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Conditional functional equations and orthogonal additivity

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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; x ⊥z). 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.

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Authors and Affiliations

  1. Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, I-20133, Milano, Italia

    Luigi Paganoni & Jürg Rätz

  2. Mathematisches Institut, Universität Bern, Sidlerstr. 5, CH-3012, Bern, Schweiz

    Luigi Paganoni & Jürg Rätz

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  1. Luigi Paganoni
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  2. Jürg Rätz
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Paganoni, L., Rätz, J. Conditional functional equations and orthogonal additivity. Aeq. Math. 50, 135–142 (1995). https://doi.org/10.1007/BF01831116

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