On $ \psi $-Hilfer generalized proportional fractional operators

Ishfaq Mallah; Idris Ahmed; Ali Akgul; Fahd Jarad; Subhash Alha; Ishfaq Mallah; Idris Ahmed; Ali Akgul; Fahd Jarad; Subhash Alha

doi:10.3934/math.2022005

AIMS Mathematics

2022, Volume 7, Issue 1: 82-103. doi: 10.3934/math.2022005

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Research article

On $ \psi $-Hilfer generalized proportional fractional operators

1.
Department of Mathematics, Maulana Azad National Urdu University Gachibowli, Hyderabad-500032, India
2.
Department of Mathematics, Sule Lamido University, P. M. B 048 Kafin Hausa, Jigawa State, Nigeria
3.
Department of Mathematics, Art and Science Faculty, Siirt University, TR-56100 Siirt, Turkey
4.
Department of Mathematics, Cankaya University, 06790 Etimesgut, Ankara, Turkey
5.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan

Received: 13 July 2021 Accepted: 22 September 2021 Published: 30 September 2021
MSC : 26A33, 34A12, 34A43, 34D20

In this paper, we introduce a generalized fractional operator in the setting of Hilfer fractional derivatives, the $ \psi $-Hilfer generalized proportional fractional derivative of a function with respect to another function. The proposed operator can be viewed as an interpolator between the Riemann-Liouville and Caputo generalized proportional fractional operators. The properties of the proposed operator are established under some classical and standard assumptions. As an application, we formulate a nonlinear fractional differential equation with a nonlocal initial condition and investigate its equivalence with Volterra integral equations, existence, and uniqueness of solutions. Finally, illustrative examples are given to demonstrate the theoretical results.
- Hilfer fractional derivative,
- generalized proportional fractional derivative,
- existence and uniqueness,
- weighed space,
- fixed point theorems
Citation: Ishfaq Mallah, Idris Ahmed, Ali Akgul, Fahd Jarad, Subhash Alha. On $ \psi $-Hilfer generalized proportional fractional operators[J]. AIMS Mathematics, 2022, 7(1): 82-103. doi: 10.3934/math.2022005

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Abstract

In this paper, we introduce a generalized fractional operator in the setting of Hilfer fractional derivatives, the $ \psi $-Hilfer generalized proportional fractional derivative of a function with respect to another function. The proposed operator can be viewed as an interpolator between the Riemann-Liouville and Caputo generalized proportional fractional operators. The properties of the proposed operator are established under some classical and standard assumptions. As an application, we formulate a nonlinear fractional differential equation with a nonlocal initial condition and investigate its equivalence with Volterra integral equations, existence, and uniqueness of solutions. Finally, illustrative examples are given to demonstrate the theoretical results.

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