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Convergence of an implicit iteration process for a finite family of total asymptotically pseudocontractive maps. (English) Zbl 1220.47106
From the introduction: In 2002, {\it Y. Zhou} and {\it S.-S. Chang} [Numer. Funct. Anal. Optimization 23, No. 7--8, 911--921 (2002; Zbl 1041.47048)] introduced the following implicit iteration scheme for common fixed points of a finite family of asymptotically nonexpansive mappings $\{T_i\}^N_{i=1}$ in Banach space: $$x_n=\alpha_nx_{n-1}+(1-\alpha_n)T^n_{n{\pmod N}}x_n.\tag1$$ By this implicit iteration scheme, Zhou and Chang proved some weak and strong convergence theorems in Banach spaces for a finite family of nonexpansive mappings. In this paper, we prove a new convergence theorem of implicit iteration (1) process to a common fixed point for a finite family of total asymptotically pseudocontractive mappings. The results extend those of [{\it S. S. Chang, K. K. Tan, H. W. J. Lee} and {\it C. K. Chan}, J. Math. Anal. Appl. 313, No. 1, 273--283 (2006; Zbl 1086.47044)].

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
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