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http://dx.doi.org/10.4134/BKMS.2005.42.4.757

ON THE STABILITY OF A JENSEN TYPE FUNCTIONAL EQUATION ON GROUPS  

FAIZIEV VALERH A. (TVER STATE AGRICULTURAL ACADEMY)
SAHOO PRASANNA K. (DEPARTMENT OF MATHEMATICS, UNIVERSITY OF LOUISVILLE)
Publication Information
Bulletin of the Korean Mathematical Society / v.42, no.4, 2005 , pp. 757-776 More about this Journal
Abstract
In this paper we establish the stability of a Jensen type functional equation, namely f(xy) - f($xy^{-1}$) = 2f(y), on some classes of groups. We prove that any group A can be embedded into some group G such that the Jensen type functional equation is stable on G. We also prove that the Jensen type functional equation is stable on any metabelian group, GL(n, $\mathbb{C}$), SL(n, $\mathbb{C}$), and T(n, $\mathbb{C}$).
Keywords
additive mapping; Banach spaces; Jensen type equation; Jensen type function; metabelian group; metric group; pseduoadditive mapping; pseudojensen type function; quasiadditive map; quasijensen type function;
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1 J. K. Chung, B. R. Ebanks, C. T. Ng, and P. K. Sahoo, On a quadratic- trigonometric functional equation and some applications, Trans. Amer. Math. Soc. 347 (1995), 1131-1161   DOI   ScienceOn
2 V. A. Faiziev, Pseudocharacters on semidirect product of semigroups, Mat. Zametki 53 (1993), no. 2, 132-139
3 V. A. Faiziev, The stability of the equation f(xy) - f(x) - f(y) = 0 on groups, ActaMath. Univ. Comenian. (N.S.) 1 (2000), 127-135
4 V. A. Faiziev, Description of the spaces of pseudocharacters on a free products of groups, Math. Inequal. Appl. 2 (2000), 269-293
5 V. A. Faiziev, Pseudocharacters on a class of extension of free groups, New York J. Math. 6 (2000), 135-152
6 V. A. Faiziev and P. K. Sahoo, On the space of pseudojensen functions on groups, St. Peterburg Math. J. 14 (2003), 1043-1065
7 D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), no. 2, 222-224
8 D. H. Hyers, The stability of homomorphisms and related topics In: Global Analysis- Analysis on Manifolds (eds Th. M. Rassias), Teubner-Texte Math. 1983, 140-153
9 D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125-153   DOI
10 D. H. Hyers, G. Isac, and Th. M. Rassias, Topics in Nonlinear Analysis and Applications, World Scientific Publ. Co. Singapore, New Jersey, London, 1997
11 D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston/Basel/Berlin, 1998
12 Y. Lee and K. Jun, A Generalization of the Hyers-Ulam-Rassias Stability of Jensen's equation, J. Math. Anal. Appl. 238 (1999), 305-315   DOI   ScienceOn
13 M. Laczkovich, The local stability of convexity, affinity and the Jensen equation, Aequationes Math. 58 (1999), 135-142   DOI
14 B. H. Neumann Adjunction of elements to groups, J. London Math. Soc. 18 (1943), 12-20   DOI
15 J. M. Rassias and M. J. Rassias, On the Ulam stability of Jensen and Jensen type mappings on restricted domains, J. Math. Anal. Appl. 281 (2003), 516-524   DOI   ScienceOn
16 L. Szekelyhidi, Ulam's problem, Hyers's solution - and to where they led, In: Functional Equations and Inequalities, Th. M. Rassias (ed), 259-285, Kluwer Academic Publishers, 2000
17 J. Tabor and J. Tabor, Local stability of the Cauchy and Jensen equations in function spaces, Aequationes Math. 58 (1999), 296-310   DOI   ScienceOn
18 S. M. Ulam, A collection of mathematical problems, Interscience Publ. New York, 1960
19 J. Aczel and J. Dhombres, Functional Equations in Several Variables, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1989
20 S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1964
21 J. Aczel, J. K. Chung, and C. T. Ng, Symmetric second differences in product form on groups, In Topics in Mathematical Analysis, Th. M. Rassias (ed), 1989, 1-22
22 G. Baumslag, Wreath product and p-groups, Proc. Camb. Phil. Soc. 55 (1959), 224-231
23 Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math. 22 (1989), 199-507
24 D. H. Hyers and S. M. Ulam, On approximate isometry, Bull. Amer. Math. Soc. 51 (1945), 228-292   DOI
25 D. H. Hyers and S. M. Ulam, Approximate isometry on the space of continuous functions, Ann. Math. 48 (1947), no. 2, 285-289   DOI   ScienceOn
26 S. M. Jung, Hyers-Ulam-Rassias Stability of Jensen's equation and its application, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3137-3143
27 G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), 143-190   DOI
28 C. T. Ng, Jensen's functional equation on groups, Aequationes Math. 39 (1999), 85-99   DOI