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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.

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)
Full Text: DOI
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[2] Osilike, M. O.: Implicit iteration process for common fixed points of a finite family of strictly pseuocontractive maps. J. math. Anal. appl. 294, 73-84 (2004) · Zbl 1045.47056
[3] Zhou, Y. Y.; Chang, S. S.: Convergence of implicit iteration processes for a finite family of asymptotically nonexpansive mappings in Banach spaces. Numer. funct. Anal. optim. 23, 911-921 (2002) · Zbl 1041.47048
[4] Bauschke, H. H.: The approximation of fixed points of composition of nonexpansive mappings in Hilbert space. J. math. Anal. appl. 202, 150-159 (1996) · Zbl 0956.47024
[5] Gornicki, J.: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Comment. math. Univ. carolin. 301, 249-252 (1998)
[6] Schu, J.: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull. austral. Math. soc. 43, 153-159 (1991) · Zbl 0709.47051