Approximate norm-additive maps on Banach spaces

Let X, Y be two Banach spaces, F : X -> Y be an epsilon-norm-additive map for some epsilon >= 0, that is, vertical bar parallel to F(x) + F(y)parallel to - parallel to x + y parallel to <= epsilon, for all x, y is an element of X. In this paper, we prove that if F is surjective with F(0) = 0, then there exists a linear surjective isometry U : X -> Y such that parallel to F(x) - U(x)parallel to <= 3/2 epsilon, for all x is an element of X. The estimate 3/2 epsilon is sharp. We also approximate standard surjective epsilon-norm-additive maps between the positive cones of continuous function spaces by linear isometries within a sharp approximation error 3/2 epsilon.