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Stability of multi-additive mappings in non-Archimedean normed spaces. (English) Zbl 1204.39027
Let \(V\) be a commutative semigroup, \(W\) a non-Archimedean space and \(n \geq 1\) an integer. A function \(f : V^n \to W\) is said to be multi-additive if it is additive in each variable. In this paper, the author proves some results concerning the generalized Hyers-Ulam stability of the multi-additive functions in non-Archimedean spaces, using the direct method.

MSC:
39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
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[1] Albert, M.; Baker, J., Functions with bounded nth differences, Ann. polon. math., 43, 93-103, (1983) · Zbl 0436.39005
[2] Aoki, T., On the stability of the linear transformation in Banach spaces, J. math. soc. Japan, 2, 64-66, (1950) · Zbl 0040.35501
[3] Badora, R., On the separation with n-additive functions, (), 219-230 · Zbl 0883.43004
[4] Bourgin, D.G., Classes of transformations and bordering transformations, Bull. amer. math. soc., 57, 223-237, (1951) · Zbl 0043.32902
[5] Brzdȩk, J., A note on stability of additive mappings, (), 19-22 · Zbl 0844.39012
[6] Brzdȩk, J.; Popa, D.; Xu, B., On nonstability of the linear recurrence of order one, J. math. anal. appl., 367, 146-153, (2010) · Zbl 1193.39006
[7] Chung, J., Hyers-Ulam-Rassias stability of Cauchy equation in the space of Schwartz distributions, J. math. anal. appl., 300, 343-350, (2004) · Zbl 1066.39028
[8] Ciepliński, K., On multi-Jensen functions and Jensen difference, Bull. Korean math. soc., 45, 729-737, (2008) · Zbl 1172.39033
[9] Ciepliński, K., Stability of the multi-Jensen equation, J. math. anal. appl., 363, 249-254, (2010) · Zbl 1211.39017
[10] K. Ciepliński, Stability of multi-Jensen mappings in non-Archimedean normed spaces, in: Functional Equations in Mathematical Analysis, Springer, New York, in press.
[11] Ciepliński, K., Generalized stability of multi-additive mappings, Appl. math. lett., 23, 1291-1294, (2010) · Zbl 1204.39026
[12] Faı̌ziev, V.A.; Riedel, T., Stability of Jensen functional equation on semigroups, J. math. anal. appl., 364, 341-351, (2010) · Zbl 1188.39028
[13] Forti, G.L., Hyers-Ulam stability of functional equations in several variables, Aequationes math., 50, 143-190, (1995) · Zbl 0836.39007
[14] Gajda, Z., Local stability of the functional equation characterizing polynomial functions, Ann. polon. math., 52, 119-137, (1990) · Zbl 0717.39007
[15] Gajda, Z., On stability of additive mappings, Int. J. math. sci., 14, 431-434, (1991) · Zbl 0739.39013
[16] Gǎvrutǎ, P., A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. math. anal. appl., 184, 431-436, (1994) · Zbl 0818.46043
[17] Gǎvrutǎ, P., On a problem of G. isac and th.M. Rassias concerning the stability of mappings, J. math. anal. appl., 261, 543-553, (2001) · Zbl 0993.47002
[18] Gouvêa, F.Q., p-adic numbers. an introduction, Universitext, (1997), Springer-Verlag Berlin · Zbl 0874.11002
[19] Eshaghi Gordji, M.; Savadkouhi, M.B., Stability of cubic and quartic functional equations in non-Archimedean spaces, Acta appl. math., 110, 1321-1329, (2010) · Zbl 1192.39018
[20] Hensel, K., Über eine neue begründung der theorie der algebraischen zahlen, Jahresber. Deutsch. math.-verein., 6, 83-88, (1899) · JFM 30.0096.03
[21] Hyers, D.H., On the stability of the linear functional equation, Proc. natl. acad. sci. USA, 27, 222-224, (1941) · Zbl 0061.26403
[22] Hyers, D.H.; Isac, G.; Rassias, Th.M., Stability of functional equations in several variables, Progr. nonlinear differential equations appl., vol. 34, (1998), Birkhäuser Boston, Inc. Boston, MA · Zbl 0894.39012
[23] Jung, S.-M., Hyers-Ulam-Rassias stability of functional equations in mathematical analysis, (2001), Hadronic Press Palm Harbor, FL · Zbl 0980.39024
[24] Khrennikov, A., Non-Archimedean analysis: quantum paradoxes, dynamical systems and biological models, Math. appl., vol. 427, (1997), Kluwer Academic Publishers Dordrecht · Zbl 0920.11087
[25] Kuczma, M., An introduction to the theory of functional equations and inequalities. Cauchy’s equation and Jensen’s inequality, (2009), Birkhäuser Basel · Zbl 1221.39041
[26] Maligranda, L., A result of tosio aoki about a generalization of Hyers-Ulam stability of additive functions - a question of priority, Aequationes math., 75, 289-296, (2008) · Zbl 1158.39019
[27] Mazur, S.; Orlicz, W., Grundlegende eigenschaften der polynomischen operationen I, II, Studia math., 5, 50-68, (1934), 179-189 (in German) · JFM 60.1074.03
[28] Miheţ, D.; Radu, V., On the stability of the additive Cauchy functional equation in random normed spaces, J. math. anal. appl., 343, 567-572, (2008) · Zbl 1139.39040
[29] Mirmostafaee, A.K.; Moslehian, M.S., Stability of additive mappings in non-Archimedean fuzzy normed spaces, Fuzzy sets and systems, 160, 1643-1652, (2009) · Zbl 1187.46068
[30] Moslehian, M.S., The Jensen functional equation in non-Archimedean normed spaces, J. funct. spaces appl., 7, 13-24, (2009) · Zbl 1177.46054
[31] Moslehian, M.S.; Rassias, Th.M., Stability of functional equations in non-Archimedean spaces, Appl. anal. discrete math., 1, 325-334, (2007) · Zbl 1257.39019
[32] Najati, A.; Moradlou, F., Hyers-Ulam-Rassias stability of the apollonius type quadratic mapping in non-Archimedean spaces, Stability of functional equations and applications, Tamsui oxf. J. math. sci., 24, 367-380, (2008) · Zbl 1170.39017
[33] Prager, W.; Schwaiger, J., Stability of the multi-Jensen equation, Bull. Korean math. soc., 45, 133-142, (2008) · Zbl 1151.39023
[34] Rassias, Th.M., On the stability of the linear mapping in Banach spaces, Proc. amer. math. soc., 72, 297-300, (1978) · Zbl 0398.47040
[35] Rassias, Th.M., On the stability of functional equations and a problem of Ulam, Acta appl. math., 62, 23-130, (2000) · Zbl 0981.39014
[36] Rassias, Th.M.; Šemrl, P., On the behavior of mappings which do not satisfy Hyers-Ulam stability, Proc. amer. math. soc., 114, 989-993, (1992) · Zbl 0761.47004
[37] Robert, A.M., A course in p-adic analysis, Grad. texts in math., vol. 198, (2000), Springer-Verlag New York · Zbl 0947.11035
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