We deal with functional equations of the form
f(x+y) = F(f(x),f(y))
(so called addition formulas) assuming that the given binary operation F is associative but its domain of definition is disconnected (admits "singularities"). The function
Flu,v) := (u+v)/(1+uv)
serves here as a good example; the corresponding equation characterizes the hyperbolic tangent. Our considerations may be viewed as counterparts of L. Losonczi's [4] and K. Domańska's [2] results on local solutions of the functional equation
f(F(x,y)) = f(x) + f(y)
with the same behaviour of the given associative operation F.
Our results exhibit a crucial role of 1 that turns out to be the critical value towards the range of the unknown function. What concerns the domain we admit fairly general structures (groupoids, groups, 2-divisible groups). In the case where the domain forms a group admitting subgroups of index 2 the family of solutions enlarges considerably.
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Vol. 18 (2004)
Published: 2004-09-30
10.1515/amsil
English