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Fixed point theorems for some new nonlinear mappings in Hilbert spaces. (English) Zbl 1315.47044

Summary: In this paper, we introduce two new classes of nonlinear mappings in Hilbert spaces. These two classes of nonlinear mappings contain some important classes of nonlinear mappings, like nonexpansive mappings and nonspreading mappings. We prove fixed point theorems, ergodic theorems, demiclosed principles, and Ray’s type theorem for these nonlinear mappings. {
}Next, we prove weak convergence theorems for Moudafi’s iteration process for these nonlinear mappings. Finally, we give some important examples for these new nonlinear mappings.

MSC:

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
47H25 Nonlinear ergodic theorems
47J25 Iterative procedures involving nonlinear operators
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