• Published : 2005.11.01


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, $\mathbb{C}$).


  1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1989
  2. 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
  3. G. Baumslag, Wreath product and p-groups, Proc. Camb. Phil. Soc. 55 (1959), 224-231
  4. 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
  5. V. A. Faiziev, Pseudocharacters on semidirect product of semigroups, Mat. Zametki 53 (1993), no. 2, 132-139
  6. 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
  7. V. A. Faiziev, Description of the spaces of pseudocharacters on a free products of groups, Math. Inequal. Appl. 2 (2000), 269-293
  8. V. A. Faiziev, Pseudocharacters on a class of extension of free groups, New York J. Math. 6 (2000), 135-152
  9. V. A. Faiziev and P. K. Sahoo, On the space of pseudojensen functions on groups, St. Peterburg Math. J. 14 (2003), 1043-1065
  10. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), no. 2, 222-224
  11. 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
  12. D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125-153
  13. D. H. Hyers, G. Isac, and Th. M. Rassias, Topics in Nonlinear Analysis and Applications, World Scientific Publ. Co. Singapore, New Jersey, London, 1997
  14. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of Functional Equations in Several Variables, Birkhauser, Boston/Basel/Berlin, 1998
  15. D. H. Hyers and S. M. Ulam, On approximate isometry, Bull. Amer. Math. Soc. 51 (1945), 228-292
  16. D. H. Hyers and S. M. Ulam, Approximate isometry on the space of continuous functions, Ann. Math. 48 (1947), no. 2, 285-289
  17. S. M. Jung, Hyers-Ulam-Rassias Stability of Jensen's equation and its application, Proc. Amer. Math. Soc. 126 (1998), no. 11, 3137-3143
  18. Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math. 22 (1989), 199-507
  19. 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
  20. M. Laczkovich, The local stability of convexity, affinity and the Jensen equation, Aequationes Math. 58 (1999), 135-142
  21. B. H. Neumann Adjunction of elements to groups, J. London Math. Soc. 18 (1943), 12-20
  22. 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
  23. 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
  24. J. Tabor and J. Tabor, Local stability of the Cauchy and Jensen equations in function spaces, Aequationes Math. 58 (1999), 296-310
  25. S. M. Ulam, A collection of mathematical problems, Interscience Publ. New York, 1960
  26. S. M. Ulam, Problems in Modern Mathematics, Wiley, New York, 1964
  27. C. T. Ng, Jensen's functional equation on groups, Aequationes Math. 39 (1999), 85-99
  28. G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), 143-190