## Strong convergence theorems for a finite family of nonexpansive mappings.(English)Zbl 1155.47314

Summary: We modify the classical Mann iterative process to obtain a strong convergence theorem for a finite family of nonexpansive mappings in the framework of Hilbert spaces. Our results improve and extend the results announced by many others.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc.
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### References:

 [1] Marino, G; Xu, H-K, A general iterative method for nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 318, 43-52, (2006) · Zbl 1095.47038 [2] Xu, HK, An iterative approach to quadratic optimization, Journal of Optimization Theory and Applications, 116, 659-678, (2003) · Zbl 1043.90063 [3] Mann, WR, Mean value methods in iteration, Proceedings of the American Mathematical Society, 4, 506-510, (1953) · Zbl 0050.11603 [4] Reich, S, Weak convergence theorems for nonexpansive mappings in Banach spaces, Journal of Mathematical Analysis and Applications, 67, 274-276, (1979) · Zbl 0423.47026 [5] Kim, T-H; Xu, H-K, Strong convergence of modified Mann iterations, Nonlinear Analysis: Theory, Methods & Applications, 61, 51-60, (2005) · Zbl 1091.47055 [6] Yao Y, Chen R, Yao J-C: Strong convergence and certain control conditions for modified Mann iteration. to appear in Nonlinear Analysis: Theory, Methods & Applications · Zbl 1068.47085 [7] Takahashi, W; Shimoji, K, Convergence theorems for nonexpansive mappings and feasibility problems, Mathematical and Computer Modelling, 32, 1463-1471, (2000) · Zbl 0971.47040 [8] Suzuki, T, Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals, Journal of Mathematical Analysis and Applications, 305, 227-239, (2005) · Zbl 1068.47085
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