Consider a finite collection of firmly nonexpansive self-mappings on a Hilbert space whose fixed-point sets intersect. It is shown that, in the finite-dimensional case, any iteration of mappings drawn from this collection converges. This resolves, for the finite-dimensional case at least, a popular conjecture concerning the convergence of the successive projection method. In the infinite-dimensional case, it is shown that if the mappings are drawn according to a certain order, called the quasi-cyclic order, then the iteration converges weakly in a sense. The quasi-cyclic order may be viewed as an extension of the well-known cyclic order in which the lengths of the cycles are permitted to grow without bound.
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