On some conditional functional equations of Gołąb-Schinzel type


We consider equation (6) in the class of continuous functions f: I→ℝ satisfying (7), where f is a non-trivial real interval and n,k are fixed positive integers. The obtained results are applied to get the solutions of the system of functional equations (3)-(5) in the class of pairs of functions f,g: I→ℝ such that f is continuous. Some connections between solutions of the equations and a class of subsemigroups of some Lie groups are established as well.

1. J. Aczél, Sur les opération définies pour nombres réel, Bull. Soc. Math. France 76 (1949), 59-64.
2. J. Aczél, S. Gołąb, Remark on one-parameter subsemigroups of the affine group and their homo- and isomorphisms, Aequationes Math. 4 (1970), 1-10.
3. N. Brillouët, J. Dhombres, Equation fonctionnelles et recherche de sous-groupes, Aequationes Math. 31 (1986), 253-293.
4. J. Brzdęk, Subgroups of the group Z_n and a generalization of the Gołąb-Schinzel functional equation, Aequationes Math. 43 (1992), 59-71.
5. R. Craigen, Z. Páles, The associativity equation revisited, Aequationes Math. 37 (1989), 306-312.
6. S. Midura, Sur la détermination de certains sous-groupes du groupe L^1_S à l'aide d'équations functionnelles, Dissertationes Math. No. 105, IMPAN, Warsaw, 1973.
7. S. Midura, Sur certains sous-demigroupes à un parametre des groupes L^1_2 et L^1_3 déterminés à l'aide d'équations fonctionnelles, Rocznik Nauk.-Dydakt. Wyż. Szkoły Ped. w Rzeszowie 7/62 (1985), 37-50.
8. P. Urban, Równania funkcyjne typu Gołąba-Schinzla. [Doctor thesis], Uniwersytet Śląski, Instytut Matematyki, Katowice, 1987.
9. P. Urban, Continuous solutions of the functional equation f(xf(y)^k + yf(x)^l) = f(x)f(y), Demonstratio Math. 16 (1983), 1019-1025.

Published : 1995-09-29

BrzdękJ. (1995). On some conditional functional equations of Gołąb-Schinzel type. Annales Mathematicae Silesianae, 9, 65-80. Retrieved from https://journals.us.edu.pl/index.php/AMSIL/article/view/14208

Janusz Brzdęk 
Instytut Matematyki, Wyższa Szkoła Pedagogiczna w Rzeszowie  Poland

The Copyright Holders of the submitted text are the Author and the Journal. The Reader is granted the right to use the pdf documents under the provisions of the Creative Commons 4.0 International License: Attribution (CC BY). The user can copy and redistribute the material in any medium or format and remix, transform, and build upon the material for any purpose.

  1. License
    This journal provides immediate open access to its content under the Creative Commons BY 4.0 license (http://creativecommons.org/licenses/by/4.0/). Authors who publish with this journal retain all copyrights and agree to the terms of the above-mentioned CC BY 4.0 license.
  2. Author’s Warranties
    The author warrants that the article is original, written by stated author/s, has not been published before, contains no unlawful statements, does not infringe the rights of others, is subject to copyright that is vested exclusively in the author and free of any third party rights, and that any necessary written permissions to quote from other sources have been obtained by the author/s.
  3. User Rights
    Under the Creative Commons Attribution license, the users are free to share (copy, distribute and transmit the contribution) and adapt (remix, transform, and build upon the material) the article for any purpose, provided they attribute the contribution in the manner specified by the author or licensor.
  4. Co-Authorship
    If the article was prepared jointly with other authors, the signatory of this form warrants that he/she has been authorized by all co-authors to sign this agreement on their behalf, and agrees to inform his/her co-authors of the terms of this agreement.