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Generalized fractional Hilfer integral and derivative

Fractional calculus, a branch of mathematics, is focused on the study and applications of the differential and integral operators of non-integer order. Although the fractional calculus is as old as the classical calculus, it has become one of the most developed areas of mathematics only in the last 40 years, not only because of the exponential growth of the number of publications in this area, but also due to its different applications and its overlapping with other areas of mathematics. This area has been developed intensively in recent years and it has found multiple applications in various fields. The classical results were basically extended in two fundamental directions: Riemann–Liouville fractional derivative and Caputo fractional derivative. As a result of the progress made in this area, numerous fractional (global) and generalized (local) operators have been appeared. These new operators give researchers the possibility to choose the one that suits best with the problem they investigate. Readers can consult the paper [2] where some reasons are given to justify the appearance of these new operators and where the applications and theoretical developments of these operators are discussed. These operators, developed by many mathematicians with a hardly specific formulation, include the Riemann–Liouville (RL), Weyl, Erdelyi-Kober and Hadamard integrals, and the fractional operators of Liouville and Katugampola. Many authors have even introduced new fractional operators generated from the general local differential operators. In this direction, a generalized local derivative was defined in [19], which generalizes both the conformable and non-conformable derivatives and that is the basis for the generalized integral operator proposed in [7], which contains as a particular case the fractional integral of Riemann– Liouville (see [31]). In fact, these new operators require a classification as they can cause confusion in researchers. Baleanu and Fernandez [3] gave a fairly complete classification of these fractional and generalized operators together with abundant information and references. For a more complete review, the readers are referred to Chapter 1 of [1], where a history of differential operators (both local and global) from Newton to Caputo is presented and where the qualitative differences between the operators are shown. Section 1.4 of [1] contains some conclusions that we want to highlight: “Therefore, we can conclude that the Riemann–Liouville and Caputo operators are not derivatives and, therefore, they are not fractional derivatives, but fractional operators. We agree with the result [27] that the local fractional operator is not a fractional derivative” (see p.24 in [1]). In this work, we present a new definition of the k-generalized fractional derivative of the Hilfer type, and we study its fundamental properties. We also present a particular case with a kernel defined in terms of the sigmoid function. The gamma function Γ (see [21,24,28,29]) and k-generalized gamma function Γk (see [6]) are defined as:


Introduction
Fractional calculus, a branch of mathematics, is focused on the study and applications of the differential and integral operators of non-integer order. Although the fractional calculus is as old as the classical calculus, it has become one of the most developed areas of mathematics only in the last 40 years, not only because of the exponential growth of the number of publications in this area, but also due to its different applications and its overlapping with other areas of mathematics. This area has been developed intensively in recent years and it has found multiple applications in various fields. The classical results were basically extended in two fundamental directions: Riemann-Liouville fractional derivative and Caputo fractional derivative.
As a result of the progress made in this area, numerous fractional (global) and generalized (local) operators have been appeared. These new operators give researchers the possibility to choose the one that suits best with the problem they investigate. Readers can consult the paper [2] where some reasons are given to justify the appearance of these new operators and where the applications and theoretical developments of these operators are discussed. These operators, developed by many mathematicians with a hardly specific formulation, include the Riemann-Liouville (RL), Weyl, Erdelyi-Kober and Hadamard integrals, and the fractional operators of Liouville and Katugampola. Many authors have even introduced new fractional operators generated from the general local differential operators. In this direction, a generalized local derivative was defined in [19], which generalizes both the conformable and non-conformable derivatives and that is the basis for the generalized integral operator proposed in [7], which contains as a particular case the fractional integral of Riemann-Liouville (see [31]). In fact, these new operators require a classification as they can cause confusion in researchers. Baleanu and Fernandez [3] gave a fairly complete classification of these fractional and generalized operators together with abundant information and references. For a more complete review, the readers are referred to Chapter 1 of [1], where a history of differential operators (both local and global) from Newton to Caputo is presented and where the qualitative differences between the operators are shown. Section 1.4 of [1] contains some conclusions that we want to highlight: "Therefore, we can conclude that the Riemann-Liouville and Caputo operators are not derivatives and, therefore, they are not fractional derivatives, but fractional operators. We agree with the result [27] that the local fractional operator is not a fractional derivative" (see p.24 in [1]).
In this work, we present a new definition of the k-generalized fractional derivative of the Hilfer type, and we study its fundamental properties. We also present a particular case with a kernel defined in terms of the sigmoid function.
The gamma function Γ (see [21,24,28,29]) and k-generalized gamma function Γ k (see [6]) are defined as: It is clear that Γ k (z) = (k) z k −1 Γ z k , Γ k (z + k) = zΓ k (z) and if k → 1 then Γ k (z) → Γ(z). Next, we present several definitions of the fractional operators, with their properties and notation which we use later (further details can be found in [15] Then the Riemann-Liouville fractional integrals (right and left respectively) of order µ ∈ C, (µ) > 0, are defined by

The above equation regarding
and similarly one can write an alternative expression for D µ − f (x).

Remark 1.1.
Another way to define these fractional derivatives is to consider them as the left (and right) inverse of the k-generalized fractional Riemann-Liouville integral with general kernel of order µ, that is

Remark 1.2. The Riemann-Liouville integral can also be expressed as
In 1967, Michele Caputo gave a new definition of fractional derivative in [5] with an interesting property: when solving differential equations using Caputo's definition, it is not necessary to define the fractional order initial conditions. This derivative has been used in the study of many applied problems (see [4]). Next, we present the definition of the Riemann-Liouville fractional derivative with respect to another function ψ, which is directly related to the Riemann-Liouville fractional derivative (see [15]).  0 be continuous on (a, b), and µ > 0, where a, b ∈ R. The ψ-fractional integrals (left-sided and right-sided) of order µ are defined by Typically, fractional derivatives are presented through a combination of integer derivatives, derivatives, and fractional integrals. Hilfer (see [10][11][12]) gives a definition that is an "intermediate" point of the classical fractional derivatives: Riemann-Liouville and Caputo.
Definition 1.5. The Hilfer fractional derivative of order µ, 0 < µ < 1, and type β, 0 ≤ β ≤ 1, with respect to x is defined by for the functions for which the expression on the right hand side exists.

Remark 1.4. In Definition 1.5, if we take β = 0 then we have the Riemann-Liouville derivative (see Definition 1.1) and the choice β = 1 gives us the Caputo derivative (see Definition 1.3).
In order to obtain a more general derivative than the one proposed by Hilfer, Hilfer et al. [13] proposed the following generalized Riemann-Liouville fractional derivative, which is associated with the Riemann-Liouville fractional integral. Definition 1.6. For n ∈ N, let µ, β ∈ R be the numbers satisfying the inequalities n − 1 < µ ≤ n and 0 ≤ β ≤ 1. The Hilfer fractional derivative of order µ and type β with respect to x is defined by for the functions for which the expression on the right hand side exists.
Below is a generalization of the Hilfer derivative specified in Definition 1.5. We call this generalization as the ψfractional integral and derivative.
The above derivatives can be written in the following form: with µ = µ + β(n − µ) and

On the k-generalized ψ-Hilfer integral
We are now able to define the k-generalized ψ-Hilfer integral as follows.
Definition 2.1. Let ψ(u) > 0 be an increasing function on (a, b] and ψ (u) > 0 be continuous on (a, b) and µ > 0, where a, b ∈ R. The generalized ψ-fractional integrals (left-sided and right-sided) of order µ are defined by We can also define generalized derivatives as follows.  0 be continuous on (a, b), and take µ > 0 and G(z, µ) ∈ AC[a 1 , a 2 ]. The generalized ψ-fractional derivative (left-sided and right-sided) of order µ are defined by 2. Under the above conditions, if k = 1 then from Definition 2.1, the k-fractionals operators proposed in [18] are obtained.
3. If ψ (t) = 1 t , k = 1 and G(z, µ) = z 1−µ then the Hadamard fractional operator (see [25]) is reproduced. 4. If ψ (t) = 1 t ρ , G(z, µ) = z 1−µ and k = 1, then we obtain the Katugampola's fractional operator defined in [14]. 5. By choosing k = 1, G(z, µ) = z 1−µ , we get the integral operator proposed in [15]. 6. By taking ψ (t) = 1, k = 1 and , we obtain the integral operator proposed in [22]. Next, we study some of the important properties of the operators specified in Definition 2.1. One of the basic questions that need to be answer for any newly defined integral operator is concerned about the boundedness. Proof. Let D = [a 1 , a 2 ] × [a 1 , a 2 ] and define C : D → R as Taking into account that C(x, t) is measurable on D (the functions ψ (t) ∈ AC[a 1 , a 2 ] and G(z, µ) ∈ AC[a 1 , a 2 ], on the compact D), we have C(x, s)h(s)ds = B(x).
If we take then by using the the Tonelli's theorem for iterated integrals (see [8] Given the generality of the operator presented, what we can say is that if the function G is additive with respect to the second variable, then it satisfies the Law of Semigroups.

Remark 2.2.
Obviously, it can be thought that the additive property of the function G is very restrictive. However, we draw our attention to the fact that all the operators presented above satisfy this certain property, but the generality of G in our definition makes possible the existence of the functions that do not satisfy it.
Next, we give a result about the continuity of the operator given in Definition 2.1. These operators have been proven useful in various practical problems (from the population growth to the shallow water wave and reaction-diffusion process). We end this paper by defining the following generalized sigmoid function (which may open a new line of research): σ(x) = 1 1 + e −g(x) , where g(x) is a smooth function.