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

We consider an $α$-relaxed projection $P_A^α:H\to H$ given by $P_A^α(x)=αP_A(x)+(1-α)x$ where $α\in[0,1]$ 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 $P_{A_k}^αP_{A_{k-1}}^α\dots P_{A_1}^α$ has a fixed point iff $α\in F$. It proves, that if $\dim H\geq 3$ and $k\geq3$ then the class of the derscribed above sets $F$ of coefficients $α$ is exactly the class of $F_σ$ subsets of $[0,1]$ containing $0$.

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