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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 $f\left(tx\right)$ is continuous in $t$ for each fixed $x$. Assume that there exists ${\Theta }\ge 0$ and $p\in \left[0,1\right)$ such that

$\frac{\parallel f\left(x+y\right)-f\left(x\right)-f\left(y\right)\parallel }{{\parallel x\parallel }^{p}+{\parallel y\parallel }^{p}}\le {\Theta },$

for any $x,y\in ℝ$. The there exists a unique linear mapping $T:{E}_{1}\to {E}_{2}$ such that $\frac{\parallel f\left(x\right)-T\left(x\right)\parallel }{{\parallel x\parallel }^{p}}\le \frac{2{\Theta }}{2-{2}^{p}}$, 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
##### Keywords:
Approximately Linear Mapping