  The block-iterative projections (BIP) method of Aharoni and Censor
[Block-iterative projection methods for parallel computation of solutions to
convex feasibility problems, Linear Algebra and its Applications 120, (1989),
165--175] is an iterative process for finding asymptotically a point in the
nonempty intersection of a family of closed convex subsets. It employs
orthogonal projections onto the individual subsets in an algorithmic regime
that uses "blocks" of operators and has great flexibility in constructing
specific algorithms from it. We extend this algorithmic scheme to handle a
family of continuous cutter operators and to find a common fixed point of them.
Since the family of continuous cutters includes several important specific
operators, our generalized scheme, which ensures global convergence and retains
the flexibility of BIP, can handle, in particular, metric (orthogonal)
projectors and continuous subgradient projections, which are very important in
applications. We also allow a certain kind of adaptive perturbations to be
included, and along the way we derive a perturbed Fej\'er monotonicity lemma
which is of independent interest.
