On two functional equations connected with distributivity of fuzzy implications (Q2822640)

The paper deals with the functional equation NEWLINE\[NEWLINE h(xg(x)) = h(x) + h(xy),\leqno(*)NEWLINE\]NEWLINEwhere \(g, h : (0,\infty) \to (0,\infty)\) are unknown functions. The main result has the following form (Theorem 5): Let \(h\) be a bijection in \((0,\infty)\). The functions \(g,h\) satisfy the Equation \((*)\) if and only if there exist a constant \(M>0\) and a bijection \(c : (0,\infty) \to (0,\infty)\) such that NEWLINE\[NEWLINEc(xy) = c(x)c(y) \text{ and } h(x) = M c(x), \quad g(x) = c^{-1}(1 + c(x)).\leqno(**)NEWLINE\]NEWLINE If additionally the solution \(h\) is continuous, then there exists a constant \(p \neq 0\) such that \(c(x) = x^p\). In the case of a continuous solution \(h\), the above result has an application for a characterization of distributivity of fuzzy implications with respect to fuzzy disjunctions (Theorem 6), where notions of fuzzy logic are based on \textit{M. Baczyński} and \textit{B. Jayaram} [Fuzzy implications. Berlin: Springer (2008; Zbl 1147.03012)]. However, Equation \((*)\) has infinitely many discontinuous (even non-measurable) solutions \(h\) (Remark 4). This can be seen from the theory of Cauchy's functional equation \((**)\) (cf. [\textit{M. Kuczma}, An introduction to the theory of functional equations and inequalities. Cauchy's equation and Jensen's inequality. Warszawa-Kraków-Katowice: Państwowe Wydawnictwo Naukowe (1985; Zbl 0555.39004)]).