Pantograph, the technological successor of trolley poles, is an overhead current collector of electric bus, electric trains, and trams. In this work, we consider the discrete fractional pantograph equation of the form ${\Delta }_{\ast }^{\beta }\left[k\right]\left(t\right)=w\left(t+\beta ,k(t+\beta ),k\left(\lambda \right(t+\beta \left)\right)\right)$, with condition $k\left(0\right)=p\left[k\right]$ for $t\in {\mathbb{N}}_{1-\beta }$, $0<\beta \le 1$, $\lambda \in (0,1)$ and investigate the properties of asymptotic stability of solutions. We will prove the main results by the aid of Krasnoselskii’s and generalized Banach fixed point theorems. Examples involving algorithms and illustrated graphs are presented to demonstrate the validity of our theoretical findings.

A human being standing upright with his feet as the pivot is the most popular example of the stabilized inverted pendulum. Achieving stability of the inverted pendulum has become common challenge for engineers. In this paper, we consider an initial value discrete fractional Duffing equation with forcing term. We establish the existence, Hyers–Ulam stability, and Hyers–Ulam Mittag-Leffler stability of solutions for the equation. We consider the inverted pendulum modeled by Duffing equation as an example. The values are tabulated and simulated to show the consistency with theoretical findings.