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Zbl 1222.47123
Wang, Lin; Yildirim, Isa; Ozdemir, Murat
(Yildirim, \D{I}sa; Özdemir, Murat)
Strong convergence of an implicit iteration process for two asymptotically nonexpansive mappings in Banach spaces. (English)
[J] An. Ştiinţ. Univ. ``Ovidius" Constanţa, Ser. Mat. 18, No. 2, 281-294 (2010). ISSN 1223-723X; ISSN 1224-1784/e

Let $E$ be a Banach space, $K$ a nonempty closed convex subset of $E$ and $S,T:K\rightarrow K$ two asymptotically nonexpansive mappings. In order to obtain a common fixed point of $S$ and $T$ in uniformly convex Banach spaces, the authors consider an implicit iterative scheme $\{x_n\}$ of the form $$x_n=\alpha_n x_{n-1}+\beta_n T^{n-1}x_{n-1}+\gamma_n T^{n}y_{n},$$ $$y_n=\alpha_n' x_{n-1}+\beta_n' S^{n-1}x_{n-1}+\gamma_n' S^{n}x_{n},$$ for which a corresponding convergence theorem (Theorem 3.3) is proven.
[Vasile Berinde (Baia Mare)]

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47H09 Mappings defined by "shrinking" properties

Keywords: Banach space; asymptotically nonexpansive mapping; common fixed point; implicit iterative scheme; convergence theorem

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