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On Solutions of a Generalization of the Goła̧b–Schinzel Functional Equation

  • Anna Mureńko
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 52)

Abstract

Under some additional assumptions we characterize solutions of the functional equation
$$f(x + M(f(x))y) = f(x) \circ f(y),$$
where \(f,M : \mathbb{R} \rightarrow \mathbb{R}\), \(\circ : {\mathbb{R}}^{2} \rightarrow \mathbb{R}\) are unknown functions and f is continuous at a point.

Keywords

Goła̧b–Schinzel functional equation Addition formulas 

References

  1. 1.
    Aczél, J.: Lectures on Functional Equations and Their Applications. Academic Press, New York–London (1966)Google Scholar
  2. 2.
    Aczél, J., Schwaiger, J.: Continuous solutions of the Goła̧b-Schinzel equation on the nonnegative reals and on related domains. Sitzungsber. Österreich. Akad. Wiss. Math.-Natur. Kl. Abt. II 208, 171–177 (1999)MATHGoogle Scholar
  3. 3.
    Baron, K.: On the continuous solutions of the Goła̧b-Schinzel equation. Aequationes Math. 38, 155–162 (1989)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Brillouët-Belluot, N.: On some functional equations of Goła̧b-Schinzel type. Aequationes Math. 42, 239–270 (1991)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Brzdȩk, J.: O rozwia̧zaniach równania funkcyjnego \(f(x + f{(x)}^{n}y) = f(x)f(y)\). Ph.D. Thesis, Silesian University, Katowice (1990)Google Scholar
  6. 6.
    Brzdȩk, J.: Some remarks on solutions of the functional equation \(f(x + f{(x)}^{n}y) = tf(x)f(y)\). Publ. Math. Debrecen 43, 147–160 (1993)MathSciNetGoogle Scholar
  7. 7.
    Brzdȩk, J.: On solutions of the Goła̧b-Schinzel functional equation. Publ. Math. Debrecen 44, 235–241 (1994)MathSciNetGoogle Scholar
  8. 8.
    Brzdȩk, J.: The Christensen measurable solutions of a generalization of the Goła̧b-Schinzel functional equation. Ann. Polon. Math. 64, 195–205 (1996)MathSciNetGoogle Scholar
  9. 9.
    Brzdȩk, J.: On the continuous solutions of a generalization of the Goła̧b-Schinzel equation. Publ. Math. Debrecen 63, 421–429 (2003)MathSciNetGoogle Scholar
  10. 10.
    Brzdȩk, J.: A generalization of addition formulae. Acta Math. Hungar. 101, 281–291 (2003)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Brzdȩk, J.: The Goła̧b-Schinzel equation and its generalization. Aequationes Math. 70, 14–24 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Chudziak, J.: Continuous solutions of a generalization of the Goła̧b-Schinzel equation. Aequationes Math. 61, 63–78 (2001)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Chudziak, J.: Continuous solutions of a generalization of the Goła̧b-Schinzel equation II. Aequationes Math. 71, 115–123 (2006)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Goła̧b, S., Schinzel, A.: Sur l’équation fonctionnelle \(f(x + yf(x)) = f(x)f(y)\). Publ. Math. Debrecen 6, 113–125 (1959)Google Scholar
  15. 15.
    Jabłońska, E.: On solutions of a generalization of the Goła̧b-Schinzel equation. Aequationes Math. 71, 269–279 (2006)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Jabłońska, E.: Functions having the Darboux property and satisfying some functional equation. Colloq. Math. 114, 113–118 (2009)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Kahlig, P., Matkowski, J.: A modified Goła̧b-Schinzel equation on a restricted domain (with applications to meteorology and fluid mechanics). Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 211, 117–136 (2002)MathSciNetMATHGoogle Scholar
  18. 18.
    Mureńko, A.: On solutions of a common generalization of the Goła̧b-Schinzel equation and of the addition formulae. J. Math. Anal. Appl. 341, 1236–1240 (2008)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Mureńko, A.: On the general solution of a generalization of the Goła̧b-Schinzel equation. Aequationes Math. 77, 107–118 (2009)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Reich, L.: Über die stetigen Lösungen der Goła̧b-Schinzel-Gleichung auf \({\mathbb{R}}_{\geq 0}\). Sitzungsber. Österreich. Akad. Wiss. Math.-Natur. Kl. Abt. II 208, 165–170 (1999)MATHGoogle Scholar
  21. 21.
    Sablik, M.: A conditional Goła̧b-Schinzel equation. Anz. Österreich. Akad. Wiss. Math.-Natur. Kl. Abt. II 137, 11–15 (2000)MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of RzeszówRzeszówPoland

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