Analytical study of $\mathcal{A}\mathcal{B}\mathcal{C}$-fractional pantograph implicit differential equation with respect to another function

This article aims to establish sufficient conditions for qualitative properties of the solutions for a new class of a pantograph implicit system in the framework of Atangana-Baleanu-Caputo ( $\mathcal{A}\mathcal{B}\mathcal{C}$) fractional derivatives with respect to another function under integral boundary conditions. The Schaefer and Banach fixed point theorems (FPTs) are utilized to investigate the existence and uniqueness results for this pantograph implicit system. Moreover, some stability types such as the Ulam-Hyers $(\mathbb{U}\mathbb{H})$, generalized $\mathbb{U}\mathbb{H}$, Ulam-Hyers-Rassias $(\mathbb{U}\mathbb{H}\mathbb{R})$ and generalized $\mathbb{U}\mathbb{H}\mathbb{R}$ are discussed. Finally, interpretation mathematical examples are given in order to guarantee the validity of the main findings. Moreover, the fractional operator used in this study is more generalized and supports our results to be more extensive and covers several new and existing problems in the literature.