## On a hybrid method for a family of relatively nonexpansive mappings in a Banach space.(English)Zbl 1166.47058

Summary: We prove strong convergence theorems by the hybrid method given by W. Takahashi, Y. Takeuchi and R. Kubota [J. Math. Anal. Appl. 341, No. 1, 276–286 (2008; Zbl 1134.47052)] for a family of relatively nonexpansive mappings under weaker conditions. The method of the proof is different from the original one and it shows that the type of projection used in the iterative method is independent of the properties of the mappings. We also deal with the problem of finding a zero of a maximal monotone operator and obtain a strong convergence theorem using this method.

### MSC:

 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H05 Monotone operators and generalizations

Zbl 1134.47052
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### References:

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