Extrapolation and local acceleration of an iterative process for common fixed point problems (Q442508)

The article deals with iterative processes for common fixed point problems including so-called cutter operators. An operator \(T:\;H \to H\) in a Hilbert space \(H\) is called a cutter operator (sometimes firmly quasi-nonexpansive or directed or even separating one) if \(\text{Fix}\, T \subseteq H(x,Tx)\) for all \(x \in H\); here \(H(x,y) = \{u \in H|\;(u - y,x - y) \leq 0\}\). The class of cutter operators includes Goebel-Reich operators (\(\|Tx - Ty\|^2 \leq (Tx - Ty,x - y)\), \(x, y \in H\)), resolvents of maximal monotone operators, orthogonal and subgradient projections and others; in its turn this class is contained in the class of Crombez (or paracontracting) operators (\(\|Tx - q\| \leq \|x - q\|\), \(q \in \text{Fix}\, T\) and \(x \in H\)). The main part of the article deals with a finite family of cutter operators \(U_i:\;H \to H\), \(i = 1,\ldots,m\), with \(\bigcap_{i=1}^m \text{Fix}\, U_i \neq \emptyset\). The authors consider the iteration \(x^{k+1} = U_{\sigma,\lambda_k}x^k\) of the operator NEWLINE\[NEWLINEU_{\sigma,\lambda}x = x + \lambda\sigma(x)(Ux - x), \quad U = U_mU_{m-1} \cdots U_1,NEWLINE\]NEWLINE where \(\lambda \in (0,2)\) is a relaxation parameter, and \(\sigma:\;H \to (0,+\infty)\) is a step size function. The main theorem describes conditions under that the sequence of iterates \(x^k\) weakly converges to a fixed point of \(U\). The general result unifies and generalizes several existing local acceleration schemes. In particular, local acceleration of the sequential Kaczmarz method for linear equations, local acceleration of the sequential cyclic projection method for linear inequalities, and local acceleration of the sequential cyclic subgradient projection method are considered in the article.