## 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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### References:

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