Constructive Solutions for Systems of Iterative Functional Equations

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 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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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.

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Correspondence to Cristina Serpa.

Additional information

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

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Keywords

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

Mathematics Subject Classification

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