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Zbl 1192.39022
Li, Yongjin; Hua, Liubin
(Li, Yong-jin; Hua, Liu-bin)
Hyers-Ulam stability of a polynomial equation. (English)
[J] Banach J. Math. Anal. 3, No. 2, 86-90, electronic only (2009). ISSN 1735-8787/e

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.
[Prasanna Sahoo (Louisville)]

MSC 2000:: *39B82 Stability, separation, extension, and related topics
39B22 Functional equations for real functions

Keywords: Hyers-Ulam stability; polynomial equation

Cited in: Zbl 1201.39012

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