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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:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H05Monotone operators (with respect to duality) and generalizations