2 - Variable AQCQ - Functional equation

  • Authors

    • M. Arunkuma GOVERNMENT ARTS COLLEGE,TIRUVANNAMALAI-606 603,TAMILNADU,INDIA.
    • S. Hema Latha Government Arts College (For Men),
    • E. Sathya Sri Vidya Mandir Arts & Science College
    2015-05-19
    https://doi.org/10.14419/ijams.v3i1.4401
  • Additive functional equations, Quadratic functional equations, Cubic functional equations, Quartic functional equations, Mixed type functional equations, Ulam - Hyers stability, Ulam - Hyers - Rassias stability, Ulam - Gavruta - Rassias stability

  • Abstract

    In this paper, the authors obtain the general solution and generalized Ulam - Hyers stability of a 2 - variable AQCQ functional equation
    \begin{align*}
    g(x+2y, u+2v)+g(x-2y, u-2v)& = 4[g(x+y, u+v) + g(x-y, u-v)]- 6g(x,u)\notag\\
    &~~+g(2y,2v)+g(-2y,-2v)-4g(y,v)-4g(-y,-v)
    \end{align*}
    using Hyers direct method. Counter examples for non stability is also discussed.

  • References

    1. [1] J. Aczel and J. Dhombres, Functional Equations in Several Variables, Cambridge Univ, Press, 1989.

      [2] T. Aoki, On the stability of the linear transformation in Banach spaces, emph{ J. Math. Soc. Japan, 2 (1950), 64-66.

      [3] M. Arunkumar, Matina J. Rassias, Yanhui Zhang, Ulam - Hyers stability of a 2- variable AC - mixed type functional equation: direct and fixed point methods, Journal of Modern Mathematics Frontier (JMMF), 2012, Vol 1 (3), 10-26.

      [4] J.H. Bae and W.G. Park , A functional equation orginating from quadratic forms, J. Math. Anal. Appl. , 326 (2007), 1142-1148.

      [5] P.W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76-86.

      [6] S. Czerwik, Functional Equations and Inequalities in Several Variables, World Scientific, River Edge, NJ, 2002.

      [7] P. Gavruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.

      [8] M. Eshaghi Gordji, H. Khodaie, Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces, arxiv: 0812. 2939 v1 Math FA, 15 Dec 2008.

      [9] M. Eshaghi Gordji, M. B. Savadkouhi, Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces, Journal of Inequalities and Applications, doi:10.1155/2009/527462.

      [10] M. Eshaghi Gordji, M. Bavand Savadkouhi, Choonkil Park, Quadratic-Quartic Functional Equations in RN-Spaces, Journal of Inequalities and Applications, doi:10.1155/2009/868423.

      [11] M. Eshaghi Gordji, S. Zolfaghari , J. M. Rassias and M. B. Savadkouhi, Solution and Stability of a Mixed type Cubic and Quartic functional equation in Quasi-Banach spaces, Abstract and Applied Analysis, Volume 2009, Art. ID 417473, 1-14, Doi:10.1155/2009/417473.

      [12] D.H. Hyers, On the stability of the linear functional equation, Proc.Nat.Acad.Sci.,U.S.A.,27 (1941) 222-224.

      [13] D.H. Hyers, G. Isac,Th.M. Rassias, Stability of functional equations in several variables, Birkhauser, Basel, 1998.

      [14] S.M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis, Hadronic Press, Palm Harbor, 2001.

      [15] Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer Monographs in Mathematics, 2009.

      [16] L. Maligranda, A result of Tosio Aoki about a generalization of Hyers-Ulam stability of additive functions- a question of priority, Aequationes Math., 75 (2008), 289-296.

      [17] C. Park, J. R. Lee, An AQCQ-functional equation in paranormed spaces, Advances in Difference Equations, doi: 10.1186/1687-1847-2012-63.

      [18] J.M. Rassias, On approximately of approximately linear mappings by linear mappings,J. Funct. Anal. USA, 46, (1982) 126-130.

      [19] K. Ravi, M. Arunkumar and J.M. Rassias, On the Ulam stability for the orthogonally general Euler-Lagrange type functional equation, International Journal of Mathematical Sciences, Autumn 2008 Vol.3, No. 08, 36-47.

      [20] K. Ravi, J.M. Rassias, M. Arunkumar, R. Kodandan, Stability of a generalized mixed type additive, quadratic, cubic and quartic functional equation, J. Inequal. Pure Appl. Math. 10 (2009), no. 4, Article 114, 29 pp.

      [21] Th.M. Rassias, On the stability of the linear mapping in Banach spaces, Proc.Amer.Math. Soc., 72 (1978), 297-300.

      [22] Th.M. Rassias, Functional Equations, Inequalities and Applications, Kluwer Acedamic Publishers, Dordrecht, Bostan London, 2003.

      [23] S.M. Ulam, Problems in Modern Mathematics, Science Editions,Wiley, NewYork, 1964 (Chapter VI, Some Questions in Analysis: 1, Stability).

      [24] T.Z. Xu, J.M. Rassias, W.X Xu, Generalized Ulam-Hyers stability of a general mixed AQCQ-functional equation in multi-Banach spaces: a fixed point approach, Eur. J. Pure Appl. Math. 3 (2010), no. 6, 1032-1047.

      [25] T.Z. Xu, J.M. Rassias, M.J. Rassias, W.X. Xu, A fixed point approach to the stability of quintic and sextic functional equations in quasi-beta-normed spaces, J. Inequal. Appl. 2010, Art. ID 423231, 23 pp.

      [26] T.Z. Xu, J.M Rassias, W.X. Xu, A fixed point approach to the stability of a general mixed AQCQ-functional equation in non-Archimedean normed spaces, Discrete Dyn. Nat. Soc. 2010, Art. ID 812545, 24 pp.

  • Downloads

  • How to Cite

    Arunkuma, M., Hema Latha, S., & Sathya, E. (2015). 2 - Variable AQCQ - Functional equation. International Journal of Advanced Mathematical Sciences, 3(1), 65-86. https://doi.org/10.14419/ijams.v3i1.4401