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http://dx.doi.org/10.4134/CKMS.2013.28.4.767

HYERS-ULAM STABILITY OF MAPPINGS FROM A RING A INTO AN A-BIMODULE  

Oubbi, Lahbib (Department of Mathematics Ecole Normale Superieure Mohammed V-Agdal University)
Publication Information
Communications of the Korean Mathematical Society / v.28, no.4, 2013 , pp. 767-782 More about this Journal
Abstract
We deal with the Hyers-Ulam stability problem of linear mappings from a vector space into a Banach one with respect to the following functional equation: $$f\(\frac{-x+y}{3}\)+f\(\frac{x-3z}{3}\)+f\(\frac{3x-y+3z}{3}\)=f(x)$$. We then combine this equation with other ones and establish the Hyers-Ulam stability of several kinds of linear mappings, among which the algebra (*-) homomorphisms, the derivations, the multipliers and others. We thus repair and improve some previous assertions in the literature.
Keywords
Hyers-Ulam stability of functional equations; Hyers-Ulam stability of additive mappings; ring homomorphisms; ring derivations;
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Times Cited By KSCI : 3  (Citation Analysis)
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1 GH. Abbaspour and A. Rahmani, Hyers-Ulam-Rassias and Ulam-Gavruta-Rassias stability of generalized quadtratic functional equations, Advances in Applied Mathematical Analysis 4 (2009), no. 1, 31-38.
2 B. Blackadar, Operator Algebras, Theory of $C^{\ast}$-Algebras and von Neumann Algebras, Encyclopedia of Mathematical Sciences, 122, Springer, 2006.
3 D. G. Bourgin, Approximately isometric and multiplicative transformations on continuous function rings, Duke Math. J. 16 (1949), 385-397.   DOI
4 L. Cadariu and V. Radu, On the stability of the Cauchy functional equation: a fixed point approach, Iteration theory (ECIT '02), 43-52, Grazer Math. Ber., 346, Karl-Franzens-Univ. Graz, Graz, 2004.
5 L. Cadariu and V. Radu, Fixed points and the stability of the Jensen's functional equation, J. Inequal. Pure and Appl. Math. 4 (2003), no. 1, http//jipam.vu/edu.au, 1-7.
6 M. Eshaghi Gordji, N. Ghobadipour, and C. Park, Jordan *-homomorphisms between unital $C^{\ast}$-algebras, Commun. Korean Math. Soc. 27 (2012), no. 1, 149-158.   과학기술학회마을   DOI   ScienceOn
7 M. Eshaghi Gordji, T. Karimi, and S. Kaboli Gharetapeh, Approximately n-Jordan homomorphisms on Banach algebras, J. Inequal. Appl. 2009 (2009), Article ID 870843, 8 pages.
8 P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approxiately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436.   DOI   ScienceOn
9 D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.   DOI   ScienceOn
10 K. Jun, S. Jung, and Y. Lee, A generalisation of the Hyers-Ulam-Rassias stability of functional equation of Davison, J. Korean Math. Soc. 41 (2004), 501-511.   DOI   ScienceOn
11 K. Jun and H. Kim, Remarks on the stability of additive functional equation, Bull. Korean Math. Soc. 38 (2001), no. 4, 679-687.   과학기술학회마을
12 K. Jun, H. Kim, and J. M. Rassias, Extended Hyers-Ulam stability for Cauchy-Jensen mapping, J. Difference Equ. Appl. 13 (2007), no. 12, 1139-1153.   DOI   ScienceOn
13 T. Miura, S. E. Takahasi, and G. Hirasawa, Hyers-Ulam-Rassias stability of Jordan homomorphisms on Banach algebras, J. Inequal. Appl. 2005 (2005), no. 4, 435-441.
14 M. S. Moslehian, Hyers-Ulam-Rassias stability of generalized derivations, Int. J. Math. Math. Sci. 2006 (2006), Article ID 93942, 8 pages.
15 L. Oubbi, Ulam-Hyers-Rassias stability problem for several kinds of mappings, Afr. Mat. Springer Verlag, 2012; DOI 10.1007/s13370-012-0078-6 (18 pages).   DOI
16 T. M. Rassias, On the stability of functional equations and a problem of Ulam, Acta Appl. Math. 62 (2000), no. 1, 23-130.   DOI   ScienceOn
17 K. Park and Y. Jung, Stability of a functional equation obtained by combining two functional equations, J. Appl. Math. & Computing 14 (2004), no. 1-2, 415-422.   과학기술학회마을
18 C. Park and J. M. Rassias, Stability of the Jensen-type functional equation in $C^{\ast}$-algebras: A fixed point approch, Abstr. Appl. Anal. 2009 (2009), Article ID 360432, 17 pages.
19 T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300.   DOI   ScienceOn
20 S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science ed. Wiley, New York, 1940.