Atsushiba, Sachiko; Iemoto, Shigeru; Kubota, Rieko; Takeuchi, Yukio Convergence theorems for some classes of nonlinear mappings in Hilbert spaces. (English) Zbl 1384.47021 Linear Nonlinear Anal. 2, No. 1, 125-153 (2016). 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 \(T:C\rightarrow H\). Reviewer: Vasile Berinde (Baia Mare) Cited in 3 Documents 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) Keywords:Hilbert space; \(k\)-acute point; attractive point; acute point; fixed point; demi-contractive mappings; hemi-contractive mappings; Ishikawa iteration; Zhou’s lemma; Maingé and Măruşter’s theorem PDFBibTeX XMLCite \textit{S. Atsushiba} et al., Linear Nonlinear Anal. 2, No. 1, 125--153 (2016; Zbl 1384.47021) Full Text: Link