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Approximation of a generalized Euler-Lagrange type additive mapping on Lie $$C^\ast$$-algebras. (English) Zbl 1361.39016
Summary: Using fixed point method, we prove some new stability results for Lie $$(\alpha,\beta,\gamma)$$-derivations and Lie $$C^\ast$$-algebra homomorphisms on Lie $$C^\ast$$-algebras associated with the Euler-Lagrange type additive functional equation
$\sum\limits_{j=1}^n f\left(-r_jx_j+\sum\limits_{1\leq i\leq n, i\neq j} r_ix_i\right)+2\sum\limits_{i=1}^n r_if(x_i)=nf\left(\sum\limits_{i=1}^n r_ix_i\right)$
where $$r_1,\dots,r_n\in\mathbb{R}$$ are given and $$r_i,r_j\neq 0$$ for some $$1\leq i<j\leq n$$.
##### MSC:
 39B82 Stability, separation, extension, and related topics for functional equations 39B52 Functional equations for functions with more general domains and/or ranges 46L57 Derivations, dissipations and positive semigroups in $$C^*$$-algebras 46L05 General theory of $$C^*$$-algebras 16W25 Derivations, actions of Lie algebras
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