A fixed-point theorem of Krasnoselskii Original Research Article

Krasnoselskiiʹs fixed-point theorem asks for a convex set M and a mapping Pz = Bz + Az such that: 1. (i) Bx + Ay ∈ M for each x, y ∈ M 2. (ii) A is continuous and compact 3. (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 ∈ M when x = Bx + Ay. The proof also yields a technique for showing that such x is in M.