  To provide generalized solutions if a given problem admits no actual solution
is an important task in mathematics and the natural sciences. It has a rich
history dating back to the early 19th century when Carl Friedrich Gauss
developed the method of least squares of a system of linear equations - its
solutions can be viewed as fixed points of averaged projections onto
hyperplanes. A powerful generalization of this problem is to find fixed points
of averaged resolvents (i.e., firmly nonexpansive mappings).
  This paper concerns the relationship between the set of fixed points of
averaged resolvents and certain fixed point sets of compositions of resolvents.
It partially extends recent work for two mappings on a question of C. Byrne.
The analysis suggests a reformulation in a product space.
  Furthermore, two new algorithms are presented. A complete convergence proof
that is based on averaged mappings is provided for the first algorithm. The
second algorithm, which currently has no convergence proof, iterates a mapping
that is not even nonexpansive. Numerical experiments indicate the potential of
these algorithms when compared to iterating the average of the resolvents.
