Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ON THE HYERS-ULAM STABILITY OF A GENERALIZED QUADRATIC AND ADDITIVE FUNCTIONAL EQUATION

  • JUN, KIL-WOUNG (Department of Mathematics, Chung-nam National University) ;
  • KIM, HARK-MAHN (Department of Mathematics, Chung-nam National University)
  • Published : 2005.02.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we obtain the general solution of a gen-eralized quadratic and additive type functional equation f(x + ay) + af(x - y) = f(x - ay) + af(x + y) for any integer a with a $\neq$ -1. 0, 1 in the class of functions between real vector spaces and investigate the generalized Hyers- Ulam stability problem for the equation.

Keywords

References

  1. J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ. Press, 1989
  2. J. H. Bae and K. W. Jun, On the generalized Hyers-Ulam-Rassias stability of an n-dimensional quadratic functional equation, J. Math. Anal. Appl. 258 (2001), 183-193 https://doi.org/10.1006/jmaa.2000.7372
  3. J. Baker, The stability of the cosine equation, Proc. Amer. Math. Soc. 80 (1980), 411-416
  4. I. S. Chang and H. M. Kim, On the Hyers-Ulam stability of quadratic functional equations, J. Inequal. Appl. 3 (2002), no. 3
  5. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64 https://doi.org/10.1007/BF02941618
  6. P. Giivruta, A generalization of the Hyers- Ulam-Rassias Stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), 431-436 https://doi.org/10.1006/jmaa.1994.1211
  7. D. H. Hyers, On the stability of the linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222-224
  8. D. H. Hyers, G. Isac, and Th. M. Rassias, 'Stability of FUnctional Equations in Several Variables', Birkhauser, Basel, 1998
  9. D. H. Hyers, On the asymptoticity aspect of Hyers- Ulam stability of mappings, Proc. Amer. Math. Soc. 126 (1998), 425-430
  10. D. H. Hyers and Th. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125-153 https://doi.org/10.1007/BF01830975
  11. K. W. Jun and H. M. Kim, Remarks on the stability of additive functional equation, Bull. Korean Math. Soc. 38 (2001), 679-687
  12. K. W. Jun and Y. H. Lee, On the Hyers-Ulam-Rassias stability of a generalized quadratic equation, Bull. Korean Math. Soc. 38 (2001), 261-272
  13. K. W. Jun, On the Hyers- Ulam-Rassias stability of a pexiderized quadratic inequality, Math. Inequal. Appl. 4 (2001), no. 1, 93-118
  14. S. -M. Jung, On the Hyers- Ulam stability of the functional equations that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137 https://doi.org/10.1006/jmaa.1998.5916
  15. Th. M. Rassias, Inner product spaces and applications, Nongman, 1997
  16. Th. M. Rassias, On the stability of functional equations in Banach spaces, J. Math. Anal. Appl. 251 (2000), 264-284 https://doi.org/10.1006/jmaa.2000.7046
  17. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300
  18. S. M. Ulam, Problems in Modern Mathematics, Chap. VI, Science ed. Wiley, New York, 1964

Cited by

  1. Stability of a Bi-Additive Functional Equation in Banach Modules Over aC⋆-Algebra vol.2012, 2012, https://doi.org/10.1155/2012/835893
  2. Stability of Pexiderized quadratic functional equation in intuitionistic fuzzy normed space vol.235, pp.8, 2011, https://doi.org/10.1016/j.cam.2010.10.010
  3. On the generalized A-quadratic mappings associated with the variance of a discrete-type distribution vol.62, pp.6, 2005, https://doi.org/10.1016/j.na.2005.04.019
  4. On a direct method for proving the Hyers–Ulam stability of functional equations vol.372, pp.1, 2010, https://doi.org/10.1016/j.jmaa.2010.06.056
  5. A Fixed Point Approach to the Stability of a Generalized Quadratic and Additive Functional Equation vol.53, pp.2, 2013, https://doi.org/10.5666/KMJ.2013.53.2.219
  6. Stability of Pexiderized Quadratic Functional Equation in Random 2-Normed Spaces vol.2015, 2015, https://doi.org/10.1155/2015/828967
  7. A fixed point approach to the stability of a generalized Apollonius type quadratic functional equation vol.31, pp.4, 2011, https://doi.org/10.1016/S0252-9602(11)60341-X
  8. A general theorem on the stability of a class of functional equations including quadratic-additive functional equations vol.5, pp.1, 2016, https://doi.org/10.1186/s40064-016-1771-y
  9. Elementary remarks on Ulam–Hyers stability of linear functional equations vol.328, pp.1, 2007, https://doi.org/10.1016/j.jmaa.2006.04.079
  10. Generalized Hyers-Ulam Stability of the Pexiderized Cauchy Functional Equation in Non-Archimedean Spaces vol.2011, 2011, https://doi.org/10.1155/2011/309026
  11. Hyers–Ulam stability of a generalized Apollonius type quadratic mapping vol.322, pp.1, 2006, https://doi.org/10.1016/j.jmaa.2005.09.027
  12. ON THE STABILITY OF THE GENERALIZED QUADRATIC AND ADDITIVE FUNCTIONAL EQUATION IN RANDOM NORMED SPACES VIA FIXED POINT METHOD vol.19, pp.4, 2011, https://doi.org/10.11568/kjm.2011.19.4.437
  13. A FIXED POINT APPROACH TO GENERALIZED STABILITY OF A MIXED TYPE FUNCTIONAL EQUATION IN RANDOM NORMED SPACES vol.32, pp.1, 2010, https://doi.org/10.5831/HMJ.2010.32.1.029
  14. –additive functional equation vol.21, pp.1, 2018, https://doi.org/10.1080/09720502.2015.1086113