Skip to main content
SpringerLink
Log in

Functional equations on restricted domains

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

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

References

  1. Aczél, J.,Miszellen über Funktionalgieichungen. I. Math. Nachrichten19 (1958), 87–99.

    Article  MATH  Google Scholar 

  2. Aczél, J.,Über die Begründung der Additionsund Multiplikations formeln von bedingten Wahrscheinlichkeiten. Magyar Tud. Akad. Mat. Kutató Int. Közl.6 (1961), 111–121.

    MATH  Google Scholar 

  3. Aczél, J.,Vorlesungen über Funktionalgleichungen und ihre Anwendungen. Birkhäuser Verlag, Basel und Stuttgart, 1961.

    MATH  Google Scholar 

  4. Aczél, J.,Lectures on functional equations and their applications. Academic Press, New York and London, 1966.

    MATH  Google Scholar 

  5. Aczél, J.,Remark (P 43). Aequationes Math.2 (1969), 369.

    Google Scholar 

  6. Aczél, J.,Problem (P 56). Aequationes Math.4 (1970), 244–245.

    Article  Google Scholar 

  7. Aczél, J.,Problem (P 141). Aequationes Math.12 (1975), 303.

    Google Scholar 

  8. Aczél, J., Baker, J. A., Djoković, D. Ž., Kannappan, Pl., andRadó, F.,Extensions of certain homomorphisms of subsemigroups to homomorphisms of groups. Aequationes Math.6 (1971), 263–271.

    Article  MATH  MathSciNet  Google Scholar 

  9. Aczél, J. andDarcóczy, Z.,Charakterisierung der Entropien positiver Ordnung und der Shannonschen Entropie. Acta Math. Acad. Sci. Hungar.14 (1963), 95–121.

    Article  MATH  MathSciNet  Google Scholar 

  10. Aczél, J. andDarcóczy, Z.,On measures of information and their characterizations. Academic Press, New York, London, and San Francisco, 1975.

    MATH  Google Scholar 

  11. Aczél, J. andErdös, P.,The non-existence of a Hamel-basis and the general solution of Cauchy’s functional equation for nonnegative numbers. Publ. Math. Debrecen12 (1965), 259–265.

    MATH  MathSciNet  Google Scholar 

  12. Aczél, J., Fladt, K. andHosszú, M.,Lösung einer mit dem Doppelverhältnis zusammenhängender Funktionalgleichung. Magyar Tud. Akad. Mat. Kutató Int. Közl.7 (1962), 335–352.

    MATH  Google Scholar 

  13. Balcerzyk, S.,Wstep do algebry homologicznej. Polish Scientific Publishers, Warszawa, 1970.

    MATH  Google Scholar 

  14. Baron, K.,On extending of solutions of a functional equation. Aequationes Math.13 (1975), 285–288.

    Article  MATH  MathSciNet  Google Scholar 

  15. Baron, K. andGer, R.,On Mikusiński-Pexider functional equation. Colloquium Math.28 (1973), 307–312.

    MATH  MathSciNet  Google Scholar 

  16. Boas, R. P.,Functions which are odd about several points. Nieuw. Arch. Wisk. [3]1 (1953), 27–32. Addendum, ibid.5 (1957), 25.

    MATH  MathSciNet  Google Scholar 

  17. de Bruijn, N. G.,On almost additive functions. Colloquium Math.15 (1966), 59–63.

    MATH  Google Scholar 

  18. Burstin, C.,Über eine spezielle Klasse reeller periodischer Funktionen. Monatsh. Math.26 (1915), 229–262.

    Article  MATH  MathSciNet  Google Scholar 

  19. Daróczy, Z.,Notwendige und hinreichende Bedingungen für die Existenz yon nichtkonstanten Lösungen linearer Funklionalgleichungen. Acta Sci. Math. Szeged22 (1964), 31–41.

    Google Scholar 

  20. Daróczy, Z.,Uber die Funktjonalgleichung φ[φ(x)y] = φ(x)φ(y). Acta Univ. Debrecen, Ser. Fiz. Chem.8 (1962), 125–132.

    Google Scholar 

  21. Daróczy, Z. andGyóry, K.,Die Cauchysche Funktionalgleichung über diskreten Mengen. Publ. Math. Debrecen13 (1966), 249–255.

    MATH  MathSciNet  Google Scholar 

  22. Daróczy, Z. andLosonczi, L.,Über die Erweiterung der auf einer Punktmenge additiven Funktionen. Publ. Math. Debrecen14 (1967), 239–245.

    MATH  MathSciNet  Google Scholar 

  23. Denny, J. L.,Sufficient conditions for a family of probabilities to be exponential. Proc. Nat. Acad. Sci. U.S.A.57 (1967), 1184–1187.

    Article  MATH  MathSciNet  Google Scholar 

  24. Denny, J. L.,Cauchy’s equation and suffcient statistics on arcwise connected spaces. Ann. Math. Statistics41 (1970), 401–411.

    Article  MATH  MathSciNet  Google Scholar 

  25. Dhombres, J. G.,Functional equations on semi-groups arising from the theory of means. Nanta Mathematica5, No 3 (1972), 48–66.

    MATH  MathSciNet  Google Scholar 

  26. Dhombres, J. G.,Interpolation linéaire et équations fonctionnelles. Ann. Polon. Math.32 (1976), 287–302.

    MATH  MathSciNet  Google Scholar 

  27. Dhombres, J. G.,On some extensions of homomorphisms. Nanta Math. (1976).

  28. Dhombres, J. G. andGer, R.,Conditional Cauchy equations. Glasnik Mat. (to appear).

  29. Dubikajtis, L., Ferens, C., Ger, R. andKuczma, M.,On Mikusiński’s functional equation. Ann. Polon. Math.28 (1973), 39–47.

    MATH  MathSciNet  Google Scholar 

  30. Dubikajtis, L. andKuczma, M.,On non-localized oriented angles. Ann. Polon. Math.25 (1972), 227–239.

    MATH  MathSciNet  Google Scholar 

  31. Erdös, P.,On the distribution function of additive functions. Ann. of Math.47 (1946), 1–20.

    Article  MathSciNet  Google Scholar 

  32. Erdös, P.,P 310, Colloquium Math.7 (1960), 311.

    Google Scholar 

  33. Erdös, P. andGolomb, M.,Functions which are symmetric about several points. Nieuw. Arch. Wisk. [3]3 (1955), 13–19.

    MATH  MathSciNet  Google Scholar 

  34. Etigson, L.,A system of functional equations with restricted argument for the cosine. Aequationes Math.10 (1974), 309–310.

    Google Scholar 

  35. Faddeev, D. K.,Zum Begriff der Entropie eines endlichen Wahrscheinlischkeitsschemas. In: Arbeiten zur Informationstheorie, Vol. I. Berlin, 1957, pp. 88–90.

    MathSciNet  Google Scholar 

  36. Fischer, P.,Sur l’équivalence des équations fonctionnelles f(x +y)=f(x)+f(y)et f 2(x +y)=[f(x)+f(y)]2. Ann. Fac. Sci. Univ. Toulouse, 1968, 71–73.

  37. Fischer, P., Remarque. Aequationes Math.2 (1969), 369.

    Google Scholar 

  38. Fischer, P. andMuszély, Gy.,A Cauchy-féle függvényegyenlet bizonyos tipusú általánositásai. Mat. Lapok16 (1965), 67–75.

    MATH  MathSciNet  Google Scholar 

  39. Fischer, P. andMuszély, Gy.,On some new generalizations of the functional equation of Cauchy. Canadian Math. Bull.10 (1967), 197–205.

    MATH  Google Scholar 

  40. Ger, R.,On almost polynomial functions. Colloquium Math.24 (1971), 95–101.

    MATH  MathSciNet  Google Scholar 

  41. Ger, R.,On some functional equations with a restricted domain. I. Fundamenta Math.89 (1975), 131–149.

    MATH  MathSciNet  Google Scholar 

  42. Ger, R.,On some functional equations with a restricted domain. II. Fundamenta Math. (to appear).

  43. Ger, R. On some functional equations with a restricted domain. Bull. Acad. Polon. Sci., Sér. Sci. Math. Astronom. Phys.24 (1976). 432–429.

    MathSciNet  Google Scholar 

  44. Ger, R.,On all alternative functional equation. Aequationes Math.15 (1977).

  45. Ger, R.,Note on almost additive functions. Aequationes Math. (to appear).

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

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

    MATH  MathSciNet  Google Scholar 

  48. Golab, S.,Einige grundlegende Fragen der Theorie der Funktionalgleichungen. Glasnik Mat.-Fiz. Astr. (2)20 (1965), 57–63.

    MATH  MathSciNet  Google Scholar 

  49. Golab, S. andLosonczi, L.,Uber die Funktionalgleichung der Funktion Arccosinus. I. Die lokalen Lösungen, Publ. Math. Debrecen12 (1965), 159–174.

    MATH  MathSciNet  Google Scholar 

  50. Grzaślewicz, A.,On extensions of homomorphisms. Aequationes Math. (to appear).

  51. Grzaślewicz, A., Powaszka, Z. andTabor, J.,On Cauchy’s nucleus, Publ. Math. Debrecen (to appear).

  52. Grzaślewicz, A. andTabor, J.,On the equivalence of two definitions of the fulfillment of the translation equation and on the extensions of the solutions of this equation. Rocznik Naukowo-Dydaktyczny WSP w Krakowie, Prace Mat.7 (1974), 47–57.

    Google Scholar 

  53. Hartman, P. andKreshner, R.,The structure of monotonic functions. Amer. J. Math.59 (1937), 809–822.

    Article  MathSciNet  Google Scholar 

  54. Hartman, P.,Problème 230, Colloquium Math.5 (1958), 236–237.

    MathSciNet  Google Scholar 

  55. Hartman, S.,A remark about Cauchy’s equation. Colloquium Math.8 (1961), 77–79.

    MATH  MathSciNet  Google Scholar 

  56. Hosszú, M.,Egy alternativ függvényegyenletröl. Mat. Lapok14 (1963), 98–102.

    MathSciNet  Google Scholar 

  57. Jurkat, W. B., On Cauchy’s functional equation. Proc. Amer. Math. Soc.16 (1965), 683–686.

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

  59. Kemperman, J. H. B., On the Levi-Civita functional equation. Aequationes Math.1 (1968), 279.

    Google Scholar 

  60. Kuczma, M.,Almost convex functions. Colloquium Math.21 (1970), 279–284.

    MATH  MathSciNet  Google Scholar 

  61. Kuczma, M.,Some remarks about additive functions on cones. Aequationes Math.4 (1970), 303–306.

    Article  MATH  MathSciNet  Google Scholar 

  62. Kuczma, M.,P 56 SI, Aequationes Math.5 (1970), 326–327.

    Article  MathSciNet  Google Scholar 

  63. Kuczma, M.,Convex functions. In: Functional equations and inequalities, Edizioni Cremonese, Roma, 1971, pp. 195–213.

    Google Scholar 

  64. Kuczma, M.,Cauchy’s functional equation on a restricted domain. Colloquium Math.28 (1973), 313–315.

    MATH  MathSciNet  Google Scholar 

  65. Lajkó, K.,Applications of extensions of additive functions. Aequationes Math.11 (1974), 68–76.

    Article  MATH  MathSciNet  Google Scholar 

  66. Losonczi, L.,Bestimmung aller nichtkonstanten Lösungen von linearen Funktionalgleichungen. Acta Sci. Math. Szeged25 (1964), 250–254.

    MATH  MathSciNet  Google Scholar 

  67. Losonczi, L.,Problem (P 19). Aequationes Math.1 (1968), 149.

    Google Scholar 

  68. Marcus, S.,Conditii de echivalentą cu o constantą pentru functii integrabile Riemann si functti cu proprietatea lui Baire, An. Univ. “C. I. Parhon” Bucureşti, Ser. Sti. Nat.22 (1959), 59–62.

    Google Scholar 

  69. Midura, S. andTabor, J.,Sur les itération avec un paramètre réel des fonctions sous-modules. Demonstratio Math.6 (1973), 271–287.

    MATH  MathSciNet  Google Scholar 

  70. Mikusiński, J. andSkórnik, K.,Macierzowa teoria ukladów optycznych. (in preparation).

  71. Moszner, Z.,P 56S 2. Aequationes Math.6 (1971), 112–113.

    Article  MathSciNet  Google Scholar 

  72. Moszner, Z.,Sur le prolongement des objets géométriques transitifs. Tensor (N. S.)26 (1972), 239–242.

    MATH  MathSciNet  Google Scholar 

  73. Moszner, Z. andPilecka, B.,Sur le prolongement des objets géométriques non-transitifs. Tensor (N. S.)28 (1974), 63–66.

    MATH  MathSciNet  Google Scholar 

  74. Moszner, Z. andTabor, J.,L’équation de translation sur la structure avec zéro. Ann. Polon. Math.31 (1976), 255–264.

    MATH  MathSciNet  Google Scholar 

  75. Ng, C. T., Representation of measures of information with the branching property. Information and Control25 (1974), 45–56.

    Article  MATH  MathSciNet  Google Scholar 

  76. Rényi, A.,On a theorem of P. Erdös and its application in information theory. Mathematica (Ciuj)1 (24) (1959), 341–344.

    Google Scholar 

  77. Sablik, M.,On proper linearly invariant set ideals. (to appear).

  78. Światak, H.,On the equation ρ(x +y)2 = [ρ(x)g(y)+ρ(y)g(x)]2. Zeszyty Nauk. Uniw. Jagiello. Prace Fiz.10 (1965), 97–104.

    Google Scholar 

  79. Światak, H.,On the functional equation ƒ1(x 1+...+x n )2=[Σ(i l ,..., i n ƒ1(x i l )...ƒ n (x i n )]2. Ann. Univ. Sci. Budapest. Eötvös Sect. Math.10 (1967), 50–52.

    Google Scholar 

  80. Światak, H.,On the equivalence of some functional equations. Publ. Tech. Univ. Miskolc30 (1970), 275–279.

    Google Scholar 

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

    Google Scholar 

  82. Światak, H.,Functional equations characterizing probability distributions. Aequationes Math.11 (1974), 303–304.

    Google Scholar 

  83. Światak, H. andHosszu, M.,Remarks on the functional equation e(x, y)f(xy)=f(x)+ f(y). Publ. Techn. Univ. Miskolc30 (1970), 323–325.

    Google Scholar 

  84. Światak, H. andHosszu, M.,Notes on functional equations of polynomial form. Publ. Math. Debrecen17 (1970), 61–66.

    MathSciNet  Google Scholar 

  85. Székelyhidi, L.,Nyít ponthalmazon additív függvény általános elöallitása, Magyar Tud. Akad. Mat. Fiz. Oszt. Közl.21 (1972), 503–509.

    Google Scholar 

  86. Tabor, J.,On the extensions of algebraic objects over a Brandt groupoid. Tensor (N. S.)29 (1975), 85–88.

    MATH  MathSciNet  Google Scholar 

  87. [88]Tabor, J.,Regular extensions of algebraic objects. Tensor (N. S.),30 (1976), 186–190.

    MATH  MathSciNet  Google Scholar 

  88. Tabor, J.,,On the translation equation occurring in the iteration theory. Demonstratio Math.9 (1976), 119–127.

    MATH  MathSciNet  Google Scholar 

  89. Tabor, J.,Solution of Cauchy’s functional equation on a restricted domain. Colloquium Math. (to appear).

  90. Tabor, J.,Continuous solutions of Cauchy’s functional equation on a restricted domain. Aequationes Math. (to appear).

  91. Vajzović, F.,Über ein Funktional H mit der Eigenschaft: (x, y) = 0 ⇒ H(x + y) + H(x − y) = 2H(x)+2H(y). Aequationes Math.1 (1968), 141.

    Google Scholar 

  92. Vincze, E.,Altemativ függvényegyenletek megoldásairól. Mat. Lapok14 (1963), 179–195.

    Google Scholar 

  93. Vincze, E.,Betrag zur Theorie der Cauchyschen Funktionalgleichungen. Archiv der Math.15 (1964), 132–135.

    Article  MATH  MathSciNet  Google Scholar 

  94. Vincze, E.,Über eine Verallgemeinerung der Cauchyschen Funktionalgleichung. Funkcial. Ekvac.6 (1964), 55–62.

    MATH  MathSciNet  Google Scholar 

  95. Vincze, E.,Über eine Klasse der ahernativen Funktionalgleichungen. Aequationes Math.2 (1969), 364–365.

    Google Scholar 

  96. Wegrzyk R.,On extending of integrable solutions of a functional equation of n-th order. Commentationes Math. (to appear).

  97. Zdun, M. C.,On the uniqueness of solutions of the functional equation ρ(x + f(x))= ρ(x)+ρ(f(x)). Aequationes Math.8 (1972), 229–232.

    Article  MATH  MathSciNet  Google Scholar 

  98. Zelazko, W.,Algebry Banacha. Polish Scientific Publishers, Warszawa, 1968.

    MATH  Google Scholar 

  99. Kuczma, M.,On some alternative functional equations. Aequationes Math. (to appear).

  100. Dhombres, J. G.,Sur quelques extensions d’homomorphismes. C. R. Acad. Sci. Paris281 (1975), 503–506.

    MATH  MathSciNet  Google Scholar 

  101. Dhombres, J. G.,Solution générale sur un groupe abélien de l’équation fonctionnelle f(x * f(y))= f(f(x) * y). Aequationes Math.15 (1977). . . . .

  102. Matras, Y.,Sur l’équation fonctionnelle f(xf(y)) = f(x)f(y). C. R. Acad. Royale Belgique. Bull. Cl. Sc. (5)55 (1969). 731–751.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Silesian University, Katowice, Poland

    Marek Kuczma

Authors
  1. Marek Kuczma
    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

Kuczma, M. Functional equations on restricted domains. Aeq. Math. 18, 1–34 (1978). https://doi.org/10.1007/BF01844065

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01844065

Keywords

Search

Navigation