Hyers-Ulam stability of hyperbolic M\"obius difference equation

TL;DR

This paper investigates the Hyers-Ulam stability of a hyperbolic Möbius difference equation, showing stability conditions based on initial points relative to specific geometric regions, extending previous results on logistic equations.

Contribution

It generalizes Hyers-Ulam stability results to hyperbolic Möbius difference equations with complex parameters, identifying geometric stability regions.

Findings

Stability holds if initial point is outside a specific disk centered at -d/c.

Stability region extends to the complement of neighborhoods around certain line segments.

Results generalize stability conditions known for the Pielou logistic equation.

Abstract

Hyers-Ulam stability of the difference equation with the initial point $ z_0 $ as follows $$ z_{i+1} = \frac{az_i + b}{cz_i + d} $$ is investigated for complex numbers $ a,b,c $ and $ d $ where $ ad - bc = 1 $, $ c \neq 0 $ and $a + d \in \mathbb{R} \setminus [-2,2] $. The stability of the sequence $ \{z_n\}_{n \in \mathbb{N}_0} $ holds if the initial point is in the exterior of a certain disk of which center is $ -\frac{d}{c} $. Furthermore, the region for stability can be extended to the complement of some neighborhood of the line segment between $ -\frac{d}{c} $ and the repelling fixed point of the map $ z \mapsto \frac{az + b}{cz + d} $. This result is the generalization of Hyers-Ulam stability of Pielou logistic equation.