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On the stability of minimum points. (English) Zbl 1084.49026

Summary: It is well known that two functions may differ uniformly by a small amount and yet their derivatives may differ widely. However, there are certain cases in which these derivatives may even be equal, provided they are evaluated at slightly different points, as in case \(n=1\) of the
Theorem: Let \(I\subset\mathbb R\) be an open interval centered at the point \(a\) and let \(f:I\to\mathbb R\) have an \(n\)-th derivative in \(I\). Suppose that \(n\)-th derivative \(f^{(n)}(a)=0\) and \(f^{(n)}(x)\) changes sign at \(x=a\). Then, corresponding to each \(\varepsilon>0\) there exists a \(\delta>0\) such that, for each function \(g\) having an \(n\)-th derivative in \(I\) and satisfying the inequality \(| g(x)-f(x)|<\delta\) for \(x\) in \(I\), there exists a point \(x=b\) in \(I\) such that \(g^{(n)}(b)=0\) and \(| b-a|<\varepsilon\).
This was proved by S. M. Ulam and D. H. Hyers in [Math. Mag. 28, 59-64 (1954; Zbl 0057.09905)].
A generalization to the case of higher order derivatives of the Ulam-Hyers theorem on the stability of differential expressions will be provided.
We define the difference operator \(\Delta_hf(x)\) by \(\Delta_hf(x)=f(x+h)-f(x)\).

MSC:

49K40 Sensitivity, stability, well-posedness

Citations:

Zbl 0057.09905
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