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General iteration algorithm and convergence rate optimal model for common fixed points of nonexpansive mappings. (English) Zbl 1116.65064
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.

MSC:
65J15Equations with nonlinear operators (numerical methods)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47J25Iterative procedures (nonlinear operator equations)
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References:
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