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

TL;DR

This paper characterizes the sets of relaxation parameters for which fixed points exist in compositions of relaxed projection operators in Hilbert spaces, revealing the topological complexity of these parameter sets.

Contribution

It provides a complete characterization of the parameter sets ensuring fixed points for compositions of relaxed projections in Hilbert spaces, especially for dimensions three and higher.

Findings

The fixed point existence depends on the relaxation parameter set F.

Sets F are exactly the F_sigma subsets of [0,1] containing 0 for certain dimensions and compositions.

The characterization applies to compositions with at least three operators in spaces of dimension three or more.

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 $\dim H\geq 3$ 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 $0$.