*(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 $p\in [0,1)$ such that

for any $x,y\in \mathbb{R}$. 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 $x\in {E}_{1}$.

##### MSC:

47H14 | Perturbations of nonlinear operators |

47A55 | Perturbation theory of linear operators |

46B99 | Normed linear spaces and Banach spaces |