Convergence of sequential and asynchronous nonlinear paracontractions. (English) Zbl 0763.65035

A continuous map \(T: \mathbb{R}^ k\to \mathbb{R}^ k\) is said to be paracontracting if for any fixed point \(y\in \mathbb{R}^ k\) of \(T\) and any \(x\in \mathbb{R}^ k\) either \(\| T(x)-y\|<\| x-y\|\) or \(T(x)=x\). The paper then considers iterative processes \(x_ i=T_{j_ i}(x_{i- 1})\), \(i=1,2,\dots\), where the maps \(T_ j\) are chosen from a finite pool of paracompact operators on \(\mathbb{R}^ k\). It is shown that the iteration converges exactly if there is a common fixed point of those operators that occur infinitely often in the sequence and that then the limit is one such fixed point. A similar theorem is proved for an asynchronous version of the process. The results are used for solving a linear system of equations subject to a convex constraint. Finally an extension to an infinite pool of operators is discussed.


65H10 Numerical computation of solutions to systems of equations
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