Kim, Hark-Mahn; Hong, Young Soon Additional stability results for quartic Lie \(\ast\)-derivations. (English) Zbl 1429.39018 Nonlinear Funct. Anal. Appl. 24, No. 3, 583-593 (2019). Let \(A\) be a complex normed \(*\)-algebra and \(M\) be a Banach \(A\)-bimodule. The mapping \(f:A\to M\) is called a quartic Lie \(*\)-derivation if it satisfies the following axioms:(i) \(f(\mu x)=\mu^4f(x)\),(ii) \(f(xy)=f(x)y^4+x^4f(y)\),(iii) \(f([x, y])=[f(x), y^4]+[x^4, f(y)]\),(iv) \(f(x^*)=f(x)^*\),for each \(x,y\in A\), \(\mu \in \mathbb{C}\), where \([x, y]=xy-yx\).The authors investigate some stability results and obtain a stability theorem of the following functional equation \[ f(ma+b)-f(a-mb)+\frac{1}{2}m(m^2+1)f(a-b)+(m^4-1)f(b) =\frac{1}{2}m(m^2+1)f(a+b)+(m^4-1)f(a) \] associated with quartic Lie \(*\)-derivation. They use a direct method and an alternative fixed point method. Reviewer: Maryam Amyari (Mashhad) Cited in 1 Document MSC: 39B52 Functional equations for functions with more general domains and/or ranges 39B72 Systems of functional equations and inequalities 16W25 Derivations, actions of Lie algebras 39B82 Stability, separation, extension, and related topics for functional equations Keywords:generalized Hyers-Ulam stability; quartic Lie \(\ast\)-derivations; quartic homogeneous mappings PDFBibTeX XMLCite \textit{H.-M. Kim} and \textit{Y. S. Hong}, Nonlinear Funct. Anal. Appl. 24, No. 3, 583--593 (2019; Zbl 1429.39018) Full Text: Link