Research Article Functional Inequalities Associated with Jordan-von Neumann-Type Additive Functional Equations

Ulam [1] gave a talk before theMathematics Club of the University ofWisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms. We are given a group G and a metric group G with metric ρ( , ). Given > 0, does there exist a δ > 0 such that if f :G G satisfies ρ( f (xy), f (x) f (y)) < δ for all x, y G, then a homomorphism h :G G exists with ρ( f (x),h(x)) < for all x G? Hyers [2] considered the case of approximately additive mappings f : E E , where E and E are Banach spaces and f satisfies Hyers inequality