Certain new weighted estimates proposing generalized proportional fractional operator in another sense
Thabet Abdeljawad, Saima Rashid, Ahmed A. El‐Deeb, Zakia Hammouch, Yu‐Ming Chu
Abstract
Abstract The present work investigates the applicability and effectiveness of generalized proportional fractional integral ( $\mathcal{GPFI}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>GPFI</mml:mi> </mml:math> ) operator in another sense. We aim to derive novel weighted generalizations involving a family of positive functions n ( $n\in \mathbb{N}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:math> ) for this recently proposed operator. As applications of this operator, we can generate notable outcomes for Riemann–Liouville ( $\mathcal{RL}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>RL</mml:mi> </mml:math> ) fractional, generalized $\mathcal{RL}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>RL</mml:mi> </mml:math> -fractional operator, conformable fractional operator, Katugampola fractional integral operator, and Hadamard fractional integral operator by changing the domain. The proposed strategy is vivid, explicit, and it can be used to derive new solutions for various fractional differential equations applied in mathematical physics. Certain remarkable consequences of the main theorems are also figured.