Hyers-Ulam stability of the first order difference equation with average growth rate

The paper proves Hyers–Ulam (HU) stability when the geometric mean of growth rates converges to a limit ≠ 1, in both the contracting-on-average case (Theorem 2.4) and the expanding-on-average case (Theorem 3.4), and shows failure under (pre)periodic average growth even if each average value is < 1 (Lemma 4.1, Theorem 4.2). In the contracting case, the model’s Part A essentially matches Lemma 2.1 and the bound derived via Lemma 2.2, yielding the K/(K−1) constant (inequality (2.1) leading to Theorem 2.4) . However, in the expanding case, the model’s Part B incorrectly claims global injectivity and a uniform K/(K−1) bound without the additional regularity on the slowly varying factors t_n used by the paper; the paper instead obtains a robust tail estimate 2/ln K (eq. (3.8)) and explains when a K/(K−1)-type bound can be recovered (Remark 3.5) . The model’s backward-inversion argument omits needed hypotheses for existence/uniqueness of preimages and uniformity on the domain S that the paper carefully enforces via difference quotients and a contraction mapping (Lemma 3.3 and Theorem 3.4) . For the obstruction under preperiodic averages, the model’s Part C aligns with the paper’s results (Lemma 4.1, Theorem 4.2) .