Hyers-Ulam stability of a polynomial equation (Q967150)

The authors prove a Hyers-Ulam type stability result for the polynomial equation \(x^n + \alpha x + \beta = 0\). In particular, using Banach's contraction mapping theorem, they prove the following result: If \( |\alpha | > n\), \(|\beta | < |\alpha|-1\) and \(y \in [-1, 1]\) satisfies the inequality \[ |y^n + \alpha y + \beta | \leq \varepsilon \] for some \(\epsilon > 0\) and for all \(y \in [-1, 1]\), then there exists a solution \(v \in [-1, 1]\) of \(x^n + \alpha x +\beta = 0\) such that \[ |y-v| \leq k \varepsilon, \] where \(k\) is a positive constant.