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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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