### Abstract

In this paper we extend the technique of computer-assisted proofs to fixed point problems in Sobolev spaces. Up to now the method was limited to the case of spaces of analytic functions. The possibility to work with Sobolev spaces is an important progress and opens up many new domains of applications. Our discussion is centered around a concrete problem that arises in the theory of critical phenomena and describes the phase transition in a hierarchical system of random resistors. For this problem we have implemented in particular the convolution product based on the Fast Fourier Transform (FFT) algorithm with rigorous error estimates.

Original language | English (US) |
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Pages (from-to) | 1-67 |

Number of pages | 67 |

Journal | Mathematical Physics Electronic Journal |

Volume | 6 |

State | Published - Dec 1 2000 |

### Keywords

- Computer-assisted proofs
- Constructive analysis in Sobolev spaces
- Discrete convolutions are convolutions of splines
- Phase transitions in random media

### ASJC Scopus subject areas

- Statistical and Nonlinear Physics
- Statistics and Probability

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## Cite this

Schenkel, A., Wehr, J., & Wittwer, P. (2000). Computer-assisted proofs for fixed point problems in Sobolev spaces.

*Mathematical Physics Electronic Journal*,*6*, 1-67.