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

Gang Lu; Yuanfeng Jin; Choonkil Park; Gang Lu; Yuanfeng Jin; Choonkil Park

doi:10.3934/math.2020445

AIMS Mathematics

2020, Volume 5, Issue 6: 6949-6958. doi: 10.3934/math.2020445

Previous Article Next Article

Research article

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

Gang Lu ^{1
,
,},
Yuanfeng Jin ^{2
,
,},
Choonkil Park ^{3
,
,}

1.
Division of Foundational Teaching, Guangzhou College of Technology and Business, Guangzhou 510850, P. R. China
2.
Department of Mathematics, Yanbian University, Yanji 133001, P. R. China
3.
Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea

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.
- Hyers-Ulam stability,
- ($\alpha,\beta, \gamma$)-derivation,
- Lie $C^*$-algebra
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:

Abstract

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.

References

[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

Your name:*

Email:*
© 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)