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Convergence theorems for some classes of nonlinear mappings in Hilbert spaces. (English) Zbl 1384.47021
Let \(H\) be a Hilbert space, \(C\) a non-empty subset of \(H\) and \(T:C\rightarrow H\) be a mapping. Denote by \(F(T)=\{x\in C:Tx=x\}\) the set of fixed points of \(T\) and by \(A(T)\) the set of attractive points of \(T\), that is, \[ A(T)=\{v\in H:\| Tx-v\| \leq \| x-v\| \text{ for all } x\in C\}. \] Let \(k\in [0,1]\). The set of \(k\)-acute points of \(T\) is defined as \[ \mathcal A_k(T)=\{v\in H:\| Tx-v\| ^2\leq \| x-v\| ^2+k\| x-Tx\| ^2 \text{ for all } x\in C\}. \] In the paper under review, the authors present some properties of \(k\)-acute points and establish relationships between \(k\)-acute points, attractive points and fixed points of various contractive type mappings \(T:C\rightarrow H\).

MSC:
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10 Fixed-point theorems
41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
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