ge 0$ and thus, we arrive at the well-known result that Solutions (74), (76), and (78) to the initial-value Problems (73), (75), and (77) with the initial condition $y_1\ge 0$ and the Hilfer, Riemann–Liouville, and Caputo fractional derivatives, respectively, are complete monotonic.

5. Conclusions and Open Problems

The main objective of this paper is to suggest a unified approach to different kinds of time-fractional derivatives of the functions of a single variable introduced so far. In the framework of this approach, we introduced a generic fractional derivative that was defined as a set of linear operators left-inverse to the Riemann–Liouville fractional integral. In particular, the Riemann–Liouville, Caputo, Hilfer, and nth-level fractional derivatives are different realizations of the generic fractional derivative associated with the Riemann–Liouville fractional integral.

Another contribution of the paper is to derive some basic and advanced properties of the generic fractional derivative that are valid for all of its realizations. In particular, we provide a characterization of its domain, null-space, and the projector operator. In the case where the null-spaces of two different realisations of the generic fractional derivative are finite-dimensional, we derived a formula that is connecting these realizations. In particular, a formula for the nth-level fractional derivative in terms of the Riemann–Liouville fractional derivative was presented in this paper for the first time.

One of the most important results derived in this paper is a generalized fractional Taylor formula valid for any fractional derivative associated with the Riemann–Liouville fractional integral. For the proof of this formula, we did not use any concrete form of a particular fractional derivative. However, the generic fractional Taylor formula can be specified for a given fractional derivative. In particular, a generalized Taylor formula for the Hilfer fractional derivative was presented in this paper for the first time.

In the last part of the paper, we dealt with an initial-value problem for the fractional relaxation equation with the generic fractional derivative and the initial condition formulated in terms of its projector operator. First, we derived a closed-form formula for its unique solution that is independent on the concrete form of the fractional derivative involved in this problem. Then, some results regarding complete monotonicity of this solution were presented.

As to the open problems related to the generic fractional derivative, it is not clear at the moment if there exist realizations of the generic fractional derivative with a finite-dimensional null-space that are different from nth-level fractional derivatives. A still open question is if there are some fractional derivatives associated with the Riemann–Liouville integrals whose null-space is infinite-dimensional. Another direction of research would be in deriving other advanced properties of the generic fractional derivative that are independent on its realizations and are based just on its definition as the set of the linear operators that are left-inverse to the Riemann–Liouville fractional integral.