Kim, Seong Sik; Rassias, John Michael; Abdou, Afrah A. N.; Cho, Yeol Je A fixed point approach to stability of cubic Lie derivatives in Banach algebras. (English) Zbl 1337.39008 J. Comput. Anal. Appl. 19, No. 2, 378-388 (2015). The authors deal with a system of “cubic” functional equations and various stability questions related to that system. The equations of the system are (1) \(f (xy) = x^3 f (y) + f (x)y^3\) and (2) \(f (x + ky) - kf (x + y) + kf (x + y) - f (x - ky) = 2k(k^2 - 1)f (y)\) and the framework are Banach algebras. Moreover, \(k\) is an integer not less than \(3\). The methods seem to be standard ones, but the reviewer has some doubts with respect to certain arguments. The authors use that every solution \(f : \mathbb R \to \mathbb R\) of (2) is of the form \(f (t) = t^3 f (1)\). It seems that this result is not contained in the paper itself nor in its references. But it is true as can be derived from J. Schwaiger [Aequationes Math. 89, No. 1, 23–40 (2015; Zbl 1323.39021)] either as a special case or by using the methods described there. Reviewer: Jens Schwaiger (Graz) MSC: 39B82 Stability, separation, extension, and related topics for functional equations 39B72 Systems of functional equations and inequalities 39B52 Functional equations for functions with more general domains and/or ranges 46L57 Derivations, dissipations and positive semigroups in \(C^*\)-algebras Keywords:cubic Lie derivation; stability; superstability; hyperstability; Banach algebra; fixed point method; cubic functional equation Citations:Zbl 1323.39021 PDFBibTeX XMLCite \textit{S. S. Kim} et al., J. Comput. Anal. Appl. 19, No. 2, 378--388 (2015; Zbl 1337.39008)