On the stability of the existence of fixed points for the projection-iterative methods with relaxation

Abstract

We consider an $\alpha$-relaxed projection $P_A^{\alpha }:H\to H$ given by $P_A^{\alpha }(x)=\alpha P_A(x)+(1-\alpha )x$ where $\alpha \in [0,1]$ and $P_A$ is the projection onto a non-empty, convex and closed subset $A$ of the real Hilbert space $H$. We characterise all the sets $F\subset [0,1]$ such that for some non-empty, convex and closed subsets $A_1,A_2,\dots ,A_k\subset H$ the composition $P_{A_k}^{\alpha }{P}_{A_{k-1}}^{\alpha }\dots P_{A_1}^{\alpha }$ has a fixed point iff $\alpha \in F$. It proves, that if $\dim H\ge 3$ and $k\ge 3$ then the class of the derscribed above sets $F$ of coefficients $\alpha$ is exactly the class of $F_{\sigma }$ subsets of $[0,1]$ containing $0$.