Stability of a Generalized Euler-Lagrange Type Additive Mapping and Homomorphisms in C*-Algebras

Abstract

Let X, Y be Banach modules over a C*-algebra and let r(1),..., r(n) is an element of R be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital C*-algebra: Sigma(n)(j=1) f (-r(j)x(j) + Sigma(1 <= i <= n,i not equal j) r(i)x(i)) + 2 Sigma(n)(i=1)r(i)f(x(i)) = nf (Sigma(n)(i=1) r(i)x(i)). We show that if Sigma(n)(i=1) r(i) not equal 0, r(i), r(j) not equal 0 for some 1 <= i <= j <= n and a mapping f : X -> Y satisfies the functional equation mentioned above then the mapping f : X -> Y is Cauchy additive. As an application, we investigate homomorphisms in unital C*-algebras.