We show that any quasi-arithmetic mean $A_{φ}$ and any non-quasi-arithmetic mean M (reasonably regular) are inconsistent in the sense that the only solutions f of both equations $f(M(x,y)) = A_{φ}(f(x),f(y))$ and $f(A_{φ}(x,y)) = M(f(x),f(y))$ are the constant ones.
Although, in general, a straightforward generalization of the Lagrange mean value theorem for vector valued mappings fails to hold we will look for what can be salvaged in that situation. In particular, we deal with Sanderson's and McLeod's type results of that kind (see [9] and [7], respectively). Moreover, we examine mappings with a prescribed intermediate point in the spirit of the celebrated Acz\'el's theorem characterizing polynomials of degree at most 2 (cf. [1]).
The distributivity law for a fuzzy implication \(I\colon [0,1]^2 \to [0,1]\) with respect to a fuzzy disjunction \(S\colon [0,1]^2 \to [0,1]\) states that the functional equation \( I(x,S(y,z))=S(I(x,y),I(x,z)) \) is satisfied for all pairs \((x,y)\) from the unit square. To compare some results obtained while solving this equation in various classes of fuzzy implications, Wanda Niemyska has reduced the problem to the study of the following two functional equations: \( h(\min(xg(y),1)) = \min(h(x)+ h(xy),1)\), \(x \in (0,1)\), \(y \in (0,1]\), and \( h(xg(y)) = h(x)+ h(xy)\), \(x,y \in (0, \infty)\), in the class of increasing bijections \(h\colon [0,1] \to [0,1]\) with an increasing function \(g\colon (0,1] \to [1, \infty)\) and in the class of monotonic bijections \(h\colon (0, \infty) \to (0, \infty)\) with a function \(g\colon (0, \infty) \to (0, \infty)\), respectively. A description of solutions in more general classes of functions (including nonmeasurable ones) is presented.
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