Rassias, Themistocles M. On the stability of minimum points. (English) Zbl 1084.49026 Mathematica 45(68), No. 1, 93-104 (2003). 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 \(\Delta_hf(x)=f(x+h)-f(x)\). Cited in 6 Documents MSC: 49K40 Sensitivity, stability, well-posedness Keywords:stability; minimum points; variational problem; isoperimetric problem; constraints; Lagrangian functional; orthonormal eigenfunctions Citations:Zbl 0057.09905 PDFBibTeX XMLCite \textit{T. M. Rassias}, Mathematica 45(68), No. 1, 93--104 (2003; Zbl 1084.49026)