an:05042734
Zbl 1099.39019
Arriola, L. M.; Beyer, W. A.
Stability of the Cauchy functional equation over \(p\)-adic fields
EN
Real Anal. Exch. 31(2005-2006), No. 1, 125-132 (2006).
00125923
2006
j
39B82 46S10 11S80 39B22
stability; additive mapping; continuity; \(p\)-adic numbers
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. \textit{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.
Mohammad Sal Moslehian (Mashhad)
Zbl 0920.11087