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A geometrical look at iterative methods for operators with fixed points. (English) Zbl 1074.65064

The authors classify operators \(T\) on a real Hilbert space \({\mathcal H}\) with fixed point set \(\text{ Fix}\;T \neq \emptyset\) in terms of the families \({\mathcal F}^{\nu} = \{ T \;:\;\langle x - Tx, z - Tx \rangle \leq {{1-\nu} \over {2}} \| x -Tx \|^2,\forall x \in {\mathcal H}, \forall z \in \text{Fix}\, T\}\) and develop geometrical results about fixed point sets centered on the real parameter \(\nu \geq 0\). In particular, Fejèr-monotone sequences \(\{ x_n \}_0^{\infty} \subset {\mathcal H}\) with respect to some \(F \subset {\mathcal H}\) are considered in the sense that \(\| x_{n+1} - x_n \| \leq \| x_n - z \|, \forall z \in F\). For a sequence \(\{ T_n \}_0^{\infty} \subset {\mathcal F}^{\nu}\) this leads to the characterization that any \(\{ x_n \}_0^{\infty} \subset {\mathcal H}\) is Fejèr-monotone with respect to a closed, convex \(F \subset \bigcap_{n \geq 0} \text{ Fix}\;T_n\) if and only if \(x_{n+1} = (1 +\nu)T_n x_n - \nu x_n\). This in turns generalizes to an algorithm for constructing common fixed points of a sequence of operators, extending results of Y. Haugazeau [Thése, Univ. de Paris (1968)], and to the study of orbits of this algorithm.

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
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