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Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces. (English) Zbl 05862809
Summary: Weak and strong convergence theorems are proved in real Hilbert spaces for a new class of nonspreading-type mappings more general than the class studied recently in [Y. Kurokawa and W. Takahashi, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 73, No. 6, 1562–1568 (2010; Zbl 1229.47117)]. We explore an auxiliary mapping in our theorems and proofs and this also yields a strong convergence theorem of Halpern type for our class of mappings and hence resolves in the affirmative an open problem posed by Kurokawa and Takahashi [loc. cit.] in their final remark for the case where the mapping $T$ is averaged.
##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H25 Nonlinear ergodic theorems 47H09 Mappings defined by “shrinking” properties