A Fixed Point Approach to the Stability of the Functional Equation f(x+y)=F[f(x),f(y)]

In 1940, Ulam 1 gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms: “Let G1 be a group and let G2 be a metric group with the metric d ·, · . Given ε > 0, does there exist a δ > 0 such that if a function h : G1 → G2 satisfies the inequality d h xy , h x h y < δ for all x, y ∈ G1, then there exists a homomorphism H : G1 → G2 with d h x ,H x < ε for all x ∈ G1?” The case of approximately additive functions was solved by Hyers 2 under the assumption that G1 and G2 are Banach spaces. Indeed, he proved that each solution of the inequality ‖f x y − f x − f y ‖ ≤ ε, for all x and y, can be approximated by an exact solution, say an additive function. Rassias 3 attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows: