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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
##### Keywords:
Approximately Linear Mapping
Full Text:
##### References:
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