Extrapolation and local acceleration of an iterative process for common fixed point problems

We consider sequential iterative processes for the common fixed point problem of families of cutter operators on a Hilbert space. These are operators that have the property that, for any point x ∈ H , the hyperplane through T x whose normal is x − T x always “cuts” the space into two half-spaces, one of which contains the point x while the other contains the (assumed nonempty) fixed point set of T . We define and study generalized relaxations and extrapolation of cutter operators, and construct extrapolated cyclic cutter operators. In this framework we investigate the Dos Santos local acceleration method in a unified manner and adopt it to a composition of cutters. For these, we conduct a convergence analysis of successive iteration algorithms.