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

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 $\dim H\geq 3$ and $k\geq3$ then the class of the derscribed above sets $F$ of coefficients $\alpha$ is exactly the class of $F_\sigma$ subsets of $[0,1]$ containing $0$.