The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation

Janusz Brzdęk

Annales Polonici Mathematici (1996)

  • Volume: 64, Issue: 3, page 195-205
  • ISSN: 0066-2216

Abstract

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Let K denote the set of all reals or complex numbers. Let X be a topological linear separable F-space over K. The following generalization of the result of C. G. Popa [16] is proved. Theorem. Let n be a positive integer. If a Christensen measurable function f: X → K satisfies the functional equation f ( x + f ( x ) n y ) = f ( x ) f ( y ) , then it is continuous or the set x ∈ X : f(x) ≠ 0 is a Christensen zero set.

How to cite

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Janusz Brzdęk. "The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation." Annales Polonici Mathematici 64.3 (1996): 195-205. <http://eudml.org/doc/269942>.

@article{JanuszBrzdęk1996,
abstract = {Let K denote the set of all reals or complex numbers. Let X be a topological linear separable F-space over K. The following generalization of the result of C. G. Popa [16] is proved. Theorem. Let n be a positive integer. If a Christensen measurable function f: X → K satisfies the functional equation $f(x + f(x)^ny) = f(x)f(y)$, then it is continuous or the set x ∈ X : f(x) ≠ 0 is a Christensen zero set.},
author = {Janusz Brzdęk},
journal = {Annales Polonici Mathematici},
keywords = {Gołąb-Schinzel functional equation; Christensen measurability; F-space; Christensen measurable function; Golab-Schinzel equation; linear separable -space; Christensen zero set; continuous solutions},
language = {eng},
number = {3},
pages = {195-205},
title = {The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation},
url = {http://eudml.org/doc/269942},
volume = {64},
year = {1996},
}

TY - JOUR
AU - Janusz Brzdęk
TI - The Christensen measurable solutions of a generalization of the Gołąb-Schinzel functional equation
JO - Annales Polonici Mathematici
PY - 1996
VL - 64
IS - 3
SP - 195
EP - 205
AB - Let K denote the set of all reals or complex numbers. Let X be a topological linear separable F-space over K. The following generalization of the result of C. G. Popa [16] is proved. Theorem. Let n be a positive integer. If a Christensen measurable function f: X → K satisfies the functional equation $f(x + f(x)^ny) = f(x)f(y)$, then it is continuous or the set x ∈ X : f(x) ≠ 0 is a Christensen zero set.
LA - eng
KW - Gołąb-Schinzel functional equation; Christensen measurability; F-space; Christensen measurable function; Golab-Schinzel equation; linear separable -space; Christensen zero set; continuous solutions
UR - http://eudml.org/doc/269942
ER -

References

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  1. [1] J. Aczél and S. Gołąb, Remarks on one-parameter subsemigroups of the affine group and their homo- and isomorphisms, Aequationes Math. 4 (1970), 1-10. Zbl0205.14802
  2. [2] K. Baron, On the continuous solutions of the Gołąb-Schinzel equation, Aequationes Math. 38 (1989), 155-162. Zbl0702.39005
  3. [3] W. Benz, The cardinality of the set of discontinuous solutions of a class of functional equations, Aequationes Math. 32 (1987), 58-62. Zbl0616.39003
  4. [4] N. Brillouët et J. Dhombres, Equations fonctionnelles et recherche de sous groupes, Aequationes Math. 31 (1986), 253-293. Zbl0611.39004
  5. [5] J. Brzdęk, Subgroups of the group Z n and a generalization of the Gołąb-Schinzel functional equation, Aequationes Math. 43 (1992), 59-71. Zbl0757.39003
  6. [6] J. Brzdęk, A generalization of the Gołąb-Schinzel functional equation, Aequationes Math. 39 (1990), 268-269. 
  7. [7] J. Brzdęk, On the solutions of the functional equation f ( x f ( y ) l + y f ( x ) k ) = t f ( x ) f ( y ) , Publ. Math. Debrecen 38 (1991), 175-183. Zbl0741.39008
  8. [8] J. P. R. Christensen, Topology and Borel Structure, North-Holland Math. Stud. 10, North-Holland, 1974. 
  9. [9] J. P. R. Christensen, On sets of Haar measure zero in abelian Polish groups, Israel J. Math. 13 (1972), 255-260. 
  10. [10] P. Fischer and Z. Słodkowski, Christensen zero sets and measurable convex functions, Proc. Amer. Math. Soc. 79 (1980), 449-453. Zbl0444.46010
  11. [11] S. Gołąb et A. Schinzel, Sur l'équation fonctionnelle f(x+yf(x)) = f(x)f(y), Publ. Math. Debrecen 6 (1959), 113-125. Zbl0083.35004
  12. [12] D. Ilse, I. Lechmann und W. Schulz, Gruppoide und Funktionalgleichungen, Deutscher Verlag Wiss., Berlin, 1984. 
  13. [13] P. Javor, On the general solution of the functional equation f(x+yf(x)) = f(x)f(y), Aequationes Math. 1 (1968), 235-238. Zbl0165.17102
  14. [14] M. Kuczma, An Introduction to the Theory of Functional Equations and Inequalities, PWN and Uniw. Śląski, Warszawa-Kraków-Katowice, 1985. 
  15. [15] P. Plaumann und S. Strambach, Zweidimensionale Quasialgebren mit Nullteilern, Aequationes Math. 15 (1977), 249-264. Zbl0375.17012
  16. [16] C. G. Popa, Sur l'équation fonctionnelle f(x+yf(x)) = f(x)f(y), Ann. Polon. Math. 17 (1965), 193-198. Zbl0141.32804
  17. [17] W. Rudin, Real and Complex Analysis, McGraw-Hill, 1974. 
  18. [18] M. Sablik and P. Urban, On the solutions of the equation f ( x f ( y ) k + y f ( x ) l ) = f ( x ) f ( y ) , Demonstratio Math. 18 (1985), 863-867. Zbl0599.39006
  19. [19] S. Wołodźko, Solution générale de l'équation fonctionnelle f(x+yf(x)) = f(x)f(y), Aequationes Math. 2 (1968), 12-29. Zbl0162.20402

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