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Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications. (English) Zbl 1203.47070
Summary: We prove a strong convergence theorem by the hybrid method for a countable family of relatively nonexpansive mappings in a Banach space. We also establish a new control condition for the sequence of mappings $\{T_n\}$ which is weaker than the control condition in Lemma 3.1 of [{\it K. Aoyama, Y. Kimura, W. Takahashi} and {\it M. Toyoda}, Nonlinear Anal., Theory Methods Appl. 67, No. 8, A, 2350--2360 (2007; Zbl 1130.47045)]. Moreover, we apply our results for finding a common fixed point of two relatively nonexpansive mappings in a Banach space and an element of the set of solutions of an equilibrium problem in a Banach space, respectively. Our results are applicable to a wide class of mappings.

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