On Generalized Fractional Operator and Related Fractional Integral Equations in Orlicz Spaces

This article aims to prove and explain novel properties of the $g$-fractional type operators, like boundedness, continuity, and monotonicity within Orlicz spaces $L_\psi$. We utilize such properties through Darbo's fixed-point theorem ($\mathbf{\mathcal{FPT}}$) and the measure of noncompactness ($\mathbf{\mathcal{MNC}}$) to study the existence in addition to the uniqueness of the solution to a quadratic integral equation in $L_\psi$. These results are new as the $g$-fractional operators are investigated for the first time in $L_\psi$. Our work generalizes and extends several fractional operators like the Riemann-Liouville, Hadamared, and Erdélyi--Kober and covers and unifies the results of many particular cases of classical and quadratic fractional problems studied in the former literature.