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Zbl 1219.39013
Gavruta, P.; Jung, S.-M.; Li, Y.
Hyers-Ulam stability of mean value points. (English)
[J] Ann. Funct. Anal. AFA 1, No. 2, 68-74, electronic only (2010). ISSN 2008-8752/e

The authors consider a few problems concerning the stability for Lagrange's and Flett's mean value points. The first result reads as follows. Let $f:\mathbb{R}\to\mathbb{R}$ be a continuously twice differentiable mapping and let $\eta\in(a,b)$ be a unique Lagrange's mean value point of $f$ in $(a,b)$ (i.e., $f'(\eta)=\frac{f(b)-f(a)}{b-a}$). It is proved that for each $\varepsilon>0$ there exists $\delta>0$ such that for each differentiable function $g:\mathbb{R}\to\mathbb{R}$ satisfying $|f(x)-g(x)|\leq\delta$, $x\in [a,b]$ there exists a Lagrange's mean value point $\xi\in (a,b)$ of $g$ such that $|\xi-\eta|\leq\varepsilon$. Other results are connected with approximate mean value points: $$\left|f'(\xi)-\frac{f(b)-f(a)}{b-a}\right|\leq\varepsilon$$ and with the stability of the equation: $$f'(x)=\frac{f(x)-f(a)}{x-a}\,.$$
[Jacek Chmieliński (Kraków)]

MSC 2000:: *39B82 Stability, separation, extension, and related topics
26A24 Differentiation of functions of one real variable
39B22 Functional equations for real functions

Keywords: stability of functional equations; Lagrange's mean value point; Flett's mean value point; Hyers-Ulam stability; Hyers-Ulam-Rassias stability

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