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

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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