Connections Between the Completion of Normed Spaces Over Non-Archimedean Fields and the Stability of the Cauchy Equation

Schwaiger, Jens

doi:10.2478/amsil-2020-0002

Abstract

In [12] a close connection between stability results for the Cauchy equation and the completion of a normed space over the rationals endowed with the usual absolute value has been investigated. Here similar results are presented when the valuation of the rationals is a p-adic valuation. Moreover a result by Zygfryd Kominek ([5]) on the stability of the Pexider equation is formulated and proved in the context of Banach spaces over the field of p-adic numbers.

References

[1] N. Bourbaki, Elements of Mathematics. Topological Vector Spaces, Chapters 1–5, Transl. from the French by H.G. Eggleston and S. Madan, Springer-Verlag, Berlin, 1987.10.1007/978-3-642-61715-7
Search in Google Scholar Back to article
[2] G.L. Forti, An existence and stability theorem for a class of functional equations, Stochastica 4 (1980), no. 1, 23–30.
Search in Google Scholar Back to article
[3] G.L. Forti and J. Schwaiger, Stability of homomorphisms and completeness, C.R. Math. Rep. Acad. Sci. Canada 11 (1989), no. 6, 215–220.
Search in Google Scholar Back to article
[4] P. Gǎvruţa, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431–436.10.1006/jmaa.1994.1211
Search in Google Scholar Back to article
[5] Z. Kominek, On Hyers-Ulam stability of the Pexider equation, Demonstratio Math. 37 (2004), no. 2, 373–376.10.1515/dema-2004-0214
Search in Google Scholar Back to article
[6] M.S. Moslehian and Th.M. Rassias, Stability of functional equations in non-Archimedean spaces, Appl. Anal. Discrete Math. 1 (2007), no. 2, 325–334.10.2298/AADM0702325M
Search in Google Scholar Back to article
[7] A. Najati and Y.J. Cho, Generalized Hyers-Ulam stability of the Pexiderized Cauchy functional equation in non-Archimedean spaces, Fixed Point Theory Appl. 2011, Art. ID 309026, 11 pp.10.1155/2011/309026
Search in Google Scholar Back to article
[8] C. Perez-Garcia and W.H. Schikhof, Locally Convex Spaces over Non-Archimedean Valued Fields, Vol. 119, Cambridge University Press, Cambridge, 2010.10.1017/CBO9780511729959
Search in Google Scholar Back to article
[9] W.H. Schikhof, Ultrametric Calculus. An Introduction to p-adic Analysis. Paperback reprint of the 1984 original, Vol. 4, Cambridge University Press, Cambridge, 2006.10.1017/CBO9780511623844
Search in Google Scholar Back to article
[10] J. Schwaiger, Remark on Hyers’s stability theorem, in: R. Ger, Report of Meeting: The Twenty-fifth International Symposium on Functional Equations, Aequationes Math. 35 (1988), no. 1, 82–124,10.1007/BF01838160
Search in Google Scholar Back to article
[11] J. Schwaiger, Functional equations for homogeneous polynomials arising from multi-linear mappings and their stability, Ann. Math. Sil. 8 (1994), 157–171.
Search in Google Scholar Back to article
[12] J. Schwaiger, On the construction of the field of reals by means of functional equations and their stability and related topics, in: J. Brzdęk et al. (eds.), Developments in Functional Equations and Related Topics, Springer, Cham, 2017, pp. 275–295.10.1007/978-3-319-61732-9_12
Search in Google Scholar Back to article

Connections Between the Completion of Normed Spaces Over Non-Archimedean Fields and the Stability of the Cauchy Equation

Abstract

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