STABILITY OF A GENERALIZED EULER-LAGRANGE TYPE ADDITIVE MAPPING AND HOMOMORPHISMS IN C∗-ALGEBRAS II

Let X;Y be Banach modules over a C∗-algebra and let r1,...,rn∈R be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital C∗-algebra: n∑j=1 f(12∑ 1≤i≤n;i≠j rixi−12rjxj)+n∑i=1 rif(xi)=nf(12n∑i=1 rixi)(0.1) We show that if ∑ni=1ri≠0;ri≠0;rj≠0 for some 1≤i j≤n and a mapping f:X→Y satisfies the functional equation (0.1) then the mapping f:X→Y is additive. As an application, we investigate homomorphisms in unital C∗-algebras.