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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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