On a hybrid method for a family of relatively nonexpansive mappings in a Banach space☆

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Abstract

We prove strong convergence theorems by the hybrid method given by Takahashi, Takeuchi, and Kubota 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.

Keywords

Nonexpansive mapping
Relatively nonexpansive mapping
Hybrid method
Approximation
Fixed point
Maximal monotone operator
Resolvent
Metric projection
Generalized projection

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The authors are respectively supported by Grant-in-Aid for Scientific Research No. 19740065 and No. 19540167 from Japan Society for the Promotion of Science.

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