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Functional equations involving certain generalized forms of the so-called 'laws of sciences' are considered. The resolution of these equations is linked to the concept of comparison meaningfulness that appears in measurement theory and dimension theory. The results obtained are stated without assuming any topological requirement. [ABSTRACT FROM AUTHOR]

The main result of the present paper is about the solutions of the functional equation F ( x + y 2) + f 1 (x) + f 2 (y) = G (g 1 (x) + g 2 (y)) , x , y ∈ I , derived originally, in a natural way, from the invariance problem of generalized weighted quasi-arithmetic means, where F , f 1 , f 2 , g 1 , g 2 : I → R and G : g 1 (I) + g 2 (I) → R are the unknown functions assumed to be continuously differentiable with 0 ∉ g 1 ′ (I) ∪ g 2 ′ (I) , and the set I stands for a nonempty open subinterval of R . In addition to these, we will also touch upon solutions not necessarily regular. More precisely, we are going to solve the above equation assuming first that F is affine on I and g 1 and g 2 are continuous functions strictly monotone in the same sense, and secondly that g 1 and g 2 are invertible affine functions with a common additive part. [ABSTRACT FROM AUTHOR]

In 1990, Benz asked whether a real additive mapping satisfying $$xf(y)=yf(x)$$ for all points ( x, y) on the unit circle must be linear. In 2005, Boros and Erdei showed that it must be so. Here we generalize the problem to a pair of additive functions f, g related by the functional equation $$xf(y)=yg(x)$$ for all points ( x, y) on a specified curve. We find that for many (but not all) types of curves this forces f and g to be equal and linear. [ABSTRACT FROM AUTHOR]