Advertisement

Aequationes mathematicae

, Volume 89, Issue 2, pp 383–391 | Cite as

Improving regularity of solutions of a difference equation

  • Witold JarczykEmail author
Open Access
Article
  • 305 Downloads

Abstract

Using some results on convex and almost convex functions defined on a locally compact Abelian group, we prove a theorem showing a “measurability implies continuity” effect for non-negative solutions of the difference equation \({\varphi(x) = \sum_{i=1}^{k}p_{i}\varphi\left(x+a_{i} \right)}\), where \({p_{1}, \ldots, p_{k} \in (0, \infty)}\) and non-zero elements \({a_{1}, \ldots, a_{k}}\) of the group are given.

Keywords

Haar measure Linear difference equation Measurable and continuous solutions Convex and almost convex functions on groups 

Mathematics Subject Classification

Primary 39A10 39B52 Secondary 43A05 28A20 39B62 

References

  1. 1.
    Aczél, J.: Lectures on functional equations and their applications. In: Mathematics in Science and Engineering, vol. 19. Academic Press, New York (1966)Google Scholar
  2. 2.
    Grinč M.: On non-negative measurable solutions of a difference functional equation. Positivity 2, 221–228 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Hewitt E., Ross K.A.: Abstract harmonic analysis, vol I. Springer, Berlin (1963)CrossRefzbMATHGoogle Scholar
  4. 4.
    Járai, A.: Regularity properties of functional equations in several variables. In: Advances in Mathematics, vol. 8. Springer, New York (2005)Google Scholar
  5. 5.
    Jarczyk, W.: A recurrent method of solving functional equations. In: Prace Naukowe Uniwersytetu Śląskiego w Katowicach, vol. 1206. Uniwersytet Śląski, Katowice (1991)Google Scholar
  6. 6.
    Jarczyk W.: Almost convexity on Abelian groups. Aequat. Math. 80, 141–154 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Jarczyk, W.: Convexity and almost convexity in groups. In: Recent Developments in Functional Equations and Inequalities, vol. 99, pp. 55–76. Polish Academy of Sciences, Institute of Mathematics. Banach Center Publications, Warsaw (2013)Google Scholar
  8. 8.
    Jarczyk W., Laczkovich M.: Convexity on Abelian groups. J. Convex Anal. 16, 33–48 (2009)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Jarczyk W., Laczkovich M.: Almost convex functions on locally compact Abelian groups. Math. Inequal. Appl. 13, 217–225 (2010)zbMATHMathSciNetGoogle Scholar
  10. 10.
    Kuczma M.: Almost convex functions. Colloq. Math. 21, 279–284 (1970)zbMATHMathSciNetGoogle Scholar
  11. 11.
    Kuczma, M.: An introduction to the theory of functional equations and inequalities. In: Gilányi, A. (ed.) Cauchy’s Equation and Jensen’s Inequality, 2nd edn., Birkhaüser, Basel (2009)Google Scholar
  12. 12.
    Laczkovich M.: Nonnegative measurable solutions of difference equations. J. London Math. Soc. (2) 34, 139–147 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Lau K.-S., Zeng W.-B.: The convolution equation of Choquet and Deny on semigroups. Studia Math. 97, 113–135 (1990)MathSciNetGoogle Scholar
  14. 14.
    Oxtoby J.C.: Measure and Category. Springer, New York (1971)CrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Authors and Affiliations

  1. 1.Faculty of Mathematics, Computer Science and EconometricsUniversity of Zielona GóraZielona GoraPoland

Personalised recommendations