Hyers-Ulam stability of elliptic Möbius difference equation

• Main Author: Young Woo Nam
• Format: Article
• Language: English
• Published: Taylor & Francis Group 2018-01-01
• Series: Cogent Mathematics & Statistics
• Subjects: difference equation; rational difference equation; stability; hyers- ulam stability; m\"obius map
• Online Access: http://dx.doi.org/10.1080/25742558.2018.1492338

The linear fractional map $$f(z) = {{az + b} \over {cz + d}}$$ on the Riemann sphere with complex coefficients $$ad - bc\ \ne\ 0$$ is called Möbius map. If $$f$$ satisfies $$ad - bc = 1\ {\rm and} - 2 \lt a + \ d\ \lt\ 2$$, then $$f$$ is called elliptic Möbius map. Let $${\left\{ {{b_n}} \right\}_{n \in {{\mathbb N}_0}}}$$ be the solution of the elliptic Möbius difference equation $${b_{n + 1}} = f({b_n})$$ for every $$n \in {{\mathbb N}_0}$$. We show that the sequence $${\left\{ {{b_n}} \right\}_{n \in {{\mathbb N}_0}}}$$ on the complex plane as well as on the real numbers has no Hyers-Ulam stability by conjugation method.