# Conditional functional equations and orthogonal additivity

Chapter

## 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 (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, 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.

## Keywords

Functional Equation Additive Mapping Aequationes Math Restricted Domain Cauchy Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. AczéL, J., A short course on functional equations. D. Reidel Publ. Co., Dordrecht, 1987.Google Scholar
2. Aczél J., Some good and bad characters I have known and where they led. (Harmonic analysis and functional equations). [CMS Conf. Proc., Vol. 1]. AMS, Providence, RI, 1981, p. 184.Google Scholar
3. Baron, K. and Kannappan, PL., On the Pexider difference. Fund. Math. 134 (1990), 247–254.
4. Baron, K. and Volkmann, P., On the Cauchy equation modulo Z. Fund. Math. 131 (1988), 143–148.
5. Baron, K. and Volkmann, P., On a theorem of van der Corput. Abh. Math. Sem. Univ. Hamburg 61 (1991), 189–195.
6. Benz, W., Geometrische Transformationen, unter besonderer Berücksichtigung der Lorentztransformationen. BI, Mannheim—Leipzig—Wien—Zürich, 1992.
7. Borelli Forti, C., Condizioni di ridondanza per l’equazione funzionale f(k(t) + h(t)) = f(K(t)) +f(h(t)). Stochastica 11 (1987), 93–105.
8. Borelli Forti, C., Solutions of a non-homogeneous Cauchy equation. Radovi Mat. 5 (1989), 213–222.
9. Borelli Forti, C. and Forti, G. L., On an alternative functional equation in Rn. In F. A. N.: Functional analysis, approximation theory and numerical analysis. World Scientific Publ. Co., Singapore, Singa-pore, 1994, pp. 33–44.Google Scholar
10. Day, M. M., Some characterizations of inner-product spaces. Trans. Amer. M.th. Soc. 62 (1947), 320–337.
11. Dhombres, J., Some aspects of functional equations. Section 4.9. Chulalongkorn University Press, Bangkok, 1979.Google Scholar
12. Drljevii, F., On a functional which is quadratic on A-orthogonal vectors. Publ. Inst. Math. (Beograd) (N.S.) 54 (1986), 63–71.Google Scholar
13. Fenyö, I., Osservazioni su alcuni teoremi di D. H. Hyers. Istit. Lombardo Accad. Sci. Lett. Rend. A 114 (1980),235–242.
14. Fenyö, I. and Rusconi, D., Sulle distribuzioni the soddisfano una equazione funzionale. Rend. Sem. Mat. Univ. Politec. Torino 39 (1981), 67–76.
15. Fenyö, I. and PaganonI, L., Su una equazione funzionale proveniente dalla teoria delle funzioni ellittiche jacobiane. Rend. Mat. Appl. (7) 5 (1985), 319–324.Google Scholar
16. Fenyö, I. and PaganonI, L., A functional equation which characterizes the jacobian sn(z, k) functions. Rend. Mat. Appl. (7) 5 (1985), 387–392.Google Scholar
17. Fochj, M., Functional equations in A-orthogonal vectors. Aequationes Math. 38 (1989), 28–40.
18. Forti, G. L., La soluzione generale dell’equazione funzionale {cf(x + y) — af(x) — bf(y) — d} {f(x + y) — f(x)-f(y)} = 0. Matematiche (Catania) 34 (1979), 219–242.
19. Forti, G. L., Redundancy conditions for the functional equation f(x + h(x)) =f(x)+f(h(x)). Z. Anal. Anwendungen 3 (1984), 549–554.
20. Forti, G. L., The stability of homomorphisms and amenability, with applications to functional equations. Abh. Math. Sem. Univ. Hamburg 57 (1987), 215–226.
21. Forti, G. L. and Paganoni, L., A method for solving a conditional Cauchy equation on abelian groups. Ann. Mat. Pura Appl. (4) 127 (1981), 79–99.
22. Forti, G. L. and Paganoni, L., Q-additive functions on topological groups. In Constantin Carathéodory: an international tribute. World Scientific Publ. Co., Singapore, 1990, 312–330.Google Scholar
23. Forti, G. L. and Paganoni, L., On an alternative Cauchy equation in two unknown functions. Some classes of solutions. Aequationes Math. 42 (1991), 271–295.
24. Ger, R., On a method of solving of conditional Cauchy equations. Univ. Beograd. Publ. Elektrotechn. Fak. Ser. Mat. Fiz. No. 544–576 (1976), 159–165.
25. Ger, R., Almost additive functions on semigroups and a functional equation. Publ. Math. Debrecen 26 (1979), 219–228.
26. Gudder, S. and Strawther, D., Orthogonally additive and orthogonally increasing functions on vector spaces. Pacific J. Math. 58 (1975), 427–436.
27. Gudder, S. and Strawther, D., A converse of Pythagoras’ theorem. Amer. Math. Monthly 84 (1977), 551–553.
28. Hednen, B. and Goovaerts, M. J., Additivity and premium calculation principles. Blätter Deutsch. Ges. Versich. Math. 17 (1986), 217–223.Google Scholar
29. James, R. C., Inner products in normed linear spaces. Bull. Amer. Math. Soc. 53 (1947), 559–566.
30. Jarczvk, W., On continuous functions which are additive on their graphs. [Grazer Ber., No. 292] Forschungsges., Graz, 1988.Google Scholar
31. Kuczma, M., Functional equations on restricted domains. Aequationes Math. 18 (1978), 1–34.
32. Lawrence, J. Orthogonality and additive mappings on normed linear spaces. Colloq. Math. 49 (1985), 253–255.
33. Matkowski, J., Cauchy functional equation on a restricted domain and commuting functions. In Iteration theory and its functional equations. Proceedings, Schloss Hofen 1984. [Lecture Notes in Mathematics, No. 1163], Springer Verlag, Berlin, 1985, pp. 101–106.Google Scholar
34. Matkowski, J., On an alternative Cauchy equation. Aequationes Math. 29 (1985), 214–221.
35. Paganoni, L. and Paganoni Marzegalli, S., Cauchy’s functional equation on semigroups. Fund. Math. 110 (1980), 63–74.
36. Paganoni, L. and Paganoni Marzegalli, S., Holomorphic solutions of an inhomogeneous Cauchy equation. Aequationes Math. 37 (1989), 179–200.
38. Rätz, J., On orthogonally additive mappings. Aequationes Math. 28 (1985), 35–49.
39. Rätz, J., On orthogonally additive mappings, II. Publ. Math. Debrecen 35 (1988), 241–249.
40. Rätz, J., On orthogonally additive mappings, III. Abh. Math. Sem. Univ. Hamburg 59 (1989), 23–33.
41. Rätz, J., Orthogonally additive mappings on free product Z-modules. To appear (1995).Google Scholar
42. Rätz, J. and Szabó, GY., On orthogonally additive mappings, IV. Aequationes Math. 38 (1989), 73–85.
43. Sablik, M., Note on a Cauchy conditional equation. Rad. Mat. 1 (1985), 241–245.
44. Sablik, M., A functional congruence revisited. Aequationes Math. 41 (1991), 273.Google Scholar
45. Sundaresan, K., Orthogonality and nonlinear functionals on Banach spaces. Proc. Amer. Math. Soc. 34 (1972), 187–190.
46. Szabó, GY., On mappings orthogonally additive in the Birkhoff -James sense. Aequationes Math. 30 (1986), 93–105.
47. Szabó, GY., Sesquilinear-orthogonally quadratic mappings. Aequationes Math. 40 (1990), 190–200.
48. Szabó, GY., On orthogonality spaces admitting nontrivial even orthogonally additive mappings. Acta Math. Hung. 56 (1990), 177–187.
49. Szabó, GY., Continuous orthogonality spaces. Publ. Math. Debrecen 38 (1991), 311–322.
50. Szabó, GY., Φ-orthogonally additive mappings, I. Acta Math. Hung. 58 (1991),101–111.
51. Szabó, GY., Φ-orthogonally, additive mappings, II. Acta Math. Hung. 59 (1992), 1–10.
52. Szabó, GY., A conditional Cauchy equation on normed spaces. Publ. Math. Debrecen 42 (1993), 265–271.
53. Szabó, GY., Isosceles orthogonally additive mappings and inner product spaces. Publ. Math. Debrecen, to appear (1995).Google Scholar
54. Szabó, GY., Pyhtagorean orthogonality and additive mappings. To appear (1996).Google Scholar
55. Tabor, J., Cauchy and Jensen equations on a restricted domain almost everywhere. Publ. Math. Debrecen 39 (1991), 219–235.
56. Vajzović, F., Über das Funktional H mit der Eigenschaft: (x y) = 0 ⇒ H(x + y) + H(x-y) = 2H(x) + 2H(y). Glasnik Mat. Ser. III 2 (22) (1967), 73–81.