On the approximation of fixed points of non-self strict pseudocontractions

Abstract

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 \(\, v(x):=\inf \{\lambda \ge 0:\lambda x+(1-\lambda )Tx\in C\}.\) 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.