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

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

Dongseung Kang, Heejeong Koh

Abstract


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.

Keywords


Hyers-Ulam stability; quartic mapping; Lie ∗-derivation; Banach ∗-algebra; fixed point alternative.

Subject classification

39B55, 39B72, 47B47, 47H10.

Sponsor(s)



Full Text:

PDF

References


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

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)

J. Brzd ̧ek, L. Cˇadariu 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)

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)

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

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

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

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

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

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)

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)

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

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

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

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

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

S. M. Ulam, Problems in Morden Mathematics, Wiley, New York, USA, 1960. (Google Scholar)

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)

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)


Refbacks

  • There are currently no refbacks.


ISSN: 1976-8605 (Print), 2288-1433 (Online)

Copyright(c) 2013 By The Kangwon-Kyungki Mathematical Society, Department of Mathematics, Kangwon National University Chuncheon 21341, Korea Fax: +82-33-259-5662 E-mail: kkms@kangwon.ac.kr