Some properties of pseudo-fractional operators

$$\psi $$-Hilfer pseudo-fractional operator: new results about fractional calculus

“…The full order on [a, b] will be denoted by . Let [a, b] (Babakhani et al 2018;Yadollahzadeh et al 2019;Hosseini et al 2016;Pap 2002Pap , 2005) A binary operation ⊕ on [a, b] is a pseudo-addition if it is commutative, nondecreasing with respect to , continuous, associative and with a zero (neutral) element denoted by 0. Definition 2 (Babakhani et al 2018;Yadollahzadeh et al 2019;Hosseini et al 2016;Pap 2002Pap , 2005…”

“…The structure ([a, b], ⊕, ) is a semiring (Babakhani et al 2018;Yadollahzadeh et al 2019;Hosseini et al 2016;Pap 2002Pap , 2005.…”

“…Definition 3 (Babakhani et al 2018;Yadollahzadeh et al 2019;Hosseini et al 2016;Pap 2002Pap , 2005 An important class of pseudo-operations ⊕ and can be constructed with the help of a monotone and continuous function g : [a, b] → [0, ∞]. For a given g, the pseudo-operations ⊕ and are defined by:…”

“…(1) Definition 4 (Babakhani et al 2018;Yadollahzadeh et al 2019;Hosseini et al 2016;Pap 2002Pap , 2005) Let X be a non-empty set and let A be a σ -algebra of subsets of set X . A set function μ : A → [a, b] is called a σ -⊕-measure if the following conditions are satisfied:…”

“…During the last years, numerous researchers have investigated new formulations of inequalities involving fractional integrals in this context (Yadollahzadeh et al 2019;Hosseini et al 2016;Agahi et al 2015). Babakhani et al (2018) investigated the properties of pseudo-fractional operators; in particular, they introduced a definition of a Riemann-Liouville pseudo-fractional derivative with respect to another function. This work opened the door to a new branch of fractional calculus involving pseudo-fractional operators.…”