Skip to main content
SpringerLink
Log in

Constructive Solutions for Systems of Iterative Functional Equations

  • Published:
Constructive Approximation Aims and scope

Abstract

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 X, Y 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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Barnsley, M.: Fractal functions and interpolation. Constr. Approx. 2, 303–329 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Barnsley, M., Hegland, M., Massopust, P.: Numerics and fractals. Bull. Inst. Math. Acad. Sin. (N.S.) 9(3), 389–430 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Barnsley, M., Harrington, A.: The calculus of fractal interpolation functions. J. Approx. Theory 57, 14–34 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  4. Barnsley, M., Vince, A.: Fractal Continuation. Constr. Approx. 38, 311–337 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cesàro, E.: Lezioni di Geometria Intrinseca. Presso L’Autore-Editore, Harvard University, Cambridge (1896)

    MATH  Google Scholar 

  6. de Rham, G.: Sur quelques courbes définies par des équations fonctionnelles. Rend. Sem. Math. Torino 16, 101–113 (1956)

    Google Scholar 

  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)

    MathSciNet  Google Scholar 

  8. Girgensohn, R.: Functional equations and nowhere differentiable functions. Aequ. Math. 46, 243–256 (1993)

    MathSciNet  MATH  Google Scholar 

  9. Jürgens, H., Peitgen, H.O., Saupe, D.: Chaos and Fractals: New Frontiers of Science. Springer, New York (1992)

    MATH  Google Scholar 

  10. Kannappan, P.: Functional Equations and Inequalities with Applications. Springer, Waterloo (2009)

    Book  MATH  Google Scholar 

  11. Kuczma, M.: Functional Equations in a Single Variable, Monografie Mat. 46. Panstwowe Wydawnictwo Naukowe, Warsaw (1968)

  12. Kuczma, M., Choczewski, B., Ger, R.: Iterative Functional Equations, Encyclopedia of Mathematics and its Applications, 32. Cambridge University Press, Cambridge (1990)

    Book  MATH  Google Scholar 

  13. Massopust, P.R.: Fractal Peano curves. J. Geom. 34, 127–138 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Massopust, P.R.: Fractal Functions, Fractal Surfaces, and Wavelets. Academic Press, Millbrae (1994)

    MATH  Google Scholar 

  15. Massopust, P.R.: Fractal functions and their applications. Chaos Solitons Fractals 8(2), 171–190 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  16. Minkowski, H.: Zur Geometrie der Zahlen. In: Gesammelte Abhandlungen, vol. 2, pp. 44–52. Chelsea, New York (1991)

  17. Pólya, G.: Über eine Peanosche kurve, Bull. Acad. Sci. Cracovie, Ser. A, 305–313 (1913)

  18. Polyanin, A.D., Manzhirov, A.V.: Handbook of Integral Equations, 2nd edn. Chapman & Hall/CRC Press, Boca Raton (2008)

    Book  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pugachev, V.S., Sinitsyn, I.N.: Lectures on Functional Analysis and Applications. World Scientific, Singapore (1999)

    Book  MATH  Google Scholar 

  21. Salem, R.: On some singular monotone functions which are strictly increasing. Trans. Am. Math. Soc. 53, 427–439 (1943)

    Article  MATH  Google Scholar 

  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)

  23. Serpa, C., Buescu, J.: Explicitly defined fractal interpolation functions with variable parameters. Chaos Solitons Fractals 75, 76–83 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Serpa, C., Buescu, J.: Non-uniqueness and exotic solutions of conjugacy equations. J. Differ. Equ. Appl. 21(12), 1147–1162 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  26. van der Waerden, B.L.: Ein einfaches Beispiel einer nicht-differenzierbaren stetigen Funktion. Math. Z. 32, 474–475 (1930)

    Article  MathSciNet  MATH  Google Scholar 

  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. Wang, H.-Y., Yu, J.-S.: Fractal interpolation functions with variable parameters and their analytical properties. J. Approx. Theory 175, 1–18 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Zdun, M.C.: On conjugacy of some systems of functions. Aequ. Math. 61, 239–254 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  30. Zwillinger, D.: Handbook of Differential Equations, 2nd edn. Academic Press, London (1957)

    MATH  Google Scholar 

Download references

Acknowledgments

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.

Author information

Authors and Affiliations

  1. Centro de Matemática Aplicações Fundamentais e Investigação Operacional, Faculdade de Ciências, Universidade de Lisboa, 1749-016, Lisbon, Portugal

    Cristina Serpa & Jorge Buescu

Authors
  1. Cristina Serpa
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jorge Buescu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Cristina Serpa.

Additional information

Communicated by Wolfgang Dahmen.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Serpa, C., Buescu, J. Constructive Solutions for Systems of Iterative Functional Equations. Constr Approx 45, 273–299 (2017). https://doi.org/10.1007/s00365-016-9349-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-016-9349-z

Keywords

Mathematics Subject Classification

Search

Navigation