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ON FUNCTIONAL INEQUALITIES ASSOCIATED WITH JORDAN-VON NEUMANN TYPE FUNCTIONAL EQUATIONS

  • Published : 2008.07.31

Abstract

In this paper, it is shown that if f satisfies the following functional inequality (0.1) $${\parallel}\sum\limits_{i,j=1}^3\;f{(xi,yj)}{\parallel}{\leq}{\parallel}f(x_1+x_2+x_3,\;y_1+y_2+y_3){\parallel}$$ then f is a bi-additive mapping. We moreover prove that if f satisfies the following functional inequality (0.2) $${\parallel}2\sum\limits_{j=1}^3\;f{(x_j,\;z)}+2\sum\limits_{j=1}^3\;f{(x_j,\;w)-f(\sum\limits_{j=1}^3\;xj,\;z-w)}{\parallel}{\leq}f(\sum\limits_{j=1}^3\;xj,\;z+w){\parallel}$$ then f is an additive-quadratic mapping.

Keywords

Jordan-von Neumann type bi-additive functional equation;Jordan-von Neumann type additive-quadratic functional equation;Hyers-Ulam-Rassias stability; functional inequality

References

  1. W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 71 (2006), 149-161 https://doi.org/10.1007/s00010-005-2775-9
  2. A. Gilanyi, Eine zur Parallelogrammgleichung aquivalente Ungleichung, Aequationes Math. 62 (2001), 303-309 https://doi.org/10.1007/PL00000156
  3. A. Gilanyi, On a problem by K. Nikodem, Math. Inequal. Appl. 5 (2002), 707-710
  4. D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224 https://doi.org/10.1073/pnas.27.4.222
  5. K. W. Jun, S. M. Jung, and Y. H. Lee, A generalization of the Hyers-Ulam-Rassias stability of a functional equation of division, J. Korean Math. Soc. 41 (2004), no. 3, 501-511 https://doi.org/10.4134/JKMS.2004.41.3.501
  6. K. W. Jun and H. M. Kim, Remarks on the stability of additive functional equation, Bull. Korean Math. Soc. 38 (2001), no. 4, 679-687
  7. K. W. Jun, On the Hyper-Ulam stability of a generalized quadratic and additive functional equation, Bull. Korean Math. Soc. 42 (2005), no. 1, 133-148 https://doi.org/10.4134/BKMS.2005.42.1.133
  8. J. Kang, C. Lee, and Y. Lee, A note on the Hyers-Ulam-Rassias stability of a quadratic equation, Bull. Korean Math. Soc. 41 (2004), no. 3, 541-557 https://doi.org/10.4134/BKMS.2004.41.3.541
  9. G. H. Kim, On the stability of the generalized G-type functional equations, Commun. Korean Math. Soc. 20 (2005), no. 1, 93-106 https://doi.org/10.4134/CKMS.2005.20.1.093
  10. G. H. Kim, On the stability of functional equations in n-variables and its applications, Commun. Korean Math. Soc. 20 (2005), no. 2, 321-338 https://doi.org/10.4134/CKMS.2005.20.2.321
  11. G. H. Kim and Y. W. Lee, The stability of the generalized form for the Gamma functional equation, Commun. Korean Math. Soc. 15 (2000), no. 1, 45-50
  12. G. H. Kim, Y. W. Lee, and K. S. Ji, Modified Hyers-Ulam-Rassias stability of functional equations with square-symmetric operation, Commun. Korean Math. Soc. 16 (2001), no. 2, 211-223
  13. E. H. Lee, On the solution and stability of the quadratic type functional equations, Commun. Korean Math. Soc. 19 (2004), no. 3, 477-493 https://doi.org/10.4134/CKMS.2004.19.3.477
  14. Y. W. Lee, On the stability of mappings in Banach algebras, Commun. Korean Math. Soc. 18 (2003), no. 2, 235-242 https://doi.org/10.4134/CKMS.2003.18.2.235
  15. Y. W. Lee and B. M. Choi, Stability of a Beta-type functional equation with a restricted domain, Commun. Korean Math. Soc. 19 (2004), no. 4, 701-713 https://doi.org/10.4134/CKMS.2004.19.4.701
  16. Gy. Maksa and P. Volkmann, Characterization of group homomorphisms having values in an inner product space, Publ. Math. Debrecen 56 (2000), 197-200
  17. C. Park, Generalized Hyers-Ulam-Rassias stability of n-sesquilinear-quadratic mappings on Banach modules over C*-algebras, J. Comput. Appl. Math. 180 (2005), 279-291 https://doi.org/10.1016/j.cam.2004.11.001
  18. C. Park, Y. Cho, and M. Han, Functional inequalities associated with Jordan-von Neumann type additive functional equations, J. Inequal. Appl. 2007, 41820 (2007), 1-13 https://doi.org/10.1155/2007/41820
  19. C. G. Park and W. G. Park, On the stability of the Jensen's equation in a Hilbert module, Bull. Korean Math. Soc. 40 (2003), no. 1, 53-61 https://doi.org/10.4134/BKMS.2003.40.1.053
  20. C. Park and Th. M. Rassias, On a generalized Trif's mapping in Banach modules over a C*-algebra, J. Korean Math. Soc. 43 (2006), no. 2, 323-356 https://doi.org/10.4134/JKMS.2006.43.2.323
  21. K. H. Park and Y. S Jung, The stability of a functional inequality with the fixed point alternative, Commun. Korean Math. Soc. 19 (2004), no. 2, 253-266 https://doi.org/10.4134/CKMS.2004.19.2.253
  22. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300 https://doi.org/10.2307/2042795
  23. J. Ratz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math. 66 (2003), 191-200 https://doi.org/10.1007/s00010-003-2684-8
  24. T. Trif, Hyers-Ulam-Rassias stability of a quadratic functional equation, Bull. Korean Math. Soc. 40 (2003), no. 2, 253-267 https://doi.org/10.4134/BKMS.2003.40.2.253
  25. S. M. Ulam, A Collection of the Mathematical Problems, Interscience Publ. New York, 1960
  26. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Uni. Hamburg. 27 (1992), 59-64 https://doi.org/10.1007/BF02941618
  27. W. G. Park and J. H. Bae On the stability of involutive A-quadratic mappings, Bull. Korean Math. Soc. 43 (2003), no. 4, 737-745 https://doi.org/10.4134/BKMS.2006.43.4.737