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Zbl 0398.47040
Rassias, Themistocles M.
On the stability of the linear mapping in Banach spaces. (English)
[J] Proc. Am. Math. Soc. 72, 297-300 (1978). ISSN 0002-9939; ISSN 1088-6826/e

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 $$\frac{\Vert f(x+y)-f(x)-f(y)\Vert}{\Vert x\Vert^p+\Vert y\Vert^p}\leq \Theta,$$ for any $x,y\in\Bbb R$. The there exists a unique linear mapping $T: E_1 \to E_2$ such that $\frac{\Vert f(x)-T(x)\Vert}{\Vert x\Vert^p}\leq \frac{2\Theta}{2-2^p}$, for any $x\in E_1$.

Display scanned Zentralblatt-MATH page with this review.
[Th. M. Rassias]

MSC 2000:: *47H14 Perturbations of nonlinear operators
47A55 Perturbation theory of linear operators
46B99 Normed linear spaces and Banach spaces

Keywords: Approximately Linear Mapping

Cited in: Zbl 1259.39021 Zbl 1246.39021 Zbl 1231.39010 Zbl 1207.39029 Zbl pre05948581 Zbl 1232.39026 Zbl 1202.39022 Zbl 1220.39031 Zbl 1175.39011 Zbl 1170.39013 Zbl 1168.39308 Zbl 1199.47170 Zbl 1158.39019 Zbl 1149.39025 Zbl 1147.39012 Zbl 1142.39023 Zbl 1182.39017 Zbl 1155.39307 Zbl 1152.39020 Zbl 1141.17302 Zbl 1140.39017 Zbl 1128.39021 Zbl 1120.39026 Zbl 1111.39027 Zbl 1146.39301 Zbl 1133.39033 Zbl 1112.39023 Zbl 1093.47040 Zbl 1053.46028 Zbl 1053.46030 Zbl 1064.47016 Zbl 1045.47037 Zbl 1036.39019 Zbl 1036.39021 Zbl 1035.46004 Zbl 0981.39015 Zbl 0894.39012 Zbl 0879.39010 Zbl 0845.39013 Zbl 0796.39012 Zbl 0795.39006 Zbl 0779.39003 Zbl 0757.47033 Zbl 0757.47032 Zbl 0739.39013 Zbl 0512.46023

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