Constructive Approximation

, Volume 45, Issue 2, pp 273–299 | Cite as

Constructive Solutions for Systems of Iterative Functional Equations

  • Cristina Serpa
  • Jorge Buescu


We formulate a general theoretical framework for systems of iterative functional equations between general spaces X and Y. We find general necessary conditions for the existence of solutions. When X and Y are topological spaces, we characterize continuity of solutions; when XY are metric spaces, we find sufficient conditions for existence and uniqueness. For finite-order systems, we construct explicit formulae for the solution. We provide an extended list of examples, including fractal interpolation functions, which are covered by our general framework.


Functional equation Explicit solution De Rham’s function Fractal interpolation Iteration methods 

Mathematics Subject Classification

39B72 39B12 28A80 41A05 



The authors gratefully acknowledge the valuable suggestions of the anonymous referees. The first author acknowledges support from Fundação para a Ciência e Tecnologia under Grant SFRH/BD/77623/2011. The second author was partially supported by FCT via UID/MAT/04561/2013.


  1. 1.
    Barnsley, M.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Barnsley, M., Hegland, M., Massopust, P.: Numerics and fractals. Bull. Inst. Math. Acad. Sin. (N.S.) 9(3), 389–430 (2014)MathSciNetMATHGoogle Scholar
  3. 3.
    Barnsley, M., Harrington, A.: The calculus of fractal interpolation functions. J. Approx. Theory 57, 14–34 (1989)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Barnsley, M., Vince, A.: Fractal Continuation. Constr. Approx. 38, 311–337 (2013)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Cesàro, E.: Lezioni di Geometria Intrinseca. Presso L’Autore-Editore, Harvard University, Cambridge (1896)MATHGoogle Scholar
  6. 6.
    de Rham, G.: Sur quelques courbes définies par des équations fonctionnelles. Rend. Sem. Math. Torino 16, 101–113 (1956)Google Scholar
  7. 7.
    du Bois-Reymond, P.: Versuch einer Classifikation der willkürlichen Funktionen reeller Argumente nach ihren Änderungen in den kleinsten Intervallen. J. Reine Angew. Math. 79, 21–37 (1875)MathSciNetGoogle Scholar
  8. 8.
    Girgensohn, R.: Functional equations and nowhere differentiable functions. Aequ. Math. 46, 243–256 (1993)MathSciNetMATHGoogle Scholar
  9. 9.
    Jürgens, H., Peitgen, H.O., Saupe, D.: Chaos and Fractals: New Frontiers of Science. Springer, New York (1992)MATHGoogle Scholar
  10. 10.
    Kannappan, P.: Functional Equations and Inequalities with Applications. Springer, Waterloo (2009)CrossRefMATHGoogle Scholar
  11. 11.
    Kuczma, M.: Functional Equations in a Single Variable, Monografie Mat. 46. Panstwowe Wydawnictwo Naukowe, Warsaw (1968)Google Scholar
  12. 12.
    Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations, Encyclopedia of Mathematics and its Applications, 32. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  13. 13.
    Massopust, P.R.: Fractal Peano curves. J. Geom. 34, 127–138 (1989)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Massopust, P.R.: Fractal Functions, Fractal Surfaces, and Wavelets. Academic Press, Millbrae (1994)MATHGoogle Scholar
  15. 15.
    Massopust, P.R.: Fractal functions and their applications. Chaos Solitons Fractals 8(2), 171–190 (1997)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Minkowski, H.: Zur Geometrie der Zahlen. In: Gesammelte Abhandlungen, vol. 2, pp. 44–52. Chelsea, New York (1991)Google Scholar
  17. 17.
    Pólya, G.: Über eine Peanosche kurve, Bull. Acad. Sci. Cracovie, Ser. A, 305–313 (1913)Google Scholar
  18. 18.
    Polyanin, A.D., Manzhirov, A.V.: Handbook of Integral Equations, 2nd edn. Chapman & Hall/CRC Press, Boca Raton (2008)CrossRefMATHGoogle Scholar
  19. 19.
    Prats’ovytyi, M.V., Kalashnikov, A.V.: Self-affine singular and monotone functions related to the \(Q\)-representation of real numbers. Ukr. Math. J. 65, 448–462 (2013)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Pugachev, V.S., Sinitsyn, I.N.: Lectures on Functional Analysis and Applications. World Scientific, Singapore (1999)CrossRefMATHGoogle Scholar
  21. 21.
    Salem, R.: On some singular monotone functions which are strictly increasing. Trans. Am. Math. Soc. 53, 427–439 (1943)CrossRefMATHGoogle Scholar
  22. 22.
    Serpa, C., Buescu, J.: Piecewise expanding maps and conjugacy equations. In: Nonlinear Maps and Their Applications. Springer Proc. Math. Stat., vol. 112, 193–202, Springer, New York (2015)Google Scholar
  23. 23.
    Serpa, C., Buescu, J.: Explicitly defined fractal interpolation functions with variable parameters. Chaos Solitons Fractals 75, 76–83 (2015)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Serpa, C., Buescu, J.: Non-uniqueness and exotic solutions of conjugacy equations. J. Differ. Equ. Appl. 21(12), 1147–1162 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Smith, H.: An Introduction to Delay Differential Equations with Applications to the Life Sciences. Texts in Applied Mathematics, vol. 57. Springer, New York (2011)Google Scholar
  26. 26.
    van der Waerden, B.L.: Ein einfaches Beispiel einer nicht-differenzierbaren stetigen Funktion. Math. Z. 32, 474–475 (1930)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    von Koch, H.: Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire. Archiv för Matemat. Astron. och Fys. 1, 681–702 (1904)Google Scholar
  28. 28.
    Wang, H.-Y., Yu, J.-S.: Fractal interpolation functions with variable parameters and their analytical properties. J. Approx. Theory 175, 1–18 (2013)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Zdun, M.C.: On conjugacy of some systems of functions. Aequ. Math. 61, 239–254 (2001)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Zwillinger, D.: Handbook of Differential Equations, 2nd edn. Academic Press, London (1957)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Centro de Matemática Aplicações Fundamentais e Investigação Operacional, Faculdade de CiênciasUniversidade de LisboaLisbonPortugal

Personalised recommendations