• Published : 2008.11.30


In this paper we characterize multi-Jensen functions f : $V^n\;{\rightarrow}\;W$, where n is a positive integer, V, W are commutative groups and V is uniquely divisible by 2. Moreover, under the assumption that f : $\mathbb{R}\;{\rightarrow}\;\mathbb{R}$ is Borel measurable, we obtain representation of f (respectively, f, g, h : $\mathbb{R}\;{\rightarrow}\;\mathbb{R}$) such that the Jensen difference $$2f\;\(\frac{x\;+\;y}{2}\)\;-\;f(x)\;-\;f(y)$$ (respectively, the Pexider difference $$2f\;\(\frac{x\;+\;y}{2}\)\;-\;g(x)\;-\;h(y))$$ takes values in a countable subgroup of $\mathbb{R}$.


  1. J.-H. Bae and W.-G. Park, On the solution of a bi-Jensen functional equation and its stability, Bull. Korean Math. Soc. 43 (2006), no. 3, 499-507
  2. M. Bajger, On the composite Pexider equation modulo a subgroup, Publ. Math. Debrecen 64 (2004), no. 1-2, 39-61
  3. K. Baron, Orthogonality and additivity modulo a discrete subgroup, Aequationes Math. 70 (2005), no. 1-2, 189-190
  4. K. Baron and Pl. Kannappan, On the Pexider difference, Fund. Math. 134 (1990), no. 3, 247-254
  5. J. Brzdek, The Cauchy and Jensen diferences on semigroups, Publ. Math. Debrecen 48 (1996), no. 1-2, 117-136
  6. J. Brzdek, On orthogonally exponential functionals, Pacific J. Math. 181 (1997), no. 2, 247-267
  7. L. Cadariu and V. Radu, Fixed points and the stability of Jensen's functional equation, JIPAM. J. Inequal. Pure Appl. Math. 4 (2003), no. 1, Article 4
  8. K. Cieplinski, On a generalized Pexider equation and the Pexider difference, Iteration theory (ECIT '06), 27-36, Grazer Math. Ber., 351, Karl-Franzens-Univ. Graz, Graz, 2007
  9. J. G. van der Corput, Goniometrische functies gekarakteriseerd door een functionaal betrekking, Euclides 17 (1940), 55-75
  10. G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), no. 1-2, 143-190
  11. N. Frantzikinakis, Additive functions modulo a countable subgroup of $\mathbb{R}$, Colloq. Math. 95 (2003), no. 1, 117-122
  12. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436
  13. P. Gavruta, S.-M. Jung, and K.-S. Lee, Remarks on the Pexider equations modulo a subgroup, Far East J. Math. Sci. (FJMS) 19 (2005), no. 2, 215-222
  14. G. Godini, Set-valued Cauchy functional equation, Rev. Roumaine Math. Pures Appl. 20 (1975), no. 10, 1113-1121
  15. D. H. Hyers, G. Isac, and Th. M. Rassias, Stability of functional equations in several variables, Progress in Nonlinear Differential Equations and their Applications, 34. Birkhaser Boston, Inc., Boston, MA, 1998
  16. K.-W. Jun and Y.-H. Lee, A generalization of the Hyers-Ulam-Rassias stability of Jensen's equation, J. Math. Anal. Appl. 238 (1999), no. 1, 305-315
  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. S.-M. Jung, On the quadratic functional equation modulo a subgroup, Indian J. Pure Appl. Math. 36 (2005), no. 8, 441-450
  19. S.-M. Jung and K.-S. Lee, On the Jensen functional equation modulo a subgroup, J. Appl. Algebra Discrete Struct. 5 (2007), no. 1, 21-32
  20. Z. Kominek, On a local stability of the Jensen functional equation, Demonstratio Math. 22 (1989), no. 2, 499-507
  21. A. Najati, Hyers-Ulam-Rassias stability of a cubic functional equation, Bull. Korean Math. Soc. 44 (2007), no. 4, 825-840
  22. C. Park and Th. M. Rassias, d-isometric linear mappings in linear d-normed Banach modules, J. Korean Math. Soc. 45 (2008), no. 1, 249-271
  23. W. Prager and J. Schwaiger, Multi-affine and multi-Jensen functions and their connection with generalized polynomials, Aequationes Math. 69 (2005), no. 1-2, 41-57
  24. W. Prager and J. Schwaiger, Stability of the multi-Jensen equation, Bull. Korean Math. Soc. 45 (2008), no. 1, 133-142
  25. S. M. Ulam, Problems in Modern Mathematics, Science Editions John Wiley & Sons, Inc., New York, 1964

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