Approximately Additive and Approximately Linear Mappings
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We have mentioned the concept of stability and given an example for the Cauchy functional equation (1.1). A definition of stability in the case of homomorphisms between metric groups was suggested by a problem posed by S.M. Ulam in 1940 (see Ulam (1960), p. 64). Given a group (G1, •), a metric group (G2,*) with metric d and a positive number η, suppose that there exists a positive numberε= ε(η) such that, if d(f (x • y), f (x) * f (y)) <ε for some f : G1 → G2 and all x and y in G1, then a homomorphism h: G1 → G2 exists with d(f (x), h(x)) <η for all x in G1. In this case, the equation of homomorphism h(x • y) = h(x) * h(y) is called stable. Theorem 1.1 with G1 = E1, G2 = E2 and with addition as the group operation in each case shows that Cauchy’s equation is stable by this definition with η= ε.
KeywordsBanach Space Functional Equation Additive Mapping Real Banach Space Topological Index
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