aequationes mathematicae

, Volume 50, Issue 1–2, pp 135–142

Conditional functional equations and orthogonal additivity

  • Luigi Paganoni
  • Jürg Rätz
Survey Papers

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)] = 0f(x + y) − f(x) − f(y) ∈ V (a suitable subset of the range) 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, z ∈ Z; 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.

AMS (1991) subject classification

39B52 

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Copyright information

© Birkhäuser Verlag 1995

Authors and Affiliations

  • Luigi Paganoni
    • 1
    • 2
  • Jürg Rätz
    • 1
    • 2
  1. 1.Dipartimento di MatematicaUniversità di MilanoMilanoItalia
  2. 2.Mathematisches InstitutUniversität BernBernSchweiz

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