Computer-assisted proofs for fixed point problems in Sobolev spaces

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.