On the Hyers-Ulam Stability of ψ-Additive Mappings

Let E1 be a real normed vector space and E2 a real Banach space. S. M. Ulam posed the problem: When does a linear mapping near an approximately additive mapping ?: E1 ? E2 exist? We give a new generalized solution to Ulam?s problem for ?-additive mappings. Some relations with the asymptotic differentiability are also indicated.