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A fixed point approach to stability of cubic Lie derivatives in Banach algebras. (English) Zbl 1337.39008

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.

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

Citations:

Zbl 1323.39021
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