Research article

Generalized ($\alpha,\beta, \gamma$)-derivations on Lie $C^*$-algebras

  • Received: 06 March 2020 Accepted: 30 August 2020 Published: 08 September 2020
  • MSC : 17B05, 17B40, 39B62, 39B52, 47H10, 46B25

  • The Hyers-Ulam stability of ($\alpha, \beta, \gamma$)-derivations on Lie $C^*$-algebras is discussed by following functional inequality $ \begin{eqnarray*} f(ax+by)+f(ax-by) = 2f(ax)+bf(y)+bf(-y), \end{eqnarray*} $ where $a, b$ are nonzero fixed complex numbers.

    Citation: Gang Lu, Yuanfeng Jin, Choonkil Park. Generalized ($\alpha,\beta, \gamma$)-derivations on Lie $C^*$-algebras[J]. AIMS Mathematics, 2020, 5(6): 6949-6958. doi: 10.3934/math.2020445

    Related Papers:

  • The Hyers-Ulam stability of ($\alpha, \beta, \gamma$)-derivations on Lie $C^*$-algebras is discussed by following functional inequality $ \begin{eqnarray*} f(ax+by)+f(ax-by) = 2f(ax)+bf(y)+bf(-y), \end{eqnarray*} $ where $a, b$ are nonzero fixed complex numbers.


    加载中


    [1] L. Aiemsomboon, W. Sintunavarat, Stability of the generalized logarithmic functional equations arising from fixed point theory, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math., 112 (2018), 229-238. doi: 10.1007/s13398-017-0375-x
    [2] T. Aoki, On the stability of the linear transformation in Banach spaces, J. Math. Soc. Japan, 2 (1950), 64-66. doi: 10.2969/jmsj/00210064
    [3] J. Aczel, J. Dhombres, Functional equations in several variables, Cambridge Univ. Press, Cambridge, 1989.
    [4] L. Cădariu, V. Radu, Fixed points and the stability of Jensen's functional equation, J. Inequal. Pure Appl. Math., 4 (2003), 4.
    [5] K. Ciepliński, Applications of fixed point theorems to the Hyers-Ulam stability of functional equations: A survey, Ann. Funct. Anal., 3 (2012), 151-164.
    [6] P. W. Cholewa, Remarks on the stability of functional equations, Aequationes Math., 27 (1984), 76-86.
    [7] Y. Cho, C. Park, R. Saadati, Functional inequalities in non-Archimedean Banach spaces, Appl. Math. Lett., 23 (2010), 1238-1242. doi: 10.1016/j.aml.2010.06.005
    [8] Y. Cho, R. Saadati, J. Vahidi, Approximation of homomorphisms and derivations on nonArchimedean Lie $C^*$-algebras via fixed point method, Discrete Dyn. Nature Soc., 2012 (2012), 373904.
    [9] Y. Cho, R. Saadati, Y. Yang, Approximation of homomorphisms and derivations on Lie $C^*$-algebras via fixed point method, J. Inequal. Appl., 2013 (2013), 415.
    [10] J. B. Diaz, 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.
    [11] H. Drygas, Quasi-inner products and their applications. Advances in Multivariate Statistical Analyis, Theory Decis. Lib Ser. B: Math. Statist. Methods, Reidel, Dordrecht, (1987), 13-30.
    [12] A. Ebadian, N. Ghobadipour, T. M. Rassias, et al. Functional inequalities associated with Cauchy additive functional equation in non-Archimedean spaces, Discrete Dyn. Nat. Soc., 2011 (2011), 929824.
    [13] B. R. Ebanks, P. L. Kannappan, P. K. Sahoo, A common generaliztion of functional equations charactering normed and quasi-inner-product spaces, Canad. Math. Bull., 35 (1992), 321-327. doi: 10.4153/CMB-1992-044-6
    [14] M. E. Gordji, Z. Alizadeh, H. Khodaei, et al. On approximate homomorphisms: A fixed point approach, Math. Sci., 6 (2012), 59.
    [15] M. E. Gordji, H. Khodaei, A fixed point technique for investigating the stability of (α, β, γ)- derivations on Lie $C^*$ algebras, Nonlinear Anal., 76 (2013), 52-57.
    [16] W. Fechner, Stability of a functional inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math., 71 (2006), 149-161. doi: 10.1007/s00010-005-2775-9
    [17] P. Gǎvruta, A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings, J. Math. Anal. Appl., 184 (1994), 431-436.
    [18] A. Gilányi, Einezur Parallelogrammgleichung äquivalente Ungleichung, Aequationes Math., 62 (2001), 303-309. doi: 10.1007/PL00000156
    [19] A. Gilányi, On a problem by K. Nikodem, Math. Inequal. Appl., 5 (2002), 707-710.
    [20] D. H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. USA, 27 (1941), 222-224.
    [21] J. I. Kang, R. Saadati, Approximation of homomorphisms and derivations on non-Archimedean random Lie $C^*$-algebras via fixed point method, J. Inequal. Appl., 2012 (2012), 251.
    [22] S. Kim, G. Kim, S. Kim, On the stability of (α, β, γ)-derivations and Lie $C^*$-algebra homomorphisms on Lie $C^*$-algebras: A fixed points method, Math. Probl. Eng., 2013 (2013). Available from: https://doi.org/10.1155/2013/954749.
    [23] G. Lu, Q. Liu, Y. Jin, et al. 3-Variable Jensen ρ-functional equations, J. Nonlinear Sci. Appl., 9 (2016), 5995-6003. doi: 10.22436/jnsa.009.12.07
    [24] G. Lu, C. Park, Hyers-Ulam stability of additive set-valued functional equations, Appl. Math. Lett., 24 (2011), 1312-1316. doi: 10.1016/j.aml.2011.02.024
    [25] G. Lu, C. Park, Hyers-Ulam stability of general Jensen-type mappings in Banach algebras, Results Math., 66 (2014), 385-404. doi: 10.1007/s00025-014-0383-5
    [26] W. A. J. Luxemburg, On the convergence of successive approximations in the theory of ordinary differential equations II, Indag. Math., 20 (1958), 540-546.
    [27] M. Montigny, J. Patera, Discrete and continuous graded contractions of Lie algebras and superalgebras, J. Phys. A, 24 (1991), 525-547. doi: 10.1088/0305-4470/24/3/012
    [28] P. Novotńy, J. Hrivńak, On (α, β, γ)-derivations of Lie algebras and corresponding invariant functions, J. Geom. Phys., 58 (2008), 208-217.
    [29] C. Park, Lie *-homomorphisms between Lie $C^*$-algebras and Lie *-derivations on Lie $C^*$-algebras, J. Math. Anal. Appl., 293 (2004), 419-434. doi: 10.1016/j.jmaa.2003.10.051
    [30] C. Park, Homomorphisms between Lie $JC^*$-algebras and Cauchy-Rassias stability of Lie $JC^*$- algebra derivations, J. Lie Theory, 15 (2005), 393-414.
    [31] C. Park, Additive ρ-functional inequalities and equations, J. Math. Inequal. 9 (2015), 17-26.
    [32] C. Park, Additive ρ-functional inequalities in non-Archimedean normed spaces, J. Math. Inequal., 9 (2015), 397-407.
    [33] C. Park, Y. Cho, M. Han, Functional inequalities associated with Jordan-von-Neumann-type additive functional equations, J. Inequal. Appl., 2007 (2007), 1-13.
    [34] C. Park, J. C. Hou, S. Q. Oh, Homomorphisms between Lie $JC^*$-algebras Lie $C^*$-algebra, Acta Math. Sinica, 21 (2005), 1391-1398. doi: 10.1007/s10114-005-0629-y
    [35] T. M. Rassias, On the stability of the linear mapping in Banach spaces, Proc. Am. Math. Soc., 72 (1978), 297-300.
    [36] J. Rätz, On inequalities associated with the Jordan-von Neumann functional equation, Aequationes Math., 66 (2003), 191-200. doi: 10.1007/s00010-003-2684-8
    [37] R. Saadati, T. M. Rassias, Y. Cho, Approximate (α, β, δ)-derivation on random Lie $C^*$-algebras, RACSAM, 109 (2015), 1-10.
    [38] S. M. Ulam, Problems in Modern Mathematics, Chapter VI, Science Ed., Wiley, New York, 1960.
    [39] Z. Wang, Stability of two types of cubic fuzzy set-valued functional equations, Results Math., 70 (2016), 1-14. doi: 10.1007/s00025-015-0457-z
    [40] T. Z. Xu, J. M. Rassias, W. X. Xu, A fixed point approach to the stability of a general mixed additive-cubic functional equation in quasi fuzzy normed spaces, Int. J. Phys. Sci., 6 (2011), 313- 324.
    [41] K. Zhen, Y. Zhang, On (α, β, γ)-Derivations of Lie superalgebras, Int. J. Geom. Methods Mod. Phys., 10 (2013), 1-18.
  • Reader Comments
  • © 2020 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(2958) PDF downloads(116) Cited by(0)

Article outline

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog