General fractional integrals and derivatives with the Sonine kernels

In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. First the Sonine kernels and their important special classes and particular cases are discussed. For the general fractional integrals and derivatives with the Sonine kernels, the two fundamental theorems of Fractional Calculus are proved. Then we construct the $n$-fold general fractional integrals and derivatives and study their properties.

They are defined as follows: I α 0+ being the Riemann-Liouville fractional integral of order α (α > 0): Other particular cases of the GFDs (1) and (2) are the multi-term fractional derivatives and the fractional derivatives of the distributed order. They are generated by (1) and (2) with the kernels k(t) = n k=1 a k h 1−α k (t), 0 < α 1 < · · · < α n < 1, a k ∈ R, k = 1, . . . , n, respectively, where ρ is a Borel measure defined on the interval [0, 1]. One of the main issues addressed in [8] was to clarify the conditions which allow an interpretation of the integro-differential operators (2) as a kind of the fractional derivatives. In particular, these derivatives and the appropriate defined fractional integrals have to satisfy the Fundamental Theorem of FC (see [13] for a discussion of this theorem). Moreover, the solutions to the time-fractional differential equations of certain types with the GFDs are expected to behave as the ones to the evolution equations. This includes the property of complete monotonicity of solutions to the relaxation equation and the positivity of the fundamental solution to the Cauchy problem for the fractional diffusion equation with the time-derivatives of this type. In [8], a class of kernels of the GFD (2) that satisfy the requirements formulated above was described in terms of their Laplace transforms.
In the meantime, some other results regarding the ordinary and partial timefractional differential equations with the GFDs of Caputo type have been already derived. The ordinary fractional differential equations with the Caputo type GFDs have been addressed in [8,10,11,23] and the time-fractional diffusion equations with these derivatives were considered in [8-10, 17, 18]. The publications [5][6][7] are devoted to the inverse problems for the GFDs and for the partial differential equations containing the general fractional integrals and derivatives.
In this paper, we address the general fractional integrals and derivatives with the Sonine kernels on the spaces of functions with an integrable singularity at the point zero. In particular, the n-fold general fractional integrals and derivatives are constructed and studied for the first time.
The rest of the paper is organized as follows. In the second section, the basic spaces of functions that are employed for derivation of our main results are shortly discussed. The third section addresses the Sonine condition and the Sonine kernels. We provide both some examples of the Sonine kernels and descriptions of several important classes of the Sonine kernels. In the fourth section, the general fractional integrals (GFI) and derivatives with the Sonine kernels are introduced and studied on the spaces of functions that can have an integrable singularity at the point zero. The first and the second fundamental theorems of FC are proved for the GFDs on the appropriate spaces of functions. Moreover, the n-fold general fractional integrals and derivatives that correspond to the Riemann-Liouville and Caputo derivatives of an arbitrary order are constructed and their basic properties are studied.

Preliminaries
In this section, we introduce the spaces of functions that are used in the further discussions and present their main properties.
The basic space of functions consists of the continuous functions on the positive real semi-axis that can have an integrable singularity of a power law type at the point zero. We denote this space by C −1 (0, +∞): A family of the spaces C α (0, +∞), α ≥ −1 was first introduced in [2] for construction of an operational calculus for the hyper-Bessel differential operator and then employed for the operational calculi for the fractional derivatives in [1,12,15,16,26] and other related publications.
Using the well-known properties of the Laplace convolution the following theorem can be easily proved: 14]). The triple R −1 = (C −1 (0, +∞), +, * ) with the usual addition + and the multiplication * in form of the Laplace convolution (10) is a commutative ring without divisors of zero.
We start with the important power law functions (see (3) for their definition): For α, β > 0, the functions h α , h β , and h α+β all belong to the space C −1 (0, +∞). The formula (13) is an easy consequence from the well-known representation of the Euler Beta-function in terms of the Gamma-function: The same formula can be applied to verify a more general relation where the function h α,ρ ∈ C −1 (0, +∞) is given by the expression Because R −1 is a ring, we can define the integer order convolution powers of a function κ ∈ C −1 (0, +∞) as follows: Say, repeatedly applying the formula (13), we get the representation In particular, we mention the well-known formula

The Sonine kernels
In the short notices [24] published in 1884, Sonine addressed a generalization of the Abel integral equation and applied Abel's method for its analytical solution. He recognized that the most essential ingredient of Abel's solution is the formula that is a particular case of (13). Note that for 0 < α < 1, the inclusions h α ∈ C −1 (0, +∞) and h 1−α ∈ C −1 (0, +∞) are valid. As a generalization of (19), Sonine suggested to consider the functions κ and k that satisfy the relation Nowadays, such functions are called the Sonine kernels and the relation (20) is called the Sonine condition. For a Sonine kernel κ, the kernel k that satisfies the Sonine condition (20) is called a dual or associate kernel to κ. Of course, κ is then an associated kernel to k. In what follows, we denote the set of the Sonine kernels by S.
Evidently, not any function defined on R + is a Sonine kernel. Say, the constant function {1} does not possess any associate kernel. The problem of analytical description of the set S of the Sonine kernels is still not completely solved. However, several important classes of the Sonine kernels have been introduced in the literature and in this section some relevant results are shortly presented.
In [24], Sonine addressed the case of the kernels that can be represented in the form where the function He showed that such functions are always the Sonine kernels and their (unique) associate kernels can be represented as follows: The coefficients b k , k ∈ N 0 are uniquely determined by the coefficients a k , k ∈ N 0 as solutions to the following triangular system of linear equations: In [24], some examples of the kernels from S in form (21), (22) were derived including the prominent pair where are the Bessel and the modified Bessel functions, respectively.
Another example of this type is the following pair of the Sonine kernels ( [27]): where the function h α,ρ is defined by the formula (15).
In [25], the condition of analyticity of the function κ 1 from (21) was replaced by a weaker condition that the series for κ 1 has a finite convergence radius r > 0. Then the function κ defined by (21) is still a Sonine kernel and its unique associate kernel is given by (22), where the series for k 1 has the same convergence radius r.
Of course, the class S of the Sonine kernels is not restricted to the functions in form (21) and includes, for instance, some functions with the power-logarithmic singularities at the origin ( [19,20]).
A completely different approach for description of the Sonine kernels was suggested in [8], where a connection between the complete Bernstein functions and the Sonine kernels has been established. More precisely, it was shown that any function k = k(t), t > 0 that satisfies the conditions K1)-K4) listed below is a Sonine kernel.
In what follows, we denote the set of the kernels that satisfy the conditions K1)-K4) by K. As already mentioned, K ⊂ S. Moreover, the solutions to the fractional differential equations with the GFDs of type (2) with the kernels k ∈ K behave as the ones to the evolution equations ( [8]). In particular, the unique solution to the Cauchy problem for the fractional relaxation equation with a positive initial condition is complete monotone and the fundamental solution to the Cauchy problem for the fractional diffusion equation with the time-derivative of this type can be interpreted as a probability density function. The Sonine kernels from K were introduced in terms of their Laplace transforms and using the Sonine condition (20) in the Laplace domain. Providing the Laplace transformsκ,k of the functions κ and k do exist, the convolution theorem for the Laplace transform leads to the equatioñ for the Laplace transforms of the Sonine kernels κ and k. However, the Laplace transforms of the Sonine kernels do not always exist. One can easily construct such kernels based on the Sonine formula (21) with the analytical functions κ 1 that grow faster than any exponential function exp(ct), c > 0 as t → +∞. One of the examples of this sort is the Sonine kernel In [3], an important class of the Sonine kernels was introduced in terms of the completely monotone functions. Namely, it was shown there that any singular (unbounded in a neighborhood of the point zero) locally integrable completely monotone function is a Sonine kernel. We denote this class of the Sonine kernels by H (H ⊂ S). Moreover, if κ ∈ H then its associate kernel k also does belong to H. A typical example of a pair of the Sonine kernels from H is as follows ( [3]): where E α,β stands for the two-parameters Mittag-Leffler function that is defined by the following convergent series: To prove this, the well-known Laplace transform formula for the two-parameters Mittag-Leffler function was applied along with the formula (28) and the known fact that the function (30) is a singular locally integrable completely monotone function ( [22]).
In the next section, we deal with the general fractional integrals and derivatives with the Sonine kernels that belong to the space C −1 (0, +∞). The set of such Sonine kernels will be denoted by S −1 : It is worth mentioning that the kernels of the conventional time-fractional derivatives and integrals introduced so far do belong to the set S −1 and thus the constructions presented in the next section are applicable both for the known and for many new FC operators. In particular, all examples of the Sonine kernels provided in this section including the general kernels (21), (22) introduced by Sonine in his original publication [24] are from the set S −1 .

Fractional integrals and derivatives with the Sonine kernels
In this section, we address the GFIs and the GFDs with the Sonine kernels from S −1 on the spaces of functions C m −1 (0, +∞), m ∈ N 0 discussed in Section 2 and on their subspaces.
Let κ ∈ S −1 . The general fractional integral with the kernel κ is defined by the formula If κ(t) = h α (t) = t α−1 Γ(α) ∈ S −1 , 0 < α < 1, the GFI (34) is the Riemann-Liouville fractional integral (6): ( For the Riemann-Liouville fractional integral, the relations hold true. Combining the formulas (35) and (36), it is natural to define the GFIs with the kernels κ(t) = h 0 (t) and κ(t) = h 1 (t) (that do not belong to the set S −1 ) as follows: In the definition (37), the function h 0 is interpreted as a kind of the δ-function that plays the role of a unity with respect to multiplication in form of the Laplace convolution. In particular, one could extend the formula (19) valid for 0 < α < 1 in the usual sense to the case α = 1 that has to be understand in the sense of generalized functions: Using the known Sonine kernels, many other particular cases of the GFI (34) can be constructed. Here we mention just a few of them: with the Sonine kernel κ defined by (30), with the Sonine kernel κ defined by (25), and with the Sonine kernel κ defined by (24). Regarding the properties of the GFI (34) on the space C −1 (0, +∞), they can be easily derived from the known properties of the Laplace convolution and Theorem 2.2. In particular, in what follows, we need the mapping property the commutativity law and the index law that are valid on the space C −1 (0, +∞). For the GFI (34) with a Sonine kernel κ ∈ S −1 , the general fractional derivatives of the Riemann-Liouville and the Caputo types are defined as follows (see (1) and (2)): where the kernel k ∈ S −1 is the Sonine kernel associate to the kernel κ.
In this paper, we mainly deal with the GFDs in form (46) on the spaces C m −1 (0, +∞), m ∈ N 0 and their subspaces. For the functions from C 1 −1 (0, ∞), Theorem 2.2 and the known formula for differentiation of the integrals depending on parameters lead to an important representation of the GDF (45): Thus, for f ∈ C 1 −1 (0, +∞), the GFD (46) can be rewritten in the form It was known already to Sonine that the operators of type (45) are left inverse to the GFI (34). However, he provided just a formal derivation that needs a justification for the given sets of the kernels and the spaces of functions. In our case, we do this in the following theorem (see [8] for the case of the GFDs with the kernels k ∈ K and [3] for the case k ∈ H). Theorem 4.1 (1st Fundamental Theorem for the GFD). Let k ∈ S −1 be a Sonine kernel associate to the kernel κ.
Then the GFD (45) is a left inverse operator to the GFI (34) on the space and the GFD (46) is a left inverse operator to the GFI (34) on the space C −1,k (0, +∞): where Proof. The proof of the formula (49) follows the lines of the Sonine derivations (t > 0): Because k, κ ∈ S −1 and f ∈ C −1 (0, +∞), validity of the formulas in the chain of the equalities above is justified by Theorem 2.2.
Thus we proved that It is worth mentioning that this characterization of the space C −1,k (0, +∞) is a generalization of the known property of the spaces of functions employed while treating the Abel integral equation (Theorem 2.3 in [21]) to the case of the GFI with the Sonine kernel κ ∈ S −1 . Now we provide the explicit formulas for the GFDs * D (k) that correspond to the particular cases (35), (39)-(41) of the GFI (34). To make the formulas more compact, we employ the representation (48). However, these formulas with the evident modifications are also valid for the GFDs in form (46). The Sonine kernel associate to the kernel κ(t) = h α (t), 0 < α < 1 is k(t) = h 1−α (t), and thus the GFD in form (48) associate to the Riemann-Liouville integral (6) is the Caputo derivative In the formula (37), the Riemann-Liouville fractional integral was defined for α = 0 and α = 1 that corresponds to the kernels κ(t) = h 0 (t) and κ(t) = h 1 (t). The convolution formula (38) suggests the following natural definition of the GFD with the kernels k = h 1 (t) and k(t) = h 0 (t), respectively: Of course, to justify the definitions (53), the suitable subsets of the GFD domain should be introduced, where the formulas are valid in some sense (see [4] for a description of these and other properties that the fractional derivatives and integrals should possess). This will be done elsewhere.
The GFD that corresponds to the GFI (39) has the Mittag-Leffler function (31) in the kernel: The Sonine kernel associate to the kernel κ(t) = h α,ρ (t), 0 < α < 1, ρ ≥ 0 is given by the formula (26) and thus the GFD associate to the GFI (40) has the form Finally, the GFD that corresponds to the GFI (41) takes the form (see the Sonine pair (24)): Now we proceed with the 2nd fundamental theorem of FC for the GFDs in the Riemann-Liouville and Caputo senses (see [8] for the case of the GFDs with the kernels k ∈ K and [3] for the case k ∈ H). Theorem 4.2 (2nd Fundamental Theorem for the GFD). Let k ∈ S −1 be a Sonine kernel associate to the kernel κ and f ∈ C 1 −1 (0, +∞). Then the relations Proof. As we have seen at the beginning of this section, for the functions f ∈ C 1 −1 (0, +∞), the GFD * D (k) can be represented in form (48). Moreover, the functions from C 1 −1 (0, +∞) are continuous on [0, +∞) (Theorem 2. 2) and f ′ ∈ C −1 (0, +∞). Then we have the following chain of equalities: For the GFD in the Riemann-Liouvile sense, we first employ the formula (47) and then proceed as follows: In the rest of this section, we consider the compositions of the GFIs and construct the suitable fractional derivatives in the sense of the 1st Fundamental Theorem of FC.
Definition 4.1. Let κ ∈ S −1 . The n-fold GFI (n ∈ N) is defined as a composition of n GFIs with the kernel κ: It is worth mentioning that the kernel κ n , n ∈ N is from the space C −1 (0, +∞), but it is not always a Sonine kernel. Say, in the case of the Riemann-Liouville fractional integral I α 0+ , 0 < α < 1, the operator (60) is the Riemann-Liouville fractional integral of the order nα, n ∈ N. Its kernel is h nα that is a Sonine kernel only in the case α < 1/n. However, the operator I α 0+ is considered to be a fractional integral for any α > 0, i.e., also in the case when its kernel h α is not a Sonine kernel and not singular at the point zero. The common justification for this actual situation is that one can define an appropriate fractional derivative, the Riemann-Liouville fractional derivative, that is connected with the Riemann-Liouville fractional integral through the 1st and the 2nd fundamental theorems of FC. In the rest of this section, we introduce an analogous construction for the GFI (34).
The n-fold GFD is a generalization of the Riemann-Liouville fractional derivative of an arbitrary order α > 0 to the case of the general Sonine kernels. Say, for the Sonine kernels pair κ(t) = h α (t), k(t) = h 1−α (t), 0 < α < 1, the two-fold GFD (61) has the kernel k(t) = h 2−2α (t) and thus it can be represented in terms of the Riemann-Liouville fractional derivative: For the n-fold GFD (61), the following recurrent formula (D n (k) f )(t) = d dt d n−1 dt n−1 (k n−1 * (k * f ))(t) = d dt (D n−1 (k) (k * f ))(t), n ∈ N holds true. This formula connects the n-fold GFD with the (n − 1)-fold GFD and allows to extend the results derived for the GFD (45) to the case of the n-fold GFD (61).
Evidently, the conditions (72) are fulfilled, for instance, in the case n = 1 (Theorem 4.2) and for the kernel k(t) = h 1−α (t), 0 < α < 1/n of the Riemann-Liouville fractional derivative (4) and then the formula (71) is valid for f ∈ C n −1 (0, +∞). In analogy to the multi-term fractional differential equations with the Caputo or Riemann-Liouville fractional derivatives, the multi-term fractional differential equations with the n-fold GFDs in the Caputo and the Riemann-Liouville sense would be worth studying. This problem will be considered elsewhere.