Asymptotic Stability of Nonlinear Discrete Fractional Pantograph Equations with Non-Local Initial Conditions

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 Δ∗β[k](t)=wt+β,k(t+β),k(λ(t+β)), with condition k(0)=p[k] for t∈N1−β, 0<β≤1, λ∈(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.