Report WCNA 2000 manuscript On λ-firmly nonexpansive mappings on nonconvex sets by W. Kaczor.

The paper contains an interesting improvement of the following result of Smarzewski.

Let X be a uniformly convex Banach space, let C = ⋃_{k=1}^{n} C_{k} be a union of nonempty, bounded, closed convex subsets C_{k} of X, and suppose T : C → C is λ-firmly nonexpansive for some λ ∈ (0, 1). Then T has a fixed point in C.

In the main theorem of the paper under consideration is obtained the same conclusion under weaker requirements both on X and on the set C.

Consequently I am pleased to recommend the article for publication in the Proceedings.