Skip to main content
SpringerLink
Log in

On an alternative functional equation related to the Cauchy equation

  • Research Papers
  • Published:
aequationes mathematicae Aims and scope Submit manuscript

Abstract

We consider the following problem: Let (G, +) be an abelian group,B a complex Banach space,a, b∈B,b≠0,M a positive integer; find all functionsf:G →B such that for every (x, y) ∈G ×G the Cauchy differencef(x+y)−f(x)−f(y) belongs to the set {a, a+b, a+2b, ...,a+Mb}. We prove that all solutions of the above problem can be obtained by means of the injective homomorphisms fromG/H intoR/Z, whereH is a suitable proper subgroup ofG.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Dubikajtis, L., Ferens, C., Ger, R. and Kuczma, M., On Mikusiński's functional equation. Ann. Polon. Math28 (1973), 39–47.

    Google Scholar 

  • Fischer, P. andMuszély, G.,On certain generalizations of the functional equation of Cauchy type. (Hungarian), Mat. Lapok16 (1965), 67–74.

    Google Scholar 

  • Fischer, P. andMuszély, G.,On some new generalizations of the functional equation of Cauchy. Canad. Math. Bull.10 (1967), 197–205.

    Google Scholar 

  • Ger, R.,On a method of solving of conditional Cauchy equations. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat.-Fiz. No. 544–576 (1976), 159–165.

  • Ger, R.,On an alternative functional equation. Aequationes Math.15 (1977), 145–162.

    Google Scholar 

  • Ger, R. andKuczma, M.,On inverse additive functions. Boll. Un. Mat. Ital. (4)11 (1975), 490–495.

    Google Scholar 

  • Hyers, D. H.,On the stability of the linear functional equation. Proc. Nat. Acad. Sci. U.S.A.27 (1941), 222–224.

    Google Scholar 

  • Kannappan, Pl. andKuczma, M.,On a functional equation related to the Cauchy equation. Ann. Polon. Math.30 (1974), 49–55.

    Google Scholar 

  • Kuczma, M.,On some alternative functional equations. Aequationes Math.16 (1977), 322.

    Google Scholar 

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

    Google Scholar 

  • Swiatak, H.,On the functional equation f(x+y) 2=[f(x)+f(y)]2. Publ. Techn. Univ. Miskolc30 (1970), 307–308.

    Google Scholar 

  • Vincze, E., Über eine Verallgemeinerung der Cauchyschen Funktionalgleichung. Funkcialaj Ekvacioj6 (1964), 55–62.

    Google Scholar 

  • Vincze, E., Beitrag zur Theorie der Cauchyschen Funktionalgleichungen. Arch. Math. (Basel)15 (1964), 132–135.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Istituto Matematico “F. Enriques”, Università di Milano, via C. Saldini, 50, I-20133, Milano, Italy

    Gian Luigi Forti

Authors
  1. Gian Luigi Forti
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Forti, G.L. On an alternative functional equation related to the Cauchy equation. Aeq. Math. 24, 195–206 (1982). https://doi.org/10.1007/BF02193044

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02193044

AMS (1980) subject classification

Search

Navigation