Hyers-Ulam-Rassias stability of generalized derivations

One of the interesting questions in the theory of functional equations concerning the problem of the stability of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to an exact solution of the given functional equation? The first stability problem was raised by Ulam during his talk at the University of Wisconsin in 1940 [18]. Given a group G1, a metric group (G2,d), and a positive number ε, does there exist a δ > 0 such that if a mapping f : G1 → G2 satisfies the inequality d( f (xy), f (x) f (y)) < δ for all x, y ∈G1, then there exists a homomorphismT :G1 →G2 such that d( f (x),T(x)) < ε for all x ∈G1? Ulam’s problem was partially solved by Hyers in 1941 in the context of Banach spaces with δ = ε as shown below [7]. Suppose that E1, E2 are Banach spaces and f : E1 → E2 is a mapping for which there exists ε > 0 such that ‖ f (x+ y)− f (x)− f (y)‖ < ε for all x, y ∈ E1. Then there is a unique additive mapping T : E1 → E2 defined by Tx = limn→∞( f (2nx)/2n) such that ‖ f (x)− T(x)‖ < ε for all x ∈ E1. Now assume that E1 and E2 are real normed spaces with E2 complete, f : E1 → E2 is a mapping such that for each fixed x ∈ E1 the mapping t → f (tx) is continuous on R, and that there exist ε ≥ 0 and p = 1 such that