Skip to main content
SpringerLink
Log in

Conditional functional equations and orthogonal additivity

  • Chapter
Aggregating clones, colors, equations, iterates, numbers, and tiles

  • 131 Accesses

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, 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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • AczéL, J., A short course on functional equations. D. Reidel Publ. Co., Dordrecht, 1987.

    Google Scholar 

  • 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 

  • Baron, K. and Kannappan, PL., On the Pexider difference. Fund. Math. 134 (1990), 247–254.

    MathSciNet  MATH  Google Scholar 

  • Baron, K. and Volkmann, P., On the Cauchy equation modulo Z. Fund. Math. 131 (1988), 143–148.

    MathSciNet  MATH  Google Scholar 

  • Baron, K. and Volkmann, P., On a theorem of van der Corput. Abh. Math. Sem. Univ. Hamburg 61 (1991), 189–195.

    Article  MathSciNet  MATH  Google Scholar 

  • Benz, W., Geometrische Transformationen, unter besonderer Berücksichtigung der Lorentztransformationen. BI, Mannheim—Leipzig—Wien—Zürich, 1992.

    MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Borelli Forti, C., Solutions of a non-homogeneous Cauchy equation. Radovi Mat. 5 (1989), 213–222.

    MathSciNet  MATH  Google Scholar 

  • 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 

  • Day, M. M., Some characterizations of inner-product spaces. Trans. Amer. M.th. Soc. 62 (1947), 320–337.

    Article  MATH  Google Scholar 

  • Dhombres, J., Some aspects of functional equations. Section 4.9. Chulalongkorn University Press, Bangkok, 1979.

    Google Scholar 

  • Drljevii, F., On a functional which is quadratic on A-orthogonal vectors. Publ. Inst. Math. (Beograd) (N.S.) 54 (1986), 63–71.

    Google Scholar 

  • Fenyö, I., Osservazioni su alcuni teoremi di D. H. Hyers. Istit. Lombardo Accad. Sci. Lett. Rend. A 114 (1980),235–242.

    MathSciNet  MATH  Google Scholar 

  • Fenyö, I. and Rusconi, D., Sulle distribuzioni the soddisfano una equazione funzionale. Rend. Sem. Mat. Univ. Politec. Torino 39 (1981), 67–76.

    MathSciNet  MATH  Google Scholar 

  • 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 

  • 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 

  • Fochj, M., Functional equations in A-orthogonal vectors. Aequationes Math. 38 (1989), 28–40.

    Article  MathSciNet  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  MATH  Google Scholar 

  • Forti, G. L., The stability of homomorphisms and amenability, with applications to functional equations. Abh. Math. Sem. Univ. Hamburg 57 (1987), 215–226.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

  • Ger, R., Almost additive functions on semigroups and a functional equation. Publ. Math. Debrecen 26 (1979), 219–228.

    MathSciNet  MATH  Google Scholar 

  • Gudder, S. and Strawther, D., Orthogonally additive and orthogonally increasing functions on vector spaces. Pacific J. Math. 58 (1975), 427–436.

    MathSciNet  MATH  Google Scholar 

  • Gudder, S. and Strawther, D., A converse of Pythagoras’ theorem. Amer. Math. Monthly 84 (1977), 551–553.

    Article  MathSciNet  MATH  Google Scholar 

  • Hednen, B. and Goovaerts, M. J., Additivity and premium calculation principles. Blätter Deutsch. Ges. Versich. Math. 17 (1986), 217–223.

    Google Scholar 

  • James, R. C., Inner products in normed linear spaces. Bull. Amer. Math. Soc. 53 (1947), 559–566.

    Article  MathSciNet  MATH  Google Scholar 

  • Jarczvk, W., On continuous functions which are additive on their graphs. [Grazer Ber., No. 292] Forschungsges., Graz, 1988.

    Google Scholar 

  • Kuczma, M., Functional equations on restricted domains. Aequationes Math. 18 (1978), 1–34.

    Article  MathSciNet  MATH  Google Scholar 

  • Lawrence, J. Orthogonality and additive mappings on normed linear spaces. Colloq. Math. 49 (1985), 253–255.

    MathSciNet  Google Scholar 

  • 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 

  • Matkowski, J., On an alternative Cauchy equation. Aequationes Math. 29 (1985), 214–221.

    Article  MathSciNet  MATH  Google Scholar 

  • Paganoni, L. and Paganoni Marzegalli, S., Cauchy’s functional equation on semigroups. Fund. Math. 110 (1980), 63–74.

    MathSciNet  MATH  Google Scholar 

  • Paganoni, L. and Paganoni Marzegalli, S., Holomorphic solutions of an inhomogeneous Cauchy equation. Aequationes Math. 37 (1989), 179–200.

    Article  MathSciNet  MATH  Google Scholar 

  • Pinsker, A., Sur une fonctionnelle dans l’espace de Hilbert. C. R. (Doklady) Acad. Sci. URSS N.S. 20 (1938), 411–414.

    Google Scholar 

  • Rätz, J., On orthogonally additive mappings. Aequationes Math. 28 (1985), 35–49.

    Article  MathSciNet  MATH  Google Scholar 

  • Rätz, J., On orthogonally additive mappings, II. Publ. Math. Debrecen 35 (1988), 241–249.

    MathSciNet  MATH  Google Scholar 

  • Rätz, J., On orthogonally additive mappings, III. Abh. Math. Sem. Univ. Hamburg 59 (1989), 23–33.

    Article  MathSciNet  MATH  Google Scholar 

  • Rätz, J., Orthogonally additive mappings on free product Z-modules. To appear (1995).

    Google Scholar 

  • Rätz, J. and Szabó, GY., On orthogonally additive mappings, IV. Aequationes Math. 38 (1989), 73–85.

    Article  MathSciNet  MATH  Google Scholar 

  • Sablik, M., Note on a Cauchy conditional equation. Rad. Mat. 1 (1985), 241–245.

    MathSciNet  MATH  Google Scholar 

  • Sablik, M., A functional congruence revisited. Aequationes Math. 41 (1991), 273.

    Google Scholar 

  • Sundaresan, K., Orthogonality and nonlinear functionals on Banach spaces. Proc. Amer. Math. Soc. 34 (1972), 187–190.

    Article  MathSciNet  MATH  Google Scholar 

  • Szabó, GY., On mappings orthogonally additive in the Birkhoff -James sense. Aequationes Math. 30 (1986), 93–105.

    Article  MathSciNet  MATH  Google Scholar 

  • Szabó, GY., Sesquilinear-orthogonally quadratic mappings. Aequationes Math. 40 (1990), 190–200.

    Article  MathSciNet  MATH  Google Scholar 

  • Szabó, GY., On orthogonality spaces admitting nontrivial even orthogonally additive mappings. Acta Math. Hung. 56 (1990), 177–187.

    Article  MATH  Google Scholar 

  • Szabó, GY., Continuous orthogonality spaces. Publ. Math. Debrecen 38 (1991), 311–322.

    MathSciNet  MATH  Google Scholar 

  • Szabó, GY., Φ-orthogonally additive mappings, I. Acta Math. Hung. 58 (1991),101–111.

    Article  MATH  Google Scholar 

  • Szabó, GY., Φ-orthogonally, additive mappings, II. Acta Math. Hung. 59 (1992), 1–10.

    Article  Google Scholar 

  • Szabó, GY., A conditional Cauchy equation on normed spaces. Publ. Math. Debrecen 42 (1993), 265–271.

    MathSciNet  Google Scholar 

  • Szabó, GY., Isosceles orthogonally additive mappings and inner product spaces. Publ. Math. Debrecen, to appear (1995).

    Google Scholar 

  • Szabó, GY., Pyhtagorean orthogonality and additive mappings. To appear (1996).

    Google Scholar 

  • Tabor, J., Cauchy and Jensen equations on a restricted domain almost everywhere. Publ. Math. Debrecen 39 (1991), 219–235.

    MathSciNet  MATH  Google Scholar 

  • 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.

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Luigi Paganoni

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

    Jürg Rätz

Authors
  1. Luigi Paganoni
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jürg Rätz
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1

    János Aczél

Rights and permissions

Reprints and permissions

Copyright information

© 1995 Birkhäuser Verlag, Basel

About this chapter

Cite this chapter

Paganoni, L., Rätz, J. (1995). Conditional functional equations and orthogonal additivity. In: Aczél, J. (eds) Aggregating clones, colors, equations, iterates, numbers, and tiles. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-9096-0_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-9096-0_8

  • Publisher Name: Birkhäuser Basel

  • Print ISBN: 978-3-7643-5243-1

  • Online ISBN: 978-3-0348-9096-0

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation