MPEJ Volume 6, No.3, 67 pp.
Received: Oct 28  1999, Revised: Mar 21 2000, Accepted: Mar 22 2000

A. Schenkel, J. Wehr, P. Wittwer:
Computer-assisted proofs for fixed point problems in Sobolev spaces


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 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.