Korean J. Math. Vol. 24 No. 4 (2016) pp.587-600
DOI: https://doi.org/10.11568/kjm.2016.24.4.587

A fixed point approach to the stability of quartic Lie $*$-derivations

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Dongseung Kang
Heejeong Koh


We obtain the general solution of the functional equation $f(ax+y)-f(x-ay)+\frac{1}{2}a(a^2+1)f(x-y)+(a^4-1)f(y)= \,\,\frac{1}{2}a(a^2+1)f(x+y)+(a^4-1)f(x)$ and prove the stability problem of the quartic Lie $*$-derivation by using a directed method and an alternative fixed point method.

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[1] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan 2 (1950), 64–66. Google Scholar

[2] N. Brillou et-Belluot, J. Brzd ek and K. Cieplin ski, Fixed point theory and the Ulam stability, Abstract and Applied Analysis 2014, Article ID 829419, 16 pages (2014). Google Scholar

[3] J. Brzd ek, L. CVadariu and K. Cieplin ski, On some recent developments in Ulam's type stability, Abstract and Applied Analysis 2012, Article ID 716936, 41 pages (2012). Google Scholar

[4] J. K. Chung and P. K. Sahoo, On the general solution of a quartic functional equation, Bulletin of the Korean Mathematical Society, 40 no.4 (2003), 565–576. Google Scholar

[5] St. Czerwik, On the stability of the quadratic mapping in normed spaces, Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. Google Scholar

[6] A. Foˇsner and M. Foˇsner, Approximate cubic Lie derivations, Abstract and Applied Analysis 2013, Article ID 425784, 5 pages (2013). Google Scholar

[7] D. H. Hyers, On the stability of the linear equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224. Google Scholar

[8] D. H. Hyers, G. Isac, and T. M. Rassias, Stability of Functional Equations in Several Variables, Birkh ̈auser, Boston, USA, 1998. Google Scholar

[9] S. Jang and C. Park, Approximate ∗-derivations and approximate quadratic ∗-derivations on C∗-algebra, J. Inequal. Appl. 2011, Articla ID 55 (2011). Google Scholar

[10] S.-M. Jung, Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis, vol. 48 of Springer Optimization and Its Applications, Springer, New York, USA, 2011. Google Scholar

[11] B. Margolis and J.B. Diaz, A fixed point theorem of the alternative for contractions on a generalized complete metric space, Bull. Amer. Math. Soc. 126, 74 (1968), 305–309. Google Scholar

[12] C. Park and A. Bodaghi, On the stability of ∗-derivations on Banach ∗-algebras, Adv. Diff. Equat. 2012 2012:138 (2012). Google Scholar

[13] J. M. Rassias, Solution of the Ulam stability problem for quartic mappings, Glasnik Matematicki Series III, 34 no. 2 (1999)243–252. Google Scholar

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

[15] I.A. Rus, Principles and Appications of Fixed Point Theory, Ed. Dacia, Cluj-Napoca, 1979 (in Romanian). Google Scholar

[16] P.K. Sahoo, A generalized cubic functional equation, Acta Math. Sinica 21 no. 5 (2005), 1159–1166. Google Scholar

[17] S. M. Ulam, Problems in Morden Mathematics, Wiley, New York, USA, 1960. Google Scholar

[18] T.Z. Xu, J.M. Rassias and W.X. Xu, A generalized mixed quadratic-quartic functional equation, Bull. Malaysian Math. Scien. Soc. 35 no. 3 (2012), 633–649. Google Scholar

[19] S.Y. Yang, A. Bodaghi, K.A.M. Atan, Approximate cubic *-derivations on Banach *-algebra, Abstract and Applied Analysis, 2012, Article ID 684179, 12 pages (2012). Google Scholar