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.

An elastic beam equation (EBEq) described by a fourth-order fractional difference equation is proposed in this work with three-point boundary conditions involving the Riemann–Liouville fractional difference operator. New sufficient conditions ensuring the solutions’ existence and uniqueness of the proposed problem are established. The findings are obtained by employing properties of discrete fractional equations, Banach contraction, and Brouwer fixed-point theorems. Further, we discuss our problem’s results concerning $\mathcal{H}$yers–$\mathcal{U}$lam ($\mathcal{HU}$), generalized $\mathcal{H}$yers–$\mathcal{U}$lam ($\mathcal{GHU}$), $\mathcal{H}$yers–$\mathcal{U}$lam–$\mathcal{R}$assias ($\mathcal{HUR}$), and generalized $\mathcal{H}$yers–$\mathcal{U}$lam–$\mathcal{R}$assias ($\mathcal{GHUR}$) stability. Specific examples with graphs and numerical experiment are presented to demonstrate the effectiveness of our results.

Pantograph, a device in which an electric current is collected from overhead contact wires, is introduced to increase the speed of trains or trams. The work aims to study the stability properties of the nonlinear fractional order generalized pantograph equation with discrete time, using the Hilfer operator. Hybrid fixed point theorem is considered to study the existence of solutions, and the uniqueness of the solution is proved using Banach contraction theorem. Stability results in the sense of Ulam and Hyers, and its generalized form of stability for the considered initial value problem are established and we depict numerical simulations to demonstrate the impact of the fractional order on stability.

Crimean-Congo hemorrhagic fever is a common disease between humans and animals that is transmitted to humans through infected ticks, contact with infected animals, and infected humans. In this paper, we present a boxed model for the transmission of Crimean-Congo fever virus. With the help of the fixed-point theory, our proposed system model is investigated in detail to prove its unique solution. Given that the Caputo fractional-order derivative preserves the system’s historical memory, we use this fractional derivative in our modeling. The equilibrium points of the proposed system and their stability conditions are determined. Using the Euler method for the Caputo fractional-order derivative, we calculate the approximate solutions of the fractional system, and then, we present a numerical simulation for the transmission of Crimean-Congo hemorrhagic fever.

A thermostat model described by a second-order fractional difference equation is proposed in this paper with one sensor and two sensors fractional boundary conditions depending on positive parameters by using the Lipschitz-type inequality. By means of well-known contraction mapping and the Brouwer fixed-point theorem, we provide new results on the existence and uniqueness of solutions. In this work by use of the Caputo fractional difference operator and Hyer–Ulam stability definitions we check the sufficient conditions and solution of the equations to be stable, while most researchers have examined the necessary conditions in different ways. Further, we also establish some results regarding Hyers–Ulam, generalized Hyers–Ulam, Hyers–Ulam–Rassias, and generalized Hyers–Ulam–Rassias stability for our discrete fractional-order thermostat models. To support the theoretical results, we present suitable examples describing the thermostat models that are illustrated by graphical representation.