  This paper discusses a more general contractive condition for a class of
extended cyclic self-mappings on the union of a finite number of subsets of a
metric space which are allowed to have a finite number of successive images in
the same subsets of its domain. If the space is uniformly convex and the
subsets are non-empty, closed and convex then all the iterates converge to a
unique closed limiting finite sequence which contains the best proximity points
of adjacent subsets and reduces to a unique fixed point if all such subsets
intersect.
