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Hyers-Ulam stability of a polynomial equation. (English) Zbl 1192.39022
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.

39B82Stability, separation, extension, and related topics
39B22Functional equations for real functions
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