A fixed-point theorem of Krasnoselskii

Krasnoselskii's fixed-point theorem asks for a convex set M and a mapping Pz = Bz + Az such that: (i) Bx + Ay is an element of M for each x, y is an element of M, (ii) A is continuous and compact, (iii) B is a contraction. Then P has a fixed point. A careful reading of the proof reveals that (i) need only ask that Bx + Ay is an element of M when x = Bx + Ay. The proof also yields a technique for showing that such x is in M.