Some results for conjugate equations

Abstract

In this paper we consider a class of conjugate equations, which generalizes de Rham’s functional equations. We give sufficient conditions for the existence and uniqueness of solutions under two different series of assumptions. We consider regularity of solutions. In our framework, two iterated function systems are associated with a series of conjugate equations. We state local regularity by using the invariant measures of the two iterated function systems with a common probability vector. We give several examples, especially an example such that infinitely many solutions exists, and a new class of fractal functions on the two-dimensional standard Sierpiński gasket which are not harmonic functions or fractal interpolation functions. We also consider a certain kind of stability.

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Acknowledgements

The author was supported by JSPS KAKENHI Grant-in-Aid for JSPS Fellows (16J04213) and for Research activity Start-up (18H05830). This work was also supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.

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Correspondence to Kazuki Okamura.

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Okamura, K. Some results for conjugate equations. Aequat. Math. 93, 1051–1084 (2019). https://doi.org/10.1007/s00010-018-0633-9

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Mathematics Subject Classification

  • 39B72
  • 39B12
  • 28A80
  • 26A30

Keywords

  • Conjugate equations
  • De Rham’s functional equations
  • Iterated function systems