Generalized Cauchy difference equations. II

The main result is an improvement of previous results on the equation f(x) + f(y) - f(x + y) = g[phi f(x) + phi(y) - phi(x + y)] for a given function phi. We find its general solution assuming only continuous differentiability and local nonlinearity of phi. We also provide new results about the more general equation f(x) + f(y) - f(x + y) = g(H(x, y)) for a given function H. Previous uniqueness results required strong regularity assumptions on a particular solution f0, g0. Here we weaken the assumptions on f0, g0 considerably and find all solutions under slightly stronger regularity assumptions on H.