Constructive Solutions for Systems of Iterative Functional Equations


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.

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  1. 1.

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

    MathSciNet  Article  MATH  Google Scholar 

  2. 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. 3.

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

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

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

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

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

    Google 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)

    MathSciNet  Google Scholar 

  8. 8.

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

    MathSciNet  MATH  Google Scholar 

  9. 9.

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

    Google Scholar 

  10. 10.

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

    Google Scholar 

  11. 11.

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

  12. 12.

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

    Google Scholar 

  13. 13.

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

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

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

    Google Scholar 

  15. 15.

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

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

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

  17. 17.

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

  18. 18.

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

    Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

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

    Google Scholar 

  21. 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. 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. 23.

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

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

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

    MathSciNet  Article  MATH  Google 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)

  26. 26.

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

    MathSciNet  Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

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

    MathSciNet  Article  MATH  Google Scholar 

  30. 30.

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

    Google Scholar 

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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.

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Correspondence to Cristina Serpa.

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Communicated by Wolfgang Dahmen.

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Serpa, C., Buescu, J. Constructive Solutions for Systems of Iterative Functional Equations. Constr Approx 45, 273–299 (2017).

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

Mathematics Subject Classification

  • 39B72
  • 39B12
  • 28A80
  • 41A05