On more general forms of proportional fractional operators

Abstract

<jats:title>Abstract</jats:title> <jats:p>In this article, more general types of fractional proportional integrals and derivatives are proposed. Some properties of these operators are discussed.</jats:p>

More properties of the proportional fractional integrals and derivatives of a function with respect to another function

“…Later, the authors in [24] presented a new type of fractional operators generated from the above-mentioned modified conformable derivatives. In addition, more generalized forms of these fractional operators were put forward in [25], and it turned out that some of these operators coincided with some operators mentioned before in [26][27][28].…”

“…In this work, orientated by the above-mentioned works, we continue our study on the proportional fractional derivatives and integrals of a function with respect to another function discovered in [25]. We present the effect of the fractional integral operators on the differential operators and vice versa.…”

“…Proposition 2.1 ( [25]) Let α, β ∈ C be such that Re(α) ≥ 0 and Re(β) > 0. Then, for any > 0, we have…”

“…The proportional derivative of a function with respect to another function;[25]) For ∈ [0, 1], let the functions κ 0 , κ 1 : [0, 1] × R → [0, ∞) be continuous such that for all t ∈ R we have…”

On existence–uniqueness results for proportional fractional differential equations and incomplete gamma functions

“…Also, the authors in [11] used particular versions of the proportional derivatives presented in [9], called modified conformable derivatives, to present the fractional counterpart pro-portional derivatives and integrals. Later, the authors in [15,16] generalized proportional derivatives and used them to generate more generalized classes of nonlocal fractional integrals and derivatives, and in [17] the authors discussed a new type of fractional operators combining proportional and classical derivatives/integrals. Besides, there have been many attempts to generate fractional operators with more complicated kernels with the hope to describe complex systems more accurately.…”