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Zbl 1116.65064
Su, Yongfu; Qin, Xiaolong
General iteration algorithm and convergence rate optimal model for common fixed points of nonexpansive mappings. (English)
[J] Appl. Math. Comput. 186, No. 1, 271-278 (2007). ISSN 0096-3003

Let there be a finite set of nonexpansive operators mapping a closed convex subset of a real uniformly convex Banach space into itself. The Banach space is supposed to fulfill Opial's condition, i.e. if $(x_n)$ converges weakly towards $x$ then $\limsup \|x_n-x\| < \limsup \|x_n - y\|$ for all $y\neq x$. The set of operators is supposed to possess a common fixed point. The authors prove weak and -- under an additional assumption -- strong convergence of a general implicit composite iteration towards a common fixed point. The Mann iteration is a special case of this general iteration. Finally, the authors study the optimal choice of the iteration parameters and the rate of convergence.
[Etienne Emmrich (Berlin)]

MSC 2000:: *65J15 Equations with nonlinear operators (numerical methods)
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47J25 Methods for solving nonlinear operator equations (general)

Keywords: nonexpansive mapping; common fixed point; iterative approximation; composite implicite iteration; Mann iteration; Banach space; convergence

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