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

Abstract

Let X, Y be Banach modules over a C*-algebra and let r(1),..., r(n) is an element of (0, infinity) be given. We prove the generalized Hyers-Ulam stability of the following functional equation in Banach modules over a unital C*-algebra: [GRAPHICS] We show that if r(1) =... = r(n) = r and an odd mapping f : X -> Y satisfies the functional equation (0.1) then the odd mapping f : X -> Y is Cauchy additive. As an application, we show that every almost linear bijection h : A -> B of a unital C*-algebra A onto a unital C*-algebra B is a C*-algebra isomorphism when h((n r)(d)uy) = h((nr)(d)u)h(y) for all unitaries u is an element of A, all y is an element of A, and all d is an element of Z. 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.