Extrapolation and Local Acceleration of an Iterative Process for Common Fixed Point Problems

TL;DR

This paper develops and analyzes accelerated iterative methods for finding common fixed points of cutter operators in Hilbert spaces, introducing generalized relaxations, extrapolation techniques, and convergence results.

Contribution

It introduces generalized relaxations and extrapolation of cutter operators, and applies local acceleration methods to improve convergence in fixed point algorithms.

Findings

Successful construction of extrapolated cyclic cutter operators.

Convergence analysis of accelerated iterative algorithms.

Unified framework for local acceleration of fixed point processes.

Abstract

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\inH, the hyperplane through Tx whose normal is x-Tx 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 convergence analysis of successive iteration algorithms.