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

Let X, Y be Banach modules over a -algebra and let be given. We prove the generalized Hyers–Ulam stability of the following functional equation in Banach modules over a unital -algebra: We show that if and an odd mapping satisfies the functional equation (0.1) then the odd mapping is Cauchy additive. As an application, we show that every almost linear bijection of a unital -algebra A onto a unital -algebra B is a -algebra isomorphism when for all unitaries , all , and all . 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.

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