On generalized Hyers-Ulam stability of additive mapings on restricted domains of Banach spaces

Let (G, .) be a group (not necessarily Abelian) with unit e and E be a Banach space. In this paper, we show that there exist alpha(p) > 0 for any 0 < p < 1 and beta(p, epsilon), gamma(p, epsilon) > 0 for any 0 < epsilon < alpha(p), such that for any surjective map f : G -> E satisfying vertical bar parallel to f(x) + f(y)parallel to - parallel to f(xy)parallel to vertical bar <= epsilon parallel to f(x) + f(y)parallel to(p) for all x, y is an element of G, there is a unique additive T : G -> E such that parallel to f(x) - T(x)parallel to <= gamma(p, epsilon)parallel to f(x)parallel to(p) for all x is an element of G satisfying parallel to f(x)parallel to >= beta(p, epsilon). Moreover, we have lim(epsilon -> 0) gamma(p, epsilon)/epsilon < infinity.