Abstract
The main result is an improvement of previous results on the equation\[f(x)+f(y)−f(x+y)=g[ϕ(x)+ϕ(y)−ϕ(x+y)]f(x)+f(y)-f(x+y)=g[\phi (x)+\phi (y)-\phi (x+y)]\]for a given functionϕ\phi. We find its general solution assuming only continuous differentiability and local nonlinearity ofϕ\phi. We also provide new results about the more general equation\[f(x)+f(y)−f(x+y)=g(H(x,y))f(x)+f(y)-f(x+y)=g(H(x,y))\]for a given functionHH. Previous uniqueness results required strong regularity assumptions on a particular solutionf0,g0f_{0},g_{0}. Here we weaken the assumptions onf0,g0f_{0},g_{0}considerably and find all solutions under slightly stronger regularity assumptions onHH.
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