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On the stability of the linear mapping in Banach spaces. (English) Zbl 0398.47040
S. M. Ulam posed the problem: Let \(E_1, E_2\) be two Banach spaces, and let \(f: E_1 \to E_2\) be a mapping, that is “approximately linear”. Give conditions in order for a linear mapping near an approximately linear mapping to exist. The author has given an answer to Ulam’s problem. In fact the following theorem has been stated and proved.
Theorem: Consider \(E_1, E_2\) to be two Banach spaces, and let \(f: E_1 \to E_2\) be a mapping such that \(f(tx)\) is continuous in \(t\) for each fixed \(x\). Assume that there exists \(\Theta\geq 0\) and \(p\in[0,1)\) such that \[ \frac{\| f(x+y)-f(x)-f(y)\|}{\| x\|^p+\| y\|^p}\leq \Theta, \] for any \(x,y\in\mathbb R\). The there exists a unique linear mapping \(T: E_1 \to E_2\) such that \(\frac{\| f(x)-T(x)\|}{\| x\|^p}\leq \frac{2\Theta}{2-2^p}\), for any \(x\in E_1\).
Reviewer: Th. M. Rassias

MSC:
47H14 Perturbations of nonlinear operators
47A55 Perturbation theory of linear operators
46B99 Normed linear spaces and Banach spaces; Banach lattices
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