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On the approximation of fixed points of non-self strict pseudocontractions. (English) Zbl 1466.47051

Summary: Let \(H\) be a Hilbert space and let \(C\) be a closed, convex and nonempty subset of \(H\). If \(T:C\rightarrow H\) is a non-self and \(k\)-strict pseudocontractive mapping, we can define a map \(v:C\rightarrow \mathbb {R}\) by \( v(x):=\inf \{\lambda \geq 0:\lambda x+(1-\lambda )Tx\in C\}\). Then, for a fixed \(x_{0}\in C\) and for \(\alpha_{0}:=\max \{k,v(x_{0})\}\), we define the Krasnoselskii-Mann algorithm \(x_{n+1}=\alpha_{n}x_{n}+(1-\alpha_{n})Tx_{n}\), where \(\alpha_{n+1}=\max \{\alpha_{n},v(x_{n+1})\}\). So, here the coefficients \(\alpha_{n}\) are not chosen a priori, but built step by step. We prove both weak and strong convergence results when \(C\) is a strictly convex set and \(T\) is an inward mapping.

MSC:

47J26 Fixed-point iterations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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