On the fixed points of affine nonexpansive mappings (Q1607795)

Let \(K\) be a closed bounded subset of a Banach space \(X\) and let \(T:K\to K\) be a continuous affine mapping. It is proved that: (1) if \(T\) is nonexpansive, then it has a fixed point; (2) if \(T\) has only one fixed point, then the mapping \(A=(I+T)/2\) is a focusing mapping; (3) a continuous mapping \(S:K\to K\) has a fixed point if and only if, for each \(x\in K\), \(\|(A^n\circ S)(x)-(S\circ A^n)(x)\|\to 0\) for some strictly nonexpansive affine mapping \(T\).