A higher-order extension of Atangana–Baleanu fractional operators with respect to another function and a Gronwall-type inequality

Abstract

Abstract This paper aims to extend the Caputo–Atangana–Baleanu ( $ABC$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mi>C</mml:mi> </mml:math> ) and Riemann–Atangana–Baleanu ( $ABR$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mi>R</mml:mi> </mml:math> ) fractional derivatives with respect to another function, from fractional order $\mu \in (0,1]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:math> to an arbitrary order $\mu \in (n,n+1]$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>μ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>,</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:math> , $n=0,1,2,\dots $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mo>…</mml:mo> </mml:math> . Also, their corresponding Atangana–Baleanu ( AB ) fractional integral is extended. Additionally, several properties of such definitions are proved. Moreover, the generalization of Gronwall’s inequality in the framework of the AB fractional integral with respect to another function is introduced. Furthermore, Picard’s iterative method is employed to discuss the existence and uniqueness of the solution for a higher-order initial fractional differential equation involving an $ABC$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>A</mml:mi> <mml:mi>B</mml:mi> <mml:mi>C</mml:mi> </mml:math> operator with respect to another function. Finally, examples are given to illustrate the effectiveness of the main findings. The idea of this work may attract many researchers in the future to study some inequalities and fractional differential equations that are related to AB fractional calculus with respect to another function.