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Zbl 1236.47074
Nilsrakoo, Weerayuth
Halpern-type iterations for strongly relatively nonexpansive mappings in Banach spaces. (English)
[J] Comput. Math. Appl. 62, No. 12, 4656-4666 (2011). ISSN 0898-1221

Summary: We note that the main convergence theorem in [{\it C.-J. Zhang}, {\it J.-L. Li} and {\it B.-Q. Liu}, Comput. Math. Appl. 61, No. 2, 262--276 (2011; Zbl 1211.65063)] is incorrect and we prove a correction. We also modify Halpern's iteration for finding a fixed point of a strongly relatively nonexpansive mapping in a Banach space. Consequently, two strong convergence theorems for a relatively nonexpansive mapping and for a mapping of firmly nonexpansive type are deduced. Using the concept of duality theorems, we obtain analogous results for strongly generalized nonexpansive mappings and for mappings of firmly generalized nonexpansive type. In addition, we study two strong convergence theorems concerning two types of resolvents of a maximal monotone operator in a Banach space.

MSC 2000:: *47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties

Keywords: firmly generalized nonexpansive type mapping; maximal monotone operator; relatively nonexpansive mapping; strongly generalized nonexpansive mapping; strongly relatively nonexpansive mapping

Citations: Zbl 1211.65063

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