On a functional equation related to distributivity of fuzzy implications

Recently in some considerations connected with the distributivity laws of fuzzy implications over triangular norms and conorms, the following functional equation appeared f(min(x + y, a)) = min(f(x) + f(y)m b). In the current paper we consider a generalized version of this equation, namely the equation f(m₁(x + y)) = m₂(f(x) + f(y)), where m₁, m₂ are functions defined on some intervals of ℝ satisfying additional assumptions. We analyze the cases when m₂ is injective and when m₂ is not injective.