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Attractive point and weak convergence theorems for normally \(N\)-generalized hybrid mappings in Hilbert spaces. (English) Zbl 1424.47123
Summary: In this paper, we introduce a new class of nonlinear mappings that contains generalized hybrid mappings [P. Kocourek et al., Taiwanese J. Math. 14, No. 6, 2497–2511 (2010; Zbl 1226.47053)], normally generalized hybrid mappings [W. Takahashi et al., J. Nonlinear Convex Anal. 13, No. 4, 745–757 (2012; Zbl 1272.47068)] and \(N\)-generalized hybrid mappings [T. Maruyama et al., J. Nonlinear Convex Anal. 12, No. 1, 185–197 (2011; Zbl 1217.47098)] as special mappings. After proving an attractive point theorem that guarantees the existence of attractive points, we establish both Baillon’s type Mann’s type weak convergence theorems of finding attractive points which are demonstrated without assuming that the domain of the mapping is closed. For the Baillon’s type theorem, even convexity is dispensable. The results in this paper simultaneously extend many existing results in the literature.

MSC:
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
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