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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, Y. Zhou and 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 ${\left\{{T}_{i}\right\}}_{i=1}^{N}$ in Banach space:

${x}_{n}={\alpha }_{n}{x}_{n-1}+\left(1-{\alpha }_{n}\right){T}_{n\phantom{\rule{10.0pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}N\right)}^{n}{x}_{n}·\phantom{\rule{2.em}{0ex}}\left(1\right)$

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 [S. S. Chang, K. K. Tan, H. W. J. Lee and C. K. Chan, J. Math. Anal. Appl. 313, No. 1, 273–283 (2006; Zbl 1086.47044)].

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties