THE q-ANALOGS OF FRACTIONAL OPERATORS CONCERNING ANOTHER FUNCTION IN THE POWER-LAW

Sabri T. M. Thabet, Meqdad A. A. Ali (), Thabet Abdeljawad, Bahaaeldin Abdalla () and Imed Kedim ()
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Sabri T. M. Thabet: Department of Mathematics, Saveetha School of Engineering, Saveetha Institute of Medical and Technical Sciences, Saveetha University, Chennai 602105, Tamil Nadu, India2Department of Mathematics, Radfan University College, University of Lahej, Lahej, Yemen3Department of Mathematics, College of Science, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02814, Republic of Korea
Meqdad A. A. Ali: Department of Mathematics, Al-Dhala University College, University of Aden, Al-Dhala, Yemen
Thabet Abdeljawad: Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia6Department of Medical Research, China Medical University, Taichung 40402, Taiwan7Department of Mathematics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul 02447, Korea8Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Garankuwa, Medusa 0204, South Africa9Center for Applied Mathematics and Bioinformatics (CAMB), Gulf University for Science and Technology, Hawally 32093, Kuwait
Bahaaeldin Abdalla: Department of Mathematics and Sciences, Prince Sultan University, Riyadh 11586, Saudi Arabia
Imed Kedim: 0Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam Bin Abdulaziz University, Al-Kharj 11942, Saudi Arabia

FRACTALS (fractals), 2025, vol. 33, issue 09, 1-24

Abstract: This paper presents new concepts of fractional quantum operators by connecting fractional and quantum calculus. First, we define the q-analogs of the higher-order derivative and integral concerning another function ψ in a new space Lq;ψ1[a,b], by the q-shifting operator aΦq(y) = qy + (1 − q)a. Then, we introduce the q-analogs of the left and right sided ψ-Riemann–Liouville (RL) fractional integral and derivative on a finite interval [a,b]. Furthermore, we investigate their important characteristics such as boundedness, continuity, semi-group, and fundamentals of fractional q-calculus theorem. Finally, to demonstrate the application of these new operators, we establish the existence and uniqueness (EaU) of the solution for a new class of nonlocal implicit differential equation involving (q;ψ )-RL fractional derivative by utilizing the Banach fixed point technique (FBT). The new operators cover the existing classical fractional and q-fractional operators; and we can deduce for the first time the q-analogs of the Katugampola, Hadamard, and conformable RL fractional operators.

Keywords: ψ-Fractional Operators; q-Calculus; (q; ψ)-Fractional Operators; Existence and Uniqueness; (q; ψ)-Fractional Implicit Differential Equation (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1142/S0218348X2550080X

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