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THE GENERALIZED HYERS-ULAM STABILITY OF QUADRATIC FUNCTIONAL EQUATION WITH AN INVOLUTION IN NON-ARCHIMEDEAN SPACES

  • Received : 2014.02.13
  • Accepted : 2014.04.07
  • Published : 2014.05.15

Abstract

In this paper, using fixed point method, we prove the Hyers-Ulam stability of the following functional equation $$(k+1)f(x+y)+f(x+{\sigma}(y))+kf({\sigma}(x)+y)-2(k+1)f(x)-2(k+1)f(y)=0$$ with an involution ${\sigma}$ for a fixed non-zero real number k with $k{\neq}-1$.

Keywords

References

  1. B. Boukhalene, E. Elqorachi, and Th. M. Rassias, On the generalized Hyers-Ulam stability of the quadratic functional equation with a general involution, Nonlinear Funct. Anal. Appl. 12 (2007), no. 2, 247-262.
  2. B. Boukhalene, E. Elqorachi, and Th. M. Rassias, On the Hyers-Ulam stability of approximately pexider mappings, Math. Ineqal. Appl. 11 (2008), 805-818.
  3. S. Czerwik, Functional equations and Inequalities in several variables, World Scientific, New Jersey, London, 2002.
  4. J. B. Diaz, Beatriz Margolis A fixed point theorem of the alternative, for contractions on a generalized complete metric space Bull. Amer. Math. Soc. 74 (1968), 305-309. https://doi.org/10.1090/S0002-9904-1968-11933-0
  5. G. L. Forti, Hyers-Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), 143-190. https://doi.org/10.1007/BF01831117
  6. D. H. Hyers, On the stability of linear functional equation, Proc. Natl. Acad. Sci. USA 27 (1941), 222-224. https://doi.org/10.1073/pnas.27.4.222
  7. D. H. Hyers, G. Isac, and T. M. Rassias, Stability of functional equations in several variables, Birkhauser, Boston, 1998.
  8. D. H. Hyers and T. M. Rassias, Approximate homomorphisms, Aequationes Math. 44 (1992), 125-153. https://doi.org/10.1007/BF01830975
  9. S. M. Jung and Z. H. Lee, A fixed point approach to the stability of quadratic functional equation with involution, Fixed Point Theory Appl. 2008.
  10. S. M. Jung, On the Hyers-Ulam stability of the functional equation that have the quadratic property, J. Math. Anal. Appl. 222 (1998), 126-137. https://doi.org/10.1006/jmaa.1998.5916
  11. F. Skof, Approssimazione di funzioni ${\delta}$-quadratic su dominio restretto, Atti. Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur. 118 (1984), 58-70.
  12. H. Stetkaer, Functional equations on abelian groups with involution, Aequationes Math. 54 (1997), 144-172. https://doi.org/10.1007/BF02755452
  13. S. M. Ulam, A collection of mathematical problems, Interscience Publ., New York, 1960.

Cited by

  1. APPROXIMATE QUADRATIC MAPPINGS IN QUASI-β-NORMED SPACES vol.28, pp.2, 2015, https://doi.org/10.14403/jcms.2015.28.2.311