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Stability of the Cauchy functional equation over $$p$$-adic fields. (English) Zbl 1099.39019
During the last three decades the $$p$$-adic number field $${\mathbb Q}_p$$ has gained the interest of physicists for their research in particular in problems coming from quantum physics, $$p$$-adic strings and superstrings [cf. A. Khrennikov, Non-archimedean analysis: quantum paradoxes, dynamical systems and biological models. Mathematics and its Applications (Dordrecht). 427. Dordrecht: Kluwer Academic Publishers. (1997; Zbl 0920.11087)]. A key property of $$p$$-adic numbers is that they do not satisfy the Archimedean axiom: for all $$x, y >0$$, there exists an integer $$n$$ such that $$x<ny$$.
The authors investigate the stability of approximate additive mappings $$f: {\mathbb Q}_p \to {\mathbb R}$$. They show that if $$f: {\mathbb Q}_p \to {\mathbb R}$$ is a continuous mapping for which there exists a fixed $$\varepsilon$$ such that $$| f(x+y) - f(x) - f(y)| \leq \varepsilon$$ $$(x, y \in {\mathbb Q}_p)$$, then there exists a unique additive mapping $$T: {\mathbb Q}_p \to {\mathbb R}$$ such that $$| f(x) - T(x)| \leq \varepsilon$$ for all $$x \in {\mathbb Q}_p$$. It seems that they do not use any essential property of $$p$$-adic numbers in their proofs.

MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 46S10 Functional analysis over fields other than $$\mathbb{R}$$ or $$\mathbb{C}$$ or the quaternions; non-Archimedean functional analysis 11S80 Other analytic theory (analogues of beta and gamma functions, $$p$$-adic integration, etc.) 39B22 Functional equations for real functions
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