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Stability of a generalized Euler-Lagrange type additive mapping and homomorphisms in \(C^{\ast}\)-algebras. (English) Zbl 1177.39038

Summary: Let \(X\), \(Y\) be Banach modules over a \(C^*\)-algebra and let \(r_1,\dots,r_n\in\mathbb R\) be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital \(C^*\)-algebra:
\[ \sum^n_{j=1} f(-r_jx_j+\sum_{1\leq i\leq n,i\neq j}r_ix_i) +2\sum^n_{i=1}r_if(x_i)=nf(\sum^n_{i=1}r_ix_i). \]
We show that if \(\sum^n_{i=1}r_i\neq 0\), \(r_i,r_j\neq 0\) for some \(1\leq i<j\leq n\) and a mapping \(f:X\to Y\) satisfies the functional equation mentioned above then the mapping \(f:X\to Y\) is Cauchy additive. As an application, we investigate homomorphisms in unital \(C^*\)-algebras.

MSC:

39B82 Stability, separation, extension, and related topics for functional equations
39B52 Functional equations for functions with more general domains and/or ranges
46L05 General theory of \(C^*\)-algebras
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