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Strong convergence of an implicit iteration process for two asymptotically nonexpansive mappings in Banach spaces. (English) Zbl 1222.47123

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.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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