Means of iterates

  • Szymon Draga
  • Janusz Morawiec
Open Access


We determine continuous bijections f, acting on a real interval into itself, whose k-fold iterate is the quasi-arithmetic mean of all its subsequent iterates from \(f^0\) up to \(f^n\) (where \(0\leqslant k\leqslant n\)). Namely, we prove that if at most one of the numbers kn is odd, then such functions consist of at most three affine pieces.


Continuous solution Iterate Polynomial-like iterative equation 

Mathematics Subject Classification

Primary 39B22 Secondary 26A18 39B12 


  1. 1.
    Draga, S.: A note on the polynomial-like iterative equations order. Comment. Math. 56, 243–249 (2016)MathSciNetGoogle Scholar
  2. 2.
    Draga, S., Morawiec, J.: Reducing the polynomial-like iterative equations order and a generalized Zoltán Boros problem. Aequ. Math. 90, 935–950 (2016)CrossRefzbMATHGoogle Scholar
  3. 3.
    Jarczyk, W.: On an equation of linear iteration. Aequ. Math. 51, 303–310 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Jerri, A.J.: Linear Difference Equations with Discrete Transform Methods. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  5. 5.
    Kuczma, M: Functional Equations in a Single Variable. Monografie Matematyczne, vol. 46. PWN, Warsaw (1968)Google Scholar
  6. 6.
    Li, L., Zhang, W.: Continuously decreasing solutions for polynomial-like iterative equations. Sci. China Math. 56, 1051–1058 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Matkowski, J.: Remark done during the twenty-sixth international symposium on functional equations. Aequ. Math. 37, 119 (1989)Google Scholar
  8. 8.
    Matkowski, J., Zhang, W.: Method of characteristic for functional equations in polynomial form. Acta Math. Sin. 13, 421–432 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Matkowski, J., Zhang, W.: Characteristic analysis for a polynomial-like iterative equation. Chin. Sci. Bull. 43, 192–196 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Mukherjea, A., Ratti, J.S.: On a functional equation involving iterates of a bijection on the unit interval. Nonlinear Anal. 7, 899–908 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Mukherjea, A., Ratti, J.S.: On a functional equation involving iterates of a bijection on the unit interval. Nonlinear Anal. 31, 459–464 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Nabeya, S.: On the functional equation \(f(p+qx+rf(x))=a+bx+cf(x)\). Aequ. Math. 11, 199–211 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Ratti, J.S., Lin, Y.F.: A functional equation involving \(f\) and \(f^{-1}\). Colloq. Math. 60(61), 519–523 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Yang, D., Zhang, W.: Characteristic solutions of polynomial-like iterative equations. Aequ. Math. 67, 80–105 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zhang, W., Baker, A.J.: Continuous solutions of a polynomial-like iterative equation with variable coefficients. Ann. Polon. Math. 73, 29–36 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Zhang, P., Gong, X.: Continuous solutions of 3-order iterative equation of linear dependence. Adv. Differ. Equ. 2014, 318 (2014)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Zhang, W., Xu, B., Zhang, W.: Global solutions for leading coefficient problem of polynomial-like iterative equations. Results Math. 63, 79–93 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Zhang, W., Zhang, W.: On continuous solutions of \(n\)-th order polynomial-like iterative equations. Publ. Math. Debr. 76, 117–134 (2010)MathSciNetzbMATHGoogle Scholar

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© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Diebold Nixdorf BPOKatowicePoland
  2. 2.Instytut MatematykiUniwersytet Śla̧skiKatowicePoland

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