On the fixed points of affine nonexpansive mappings
Hülya Duru · International Journal of Mathematics and Mathematical Sciences · 2001
Let K be a closed convex bounded subset of a Banach space X and let T : K → K be a continuous affine mapping. In this note, we show that (a) if T is nonexpansive then it has a fixed point, (b) if T has only one fixed point then the mapping A = (I + T)/2 is a focusing mapping; and (c) a continuous mapping S : K → K has a fixed point if and only if, for each x ∈ k, ‖(An∘S)(x) − (S∘An)(x)‖ → 0for some strictly nonexpansive affine mapping T.