On some equation related to the distributivity laws of fuzzy implications. Jensen equation extended to the infinity

Recently, in several articles related to 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), b), where a, b are finite or infinite nonnegative constants. In our earlier papers we have considered a generalized version of this equation, i.e., the equation f(m1(x+y)) =m2(f(x) +f(y)). Firstly, we analyzedthe situation when both functions m1, m2 are defined on some finite intervals of R. We also investigated the situation when both functions m1, m2 have finite or infinite domains and codomains, but they satisfy several additional assumptions. In this article we consider the above equation when m1, m2 are defined on some finite or infinite intervals and satisfy only one additional assumption: m2 is injective. Our proofs are based on the Jensen equation, therefore we also present the detailed analysisof this functional equation when the domain or codomain are extended to the infinity and/or are bounded.