Cho, In-Goo;Koh, Hee-Jeong

  • Received : 2010.01.18
  • Published : 2011.01.31


In this paper, we investigate the stability using shadowing property in Abelian metric group and the generalized Hyers-Ulam-Rassias stability in Banach spaces of a quadratic functional equation, $f(x_1+x_2+x_3+x_4)+f(-x_1+x_2-x_3+x_4)+f(-x_1+x_2+x_3)+f(-x_2+x_3+x_4)+f(-x_3+x_4+x_1)+f(-x_4+x_1+x_2)=5{\sum\limits_{i=1}^4}f(x_i)$. Also, we study the stability using the alternative fixed point theory of the functional equation in Banach spaces.


shadowing property-stability;generalized Hyers-Ulam stability;quadratic mapping


  1. J.-H. Bae and K.-W. Jun, On the generalized Hyers-Ulam-Rassias stability of a quadratic functional equation, Bull. Korean Math. Soc. 38 (2001), no. 2, 325-336.
  2. P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math. 27 (1984), no. 1-2, 76-86.
  3. S. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59-64.
  4. J. B. Diaz and B. Margolis, A fixed point theorem of the alternative, for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 74 (1968), 305-309.
  5. Z. Gajda, On stability of additive mappings, Internat. J. Math. Math. Sci. 14 (1991), no. 3, 431-434.
  6. P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl. 184 (1994), no. 3, 431-436.
  7. D. H. Hyers, On the stability of the linear equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222-224.
  8. S.-H. Lee, H. Koh, and S.-H. Ku, Investigation of the stability via shadowing property, J. Inequal. Appl. 2009 (2009), Art. ID 156167, 12 pp.
  9. A. Najati and C. Park, Stability of a generalized Euler-Lagrange type additive mapping and homomorphisms in $C^{\ast}-algebras$, Adv. Difference Equ. 2009 (2009), Art. ID 273165, 22 pp.
  10. Th. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), no. 2, 297-300.
  11. I. A. Rus, Principles and Applications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979.
  12. F. Skof, Local properties and approximation of operators, Rend. Sem. Mat. Fis. Milano 53 (1983), 113-129.
  13. J. Tabor and J. Tabor, General stability of functional equations of linear type, J. Math. Anal. Appl. 328 (2007), no. 1, 192-200.
  14. J. Tabor, Locally expanding mappings and hyperbolicity, Topol. Methods Nonlinear Anal. 30 (2007), no. 2, 335-343.
  15. S. M. Ulam, Problems in Morden Mathematics, Wiley, New York, 1960.

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