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Conditional functional equations and orthogonal additivity. (English) Zbl 0876.39008

Summary: Some examples of classes of conditional equations coming from information theory, geometry and from the social and behavioral sciences are presented. Then the classical case of the Cauchy equation on a restricted domain Ω is extensively discussed. Some results concerning the extension of local homomorphisms and the implication “Ω-additivity implies global additivity” are illustrated. Problems concerning the equations

[cf(x+y)-af(x)-bf(y)-d][f(x+y)-f(x)-f(y)]=0,[g(x+y)-g(x)-g(y)][f(x+y)-f(x)-f(y)]=0,f(x+y)-f(x)-f(y)V(asuitablesubsetoftherange)

are presented.

The consideration of the conditional Cauchy equation is subsequently focused on the case when it makes sense to interpret Ω as a binary relation (orthogonality):

f:(X,+,)(Y,+);f(x+z)=f(x)+f(z)(x,zZ;xz)·

A brief sketch on solutions under regularity conditions is given. It is then shown that all regularity conditions can be removed. Finally, several applications (also to physics and to the actuarial sciences) are discussed. In all these cases the attention is focused on open problems and possible extensions of previous results.


MSC:
39B52Functional equations for functions with more general domains and/or ranges
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