axioms Article On a New Generalized Integral Operator and Certain Operating Properties Paulo M. Guzman 1,2,†, Luciano M. Lugo 1,†, Juan E. Nápoles Valdés 1,† and Miguel Vivas-Cortez 3,*,† 1 FaCENA, UNNE, Av. Libertad 5450, Corrientes 3400, Argentina; [email protected] (P.M.G.); [email protected] (L.M.L.); [email protected] (J.E.N.V.) 2 Facultad de Ingeniería, UNNE, Resistencia, Chaco 3500, Argentina 3 Facultad de Ciencias Exactas y Naturales, Escuela de Ciencias Físicas y Matemática, Pontificia Universidad Católica del Ecuador, Quito 170143, Ecuador * Correspondence: [email protected] † These authors contributed equally to this work. Received: 20 April 2020; Accepted: 22 May 2020; Published: 20 June 2020 Abstract: In this paper, we present a general definition of a generalized integral operator which contains as particular cases, many of the well-known, fractional and integer order integrals. Keywords: integral operator; fractional calculus 1. Preliminars Integral Calculus is a mathematical area with so many ramifications and applications, that the sole intention of enumerating them makes the task practically impossible. Suffice it to say that the simple procedure of calculating the area of an elementary figure is a simple case of this topic. If we refer only to the case of integral inequalities present in the literature, there are different types of these, which involve certain properties of the functions involved, from generalizations of the known Mean Value Theorem of classical Integral Calculus, to varied inequalities in norm. Let (U, ∑ U, u) and (V, ∑ V, m) be s-fìnite measure spaces, and let (W, ∑ W, l) be the product of these spaces, thus W = UxV and l = u x m. In a general sense, an operator I : A ! B is called integral if there exists a l-measurable function K(u, v) (with u 2 U, v 2 V) such that, for every x 2 A the image R y = I(t) 2 B of x is y(u) = V K(u, v)x(v)dm(v) being U denotes an FS on (V, ∑ V, m) and V is an FS on (U, ∑ U, u). The function K(u, v) is called the kernel of I. It is easy to see that the kernel is finite almost everywhere relative to l. In particular, we will deal with real integral operators defined on R. It is known that from the XVIIth century the study of problems dealing with derivatives and integrals of fractional order began. The first works that are registered deal with this subject from a theoretical point of view, however over time, until today, its applicability is undeniable. Important mathematicians, such as Euler, Laplace, Fourier, Abel, Liouville and Riemann, worked on this topic (see [1]). Fractional derivative and fractional integral are generalizations of those always present in the ordinary calculation, considering derivatives of real or complex arbitrary order and a general form for multiple integrals. The principle used to find models for fractional derivatives has been to define, first, a fractional integral. Applicability in areas such as physics, engineering, biology, has managed to establish its usefulness and many important results have appeared in the literature. New definitions of differential and integral fractional operators have emerged in recent decades, and around this many researchers wonder what type of operator to choose from a possible problem considered. An attractive characteristic of this field is that there are numerous fractional operators, and this permits researchers to Axioms 2020, 9, 69; doi:10.3390/axioms9020069 www.mdpi.com/journal/axioms Axioms 2020, 9, 69 2 of 14 choose the most appropriate operator for the sake of modeling the problem under investigation. In [2] a fairly complete classification of these fractional operators is presented, with abundant information, on the other hand, in the work [3] some reasons are presented why new operators linked to applications and developments theorists appear every day. These operators had been developed by numerous mathematicians with a barely specific formulation, for instance, the Riemann–Liouville (RL), the Weyl, Erdelyi–Kober, Hadamard integrals and the Liouville and Katugampola fractional operators and many authors have introduced new fractional operators generated from general classical local derivatives. In addition, Section 1 of [4] presents a history of differential operators, both local and global, from Newton to Caputo and presents a definition of local derivative with new parameter, providing a large number of applications, with a difference qualitative between both types of operators, local and global. Most importantly, Section 1.4 (p. 24), dealing with limitations and strength of local and fractional derivatives, concludes: “We can therefore conclude that both the Riemann–Liouville and Caputo operators are not derivatives, and then they are not fractional derivatives, but fractional operators. We agree with the result that, the local fractional operator is not a fractional derivative”. As we said before, they are new tools that have demonstrated their usefulness and potential in the modelling of different processes and phenomena. In the literature many different types of fractional operators have been proposed, here we show that various of that different notions of derivatives can be considered particular cases of our definition and, even more relevant, that it is possible to establish a direct relationship between global (classical) and local derivatives, the latter not very accepted by the mathematical community, under two arguments: their local character and compliance with the Leibniz Rule. However, in the works [5–9], various results related to the existence and uniqueness of solutions of fractional differential equations and integral equations of the Volterra and Volterra–Fredholm type are investigated, within the framework of classical fractional derivatives, that is, using global operators, although they are important results because their applicability, they are not related to the operators used in our work that are of local type. To facilitate the understanding of the scope of our definition, we present the best known definitions of integral operators and their corresponding differential operators (for more details you can consult [10]). Without many difficulties, we can extend these definitions, for any higher order. We assume that the reader is familiar with the classic definition of the Riemann Integral, so we will not present it. One of the first operators that can be called fractional is that of Riemann–Liouville fractional derivatives of order a 2 C, Re(a) ≥ 0, defined by (see [11]). Definition 1. Let f 2 L1((a, b); R), (a, b) 2 R2, a < b. The right and left side Riemann–Liouville fractional integrals of order a > 0 are defined by Z t RL a 1 a−1 Ja+ f (t) = (t − s) f (s)ds, t > a (1) G(a) a and Z b RL a 1 a−1 Jb− f (t) = (s − t) f (s)ds, t < b. (2) G(a) t and their corresponding differential operators are given by d 1 d Z t f (t) a ( ) = RL 1−a ( ) = Da+ f t Ja+ f t ds dt G(1 − a) dt a (t − s)a d 1 d Z b f (t) a ( ) = − RL 1−a ( ) = − Db− f t Jb− f t ds dt G(1 − a) dt t (s − t)a Other definitions of fractional operators are as follows. Axioms 2020, 9, 69 3 of 14 Definition 2. Let f 2 L1((a, b); R), (a, b) 2 R2, a < b. The right and left side Hadamard fractional integrals of order a with Re(a) > 0 are defined by Z t a−1 a 1 t f (s) Ha+ f (t) = (log ) ds, a < t < b, (3) G(a) a s s and Z b a−1 a 1 s f (s) Hb− f (t) = (log ) ds, a < t < b. (4) G(a) t t s Hadamard differential operators are given by the following expressions. Z t −a−1 H a d a −G(a + 1) t f (s) ( Da+ f )(t) = t Ha+ f (t) = (log ) ds, a < t < b dt B(a, 1 − a) a s s Z b −a−1 H a d a G(a + 1) s f (s) ( Db− f )(t) = −t Hb− f (t) = − (log ) ds, a < t < b dt B(a, 1 − a) t t s In [12], the author introduced new fractional integral operators, called the Katugampola fractional integrals, in the following way: Definition 3. Let 0 < a < b < +¥, f : [a, b] ! R is an integrable function, and a 2 (0, 1) and r > 0 two fixed real numbers. The right and left side Katugampola fractional integrals of order a are defined by r1−a Z t sr−1 a,r ( ) = ( ) < Ka+ f t r r − f s ds, a t (5) G(a) a (t − s )1 a and r1−a Z b tr−1 a,r ( ) = ( ) < Kb− f t r r − f s ds, t b. (6) G(a) t (s − t )1 a In [13], it appeared a generalization to the Riemann–Liouville and Hadamard fractional derivatives, called the Katugampola fractional derivatives: a Z t r−1 a r 1−r d s (Da+ f )(t) = t r r f (s)ds, a < t, G(1 − a) dt a (t − s )a −ra d Z t sr−1 ( a,r )( ) = 1−r ( ) < Db− f t t r r f s ds, t b. G(1 − a) dt a (s − t )a The relation between these two fractional operators is the following: a,r d 1−a,r a,r d 1−a,r (D f )(t) = t1−r K f (t), (D f )(t) = −t1−r K f (t). a+ dt a+ b− dt b− There are other definitions of integral operators in the global case, but they can be slight modifications of the previous ones, some include non-singular kernel and others incorporate different terms.