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Additional stability results for quartic Lie \(\ast\)-derivations. (English) Zbl 1429.39018

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.

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
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