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http://dx.doi.org/10.5666/KMJ.2010.50.2.315

Hyers-Ulam Stability of Cubic Mappings in Non-Archimedean Normed Spaces  

Mirmostafaee, Alireza Kamel (Department of Pure Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad)
Publication Information
Kyungpook Mathematical Journal / v.50, no.2, 2010 , pp. 315-327 More about this Journal
Abstract
We give a xed point approach to the generalized Hyers-Ulam stability of the cubic equation f(2x + y) + f(2x - y) = 2f(x + y) + 2f(x - y) + 12f(x) in non-Archimedean normed spaces. We will give an example to show that some known results in the stability of cubic functional equations in real normed spaces fail in non-Archimedean normed spaces. Finally, some applications of our results in non-Archimedean normed spaces over p-adic numbers will be exhibited.
Keywords
Hyers-Ulam-Rassias stability; Cubic functional equation; fixed point alternative; non-Archimedean normed space;
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Times Cited By KSCI : 2  (Citation Analysis)
Times Cited By SCOPUS : 0
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1 T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2(1950) 64-66.   DOI
2 L. M. Arriola and W. A. Beyer, Stability of the Cauchy functional equation over p-adic fields, Real Analysis Exchange, 31(2005/2006), 125-132   DOI
3 D. G. Bourgin, Classes of transformations and bordering transformations, Bull. AMS, 57(1951), 223-237.   DOI
4 Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc., 72(1978), 297-300.   DOI   ScienceOn
5 S. M. Ulam, Problems in Modern Mathematics (Chapter VI, Some Questions in Analysis: 1, Stability), Science Editions, John Wiley & Sons, New York, 1964.
6 M. Mirzavaziri and M. S. Moslehian, A fixed point approach to stability of a quadratic equation, Bull. Braz. Math. Soc., (N. S.), 37(2006), no. 3, 361-376.   DOI
7 A. K. Mirmostafaee, Approximately additive mappings in non-Archimedean normed spaces, Bull. Korean Math. Soc., 46(2009) no.2, 378-400.   과학기술학회마을   DOI   ScienceOn
8 A.K. Mirmostafaee, Stability of quartic mappings in non-Archimedean normed spaces, Kyungpook Math. J., 49(2009), 289-297.   DOI   ScienceOn
9 A. K. Mirmostafaee and M. S. Moslehian, Fuzzy approximately cubic mappings, Inform. Sci., 178(2008), no 19, 3791-3798.   DOI   ScienceOn
10 M.S. Moslehian, The Jensen functional equation in non-Archimedean normed spaces, J. Funct. Spaces Appl., 7(2009), no. 1, 13-24.   DOI
11 M. S. Moslehian and T. M. Rassias, Stability of functional equations in non Archimedean spaces, Appl. Anal. Discrete Math., 1(2007), 325-334.   DOI
12 M. S. Moslehian and G. Sadeghi, Stability of two types of cubic functional equations in non-Archimedean spaces, Real Anal. Exchange, 33(2008), no. 2, 375-383.   DOI
13 A. Najati, The generalized Hyers-Ulam-Rassias stability of a cubic functional equa- tion, Turk J. Math., 31(2007), 395-408.
14 J. M. Rassias, Solution of the stability problem for cubic mappings, Glasnik Math., Vol. 36, 56(2001), 63-72.
15 L. Narici and E. Beckenstein, Strange terrain|non-Archimedean spaces, Amer. Math. Monthly, 88(1981), no. 9, 667-676.   DOI   ScienceOn
16 J. M. Rassias, Alternative contraction principle and Ulam stability problem, Math. Sci. Res. J., 9(7)(2005), 190-199.
17 V. Radu, The fixed point alternative and stability of functional equations, Fixed Point Theory, 4(2003), no. 1. 91-96.
18 L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Grazer Math. Ber., 346(2004), 43-52.
19 J. B. Diaz and B. Margolis, A fixed point theorem of the alternative for the con- tractions on a generalized complete metric space, Bull. Amer. Math. Soc., 74(1968), 305-309.   DOI
20 P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184(1994), 431-436.   DOI   ScienceOn
21 K. Hensel, Uber eine news Begrundung der Theorie der algebraischen Zahlen, Jahresber. Deutsch. Math. Verein, 6(1897), 83-88.
22 D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci.U.S.A., 27(1941), 222-224.   DOI   ScienceOn
23 D. H. Hyers, G. Isac and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Basel, 1998.
24 S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.
25 S.-M. Jung and T.-S. Kim, A fixed point approach to the stability of the cubic func- tional equation, Bol. Soc. Mat. Mexicana, (3) 12(2006), no. 1, 51-57.
26 Y.-S Jung and I.-S. Chang, The stability of a cubic type functional equation with the fixed point alternative, J. Math. Anal. Appl., 306b(2005) 752-760.
27 Z. Kaiser, On stability of the monomial functional equation in normed spaces over fields with valuation, J. Math. Anal. Appl., 322(2006), No. 2, 1188-1198.   DOI   ScienceOn
28 K.W. Jun and H.M. Kim, The generalized Hyers-Ulam-Rassias stability of a cubic functional equation, J. Math. Anal. Appl., 274(2002), 867-878.   DOI   ScienceOn