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Strong convergence theorems for the implicit iteration process for a finite family of hemicontractive mappings in Banach space. (English) Zbl 1179.47059

Summary: The purpose of this work is to study the following implicit iteration scheme \[ x_n = \alpha _{n}x_{n-1} + (1- \alpha _{n})T_{n}x_{n},\quad n \geq 1, \] where \(T_n = T_{n\bmod N}\), and to prove several strongly convergent theorems of the iteration for a finite family of hemicontractive mappings in Banach space. Our results extend a recent result of H.-Y.Zhou [Nonlinear Anal., Theory Methods Appl.68, No.10 (A), 2977–2983 (2008; Zbl 1145.47055)] and H.-K.Xu and R.G.Ori [Numer.Funct.Anal.Optimization 22, No.5–6, 767–773 (2001; Zbl 0999.47043)], and we also prove that the sequence \(\{x_n\}\) converges strongly to a common fixed point of a finite family of hemicontractive mappings \(\{T_i\}^N_{i=1}\).

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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References:

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