Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping.

Let X, Y be Banach modules over a [image omitted] -algebra and let [image omitted] be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital [image omitted] -algebra:[image omitted] We show that if [image omitted] and an odd mapping [image omitted] satisfies the functional equation (0.1) then the odd mapping [image omitted] is Cauchy additive. As an application, we show that every almost linear bijection [image omitted] of a unital [image omitted] -algebra A onto a unital [image omitted] -algebra B is a [image omitted] -algebra isomorphism when [image omitted] for all unitaries [image omitted] , all [image omitted] , and all [image omitted] . The concept of generalized Hyers-Ulam stability originated from Th.M. Rassias' stability Theorem that appeared in his paper: On the stability of the linear mapping in Banach spaces, Proc. Amer. Math. Soc. 72 (1978), 297-300. [ABSTRACT FROM AUTHOR]